Quantum cohomology: lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 30-July 8, 1997

Lecture Notes in Mathematics Editors:

J.-M. Morel, Cachan R Takens, Groningen B. Teissier, Paris Subseries: Fondazione C.I.M.E., Firenze Adviser: Arrigo Cellina

1776

Springer Berlin

Heidelberg New York

Barcelona

Hong Kong London Milan Paris

Tokyo

K. Behrend

C. Gomez

V. Tarasov

G.Tian

Quantum Cohomology Lectures given at the C.I.M.E. Summer School

held in Cetraro,

Italy, June

Editors: P. de Bartolomeis B. Dubrovin

C. Reina

MM

Fondazione

C.I.M.E.

Springer

30

-

July 8,

1997

Authors Dept. of Mathematics

Vitaly Tarasov St. Petersburg Branch of

University of British Columbia

Steklov Mathematical Institute

1984 Math Rd.

Fontanka 27

Kai Behrend

Petersburg 191011, Russia [email protected]

Vancouver, BC scV6T 1Z2

St.

E-mail. [email protected]

E-mail.

C6sar G6mez

Gang Tian Dept. of Mathematics

Instituto de Materniticas y Fisicas Fundamental

Consejo Superior de Investigacion

M.I.T.

Calle Serrano SC123 28006

Cambridge, MA 02139, USA

Madrid, Spain

E-mail.

E-mail: [email protected]

[email protected]

Editors Boris A. Dubrovin

Paolo de Bartolorneis Dipartimento di Maternatica

Cesare Reina

Applicata

C.R.

"G. Sansone" Via di S. Marta, 3 50139

34100

Firenze, Italy

E-mail:

-

SISSA

Via Beirut, 4

Trieste,

Italy

E-mail. [email protected] E-mail., [email protected]

[email protected]/i.it

Cataloging-in-Publication Data applied for Die Deutsche Bibliothek

-

CIP-Einheitsaufnahme

at the CIME Summer School, held in Quantum cohomology: Cetraro, Italy, JunY 30 July 8,1997 / K. Behrend Ed.: P. de Bartolomeis.... Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Tokyo: Springer, 2002 (Lecture notes in mathematics; VOL 1776: Subseries: Fondazione CIME)

lecture

given

-

-

ISBN 3-540-43121-7

Mathematics

Subject Classification (2ooo): 53D45, 14N35, 81T30,

83E30

ISSN 0075-8434

ISBN 3-540-43121-7

Springer-Verlag Berlin Heidelberg New York

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Table of Contents

Quantum Cohomology Introduction

..................................................

1

Localization and Gromov-Witten Invariants K. Behrend

..................................................

1.

Introduction

2.

Lecture 1: A short introduction to stacks

3.

...............................................

2.1

What is

2.2

2.3

Algebraic spaces Groupoids

2.4

Fibered

2.5

Algebraic

variety?

a

3

.....................................

3

.......................................

5

............................................

8

products stacks

of

groupoids

..........................

9

.......................................

11

3.3

Equivariant intersection theory theory Equivariant theory Comparing equivariant with usual intersection theory

3.4

Localization

....................

3.2

3.5 4.

Intersection

15

....................................

15

....................................

16

......

19

..........................................

20

The residue formula

...................................

21

Lecture III: The localization formula for Gromov-Witten invariants 4.1

.................................................

The fixed locus

........................................

4.2

The first step

4.3 4.4

The second step The third step

4.5

Conclusion

Fields, Strings

.........................................

.......................................

Introduction

Chapter 1.1

1.2

I

Dirac

The 't

35

37

and Branes ..............................

39

...............................................

39

.................................................

42

Monopole

.......................................

Hooft-Polyakov Monopole

1.3

Instantons

Dyon Effect Yang-Mills Theory

........................

............................................

...........................

1.5.1 The Toron Vortex

1.5.2't Hooft's Toron

1.6

32 33

........................................

1.4

1.5

25 28

...........................................

C6sar G6mez andRafael Hern6ndez

I.

3

......................

Lecture II: 3.1

3

!

...............

T4

42

43 46 49

...............................

50

.....................................

53

on

Configurations

Instanton Effective Vertex

.........................

54

..............................

56

Table of Contents

V1

1.7

Three Dimensional Instantons

1.7.1 Callias Index theorem

1.7.2 The Dual Photon 1.8 1.9

as

...........................

58

..................................

60

Goldstone Boson

....................

...................

62

N=2

................................................

65

=

1

1.9.1 A Toron

Computation

.................................

67

............................................

68

2.

Chapter II

2.1

Moduli of Vacua

2.2

N = 4 Three Dimensional

2.3

Atiyah-Hitchin -Spaces

2.4

Kodaira's Classification of

2.5

The Moduli

2.6

Effective

3.

Chapter

......................................

Yang-Mills

69

.................................

77

Elliptic Fibrations

.............

78

=

..............................................

Superpotentials III

Bosonic

87

...........................................

94

String 3.1.1 Classical Theory 3.1.2 Background Fields 3.1.3 World Sheet Symmetries 3.1.4 A Toroidal Compactificati,on 3.1.5 a-Model K3 Geometry. A First Look To A Quantum Cohomology 3.1.6 Elliptically Fibered K3 And Mirror Symmetry 3.1.7 The Open Bosonic String 3.1.8 D-Branes

........................................

94

......................................

94

.....................................

96

...............................

97

............................

97

..........................................

98

............

104

...............................

105

.............................................

106

...................

107

...................................

109

Superstring Theories 3.2.1 Toroidal Compactification Theories. U-Duality 3.2.2 Etherotic String 3.2

3.2.3 Etherotic

81

...............................

3.1.9 Chan-Paton Factors And Wilson Lines

4.

68

.....................

2 Space of the Four Dimensional N Supersymmetric Yang-Mills Theory. The Seiberg-Witten

Solution

3.1

61

Supersymmetric Gauge Theories Instanton Generated Superpotentials in Three Dimensional N

of

Type

iia and

Type

iib

...................................

111

.......................................

116

Compactification

to Four Dimensions

...........

Chapter IV M-Theory Compactifications 4.2 M-Theory Instantons D-Brane Configurations in Flat Space 4.3 D-Brane Description of Seiberg-Witten Solution 4.4 4.4.1 M-Theory and Strogn Coupling 1 Four Dimensional Field Brane Description of N 4.5

119

...............................................

122

............................

122

..................................

125

....................

128

...........

136

.........................

143

.............................................

147

4.1

=

Theories

4.5.1 Rotation of Branes 4.5.2

QCD Strings

4.5.3 N

=

....................................

and Scales

2 Models With

................................

Vanishing Beta

Functions

............

150 151 155

Table of Cantents

4.6 4.7

4.8 4.9 4.10 4.11 4.12

4.13

M-Theory and String Theory Local Models for Elliptic Fibrations Singularities of Type b4: Z2 Orbifolds

...........................

......................

.............................

..............................

.....................

M(atrix) Theory The

A.2

Toroidal

and

M(atrix) Theory

AAAcknowledgments

q-Hypergeometric Vitali Tarasov

Quantum

Directions

................

........................

208

........................................

208

hypergeometric Riemann identity The

............................

The

.

hypergeometric integral hypergeometric spaces and the hypergeometric pairing The Shapovalov pairings Thehypergeometric Riemann identity

...............................

....................

on

the

hypergeometric

Bases of the

4.2

Tensor coordinates and the

218 219

.................

221

............

223

.

224

........

225

........................

227

5.3

Discrete local system associated with the

hypergeometric

..............................................

of the

homological

maps

228

bundle via the

................................

hypergeometric

216

.........

Discrete Gauss-Manin connection

Asymptotics

215

..........

5.2

hypergeometric integral

214

......................

Discrete local systems and the discrete Gauss-Manin connection Discrete flat connections and discrete local systems 5.1

5.4

211

216

hypergeometric spaces hypergeometric maps Difference equations for the hypergeometric maps Asymptotics of the hypergeometric maps Proof of the hypergeometric Riemann identity

Periodic sections of the

210

spaces and the

.......................................

4.1

integrals

193 196

3.3

4.5

193

200

3.2

4.4

184

..........................

Basic notations

4.3

182

.........................

3.1

maps

177

Representation Theory

...............................................

hypergeometric

6.

174

.......................................

The

5.

170

.........................

180

Functions and

Tensor coordinates

107

.............................

3.

4.

167

..............................

2.

3.5

165

177

One-dimensional differential example One-dimensional difference example

3.4

161 163

168

................................................

Introduction 1.

157

...................

...........................................

Holographic Principle Compactifications

A.1

A.3

...................

Singularities of Type A,,-, Singularities of Type bn+4 Decompactification and Affinization M-Theory Instantons and Holomorphic Euler Characteristic O-Parameter and Gaugino Condensates

4.14 Domain Walls and Intersections

A.

Vil

......................

234 237

Table of Contents

VIII

7.

7.1 7.2 7.3

7.4

8.

spaces

245

.......

239

.........................

240

.......................

240

........................

243

...............................................

group Ep,-y(-612) the elliptic quantum group

The'elliptic quantum 8.1

9.

loop algebra Uq'(j(2) and the qKZ equation Highest weight Uq(-612)-modules The quantum loop algebra Uq'(i(2) The trigonometric qKZ equation Tensor coordinates on the trigonometric hypergeometric

The quantum

Modules

over

.........................

8.2

Tensor coordinates

8.3

The

on

hypergeometric

Asymptotic

the

maps

solutionss of the

A. Six determinant formulae B. The Jackson

integrals

elliptic hypergeometric

.........

spacess

...

..............................

qKZ equation

...................

...................................

via the

Constructing symplectic Gang Tian

Ep,.y(.512)

hypergeometric integrals

..........

Introduction Euler class of

2.

Smooth stratified

2.2

2.3 2.4

2.5 2.6 2.7

3.1 3.2

261 264

269

270 270

...................................

272

............................

273

..........................

.............................................

Stable maps Stratifying the space of stable maps Topology of the space of stable maps

..........................................

......................

276 283 283

285

.....................

287

..........................

289

............................

299

.....................

302

Compactness of moduli spaces Constructing GW-invariants Composition laws for GW-invariants Rational GW-invariants for projective

Some

257

............................

Construction of the Euler class

GW-invariants

254

.....................

1.4

2.1

3.

weakly

Fredholm V-bundles

1.3

1.2

251

269

.....

orbispaces Weakly pseudocycles Weakly Fredholm V-bundles

1.1

248

invariants

...................................................

1.

248

..............

303

...................................

303

..................................

304

.......................

306

simple applications Quantum cohomology Examples of symplectic manifolds

spaces

Introduction

The progress

of the

theory

string

in the

last

decade

strongly

development

of many branches of geometry. In particular, in the enumerative and geometry symplectic of physicists venture a joint and mathematicians.

researches created

as

striking

achievements

of this

period

influenced new

topology

of

have been

Among the

the

the

directions

most

description theory on moduli spaces of Riemann surfaces in terms of the Korteweg de Vries integrable hierarchy of PDEs, and the proof of mirror conjecture for Calabi Yau complete intersections. One of the essential of these beautiful mathematical ingredients theories is a bunch of new approaches to the problem of constructing invariants of aland of compact symplectic manifolds gebraic varities known under the name quantum cohomology. Physical ideas from topological gravity brought into the problem of invariants new structures of the theory of integrable systems of differential The discovery of dualities between different equations. physical theories suggested existence of deep and often unexpected relationships mention

we

of the inter-

section

-

-

between In

ideas

different

order to

School

to

of invariants.

types

by

present,

young researchers, under the general

place at Calabrian organized in four theories.

course

we

mathematicians have decided

various

Notes contain

of Kai Behrend

aspects

the extended

"Localization

the approach to enumerative invariants the Bott residue formula has been developed. Gromov

Witten

The lecture written in

of this

exposition -

course

approach

with

that

text

new

of the lecture

of

algebraic

Behrend

the

-

courses.

Witten

varieties

Invari-

based

essentially particular

on

self-

gave

important

was

mathematical

case

of

spaces.

be

Nevertheless a physical one. we working in the area of quantum against reading physical papers will be to

mathematicians

those

cohomology who have

of these

Strings and Branes" by C4sar G6mez, HernAndez, collect some ideas of duality the development of quantum cohomology.

for string theories important of the looks design presentation

confident

took

Rafael

The are

new

"Fields,

of,urse

in collaboration

for

of projective

invariants

these

CIME Summer

a

and Gromov

ants"

consistent

physicists,

organize

Cetraro

covering

courses

and

to

The School "Quantum Cohomology". from June 30 to July 8, 1997. It

title

resort

sea

These Lecture

In the

both

no

prejudices

benefitted. The lecture

notes

of

Vitaly

Tarasov

"q-Hypergeometric

Functions

and

Theory" Representation of integrable in the theory of form factors in massive systems originated of models field This branch into now developed integrable quantum theory. of representation of affine and of the a part cortheory quantum algebras vertex how to compute the matrix operators.Tarasov responding explains elements of the vertex in the terms of solutions to the quantized operators introduces

ory

the reader

to

another

branch

of the the-

Bartolomeis,

P. de

Knizhnik these

Zamolodchikov

-

Dubrovin,

C. Reina.

equation,

and derives

B.

integral

representations

for

solutions.

The

plectic

of

course

topology

Gang

applies

class

the

to the

of Gromov

-

techniques Witten

of symof

invariants

tool is the theory of virtual Tian curves. pseudoholomorphic spaces definition of quantum cohomology of symplectic of certain nontrivial examples of symplectic The main technical

moduli

to the

technique

this

manifolds

on

the reader

in the construction

manifolds.

compact symplectic fundamental

Tian introduces

involved

and to constructions

of

manifolds. We believe express

and their

the

gratitude availability

We also invitation

that

to

our

to

thank

for Prof.

organize

School the

was

successful

speakers

discussions R-Conti

for

during

the

reaching high quality

its

in

Boris

Dubrovin,

and

we

lectures

the School.

and CIME Scientific

Committee

the School.

Paolo de Bartolomeis,

aims,

of their

Cesare Reina

for

the

Localization

and Gromov-Witten

Invariants

K. Behrend of British

University

Weexplain

Summary. maps.

This

leads

space in terms

1.

to

of

a

Vancouver,

apply the

how to

integrals

Bott residue formula to stacks of stable Gromov-Witten invariants of projective

expressing

formula over

stacks

Canada

of stable

curves.

Introduction

The

is divided

course

stacks.

Wetry

the definition not

Columbia,

to

of

require

any

Lecture

II

give

a

few ideas

algebraic knowledge

introduces

lectures.

three

into

stack

Lecture

about

of finite

the

type

I is

a

philosophy over

a

short

introduction

of stacks

field.

and

Our definition

we

to

give does

of schemes.

equivariant

intersection

The basic

constructions

theory as constructed by in a rather explained The localization case. (in the algebraic context also easy special property due to Edidin-Graham. is mentioned and proved for an example. We set [6]) framework for the localization a general using property to localize up integrals subvarieties to the fixed or locus, (substacks) containing the fixed locus. III In Lecture the localization formula to the stack of stable we apply P'. deduce We formula to the Gromov-Witten a of IF invariants giving maps in of terms stacks of stable over curves (for any genus) integrals Mg,n- The if sometimes At the same proof given here is essentially complete, sketchy. time these lectures Graber and Pandharipande were given, [12] independently from ours. Weavoid proved the same formula. Their approach is very different the consideration of equivariant obstruction on which entirely theories, [12] relies. The idea to use localization to compute Gromov-Witten invariants is, of

[5].

Edidin

and Graham

course,

due to Kontsevich

2.

Lecture

What is

We will

a

(see [13],

A short

I:

are

where the genus

introduction

to

zero

case

is

considered).

stacks

variety?

explain

Grothendieck's

point of view that a variety is a functor. X3. According to example the affine plane curve y2 X3 is nothing but the 'system' the variety of all solutions Grothendieck, y2 X3 in all rings. Werestrict of the equation y2 and fix a ground filed slightly k and consider instead of all rings only k-algebras of finite type (in other of polynomial words quotients over rings in finitely k). So, many variables Let

us

consider

for

=

=

=

K. Behrend, C. G´omez, V. Tarasov, G Tian: LNM 1776, P. de Bartolomeis et al. (Eds.), pp. 3–38, 2002. c Springer-Verlag Berlin Heidelberg 2002


K. Behrend

4

Grothendieck, X3 of y2 all solutions

following

we

hv

(f.g.

:

associate

k-algebras) A

hV is actually

that

Notice

k-algebras

of

O(X)2

0(y)3

=

.

a

this

=

justified

(covariant)

The

as a

subcategory

with

the functor

functor

of

finite

a

isomorphic

3)

X

,

to the

the

and for

every

I(X,Y) Terminology: we

get

a

larger

for

of solutions:

the

variety

affine

We

V C A2 this

varieties

is

stands is

k-algebras),

(sets))

for

the

(sets),

to

faithful

fully

of functors:

category

morphisms we

may think

(sets))

k-algebras),

and

objects

k-varieties)

(affine

of

are

transforma-

natural

are

identify

variety

the

V

affine of

affine

2

E A

=

HOInk-alg (k[V],

coordinate

k-algebra

1 Y2

A =

The functor

we

21

of the

ring

=

curve

y

2

3 X

=

Y] / (Y2

Homk-alg(k[x,

hV is the functor

-

k[x, y]/(y2

is

by V.

k-varieties) the former still

X3) A). ,

represented

(affine

(sets)) we may enlarge than (affine k-varieties),

category

A)

have

have embedded the category

k-algebras),

Funct((f.g.

(O(x), 0(y))

by 'system'

morphism

a

2 E B satisfies

variety V there are many ways to write it as the zero in some affine So one gets many polynomials n-space. because all these functors is not a problem, are canonically functor ring k[V] of V: given by the affine coordinate

an

set

example,

Once

B is

-+

hv

hv (A) For

A

X31

hV.

hV. This

functors

:

=

lemma.

Funct((f.g.

Rmct((f.g.

of

Given

Note 2. 1.

locus

we mean

of Yoneda's

corollary

k-algebras)

Because this

tions.

X3,

0

If

then

point hV. At least

functor

Here Funct

(f.g.

from

functor: =

1 Y2

2

E A

of view is that

k-varieties)

faithful.

fully

I (X, Y)

what

V is

A,

k-algebra

generated

functor

(affine

functors

(sets)

--+

y2

This makes precise Grothendieck's

X3 is by y2 the following by

defined

-+

(covariant)

E A2satisfies

functor.

this

mean

(x, y)

and

finitely

to every

in A2:

=

into

inside

consisting

the latter of

to

'geometric'

objects. rise

For

example,

to

the functor

every

hspecA

:

(f.g.

finitely

generated

k-algebras) R

The functor

k-algebra

A,

reduced

--4

(sets)

--+

HOInk-aig(A,R)

or

not,

gives

Localization

(f.g.

hsp

k-algebras) A

fully

and

(f.g.

k-algebras).

The above

the

equivalence

rings. place

Yoneda's

keeping

lemma is

F-4

hsPec

completely The

it

proof

represents,

Spec A : (f.g. for

affine formal

lemma for

the

lemma follows

category

from this

k-varieties

and their

and holds

for

and

coordinate in

category

every

simple exercise in category theory. oba geometric philosophy of identifying is

a

we

write

k-algebras)

(sets)

--+

hSpecA, and call it the spectrum of A. The full subcategory of of functors k-algebras), (sets)) consisting isomorphic to functors

the functor

Funct((f.g. of the

Spec A

form

the

To construct

Unless

is

called

the

of affine

category

k-schemes

of finite

type,

(aff/k).

denoted

one

k-scheme

functor

knows scheme

of finite

hx

hV for a general k-variety theory. Then it is easy, and

is then

variant)

(f.g.

:

(f.

:

t

little

a

do it

tricky. for

any

(sets)

k-algebras)

less trivial slightly functor fully faithful

h

V is we can

type X:

Hom.,chemes (Spec A, X)

A It

(sets))

k-algebras),

of Yoneda's

between

Invaxiants

A

is Yoneda's

This

Grothendieck's

with

the functor

with

Funct((f.g.

corollary

categories

k-algebras).

(f.g.

of

In

ject

of

---+

faithful.

is contravariant

and Gromov-Witten

-

than just

k-schemes)

Yoneda's

((f.g.

Funct

X

hx

lemma that

k-algebras),

one

gets

(co-

a

(sets))

.

(This is,

in fact, part of what is known as descent theory.) largest subcategory of Funct((f.g. k-algebras), (sets)) which still sists of 'geometric' objects is the category of finite type algebraic spaces k. Wewill now describe this category (without using any scheme theory). The

Algebraic First

of

all,

conover

spaces

to

get

a more

'geometric'

picture,

we

prefer

to

think

in terms

of

Thus we (aff/k) rather than the dual category (fg. k-algebras), replace Funct((f.g. k-algebras), (sets)) by the equivalent category where Rinct* refers to the category of contravariant Funct*((aff/k), (sets)), functors. Grothendieck calls Dinct* ((aff/k), (sets)) the category of presheaves on (aff/k).

the category

Westart

by considering

the covariant

h:(afflk)

Funct*((aff/k), X

where hX (Y)

=

functor

HOInk-schemes (y, X)

hx =

(sets))

,

HOMk-alg (k[X],

k [Y]).

K. Behrend

fibered containes The category (aff/k) is and tensor a final k-algebras) product)

products (the dual concept object Spec k. The same is

Note 2.2.

(f.g.

in

(sets)).

((aff/k),

Funct*

for

true

Given

diagram

a

Z

X

((aff/k),

in Fanct*

(sets))

W(SpecR)

Ig

f

>Y

product

the fibered

f (Spec R) (x)

((aff/k),

of Funct*

object

left

Z is

given by

=

R) (z)

g (Spec

E Y (Spec

is the constant

in

set

exact.

also contains direct sums (called The category (aff/k) If X Y and k-schemes then their affine are context).

disjoint disjoint

Note 2.3.

this

in

R)}

functor Spec R -+ 101. place of 101 will do. Moreover, the One says that h products and final. objects.

(sets))

of course, any one-element h commutes with fibered functor

Here, is

x y

X(SpecR) XY(SpecR) Z(SpecR) I(X,z) E X(SpecR) x Z(SpecR)

=

=

A final

W= X

sum

contains AX x Ay. Also, (aff/k) ring AZ the affine whose coordinate the zero is initial an empty scheme, object, ring in the Funct* consider notions Wedo not corresponding ((aff/k), (sets)), ring. h does not commute with disjoint the functor sums anyway. Z

X LI Y has affine

sums

=

Definition X

over

We call

Ili,E,rXi

means

X is

--*

Now that

we

object

an

each Xi

that

defines

This

Remark 2.1.

=

of (aff/k)

(Xi)iEi

family of objects morphism Xi -4 X). and the induced morphism a covering of X, if I is finite flat, i.e. flat and surJective. faithfully Let X be

2.1.

(which (Xi)iEi

coordinate

topology

covering,

of

notion

and

endowed with

Grothendieck

a

have the

comes

a

a

on

we can

(aff/k). define

the

notion

of

sheaf. Definition

(i.e.

a

of

ering 1. 2.

if

(aff/k)

on

an

the satisfying object U of (aff/k),

X, y E X (U)

x

=

if

xi

I

A sheaf

2.2.

presheaf),

x

xJUi

y

,

(Here

E X (Ui),

1, (tTij =

xi,

x

I Ui denotes

Ui for all =

we

elements

are

i E

is

two

1,

x u

are

Uj)

i E I.

object

axioms:

image of

given

X

of Funct* ((aff/k),

(sets))

(Ui)iEI

Whenever

is

a cov-

have

such that

the

then

an

sheaf

such that

there

exists

x x

I Ui

=

y

I Ui, for

under X (U) xi an

I Uij

= xj element

-+

I Uij, x

all

i

G

1, then

X (Ui).)

for all (i, i)

E X

(U)

E

such that

Localization

It

is

basic

a

this

flatness

of faithful

terms

statement

Definition

(aff/k)

theory sheaf

a

is the

that

2.

there Let

An

2.3.

us

this

X

algebraic

is

'45

affine

an

try

explain

to

X

4

-

exists

So let

context.

(aff/k) Y(U)

general

most

every

(affine)

of

k-scheme

of

covering

in

of

The notion

notion

type)

(of finite

space

covering

makes

that

k is

over

a

sheaf

X

on

the

f

:

affine

Funct*((aff/k),

(sets)) is

or

X.

-+

and smooth

product U

then

not

a

V is

finite

I

f

X

V is

U

epimorphism morphism of sheaves on the map f (U) : X(U) -+ scheme and U -+ Y is a morphism and

I say that

epimorphism

smooth

a

meaning of quasi-affine X -+ Y be an injective all objects U of (aff/k)

(this means that for If U is an injective).

form the fibered

quasi-affine,

X is

x

scheme U and

V

in

for

(aff/k).

on

7

true.

diagonal

the

we

descent

hX is

Invaxiants

such that

1.

in

from

fact

type X, the functor

anite

and Gromov-Witten

a

>Y

subsheaf

of U. Thus it

of affine

union

makes

subschemes

sense

to

of U. Now the

if for all affine schemes U and for f : X -+ Y is called quasi-affine, injection for all elements of Y(U)) the pullback all morphisms U -4 Y (so equivalently V C U is

finite

a

Now let This

implies

then

the fibered

product every

by finitely the morphism is surjective.

be covered

for

each

i

-+

V

course

all 2.4.

k-varieties

subschemes of U.

(aff/k)

on

whenever

if for

can

Definition

of affine

sheaf

of the above

epimorphism,

Of

a

that

the situation

IIWi

union

X be

such that

U xX V is

definition, affine

a

the

-+

union

-4

smooth

and k-schemes

A k-scheme

is

an

of affine

algebraic

are

quasi-affine. over

schemes. Now,

X is called

X the fibered

Zariski-open

V is

is

schemes U and V

morphism U ---

scheme V

many affine

Wi

finite

diagonal

the

have two affine

we

product

a

X, in

smooth

U xx V

subschemes Wi such that and the induced morphism

algebraic k-space

k-spaces.

X, which

is

locally affine

in

affine topology Un and open immersions of algebraic U1,...' spaces Ui -+ X such that IJ Ui -+ X is surJective. immersion n of algebraic spaces X -+ (A open scheme U -+ Y the pullback Y is a morphism such that for every affine X xy U --+ U is an isomorphism onto a Zariski open subset.) A k-variety which means is a k-scheme which is reduced and irreducible, that the Ui in the definition of scheme may be chosen reduced and irreducible the

Zariski-

schemes

with

dense intersection.

an

scheme.

This

means

that

there

exist

k-

K. Behrend

One an

can

affine

that

prove

an

This

scheme.

algebraic that

means

space X is

affine

locally U1,

in the

schemes

.

.

.

6tale topology U,, together with

,

X can be found, such that 6tale morphisms Ui -LI Ui (The notion of 6tale epimorphism is defined epimorphism. smooth epimorphism, above, using fibered products.) Using such 6tale (or smooth) covers, one can do a lot algebraic spaces. A vector bundle, for example, is a family Ei / Ui, together with gluing data Ei I Uij c- " Ej I Uij.

X is

--+ as

an

the

6tale

of

notion

of geometry cm bundles

of vector

Groupoids Definition

A

2.5.

groupoid

is

a

in which

category

all

morphisms

invert-

are

ible.

Example as

2.

objects a

and

Let X be

a

X, and for

G-set.

set.

Wethink

of X as a groupoid by taking X morphisms to be identity morphisms. the groupoid BG to have a single object

all

group G. Then we define

objects

two

is called

groupoid

This

a

declaring

We define

group.

automorphism

with 3.

1.

of

G be

Let

Let X be

2. 1.

set

x, y E X

the we

groupoid let

transformation

the

XG to have

Hom(x, y) groupoid

=

Ig

of

set

E G

I

objects

gx

=

yJ.

given by the action

of G on X. 4.

on the set X. Then we define equivalence relation an groupoid by taking as objects the elements of X and as the elements of R, where the element (x, y) E R is then a morphisms unique morphism from x to y.

Let R C X

x

X be

an

associated

groupoids

the same' if they are equivalent 'essentially is rigid if every object has trivial groupoid categories. if all and connected are isomorphic. objects Every rigid automorphism group, A groupoid the to is an equivalence relation. equal groupoid given by groupoid if and only if it is equivalent to a groupoid set as in is rigid a by given to Example 1, above. A groupoid is connected if and only if it is equivalent of type BG, for some group G. All these follow a groupoid easily from the well-known equivalence criterion. following Wethink

Proposition between

categories

1. 3.

as

a

2. 1. Let f : X -+ Y be a morphism of groupoids (i.e. a Junctor X and Y). Then f is an equivalence categories underlying of and essentially if and only if f is fully faithful surJective.

the

Remark 2.2.

groupoids 2.

of two

We say that

as

Groupoids

consists

form

a

2-category.

This

means

that

the category

of

objects: groupoids morphisms: functors between groupoids 2-morphisms, or morphisms between morphisms: between functors.

natural

transformations

of

Note that

this

special

a

X, Y

objects

two

is

think

One should

vertible.

type of 2-category, of such

as

Hom(X, Y)

2-morphisms

all

since

2-category

a

morphisms

the

Invariants

and Gromov-Witten

Localization

a

form

category not

a

in-

are

where for but

set

any

rather

a

gioupoid.

spaces objects: topological morphisms: continuous maps 3. 2-morphisms: homotopies up

invertible

with

example of a 2-category Example 2.2. Another important the is homotopy category: (truncated) morphisms

2-

1.

2.

of

One may think

groupoids

generalized

as

of sets and groups. If we alization of algebraic space by the 2-category is not

a

from the

completely

trivial

that

fact

reparametrization.

to

replace

(groupoids),

is

a

if

it

or

rather

we

(sets)

a common gener-

in the definition

get algebraic

This

stacks.

arising complications like than a 1-category,

because of the

generalization

(groupoids)

sets,

the category

rather

2-category

(sets). We call

a

groupoid

objects and every groupoid X we define of

finite, object its

has

'number

#(X) where the

sum

is

taken

over

a

classes

finitely many isomorphisms automorphism group. For elements' by

has

a

finite

of

E # Aut

a

finite

-L

=

set

x'

of representatives

for

isomorphism

the

classes.

The fibered of groupoids. product is products of for the that is not only basic groupoids and stacks, theory of the of 2-categories. philosophy good example

Fibered

a

construction but

is

also

a

Let Z

X

19

f

>Y

groupoids and morphisms., Then the fibered product W groupoids defined as follows: Objects of Ware triples (x, 0, z), is a morphism in Y. A -+ g (z) where x E ob X, z E ob Z and 0 : f (x) where a : x -+ x, morphism in X from (x, 0, z) to (x, 0', z') is a pair (a,#), such that the diagram and z -+ z' are morphisms in X and Z, respectively, diagram

be

a

X

x y

of

Z is the

f W

0 >

W)

A-)

f W)

gW

>

g W)

K. Behrend

10

commutes in

Y.

with two morphisms groupoid Wcomes together W -+ X and W-+ Z given by projecting onto the first and last components, respectively. Moreover, Wcomes with a 2-morphism 0

The

W

I

>Z

0t

X

making from

diagram '2-commute',

the

the

>

f

which

-+ Y composition 0 is 2-isomorphism given by O(x, 0, z)

the very

W-+ X

definition

-q

to

(2.1)

Y

just

means

the

that

composition It

is

a

0 is

an

isomorphism

W-+ Z

natural

--*

Y. The

transformation

by

of W.

Example 2.3. If X, Y and Z are sets, then Wis (canonically isomorphic to) product I (x, y) E X X Yf (x) of sets. g (y) I in the category

the fibered

=

The 2-fibered

a universal product Wsatisfies mapping property in the 2category groupoids. Namely, given any groupoid V with morphisms V -+ X and V -+ Z and a 2-isomorphism from V -+ X -4 Y to V -+ Z -+ Y (depicted in the diagram below by the 2-arrow crossing the dotted arrow), there exists V -+ Wand 2-isomorphisms a morphism from VX to V W X and

of

V

W

Z to V

Z such that

the

diagram

V

W

Z

I X

which

commutes,

I

-q

Y

f

of the various 2compatibility image this diagram as lying on the surface of a sphere.) The morphism V -+ Wis unique up to unique isomorphism. Whenever a diagram such as (2.1) satisfies this universal mapping propbecause in a 2-category, erty, we say that it is 2-cartesian (or just cartesian, 2-cartesian is the default value). In this case, Wis equivalent to the fibered above. product constructed

isomorphisms

If X is

a

amounts

involved.

G set,

then

to

(One

we

a

certain

should

have two fundamental X

1

cartesian

diagrams:

pt

1 BG

(2.2)

Invariants

and Gromov-Witten

Localization

11

and 0'

GxX

>X

PI "'

groupoid

the

with

(2.3)

XG

'

Here pt denotes

1

.

and

object

one

one

morphism (necessarily

set, we mean the set object). morphism identity and the projection, the action denote and we a of as a groupoid. By p thought respectively. the universal Hence XG satisfies Diagram (2.3) is moreover 2-cocartesian'. Note of in the X G of of a groupoids. category by quotient mapping property the cocartesian set XIG satisfies of sets the quotient that in the category but not the cartesian (unless the action of G on X is free, property property, to the groupoid is quotient in which case the set quotient equivalent XIG better much have of the in taken Thus groupoids category quotients XG). For of we have sets. the in taken example, than category quotients properties If

of the

the

write

we

a

#X

#(XG)

=

#G

if X and G are finite.

groupoid and let X0 be the set of objects of X and X, the set with each of all morphisms of X. Let s : X, -4 X0 be the map associating with each the map associating morphism its source object, and t : X, -+ X0 Then the diagram morphism its target object. Let X be

a

t

X,

>

X0

SI

7r

Ir

X0

>

X

where 7r : X0 and cocartesian, be thought of as the groupoid may

is cartesian

Thus

a

the definition

subdivide

of

The notion

of 2-cocartesian

is

The correct

definition

simply

above.

explained is

sufficient

which reduces

morphism. set by the

canonical

object

of its

stacks

Algebraic Wewill

X is the

quotient

morphisms.

of the

action

-+

axe

to

to

rigid

It

is not

involves,

remark

that

groupoids,

the usual

notion

or

algebraic

more

subtle

the dual

instead

(2.3)

is

even

just

stacks

than notion

into

one

three

might

to the

be led

sets.

For such text

to

2-cartesian

For of a square, a cube. with respect cocartesian

of cocartesian.

steps.

objects,

our

to

believe.

property purposes test

objects

2-cocartesian

it

K. Behrend

12

Prestacks

Prestacks.

(aff/k)

functors

Definition

1.

2.

for for

This

every

every

X(V) 3.

A

2.6.

(groupoids).

generalization

a

are

of

(i.e.

presheaves

contravariant

(sets)).

-+

prestack

is

that

means

(lax)

a

X is

Junctor

contravariant

by

given

the

X

(aff/k)

:

data

affine k-scheme U a groupoid X(U), morphism of k-schemes U -+ V

a

morphism

of groupoids

U -4 V

Wa natural

X(U),

-+

of morphisms of k-schemes

for every composition 0: transformation

X(W)

-+

X (V)

"

I XM

(this X(U) This 1.

2.

the

to

data

0 is

a

the

to

transformation from the Junctor X (W) functors X(W) -+ X(V) -+ X(U).

natural

of

composition

subject

is

the

conditions

then so is X(U) -4 X(U), if U -4 U is the identity, a 2-cocycle for each composition U -+ V -+ W-+ Z in (afflk) 0 the the have various to compatibilities expressing satisfy. is not difficult examples below as guide, this 2-cocycle condition down.

to

be

-+

(category

(U

V)

1---+

V

W)

-

of vector

pullback 0

:

of vector

the canonical

via

steps In this over

example all the 0

to

write

is

a

in this

lax

case.

stack:

bundles

:

(aff/k)

(U

-+

isomorphism

Let

of

pullback

pullback

with

in two

V.

G be

an

algebraic

(groupoids) B(G(U))

U

V)

--4

the

morphism of groupoids

B(G(V))

-+

morphism of

U

only)

directly

again.

n over

bundles

the intermediate

trivial

are

of rank

the functor

k and consider

preBG

(sets)

-+

identities

are

prototype

as a

from Wto U

3.

(aff/k) 0

isomorphisms

with

-+

the

Using

(groupoids) U

(U

of

thought

(aff/k)

Vectn:

(presheaf) All (groupoids).

functor

(aff/k)

might

following

condition

the reader.

Each actual

1.

(prestack)

functor The

this

We leave

Example 2.4. 2.

that

means

B(G(U)) groups

induced

G(V)

--+

by

the

G(U)

group

and Gromov-Witten

Localization

Let

the

denote

us

phisms an

k-scheme,

affine

X and

lax functor

a

groupoids). We leave

it

get

we

object

an

the

(i.e.,

morphism every

associates

x

about Hom.*

fact

A basic

ucts,

of the

x

mor-

X(U),

groupoid

where U is

X

--

transformation).

natural

a

U

X:

i.e.

the

We denote

morphism by

this

letter:

same

The

2-category. explicate

to

ob-

Its

morphism

induced

an

U of lax functors

(aff/k)

from

functors

is, of course, a to to the reader

It

2-isomorphisms.

and the

Given

lax

of contravariant

category

by Hom* (aff/k, (groupoids) have just defined. jects we

13

Invariants

V

to

--+

X.

-+

pullback

U the

groupoids)

(aff/k,

is that

xIV. it admits

prod-

2-fibered

diagram Z

f

X can

be

completed

to

diagram

cartesian

a

W

I is

//'

simply

as

Stacks.

satisfies

of

The notion

of stacks

generalizes

1.

A

2.7.

prestack

following

the

If U is presheaf

an

affine

by defining

X

two stack

X(U)

:

(afflk)

:

is

sheaf satisfies

a

W(U), for U an Z(U) over Y(U). of sheaf

the notion -+

(groupoids)

is

k-scheme,

affine

on

called

(aff/k). a

stack

if

it

axioms.

(afflU) V

2.

and

scheme and x, y E

Isom(x, y)

Y

>

essentially product

Definition

-q

f

the fibered

accomplished

Z

'

X This

19 >Y

X(U)

are

objects

of X(U)

--+

(sets)

--4

ISOM(XIVYIV)

then

the

(afflU).

on

scheme U, with Given an affine a property: and given objects E xi 2-1) (Ui)iEI, of Definition -4 xjlUij, for all X(Ui), for all i E I and isomorphisms Oij : xilUij such that the (Oij) C- I x I, satisfy the obvious cocycle condition (i,j) (for each (i, j, k) E I X I X I), then there exists an object x E X (U) and isomorphisms Oi : xi -+ xJ Ui, such that for all (i, j) E Uij we have

X

cover

(in

Oj luij

-

the

the

Oij

=

descent

sense

Oiluij

-

K. Behrend

14

(xi, Oij) is called if (x, Oi) exists,

The data

(Ui);

covering

second stack

descent

a

datum for

descent

the

axiom may be summarized

X with

datum is called

by saying that

respect

effective. descent

every

the

to

So the

datum

is effective.

Example

2.5.

axioms

prestack

The

2.

Of

1.

stack

the

course

for

sheaf is in

every

presheaves is a stack,

Vectn

a

reduce since

natural

to the

way

a

Note how

stack.

sheaf axioms. bundles

vector

satisfy

decent

the

property. 3.

The

prestack

spect

to the

stack

(similar

preBG is not a stack. A descent datum for preBG with recovering (Ui) of U is a 6ech cocycle with values in G. It is efif it is a boundary. fective Thus the Cech cohomology groups H1 ((Ui), G) the obstructions contain to preBG being a stack. Thus we let BGbe the prestack whose groupoid of sections over U C- (aff/k) is the category of G-bundles over U. This is then a stack. There is a general proprincipal to a prestack cess associating called a stack, passing to the associated the

Algebraic

sheafification).

to

This

stacks.

Definition

The stack

BG is the stack

associated

to

preBG.

prestack

A stack

2.8.

generalizes

notion X

:

(aff/k)

-+

the notion

(groupoids)

algebraic

of

is

an

space.

algebraic

k-stack

satisfies

it

A

diagonal

1.

the

2.

there

exists

an

such U is called

The first

property

:

X

-+

affine a

X

x

X is

of

presentation is

representable

scheme U and

separation

a

of finite epimorphism and

smooth

if

type, U -+ X.

Any

X. It

be

in terms interpreted stack axiom. It says isomorphisms occurring that all these isomorphism sheaves are algebraic spaces of finite type. (The definition of representability is as follows. The morphism X -+ Y of stacks is if for all affine U -+ Y the base change X xy U is an algebraic representable

of the

sheaves

a

property.

of

in

can

first

the

space.)

The second property says one can do 'geometry'

Thus

bundle

E

where

U is

that, on

locally, every algebraic

an

stack stack.

is

just

For

an

affine

example,

scheme. a

vector

algebraic stack X is a vector bundle E' on such an affine presentation U, together with gluing data over U xx U (which is an algebraic space by the first For another example, an algebraic stack property). X is smooth of dimension a smooth presentation U -+ X, n, if there exists over

dimension

'locally', spaces,

make

an

smooth

k.

by pulling

above.)

of dimension

n

+ k and U

(Smoothness

--

X is

smooth

of relative

of representable back to affine schemes,

Note that

according

to

morphisms of stacks is defined to the case of algebraic similarly this definition, negative dimensions

sense.

Example 2.6.

1.

Of course,

all

algebraic

spaces

are

algebraic

stacks

and Gromov-Witten

Localization

Invariants

15

of The isomorphism Vect,, is algebraic. spaces are just twists take Spec k -+ Vect", For a presentation, algebraic. GL, and therefore bundle k' over Spec k. This is a smooth morvector given by the trivial 2 scheme U with rank relative since for any affine dimension n phism of U E bundle the back induced morphism -+ Vectn pulls over U, n vector hence and to the bundle of frames of E, which is a principal GLn-bundle, dimension this makes Vectn a smooth smooth of relative n 2. Note that The stack

2.

,

stack

of dimension

2.

-n

G is assume that algebraic group over k. To avoid pathologies BG the if k Then is is char case an always algebraic 0). (which The proof of algebralcity stack. is the same as for Vectn, after all, Vectn Whenever P is a G-bundle over a scheme X, is isomorphic to BGL,,. then we get an induced morphism X --+ BG, giving rise to the cartesian diagram > Speck P Let G be

3.

an

smooth

=

BG

X

Therefore,

Speck

smooth of dimension If

4.

G is

then

(smooth)

a

we

define

an

a

principal

that

algebraic algebraic

X

-4

XIG XIG

the

G-bundle.

universal

Moreover,

BG is

dim G.

G-bundle

One checks

morphism diagrams

-

XIG(U)

U, the groupoid is

BG is

-+

has and

is

an

is

a

acting

group

XIG

stack as

objects

0 algebraic :

P

on

all X is

-+

stack

presentation)

the

follows.

as

algebraic For

an

X,

space

affine

scheme

pairs (P, 0), where P -+ U a G-equivariant morphism. (for example, the canonical

and that

GxX

X

1

1

X

XIG

there

are

2-cartesian

(2.4)

and

Spec k

X

1

1

XIG

Lecture

3.

Intersection For

of

a

Equivariant

BG

intersection

theory

theory

k-scheme

k-cycles

II:

(2.5)

A,, (X) k Ak (X) where Ak (X) is the Chow group rational equivalence tensored with Q. Readers not familiar

X let

up to

=

,

K. Behrend

16

Chow groups

with

may

H2Bkl (X'11)

Ak (X) the strong

=

of

Let also A*

relative

space

a

(X)

=

(Dk

homology

Moore

as

Ak(X)

A.,

scheme

every

being

X,

be the also

take Ak

ogy with Q-coefficients. The most basic properties

for

the results

(see [9]),

of Fulton-MacPherson

C and work with

(X)

(X)

is

a

Everything

Chow cohomology

operational

tensored =

space with i.e. relative

weaker.

are

Q.

with

H2k(Xan)Q,

of A* and A.

and A.

is

analytic for Borel-Moore homology, one-point compactification.

its

to

A., although

this

works with

ground field

the

Here X" is the associated

and BMstands

topology

homology

that

assume

Q instead.

If

(singular)

usual

groups

with

working

Borel-

cohomol-

(X) is a graded Q-algebra, the operation (X)-module,

A*

are:

graded

A*

product

cap

Ak (X)

A,,(X) (a, -y)

x

--+

A,,,-k(X)

--+

a

n -y

-

A* and A.

exist for Deligne-Mumford stacks. more generally stacks be should conby A. Vistoli [16]. Deligne-Mumford sidered not too far from algebraic schemes or (especially spaces concerning their all over properties cohomological Q. Many moduli stacks (certainly of are Deligne-Mumford type. #)) (X, Hg,,, stack is an algebraic A Deligne-Mumford that is locally k-stack an affine to the 6tale Thus a Deligne-Mumford scheme with respect stack topology. such that p is 6tale. X admits a presentation This p : U -+ X (U affine) for example, that all automorphism conditions and implies, groups are finite Note that

This

was

shown

reduced.

Equivariant Let

theory

algebraic group over k. To work G-equivariantly of algebraic G-spaces (i.e. algebraic k-spaces of categories there is an equivalence Now G be

in the

an

category

(algebraic

G-spaces)

(algebraic XIG

X Here

G-spaces)

(algebraic

(algebraic Y

-+

over

spaces

(algebraic

BG which

1BG)

is

an

category are

morphism of

algebraic

algebraic

and 77

algebraic

spaces

stack

X

(algebraic

-+

stacks

of

1BG)

(3-1)

k-spaces

1BG)

is

the

with

together

with

spaces

1BG)

of al-

category

BG. So

over

G-action

a

object of representable an

from X

-+

pairs (f 77), f 2-morphism making the diagram

of a

to work

G-action).

.

representable

BG. A morphism in class BG is an isomorphism

morphism X to

is the

morphisms,

and equivariant gebraic stacks

spaces

means

with

,

where

:

X

-4

BG

Y is

a

and Gromov-Witten

Localization f

X

Invariants

17

Y

.

\

I BG

of is

of the

The inverse

commute.

Diagram (2.5). Defining equivariant to defining equivalent

XIG,

the form If the

and

A (X)

XIG

stack

quotient A*

(XIG).

is

an

the

In

is defined

using the

A (X) and A (X), for A* (XIG) and A, (XIG)

Chow groups

Chow groups stacks. quotient

i.e.

=

(3.1)

functor

algebraic general

the

case,

a

G-space X,

for

A (X)

then

space,

construction

=

construction

stacks

A,, (XIG) due to

is

[5]. They proceed as follows. Assume that G is linear in positive separable, to avoid certain pathologies characteristic). First define AG(X) for Choose a representation Ap (XIG) p fixed.

Edidin-Graham,

=

P

GL(V), V

on

such that

there

complement

Z

=

V

-

acts

on

X

V

x

UIG

V)

> dim X

V of G associates

XIG. by (x, v) g over

It

-

I XXGU The vertical

maps

inclusions

on

dim X

the

by

principal It

(the space). is

a

G-bundle

is

X

=

not

a

x

space,

X

VIG, but

-+

XIG

where G

the

open

morphism X XG U -+ UIG Thus we have the following

-

are

the left

dim G

-

principal

C

.Xxv

>

I

C

X

I

X XG V

G-bundles,

>

XIG

hence smooth

epimorphisms.

The

with complement of codimension open immersions bundles are vector p. The horizontal maps on the right are

dim V.

Having

chosen V and U C

Ap (XIG) which

space

p

-

X xG V

(xg, g`v).

=

XXU

tient

and such that

G

diagram.

cartesian

>

space)

a

dim G

-

the

to

given

is

X XG U C X XG V certainly and UIG is already representable

of rank

is

U in the vector

(and

U has codimension

substack is

open subset

such that

codim(Z, The representation bundle a vector

G-invariant

a

(i.e.

freely

G acts

which

exists

of

makes sense, because for have we should

V,

we now

=

a

Ap+dim

define V

reasonable

(X

XG

theory

U)

,

of Chow groups

stacks

Ap (XIG)

=

Ap+dim

V

(X

XG

V)

I

for

quo-

K. Behrend

18

the Chow group

since

of

Ap+dim

bundle

vector

a

equal bundle,

should

V

(X

XG

V)

Ap+dim

:'--:

V

(X

XG

Chow group

to the

be

the rank of the vector

by

base, but shifted

of the

and

U)

I

dim X X G Z < p + dim V, and cycles of Ak is justified This definition by giving rise to an adequate theory. For exof the choice of V and U C V, as long is independent ample, the definition This is proved by the 'double is satisfied. codimension as the requirement fibration see [5]. argument', As an example, let us work out what we get for XIG BG_. Consider of Gm on A, given by scalar the action Gm x A' -+ A, multiplication A' for U exists f Q and Z JO} (t, x) -+ tx. A principal bundle quotient is Thus this has codimension n. good enough to calculate representation > -I we have for -n. Moreover, by definition, p p Ap(BGm) for n >

complement

the

since

has dimension

< k should

dimension

affect

not

-

=

=

=

-

-

all

p >

-n

Ap(BGm) In

Ap+n (]pn-1).

=

particular,

Ap(BGm) A-i(BGm) A-2(BGm)

for

=

0,

=

An-1

=

An-2

all

p > 0

(]pn-1) (pn-1),

etc.

for various see how these n, groups fit together An' --* An. This induces the projection projection An'-n from F)n'-1 to I?n-1. An)

n'

let

To

>

with

a

and consider

n

ker(An

center

,

-4

=

C

U

I I?n-1 Here the

vertical

map is the

Thus

we

map is

have for

all

Ap+n (]?n-1) So

we

case

have

p > =

Ap+n+nl of

fibration

Under the identification in

]?n-1

corresponds

intersections

[H]k.

of the

n'

of

horizontal

and the

n

-

center

Pn'-n-1.

projection

-n

independence

of the double

of rank

bundle

vector

a

of the complement

inclusion

to the

We write

-n

(U)

=

Ap(BGm)

Ap+nl (U)

on

argument.

Ap+n (pn-1) h

=

[H]

Ak(BGm)

=

[H]

hyperplane

the

choice

Ap+nl

(]?n'-I).

of

This

Ap+nl (]?n'-l) in

pn'-I.

and thus =

=

(Qh-l-k,

we

n.

the

The have for

a

special

[H]

hyperplane

same

all

is

is true

k E Z

for

all

Localization

where

and Gromov-Witten

Invaxiants

19

all negative powers of h are 0. A* (XIG) are defined analcohomology groups A*G(X) ogously to the usual A*, namely by operating on AG(y), for all equivariant Y -4 X, where Y is a space (or equivalently all representable Y -4 XIG, we

The

where Y is In

stack).

a

of the

Gm-space

a

=

example BG_

our

Chern class is

that

agree

equivariant

we

get A* line

universal

(BGm)

bundle

A _ (pt)

=

Q[c],

=

where

c

is the

degree +1. Whenever X of Gm a line bundle representation and is in

get via the standard line bundle X x A' over X). The an equivariant is through on A* (X/Gm) the Chern class of this We have c hk hl", and so we see that A* (BGm) is a free Q[c]-module on ho E A_j(BGm)- We may think of hO as the we

XIGm (or equivalently of c E A* (BGm) operation over

line

bundle.

=

-

A*(BGm)

=

of BGm (it corresponds to [?'-'] under any realization An-, (pn-i).) More generally, if T is an algebraic with character torus group M, then A* (BT) (Note how c comes from the canonSymQMQ=: RT canonically. ical character id : Gm -4 Gm.) Moreover, A* (BT) is a free RT-module of rank

class

fundamental

A-, (BGm)

=

=

one

on

,

the generator be only

We shall

Then for

[BT]

degree

in

interested

in

-

dim T.

the

case

where

the

G

group

=

T is

a

T-spaces X, we have that A*T (X) is an RT-algebra and AT(X) is an RT -module. Therefore, RT is the natural ground ring to work As in the usual case (the non-equivariant over. case, where one passes from A* (pt) Z to Q) we want to pass from RT to its quotient field. However, loose the grading, localize so as to not at the multiplicative we only system of homogeneous elements of positive degree, and call the resulting ring QT. Then we may tensor all A*T (X) and AT (X) with QT- Still better, though, is to first of RT at the augmentation pass to the completion ideal, RT and then the homogeneous elements of positive invert degree to obtain QTtorus.

all

=

Comparing For

a

equivariant

G-space X,

dimension

of relative

pullback pullback

is

a

usual

a

A*G(X) The

-+

theory

morphism X -+ XIG, which is smooth G-bundle. Thus flat fact, a principal A (X) -+ A* (X) of degree dim G. 'Usual' A* (X) preserving degrees.

is, homomorphism

defines

intersection

canonical

dim G. It

defines

Lemma3.1. an

there

with

top-dimensional

in

map

AGM X-dimG(X) di

Adim

X

(X)

is

isomorphism.

Proof.

By using the definitions, the top-dimensional

of spaces,

This isomorphism AG dirnX-dimG(X)* Note 3.1.

If

one

defines

works with

this

reduces

Chow-groups the fundamental

cohomology

Hb (X, Hi (G))

= ,

one

to

proving

that

for

of

XIG

a

G-bundle

agree.

class

gets

a

[XG]

Leray spectral

H'+j (X, Q.

in

sequence

K. Behrend

20

Localization X be

Let U

X

=

T-space

a

pushforward by

induced

and Y C X

map

t

After

3.1.

Y

:

subspace such that

T-invariant

fixed

Then

points.

on

have the proper

we

AT(X)

AT (y)

the inclusion

Proposition

closed

a

without

T acts

Y the torus

-

X.

-4

QT

with

tensoring

AT (y) ORT QT F-+ AT(X) ORT QT is

isomorphism.

an

Proof.

Reduces the

AT(X)

ORT QT than

Rather

the

details,

studying

0 and X

Y

case

0. For

=

proof of this

the

U, when the claim

=

that

is

[6].

see

let

proposition,

us

study

an exam-

ple. Consider A*T (pt)

class Let

=

the A*

of BT X

=

(BT)

by

ith

=

JPo,..

(-TO)

FnJ,

-,

-

of T

Xn)

(Ao (t).To

-,

=

7

...

7

=

An and

tQ[AO,

An]

-

An (t)-Tn)-

0),

I

being

may translate

this

0, 1, 0'...,

=

tRT

1

fundamental

the

given by

pn

on

(0,..., (Proposition

where Pi

Then localization

position.

=

Ao

basis

denote

=

the action

-

with

An]. Let us Q[Ao, have A*T (pt) A* (BT)

we

]pn and consider t,

and M

-

RT

Then

t.

I

Take Y

=

n+1

G

T

torus

3.1)

the

the

in

says that

n

(])A T(fp

il)

0

QT -+ AT(pn)

(9

QT

i=O

is

an

isomorphism.

statement

about

Since everything cohomology:

smooth,

is

we

into

a

n

(DA*

ti:

T

(JP1J)

0

QT -*

A*T (pn)

0

QT

i=O

is

an

isomorphism

To understand

of this

namely the projective given by the action of

degree +n. isomorphism note that bundle corresponding to T

on

An+'.

Hence

A*T (1pn)

A* A* =

(BT) [ ]gn+l

(Q[Ao,

-

-

-,

-

we

IT

the

-+

BT is

vector

a

bundle

]?n

-bundle,

E

on

BT

have

(1pn IT)

ci(E) n

An] [6]/6n+l

]pn

+... _...

+

+ (_l)n+l Cn+1 (E) (-l)n+l Cn+1 (E).

(3.2)

and Gromov-Witten

Localization

Invahants

21

a sum of line bundles, each associated to one of the characters An. Hence we have ci (E) An), the symmetric function of ai (Ao, i in Ao, An. In other words,

Now E is

A0, degree .

.

,

.

=

.

.

.

.

.

,

.

,

n+1

n

E(-1)'ci(E) n+i-i

jj(

=

so

-

A,),

i=O

i=O

that n

A*(pn) T

Q[AO

...

7

7

Ai)

An7 i=O

Hence

we

have

=QT[ I/jji--O( -Ai) i= Ai) ,in =0 QT[611(6 rIni=0 QT 11ni=0 A*T (PO ORr QT7 n

A T(Fln)ORTQT

-

by

remainder

the Chinese

theorem.

This

map n

A*T (pn)

A*T (A)

ORT QT

ORTQT

i=O

at

Pi,

by

0 and induced

degree

is of

is the

which

character

0.

(Note

6

that

of the action

of T

cl(0(1))

=

on

back to Ai If we 0(l)(Pi).)

pulls

the fiber

compose with n

n

A*T (Pi)

(9

which

we

weights

(Aj

The residue Let

us

V

-+

t

:

inverse

flj:oi

Ai)j,4-i

QT

tangent tangent

space

The

(Aj

-

Ai)

to

Y

-4

the

in the ith

normal

Tp. (Pi) has component.

of Proposition the 3.1. Moreover, assume that the pullback of a regular immersion T-equivariantly

W x

91 1 V

we

(i.e.

setup

X is

Y

Then

space

formula

return

inclusion

of the above map tI. and so we divide by

get the -

of the

by the tops Chern class

division

is

(9

i=O

i=O

bundle)

A*T (Pi)

QT

have the self v t.

(a)

intersection =

(3.3)

W.

formula

e(g*Nvlw)a,

for

all

a

E

A*T(y),

22

K. Behrend

where

e

A*T (Y)

for

stands

the

we

Euler)

(i.e.

Chern

top

QT is invertible,

0

e,(g*Nvlw)

So if

class.

E

have

v!t*a

e(g*N)' and

have identified

we

1

1

v

C(g;N)e(g*N)

That

invertible, weights of g*N

the

that

X is smooth

and

non-zero

Let Then

us we

and

=

e(N)

so

from

t

fixed

v, then

it

is

theorem

a

one

always invertible.

is

all

E

e(g*N) is, indeed,

that

non-zero.

are

weights

these

that

namely

has to check

just

T

of X under

points

t*,

QT-

verified,

easily

practise

the

at

now assume

have for

QT --+ A*T(y)

T

:A* (X) is in

is

isomorphism

of the localization

the inverse

invertible

in

are

If

always

AT (y) (&QT-

A*T(X)

Vio e(g*N) If X is smooth and

v,

t

we

will

IXTI So if

a

E A*T

(X)

we

[XT]

E

A*T(X):

e(Nylx)

have t* t*

(a) [YT]

e(Nylx)

A*T(X). Now assume that

proper

QT

=

pushforward U11 and

fx

T

an

to

IYTI

t*

a[XT] in

apply this

want to

gives

we

a:=

X is

Now consider

proper.

T

deg (a[XTI)

the

=

(DRTQT cartesian

degT(

e(Nylx)

flat

XIT --* BT is proper and AT (X) OQT-+A* T( Pt) (&

pullback commutes diagram

Tf

0

(a)

e(Nylx)'

=

XIT commutative

T

tilldiagram x

Since

Then

homomorphism deg

get

in AT(P t)

equation

moreover

a

with

>

>

proper

Pt

BT.

pushforward,

we

get

an

induced

and Gromov-Witten

Localization

23

Invaxiants

deg

A.,(X)

(3.4)

0

degT

AT(X) where the

-

homomorphism 0 : QA...... Diagram (3.4) fits into

A,,]

is given by sending diagram

larger

the

and the Ai to 0.

I

t to

deg

A* (X)

I

degT

AT (X)

t9jZ

>

(3-5) T

A* (X)

(9

QT

0

QT-

I

degT

_---'-de

AT(y)

3.1 (Residue Corollary Formula). X If a E Adim (X) comes from a

Tfy is

contained

fx

=

deg a[X]

(X),

E

t IZ

(a)

e(Nylx) Q and

=

0

deg

T

t

=

v.

then

have

we

a[XT]

=

0

Tfy

(a) e

(Nyl X)

*

factor of t. element P T(X). for corresponding element of of A* (X). Let a E A*T (X) and write a for the corresponding A* (X). Then if deg dim T, then deg b 0 and deg a deg a The 0 in

2.

a

Assume X is smooth and

1.

dim X

E AT

0

the submodule

in

U111

General

formula

this

case.

only

Assume

to

serves

E A

remove

Write

the

b

the

a

Again,

this

=

0

=

0

is to be

a

deg

T

=

T

Odeg

(t*

(a)

interpreted

contained

in

Q and after

n

to

Tf,!,a is

=

-

-

a

n#

=

OdegTa

=01

e(g*N) mean

,io

that

Oa e

(g N) *

removing

t

n t*

we

get

fb

a.

e(g*N) t*a

e(g*N)

*

(3-6)

24

K. Behrend

Proof. degrees.

This

just

is

Remark 3. 1. at

element

an

Tfy

can

/

e(Ny/x) of T to

one-parameter

X)

by evaluating

be calculated

Y is the fixed

Assume that A*T (Y)

Tf Y

action

locus

at

D A*

(Y)

a

=

E RT c A*T (Y),

(a)

-

Tf ,!,3

Then AT (y)

locus.

&Q RT If t* t*a

t*

e(g*N)=

(a)

Tf,!,3

This

p.

e(lVy/x) in practise. evaluates 2. The standard way to ensure that a comes from bundles. vector in Chern classes of equivariant I

0

zero

the

of T and of the one-parameter subgroup will be " .() will not vanish at P. Then of and the denominator (N-y

t*(a)

TfY

degree

corresponds to restricting subgroup. For a generic one-parameter

corresponding subgroup the fixed same

of

of

track

keeping

and

M'

E

[t

the

the

(3.5)

using

function

the rational

Evaluating

1.

chase

diagram

simple

a

how

also

is

one

polynomials

is to take

A,, (Y) &QA. (BT) and then

e(g*N)

formula. by the projection with line bundle has a filtration on Y, and Nylx Also, if T acts trivially where + Xi), c(Li) E A* (Y) is rli(c(Li) Li, then e(Nylx) quotients This gives a T of the on Li. the Chern class of Li and Ai rr- RT weight formula. form of the Bott residue very explicit

4.

=

Example

of T.

points

E(oo) A,

are

we can

to

E be

Let

on

calculate

Let

on

a,,...,

P1,

P1,

in such

equivariant

an

of representations T of the weights

p, T acts

....

which Roch:

Let T operate

3. 1.

i.e.

t

T. Let on -

the

weights

a,,

be these

1

a

way that

bundle

vector

A,

0 and on

00 are

A, be the weights,of

....

the fixed

V. Then E(O) and T

on

E(O)

and

through E(oo). Also, 0. Then Assume that HI (1?1, E) w(t) Riemannof T on HO(1?1, E) by equivariant Riemann-Roch have Then we (apply weights. let

w

be the

character

=

=

-

VIT -14 BT):

ch(HO (IF', E))

=

degT (ch(E) td(T,)

n

[IF' T

or n

e'i

by localization. and

on

Tp-,(oo)

td(Tpi(0)) C1 Mp (0))

ch(E(O)) =

Now since is -w,

we

ch i-

td(x)

and the

i

weight

get

n

eai

or

(E (oo)) td (Tp (oo)) C1 Mp (00))

e-w

+ch(E(oo))

ew

of T

on

Tpi(O)

is

W

Localization

and Gromov-Witten

Invariants

25

n

ei

+

e-W

in

QT.

Note that

we

have

determines

This

the ai (which holds

is the formula

ew

uncapped with [BT]. Useful to calculate uniquely. for

eaw ew

all

a, b E

context

Z) b

ebw

+

the ai in this

=

e-'0

E enw n=a

where for

a

> b + I we set

Lecture

4.

Eb

The localization

III:

Gromov-Witten

invariants

Using

formula

the localization

calculate

Gromov-Witten

associativity

of the quantum genus. The idea of

low)

very

[13].

to Kontsevich

of Mirror

variety

invariants

n

to we

Let stack

one

the Bott

the

a

will

calculation treat

the

marked points.

Vg,.

and

degree

For

affine

an

in toric

a

of projective be of characteristic

maps of

[11]

in this

verify

to

is due

context

the

the

still

predictions

varieties.

under finitely many fixed points the calculation of its Gromov-Witten

d to

combinatorial

space P'. 0. Let

case

ground field

of stable

have to

in has

reduces

on

formula

intersections

interested

we

WDVV-equations (i.e. analogues for higher (but

and its

by Givental

formula

methods

the

applying

we are

for

of the most useful besides

product)

eiW

formula

It has been used

the Bott

action,

lecture

is

invariants,

symmetry for complete

If the torus

Ea-1 i=b+l

en,

n=a

?I, whose

k-scheme

source

U the

problem.

In

Mg,,,(?1,d)

denote

is

curve

a

genus g

a

this the

with

groupoid

Mg,n OF, d) (U) is the

groupoid

of such stable

maps

C

parameterized ----

f-->-

by

U. These

are

diagrams

]F)r

7rI U

U is a family 7r : C -+ of prestable with n sections and f is curves condition family of maps of degree d, such that the stability is satisfied (see, for example, [13], [14], [10], [4], [2]). Evaluation at the n marks defines a morphism ev: M9,n (]pr, d) --- (1pr)n.

where a

Gromov-Witten

invariants

are

the induced

linear

maps

K. Behrend

26

A*(]Fr)(gn a, (9

For

>

g

Vg,n(Pr,

f

0 an

...

ev*

[Mg,n(PI, d)] is d) (see [2], [3], [1] or [15]). This

the

cycle

0 the

(a,

(D

an).

...

fundamental

'virtual

the

Now consider

canonical

generators

QT C Q(AO)

AT)

)

...

-

The torus

(t (X0, i

induced

an

X

Ao

....

]?r

on

Mg,n (pr, d): given f

C

>

general,

M, whose

Q[Ao,...,

Ad

and

Ar W Xr)

(,\O WX0,

-4

so-called

in

dimension

group =

a

pr

___

of T

action

character

with

An. Then RT on pr by

T acts

Xr))

i

Gm'+'

=

denoted

T

We get

T

torus are

cycle giv-

constructed

a

of

class'

carefully (i.e., theory of Gromov-Witten invariants ing rise to a consistent field theory, [14]). The usual fundamental cycle is, cohomological not even in the correct degree, as 'Hg,n (IF', d) may have higher because of the presence of obstructions. than expected, is

t E T (U)

and

pr

7rI U in

Mg,n (Pr, d) (U)

we

define

t

-

C-

(C, f(C, (7r,f)

t

>

UX?1

turn

this

f ),

o

where

(C,

t

f ) stands for

o

t

7rI U We leave

it

as

an

exercise,

to

into

an

action

of the

group

T(U)

i.e., actions on the morphism and object groupoid Vsq,n(P,d)(U), under with all the groupoid structure sets compatible maps. Compatibility change of U gives the action of the algebraic group T on the algebraic stack

on

the

Mg,n (P', d). The

same

fundamental

general of

class

fMg,n Rr, d)T]

mental

class

virtual

fundamental

Formula

If A*

(pr),

a,,

(3.6) .

.

.

with ,

an

class =

allow

give

rise

T(Vg,n(Pr,

[Mg,n (1 r,d)] and a,,

Gromov-Witten

.

G

.

the to

,

an

equivariant

which

pulls

virtual funda-

We shall

apply

=

are

invariants

of the virtual

back to the usual

A*(Vg,n(1P,d)). FMg,n (?', d)].

and b .

construction

an

d)),

E A

FMg,n (yr, d)TJ

E A*T (]?')

the induced

that

arguments

Vg,n(P',d)

the are

corresponding given by

classes

in

and Gromov-Witten

Localization

f (-M,,,,,

(a,

ev*

(D

Invaxiants

27

an)

0

(4.1)

(Pl,d)]

I

0

(a, (9 e(g*N)

0 ev*

an)

&

...

vI[_M_,,,,(P'I,d)T] at

if the ai this

least

diagram

such

Ei'_1 deg ai

homogeneous and we need to formula,

are

apply

To

construct

(4.2) V

V

as

is

v

regular

a

Mg,n (IF', d).

namely equal of more general

problem fixed

step

in

follow

locus. such

what

As stable

several

into

the

a

shall

we

invariants

the

integrals approach has

(See [11],

We shall

the

right

hand side

involved. and

1.

e(g*N)

fixed

locus

evaluate

of

(4. 1)

i.e.,

we

restricted

on

all if

the fixed

can

is

be described

turn

locus.

a sum over

all

in

,

terms

out

i.e.,

for

each

to

the

fixed

locus

ev*

(al

reach

we can

of stacks

be non-trivial.

to

The connected

graphs

of

marked modular

(,r, d, - )

the

component

classes

Moreover, this Still, invariants.

components

(-r, d, -Y).

marked

VI

of

Thus the

graphs

components given by different

determine

we

the computation of GromovM,,,.. Since the fixed locus has

by marked modular

the fixed

Y until

be chosen at each

on

fixed

of

small

class.

reduces various

as

The point decompose the

us

[-M,,,, (IF, d)TI

of vI

points Y

M,,,, (PI, d).

Wwill

-+

take

we

too. Mg,n are non-trivial, in determining Gromov-Witten more details.)

the

indexed

V

:

track

combinatories

determine

are

v

fundamental

on

Wecan treat

separately,

keep

we can

has to

next

locus

fixed

of T

locus

immersions

been very successful [12] or [7], [8] for

[13],

results

useful, because it lets We pass successively to smaller

computation

a

best

still

Y is

(4.1)

the

one

get

the fixed

the virtual

the

and Y contains the

Thus Formula to

components,

many

to

see,

Hg,n.

we

to

steps.

way that

W

immersion

regular

The

happens

curves

Witten

we

closed

Of course,

possible,

of view

T-equivariant

(Yr, d)

M,,,,,

-

--

91

T in

deg[M_g," (1PI, d)].

cartesian

(3.3):

as

Y

where

=

a

(,r, d, 7) graphs

FM-g,n (IF, d)T]

given by (-r, d, -y).

Then

have 0

(9

...

(4.3)

an)

t7MWg,n(PI,d)] T

0 VI

FMg,n (Pl,d)T]

b* ev*

(a,

an)

K. Behrend

28

The fixed

Recall

locus

that

prestable

graphs are They

modular

marked

F, (which

either

can

be tails

graphs that

the

of

consist

curves.

pair

or

edges),

up to

markings of the vertices, giving the vertices and edges E,. The set of flags connected with A vertex

(the

is

(-r, d,,y)

be

least

at

of

-r'

a

1

non-negative Tails

the vertex

v

graph

is

a

valence valence

by contracting

is obtained

The set of vertices

vertices.

integer S, denoted F, (v). denoted

are

genus is one and its its genus is 0 and its

or

modular

and

genus.

a

2, its

is at least

degeneracy type of V,, a set of flags

the

of the stabilization

see [4]. details, marked modular graph of the following

For

vertices.

type.

modular

T' graph which is connected and whose stabilization the is and to the of set tails Moreover, empty. equal genus g(,r) g S, f 1'.. nI. d : V, -+ Z >0 a marking of the vertices such that by 'degrees', a

-r:

Note that

degree

:

V,8

the

i-th

-Y

Vu

:

and

x

C)

-y

F,

:

These data

Every edge

3.

-y is constant if v is a stable

4.

if

is

v

a) b) Fix

-y

(i)

y (i),

such

1.

Wr")

2.

M(P'I

an

a

for

V,,

d for

the

where Pi y associates

..,

=

to

(0,..., every

1,...,0), stable

7-,

the vertex

and

r

<

=

being

1

of

...

r

ly,

y in

PJ;

so

to the

-f associates

following

list

vertex,

i.e.,

unstable

to

of

every

flag

compatibility

no

edge

for

all

connects

-y(v)

then

vertex

vertex

a

fixed

to

in

fixed

a

...

10)

every

point.

requirements: stable

vertices,

for

all

i E

F, (v)

=

7(i),

i E

F, (v),

then

F, (v),

i E

are

marked modular

distinct.

graph (,r, d, -y).

The

following

rIVEV,' Mg(v),F,(v)i

d),

graph

i :5 rJ, where Lij (0.... 1XI the j-th so -y associates position; one-dimensional orbit closure, < i

in what follows: =

of the

marking

edges,

on

E 7 (v),

all

important

has

unstable

an

a

subject

1.

E

graph.

i-th,

JPo,.

2.

letter

same

so

10

vertex -+

are

fLij

-+

v

maps:

is in the

unstable

vertex

IPO,...,P,},

-+

position; of T on F',

point -y

the

of the

of three

y consists

b)

S, is

d.

=

we use

the total

stable

every

=

a)

is not

-,

b, for a) d(v) b) EvE V d(v)

3.

bounds)

it

unstable edges containing equal to the set of stable

=

2.

genus is

3. The stabilization

Let 1.

its

flags

number of

at least

all

if

stable

is

give

of vertices

set

a

which is defined

as

the fibered

product

stacks

will

be

Localization

MR',

1

and Gromov-Witten

(4.4) (pr

where the vertical

V(P'

d; ^ ),

'r,

incident

with

=

y(i)

i,

fe(j)

the mark of the is

closed

a

Stacks

defines

.

is the

d)

-r,

fo(j)

indexed

by

a

in

a

has

-r,

d)

element for

with

xi,

we

words,

maps

as

in [4]. Given a colwe can M(?',T,d), i2} Of 7-7 every edge jil,

of

xj2.

Doing this

in families

(4.5)

as

a

d)lAut(T,

no

edge

construct

be

a

a

k-valued

of stable

Aut

to

d)

Then

of

(-r, d))

group acts

on

degeneracy

as

because

is true

the result

every

stable

vertices. -+

H(IF,

-r,

marked curves,

H(,r').

of

point

stable

7W(-rl)

morphism

collection

(up This

connects

Aut(-r,

7Vg,n (IF', d)

map in

uniquely

fv)VEV,,iGF,.

Xi)

automorphism

Then

Hg,n (P', d).

-4

any stable

that

A,

d)

the

produce

xv a

d). :::::

(Xi)iEF,

collection

(v) I in of stable

follows: v

,y(v) for

shall

xv)vEv,,

for

-r,

Mg,,, (IF", d).

-4

be written

can

0 and

degree

other

2.

T,

One has to prove

(-r, d) or worse, gluing a collection

type

I.

is

d; -y)

finite

H(P'7

Next,

and xi

H(pr,

Clearly,

i.

detail

great

an

by identifying

morphism such

immersion.

Let(Cv,

is the vertex

vertex

d).

4. 1. Let Aut (-r, d) be the subgroup of Proposition the degrees d. of the modular graph -r preserving induces a closed immersion M(IF, -r, d) and (4.5)

vertex

by requiring

0(i)

(4.5), giving rise to a boundary component morphism. But because of the special nature (4.5) is actually a finite 6tale morphism followed by More precisely:

general,

M,,,, (?I, d) is only (-r, d) in our context,

of

defined

morphism

of

Proof.

d)

-r,

.

studied

are

V(]pr)

for all i E F, Here map indexed by this

d) by gluing,

Ca(i,)

and

Pr},

]pr)E,,

X

maps,

of

representing

map in

Ca(i,)

the

closed

,

of

]W(]?r,

of T,

In

a

.

stable

M(Fr,

of

.

curve

source

substack

stable

a

curves

substack

jPo,

E

of type

associate

evaluation

are

is the

(CV,XiJV)VEV-iEF,,

lection the

maps

which

fo(j) (xi)

that

29

d)

-r,

(]pr)E, I

Invaxiants

E

E v

E

V,

jP0,

a -

-

stable -,

let

vertex,

fv

:

C,

--+

Pr

by

PrI,

Vr unstable,

Cv

let

fV

:

?I Z

=

PI and fv be --+

P,

=

F-4

-Y(V) d(v) z

C Pr

the

constant

map to

K. Behrend

30

marks

Then put 0 = (1, 0)

Xv

morphism

This

7

A)

an

let

xi

way such that

unique of

element

finite

a

F,(v)

each i E

V(Ipr,,r,

E Cv be equal f, (xi) -y (i).

d) (k). Again,

morphism M(r') covering followed by

6tale

to

=

the desired

obtain

is also

for

in the

E V,,

v

we

Pl:

=

(0, 1),

=

and

done in families

C,

on

oo

A)

defines

This

or

-+

this

d).

-r,

closed

a

be

can

M(F',

immer-

sion:

Proposition

d(v)-th

of

=

rIvEVI

Let p act

trivially

Let

4.2.

roots

1.

p

/-td(v)

where Ad(v)

i

on

V(-rs).

cyclic

the

is

Then

have

we

a

of

group

closed

im-

mersion

M(-r')1p

M(]?'l

--+

(4-6)

d).

-r,

n

We can say more, because, phism (4.6). More precisely,

Proposition and

(4.6)

The semidirect

4.3.

induces

a

closed

fact,

IG

Aut(r,

group

G

product

-4

V(?',

=

-r,

d) /

together,

4.1 and 4.3

Propositions

the

d)

acts

Aut(T, d)

p

the

on

acts

on

mor-

-H(,rS)

immersion

V(-r')

Putting

in

Aut

(T, d).

we

obtain

the composition

(4.7)

V(P')

V(-r-')IG which is

a

closed

T7 d) /

Vg,n (Pr d),

(r, d)

Aut

,

immersion.

Consider

4.4. Proposition classes Of Vg,n(]Pr, the image Of 4 (7-,d,-y)

d)(k). (k), for

the group some

T(k) acting of

An element

this

marked modular

of isomorphism fixed if and only if it is in graph (-r, d, -Y) as described on

set is

the set

above. In this

sense,

the image of U P is the fixed locus Of Mg,n (pr, the fixed component calling the image Of P(r,d,-y)

justified (r, d, -y). But if we endow M(Ts) IG To make it so, is not T-equivariant. in

are

Consider "

on

finite

V

the

'

Evev.

=

character

MQ,

Av 177V

C

Lij

7(v).

=

action

we

by

!P(,-,d,7)

of T, then

torus. have to pass to a larger M C MQ= M(9z Q and let H = M+

be the torus

character

with

of T

character

through group

T. Wecan view passing from homomorphism T by d(v). A, divisible

make the character

Thus

indexed

we

group where A, is the

Let

the trivial

with

d).

T to

which

T acts

We have as

a

a

way to

and Gromov-Witten

Localization

T

The torus

acts

2-isomorphism

a

on

V,,. (PI, d) through id xP

jw(e)

>

1

proj

Let

describe

us

0

So for

morphism

0

t

:

":--

points.

for

v

E

VI,

we

let

v

E

V,",

we

let

Cv C,

Ov 0,

--+ =

Y on

=

to think

T, acting

back to

A-

a

have

Then

]?'r

>

It

C

7 (V)

>

image Of 4 (7-,d,,y)

jFr.

as a

fixed

component of

d).

Diagram (4.2),

leads

This

C

IAv

)d( ) >

of the

o

7

we

(t)Z'

d(v)

-Y(V)

on

V(7-8) IG(r,d).

lj(,r,d,,y)

M(-r').

trans-

define

diagram

fv=(.

CV

Going

natural

=

f

I

0-

is better

a

Cv be the identity, F1 -4 C, V be given by

the commutative

into

=

7

CV

than

define

We need to

Z -_4

Thus it

d).

-

for

rather

Mg,,, (P, d)

M,,,.(pr,

>

7

1.

fits

construct

(t, (C,, x,)) of T(k) x V(,r8) (k) we need to !P (C, x.,) -+ 0 (C, x,). as above, Using notation (Cty Xv MvEv, and t O(C, xv) (Cv xv, t fv).

2.

which

x

each

-

C(Cv) Xv)vEV,)

now

1action

4i

k-valued

on

T

4

H(T.) formation.

T. We can

31

diagram

0 in the

X

Invaxiants

The

to the

we can

integrals

now

Y is.

IG(r,d)

will

We shall

use

be evaluated

factor

correction

11

X

.L -

-

# Aut(

#G(r,d)

say what

on

'r,

d)

v

E V,-

X

d(v)

We shall next show how to obtain regular immersions v : V -+ Was Diagram (4.2). As mentioned above, we can treat each fixed component Wewill proceed in several to the following separately. steps, corresponding in

factorization

of -P:

IG

-114

Aut

(-r, d)

M(-r') where A

determine

=

vI

[M-.,,,,,

_M(F", -

-r,

d; -y) /A

For each step

(?", d) T]

and

- 14 we

1, e(g*N)

M(IF",

shall

-r,

d) /A

construct

-,-*+ -Mg,,, a

suitable

(IF, d), v

and then

K. Behrend

32

The first

step

diagram

following

Weuse the

V(Pr,

for

(4.2):

Aut

(T, d)

d) /

-r,

91

(pr, d)

1

9R(,r)/Aut(-r,D)

and closed

diagram

this

Wenote that

the

in

n

product.

cartesian

fixed

locus

M(Pr '-r, d) I Aut(,r,

but

cartesian,

is not

(T, d, 7)-component of the 9Jtg,n stands for the (highly genus g with

9Rg,n

Since

we

non-separated)

only

are

Artin

of

stack

is open in the

is sufficient.

moment, this

at the

d)

interested

prestable

Here of

curves

Moreover,

marks.

9x(-r)

II

=

VEZ,

morphism 9X(,r)

and the ,r.

The vertical

9R,,,,,

with but

immersion,

[16]),

which is

It

of the

way to to

classes

in

one

bundle

regular our

We also local

that

note

(for

immersion

not

closed

a

terminology

see

purposes.

define

fMg,n (Ipr d)T]

=-

,

the

virtual

bundle

summand for

of

7-,

classes, along

proof

used in the v

preserves

virtual

d)

set

9A(r)

in

edge

7%

of

into

For the

V(Ipr,

-r,

of virtual

product

splits

Vg,,, of

d)].

d)/Aut(,r, class

A of the

via

each

rM(]Pr,

fundamental

Gysin pullback Diagram 4.4.)

the

a

direct

edge jil,i2j,

is to

it

fundamental

sum

of line

bun-

the normal

line

is

X

Tj

(W')

(9 X

22

of the universal and xj2 are the sections where xil the flags il and i2 of T and w is the relative dualizing bundle. w' relative is the whose tangent dual, curve, the

is

v

this

classes:

The normal

dles,

a

for

is sufficient

VI

(One

T-action.

trivial

the

certainly

fundamental fact about virtual a general that the Gysin pullback WDVV-equation,

fundamental

equal

to the edges of is given by gluing according the prestable given by forgettinglhe map, retaining if we endow 9X(T) The diagram is T-equivariant,

are

stabilizing.

without

curve,

and

maps

Chern class

of the line

bundle

x (w)

on

9X(-r).

fil,i2JEE,

-Cil

corresponding

to

sheaf of the universal We use notation

ci

for

Then

JL

e(g*N'

curves

Ci2

(4-8)

Localization

The second

shall

33

V(?',

T,

d; -y) / Aut (T, d)

M(F')

T,

d;,y)

T

stable,

V(?')

--

d) /

T,

Aut

(T, d),

we

consider

Wecall we

Invariants

step

considering

of

Instead

and Gromov-Witten

an

call

tail)

edge (flag,

Weshall

unstable.

it

of

if it

S is

finite

a

ments) and Mo, s (IF', d) f (xi) -y(i),

-+

-y is the closed

for

=

Lemma4.1.

#S

For

follows

in

Vo,s (P',

d;

we

have

M(Fr

d; -y)

=

-

d; 7(S))

-

X2))

=

11 Vo,F,

x

(,)

V,'

fits

d; 7)

-r,

E

(v) +

d; -y)

2 ele-

is

smooth

0, for

a

stable

(,)

the

ex-

map

f

(I?r, d(v); -YF, (v)),

is smooth of the

dim

Hg(v),F,

(v)

'expected'

(I?r, d(v))

dimension

r#F.

-

r

into

the

T-equivariant

cartesian

diagram

_rTV 11 EVMg( v),F,

-r, d

Axid

(pDr)EX(]pr)S,X(]pr)V,'

(v)

(IFIAv))

Iexp

9

>

of

E Vu

M(FrI

g,

or

V E Vu

v

morphism (4.9) -r,

S has 1

that

cases

(S)).

E V,'

Now the

M(]pr

*

Vg(v),F,

dim v

Ro,s (]?',

H1 (C, f Tp, (-xi

V(F',

that

particular,

the

a

r#S.

-

J1 Vg(,),F, VE

and in

Otherwise, following type:

vertex.

of the

map. The stack Ho,s(?', d; -y(S)) C of stable maps f , defined by requiring that

is

2, the stack

< -7,

from

Note that r,

stacks

d; -y(S)),

consider

Mo,s (Pr, d)

dim

This

C -* F'

substack

(4.9)

i E S.

all

pected dimension

Proof.

(we only JP0,..., P,,}

set

S

:

d).

T,

meets a stable

need to consider

Ho,s(]?', where

V(P'l

--+

>

(ffDr)F,

(]Pr)V,'

X

q

(Ipr)Fu The

morphism

e x

p is the e :

11 jW9(v),F,(v)(]pr vEV,

and the projection

product

of the evaluation

,

d(v))

___

morphism

(]pr)F,

X

(1pr) V,. (4.10)

K. Behrend

34

VE

V.,

(IF', d(v)) VE

v

x

is

just

identity a

of the

of the

identity

X

(]P'r)v-'.

x

change

product

is the

(IF")S,

on

base

(]?r

Pr) -?4 (-]Fr)V"

E V,'

product

id is the

(I?r)E, and the

X

V,'

morphismA

The

X

of the

diagonal

]?r)E,

=

(yr)F,-S,

The square to the upper right of (4.10) square of V(Pr, T, d) The morphism

defining

v

-

(pr)S.r

X

morphism

and the

Pt

,Y:

(I?r)E!r

-+

X

(Ipr)Sr'

I*

X

(Ipr)V,"

marking -y on the graph (T, d). The morphism g is given by to stable flags and is, in fact, constant. points corresponding out the factors The morphism q projects to stable flags. corresponding Finally, vi is given, again, by -y. is smooth, but not of the of (4.10) The stack in the upper right corner It has a virtual class given by fundamental dimension. 'expected' by

induced

the

at the

evaluation

11 (H (v)

v

e

2Tp,

(-y(v)))

[

H(v)

Here

is the

is

(,)

(IF', d(v)) TI

corresponding then

curve,

general

H(v)

to the

vertex

7rv,,(WcJ,

=

of virtual compatibilities the virtual x id)l gives (,A

fundamental

vI

fM-(P')

-r,

d) TJ

by Lemma4.1, the big (total)

vi(the

Thus

vI [-M(I?')

T,

(

:

C, is the

class

(4.11))

=

vit (the

square

is

equal

in

classes

fundamental

M(?"I

But

If 7rv

v.

where we,,

(4.11) pulled in T, d). Now because there is no excess intersection of (4. 10), we get the same class in M(P'l T, d; -y) by pulling steps via (,A x id)l and vI or in one step via vi. Thus back via

that

either.

(4.11)

sheaf.

of the

part

bundle'

universal

dualizing

relative It

'Hodge

is the

Mg(v),F,(v)

11 Vg(,),.p, VEV,

V CS VI

class

(4. 11)).

(4.10)

has

the lower

class

rectangle

(4. 11)

back

no excess

in two

intersection

to

d)TI

II e(H(v)v 2Tp,,(-y(v)))

fM-(1?',,r,d;Y)TI

VEV,'

II

II (Ai-Ay(v))9Mct(H(v))Jt=

vEV,'i:A,y(v)

XT=-I

rM-(]?')

T,

of

d; 'Y) TI

and Gromov-Witten

Localization

Because

the

e(g*(N))

is

35

Invahants

g*(N) is constant morphism g in (4.10) is constant, just the product of the weights of T on g*N. Thus

and

1

so

(4.12)

e(g*N) 1

(Tp,- (7 (j

e

jEE,u

(Tp, (-y (i)

e

jESu

H e(Tp-(-y(v))) iE V.

1

The third

M(-r')

Ai

VEV,' 0'--f(v)

-

A-Y(V)

step consider

Weshall

A-'(j)

Ai

i:A-y(i)

jEEruUSru

/p

--+

morphism.

the M(?',

d; 7)

-r,

M(,r')

=

11 VO,F,

x

(]?', d(v);

(v)

yF, (v)),

V E Vu

which

we

stacks

without

T-equivariant

the

into

may insert

of smooth

diagram

cartesian

intersection

excess

H(TSVP

>

]W(TS)

X

11 VO,F'

(v)

(pr, d(v); 7F, (v))

V E Vu

I

9 ,(

VO,F,

BPd(v) follows

that

To calculate

thus

reduce

to

vl[M(]?''-r,

d; 'Y) T]

considering

notation,

let

us

case

of

v

the

(v)

v

into

#Vl morphisms and

(IF', d(v); -yF, (v)).

positive

integer

---4

]WO,2 (Ipr

,

1

leave

consider

having valence

by requiring

of v, factor

VO,Fr

---4

MO,2 (]?r is defined

[M

morphism

the

BILd

(the

=

bundle

normal

the

BAd(V) To fix

(P', d(v); -yF, (v)).

V E Vu

V E Vu

It

(v)

(4.13)

a

,

we

d; Po, Pi)

(4.14)

d; Po, Pi)

to the C

d and

reader).

The stack

VO'2 (1pr d) ,

marked point the image of the first to be P1 E pr.

image of the second marked point stable map The particular

to be

Po

E Pr and

K. Behrend

36

f:?'

___+

z

(where

x,

=

0 and X2

z

MO,2 (pr, d; PO, PI) and gives bundle to (4.14) is the tangent space T

on

equal

(4.15)

d

the marks

are

oc

=

P'=LolC?r

-+

on

V)

is the unique fixed point of normal morphism (4.14).The VO,2 (?I, d; PO, PI) and hence

the

rise

to

to

(4.15)

in

to

HO(?1, f *Tp,(-O Wecalculate

the

using Example

(ai)

Let acts

the

oo)(0)

of T w

f*Tp,(-O

and

(Ai

weights

has

weights character

the

denote

-

HO(V, f *Tp,-(-O-

of

oo))

-

and

(4.16)

oo)).

HI(?1, Tpi(-O-

00))

3.1.

V via

on

f*Tp,(-O Tp,(Po)

weights

oo)) / HO(?1, Tpl(-O

-

-

on

H'(71,

L

=

f *Tp,(-O

dAO.

We also

oo)(oo). and Tp,(Pl)

To calculate

-

Ao)ioo

weights

has

oo)).

-

The torus

need the

of

weights

these,

that

note

(Ai -Aj)j:Aj. (-00) changes

The

the by (-0) Twisting But has respectively. Tpi(oo), Tpi(O) weights by Tpi(O) weight AldA has weight and Tpi(oo) oo) are AOdA'. Thus the weights of f*Tp,(-O Then by Example 3.1 at (oo). A, + uj)it-l Ao (Ai (,))jOo at (0) and (Aj holds

same

f*.

applying

after

and

and

0

-

we

-

-

-

have e'i

I-e-I

Ej:AOe'\i-AO-'+ Ej eXj

I+

+

'useful

formula'

the

that

HO(PI,T]pi(-O Similarly, the weights of (4.16)

-

for

(

(1-d). e

+

I-e-

Ed-' n=1 e

-+-=d

e'+,\O-xl 1-el

e-' I-e-w

(4.17)

e-nw

Ai_RAI-nIAO d d

n,m:AO

in Exercise

oo))

+

3.1.

is one-dimensional

and has

weight

0,

so

are

Ai

We deduce that

e-'

Ej e,\i-Xo

mentioned

by

-Xo

Eri==o F,

+

(1-e-,

Ai -AO

=1+Eallie

Fj:AjeAi-,\1+'

4

1-el

_

%,

m

_

the normal

-Ao d

d

bundle

)

.+,n=d n,->O

N of the

morphism

v

in

(4.13)

we

have

(4.18)

e(g*N)

II E V. ,

jvj=2

-y(,)=Lab

i=0

ri n+m=d(v) n,-:j6O

Ai

-

"Aa d

-

1"Ab d

Localization

1

H E V.,'

and Gromov-Witten

Aa

n+m=d( ,,,)

"/\a d

Ab

+ _= d( )

n,-00

1v1=1

37

I

ri

M/\b d

-

Invariants

n d

Aa

Ab

M d

.960,1

-y( )=Lab

i=O

Ai

n+-=d(v) n:00

i0a,b

nAa d

-

-MdXb

-

Conclusion Wehave

now

completed

VI[Mg,n(Fri

d)T](,r,d,-I)

the computation

of the

right

hand side

of

(4.3).

We

have

H 11 VEVI .r

and

(4.18).

A.,(,))g(')

-

ct(H(v))It=

fM_(-r')1G(7-,d)1

i

_

Ile(g*N)(,r,d,-y)

and

(Ai

i: 4_&) is

When pulling

the

product

of the

three

-

Ji142JEE-r

Ci

(4.8),

contributions

(4.8),

back the contribution

which

(4.12)

is

Ci2

-

replace -ci, for an unstable flag i E F, by the weight of T on where f i, jJ is the edge containing i. weight is d Thus we finally arrive at the localization formula for Gromov-Witten inof F. Our graph formalism variants is well-suited for our derivation of the formula. To actually it is more convenient to translate perform calculations, our formalism into the simpler graph formalism introduced by Kontsevich [13]. But this, of course, just amounts to a reindexing of our sum. to

M(-r'),

Tpi (xi).

we

This

Bibliography 1.

K.

Gromov-Witten

Behrend.

127(3):601-617, 2.

Algebraic

K. Behrend.

ings of the conference 3.

K.

Behrend

128(l):45-88, 4.

K. Behrend ants.

5.

D.

and B.

Edidin

in

algebraic

geometry.

Invent.

Math.,

Gromov-Witten on

Algebraic

Fantechi.

invariants.

Geometry, The intrinsic

In M.

Reid,

Warwick

1996.

normal

cone.

editor, 1999. Invent.

Proceed-

Math.,

1997.

and Yu. Manin.

Duke Math.

131:595-634,

invariants

1997.

and W. Graham. 1998.

Stacks

J., 85(l):1-60,

of stable

maps and Gromov-Witten

invari-

1996.

Equivariant

intersection

theory.

Invent.

math.,

K. Behrend

38 6. 7.

8.

and W. Graham.

D. Edidin

Localization

in equivaxiant intersection theory aad 1998. Math., 120:619-636, and Gromov-Witten theory. Hodge integrals

Amer.

the Bott

residue

formula.

C. Faber

and R.

Pandhaxipande.

Preprint,

math.AG/9810173.

J.

Pandharipande. partition Hodge integrals, math. AG/990805 2. Preprint, Intersection W. Fulton. Theory. Ergebnisse der Mathematik biete 3. Folge Band 2. Springer-Verlag, Berlin, Heidelberg, C. Faber and R.

matrices,

and the

und ihrer

GrenzgeTokyo,

A, conjecture.

9.

New York,

1984. 10.

11. 12.

and R. Pandharipande. Notes on stable W. Fulton maps and quantum cohoAmer. Math. Cruz 1995, pages 45-96. geometry-Santa mology. In Algebraic Soc., Providence, RI, 1997. Math. Res. invariants. Internat. Gromov-Witten A. Givental. Equivariant 1996(13):613-663. Notices, Localization of virtual classes. Invent. T. Graber and R. Pandhaxipande. math.,

1999.

135:487-518, 13.

M. Kontsevich. space

of

curves

of rational

Enumeration

(Texel Island,

1994),

curves

via torus

pages 335-368.

In The moduli

actions.

Birkhiiuser

Boston,

Boston,

MA, 1995. 14.

Gromov-Witten

and Yu. Manin.

M. Kontsevich

and enumerative

geometry.

Communications

classes,

quantum cohomology, Physics, 164:525-

in Mathematical

562,1994. 15.

J.

Li

and G. Tian.

algebraic 16.

A. Vistoli. Invent.

varieties.

Amer.

Intersection

math.,

moduli

Virtual

J.

97:613-670,

theory

Math. on

1989.

cycles and Gromov-Witten 1998. Soc., 11(l):119-174, algebraic stacks and on their

invariants

moduli

of

spaces.

and Branes

Strings

Fields,

C6sar G6mez1 and Rafael

Spam 2Instituto blanco,

CSIC Serrano

What is your To show the

aim in

fly

Canto-

phylosophy?

the way out

Philosophycal

Wittgenstein.

123, 28006 Madrid,

Aut6noma de Madrid

Universidad

C-XVI, Te6rica, Madrid, Spain

de Fisica 28049

2

Fundamental,

y Fisica

de Matem6ticas

Insiituto

HernAndez

of the

fly-bottle. 309.

Investigations,

Introduction

physics is finding a consistent high energy theoretical time the For theory is the best being, string quantum gravity. theory to quantum gravity think that the solution candidate at hand. Many phisicists in our daily way of doing physics-, if any, practical will have little, implications less with a or approach to science, practical more simply optimistic, others, will of the that provide a new theory quantum gravity forthcoming hope is an easy At of theory of string present, quantum physics. thinking way is yet evidence of as no criticisms for the experimental pragmatics, target construction rich and also is but it a conceptual deep however, available; theoretical problems in quantum field where new ways of solving longstanding in string Until to theory is mostly are now, progress starting emerge. theory in evolution the to similar one evolve to underlying a "internal", very way what in decadence of not This is a mathematics. necessarily symptom pure considered as an experimeantal is traditionally science, but maybe the only in the quantum realm. possible way to improve physical intuition to permost of the work in string Until very recently, theory was restricted Different point perturbative turbation string theories are, from this theory. set field a certain of view, defined theories, satisfying by two dimensional The

challenge

great

of

of

such

of constraints in

the

conformal

dimensional and string surfaces.

as

conformal

perturbative

string

amplitudes This

set

and modular

expansion field

theories

become

of rules

good

constitutes

are on

obtained

invariance.

Riemann surfaces

measures

what

we

Different

by working

out

orders these

two

of different

genus, on the moduli space of these the "world-sheet" now call ap-

point of view, we can think theory. From this perturbative field conformal as many as two dimensional of many different string theories, deterwhich is central the of value with an appropiate extension, theories, should define good measures that amplitudes mined by the generic constraint of of Riemann surfaces. on the moduli Among these conformal field theories,

proach

to

string

K. Behrend, C. G´omez, V. Tarasov, G Tian: LNM 1776, P. de Bartolomeis et al. (Eds.), pp. 39–191, 2002. c Springer-Verlag Berlin Heidelberg 2002


40

special

C6sar G6mez and Rafael

interested

the

Hern6ndez

whJC'h possessing a spacetime interpretation, the dynamics of strings as describing moving in a definite target spacetime. Different string theories will then be deEned as different types of strings moving in the same spacetime. Using this definifour different theories tion, we find, for instance, types of closed superstring and one open su(type IIA, type IIB, E8 x E8 heterotic and SO(32) heterotic) However, this image of string theory has been enormously modified perstring. in the last few years, due to the clear emergence of duality These symmetries. of two different and non perturbative, resymmetries, species, perturbative late through relations a string equivalence theory on a particular Spacetime to a string theory on some different spacetime. When this equivalence is perit can be proved in the genus expansion, which in practice means turbative, a general type of Montonen-Olive duality for the two dimensional conformal These duality field theory. A symmetries are usually refered to as T-duality. relation between string more ambitious is known as theories type of duality where the equivalence is pretended and to be non perturbative, S-duality, from strongly where a transformation to weakly coupled string theory is involved. Obviously, the first thing needed in order to address non perturbative of string duality symmetries is searching for a definition theory beyond perturbation theory, L e., beyond the worldsheet approach; it is in this direction where the most ambitious in string theory is focussing. program of research An important from the discovery comes of course step in this direction of D-branes. These new objects, which appear as necessary ingredients for to open strings, for the Ramond fields in are sources extending T-duality at the worldsheet string theory, a part of the string spectrum not coupling, and that are therefore not entering the allowed set of level, to the string, the definition in used of the two dimensional conformal field backgrounds theory. Thus, adding this backgrounds is already going beyond the worldsheet constitutes the desired non an open window for point of view and, therefore, of definition perturbative string theory. Maybe the simplest way to address the problem of how a non perturbative definition of string theory will look like is wondering about the strong coupled of strings. This question becomes specially behaviour neat if the string theory chosen is the closed string of type IIA, where the string coupling Constant of eleven dimensional to the metric can be related the so that supergravity, be understood can eleven dimensional a as new theory coupled strongly string When thinking about the relation between D-branes and theory, M-theory. M-theory or, more precisely, trying to understand the way D-branes dynamics should be used in order to understand the eleven dimensional dynamics the strong coupling describing regime of string theory, a good answer comes for a while, relation between type IIA strings and again from the misterious, eleven dimensional the Kaluza-Klein modes in ten dimensions supergravity: D-Obrane sources for the Ramond U(I) field. are the What makes this, suKaluza-Klein perficially ordinary modes, very special objects is its nature means

that

we can

are

interpret

ones

them

Fields,

Strings

and Branes

41

powerful enough to fact, D-branes are sources for strings, string spectrum. under of these D-Obranes comes recently A very appealing way to think the name of M(atrix) theory ground for M(atrix) theory. The phylosophical based on black hole bounds on quanprinciple, goes back to the holographic tum iiformation packing in space. From this point of view, the hologram of eleven dimensional theory for the peculiar set M-theory is a ten dimensional of ten dimensional degrees of freedom in terms of which we can codify all that D-Obrane eleven dimensional theory is the conjecture physics. M(atrix) dynamics, which is a very special type of matrix quantum mechanics, is the We do not correct M-theory. hologram of the unknown eleven dimensional but it seems we have know the non perturbative region of string theory, already its healthy radiography.

oi D-branes.

provide

In

the whole

The audience. adressed to mathematics were originally along them is of course only a very small part of the huge amount of material growing around string theory on these days, and needless References the personal it reflects to say that point of view of the authors. for this in advance. we apologize so that not exhaustive, are certainly and particiLast, but not least, C. G. would like to thank the organizers most of and interesting questions, pants of the CIME school for suggestions These

content

them yet

lectures

covered

unanswered

in the

text.

C6sar G6mez and Rafael

42

Chapter

1.

I

Monopole.

Dirac

1.1

HernAndez

Maxwell's

in the

equations

absence of matter,

VE V under

invariant

are

x

aE

B

at

duality

the

0,

VB =

0,

V

aB

E+

x

(1.1)

0,

at

transformation E

B,

-

(1.2)

E,

B or,

0,

equivalently, P"'

*F"' with

*FO'

In the

P4'

=-

pressence

=

!,EO'P'FP, 2

of both

-4

*Fmv,

----4

-FO',

auAv avAl-'. Hodge dual of F4v magnetic matter, Maxwell's equations

the

electric

(1-3) =

and

-

become

0,F,'v

-jl-t

0,*F4'

(1.2)

and

generalized

must be

F"'

*F"' As is clear

from

the

with

-+* -+

transformation

a

F4v -Fl"

definition

(1.4)

-P, law for

j"

-+

V

V

-+

-jl-t.

the

currents,

(1-5)

F"v, the existence of magnetic sources The appropisingular vector potentials. (monopoles) [1] requires dealing for these ate mathematical vector is that of language describing potentials of

with

fiber

bundles To start

noting

H:

defined

by

[2]. with, the

we

two

will

U(1)

consider

hemispheres,

9

U(1)

valued

functions

on

the two

o the

U(1)

bundle.

equatorial Notice

angle, that

n

and

n

=

the

two

sphere S2.

S1,

the

U(1)

S,

is

(1-6) and such that

on

the S' equator

integer number characterizing winding number of the map

_4

De-

bundle

e

some

the :

on

e ' Y'O-,

=

defines

einW

=

hemispheres

e'+ with

bundles

H+ n H-

with

U(1),

(1-7) the

(1-8)

Strings

Fields, classified

under

the first

homotopy

group

-U,(U(I))

g,

A' (1.7)

we

easily

get,

=

Stokes

and, through

fS

F

2 -7r

2

identifying

the

In quantum

rule

tization

ued,

for

function, we

[f

theorem, dA+

we

+

H+

f

=

connections,

pure gauge

F

get

dA-]

27r

fs

A+

-

A-

=

(1.12)

n

winding number n with the magnetic charge of the monopole. mechanics, the presence of a magnetic charge implies a quanthe electric charge. In fact, as we require that the Schr5dinger be single valfor an electric field in a monopole background,

get

a non

A

(1.10)

A- +

ie

1

define

g-'alj.

1.Tj_

exp

with

we can

(1.9)

Z.

-

the equator,

on

A+

wave

(SI)

H as follows:

on

From

I,

--

Using the U(1) valued functions

A,

43

and Branes

we

loop.

contractible

get Dirac's

quantization

IFA

T7 h_ In

the rule

em

=

=

(1.13)

1,

presence

of

a

magnetic

charge

m

[1], nh.

to the definition (1.14) is'equivalent (1.12) as minus winding number or, more precisely, bundle on S2. In fact, Chern class of a U(1) principal the single the first of the Schr6dinger is equivalent valuedness to condition wave function (1.7), for the transition in order to where we have required n to be integer function, The gauge connection used in (1.12) defined was implicitely get a manifold. for the physical A standing as eA, with appearing in the gauge configuration 1. Schr6dinger equation. From now on, we will use units with h The main problem with Dirac monopoles is that they are not part of the as a dynamical spectrum of standard QED. In order to use the idea of duality in symmetry, we need to search for more general gauge theories, containing the spectrum magnetically charged particles [3, 4, 5].

the quantization

Notice

that

of the

magnetic

charge

as

rule

a

=

Usar G6mez and Rafael

44

Hooft-Polyakov

The 't

1.2

Let

,C with

=

IF,

-

"

4

F,,

and

A > 0 and A classical

a

arbitrary

V(O)

now

define

the

a

in

finite

in the

this

is

case

manifold

a6a3,

Aa

manifold

V

A

DjOjs2

S2)

=

(1-16)

(1-17)

0.

=:

as

(1-18)

for equal a. A necessary condition the field values infinity, 0 takes Higgs

of radius at

Maps of

0.

which

V

___

the

type

for

the

(1.19)

are

Once

we

is

3

47ra

S2

impose the finite

energy

f4

is

second

the

(with

ako). D40IS2

condition

V

These maps

(1.20) =

0, the

gauge field

an

arbitrary

A

a2 9

function.

a4O +

The

aOf

(1.21)

corresponding

stress

tensor

is

given

by Fa4v

which

OaF"v

Oa

that

the

magnetic 1

M=

0 09t'O -

a2 9

a

a

implies

=

are

given by A"

where

by

model

numbers.

dS'l'E2ijkO'((9j0A

1

classified

Georgi-Glashow

non

infinity

9E(IbAlb 10c

V,

N ==

at

-

2)2.

and equal to the set of integer trivial, characterized by their winding number, is

-

=

is that

H2(V),

group,

aAOa

=

10, VW 01,

==

00

homotopy

(1.15)

1, 2,3,

=

given by

is

S2 and that

a

-

4

2-sphere

a

configuration

energy vacuum

'A(02

=:

=

vacuum

V which

a

constants.

Oa We can

V(O),

-

DAO,, Higgs potential,

the

configuration

vacuum

SU(2),

for

representation,

V(O) with

D,,o

-

2

adjoint

the

the covariant

[6]

model

IDI'o

+

Higgs field in derivative,

the

Monopole.

Georgi-Glashow

the

consider

us

Herndndez

-

2ga 3

fS

A

avO)

+

&V -,9"f "

4

(1.22)

charge

EijkO *pjo 2

A

j9k O)dS',

(1.23)

for

a

finite

(1.20)

as

configuration

energy

given

is

in

and Branes

Strings

Fields,

winding

of the

terms

45

number

[7] 4-7rN

(1.24)

IM

9

In

order

to

U(1)

the

(1.24)

combine

charge.

electric

Dirac's

with

A'

charge of

Thus, the electric

field

a

(A"

=

(1.24)

From

H,

and

a

the

vacuum

(1.26)

we

manifold

with

should

define

(1.25)

-.

a

spin j is given by

(1.26)

gi.

for

recover,

Higgs model,

generic

For

=

we

by

I

0)

-

is defined

field

of isotopic e

rule

quantization

U(1) photon

The

j

1,2

=

Dirac's

rule.

quantization

G spontaneously

gauge group

broken

to

given by

V is

V

GIH,

=

(1.27)

with

H2(GIH) where

G,

H1 (H)

which The

mass

a

is the

of the

monopole

monopole,

static

in the

the

f

which

implies

(k )2

the

if Bak

=

f

d

+

a

mass

d

be contracted

the form

to

a

point

in

(1.9).

+

(D'Oa )2]

+

V(O).

(1.29)

becomes

a

[8]

3X1 [((B'

a

2

Bogomolny [9]

can

in

(DOOa)2

3X1 [(B' )2 2

DkOa, which

(1.28)

given by

is

Prasad-Sommerfeld M

saturated

+

a

H that

in

condition

Dirac's

M

then,

of paths

set

contains

d3XI [(E )2 2

M

For

G

again

H,(H)G)

--

+

(D'Oa )2]

limit

A

+

D'Oa )2

=

bound M > are

known

as

0

+

V(O);

(see equation

-

2B'D'Oal, a

am.

the

(1-30)

(1.16)),

we

get

(1-31)

Bogomolny bound Bogomolny equations. The

is

C6sar G6mez and Rafael

46

Hernindez

Instantons.

1.3

Let

us

consider

now

SU(N) Yang-Mills

pure

1

L In euclidean

sphere S3

R',

spacetime A necessary

.

the

=

-

4

FattvFa'.A

region

condition

theory,

infinity

at

finite

for

(1-32) can

be identified

euclidean

action

of

with

the 3-

configurations

is

IS3

Fa t"

equivalently,

or,

configuration

the gauge

that

A41S3 Hence, finite

euclidean

topologically

are

=

A" at

:

S3

classified

winding

n

12

=

24,7r

As for the equator.

depending

bundles

the

simplest

the

value

of n;

maps

(1-35) homotopy

is

Vjg(X)9-1 we can

(1.37)

the map g in order function

the transition

we

will

n

=

for

given by

Vkg(X)l

use

g defines

particular,

group,

(1-36) by (1.35)

SU(2),

group, in

with

Z.

--

tr[9-1 Vig(X)9-1

So, for on

associated

of the third

in terms

monopole construction, on S4. In this case,

the Dirac

SU(N)

define

d3 XEijk

gauge,

pure

a

SU(N),

-+

number of the map g defined

fS3

is

(1-34)

are

H3(SU(N)) The

infinity

g(x)-1o94g(x).

configurations

action

g

which

(1.33)

0,

=

-

have different

1,

we

obtain

to on

bundles, Hopft

the

bundle

S7 S4

Interpreting a

gauge

topology

as

the

configuration of

S3,

we

S4.

___

compactification on S4 such that

A+ and A- the

now

R4,

which

we can now

define

has the

gauge

+

g-1 a07

configurations

on

the

(1.39) two

hemispheres.

Using

relation

tr(F,,,,.P4') we

space

equator,

have

M

the

the

on

A+= gA g-_1 with

(1.38)

of euclidean

dtr(F

A

A

-

1AAAAA),

(1.40)

3

get I

7r2

f

tr(F,,,P1"') . 4 S

2472

f

Eijktr[g-1o9jgg-1,9jgg-'o9kg] S3

=

n,

(1.41)

Strings

Fields, which

is the

tween

the

above beto S' of the relation we have derived charge of the monopole and the winding number of the charge defining the U(1) bundle on S2 The topological action. In fact, the total euclidean a bound for

generalization

magnetic function

transition

.

by (1.41)

defined

4

47

and Branes

f

is

F aav F a, )v

f

1 = -

2

configuration

The instanton

tr(F4'F,,,)

2

be defined

will

f

tr(FO'-P,,)

(1.42)

gauge field

by the

saturating

the

(1.42),

bound

Fm,,

(1.43)

FP

=

DF 0, together charge equal one. Bianchi identity, DP condition self the in with the field duality fact, 0; implies equations, euclidean with If start the we related to be Bogomolny equation. (1.43) can dimensions to three and reduce dimensionally through the defYang-Mills, dimensional three the inition lagrangian. Yang-Mills-Higgs A4 =- 0, we get (1.43) becomes the Bogomolny equation. Then, the self duality relation for SU(2) was discovered A solution to (1.43) by Belavin et al [10]. Inthe bare the on coupling constant g, dependence explicit cluding

topological

and with

=

=

Fm,, the BPST solution

to

OmA,, -0,,Aj,

=

(1-43)

satisfying

4i

Nttv

(-1)

configuration

is

Caij7

9

Notice can

we can

self

dual

that

the

tions

xt'

that be

87r2

the

-4

x"

is

gauge

invariant

+ a".

This

f

Fa,

the classical zero

around

fluctuations action

as

2

solution

the instanton

interpreted

consider

-?7auv7

=

and

(1.46)

2

number

327r

that

77aiO

=

9

Pontryagin

(1.45)

+ P2)21

(X2

bai) 77attv 1, 2, 3.

S with

2

j take values

=

=

6'0 +6"0

The value

?IattvP

g

77aij ?Iapv, where a, i, of the action for this nattv

(1.44)

AJ,

21 g X2 + P

Fm'a,

7?auv

7

77alwXv

2i

IL

=

g[A,,

given by

is

Aa

with

+

under means

=

(1.47)

1.

(1.45)

depends on a free parameter p, In particular, configuration. i. e., small the instanton solution,

size of the

modes of

the

4X

F al"'d

instanton

solution.

changes

of the

that

will

we

From size

p,

have five

(1.45),

it

and under

independent

is clear

translagauge

Usar

48

G6mez and Rafael

The number of gauge zero modes is called, the dimension of the moduli space of self

modes.

zero

literature,

cal

Herndndez

computed [11, 12, 13] using SU(N) instantons on S4 is

number for

be

can

dim Instanton k the

with

Pontryagin

the five

recover

X and

1

4nk

=

the

(N

-

2

2 _

boundary

with

define the

of

configurations

vacuum

g

different

a

a

g,

vacuum

map from

the

interpret between

Moreover, formations under

all

> and

the

with

we

of on a

(1.49)

in

>, characterized

temporal

=

Ai(t

=

oo)

-

SU(N),

configurations,

SU(N).

AA

We can

=

now

winding number configuration boundary conditions:

the

of

instanton

o,

laigi,

gi

=

of

an

following

the =

+oo)

by

gauge,

satisfies

one

(1.51)

(1-51)

number equal

winding

configuration

11

(1-50)

0,

-+

gauge are pure gauge the gauge group

the

Ai (t

as

defining

a

Wecan

one.

tunnelling

now

process

> vacua.

vacuum

non

2

=

1) [X

S' into

states

S' into

instanton

10

the

n

condition

in this

map from

corresponding map g. In Pontryagin number equal

with

1 and

=

of the manifold A4. signature of instantons, it is convenient interpretation R1 to gauge [15, 16, 17]. If we compactify

Aj(r)Ijrj-+co g-1,9,,g,

k

number and the

=

the

For

(1.48)

1,

to instantons

to

S' by impossing

+

result

The

get a clear physical work in the A0 0 temporal

to

2

This

the

and dilatations

the Euler

-r

In order

n

to translations corresponding of equation generalization (1.48)

dim

with

-

instanton'.

number of the

Mis

manifold

4nk

=

[14];

theorems

index

mathemati-

solutions.

modes

zero

(1.45)'.

the solution

Moduli

the

in

dual

states

vanishing

gauge transformations

10

In

>

are

winding

not

A

would be defined

by

>=

Ee

ino

In

in

the

under

invariant

number.

vacuum

gauge trans-

state

the coherent

invariant state

(1-52)

>,

n

with

0

a

free

transformations

parameter of

winding

taking

values

number m, the

U(g.,,)

In

>=

In

interval

vacuum

+

m >,

[0, 27r]. In >

states

Under

gauge transform as

(1-53)

the irreducibility condition k > 2. This condition must hold if satisfy 2 to be irrJuCible, i. e., that the connection require the gauge configuration be obtained of a smaller can not by embedding the connection group. Observe that the total number of gauge zero modes is 4, and that n 2 I are of the instanton simply gauge rotations configuration. k must

we

2

_

Strings

Fields, and therefore

the 0-vacua

will

transform

U(g,,,)10 which

means

invariance

49

as

ei'010

>=

the projective

in

and Branes

(1.54)

>' i.

sense,

e.,

on

the Hilbert

space of

rays.

The

functional

generating <

010

now

E < Oln

>=

becomes

inO

>

f

=

n

the

with

Yang-Mills

0-topological Notice lagrangian. The

a,u

--F 4

=

term

in

that

if

v

(1.56)

us now

(1.15).

(f

1 -

4

add the

At this

level,

7r2

(1.56)

'V

explicitely

i

vF4av

Faja

87r2

the

+

0g2

327r2

CP invariance

functional

of the

integral

FP d 4X,

(1-57)

(1-58)

+ io.

2

electric

(1.56)

Georgi-Glashow model the O-angle as an extra cousimply considering the topological density FP. In order to define can simply apply Noether's theorem for a gauge 0-term

we are

multiplying charge,

constant,

U(1)

topological

we

in

the

unbroken

transformation

in

the

0

U(1)

direction

of

corresponding

Noether

given,

after

N= ag

the 0-term

f

d

3

x0i (0

An infinitesimal

gauge

by

D,,O,

(1-59)

0.

charge,

N=

be

[18].

be defined

ag

60

to the

direction

would I

will

F attvFav.

-

becomes

action

transformation

The

(1-55)

,

Dyon Effect.

1.4

pling

(if L(A))

the euclidean

9

the

2

breaks

S=

Let

Og2

F,a'v +

consider

we

dAexp-

euclidean

the instanton

-

lagrangian L

f

dA exp

-

6'C

J,9OA is

R0i)

6L

.6A +

6,900

included, +

Og 8,7r2a

.01

(1-60)

by

f

d

3XO,(Ol 2 'EiikFjk)

(1.61)

C6sar G6mez and Rafael

50

or,

in

charge,

electric

of the

terms

HernAndez

as

Og

e

N --

+

-

Notice

(1-61)

from

ground

of the

rotation,

we

that

If

e

which electric

that

implies

charge

lagrangian,

if

we

27ri( '+jt2 'Tm) 9

ng

N = f.

9

e27riN

as

for

(1-65)

vacua

charge

of

with

implies respect

>=

that to

21riN

1.5

of exp,

of the order

We will

length invariant

consider ao, a,,

-

now

a2, a3.

a

dyon

with

incuding

without

0 term

a

in

the

eiolTn the

(1.65)

>,

monopole

transforms

state

of non gauge transformations be continously connected with induced not

822 [18, 19].

electric

suppresed

under

vanishing the iden-

charge of the by a tunnelling

9

Theory

Yang-Mills

m becomes

monopole state,

the

can winding number. However, e tity which, in physical terms, means that the and is monopole is independent of instantons,

factor

(1-64)

2

(1.63),

result,

same

require,

Equation

the 0

U(I) (1-63)

17

Og2M,

g7-r

monopole

e27riN Im for

backa

[18]

to

-

=

the

under

m.

the

We can reach

=

magnetic

a

092,, 87r

-

e

=

becomes equal

charge

and the electric

27riN

N in

we now

get e

to only contributes invariance require

0-term

the

monopole field.

(1.62)

2M.

81r

g

on

V.

on a 4-box pure Yang-Mills [20], with sides of for gauge impose periodic boundary conditions

SU(N)

Let

us

quantities, A"

(xO

+

aO,XI,X2,X3)

QoA"(xO,X1)X2iX3)7

+a,,X2,X3)

OjA4(xO,XI,X2,X3),

A,'(xo,xl

A"(xo,XI;X2+a2)X3) A"

(xo,

X1

i

X2 7 X3 +

a3)

02 A" (xO,

x 1,

X2)

X3)

03 A" (xo,

X 1)

X2 7

X3);

7

(1-66)

where

f2pA4

_=

QpA4j?P-1

As the gauge field transforms the existence of Z(N) twists, 0 A j?v

in the

=

+

S?P-'01-'S?p

adjoint

f2vj?4e27rin,,1N

representation,

(1-67) we can

allow

(1-68)

and therefore

51

T4 by the topo-

in

can be internumbers, In order in the 3, 2 and 1 directions, fluxes respectively. as magnetic numbers the introduce these magnetic fluxes, we characterize

logical preted to

configurations

different

characterize

we can

and Branes

Strings

Fields,

Three of these

numbers n/.tv.

Mi

These

magnetic

due to the fact

n12, n13 and n23,

fluxes

(1.69)

6ijknjk,

:-::::

Z(N) charge,

carry

topological

and their

stability

is

that

H, (S U(N) / Z (N))

(1.70)

Z (N).

--

let us Hilbert the physical In order to characterize space of the theory, box T', 0. For the three dimensional again work in the temporal gauge A' to magnetic flow m corresponding we impose twisted boundary conditions, (Ml M2 M3). The residual gauge symmetry is defined by the set of gauge We may distinguish these boundary conditions. transformations preserving different the following types of gauge transformations: =

=

,

i)

Periodic

gauge

winding

IT3(SU(N))

number in

ii)Gauge

which

transformations,

f2(xi

+ a,,

X2,

an

X27

X3)e

27riki

IN

f2(X1,X2,X3+a3)

O(X1,X2,X3)e

27rik3

be denoted

ii-Vhose

7

center:

27rik2/N

by

characterized k

extract

f2(XI

X3)

in the

elements

up to

O(Xl,X2,X3)e

are

=

by Qk(x).

by their

characterized

D(XI,X2+a2,X3)

These transformations

and will

are

Z.

2-,

periodic,

transformations

usual

as

(ki, k2, k3)

Among this

IN

the vector

(1.72)

7

type of transformations

we can

classification:

extra

such that

(9 k(X))N

such that

(j?k(X))N

is

periodic,

with

vanishing

Pontryagin

num-

ber.

ii-20hose

is

periodic,

with

non

vanishing

Pontryagin

number. In terms

then,

the

temporal

of unitary we

gauge,

operators.

we can

Let

jTf

represent > be

get

f?k(X)ITf

>=

e

27r

a

i-kN

e

the

transformations

state

in

the

i0k.-N

ITI

>,

Hilbert

in space

ii-2) in 1-1(m);

(1.73)

is Notice that the second term in (1.73) parameters. effect described to the Witten for Z(N) magnetic vortices, dyon equivalent, In fact, we can write section. in the previous (1.73) in terms of an effective

where

e,ff,

e

and 0

are

free

C6sar G6mez and Rafael

52

Hern6ndez

Om

eeff

Moreover, as 0 -+ 0 + 27r, Pontryagin number of determined conditions, by

change

we

the

a a

set

f

9

eef f +

-+

M.

configuration

On the other

with

hand, boundary

twisted

given by [21]

is

n,,,,

(1.74)

27r

eef f

gauge field

167r2

+

e

=

tr(Fl"P,,)d4X

k

=

n -

j

(1-75)

I

A simple way to understand the origin of the fractional -!nkjvn1,jv4 piece in the above expression is noticing a twist that, for instance, n12 corwith value 27rnj2 responds to magnetic flux in the 3-direction, which can be N 27rnj2 and a twist n03, which corresponds formally described by F12 to an Naja2 where

=

n

,

-

,

electric

field

3-direction,

the

in

is described

by F03

27rn03

.Using now the integral representation we easily Pontryagin get the fractional piece, with the right dependence on the twist coefficients (see section 1.5.2). Moreover, (S?k(X))N acting on the state ITI > produces of the

(X))NIT,

(Q which

k

that

means

-

configuration

(f? k(X))N.

number

will

n we

get,

is

the

For

a

m

as

Using (1.71),

it

tion

of the

0-term,

is easy to

nothing

are

parameter

is that

of

an

e

C

a

path

in the

we

n) I T1

see

means

e

i0k-m

but

flux.

=

3-direction.

ITI

(1-76)

>'

(1-77)

>

characterizing The

twists.

(1.73),

fact,

in

we can

the

the residual

physical

very

define

the

same

gauge

interpretaway

Wilson

Under

flk(x),

A(C)

as

the

loop

fe igA(6)d6,

trexp

_+

ein9o/

the n0i in

In

yl

=

the k's

that

(1-78) transforms

as

e27rik3/N A(C);

get

f?k(:x)A(C)jT1 which

>

introduced

electric

A(C) therefore,

>=

number of the periodic Pontryagin gauge with Pontryagin generic gauge configuration

else

A(C) with

Na0a3

usual, P (x;

transformations

_

number

that

A(C)

creates

>=

a

unit

e

27ri

N

A(C)IP

of electric

flux

(1-80)

>, in

the

3-direction.

Strings

Fields,

non

can

have

in

order

vanishing magnetic

flux.

magnetic flux

for

this

goal,

achieve

to

It

classical

a

is

to

Z

them to define

find

two

a vacuum

come as a

configuration

configuration.

What

matrices

constant

that

surprise

the

in

we

need,

we

gauge

=

QPZ of the group.

in the center

exist,

in the in two directions boundary conditions A satisfies 0 automatically these boundary classical a non vanishing vacuum with magnetic =

and we will get a conditions, by the center element flux, characterized SU(N) those matrices exist; they are 0

If such matrices

twisted

configuration

The trivial

box.

element

trivial

a non

we can use

priori

a

vauum

PQ with

consider

now

may

53

[21]

such that

group,

We will

Vortex..

The Toron

1.5.1

with

and Branes

(1.81).

-7 in

For the

gauge group

1 1

0 P

0

e21ri/N

e7ri(l-N)IN,

Q

e

satisfying

QPe21ri/N

PQ

A(xi

impose twisted

we

X2,

X3)

=

PA(xi,

X2

X2 + a2,

X3)

=

QA(xi,

X2,

=

A(x,,X2,X3),

+ a,,

A(xj,

A(x,,X2,X3+a3) in

the

sector

gauge A'

temporal with

Classical

non

g(xj

X2 i

X3)

9(XI,X2+a2,X3)' ,q

for

generic

be written

(XI)

X2 7 X3 +

a3)

(ki, k2, k3). Now,

Ai(x)

X3) P-1

X3)Q-l (1-83) vacuum

=

g-'(x),9jg(x),

g(x)

A

=

0 is

in

the

=

Qq(Xl7X27X3)Q 9 (X1

7

X2,

any gauge transformation

Tk'T k2T 3k3 97 1 2 -

==

-1

X3) e2-7rik3

as

(1.83),

X3)P-1 .27riki IN,

Pq(X1

X2)

satisfying

satisfying

=

=

9

conditions,

=

gauge transformations

+ a,,

2

0, then the classical 1magnetic flux, M3

vanishing configurations, by

boundary

=

vacuum

would be defined

27ri(N-1)/N

If

.

(1.82)

e27rik2/N, ,

IN

(1.84) satisfying

(1 .84)

can

Hern6ndez

G6mez and Rafael

Usar

54

with

TI

(1.84),

satisfying

and

Aj

0 >,

we

k,

with

P-17

T2

k3

k2

=

T,1Ai=0>

lAi=o>,

T2lAi-:::::O>

jAj-,::=0>7

that the different using (1.73), implies, with T3, other hand, we get, acting

T3k3 jAj and, therefore,

we

I

63

N

0,...,

=

>=

Tk3 je3 3 which

from

we

>=

N

T3N

Of

definition M3

get

is

periodic,

je3

"

2-7ri

=

e3

=

6-2

=

0.

On

(1-88)

0; k3 >;

jAj

N

have el

by

=

(1-89)

0; k3 >i

e

k3

T3

with

2703" N

e

je3

on

i0k3-3 N

>,

le-3

we

get

(1.90)

>;

that

observe

I

e.,

e

now

T3 i.

vacuum

k3

Acting

1.

-

the

on

(1-87)

vacua

defined

vacua

E

j

jAj

0 >=

=

get N different

Je3 with

Acting

0.

=

which the

(1-86)

(1-86),

from

get,

Q)

-.,:::

>=

e-3

> we have

io

je3

(1.91)

>

equal the 0-parameter number

winding

with

e

included

one.

Notice

and the

with the boundary conditions (1.83). 1, associated two basic things: discussion we learn From the previous

that

in

magnetic

the flux

=

zero

fluxes

energy

are

boundary

secondly,

conditions

above is the well

with

states,

parallel;

(1.83)

both that is

known Witten

magnetic

and

electric

number of

the

equal to index,

N. In tr

vacuum

states

with

what has been

fact,

(_I)F

first, that we can flux, provided both twisted

computed

[22].

Wewill now try to find configConfigurations.. the equations with fractional number, satisfying urations on T Pontryagin discovered this of by 't Hooft of motion. type were initially Configurations choose a first this we describe In order to configurations, for SU(N) [21]. be the 1 N. Let k with w of + SU(N), subgroup SU(k) x SU(1) x U(1) x of x the to U(1), SU(1) SU(k) matrix U(1) generators corresponding 1.5.2

It

Hooft's

Toron

4

=

1

w

=

27r

1

(1.92)

-k -k

Strings

Fields, with

tr

w

=

configuration

0. The toron

is defined a

A,, (x)

=

55

by

X'\)

It,\

_W

and Branes

(1-93)

a,\ aju

with

attv

-

av

(2)

(1)

NWIF

Nk'

(1.94)

and

n(l)

n,,, The stress

(1.93)

configuration

for

tensor

we

consider

the

simplest

n

(2)

(1.95)

/IV

given by

is

attv

F1, If

+

gV

avg.

-

(1-96)

a,,a,

case,

n12

=

n(l)12

1, and

n30

n

(2)

1,

30

we

will

be led to

F12

+W

F30

-W

Nkaja2' 1

(1.97)

Nla3a4'

and therefore

f

tr(Fjv_P4v)

impose the self duality

condition,

9

167r If

we now

which

2

aja2

1

a3a4

k

the relative

constrains

sizes

-L, IV

equal

SU(N).

In this

we

sense,

we can

think

get

k

-

(1.99)

k

of the box.

(1.93)

configuration -r

=

=

T4

0 for

only get four translational

we

(1-98)

-

N

N

The gauge zero modes for the toron from the general relation (1.49), with k number

1 =

as

can

Thus,

having

a

size

be derived

Pontryagin

for

modes for

zero

of the toron

.

gauge group

equal

to the

of the box.

size

The toron for

the

JM3

=

Let terized

Pontryagin number equal as a tunnelling instanton, process 1, k + (0, 0, 1) >.

1 can N

of

us

fix

a

by

e

and

concrete m.

distribution

of electric

The functional

be

between

integral

and for

this

interpreted,

states

JM3

as =

we

did

1, k >. and

fluxes, characbackground is given by

magnetic

[21] <

e,mle,m

>=

E k

where

e

27ri

k"

NW (k,

m),

(1.100)

G6mez and Rafael

Usar

56

W(k,m)

HernAndez

f [dA]k, ,,

=

-f

exp

(1-101)

L(A),

the in (1-101) over satisfying configurations integral gauge field consider We twists the defined conditions can m). (k, by boundary for the toron action the particular case m (0, 0, 1) to define the effective configuration, the

with

twisted

=

87r2

S A

generalization equal -L, N

possible

tryagin

number

(1.102)

becomes

is obtained

It

must

be noticed

Next,

we

will

consider

=

using configurations

(kj, k2 1) 1

27ri(k

N

we

(1.102)

N

In

-

this

with

case,

the

e)

-

(1-103)

N

have not

with

a

Ponaction

included

in

(1-102)

the

effect

of

0,

L'

factor

N

Vertex.

of instantons

effect

the

work

being, we will matrices satisfy

time

27rie3

when

k

87r2 =

action

Effective

Instanton

1.6

that

to the

contributes

with

but

S

which

+

92N

compactified

on

on

[15, 11].

fermions

euclidean

spacetime,

For the

S4. The Dirac

(1.104) and the chiral

operator

-y5,

75

=

(

0'Y1'Y2^f3

splits

fermions

The space of Dirac

'Y

into

-/50 Let

us

work

configuration.

massless

with

b+

=

As with v-,

a

consequence

of the index

chirality, positive is given by

the

by the topological change of chirality

chirality,

coupled to

to

Dirac's

an

instanton

gauge

equation,

(1-107)

0.

the number v+ of solutions to (1.107) with negative chirality,

theorem,

minus the number of solutions

V+

i..e.,

=:

opposite

(1.106)

solutions

,y,"D,,(A),O

of

-

fermions

Dirac

(1.105)

1

two spaces

normalized

Weconsider

0

-

V-

charge induced

92Nf

f

- 21r2 of the

by

an

, AQ5

F,,vpAvad 4X,

instanton

instanton

=

2Nf k,

(1-108)

Thus, gauge configuration. is given by configuration

(1-109)

Strings

Fields, with

57

massless Dirac Pontryagin number, and Nf the number of different in the fundamental of the gauge group. transforming representation generalize equation (1.108) to work with instanton on configurations

k the

fermions, We can

dimensional

four

generic

a

and Branes

euclidean

M. The index

manifold

theorem

then

becomes N V+

V_

-

24

tr(R

8-7r

-

A

92 Nf

R)

f

-

327r2

M

FM'v_pizvad

4

(1.110)

X,

where

fermions in the fundamental again we consider of representation Equation (1-109) implies that instanton induce efconfigurations fective with change of chirality In order to comvertices, given by (1.109). we will use a semiclassical pute these effective vertices, to the approximation functional, generating

SU(N).

Z(J, j)

=

f [dA] [d ]

around

the

gration

of fermions

ZPI J)

instanton

=

in

[do]

exp

f

L (A,

Let

us

-

configuration. (1.111):

f [dA]det'4 A) f j(x)G(x, 11 f O(n) (x) (x) f exp

J

d

4

, 0) first

J

+

0J,

+

perform

the

A)J(y)dxdy

y;

-

(1.111) gaussian

exp

-

f

L(A)-

j(&( on ) (y)d 4V,

X

inte-

(1.112)

n(A)

O(n)

where the

0

are

regularized

the fermionic

modes for

zero

and G(x, y;

determinant,

A, det'P(A)

configuration regularized

the

A)

is the

Green's

tion,

.P(A)

G(x, y;

A)

=

_j(X

_

Y)

+

E On(X)V)n(Y). 0

0

n

In semiclassical

approximation

around

instanton,

87r2

[dQ] det'V (Ainst)

Z(J, J)

the

exp

exp

-

we

get

i(x)G(x,y;A)J(y)d

4

xd'y.

M

C'0 (A inst )Q2

exp

O(x)J(x)d 4X

where

,C for that

CO the

-1FapvF,',, 4

only

non

,0

and

(JJ(x,)J,f(xl)

4Y,

(1.114)

JAJA)A=Ai, , t j2,CO

'(A inst)

vanishing

i(y)00(y)d

Q the small amplitudes

j2mZ(j ...

fluctuation. are

those

j) JJ(XM)6j -(Xm)-)

(1.115) It

is

clear

from

(1.115)

with

,

I

J=J=O

is

func-

(1-116)

Herndndez

Usar G6mez and Rafael

58

for

m

v+

=

consider factor

as we

have 4N

I

C

-

9

01

where

(1.117)

P3Nf

is

a

comes

#-function,

(A

ing 0 is

<

comes

...

be stressed

measure.

The factor

( W2 )1/2 Nf

in the

I

=

case).

The

for fermions,

from the determinants

ghosts.

was

00(XI)

result

the

must

It

p.

is obtained

condensate

computation

previous simply

way that

a

2

.

p2)3/2

+

and Faddeev-Popov

gauge bosons

The

(X2

(1-117)

zdp,

modes3,

zero

(1.117)

C in

factor

such

point

P

symmetry breaking

chiral

proportionality

in

invariant

-

a

i=1

3/2

07,0

4 0

and dilatation

from the fermionic

with

>=

renormalization

translation

need to

get

[p])31 jj( O')d0

g2

of the of the

we

Nf

8,7r2 exp

P

independent

is

we

Q,

over

mode contributes

zero

O(X.)V)(X.)

...

1

coefficient

the

is

d4zdpp-'

that

)4N

modes,

zero

P3Nf 5

integration

Each gauge

O(x0V)(X1)

<

the

perform

to

modes.

zero

gauge

.1. So, 9

order

In

+ v-.

the

carried

00(x"')

for

out

>0=<

...

0

=

0. The effect

of includ-

-eiO

>0=0

size in the instanton the integration over contribufinite instanton order to in get thus, (1.117) cut off the integration size, something that can be implewe should tions, are known mented if we work with a Higgs model. The so defined instantons It

is

important

constrained

as

1.7

[11].

instantons

Dimensional

Three

An instanton

implies, necesarily homotopy group of the for

pure

Instantons.

dimensions

three

in

This

be realized

that

stress

to

divergent;

is infrared

is

finite

a

in order

to have

vacuum

manifold

gauge

theories,

as

euclidean

topological

stability,

is different

H2 (SU (N))

configuration.

action

from

that

0,

so we

from

the

-_

will

=

=

are

but

nothing The first

photon 3

In

is

fact,

fermionic

't

thing

a

scalar

P3Nf zero

.

be noticed

not a

instantons

monopoles (see table).

Hooft-Polyakov to

can

consider

G gauge group SU(N) and H =

Higgs model with spontaneous symmetry breaking a subgroup H, such that 172(GIH)) 0 0. Think of G ZN-1 Thus, we see that three dimensional then H2 U(I)N-1,

to

the second

This

zero.

in

three

dimensions

is that

the

dual

to

the

for

the

field,

is the factor

modes.

that

appears

in the fermionic

Berezin

measure

Strings

Fields,

Energy Density

Energy

Action

3+1

I-TO 17,

HO Iii H2

Hi H2 IT3

Name

Vortex

Monopole

Instanton

Dimension 1 + I

2+1

In the weak

coupling

anti-instantons

=

Ht,

=

regime,

Z

n

Ef

=

exp

n+ +

n-

I

47r

2

g

-

P(X)

1 exp

-

2

dx+dx-

H n+ln-I %

2

) f

photon [23]:

( )2 Id

3xd 3YP(X)

g

gas of instantons

function

is

and

given by [23]

[exp -SO]n++n-.

'

(_ ) p(y)d 1

P(X)

-

1 _

92

3xd 3y,

92

action,

X- )

admits

term

of the dual 47r

the dilute

i=1

E 6(X

=

The Coulomb interaction in terms

FP,

(1.120)

describe

So the instanton

n,

=

2

59

Omx.

we can

n

with

I6ppa

*Fp,

Coulomb gas. The partition

a

as

HO

and Branes

-

and p the instanton

E 6(X

f [dx]

P (y)

X-).

-

following

the

density,

(1.122)

gaussian

exp

-

f

representation,

I pX) 2

2

4,7riXp.

+

9

(1-123) When we effective

up the

sum

lagrangian

and anti-instanton

instanton

for

contributions,

we

get the

X,

Lef

f

(X)

1 =

2

(,gX)2

+

e-so

cos

47rX

(1.124)

e

which implies the dual photon X equal to e-so. That X is the dual a mass for in (1.123), photon becomes clear from the x between X and the p coupling magnetic density p. The generation of a mass for the dual photon in a dilute is a nice example of confinement in the sense of dual Higgs gas of instantons phenomena. -

The inclusion

picture.

In

particular,

Goldstone vertices ones

tive

boson induced

studied

in

interactions

fermionic

of massless

zero

[24]. by

will

as

This the

previous in three

will

three section.

fermions be

shown,

will the

drastically photon

change will

be due to the existence

dimensional In order

instanton, to analyze,

of effective of similar

instanton

dimensions, we should first consider background of a monopole.

modes in the

physical

the

become

a

massless

fermionic

type induced

the

to

the

effec-

problem of

Usax G6rnez and Rafael

60

dimensional

Consider

Theorem..

Index

Callias

1.7.1

three

Hern6ndez

iy

ly

+

-yj7'

=

of (1.125) We can get a representation of dimension euclidean for space general,

2(n-l)/2

2P.

(1.125)

using

2

constant

x

we

define

Dirac

the

L

are

constant

Ai

=

gT'Ail,

i-y',9i

=

4i(x)

and

71

operator,

=

-y'Ai

+

for

the

T'

Wecan

of the

generators consider

now

a

Dirac

+

Then, Dirac's

consider

in

(

0

L

L+

0

V; (x, t)

fields

fermion

solutions

to

=

)(

0+ 0-

V) (x) eiEt

(1.128)

generalizing formula

for

respectively, Atiyah-Singer

k+

the

our

case,

(

0+ 0-

)

k-

of L. If

Dirac

equation

we

in

three

(1-130)

k+, fo

Ker(L) and Ker(L+). By [25] got the following

Callias

theorem,

index

adjoint

(1.129)

euclidean

-

(1.128)

,

get

the dimensions

(n2 1)

dimension

(87r

n

=

fs-l

2

3. In terms

tr[U(x)(dU(X))n-1],

spacetime,

of euclidean

U(X) In

we

of the

=

E

I(L):

2 n

0,

0,

n-1

with

=

L+O+

are,

the

=

0,

I(L) and

)

Lo-

e.,

where k-

becomes

and where L+ is the

,

E

with

0- and 0+ are zero modes defined by (1.126). dimensions, Now, we can define the index

i.

(1.127)

3 + 1 dimensions

equation

gauge

fermion

component spinor,

four

a

(1.126)

4(x),

+

0'(x)T',

representation. group in some particular This is 3 + I spacetime. in Minkowski

for

In

2 matrices.

-Y'

corresponding

n, the

matrices.

Now,

with

euclidean

in

matrices

Dirac

spacetime,

=-

of the

(1-131)

and

OWl'CX). magnetic

charge

(1-132) of the

monopole,

(1.20),

Strings

Fields, N

where we

have normalized

we

flEijk0i'90190')

87r

a

-T(L) index

is

infinity,

the

are

more

we

case,

fml

largest

the

(j(j

=

smallest

eigenvalue, fundamental representation to observe

+

(1-131)

dimensions,

the

because

appears

at

also

(1-135)

rn.

a

1m)(Iml

-

O'T'

of

one.

,mj

have

we

Photon

The Dual

1.7.2

1)

+

+

smaller

Thus,

than for

-1,2

=

(1.136)

1))N,

and

I(L)

there

if

or,

rn

massless

is

fermions

=

N. It

is

=

impor-

(we the

2

-

I(L)

we

get

as

Goldstone

=

no

the

in

the bare mass, the index also changes 1) Thus, for Tn > 1, and fermions in

by changing

that

representation,

fundamental

in odd

in

with

O'T'

minus

normalization

the

using

are

=

eigenvalue

the

such

tant

that

(1.131),

from

get,

I(L) with

0

for

special boundary conditions the ones defining the monopole configuration. Wecan general case of massive fermions replacing (1.127) by P

In this

(1.127)

using

(1-134)

Notice

representation.

compact spaces. a non compact space,

in

which

consider

and

2N,

=

The contribution

for

zero

working

we are

adjoint

in the

fermions

for

(1-133)

(1.20),

equation

1 in

=

61

SU(2),

for

get,

and Branes

0.

We will

Boson..

consider

the

SU(2) lagrangian 1

Ic where

we

4

an

instanton,

0+ (0-)

ip

that

the

the

while

V)+

of

zero

V(O)

+

(1.127), 0-

-+

0+

-

coupled

in the

OT, t

to

induce

go)O_'

Dirac

(1-137)

(1.126).

operator

(1-138) of SU(2). adjoint representation 0- fermions to OTyo' through in

the

anti-instanton

symmetric

is zero,

effective

La-

trasnformation

spherically

configuration

These vertices

U(1)

couple

instanton

modes for

V)+(ip+

e'OV)-, eioo+.

0 transform is

+

and the

the

under

induced

and the two mass

terms

for

(the

case

monopoles zero

the

in

modes

fermions

These

mass

terms

clearly

the

Coulomb

interaction

between

becomes break

the

U(1) symmetry (1.138).

are

with

O(e-SO)4.

Now, we should include lagrangian then, the effective 4

M0)2

2

(anti-instanton)

0 and ?pTro. mass

+

"

is invariant

assume

instanton

2

have used notation

Using (1.134), number

F

-

(1.137)

grangian

Wewill

=

instantons;

C6sar G6mez and Rafael

62

L that

so

the

=

2

HernAndez 47riX

4,ix

(aX)2 +,MOT, O

e-g

1-

0-

-

old

the

now

instanton

clear In

(1-139)

+

g

where 0 to OT-yo become vertices dual and the to OTyo photon X5. couple V) 4-

anti-instanton

or

(1.139) it is now symmetry [24] (1.138).

MO+70e- -7- 0+

coupling

vertices

From

+

X becomes

-that

fact,

boson for

Goldstone

a

(1.138)

under

L is invariant

the

U(1)

if

9 20

27r

that

Notice

anomalous

a

This

we

in

in

N = I'

is

of

point

Yang-Mills the superfield, containing real will be represented a Majorana by is given The lagrangian representation. the

of

terms

gluino adjoint

the

different

a

of X.

consider

The

gluino. transforming

spinor, by

will

from

explains,

which

by

Gauge Theories.

model is defined

and the

gluon

dimensions,

boson nature

example,

first

theory.

2 + I

Supersymmetric

N=I

1.8

in

Goldstone

the

view,

As

now

effects.

instanton not

for X is generated and that no potential X is massless, that the symmetry (1.138) to stress It is also important

of pure

extension

vector

a

2

L As it

=

be

can

apv

--F 4

A'-y ,D"(A)A' 2

F aV +

(1.141.)

checked,

easily

is

Og

+

y ,-2IT

.

F altvF

(1.141)

a

/.tv

,

the

under

invariant

supersymmetry

transformations JA a

6 Aa

4

real

[-yl, 7v]aF

-di

ga with

Aa,

id7

1A

4

al-tv

(1.142)

[-yp, -y,,]F aav,

we can use either Majorana spinor. Notice that, for A in (1.141), or complex Weyl spinors. Majorana for (1.141) effects Wewill now study instanton [27, 28, 29, 30, 31, 32]. For a

SU(N)

a

a

gauge group,

the total

number of fermionic

#zero modes with

(1.143)

2Nk,

=

Pontryagin number of the instanton. of dimension 2j isospin representation,

For S U(2)

k the

in the

modes is

zero

+

and Dirac

fermions of

(L 108)

the wilsonian

sense,

1, the generalization

is The effective but

simply

(1.139) lagrangian the generating

as

will

not

be

functional

interpreted

of the

effective

instantons. 6

For

a

complete

reference

on

supersymmetry,

see

[26].

in

vertices

induced

by

Fields, 2 V+

which

from

certainly

we

The 2N

zero

lies,

(j

=

3

get (1.143)

modes for

where the instanton

V_

-

k

1

=

+

for

1)(2j

j

+

Strings

and Branes

I)k,

(1.144)

1, using Majorana

=

decompose,

relative

63

to the

fermions.

SU(2) subgroup

into

triplets,

4

2(N

2)

-

doublets.

(1.145)

The

clear from supersymmetry. zero modes is quite meaning of the 4 triplet Namely, two of them are just the result of acting with the supersymmetric For N I we have four supersymcharges on the instanton configuration. metric The two charges, two of which anhilate the instanton configuration. other triplets from superconformal result transformations instanton. on the In fact, under supersymmetry, but lagrangian (1.141) is not only invariant also under the superconformal Now, we can repeat the computation group. of section 1.6. The only non vanishing will be of the type amplitudes =

AA(xi)

<

Impossing

the

and dilatation

instanton

measure

invariant,

we

1

T

(see table).

point,

p

2N

comes

*

AA(XN)

collective

>

(1.146)

-

coordinates

to be translation

d4 zdpo 2N

105

from the 2N fermionic

Wemust include

the instanton

action,

zero

modes,

that

scale

as

and the renormalization

p, 4N-

A

where the

power of y is

Majorana

each

fermionic

given zero

A

and

on

*

get

f where the factor

*

using

the

0-function

872

2N 2

exp

by +1 mode.

==

A exp

g(p)2'

for

each gauge zero Defining the scale,

f

-

g3 167r

mode,

dg' 0 09

2

3N,

and -1

2

for

(1.149)

,

SU(N) supersymmetric

for

O(g') (1-148)

(1-148)

-

Yang-Mills,

(1-150)

becomes

A3N, with

A

87r2 =

p exp

-

3Ng(,U)2

(1-152)

HernAndez

C6sar G6mez and Rafael

64

all

Combining

pieces,

these

E

d4zd pp2N

AA(XN)

/\/\(Xl)

<

get

we

A3N

P5

(- 1) Ptr (Ai, Aj2(xj))

(XN))

tr(Ai2N-lAi2N

...

(1-153)

-

permutations over perform the integration the modes for zero given expression

order

In

to

need the

triplet triplet

Supersymmetric Superconformal

_

P (f

Doblets

The fermionic

2

perform

integration <

which

is

the

very

a

(1.155)

to

is that

same

order

(1.155),

in

P

Using

position.

instanton

the

over

n

the

AA(XN)

AA(xi)

I X1

-

0,

.

.

,

of

N

I

XN

mass

we

by

e

This

The

reason

leading

with

size

of

decomposition

cluster

27rin/N

(L 156)

(1-156) generated by instanton the or, equivalently,

is not

clustering

assuming

(1-155)

instantons

we now use

3

we can

,

result.

If

above,

3N

surprising

result

theory.

gap in the

given

is saturated

constantA

get it

(1-154)

the result

constantA

>-

distance.

that

1. Notice

and that

configurations, existence

instanton

Z)"

-

expressions

get

-

.

77"(X 11

the

amusing and, a priori, the integral (1.153)

as

we

=

given for the

are

and p, to obtain

z

< AA >-

with

above table

g(X-Z)2+p2

-

A

the

2

gauge,

A"'t z

1/

(X))

we

the table7.

in

2

modes in the

zero

singular

in the

with

P2 (f (X)) PX(f(X))2

-

coordinates,

collective

the

now

map gap should

be

interpreted

as

confinement.

approach for computing the QCD, and requires supersymmetric

A different massive

SU(2)

for

with

one

flavor <

with

A the scale

of

mass m we

A/\(Xl)A/\(X2)

of the N

=

I

get,

>-

QCDtheory.

< a

AA > condensate

decoupling

from the instanton

constant

Relying

5

now

The function

f (x)

is the

instanton

factor

f (x)

')+P,

with

starts

So, computation m --+

00.

(1-157)

A M,

get 7

limit,

upon

clustering,

we

Strings

Fields, < AA >-

We can

now

take

the

limit,

oo

cluster

only difference decomposition

Until

now we

Wewill

zero.

the

with

show the

computation

definig <

given by Shiffnan anomaly is given by

means

under

that

the

lagrangian

changes

chiral

< AA > at

a non

1

now we

for

vacuum

condensate For

perform

on

SU(N)

angle 0 equal 0, through an

gauge group,

the

(1-160)

2

transformation

e"A,

-+

zero

aN

L +

-

value

167r2

FP.

of 0 is the ==

(1.162)

same as

<

A'A'

>0=0,

with

eia. '

(1.163)

now

27ra

Hence

=

0.

(1-164)

[33], <

Instanton

1.9

that

N --FF.

167r

A' where

=

as

L

Thus,

of the

and Vainshtein.

A

the

N

limit.

AA > condensates

,9-5 This

A of pure

scale

is

decoupling

the

dependence

argument axial

the

A5/27nl/2.

_

previous

before

have consider

now

and define

(1.158)

as

A3 The

65

A512 M1/2e27rin/2.

constant

m -+

Yang-Mills

supersymmetric

and Branes

AA >0=0=< A'A'

Generated

>o=o=< AA >o=o

Superpotentials

in

e'

Three

0 n .

(1-165)

Dimensional

N=2. To start to

three

discussed

with,

we

dimensions. in section

will

consider In this

1.7.

dimensional

case,

we

We can then

arrive

define

of lagrangian Higgs lagrangian complex Higgs field,

reduction to a

the

(1.141) in

with

2 + I

the

given by the fourth component of A/_, in 3 + 1, and the imaginary the part by photon field X. If, as was the case in section 1.7, we consider < 0 for the real Higgs field, >= then we automatically 0 break superconformal and the for invariance, SU(2) case we will find only two fermionic zero modes in the instanton background ('t Hooft-Polyakov monopole). The action of the real

three

part

dimensional

instanton

is

Sinst

=

41ro 92

(1-166)

Hern6ndez

C6sar G6mez and Rafael

66

1

L

2

((,gX)2

(090)2)

+

47r me

where

the dual

(1.167)

X. In

photon

o/,q2

order

which

pling,

order

X. In a

the

that

notice

to

superpotential

the

(1-167)

as

induces

a

e.,

generating fields

0 lagrangian,

as expected potential, (1.170) is at 0 potential

exp -!P

=

+

0 of

for

for

=

to

a

Yukawa dual

the we

cou-

photon

need to add

[24]

&PY F

no

0, and

(1.168)

iX,

certainly coupling supersymmetric

-

the

+

the real Higgs field complex Higgs field,

oo

the

=::

dual

(1.169)

hc,

the type

19W OW

VW i.

a

The

(1.167)

is

potential

effective

an

+

i47rX/92

both

define

0

V) T-fooe

i47rX/g2

vertex

W(floo which

for

=

of the type

term

term

instanton

but

nothing

is

to write

e-

we can

4 in

T

70

the kinetic

have included

we

ioo +,rne-47ro/92

+

Higgs field'.

of the

value

0 standing for the vacuum expectation effective (1-139) becomes lagrangian

with

exp

(1-170)

-0,

photon field,

X.

The minima for

9-

first of the previous computation: with instanton the action, by simply given all, 0 term in four dimensions. of a topological the extra effective the are the in fermions the lagrangian, (1.167), appearing Secondly, the 2 N of the superpotential Finally, the in theory. ones hypermultiplet It

is

important

stress

to

some

(1.169) superpotential ,4 term '22( the analog 9

the

of

aspects

is

'

=

45 is defined

for

on a

flat

direction.

dimensional case to the four of the previous picture generalization flat have not that in we case directions, as not straightforward, certainly in terms of chiral be written superfields can not and the effective lagrangian the but the pair. gluino-gluino gluino, containing The

is

8

Notice

9

The reader

the

that

coupling

gauge

constant,

in

three

dimensions,

has

length-

1/2

dimensions.

field.

noticing

that

dimensional flat as

might

The crucial

direction

the

be

for

surprised

the

correct

N=

reduction

(in

(1.170) for the Higgs concerning potential of this potential requires understanding dimensional 2 three theory has been obtained through which contains a I four dimensional of N Yang-Mills, more precisely, chapter we will define these flat directions

slightly

issue

next

Coulomb branches

=

of moduli

of

vacua).

Strings

Fields, A direct

Computation-

A Toron

1.9.1

obtain

way to

Yang-Millsis

and Branes

67

AA > conden-

<

configuraIn subsection 1.5.2 we have described Pontryagin number -1" [34]. N these configurations. The main point in using these torons is that the number of fermionic reduces to two, which we can identify zero modes automatically defined with the two triplets of one inby supersymmetry transformations Wewill per-form the computation stanton in a box, sending at configurations. The size of the box is the size of the toron, the end its size to infinity. but we will avoid the dilatation zero mode and the two triplet zero modes defined by transformations. The toron measure now becomes, simply, superconformal sates

in four

tions,

with

dimensional

N= 1

f for

translation

the

collective

by the four translation

we

A3. Now,

have included we

obtained

are

1

P-93),

T 7,

final

result

in

of p,

power

a

means

(1.172) -

(1-142)

transformation each fermionic

that

of L should

given

modes,

zero

g(p)2N'

action

toron

powers

no

have

the

in

modes

the

toron

mode behaves

zero

be included

over

simply

is zero

measure.

as

The

is

3.2irie/N

AA >- constantA

<

with

agreement

preted?

which

we

87r2 -

integrate by the supersymmetry

and therefore

Now,

two fermionic

8`2 Notice that (1.172) 92N the box of size L. The two fermionic

the z over

configuration

(1.171)

coordinate.

3exp

gauge

d4Z

modes, and

zero

P

where

using self dual

of

First

the

all,

cluster

the

(1.173) How should

derivation.

(1.173)

value

expectation

this

result

corresponds

be inter-

to the

ampli-

tude

(0,0,I)jAAje,m=

< e,m=

(0, 0, 1),

< k +

Then,

the

(1.173)

in

e

m=

of N different

the set

Notice

that

a

(0, 0, 1)IAAlk,

m=

and the different

is e-3, vacua

change 0

-+

described

e.,

a

vacua.

Z(N) Let

rotation.

us now

In

try the

>

values

in

0 + 27r in equation

other same

words,

AA >o

0

-4

>=

(0, 0, 1)

subsection

in

< AA >9-+<

i.

(0,0,1)

e

0 + 27r

argument for (1.174).

(1.174)

.

(1.173)

correspond

to

1.5.1.

(1.165)

27ri/N e

21ri'*(O'O'1)N

produces

a

change

(1.175) exchanges the different Using (1.74), we observe

that It should to

already

< AA >

< AA > mass

gap.

are

was

be noticed most

derived

that

probably through

topological the a

configurations directly contributing for confinement, as configurations of a argument assuming the existence

relevant

cluster

C6sar G6mez and Rafael

68

< AA >-

in

agreement

eef f

m.

Chapter

2.

Moduli

2.1

In

with

to eef f +

this

part

tentials

with

definition

the

(1-165).

Hern6ndez

A3e2wieff

IN

So,

0

under

for

that

Notice

=

-+

A3e 2-7rielN

e'OIN,

0 +

the toron

21r, we compuation

go,

(1.176)

(1.74),

using

using

we are

from m=

1.

II of Vacua.

lectures,

of the flat

of moduli

consider

will

we

directions.

gauge theories of flat potentials

The existence

of vacua,

which

will

we

understand

possessing will

the

as

po-

motivate

quotient

manifold

.A4

=

V19,

(2.1)

modding of the vacuum manifold V by gauge symmetries. has already been discussed, an example chapter, namely three 2 Yang-Mills, N defined dimensional of N reduction 1 as dimensional in four dimensions. A' fourth the Yang-Mills Denoting by 0' component 4 of the gauge field, the dimensionally reduced lagrangian is from the

obtained In

first

the

=

=

=

4

F a F aii ij

1 +

2

Di OaD' Oa

+

iXa -yiD'X

+

ifab

c/

VbXcoa.

(2.2)

in the Prasad-Sommerfeld limit Yang-Mills-Higgs lagrangian value for the field level, the vacuum expectation 0 is unat the classical level we can define a moduli of (real) therefore, determined; the different dimension values of < 0 >. As we already one, parametrizing to the scalar know, in addition 0 we have yet another scalar field, X, the No potential dual photon field. for X, neither can be defined nor classically If we took into account the action of the Weyl group, quantum mechanically.

the

This

is

V(O)

=

0. At tree

X

-X,

the classical

moduli Rx

The fields

and X

As discussed

in

can

chapter I, a potential

should

be

SI/Z2-

be combined instantons

manifold

into

(2-3) a

generate

complex scalar, a superpotential

P

of type

+

iX.

e-1p,

with its minimum at oo. This 0 fields of The vacuum expectation vacua. potential degeneracy remains undetermined, value of X still but can be changed by just shifting the coefficient of the topological term. The physics of this first what is we example vacuum grounds: quantum effects breaking the classical expect from physical degeneracy. However, there are cases where the amount of supersymmetry the generation of superpotential a priori, prevents, terms; it is in these cases, where we should be able to define the most general concept of quantum moduli [35, 36], where quantum effects will modify the topology and geometry

which

induces lifts

of the classical

the

for

classical

moduli

manifold.

the

Strings

Fields, 2.2

N

of N

with

A'.,

dimensional

4 three

=

reduction

as

Dimensional

N = 4 Three

i

in the

are

with

vector

a

directions.

Yang-Mills

1, 2, 3, corresponding adjoint representation

=

SU(2)R

double

consider transform

cover,

the

in

rotation

[37].

Yang-Mills the

y

through

3, 4,

=

5 vector

model i.

will

e.,

we

the

potential

are

in the

have used

R',

space,

T tr

4g2

possesses

Cartan

subalgebra

of the gauge group.

ai

we now

fermions

then

By dimensional

[0j, 0 j]2,

(2.4)

lagrangian obtained directions,

flat

the

to

i<j

dimensional

six

a

(2.4)

of

components

with respect transform, half If one spin particles.

as

of euclidean

" =

scalars

real

of the gauge group, and will transform in the 3,4,5SO(3)R group of rotations

group

V (0)

dimensional

The three

while scalars, again as doublets, O , are singlets. for the 0j: we get the following potential

reduction,

where

be defined

will to

doublets,

as

SU(2)E

the

the

to

respect

The fermions

69

Yang-Mills.

dimensional

1 six

=

and Branes

15F.1,F141.

-

Obviously, Oi fields

4g

as

For

those

whose

SU(N),

we

will

get

1

(2-5)

a a'

with

flat

of

coordinates

SU(N) with

j

3r +

r

will

to

=

moduli,

that In

N 4r.

the

required.

be

U(1)N-1.

=

3(N

so

(2.4).

0,

directions

A

simplest classically

case

case

a

SU(2),

has

a

a a

point

(2.5)

value

like

dual

photon

dimension

total

corresponds

to

a

four

the

in

of rank

gauge group

expectation

moduli

of

of

has associated

1, the classical

-

characterize

parameters

vacuum

U(1)

As each

The

which

1) general

3,r

r,

breaks

field, Xj, equal to

dimensional

is

(R

M=

3

X

S1 )/Z2-

(2-6)

the Z2 Weyl action, and S' parametrizes the expecphoton field. The group SU(2)R acts on the R3 piece of (2.6). In the same sense as for the N 2 theory considered in chapter I, the N 4 model posseses instanton solutions to the Bogomolny equations which are simply the dimensional reduction to three dimensional euclidean

Wehave

tation

quotiented

value

by

of the dual

=

=

space of four

dimensional

Prasad-Sommerfeld three

available

(we

self

instantons will

therefore

dual

Yang-Mills only

involve

choose

one

equations. of

These

field, them, say 03),

one

scalar

Bogomolny-

0j, and

out

of the

satisfy

the

equation F

=

*D03-

(2-7)

Usar G6mez and Rafael

70

Once

we

break

SU(2)R

choose

particular

a

to

an

0, and 03 different the BPS instanton.

HernAndez

value for the Oi fields, we expectation we can choose 01 particular, 02 from zero, with 03 the field used in the construction of The remaining around the 03 U(1)R stands for rotations vacuum

U(1)R subgroup.

In

-:::

:

direction.

Now, case,

fermionic

transformations

symmetry

addition, should

consider

will

have four

we

above, the BPS instanton

discussed

as

We first

vertices.

the

do not

know, from the results

we

annihilate

the

instanton

chapter

I, that

the

in

X the

dual

03

T

transform

field,

photon

and I

under

super-

solution.

effective

In

vertex

this

a

(2-8)

classical

the

the

-

instanton

quantum effect,

that

of

(0)

Goldstone

a

under

boson,

U(I)R

(2.10)

important

consequences

the

working,

U2 be two

define it

S' bundle

trivial

a non

useful

is

to

use

complex variables,

on

spinorial satisfying the

x

S1. In this

S2. In order

region to

see

on

if

topolthe R3

we

the

work

infinity,

at

Henceforth,

notation.

the

on

the

topological meaning of boundary of R' x S, namely S2

[371 on (2.10) will

com-

-

2

U(1)R is not acting only by (2.10)) on the S' piece. (2.10) can be better understood

The

to

as

-

have

will

V) fermions

(2.9)

X transforms

In fact, moduli. ogy of the classical also of but A4, (as expressed part

behav-

action, the

as

iO/20.

plays the role

behaviour;

anomalous

e

x-4x-4 is

iX),

(2-7) breaks the U(1)R symmetry, U(I)R subgroup of SU(2)R like

However, the dual photon field pensate

+

The term

.

0

is

In this

behave like

ing like

This

fermionic

extra

no

O,OOV) exp -J with

effective

induce

can

hypermultiplets. to the four modes, corresponding

zero

that

of

case

way this

let

ul

and

JUI 12 +JU2 12 This as

defines

sphere S. Parametrizing

the

points

S2 by

in

a

vector

n,

defined

follows: n

with

o,-

the Pauli

matrices,

to u, u,

in

and

S3

,

e'ou,

a

point yield

n

S3 in

the

S2

we can

define

S2,

_

.

value

S31U(j),

a

projection,

(2-13)

S2 The fiber same

=

(2.12)

iiau,

(2.12),

and using 4i:

associating S'. In fact,

=

projection (2.13) we conclude therefore,

of the

of n;

(2.14)

is

Strings

Fields, with

U(1)

the

U(1)

around

rotation

the

point

eioUce-

X to

now

X

4

-

(2.16)

0 in

-+

(t).

So,

2

U(I)R

is the

the

infinity

of

with

U(1)

the

e

iO/2

X

changes

u as

(2.16)

Ua,

angle.

rotation

Moreover,

quantum moduli

our

(S3

(2.15) n, but

preserves

n

u,

where

looks

it

changes

also

like

S1)1U(j)'

(2.17)

action -+

u.,

The space (2.17) is the can be defined through

well

(2.17),

e'ou,

X -+ X

(2.18)

40.

-

known Lens space L-4. Generically, with the U(1) action defined by

L, spaces

(2.19)

X -4 X + 80.

The spaces

L,

also

can

be defined

as

L, with

the

u,

include

S31Zs'

=

Z, action

To finish

the construction the

Weyl action,

so

e

27ri/sU

infinity

of the

Z2,

-+

that

(2.20) (2.21)

of the

we

quantum moduli,

we

equivalently, Weyl

and the

S'IF,

(2.22)

where F, is the group

generated

by 0 given

in

(2.21),

action

(Ul U2)

a:

which

need to

get

L,/Z2 or,

71

action Ua -*

A

and Branes

reproduces

n -+

-+

The relations

-n.

a

i

2

=

#8

==

1,

(fi2) -Ul)) defining a#

=

#-'a.

'(2.23) the group

F,

are

(2.24)

for s 2k, F, is the dihedral group D2kentering into a more careful analysis of the moduli space for these N=4 theories, let us come back to the discussion on the SU(2)R symmetry, and the physical origin of the parameter s. In order to do so, we will first down to four dimensions. The six dimensional compactify super Yang-Mills N 2 supersymmetric theory has only two scalar fields, Yang-Mills resulting becomes (4,5)-plane 01 and 02. The rotation symmetry in the compactified which is the well known U(1)R symmenow a U(1) symmetry for the Oi fields, As in the case for four dimensional Yang-Mills. try of N 2 supersymmetric in four dimensions instantons the N 4 theory in three dimensions, generate

Moreover,

=

Before

=

=

=

C6sar G6mez and Rafael

72

effective

which break this U(I)R symmetry. Following the computations presented in chapter I, we easily discover Of U(I)R to Z8 In the case of N 2 in four dimensions, the vertices

of the instanton

steps a

fermionic

Hernindez

breakdown

potential

=

-

obtained

from dimensional

reduction

V(O) direction,

the flat

with

coordinate

field.

complex

The

(

=

that

(2.26)

-ai u

tr02'

=

transformation

Z4 invariant,

is

U

(2.25)

ai

therefore

is

U(I)R

0 so

02]2,

tr[ol,

SU(2),

for

Oi The moduli

4g 2

is

where

0

is

we now

define

0

as

a

given by

e2iceo,

-+

and the

of

(2.27)

Z8 symmetry

acts

as

Z2

on

the

moduli

space.

The difference

of

a

with

Goldstone

three

the

does not

contribution

boson.

dimensional

any dual

contain

From the three

is that

case

photon field, dimensional

now

that

can

the instanton

play

of view,

point

the

the

role

U(1)R

symmetry of the four dimensional

theory is the rotation group acting on fields be identified with the U(I)R part of SU(2)R 01 and 02, and can therefore We should wonder about the relationship fixing the 03 direction. between the effect observed in three dimensions, and the breakdown Of U(I)R in four dimensions. A qualitative answer is simple to obtain. The breakdown Of U(1) R in four

dimensions

to

oo),

u -4

we

get

an

since

can

the

effective

be studied

theory

We have

0000

only four fermionic

conformal

invariance

transformation

is

>0

not

equivalent

of transformation

come as

a

to

surprise;

rule in

fact,

It

-+

0

-

Now, this

(2.10),

for

the four

327r

by dimensional

coupling free,

(corresponding

limit

using instantons.

Then,

reduction,

0000

since is

>0

for

clear,

u

(2.28)

-

34

from

0

we

break the superthat a U(I)R

(2-28),

change

the

io

produces,

==<

modes,

zero

UMRparameter.

the

a

analog

eio

of the instanton.

0 with

weak

of the type

vertex <

in the

assymptotically

is

4a,

(2.29)

change

0 is, in fact, the perfect This should photon field. dimensional term topological

the

in

dual

2F*F the three

(2-30) dimensional

topological

term

Strings

Fields, io

M3

Zr2 EijkFik This

is

with

the

precisely

topological

coupling

of

the type

charge,

again

we

73

(2.31)

-

of the dual

and thus

and Branes

photon,

in three

recover

the result

dimensions, of section

1.7.

From the

we can previous discussion, a physical interesting point of view. derived fermionic zero counting instanton is a one loop describing pure perturbative

discover

from

something

The transformation

else,

specially

law of X was the effect we are

modes; however, effect, as is the U(1)R anomaly in four dimensions, Consider the wilsonian [38, 39] effective coupling constant for the N 2 theory, without Recall that in the wilsonian hypermultiplets. approach [40], the effective coupling constant is defined in terms of the scale with wave length smaller than that scale out fluctuations we use to integrate is the the to Kadanoff equivalent (this approach for lattice models). In a the natural scale the is value vacuum of the Higgs Higgs model, expectation field. the above the wilsonian in the four constant Using notation, coupling dimensional model is with u the moduli parameter defined by tro'. F 1_)__T as follows: Now, let us write the lagrangian =

,

L with

Im

647r

f

r(F

+ i

*

F)2'

(2-32)

by

defined

-r

1 =

0

i87r +

7-

(2-33)

_.

7r

Using

F2

2

-*F

=

we

space,

L

Now,

we use

the

one

(2.32),

get, from

=

f

12FF

4

loop effective

2 =

g(U)2 general, theory, In

if

add

we

n

the in

infrared

freedom

well

the

n

> 4.

2F

*

in Minkowski

(2-34)

F.

for

the

theory,

0(1).

(2-35) for

the

four

we

get,

u

+

0(l),

=

2

supersymmetric

dimensional

-

known result

dimensions, for

+

U

4 n -In 27

=

for

N

of finiteness For

perturbative computation in the assymptotic infinity. Now, let us perform a rotation

that u

four

In

7r

87r

theories

327r

hypermultiplets,

g(U)2 recovering

0 +

beta function

87r

lagrangian

the standard

n

of the

(2.36)

SU(2)

gauge and

theory when n 4, is assymptotically < 4 the theory free, (2.36) is only valid at small distances, on

u -+

=

so

for

u,

e27riU

(2.37)

C6sar

74

(2.35)

From

for

get,

we

(2.33),

using

so,

HernAndez

G6mez and Rafael

we

n

0,

=

8-7r

8-7r

g(U)2

RT

perfect

(at

case

0

(2.39)

47r,

-

(2.29). Thus, we observe that we have hypermultiplets)

without

the

that

equation

with

agreement for the

least

(2.38)

4i,

get 0

in

+

discovered

term

s

above

is exactly effects, given by the one loop order But what about higher theory. but the non is have As the we effects? nothing presented argument loop the U(I)R action theorem [391 in supersymmetric renormalization theories, of the coupling constant to scale u forces the renormalization on the wilsonian which is of the lagrangian, with the U(I) anomalous behaviour be consistent theorem [41] to be exact at one loop. What determined by the Adler-Bardeen of of all, and from the point First happens as we include hypermultiplets? view of the three dimensional theory, the instanton effect will now be a vertex

dimensional

using

three

effect

of the

instanton

dimensional

four

N= 2

of type, 2Nf

11 xe-(,+'X),

0000 with

2Nf

the

fermionic

index

theorem

get

=

s

equivalently, diagram we

(1.136),

[25],

2Nf, Dynkin

a

dimensional The idea

RI

=

Let

Requiring, we

as

in

A(X4)

saying field

=

that

a

is in

U(I),

We are not stable at

this

moduli

'4b, 7rR

we

in

point

the

account

group, and the

theory,

euclidean

this in deriving that Z2. Weyl action, the moduli characterizing beta

function

W. The massless

space,

we

or,

for

four

the

grounds. theory on

fields,

from

the

of view, contain the fourth component of the photon dual photon X in three dimensions. and the standard

with

have

b

non

sense

of the

to be the

game is

independent

be

(2.41)

lines

for

case

magnetic =

b E

variable,

Wilson

(U(I))

to

A4+a4A(X4)7

-4

angular

to the

of H,

fields

of the type

gauge transformations

trivial

happens

impossing,

parameter.

an

all

compactification,

Kaluza-Klein

as

the

group Notice

P, of

can

A4 with

DNf.

From

be put on more solid geometrical work with the N = 2 four dimensional

have residual

still

jTnj

dihedral

(2.40), type D2N,-4

-1/2.

=

point

dimensions,

four

us

on

1/2

of Callias

consequence

a

as

and

dihedral

N= 4

theory

of

instead

dimensional

Of X4,

2

into

the

between

simple.

is

SI,

x

three in

N

a

appearing

of type

taken

dimensional

three

j

=

means

diagram

already

have

The connection

of the

for

which

-4 +

modes

zero

(2.40)

Z.

completely

flux

[0, 2r].

in the a

generic

through

This

is

equivalent

S' direction,

the

value

S1,

of u".

to be

Now,

at

topologically

point is that the and in that undetermined,

The crucial

to

if the gauge

value

of b

sense

is

a

Strings

Fields,

and Branes

75

N 2 moduli, each point u in the four dimensional we have a two torus E, parametrized by the dual photon field X, and the field b. This Eu is obtained with b. Its volume, in with X, and the S' associated from the S' associated =

defined

units

by the

coupling

dimensional

three

1

Vol E In

(2.42)

-

R

the volume is 1 -g 32, where g2, the three dimensional 3 R of the S' associated to the dual photon (notice

fact,

W1 R2

is of order

constant,

coupling

constant,

coupling Equation (2.42) Eu goes to zero volume. Now, we of the theory in R3 x SIR, if we keep R finite, have a picture namely that of the fibration vacuum over the u-plane, an elliptic expectation parametrizing values of the N 2 four dimensional theory. If we keep ourselves at one particular point u, the torus E,, should be b and X. There for the fields the effective for the target lagrangian space this lagrangian is a simple way to derive by means of a general procedure, To show the steps to follow, that we will now describe. called dualization, we will consider the four dimensional lagrangian (2.32). In order to add a dual photon field, let us say A',D we must couple A4D to the monopole charge, is the

that

size

has units 932, in three dimensions, shows how, in the four dimensional limit,

length).

of inverse

constant

the

=

COijk aiFjk Thus,

we

add

f

4

the

same

notation

I

47r so

that

our

L=-' After

f

647r

gaussian

f

e.,

Now,

that

we use

X. Start

with

81r

f

we

(2.44)

*FDF.

get

f (*FD

iFD)(F

-

+ i

*

F),

(2.45)

is 1

-r(F+i*F)2+

(2.32), these

the

Re

=

integration,

lagrangian

account

(2.43)

(X).

47r

(2.32),

in 1

*FDF

L

i.

A"D E1.jvpu,9"FP' as

lagrangian in

47r6 (3)

term

a

I

Using

=

we

-Ini 64-x with

the four

trick

finally

Re

-

replaced integrations

-r

to

get

dimensional

f (*FD

-

iFD) (F

+ i

*

(2.46)

F).

get

f (-I)

I

=

gaussian

same

87r

an

(*F by

D

'r

are

effective

lagrangian,

-iF

D)2'

(2.47)

The reader

rather

formal

lagrangian

for

should

take

into

manipulations. the fields

b and

C6sar G6mez and Rafael

76

Hernindez

L=f where

work

we now 3

2,7rRd

L

Now,

as

W

euclidean

in

d

3

1 X

jdb 12

7rRg2

before,

did

we

f

=

we

=

6ijk Fjk

,

get

to

a

7rR

+

couple

0jH' H'

dimensions,

three

In

(2.48)

F*F,

327F2

space.

2g2

the

charge, with

io

FF+

d'x

using

get

we

X,

1

=

Fi2j +'0T7r

62*3k Fj kakb.

photon

dual

47r6 (3)

2

field

to

(2.49) the

monopole

(X),

(2.50)

term

ZCijk FjOiXi

(2.51)

87r

that

so

we can

perform

d3X

L

What

we

get is precisely

of moduli

E,,

is

the

four

1

7rRg

a

given by (2.33).

given

three

four

T

gaussian

a

in

of the

terms

2jdb 12

dimensional

R

-0db 12.

(2.52)

7r

fields, which is the torus of the torus complex structure dimensional coupling constant g [37], and while its volume, (2.42), depends on the

four

93

X and b

the

that

,

acts

as

unit.

When we go to the

volume becomes zero, but the limit, The fact that the complex structure same.

-+

this

oc

the

remains

JdX

7rR(87r )2

Observe that

0-parameter, coupling constant

dimensional

+

space for

target

dimensional

structure

integration,

complex of E" is

effective given by the four dimensional coupling will make more transparent of In oo for equation (2.36). fact, the monodromy around u meaning is matrix a (2.36) given by

the

=

( with

so

-r

transforming

that

formation

for

n

=

(2.53)

0

a

b

c

d

-n+4

I

0

(2-53)

-1

as

we

is

group characterization in order to do that

a-r

+ b

c,T

+ d'

(2.39).

get transformation

precisely we

what

we

(2-54)

need,

in

Next,

we

order

to

of the N = 4 three

dimensional

need

on

a

few words

will

moduli

Atiyah-Hitchin

see

that

match the space; spaces

trans-

dihedral

however,

[42].

Strings

Fields,

Atiyah-Hitchin

2.3

equations

77

Spaces.

Atiyah-Hitchin spaces monopole configurations. (2.7), which are simply for

and Branes

study of moduli

in the

appear

Static

solutions

are

the dimensional

Next,

reduction

for

spaces

defined

by

multi-

static

BPS equations,

the

R' of euclidean

to

self-dual

some of the relevant resimply sults on Atiyah-Hitchin for refer our the interested reader problem (we spaces to the book by M. Atiyah and N. Hitchin, [42]). First of all, the Atiyah-Hitchin manifolds of dimension are hyperkdhler spaces 4r, on which a rotation SO(3) is acting in a specific This of is what the moduli we need to define part way. space of N 4 three dimensional Yang-Mills theory for gauge group of rank In fact, N 4 supersymmetry in order to define r. on this space, interpreted of the low as a a-model effective we have target space lagrangian, energy to require Recall here that hyperkdhler structure. hyperkdhler simply means that we have three different anf J complex structures, 1, K, and therefore three different Kdhler forms, wi, wj and Wk, which are closed. Following the notation used by Atiyah and Hitchin, we define Nk as the moduli space of a k monopole configuration. The dimension of Nk is 4k 1 we 1, so for k of the monopole center. to the position If we get dimension 3, corresponding mode out by the translation of the center of mass, we get the space

instantons.

summarize

we

=

=

=

-

Nk /]R3, U

MO k

=

4(k

of dimension spaces

Mko

are

1).

-

For

generically

monopoles, we get dimM20 simply connected,

two

non

0

ITI (Mk so

following:

the

L-4/Z2, for

define

we can

the

case.

the

and

k-fold

its

spaces

L-2/Z2,

moduli

M20

Nf

=

and

0

M-2'

0 case, can

Now,

the

(2.56)

The known

results,

for

k

infinity,

2,

=

are

of type candidate

respectively a good and M21 is the adequate for the Nf be represented by a surface in C3 defined

strongly

1 f2o

the spaces

4.

=

Zk,

Mk.

covering

which

of the

Moreover,

(2.55)

at

are,

indicates

that

M20 is

=

I

,

by y

The space M20 so that we get

=

M20/Z2,

can

2

The spaces of spaces

M2,

defined

by (2.55), y

where 1 should, well known in

singularity

obtained

CII.V,

according

from

in

to the

our

case,

following

=

a

table

X

be

X

=

X2 and y

2V

+

discrete

=

x,

(2-58)

interpreted

as a

of the

limit

family

V1,

(2.59)

with

they give

[43],

variables

X2V + X.

can

2

using

be identified

theory;

with.V

=

(2.57)

+

V

be obtained

y2 -0

:=X2

Nf

-

1.

(2.59)

Surfaces

rise

to

the

subgroup

of

SO(3),

type

of

and

are

singularities are

classified

C6sar G6mez and Rafael

78

r

Name

HernAndez

Singularity

Z".

A,,-,

V, + XY

D2n

D,,+2

V

T12

E6

024

E7

160

E8

X2

V4 V3 V5

==

0

Vn+1 + Y2 X3 + Y2 0 VX3 + Y 2= 0 X3 + Y2 0

+

0

=

_

=

+

+

=

table, the manifold (2.59) corresponds to a Dn+2 2, and dihedral group D2Nf-47 i. e., the group F in the previous section. we have discussed in It is important to stress that the type of singularities we are describing rational the above table are the so called singularities [44]. The geometrical of the meaning of the associated Dynkin diagram is given by the resolution As

can

be

singularity,

from this

seen

with

n

Nf

=

-

corresponding

as the singularity by blowing up the diagram corresponds curve Xj, with self dual

mode of rational

Xi.Xj

the intersection

In

the

elliptic coupling an

try

to

previous fibration,

But

According ,A, where

dimensional

to

Kodaira's

A will

singular given by be

equal

[45],

notation

be chosen

for

19p, more

types:

1, R

N

=

4 moduli

and moduli

a

to

-r

space

as

by the

given Next,

at

zero.

some

irreducible

details),

we

define

an

elliptic

compact Riemann surface. The elliptic fibration,

discrete

curves.

all

Fibrations.

as a

:

V

set

posible

fibration In

we

singular

take

(2-60)

A, ap.

The

singular

Cap,

fibers7

(2.61)

to Kodaira's

of

V onto

general,

n,(9p,,

According types

-*

of points,

Cap with

is

and each line

=

Elliptic

of

!P

are

-2,

=

components.

the

of volume

each

group,

Classification

. A to be of genus

will

irreducible

have modelled

com-

which

component,

Xj.Xj

intersection

irreducible

interpretation,

singularities.

Kodaira's

2.4

irreducible

an

of the

In this

2 gauge theory. N we will the Atiyah-Hitchin characterizing space with the monodromy at infinity 4 moduli, of the elliptic before doing this, we will briefly review Kodaira's theory

dihedral

the N =

to

E-.,,

fiber

of the four

the

modulus of E.,,.

elliptic

with

we

matrix

singularity.

the

different

between

section

constant

connect

describing on

intersection

obtained

ponents

theorem

curves

are

(see of the

section

4.7

following

Strings

Fields,

In+1 where (9i

(e

are

192)

1

=:

*

A,, affine

The

1

=

7

(9,

+

...

with

curves

(2.62)

3,

+ 1 >

n

79

((9o, (91)

intersections

=

-

diagram

Dynkin

+

rational

(19n eO)

::--

Oo + 19,

:=

singular

non

*

*

Cap

:

and Branes

be associated

can

I,,+,

to

-

Different

cases

are

i) Io, ii)Ij,

with

with

Cp Cp

and 190 and i9o elliptic and rational (90 (90 a

Cp

eo

non

singular. with

curve,

one

ordinary

double

point.

iii)[2,

with

Notice -

Singularities

with

:

+

i.

=

that

'n"-4

el and eo and el

singular rational points, points. (eo, 01) e., pi + p2, I, and 12 correspond to diagrams A0 and A,, respectively. of type 'n-4 are characterized by

intersection

Cp

=

190

+

191

((90, (94)

intersections

((95; (96)

1,

192

+ =

03

+

2194

+

(191; (94)

=

singularities

these

with

non

two

+

2(95

(192094)

+

to

the

(2.63)

20,,

(193) (94)

=

correspond

+

...

=

D,,,

(041 (95)

Dynkin

=

dia-

gram. -

Singularities E8

of type

In addition -

-

II*,

correspond

and IV*

III*

types

to

E6, E7 and

-

to these

singularities,

we

rational

have also

the types

cusp Cp (90, rational with eo + el, with (90 and (91 non singular curves, Cp intersection 2p. (i9o,(91) 'TV : Cp 190 + (91 + (92 with Oo, (91 and 192 non singular rational curves, with intersections (02 (90) ((917 192) (0o, 01) P.

11 III

:

with

=

:

00

a

with

curve

a

-

=

=

-

=

,

=

In contrast

to the

these all

singularities these singularities

are

singularities

described

associated we

=

to affine

in the rational

case

in last

section

(the

Dynkin diagrams.

rational

ones),

Observe that

for

have

C-C while

=

7

the

corresponding C.C

the affinization

=

(2.64)

0,

=

-2.

maximal

cycle

satisfies

(2-65)

fibraDynkin diagram is the elliptic rational of ADE type in singularity of the Dynkin diagram whenever there is surface, and get the affinization fiIn the case of an elliptic singular curve passing through the singularity. fiber itself. this curve is the elliptic So, the extra node in the Dynkin bration, fiber. This can be seen more clearly as the elliptic diagram can be interpreted of the surface. In fact, for the elliptic the fibration as we compute the Picard the fiber, the basis, and the contrito the Picard contribution comes from bution from each singularity. from each to Picard Now, in the contribution

The

tion

origin

structure.

for

In

fact,

we can

of the

think

of

a

singularity, taken

we

into

should

limit,

the

not

consider this

L e.,

the extra

us

as

corresponding

4 gauge theory. and therefore

define

then

(U)

'T

volume, as

should

f.X'Yj 11

--

p(u)

with a

holomorphic

the

holomorphic

j(-r(u)),

on

form

one

function

of

on

Next, half plane,

the upper

we

u.

A is the discriminant

where

A

92

become

T(u)

Defining

(that Ap,

is,

an

j(w(u))

-=

function.

To each

element

as

a

pole

a

F(u),

of

bp,

of order

Ap-r for to

some

bp.

AP) of finite

The matrix

singularities it is always possible which

can

then

S

-q

with

ps

singularities. the table

-

1.

qr

The below.

-r

p

function

we

Ap7If ap is

that

modular

(2.67)

-4

cT

+

it +

it

turns

out

be

to

a mero-

contractible

associate

+ b

T

A, non

a-r

then

(2-68)

+

to

-+

functions

the modular

1n27)4"' 1n2T)6

of

want

-F(u)

function,

+

pole ap, and each

ffj(zA)),

in

follows

it

3

(n,

EZ

ni,n2

morphic

(2.66)

with

(n,

EZ

E

140

=

27g2,3

-

nl,n2

93

and the

work out the monodromies

Cu. From (2.66), define the elliptic

E

60

=

Thus,

curve.

A

g'2

=

we can

elliptic rational,

an

go to In this

we

(U)

(P

.172892

j(T(U))

when

(P(U)

X72

is

=

been

Picard.

happens

L e.,

singularity

already

in the

N

Dynkin diagram is not affine. let us However, before entering that discussion, for the elliptic of Kodaira's fibrations classification. We will

element

what

dimensional

compact torus,

a

has

this

since

an

becomes of infinite

E,,

anymore

the

as

telling

already

is

node,

the fiber

count

the three

to

fiber

it

Herndiadez

count we

discussion

elliptic

limit

not

when

account

The previous R = 0 limit,

the

in

G6mez and Rafael

Usar

80

a

path -Yp in A monodromy matrix,

(2.69)

d' can

be

proved that

Ap

bp)

is of

type

(2.70)

order, Am 1, for some m, corresponds P Moreover, if Ap is of infinite order, =

be removed. to find

)(

Next, we classification,

numbers p, q,

a

b

c

d

relate

)(

and

such that

s

P

r

1

bp

q

s

0

1

matrices

according

r

to

Ap

with

Kodaira's

the

)

(2.71)

'

different

work,

is

types as

of

shown in

Fields,

Strings

and Branes

81

Type of singularity

Matrix

0

1

b 0

Ib

1

b

1

0

-1

1

1

-1

0

0

1

-1

0

01

1

IV

1

Now, we can compare the monodromy (2.53) with the ones in the table. It of type Ib*, with b n 4, corresponds to the one associated with a singularity to a i. e., a Dynkin diagram of type D,,. In the rational case, this corresponds the number of flavors, so that n represents dihedral group D2n-4- In (2.53), dihedral we get the Atiyah-Hitchin space. group of the corresponding Summarizing, we get that the dihedral group of N 4 in three dimensions of the at infinity with the type of elliptic is the one associated singularity In other dimensional 2 four defined fibration theory. by the N elliptic in the R -+ 0 three dimensional words, the picture we get is the following: When of type C'ID2Nf-4. at infinity, limit a rational we have, singularity with at infinity, the R -+ oo limit an elliptic we get, we go to singularity of one respectively, describe, singularities Dynkin diagram DNf. Both types N 2. 4 and four dimensional N loop effects in three dimensional =

-

=

=

=

=

2.5

The Moduli

Supersymmetric

N 2 Space of the Four Dimensional Theory. The Seiberg-Witten Yang-Mills =

Solution. From

of the

our

discussion,

previous

moduli

space

of three

we

have observed

dimensional

N= 4

that the complex structure Yang-Mills supersymmetric

fibration on the moduli space of the four dimengiven by the elliptic with the effective modulus is identified theory, where the elliptic complexified coupling constant -r, as defined in (2.33). This result will in pracN 2 theory can to the four dimensional tice mean that the complete solution of the Atiyah-Hitchin read out form the complex structure be directly spaces (2.59), with 1 Nf 1. In previous sections, we have already done part of this oo, i. e., in the assymptotic job, comparing the monodromy of T around u of the the infinity with the dihedral freedom regime, group characterizing

theory sional

is

N= 2

=

=

-

=

dimensional

three the

structure

review In this section, we will briefly difor four 51, 52, 50, 49, 48, 531 47, 46, 36, [35, theory, and compare the result with the complex

N = 4 moduli

solution

Seiberg-Witten

mensional

Yang-Mills Atiyah-Hitchin

N= 2

of

HernAndez

G6mez and Rafael

Usar

82

spaces.

and therefore

hyperkdhler, complex structure

space.

possess

the

that

Recall

different

three

Atiyah-Hitchin spaces are The complex structures.

is one N 2 solution by the four dimensional space namely the one where the Atiyah-Hitchin of Seiberg and Witten was origiThe analysis by u argument: the moduli space parametrized nally based on the following 0 the Nf of all consider first to a sphere (we will should be compactified A is taken Kodaira's to notation, for According group). SU(2) gauge case, of T at u oo is directly Next, the behaviour to be of genus equal zero. this leads from the one loop beta function (see equation (2.36)); obtained of the type (2.53). Next, if -r(u) is a holoto a monodromy around infinity mathematical fibration of u, which is clear from the elliptic morphic function and is a direct consequence of N 2 supoint of view (see equation (2.66)), As then the real and imaginary parts are harmonic functions. persymmetry, is the imaginary T(u), constant the coupling part of the complex structure with an elliptic we are dealing which is on physical grounds always positive, That some exknow all posible so we already types of singularities. fibration, is clear form in addition to the one at, infinity, should exist, tra singularities but in it is that and the fact of Im-r(u), positive, the harmonic properties

determined

=

complex structures, fibered. becomes elliptically of these

=

=

=

principle The

we

answer

do not to this

how many of them question can not, in

we

should

principle,

expect,

and of what.

be derived

from

type.

Kodaira's

approach, using theory. In fact, all what we can obtain from Kodaira's bundle K of the between the canonical is a relation formula, adjunction and take which as IP1, we can the K of the base space, fibration, liptic type of singularities, Kv where the aj,

for

each type

Singularity

ai

11

1/12 1/2 + b/12 b/12 1/6 1/4 1/3 5/6 3/4 2/3

Ib* lb 11

111 IV

IP

111* IV*

=

of

17*

(KA

singularity,

+

E aiPi), are

given below.

the elthe

(2-72)

Strings

Fields,

However, (2.72) which V manifold, according to physical The

like

what

of to

have

: - as looking

we

We will

instead,

(2-47)),

performed

the for

singularities

are

theory,

of the

space

an

S=

Thinking analysis

for.

do not

know the

therefore

proceed

strong coupling hopeless to try to use a on a duality we can rely approach. In behaves constant the effective coupling with S transformation,

for

looking

analysis;

we

since

point,

looking

we are

(see equation

L e.,

this

at

arguments.

moduli

variables

dual

useful,

not

we are

perturbative

naive

is

singularities of the

regime

is

83

and Branes

(

it

so

in the

is

0'

, 1

(2.73)

-r"9, magnetic coupling, of monodromies type perturbative effective

,Tmag -+,Tmag

we can

reduce

our

(2.74)

+ b.

of Kodaira's type is related to a monIndeed, we know that any singularity (see equation (2.71)). odromy of type (2.74), up to a unitary transformation, of the type Now, and on physical grounds, we can expect a transformation of an constant effective the for coupling (2.74) as the monodromy singularity massless of number the hypermultiplets. effective U*(1) theory, with b equal to with n hypermultiplets, for the U(1) N 2 theory, the beta function In fact, -=

is

given by

ik

Fmag (U) k the number of massless

with

or,

notation,

Kodaira's

in

between

a

the

2-7r

hypermultiplets. I

k

0

1

monodromy type D, and

(2.75)

In(u), This

yields

the

monodromy

(2.76) of type Ak-1. Notice that the type A monodromies,

the

dif-

reflects

sign and type D (that free theories, type A for infrared obtaining free theories (notice the sign in is Do, D1, D2, and D3) for assymtotically (2.75)) [54]. Now, we should wonder about the meaning of (2.75). Recall that so the meaning of coupling constant, our analysis relies upon the wilsonian i. the vacuum e. the in scale the related to be must theory, U(1) u in (2.75) more propthe field in scalar the for value multiplet photon or, expectation value gives a This vacuum expectation erly, in the dual photon multiplet. so the through the standard Yukawa coupling, mass to the hypermultiplets to the with at u proportional be should u of 0, expected (2.75) singularity we do know which hypermultiplet Fortunately, mass of the hypermultiplet. the one defined consider: by the monopole of the theory. In fact, we should

ference that

in

we are

=

we

should

rewrite

(2.75)

as

,rmag

(U)

ik 2 -x

In

(M (u)),

(2-77)

C6sar G6mez and Rafael

84

M(u)

with

the

of the

mass

the point

around

Hermindez

monopole,

M(Uo) Therefore,

(2.77)

and consider

perturbatively

uo, where

(2.78)

0.

=

conclude

that a singularity of Ao type will appear whenever monopole equals zero. The nature of the point uo is quite clear from a physical point of view: the magnetic effective coupling constant is zero, as can be seen from (2.77), the dual electric so that coupling should become infinity. But the point where the coupling is infinity constant is by definition the scale A of the theory; A. then, uo of Ao type are there. Now, it remains to discover how many singularities In principle, a single point where the monopole becomes massless should be A point); expected (the uo however, as mentioned in section 2.2, the U(1)R is symmetry acting on the moduli space as a Z2 transformation. Therefore, in order to implement this symmetry, an extra of Ao type must singularity The simplest exist. solution for the Nf 0 theory, with SU(2) gauge group, fibration to an the over corresponds elliptic with IP', compactified u-plane, three singular of type points, the

we

mass

of the

=

=

=

Ao, Ao,

Do; with

Do the singularity

A,

with

What about

(2.79)

is

alent

to

at

A the scale the

infinity, theory.

and the two

of flavors?

inclusion

The

case

and

singularities Another

at the

points

A,,

should

we

know that

we

clear,

A,..

(2.80) become is

=

the

free into

monodromy

theories

can

around

the

be obtained

origin. through

a

trivial

mass-

mon-

The two other

decoupling arU(1)R symmetry. The results

the residual

account

Do in D2 is equiv-

expect

of A, type indicate that two hypermultiplets with Nf case is that 4, where there now

as

[54]

Now, ture.

this case, 2 should be

In =

simple

odromy D4, which is cases of assymptotically guments, and taking are

Nf

therefore,

D2; The

Ao singularities

of the

replaced by DNf. two A, singularities

less.

(2.79)

As

with we

singularities we

know that

for

the

these

correspond this

corresponding

Ao, Ao, Ao,

D3

Ao7 A3-

fibrations,

elliptic

know from

D,

Kodaira's to

a

we

(2-81)

shoud consider

for

argument

rational

curve

with

the a

singularity appears at u complex structure is, with A

double

Y2

=

X

3-

X

2U +

X.

Nf

the

complex

struc-

Ao singular point; as A, the simplest guess 1, =

0 case,

the

double

(2.82)

Strings

Fields, The

for

(2.83),

curve

for generic by f (x, y; u)

defined

a curve

u, does not

F

=

F. and Fy the derivatives

with

genus of the

curve

F

0,

Fy

0,

with

be obtained

can

(n

9-

that

such that

(2-83)

respect

to

2)

-

those

are

85

Recall

points.

points

rp

(rP

2

The

respectively. theorem,

and y,

x

using Riemann's

1)(n

-

singular

have

0, the singular

=

and Branes

-

(2-84)

2 P

where the in

(2.84)

u,

we

is

sum

singular

over

degree

is the

of the

get, for (2.82), 2, Now, for u

g

=

L

have

we

=

and n points, rp is the order of the singularity, polynomial F, defining the curve. So, for generic

singular

a

Y

=

0,

X

=

-

(2.83),

satisfying

point

namely

U

This

is

double

a

AO type.

and

point

classification,

daira's

that

also

Notice

therefore,

has

curve

get genus

now

derived

0. From Ko-

X

_

y

=

2U

=

curve

(2-86)

.

0 for

generic

(2.82)

u.

satisfies

Using (2.84), we the properties

all

point point

ZY2

region we

(2.87),

a

at

x

=

y

infinity.

at

=

oo.

This

In can

order

to

be done

compactify going to the

curve

at

dimensional of

has

add the

must

we

projective

Next,

point at x equal zero. Thus, the

(2-82)

curve

curve,

The

=

above.

The the

X3

=

double

a

get g

this

at

=

there

Y2 This

we

of points we get two singularities 2 X3 -4- X, 0, we have the curve y origin, u are no singular points. Moreover, if we take

at the

which is of genus one, since A = 0, we get the curve

(2.84),

using

know that

we

(2-85)

*

2

infinity

R

will i.

see

e.,

0

-+

of this

limit,

that

the

limit with

points

ZX

_

2U

+

Z2X.

is defined

curve

can

this

X3

=

0. The curve, in the three by z 00. by (2.82), but with Vol(E") the points at infinity to deleting equivalent for z 0 0 we cab define a new 0. In fact, =

be described is z

(2-87) =

=

variable, V

and write

(2.87)

(2.89)

interpret

is in fact

X

-

(2-88)

zu,

as

ZY

We can

=

(2.89) the

2

defining Atiyah-Hitchin as

=X2V + Z 2X. a

surface

space in

(2.89)

but projective space ]p3 coordinates. Thus, homogeneous in the

,

C6sar G6mez and Rafael

86

conclude

we

the

that

R

-+

Herndndez

0 limit

equivalent

is

deleting

to

the

points

at

E.,, by (2.82). The representaphenomena in a different way as follows. of the tion is as an so that we elliptic Atiyah-Hitchin fibration, (2.82) space structure. have selected in rotate the space can we one complex However, yet the one selected of complex structures, fibration. by the elliptic preserving This U(1) action must act on E.,,; however, this is This defines a U(1) action. impossible if E,, is a compact torus. But when we delete the point at infinity,

infinity

of the

We can

defined

curves

see

this

and pass to the

projective

(2.89),

curve

we

have

a

U(1)

defined

well

action

[37], \2

X

v

Only

Z2 subgroup

a

of this

action

which

A4

e.,

simply, A2

at

Notice

A

=

0

or

2

z

0,

i.

and the

,

also we

=

=

the

1. This

surviving

action

A-2

=

I

=

U(1)

the

e.,

at

A-2

=

Z2 action

when

we

U'

(2.91)

infinity,

=

(2.92)

A,

moves u -+

and is the

-u,

work in the four

dimensional

the projective sense, becomes

in

v

==

only part

x,

of

More

limit.

and

we

get

Z4 symmetry of (2.82)

relation

y

-4

X

-+

U

-+

between

zy, -

X,

(2.93)

-U.

A and the

breaking

of

U(1).

In

fact,

for

have Y

which

on u:

A-2V

_

(2.90)

means

A2 i.

survives

A2X

U _+

X,

Ay, A -2V.

y

is invariant

2

=X

3 _

X2U,

(2.94)

under

X

A3Y' A2X,

U

A2U.

Y

---

(2.95)

Fields,

and Branes

87

Superpotentials.

Effective

2.6

Strings

solution derived from the Seiberg-Witten result spectacular of electric first the theories is 2 supersymmetric to N dynamical proof need first this to go understand In order to properly confinement. proof, we The confinement. of simplest physical picture of through the recent history BCS is that of dual confinement theory [23, 55, 56]. In that superconductivity the dual of the standard be is to as vacua a confining represented picture, of Cooper the condensation characterized which is by superconducting vacua, efof the Meisner under name In we find, ordinary superconductivity pairs. In confinement. for mechanism a the superconducting vacua, a magnetic fect, confines them. flux tube that creates a magnetic pair monopole-antimonopole first of was The relativistic superconductivity description Landau-Ginzburg the in vortices where Olesen and Nielsen introduced phase Higgs [57], by as Meisner are interpreted magnetic flux tubes. The order parameter of the in this value of the Higgs field; phase is the standard vacuum expectation field. The the conscalar to a electric-magnetic coupled U(1) model, simply and the magnetic fined monopoles would be U(1) Dirac monopoles, string is characterized by the Higgs mass of the photon. The dual version of this We simply consider fact a dual in is photon, or dual picture easy to imagine. the field a to now representing matter, Higgs coupled magnetic theory, U(1) dual for and look with a we Higgs charge, U(1) monopoles magnetic magnetic will value of the monopole field, mechanism that, by a vacuum expectation This mass gap will charinduce a Higgs mass for the dual magnetic photon. the confinement acterize phase. As the reader may realize, this whole picture is based on Higgs, or dual Higgs mechanisms for abelian of confinement gauge however, in standard QCD, we expect confinement to be related to theories; of the gauge groups. nature the very non abelian Indeed, only non abelian slavwould and theories are free, assymptotically possess the infrared gauge theabelian in a non or Moreover, phenomena. confinement, pure gauge ery, 't define stable to have the do not Hooft-Polyakov we right topology ory, to the abelian picture monopoles, so the extesion of the superconductivity N 0 pure Yang-Mills theory, or standard QCD, is far from being direct. some Along the last two decades, with 't Hooft and Polyakov as leaders, have been sugested. for confinement Perhaps, the main steps in the pictures

Maybe the

most

=

=

story

i) 2 ii)'t hilt ivJt

are

Polyakov

+ I

Z(N) duality

Hooft

twisted

Hooft

abelian

Concerning I.

Let

us

idea is

namely

i),

we

therefore

dealing

[23].

quantum electrodynamics

Hooft

with

relations

boundary projection have now

the

[56].

conditions gauge

[20].

[58].

already

described

the

consider

the other

points.

topology

underlying

dynamics in chapter Concerning ii), the general SU(N) Yang-Mills theory,

relevant

pure

HernAndez

C6sar G6mez and Rafael

88

I

H, (S U(N) This

is

the

for

condition

the

U(1))

magnetic analog a Z(N) magnetic flux for confinement, criteria A(C) going like reproducing behaving like the perimeter, The duality to confinement. is equivalent

loop B(C) defined for creating Hooft

reduce

is the

to

A(C)B(C') v(C, C')

where

=

e

along

tube the

again

the

was loop A(C), The Wilson C. path

now

that

picture

in

B(C)

dual

Higgs

dual

its

by

established

relations

27riv(C,C')IN

the

has

area,

The 't

vortices.

and

Wilson

of the

't

B(C')A(C),

number between the loops

is the link

Z(N)

magnetic

of

existence

(2-96)

Z (N).

-

Hooft

(2.97)

C and C'.

From

(2.96),

A way with duality were obtained. posible phases compatible introduced was also the previous by 't picture to make more quantitative main of the in a box. Some Hooft, by means of twisted boundary conditions in chapter introduced I, but we will come back to were already ingredients in of this section we will mainly be interested them later on. In what follows the abelian projection gauge. that of defining a The idea of the abelian projection gauge was originally it is do to The simplest way unitary gauge, i. e., a gauge absent of ghosts. first reducing the theory to an abelian one, and then fixing the gauge, which if easier task. Using a formal notation, is (in the abelian theory) a certainly abelian maximal its is L subgroup, G is the non abelian gauge group, and then the non abelian part is simply given by GIL, so that we can take, as the where degrees of freedom for the abelian gauge theory, the space RI(GIL), of the whole space Now, the R generically gauge configurations. represents fix the and we can is an abelian theory, gauge, theory defined by RI(GIL), L\RI(GIL), by RIG to the unitary gauge, characterized going finally

different

the

=

Now, theory,

two

questions

RI (GIL),

concerning and the more important arise,

to fix

be defined.

In order

piece GIL,

't

Hooft

a

functional

think at

of

as

the end.

We will

the

used the

X(A)

point

abelian

non

following

part

trick

[58]:

of the intermediate

of how such

a

theory

abelian

should

of the gauge group, i. e., the let X be a field that we can

A, X(A), or an extra field that will be decoupled X (A). being, we simply think of X as a functional, i. under the adjoint transform representation, e.,

of

For the time

require

the content

to

X(A) Now, the gauge condition

that

-+

fixes

gX(A)g-1. the

non

(2-98)

abelian

part

of the gauge group

is

Al

(2.99)

X(A) AN

Fields,

Strings

and Branes

89

Indeed, if X (A) is diagonal, the residual group is just the maximal abelian subgroup. Notice that X(A) is playing a similar role to a Higgs field in the and (2.99) is what we will adjoint representation, as a vacuum exinterpret pectation value, breaking the G symmetry to its maximal abelian subgroup. As in the standard Higgs mechanism, now the degrees of freedom are the diagonal parts of the gauge field, A('j), that transform as U(1) charged particles. In addition, fields we have the N scalars A, appearing in (2.99). Summarizing, the particle content we get in the maximal abelian gauge is

i) N I photons, A("). ii) 1-N(N 1) charged particles, 2 iii)1V scalar fields, Ai. -

AW)

-

Notice

(2.99)

that

depending that, by in principle,

on

is

values

are

a

does not

in fact, require the Ai to be constant; Ai are fields Another important spacetime position. aspect of (2.99)

the

of this

means

any form

maximal

potential

abelian

for

gauge

fields,

the Ai

undetermined.

priori

of type ii) particles to Ai proportional The spectrum

of

can

Aj, i), ii) -

be as

and

that

Concerning considered formally

the previous

is the

standard

iii)

in the

case

is not

massive,

complete.

not

we are

so

Extra

their

introducing, expectation

spectrum, with

Higgs

the

charged being

mass

mechanism.

correspondof the maximal abelian gauge, (2.99), ing to singularities is also allowed. These singularities correspond to points in spacetime, where Ai (x) Ai+1 (x), i. e., where two eigenvalues coincide. Wehave impossed that Ai > Ai+,, i. e., the of (2.99) These singularities are ordered. eigenvalues in three are point-like dimensions, and d 3 dimensional for spaces of dimension d. It is easy to see that these singularities of the gauge (2-99) are 't Hooft-Polyakov monopoles. spectrum,

=

-

Once

we

have this

set

of

degrees

of freedom

to

describe

the

non

abelian

the-

the phenomenum of confinement, ory, we may proceed to consider following in essence the same philosophy as in abelian 't Hooft's rules superconductors. of construction

RIEliminate

grangian,

are:

the

electric

where the

charges. "massive"

This

electric

means

particles

an effective laconstructing A(") have been integrated

loops. on the effective duality transformations lagrangian obtained upon the above integration of the electric charges, going to dual photons. These dual photons should interact with the charged monopoles by ordinary vertices, coupling the dual photon to two monopoles. The interaction between not reduced to the the single monopoles is certainly exchange of dual link a missing photons; there is in practice the dual photonconnecting and the effective monopole vertices, and which is played by the lagrangian, A-fields: the Ff f action that have Yukawa depends also on the A-fields, coupling with the charged AW) particles, inside the loop. As we running dualize, we should also take into account duality on these fields Ai. In fact, this should be the most relevant is the potential as it part of our story, out

inside

R2?erform

G6rnez and Rafael

Usar

90

between

interaction

monopoles and the dual Ai fields

value R3rhe expectation In fact, this computed. the theory

minimizing In

this

naturally

leads

< M

>, for

vacuum

with

theory

the to the

in

R2,

must

be obtained

should

Ai field

values.

expectation

respect

obtained

value

be

after

't Hooft's physical structure underlying approach, standard use in QCD or pure being of practical is being made However, progress in lattice computations

of the beatiful

spite

far

is

program

Yang-Mills at

what

rule.

next

to

HernAndez

theory.

from

present. After

introduction

this

back to the

of the

validity similarities

in

abelian projection 't Hooft's gauge, let us come of N = 2 pure Yang-Mills theory to find out the reader wil have already The careful found some

to

example

simpler

above rules.

and the

discussion

our

Seiberg-Witten In fact, presented. the as Higgs field

way the

for

solution

N 2 supersymmetric Yang-Mills in the adjoint, theory, the X field can simply be interpreted the moduli of to U(1) on generic points (for a group of higher breaking SU(2) have the spectrum also down is to the we U(1)'). Moreover, breaking rank,,r, to of 't Hooft-Polyakov monopoles and, according degrees of freedom, we are abelian the close to however, we should picture; projection certainly quite In it was not assumed Hooft's abelian 't be careful at this projection, point. defined massive with well we must be at a Higgs at any moment that phase in find of the abelian The we monopoles projection monopoles. type gauge size massive in the usual sense and, moreover, are not they have not finite but are simply point like singularities. solution Rule RI is almost accomplished through the Seiberg-Witten the effective from consider obtained In we can lagrangian fact, [35, 36]. AO and the in field the scalar is where a photon, represents F,f f (AO, a), effective that this constrained is the N 2 hypermultiplet lagrangian (notice

has been

=

to

be N

=

2

invariant).

value

expectation

For

of the field

each value a, in the

a(u) The effective

(see equation

lagrangian

(2-36)),

contains

and instanton

=

of

u

=

perturbative

.1 2

<

tro'

regime,

vfKu.

only effects.

>, is

in

the

the

N= 2

vacuum

simply

(2.100)

one

loop logarithmic

The instanton

contributions

and multiinstanton

four fermionic the zero modes, as we kill transformations. The expansuperconformal and non perturbative in perturbative effects sion of the effective lagrangian weak coupling can be done in the regime and, if we know how to perform information the duality non trivial we can start trasnformation, obtaining denote through Ff f (A', aD) on the strong coupling regime. Let us formally In the dual perturbative the dual effective regime, the effective lagrangian. in one loop terms, to light is an expansion corresponding lagrangian magand non perturbative netic monopoles, higher order terms. From the moduli expansion should appear as a good space point of view, the dual perturbative contributions

four

zero

contribute

modes associated

each with with

Fields, of the infrared

description

region,

i.

for

e.,

Strings

values

of

u

and Branes

constant

at

the

of

dual aD

very

same

monopoles.

way

as

the

mass

given by a. We can charged particles, magnetically

nism,

is

Then,

of W---' -

we

have

particles, a general

In,a

+

n,,,aDj-

(2.102)

equation

monopole, Higgs for electrically

far

from

physical

require the

on

u

distinguished sional

back to some

on

aD(u),

discovering

of

Kdhler

geometry.

fact,

In

a

we

proper

(2.102)

but, goes

description

know that

the met-

Kdhler with respect to the complex structure moduli, is certainly of the N 4 three dimenfibration representation by the elliptic --::::

moduli

potential

problem

our

results

and

arguments

will

Coming will

in the

mecha-

(2.102)

and supersymmetric meaning of see, the mathematical beyond the scope of the simple argument we have used.

as we

ric

=

(2.101)

standard

formula

write

now

only motivated

the

be

theory

of the

mass

in

M(n, n,) Here,

aD is the

should

the dual

=

for

A, with A descrip-

--

u

the

of Yukawa type

the electric

such that

are points neighbourhood large, dynamically generated scale. To complete the equivalent to expression tion, the equivalent (2.100) for the dual variable 2 N that we obtain constructed; impossing supersymmetry, has a coupling aDMM,

which

is

91

can

If

space.

be defined

it

has

a

Kdhler

the

structure,

corresponding

Kdhler

through 9UU-

==

Im

(

a2 K auaft

)

(2.103)

-

N out from the effective 2 low energy can be read potential the metric moduli space is on the fact, as a general statement, low energy lagrangian. terms of the effective Now, for given by the quadratic in terms of the so called N 2 the lagrangian can be written as prepotential This

Kdhler

action.

=

In

=

follows:

f

,C where A is

language, ,T

which superfield, chiral fields. on depends only an

N= 2

as

K

from which

(2-103)

we

=

holomorphic

is

The Kdhler

(2.1Q4) or,

in

potential

supersymmetric is derived

from

Im.

(aA -A),

(2.105)

IM

(OaD ad),

(2-106)

(9.T

becomes

guiz

where

d 4O-T(A),

=

au Dii

have defined aD

=

o9a

(2.107)

C6sar G6mez and Rafael

92

in the

sense

of lower

HernAndez

Using (2.104)

components.

metric,

d'S2 and therefore

IM

a2.F

identify

we can

=

that

Notice

equation

(2.109)

of

as

definition

the

aD,

is it

o9a2

(2-105)

we

for

get,

equivalently,

or,

daD

(2.109)

=

da

perfectly provides

the

(2.108)

dada,

T(u)

with

=,, 8a

T(U)

for

a2y

and

with

consistent the

of the

mass

what

we

expect

monopole. In the a Therefore,

regime, we know that it behaves like ImT and relation thanks to (2.109) is the right generalization. Fortunatelly, the of in terms of definite a Seibergwe get aD (U) representation fibration solution, elliptic

perturbative

(2.107) (2.66), Witten

-

i

daD =

-

du

P

Now that

lowing

we

have

candidate

a

AP,

of

for

E.,

which

given by

is

dx

(Z; U)

=Y

aD

aD(U)

(in fact,

rules

Hooft's

't

(2.110)

(z; u) dz,

Y

differential

holomorphic

where W is the

o

-

(U),

or,

(2.111)

.

we can

continue

equivalently,

our

analysis

fol-

the dual

how to define

a missing program). Next, we want part in 't Hooft's dynamics of the monopoles Until now, we have used N 2 and lagrangian, dynamics, so that the fields a and aD are part of our original abelian projection as in 't Hooft's not a gauge artifact, gauge. However, if we softly break N 2 to N 1 [35] adding a mass term for the. scalar fields,

field

scalar to

was

work out the

=

-

=

=

mtr,p2, then

for

large

enough

m the

low energy

(2.112) theory

is N =

1, where the interpre-

of and aD should become closer and closer to the fields 't should reproduce The soft breaking term (2.112) the abelian projection. In fact, there is a simple prohidden dynamics governing the A-fields. Hooft's of the fields

tation

cedure, on

the

lower

a

by Seiberg description

discovered low

energy

component

u,

such that

The effect to do that. and Witten, is to add a superfield of the theory <

u

>=<

tr02

>, with

of

superpotential

(2.113)

W=MU. This

for,

extra

term

so we can

contains

write

in fact

(2.113)

the

dynamics about

(2.112) U, with

aD

fields

we are

looking

as

W=

U(aD),

m

(2.114)

Strings

Fields, and interpret

it

by

controlled

lagrangian superpotential

first

rule

R3,

aDMI l

with

two minima

+

U(aD)

m

monopole

coupling. superpotential

the

)

5-aD is

the

of confinement.

proof

desired

value

expectation

vacuum

aU

which

is then

(2.115)

,

N = 2 Yukawa

need to minimize

only

we

the

is

term

monopole dynamics

The

aD.

93

of type

W==

where the

for

terra

as a

a

and Branes

Now,

in

order

to

(2.115).

Clearly, given by

fulfill we

get

1/2

(2.116)

't

Hooft's

is

program

then

com-

this

we approach to non supersymmetric theories, trick of adding a mass term for the X field; use the the can still however, of such procedure because of the lack of holomorphy, in the no translation form of (2.114) is possible. of using the relation for U(aD), we can try to get a more direct Instead of (1.103): let us work with the curve (2.82), and interpretation geometrical A and B with y the points consider 0,

pleted.

In order

extend

to

=

X2 Now,

we can

define

of this

function

is

(2-117)

0.

A X

=

(2-118)

+ X

giving

U(x)

Obviously,

points.

crossing

=

the function

U(X) The purpose

2

xu+A

_

value

a

posseses x

=

of

U, such that

minima,

two

V

is

one

of the

at

(2.119)

A,

rnU has two minina, at A1, with A, the superpotential theory. Of course, the minima of U(x) take place when the A and B coincide, L e., at the singular nodal curves. Now, we can tow points heuristic find the what to out use following happens in the three argument In projective dimensional R -- 0 limit. the of coordinates, region at infinity and therefore

scale

the

of the N

(2-82)

=

1

is ZY

at

z

=

curve

0. If

we

C defined

(2.120)

we

get,

delete

the of

=

infinity

by (2.120) instead

2

X

3

point,

and H,,

(2.118)

2

AN=2

the N

=

2 three

2U

+

I (x,

(2.120)

the intersection

L e.,

=

A2XZ2'

y,

0) 1,

and

we

of the

then

projective

put x3

=

0 in

[37],

U3D(X) with

ZX

_

`

dimensional

A2N=2 1

X

scale.

(2.121)

G6mez and Rafael

Usar

94

III

Chapter

3.

Taking

into

mass

stablishing

and

notation

some

where hO'O is the

VO, imply

=

__

2

worldsheet

(3.1)

The parameter

T in

with

tension,

Using

and

this

fundamental

books

section

to

relations

as

string

Weyl

the

d

uVh_h'Oa,,X,90X,

2

2

of

gauge,

light

-

-

solution

we

of motion,

with

respect

Vfh-

MOO

of

squared

will

closed

mass,

and

can

be identified

I =

(3-3)

'

2,ira'

(3.1),

the gauge

01

n,,3

=

(3.2)

0.

(3.4)

1

of motion

the equations =

0

for

(3.1)

become

0.

(3-5)

+ a,

(3.6)

coordinates,

cone

to

(3.5)

can

X/-'

of the

=

-

T

o-+

Now,

6S

1

-

0 X

generic

(3-1)

The equations

has units

invariance

In this

be chosen.

Defining

bosonic string classical considering by physical system is characterized

This

metric.

h,,3

case

in

motivating

start

T

the

reviews

that

T""3

the

f

T

'C

can

good

String.

Let us Classical 3.1.1 Theory.. theory in flat Minkowski spacetime. the lagrangian

to

reduce

will

we

ourselves

of

amount

enormous

theory,

string

in

formulas.

Bosonic

3.1

the

account

[59, 61, 60, 621 simply

Herndndez

introduce bosonic

::=

-r

=

be written

XRIP'(07-)

as

+

open and closed

string;

in

this

A XLP (0' +).

Wewill

strings. case,

(3.7)

we

first

work out the

impose periodic

boundary

conditions, XP (T, The solution comes

to

(3.5),

9)

=

compatible

X"

(T,

with

9

+

(3-8)

7r)

these

boundary

conditions,

be-

Strings

Fields, X1,

-

2

XLII

Using

2

this

+

X

XP +

-

2

2

(2a)p'(,r

(2a')p/(7-

+

decomposition

Fourier

a)

-

we

2'

+

9)

for

W

H

E _00

where

we

0

now

(3-2),

we

(3.2)

The constraint

equal.

2

1:

=

hamiltonian,

61m-ndn]

Using

also

implies

the standard

into

account

+

the left

that

right

=

MJrn+n?74",

n

(3.13)

in", ordering

factors

we

Y"

d-ndn

=

things are left free in deriving point energy, and the number of

oscillations.

these

From

all,

(-8).

we

(3.14), have

we can a

tachyon

The massless

for

degrees

the

easily with

modes

is

constants

physical

closed

and the number of dimensions

one

of

fix

gauge, where The result,

are

of

particles: part

(3.14)

(3.14),

the

impossing

Lorenz

of freedom

are

bosonic

a,

defining

target

space.

constant

of the

string,

invariance

reduced is

that

the The

in

the

to transversal a

should

equal

be 26.

deduce the spectrum of massless states. First oscillator modes, and squared mass negative

no

of the type >

(3.15)

meaning of these modes, we can see the way they transin the light cone gauge; then, we get three different types for the symmetric and traceless for the gravitons part, a dilaton the antisymmetric and, finally, part.

To discover

trace

2)

E Ce-nOln-

-8a + 8

dimensions

should

A A a-ia-,IO

form under

a'

n=1

Two

way to

for

get,

0',

n=1

cone

(3.12)

to

01

=

the normal

-8a + 8

contributions

rules,

,X,

M2

(3.12)

a-ndn)

and

quantization

m

taking

(3.11)

(a-nan

IXA, P'l

classical

(3-10)

formula

mass

[a,m", anv]

light

(3.9)

n=1

1 4', dn'1

zero

2in(T+a). n

n:AO

00

a

and

2in(r-a) a/Aen

r2la-P

=:

get the classical

M2

are

E

95

have used the notation

aA

Using

n

n:AO

the

"0

am-nan +

I

r2

+

get,

1

and Branes

the

SO(24)

Hernindez

C6sar G6mez and Rafael

96

simplest generalization to including background fields. spacetime,

Background Fields.. (3-1) corresponds lagrangian of the target

the GA' metric

T

S1 However,

Weyl

not

any

invariance

on

f

d

uVh-h,,8GA'(X)c9,,X400X,.

2

G4'

background

allowed,

is

is

(3.16)

since

we

want

to

preserve

for the two dimensional Scale invariance, from the quantum field is equivalent, theory point

a

At

0-function.

vanishing

given by

is

worldsheet

The obvious

the worldsheet.

(3.16)

system defined by of view, to requiring

(3.16)

2

of the

The

3.1.2

the

loop,

one

0-function

for

1 21r

a'

for

and with

2

condition

the first

manifolds.

flat

Wewill

=

-

2

where R(2) last

term

the X11

(3.18)

in

due to

field,

Notice

Ricci on

which for

that

has a

to

the

powers

of

in

f

-

background generalizes

Weyl loop, they

=

a'

does not in

terms

(3.18)

appear

(3.18)

in

the

contain

fields to

(3.18)

in

term

simply

is

(3.19) genus,

g,

for

a

Riemann

generic

2

-

(3.20)

2g. function

behave

like

2

-

2g. This is equal

theory: string needed to build strings, up a leads to a precise physical naturally meaning it is the string coupling constant,

in

it

in

elp.

=

(3.18)

(3.21) have been

0-funtions

vanishing

added, G,

for

the

condition

of

At

one

B and !P.

are

Rm,

+

-H\PH,,\p 4

4(DA p) =

2-

4DADA fi

alBvp +,OpB,,v

+ R+ +

-

12

-

/-'

D,\H'\

Hpv,,

namely

given by

g

invariance

(3.16),

-P,

partition

of genus g. This background field:

Once the

to

string,

(2), aVh-!P(X)R

2

two

number possesses a nice meaning three closed number of vertices joining

of the dilaton

d

the last

of the

terms

Riemann surface

where

4

first

field,

simply

the

!P in

closed

curvature.

dilaton

X

Thus, the topological

manifolds

is to be Ricci

units).

length

number is

the Euler

I +

(the

reasons

constant

number;

Euler

X the

surface

of extra

worldsheet

is the

dimensional

backgrounds

of the bosonic

particles

Therefore,

spacetime.

target

spacetime

the addition

allow

X

with

of the

tensor

allowed

fd 2aeaO a,,XAa,6X'BA,(X)

T

S,

R the

require

of massless

the spectrum

S

we

-

AV

HA,pHIIvP

o9vBp1_L,

2DA D,

P

2(D,\!P)H\,

A

(D

26)

+

-

=

0,

=

0,

=

0,

(3.22)

Strings

Fields, World

3.1.3

bosonic

parity

Sheet Symmetries.. Before ending this quick string, let us mention an aspect of worldsheet symmetries. left and right acts exchanging oscillators, Q : a,' n

Among massless invariant

under

to

invariant

states

torus,

Toroidal

3.1.4 can

be used

S',

where the

dimension

this

on

we now

for

get,

S',

must

25

25

=

_x

the

right

and left

(2R

M

M2 =4

-

-

tively.

thing

The first

be of radius

that

R"

case,

x

Then, the

R.

(3.24) (3.9)

mode expansion

the

we

nR,

_-

-

2R =

-

2R

+

(3.25)

nR,

(2R

M

+8(N-I)=4

9 the total

N and

with

manifold

simplest

becomes

nR

-

generate

bottle.

flat

Ricci

the

to

in

M

formula

a

to

Klein

momenta,

PR mass

is

a

space

the opposite

orientation, to

is

worldsheet

+ 27rnR.

7n

the

the

satisfy

(3.24)

PL

while

the

on

in two ways:

consider

is taken

identification

include

the

A torus us

Hilbert

the

of this

loop surface can be defined can be glued cylinder preserving an Q trasnformation, giving rise

x

If

effect

a

x", living

coordinate

D. The inmediate

S'

on

Worldsheet

(the graviton)

part

reduce

now

a one

compact

survey

(3.23)

We can

Compactifications.. Let target spacetime.

as

97

d-"n-

44

only the symmetric

transformation. under

up to

or

(3.15),

states

this

geometry is that S' boundaries of a

and Branes

to

of left

level

be

right

and

noticed,

2

+ nR

8(N

-

1),

(3.26)

moving excitations,

(3.25),

from

+

is the

respec-

invariance

under

the

transformation T:

R

2R'

m

A nice

(3.25)

way to represent

be referred

to

as

F1,1.

This

is

is

an even 2

PL

If

H is

froms a "

a

the

spacelike

012 angle

negative

angle,

I-plane

with

-0,

0 is the coordinate

the

and

using lattice,

2

_

PR

=

where PL

positive

changes

parametrizing

(3.27)

n.

axis

in

a

lattice as can

of

(1, 1) type,

be observed

which

(3.28)

2mn.

lives,

then

of the

R, which

the radius

will

(3-25),

from

Fl,' are

PR E

lattice, simply

of the compact

Hj-.

fact,

In

while

PL

PR forms

changes dimension.

in

0

(or

C6sar G6mez and Rafael

98

Hern6ndez

r1,1

hyperbolic space), are changes in the target and therefore are what can be called condition, space preserving Of course, no change arises in the spectrum the moduli of the a-model (3.16). of the H and HI planes. We have now obtained a good upon rotations a-model on a simple S' of the moduli space for the string characterization in 17 and 171, we should also take to rotations torus. However, in addition of the FIJ lattice. rotations into account the symmetry (3.27), representing discussion to compactifications The previous can be generalized on higher d 26-d dimensional x T d). tori, T (i. e., working in a background spacetime R r d,d and the moduli space will In this case, (PL PR) will belong to a lattice be given by [64] 0(d, d; Z)\O(d, d)10(d) x 0(d), (3.29)

Lorentz

the

in

rotations

0

the

=

0

,

7

where

0(d, d; Z) piece generalizes

the

the

(3.27)

T-transformations

to

T d.

T-duality [65]. d, which is the number of massles degrees of freedom that have been used to define the background fields of The manifold the u-model (3.18). (3.29) is the first example of moduli of a these moduli spaces will be compared, in next section, a-model we find; to From

the

now we

dimension

will

a-Model

K3

The concept the u-model (3.18), be

Notice

transformations

(3.29)

d

is

also

that

-

described.

ogy..

(3.29),

these

moduli

of the

the K3 moduli 3.1.5

call

Geometry.

of moduli

A First

space

when the target

generalized

Look at

introduced space is

a

in

Quantum Cohomolfor previous paragraph,

T d torus,

leading

to

manifold

complicated

spacetime geometries satisderived from conformal invariance, namely Ricci flat fying the constraints of moduli spaces This is a physical manifolds. way to approach the theory is a string where, instead of working out the cohomology of the manifold, whicb allows of the so to wonder about the moduli forced to move on it, field theory. In order to properly defined conformal use this approach, let us review some facts about K3 geometry. first the relation between supersymmetry and the number of recall Let us first Let think of with a a-model, us complex structures. target space M. Now, we

can

want

mations. is

a-model

this It

turns

out

to

that

to

more

be invariant in order

under

some

to make the

supersymmetry

a-model,

transfor-

whose bosonic

part

given by 77

(3.30)

ttv

and g the metric invariant on spacetime, on the target, the manifold to be Kdhler we have to require supersymmetry and, in order to be N 4, to be hyperkdhler. of the K3 manifold the description Let us now enter [66, 67, 681. To characterize K3, we will first obtain its Hodge diamond. The topologically of K3 is that the canonical first class, property

with

under

71 the

metric

N= 2

=

K

=-

-

cl

(T),

(3-31)

Fields, with

(T)

cl

the first

Chern class

of the tangent K

Strings

bundle,

and Branes

T,

is zero,

(3.32)

0.

=

99

Equation (3.32) implies that there exists a holomorphic 2-form fl, everywhere are vanishing. Using the fact that only constant holomorphic functions defined, we easily derive, from (3.32), that globally non

dim H2,0

h 2,0

=

=

(3.33)

1.

2-forms fact, if there are two different fl, and S?2, then and and therefore constant. holomorphic defined, globally The second important K3 is property characterizing In

H,

01102

be

(3.34)

0,

=

will

so'that

h1,0 as

b,

=

hl,'

==

The Euler

hO,1

the

hO,1

(3.35)

0,

=

0, because of (3.34).

=

number

and property (3.32), of the Euler number

=

be

can

and it as an

using Noether-Riemann theorem, Using now the decomposition sum of Betti numbers, we can complete

derived

now

turns

out

alternating

be 24.

to

Hodge diamond, 24

which

implies

=

bo

-

b,

+

b2

-

b3

b4

+

2

therefore,

from

(3.33),

to

the

-

0 +

b2

0

(3-36)

+1)

Hodge

=

(3.37)

22,

get

we

dim

leading

1

that

dim H and

=

H1,1

h1,1

(3.38)

20,

=

diamond

0

0

20

1

(3.39)

1

0

0 1

Using sional

Hirzebuch's

2 space H

.

In

we can give an pairing, homology terms, we have

ol,

with

a,,

From the

2

a2 E H

signature

(X, Z),

and

complex,

a2

=

#(a,

#(a, na2)

n

inner

product

to the

a2),

the number of oriented

22 dimen-

(3.40) intersections.

Hernindez

C6sar G6mez and Rafael

100

-r

dual,

to be self

f

I

3

X

(c2l

i.

e.,

3 of

lattice

a

there

f

2

2C2)

-

HI (X, Z) is

know that

we

=

exits

ai

ce

aj

-

lattices Fortunatelly, fact, the (3, 19)

In

with

E8 U the

with

signature, be at

will

hyperbolic defined by the

of K3.

metric

Recall

and E8 the

of

lattice

(0, 8)

of E8. The appearance of ES in K3 between K3 and string relations theory,

string. the

characterize

this

that

(3.44)

algebra

the heterotic

separetely

unique

are

up to isometries.

as

(1, 1),

lattice

of future

with

out

(3.43)

(X, Z).

U I U I U,

I

with

Cartan

core

very

mainly in connection Next, we should

E8

I

plane, the

2

Va E H

represented

be

can

turns

(3.42)

characteristics

these

lattice

The lattice

6ij,

=

2Z,

G

a

-

(3, 19).

(3.41)

-16,

such that

and even, a

3

X

signature

basis

a

2.24 C2

what

exactly

is

complex we

did

and the

structure

in

our

study

of the

Conthree dimensional Yang-Mills supersymmetric Torelli's used is the proper tool to be theorem, cerning the complex structure, is comof a K3 marked surface" that the complex structure that stablishes Q. Thus, the 2-form, pletely determined by the periods of the holomorphic is fixed by complex structure of N

moduli

i) ii)

=

theories.

4

holomorphic marking.

A

Q.

form

The

S? E H2,0

To characterize

(X, C),

we can

S?

and y in know that

with

x

H2(X,

R),

that

fX fX and

13

we

By

we

+

X

identify

Q A fl

0 A

f2

(3.45)

iy, with

=

0,

>

0,

the

space

R3,19

.

Now,

we

(3.46)

derive

a

(3.44),

=

write

marked K3 surface

that

we

will

denote,

X-Y

=

0,

X.X

=

Y.Y.

we mean a

from

now

specific map of on, F3,19.

(3.47) 2

H

(X, Z)

into

the

lattice

Strings

Fields,

Therefore,

associated

(3.46),

due to

S7,

with

space-like,

is

define

we

i.

(3.45)

of

fixes

complex conjugation. will reduce to simply describe

this

space,

the

of

( )+

use

working

with

the result

turns

for

the part

particular

by O(F3,11).

group

be

to

out

an

(0(2)

(3.49)

0(1, 19))+'

x

preserving orientation. If, instead marking we have been using, we change it, let us refer to this isometry of the F'," lattice; becomes

then

Gr10+ pr3,19).

=

analog to complex structures is the

space of

.

[67],

of the group

The moduli

O(F3,19)

plane, that changes upon of K3, complex structures space-like 2-planes in R3,19 To

space of

Grassmanian

a

MC The group the moduli

+my which,

(0(3,19))+ =

the

stands

nx

(3.48)

of oriented

space

Gr where

=

of the two

moduli

the

we can

v

> 0.

orientation

an

Thus,

vectors

101

e., V-V

The choice

plane of

a

and Branes

(3-50)

the modular for

a

when we work out group, Riemann surface (Sl(2, Z) for a

torus). Let we

us now

make

have used in

theories.

form,

some

the

of the

study

complex

This

comments

distinguished of the

is such that

structure

and is characterized

the

on

moduli

complex

three

the the

structure

dimensional

elliptic

curve

is

N a

=

du A

dx

entering

holomorphic

the

Y

Once

of metrics.

a

fiber. elliptic However, before issue, let us consider the question has been introduced, we have a Hodge on

discussion

complex

of H2 ,

decomposition

differential

detailed

a more

(1, 1)-

(3-51)

-,

y

4-x

4

by the 2-form S?

with

::=

on

structure

the

this

as 2

H

=

H2,0 (D H1,1 ED Ho

to a complex structure characterized Thus, relative HIJ is orthogonal to S?, and such that

2

(3-52) by fl,

form

the Kdhler

J

in

Vol which

fore,

means

together

of R1,11.

by

that

Yau's

J and

position

with

fl,

to

i.

the

J is

represented

=

fX by

(3-53)

J A J > 0,

a

space-like

vector

in R3,

"

and, there-

S?,

spans the whole three dimensional space-like subspace theorem now shows how the metric is completely determined

e.,

by

a

space-like

characterization

3-plane of the

in

R1,11.

moduli

Thus,

we are

space of complex

in

a

smilar

structures,

C6sar G6mez and Rafael

102

and

end up with

we

HernAndez

manifold

Grassmannian

a

of three

planes

space-like

in

R3,19

Gr

Now,

we

modular

Gr with

change by dilatations,

corresponding

part,

ingredients.

complete can

two extra

of

isometries

to

MM O(F3,19)\Gr Hence, the moduli of the a-model (3.18), the moduli

of Einstein

metrics

(3.26)

of manifold

Now, the dimension

the

on

so

other

that

again the

is

finally

get

we

R+.

(3-55) on a

K3

surface,

will

con-

(see equations

(3.54) and (3.55)). a-model (3.18) we must

is 58. For the

B-backgrounds.

of

moduli

_V3,19,

defined

K3

One is the volume

and the

x

=

tain

0(19).

x

that

need to

manifold,

of the

0(3,19)/0(3)

=

In the string action, which now becomes worldsheet, is given by the second of K3; thus, the moduli of B-backgrounds a 2-cycle field 4 which is 22. Finally, the dilaton number of the K3 manifold, Betti if P is constant, As mentioned, has to be taken into account in (3.18). as we it counts the number of loops in the perturbation will require, series, so we More precisely, it as an extra moduli. will not consider we will probe the K3 the Under these conditions, geometry working at tree level in string theory. take

also

what

a

we

into

account

moduli

space is

f

integral,

have is the

B,

[69]

of dimension

58 + 22

and the natural

the

over

=

(3.56)

80,

guess is the manifold

A4'

=

0(4,20)/0(4)

0(20).

x

this is not the final as Naturally, answer, trasnformations the to T-duality equivalent of the H2 (X. Z) lattice, for K3, isometries

have not

we

the

in i.

(3.57) divided

toroidal

yet

case,

the

answer

from

symmetry

portant

being missed: of mirror Let

fined

final mirror

symmetry us

consider

is

the

[70], curves

In order

the

to

to

(3.57)

of

quotient

need first

inside

(3.58)

of view

point

symmetry. we

the

not

are,

e.,

0(j,3,19). However,

by the

which

by (3.58), as an imfield theory is yet geometrical understanding

of conformal

get

a

define

the Picard

K3 manifold.

lattice.

The Picard

lattice

is

de-

as

Pic(X)

=

Hl,'(x)

2

n H

(S' Z),

(3-59)

embedded in X. By holomorphically (i. e., 2-cycles) of defines sublattice a H'(S; Z). This Picard lattice (3-59), Pic(X) fibration Let us an as consider, example, an elliptic (18, t). signature

which

means

curves

definition has

2-cycle B, and 2-cycles is given by

where the base is

by

these

two

a

F is the fiber.

The Picard

lattice

defined

Strings

Fields,

which

is

=

-2,

B-F

=

1,

F-F

=

0,

(1, 1) type.

of

lattice

a

B-B

Self

and Branes

103

(3.60)

intersections

by

given

are

the

general

expression

2(g

C C -

(3.61)

1),

-

for g 0, the base space, we get -2, and for The intersection 1, we get 0 for the intersection. of the fibration. the nature B F, reflects between the base and the fiber,

where g is the

genus,

fiber,

elliptic

the

that

so

with

g

=

-

that

Notice

(3.61)

expression

that it is clear Now, from (3.59), Pic(X) depends on the complex structure. ask ourselves

can a

sublattice;

Picard

given

fibrations space of elliptic in elements are H','(X),

will

for

looking 2-planes in are

preserving they should

be defined

R2,19-t,

i.

group

lattice,

the Picard

lattice.

value at

the

of t for

posibility

the

Picard

the

of mirror to

define

t)/0(2)

-

0(19

x

t),

-

(3.62)

(3.63)

At this comes

complex given by the of the discussion, a question the to our mind, concerning

point naturally

X* whose Picard the

terms,

moduli

of the

reduces

group,

lattice,

these

GrPIO(A).

dimension

the

manifold In

=

Picard

symmetry a

(71].

A of X

lattice

0(2,19

(3-62),

from

preserving

core

=

.Mp

clear

structures

we

space-like

of

is

group

As is

Pic(X)

moduli

the

so

moduli

will

scendental Picard

0,

to

Grassmannian

of the

terms

As

modular group. This by the corresponding again quotient of the lattice be given by isometries A, called the trandefined and is simply as the complement to orthogonal F',19-' moduli the and of A is preserving the Thus, type,

should

we

modular

in

orthogonal

we

preserving the

in

of the fibration.

the structure

have in

account,

structures

be interested

we can

be

into

e.,

Grp where

instance,

for

Taking complex

space of

moduli

the

we

curves

fact

this

of the lattice

nature

even

number of

the

F3,19.

about

the

with

is consistent

answer

in

an

is

transcendental

is the

group

of

space

amount

clearly

negative,

as

the

(2, t), (1, t), signature the concept or generalize passing from A to a (1, t) lattice, of signature of Picard lattice, (2, t). It turns out that both admiting lattices has the second but more a are physical flavor; in order equivalent, approaches what we can do is to introduce an isotropic to get from A a Picard lattice, define lattice the and in new vector through A, f Picard

we

lattice

is

and A is of signature

of

19

-

so

that

need either

f which

is

manifold

of

(1,

18

possesing

-

t) type; as

Picard

now,

lattice

-LIf,

the

(3.64)

mirror

the

one

manifold defined

X* is

by (3.64).

defined

as

the

The moduli

C6sar G6mez and Rafael

104

(3-62), Then,

Gr*p we

and that

observe

is therefore

manifold

space of the mirror

HernAndez

=

0(2,

t +

the dimension

that

the dimension

of the moduli

given by the equivalent

1)/0(2)

x

O(t

+

to

expression

1).

of the two moduli

(3.65)

spaces

space of the mirror

sums

manifold

20, exactly

up to

is

moduli space. given by the rank t + 1 of the Picard of the original A different approach will consist in definig the so called quantum Picard of signature lattice. Given a Picard lattice (1, t), we define its quantum analog the of t lattice as (2, + 1), obtained after multiplying signature by the hyperF','. bolic lattice So, the question of mirror will be that of given a manifold lattice A, finding a manifold X* such that its quanX, with transcendental A. Now, we observe that the quantum Picard is precisely lattice tum Picard of X and X* produce a lattice lattices of signature (4,20). The automorof result will the transformations and phisms O(V4,20) T-duality compossing back mirror and to mirror including (3.57), symmetry. Coming symmetry, we the of moduli a-model as on K3, get, space

0(4,20; This

in

analysis

our

of a-models

x

on

0(20).

(3.66)

K3.

Elliptically

Fibered K3 and Mirror We are now Symmetry.. in the K3 manifold. Let C be a rational curve singularities the K3 manifold; C C -2. If the curve C then, by equation (3-61), embedded it will be an element of the Picard lattice. Its holomorphically

3.1.6

going is

concludes

Z)\0(4,20)/0(4)

consider

to

-

volume is defined

as

Vol(C) with

J the Kahler

goes zero,

i.

e.,

=

class.

whenever

A

singularity

the

Kdhler

=

i

will

class

-

C'

(3.67)

appear whenever the volume of C J is orthogonal to C. Notice that

to the whole 3-plane defined by f? implies that C should be orthogonal and J, as C is in fact (1, 1), and therefore orthogonal to S?. Now, we can define the process of blowing up or down a curve C in X. In fact, a way to blow up is simply changing the moduli space of metrics J, until from zero. The opposite J C becomes different is the blow down of the curve. The other way to get rid off the singularity is simply changing the in such a way that the curve is not in H,', i. e., the curve complex structure

this

-

does not exist

anymore. We can have different

types of singularities, according to how many rato J. The type of singularity will be given orthogonal would be generated by these Ci curves. Again, these lattices by the lattice characterized by Dynkin diagrams. Let us now consider fibered K3 manifold, an elliptically

tional

curves

Ci

are

E

-+

X

-+

B.

(3.68)

Strings

Fields,

Now, sented

chapter

in

analysis

back to Kodaira's

come

we can

fibrations,

elliptic

on

of Kodaira

singularities

Elliptic

II.

105

and Branes

type

pre-

as

characterized

are

singularities. Xi of the corresponding lattice the r',' contains fibrations genelliptic of each singularity as erated by the fiber and the base, and the contribution number as Picard the formula p(X) Shioda-Tate the Defining [71]. given by

by the

of irreducible

set

The Picard

I + t for

a

components

these

for

lattice

(1, t)

of type

lattice

Picard

p(X)

we

E a(F,),

2 +

=

get

(3.69)

V

the set of singularities, over n + 4, o,(E6) 1, o,(Dn+4) a(A,,-,) 0. Equation 1, a(11) 2, a(M) u(IV) is trivial. Mordell-Weyl group of sections the mirror in the previous As described section, of type (1, t), to X*, with X, with Picard lattice the

where

sum n

=

=

=

=

equivalently,

p(X)

Through mirror,

we can

p(X)

number

Picard

=

=

of type A0, to 16 singularities

a

=

then pass from 2, which should

A, type,

some

or

an

=

for

other

6, o,(E7)

(3.69)

is

=

by

(1,

lattice

a

8,

=

provided

true

map goes from

Picard

the

manifold

18

-

t)

or,

(3.70)

20.

elliptically

fibered

K3

have all

instance

number

of Picard

K3 surface

of

p(X*)

+

is given o, 7, a(E8)

and where

is

-

p(X*)

=

combination

its

surface,

with

singularities

18, which should have of singularities.

Repeating previous comments on closed String.. The only crucial point is deciding strings for the open case is straightforward. From we to be imposed. (3.1), get boundary the type of boundary conditions Open Bosonic

3.1.7

The

terms

of the form

IT 2

0,,

with

the

boundary

normal

away form the

string,

to

In order

open

these

boundary

conditions

to

avoid

momentum flow

imposse Neumann boundary

0'XI, Using

(3.71)

ax'-a"X"

derivative.

is natural

it

f

=

conditions,

(3.72)

0.

the mode

expansion

(3.9)

becomes, for the

string,

X,"(a,,T)

=

and the quantum

x" +

mass

2a'p"-r

+

a/-te

-in-r

n=AOn (3.14)

formula

M2

iv'2a'

=

is, for a'

-2 + 2

cos

(3.73)

na,

-

21

(3.74)

01-nann=1

Now, the first metry,

(3.27),

surprise

to the

open

arises

string

when case.

trying

to

generalize

the

T-duality

sym-

C6sar G6niez

106

and Rafael

By introducing

D-Branes..

3.1.8

HernAndez

Z

with

a

2

=

X"

Let

(3.73)

ir,

(a, -r)

x"

=

us now

ia'p"

-

ln(z. )

+

R

-+

we 1

will

work out

transformation

R

(3.75)

ia,

r l'

a-

E -a" n

n:AO

moving

direction

0 nX25

Now,

coordinate

as

string

compactified

in the

0,2

+ i

the open

consider

ary conditions

=

be rewritten

can

complex

the

n

(z-'

R"

in

x

(3-76) S1. Neumann bound-

are

(3-77)

0.

==

the way these boundary conditions modify under the To visualize the will consider the we [721. answer, closed both from a time the closed and evolving string,

cylinder swept out by pictures (in the open string picture the cylinder can be understood open string with both ends at the S' edges of the cylinder). as an open string In fact, from the open string of the string is at tree point of view, the propagation level, while the open string approach is a one loop effect. Wewill now assume that the S' boundary circles of the cylinder are in the 25 direction. Recalling then what happens in the closed string the mode case, under change (3.27), to the change expansion (3-9) turns (3.27) equivalent a25 In the

n

=

0

case

we

a

What this to

a

theory

means on

y25

coordinate

is that

a ,

25 0

circle

defined

(3.11)

from

get,

M =:

_

2R

nR

theory

the

_d25

_+

n

(3.78)

n

(3.25)

and nR

-+

(with

M =

-

2R

in the dual

circle

oz'

2

_d25. 0

(3-79)

of radius

is

R

of radius

in terms of R, but written from X25 by the change (3.78). Now,

a

it

equivalent new

space

easy to

see

that

aaY25

=

6coOx25.

(3-80)

the cylinder now to Returning above, let us consider image described in the open string From the closed string boundary conditions picture. apas proach, they will be represented

19, X25 Now, after plies

performing

the

duality

that, ditions,

so

the

open

that

the

string extreme

(3-81)

0.

transformation

19, y25 from

==

point

points

of

=:

(3.27),

equation

(3.80)

im-

(3-82)

0,

view,

looks

as

of the

open

string

Dirichlet do not

boundary

con-

in

time

move

Fields,

Strings

and Branes

107

Neumann Summarizing, we observe that under R -+ fL, R for the open string are exchanged. Besides, boundary conditions do not move in the 25 of the open string the picture we get if the end points where with fixed 25 coordinate, is that of D-brane hypersurfaces, direction should end. the open string of these D-brane of the dynamical nature For a better understanding and their physical meaning, the above approach must be genhypersurfaces, the tool needed comes from several D-brane hypersurfaces; eralized to include the as a meson model: the old fashioned string theory, interpreted primitive

in

the

25 direction.

and Dirichlet

[73].

factors

Chan-Paton

Lines.. Chan-paton factors are with labels i, j, with of the string points open encoding simply defined as I k; i, j > be will states The N. string corresponding 1, i, j and unitary, A'NxN1 hermitian of N x N matrices, a set Let us now define of U(N). Wecan now define the open which define the adjoint representation string state 1k; a > as Chan-Paton

3.1.9

=

.

.

.

and Wilson

Factors

the end

defined

-

,

1k;

>=

a

E Aia ,11k;iJ

(3-83)

>

ij

of in the language interpreted the abelian use projection that, again Ji, i > previous chapter. In the abelian projection gauge, states while states > to components diagonal (non IiJ correspond U(1) photons, The way they of the gauge field) correspond to charged massive particles.

The

string

transform

IiJ

states

gauge theories. in introduced

under

>

In order

the

now

can

the abelian

easily

we

U(1)N

abelian

IiJ for

be

do

to

>-

-will

is

group e

i(aj-ai)Ji'j

transformation e

eiCVN As discussed

(3-84)

>'

in

chapter

11,

X must be chosen to transform

)

(3.85)

projection gauge, a field then, the gauge adjoint representation; A simple example of field X is a be diagonal. in R" x S1, and define X as we are working to

define

an

abelian

in the

through imposing X to line. So, let us assume direction. the Wilson line in the 25 compactified Choosing X diagonal means 25 abelian the in A a diagonal Wilson line is obtained taking group U(1)N ; is fixed

Wilson

from

A25

1

01

=

2-7rR

ON

)

1

(3-86)

1

C6sar Gomez and Rafael

108

corresponding

to

a

pure

25

A

gauge

025A

==

HernAndez

925

=

01

x25

(3.87)

...

27rR

ON

Now, 101, ON} are the analogs to JA1, ANI, used in the standard The effect of the Wilson line projection. (3.86) on a charged state it in the way (3-84) li, j > is transforming defines, which in particular means that the p25 momentum of the li, j > state becomes abelian

When moving from R to R'

=

The

dual

+

-

Oi

(3-88)

27rR

the momentum (3.88)

2R

(OjR'

meaning of (3.89)

geometrical the

-

R

2nR'+

around

Oj

n

25

P

-

OiR')

quite

is

into

turns

(3.89)

r

clear:

the open

string

R' any number of times, but its the R R' duality transformation,

circle

winding,

a

of radius

can

wind

end

points expected after to be in OjR' and OiR' positions. Thus, the picture we get is that of several D-brane fixed in the dual circle to be at positions hypersurfaces OIR', ONR', and the string states of type li, j > are now living between the ith and /h D-brane hypersurface. Using mass formula (3.26), and equation (3.88) for the momentum, we observe that only a` Ili, i > states can be massless (the U(1) photons), and

fixed,

are

as

-

the

mass

A of the a-,

have the

kinematical

consider

the

li, i

index

massless

for

a

Therefore,

U(N)

p in

on

this

the

two

metrical D-brane.

is

of D-branes

line

the

Both of these

.

directions.

li, i>,

which

states

We can also can

be inter-

by the D-brane space defined abelian projected gauge spec-

defined

on

the

D-brane

hypersurface.

arise,

represents

theory, where a U(N) Wilson S'. compactified The distribution

Ce25 -1

states,

spectrum

of D-branes

F

24 dimensional

gauge

2

( (Oi-0j)R' )

uncompactified

theory, now complementary pictures

-The distribution

-

the

Kaluza-Klein

preted as scalars living hypersurface. However, trum

goes like

> states

-)

-

a new

type of background

has been introduced

provides,

for

the

of a gauge theory living representation Moreover, the spectrum is presented

in

massless on as

the

for

spectrum, abelian

or

geoof the

a

the worldvolume the

string

internal

projection

spectrum. Of course,

string doing

this

second

approach

only

takes

into

account,

theory, low energy degrees of freedom. Properly is embedding the gauge theory into string theory

speaking, in

as

is

usual

what

a new

way.

in

we are

Strings

Fields, first

To end this

[60],

and references

tion

possed

above

will

be obtained

therein)

we

on

(for

D-branes

should,

the

details

more

the ques-

simplest through the

interactions tension

for instamce,

answer

The

of D-branes.

nature

see,

qualitatively,

at least

dynamical the gravitational analizing mass density, leading to the

of the

tation

with

contact

109

and Branes

answer

compu-

hypersurface. D-brane, defining

of the D-brane

an string state can couple a be D-brane the to the can graviton coupling Withcircle its of on in terms boundary. ending interpreted open strings know something on the order of we already out performing any computation, determined is it the of a by the topology of a disc, process magnitude process: order in the string so the with half the Euler number of a sphere, coupling

A

which

graviton,

interaction

defined

constant, A

more

a

closed

The disc

in

detailed

(3-21),

0(').9

is

discussion

needs the

D-branes

on

theories),

(superstring

theories

string

is

vertex.

is what

which

use

we

of

more

discuss

will

general in

next

section.

3.2

Superstring

Superstrings

(3.1).

This

is

Theories.

correspond to done adding

the

supersymmetric

the fermionic

SF

=

f

term

d 20ri AoceacV).,

(3-90)

to the worldsheet, relative Oil are spinors, Lorentz to the spacetime 1). SO(1, D group, Dirac matrices the and p', a Majorana spinors,

where

-

0

P0

fp',p,31

The supersymmetry

with

e a

constant

transformations

i

i

0

are

(3.90) defined

respect are

real

by

(3.91)

(3.92) by

defined

N)"

J01,

-iP,09,X'"C' spinor.

0, 1,

are

with

-277"13.

6XI,

anticonmuting

Spinors

in

0

0

=

and vectors

-i

i

P

satisfying

of the 0'-model

generalization

(3.93)

Defining

the

components

A

(3.94) +

the fermionic

lagrangian

(3.90)

can

be written

as

C6sar G6mez and Rafael

110

SF

o9 specify

with to

!-(o9, 2

=

,9,).

Hern6ndez

Jd

=

As

2

or

-

was

the

Ramond

the mode

produce

+

for

the

fermion

there

:

the

of

case

boundary

the

mass

formulas

6

string

we

we

the

now

open

-r),

(3-96)

-in(-r=Fo,)

(3.97)

n

+2

either

impose

can

obtaining

the critical

ordering

correlators

JL)

-

easily

=

=

2(NR

periodic Ramond

we

or

and that

JR)y

-

get the massless

Neveu-

similar

get,

(3.98)

the R sector.

0 in

antiperi-

or

(R)

following dimension is 10, are given by

quantization

Using this

spectrum.

formula,

For the closed

get NS-NS sector

V

NS-R sector

V

1121S

IS

>

R-R sector

IS

The state

Y V'_

V2_

fermions,

and J

NS sector,

and the GSOprojection,

need

d n' e

that

2(NL

=

=

After

case,

and normal

in the

2

the

Ofields.

M2 with

we

for

in the bosonic

to those

steps

both

for

in

posibilities:

=

I

O 'T

:

strings,

conditions

(NS)

Schwarz

closed

two

we

both

0" (7r, -F),

=

Y']

Neveu-Schwarz

odic

string,

fields,

expansions

Ramond

In

(3-95)

bosonic

are

O '(7r,,r) O '(7r, -r)

:

0"),

OP

the

case

for boundary conditions string case. For open strings,

Neveu-Schwarz

OIL

+

the

and closed

which

(01-119

>

corresponds

1/2

V

10

1/2

>1

OIS

to the Ramond vacua

(3.99)

>.

(recall

J

=

0 in the Ramond

sector)The

do'

oscillators

in

(3.97)

define

a

f dol', do'}

Clifford =

algebra,

(3-100)

?f',

the IS > vacua can be one of the two 8S, 8S, spinorial SO(8). Depending on what is the spinorial representation different from theories. In the chiral two (3.99), superstring get,

and therefore tations we

choose the sectors.

represen-

of

same

This

R-R sector

chirality

will

we

get,

lead for

for to

the two fermionic

gravitinos chirality,

two

same

8s

x

8s

=

I

of

states

35s,

case,

we

in the NS-R and R-NS

equal chirality.

28 T

chosen

Moreover,

in

the

(3.101)

Strings

Fields,

corresponding field,

scalar

field

being

4-form

field.

Wewill

identified

with

and Branes

ill

antisymtheory type 1113. superstring for the spinor representations chiralities In case we choose different associated with the Ramond vacua, what we get is type IIA superstring theory, which is but this time with two gravitinos also an N=2 theory, of different chirality;

metric

now, the

to

a

and

a

R-R sector

a

vector

field

that

we

theories 3.2.1

Toroidal

ories.

U-duality.. theories,

string

and

will

axion,

an

3-form.

a

(3.102)

8V G 56v,

=

These

the first

are

two

types

of

superstring

consider.

Compactification Before considering will

we

number of allowed

the

this

contains

8S 0 8S, L e.,

call

first

review

supersymmetry,

Spinors should be considered have dimension representations

as

Type

of

different

IIA

and Type compactifications

11B The-

of super-

maximum on the general results depending on the spacetime dimension. of SO(1, d- 1). Irreducible representations some

+9-1,

2 1d2

where

for

stands

spinor

can

be

real,

Using (3.103) in the

table

the integer part. Depending complex or quaternionic,

and

below

the

=

1, 2, 3 mod 8,

C,

if

d

=

0 mod 4,

H, if

d

=

5,6,7

we

get the number of supersymmetries

(3.104),

larger

(3.104)

mod 8.

listed

14

11

1

R

10

2

R

R1r3

Representation

32 16

9

2

8

2

C8

7

2

H8

6

4

H

4

4

H

5

4

4

8

C2

3

16

R

table

dimension,

d

Irreducible

> 2 do not

the

if

N

This

on

R,

Dimension

14

(3.103)

is

2

constrained

appear.

by

the

physical

requirement

that

particles

with

spin

C6sar G6mez and Rafael

112

Hernindez

The maximum number of From the table

16.

pactification, also

can

four

N= that

notice

supersymmetries that through dimensional

six

2,

dimensional

ten

N= I

dimensional

and three

three

in

dimensions

standard

clear

with

starting

dimensional

four

is also

it

then

is

Kaluza-Klein

com-

leads

supersymmetry

to

N = 4 supersymmetry. We to N= 4 supersymmetry in

N= I leads

dimensions. It

that

be stressed

must

the

slightly

of supersymmetries

counting

after

dimen-

with compactify the is the of interthe Here, adequate concept holonomy topology. recall let us therefore nal manifold; some facts on the concept of holonomy. Given a Riemannian manifold M, the holonomy group HMis defined as the set of transformations My associated with paths -y in A4, defined by parallel in The connection the tangent bundle. of vectors used in this defitransport In general, for a vector budle E -+ M, is the Levi-Civita connection. nition the holonomy group HMis defined by the paralell of v in the fiber, transport sional

reduction

with

holonomy

Manifolds

=

=

HM

=

-

H.M

-

if

we

on

manifolds

connection

be classified

can

Ambrose-Singer

The

E.

on

by the curvature. according to its holonomy

generated

is

=

The

O(d),

U(4),2 SU(A), 2 Sp(4),4

manifolds

for

real

for

Kdhler

for for

Therefore,

flat

Ricci

Kihler

manifolds.

manifolds". of what the role

question

surviving dimension

d,

1). Now, the theory is compactified in d2 down to d2 = d d, Supersymmetries

are

-

internal

-

1),

1)

-

into

,

so we

HM,1

manifold

would be associated

manifold.

Let

us

consider

1)

-

singlets the simplest =

SO(dj).

of

with

SO(4)

manifold

associated

the

SU(2)

in

are

of dimension

dj,

with

Good spinors

with

0

quite

is

representarepresentation group of the in d2 dimen-

holonomy group of the internal d, 4; then,

of the case,

the count-

spinors

SO(di).

x

be part

will

that

decompose an irreducible Now, the holonomy

need to

SO(1, d2

holonomy is in compactification

so

on a

-

of SO(1, d2

of after

-

of SO(1, d

d.

of dimension

hyperkdhler

to the

answer

SO(1, d

sions

group.

shows

manifolds.

ing of the number of supersymetries simple: let us suppose we are in

tions

theorem

[74]

get

H.M H.M

-

the

to

respect

how the

-

subtle

more

trivial

non

we

is

=

SU(2)

(3-105)

flat the holonomy will Ricci and Kdhler, be one of need a singlet with respect to this we will Therefore, with SU(2). As an example, let us consider the spinor in ten dimensions, N 1; as we can see from the above table, it is a 16, that we can decompose if

and,

these

our

manifold

SU(2)

is

factors.

=

with

respect

to

SO(1, 5)

x

SU(2)

16 1'5

Notice

that

any

hyperkiihler

=

x

SU(2)

(4,2,1)

manifold

0 is

as

(4,1,2).

always Ricci

(3-106) flat.

Strings

Fields,

Therefore, a general

only get

and Branes

113

This supersymmetry in six dimensions. surviving dimensional ten manifold a on a compactify theory with SU(2) holonomy, we will get a six dimensional of dimension four, thewith if the is one a on However, only compactification supersymmetry. ory with trivial obtained two maxitorus are holonomy, supersymmetries (the mumnumber of supersymmetries available). we

if

result:

is

As the first

on

in

supersymmetries do not take

we

exactly

the

one

dimensions

account

0(4,4;

T-duality

the

table,

we

will

we

then

learn

the moduli

consider

let

with,

its

work in

us

the number of

that

T 4 is trivial.

holonomy of

the

as

fields,

of the

If

a-model

string

3.1,

section

in

Z)\0(4,4)/0(4)

0(4,4; with

4,

is

the R-R

described

theory

T d. To start

torus,

4. From the above

=

six

into

string

d-dimensional

a

d

case

type IIA

with

contact

compactification the particular

is

one

we

x

Z) corresponding

0(4),

(3.107)

changes of

to

wL

Ri

the type

Ri

S' cycles

the four

becomes different compossing the torus. The situation In such a case, we should take into account if we allow R-R background fields. of including the possiblity Wilson lines for the A,-, field (the 8V in (3.102)), of and also a background for the 3-form A,,,p 56V (the (3-102)). The number of Wilson lines is certainly 4, one for each non contractible loop in V, so we

for

need to add 4 dimensions

A,,p

background, 4 extra

implies

to the

16-dimensional

corresponding Finally,

the

moduli

dimension

the

parameters.

16 + 4 + 4

Now,

extra

a new

It

is

to

a-model

important

moduli

to

Adding (3.25), that

dimension

the the

proposal

of moduli

dilaton

as

to

for

approach has not

a

field

dilaton

the

section

we

get

This and

constant,

differentiation

this

(3.108),

previous

been considered.

coupling

string

be added.

must

in

a

rather

is

moduli

space of

as

Z)\0(5,5)/0(5)

(3.109)

equals

(3.108)

Anyway,

moduli

be written

can

in

Concerning an H3(T 4), which

24.

moduli

string.

the dilaton

0(5,5; The

fact:

dilaton

space the

corresponds interpreting allowing changes only in cumbersome.

this

stress

to

=

coming form the

dimension

here

space (3.107). is determined by

type IIA

x

on

0(5).

(3.109)

T' already

contains

a

lot

of

all, group 0(5, 5; Z) In fact, relative to the 0(4,4; resting Ramond fields. Z) T-duality of toroidal we have now an extra compactifications, symmetry which is Sduality [5, 75, 767 77, 78, 79, 80, 81, 82, 83, 84, 85, 86], novelties.

First

of

the

modular

now

acts

on

the

dilaton

and the

1

9-+

9

(3.110)

C6sar G6mez and Rafael

114

with the

g the

group was 0(r3,19; enhancement to O(F',"; Z)

the

transformations.

have mirror

the

include

Z)

0(4,4;

new

physical on

The dilaton

any

the moduli

new

Z),

IIA

of type

0(4,20;

in

mirror to

symmetry

T-duality,

we

of type IIA on T1, it is because we dilaton that the modular symmetry

To

of the

symmetry. In spite this, let us apreciate

added,

be

can

fact, recall K3 is simply In

on

and quantum in addition

where,

and the

moduli

moduli.

symmetry is called

case

U-duality

the

to

is different.

meaning

K3.

producing

In the

backgrounds

R-R

is enhanced

modular

The

modular

creates

IIA

from mirror

much what arises

This

phenomena found here resembles symmetry in the analysis of K3. There, the

literature

"classical"

the

coupling constant. U-duality [78].

string

physics

very

HernAndez

that

Z)\0(4,20)/0(4)

x

but

the

H, (K3)

0(20)

x

now

analogies,

consider

R-R fields

0, and H3

=

type not

are =

0,

so

(3-111)

R,

and the modular group not acting the dilaton, R parametrizing on it. the moduli (3.109) goes under the name of M-theory. The way to interpret the basic idea is thinkof M-theory, definition Before entering a more precise on V; howcompactification ing of (3.109) simply as the moduli of a toroidal

with

N 4 theory, with to obtain we need to start a six dimensional The theory satisfying this is M-theory, theory living in 11 dimensions. is well understood: whose low energy supergravity it a theory description it gives should be such that through standard Kaluza-Klein compactification but this a theory known as eleven the field theory limit of type IIA strings; dimensional type IIA supergravity. the construction of the type IIA string Once we have followed theory moduli on T 4, let us consider the general case of compactification on T d. The in order

ever,

=

some

of the moduli

dimension

dim where d 2 is the

is

=

d 2+ I + d +

the

dual

has to be to

for

the Morm Al-tvp -

1)(d

-

is

d >

5, by including The result

scalar.

a

2)(d

d(d

-

3)(d

-

4)

-

1)

(d

...

-

6)

7

For should

spaces,

torus,

are

according listed

in the

practitioners, supergravity be a surprise.

not

-

2)

(3.112)

3

ri

ification

1)(d

3

completed,

d(d

The moduli

-

NS-NS contribution, the I sumand comes form the dilaton, from the Morm A1jvp* The formula lines, and d(d-1)(d-2)

d from the Wilson

(3.112)

d(d

to

the value

table

dual

scalars.

For d

=

5,

is

duals

to

Al_lvp7

duals

to

A,,.

of the dimension

(3.113) of the

compact-

below.

the appearance

of E6 and E7 in this

table

Strings

Fields, Dimension

Moduli

d

=

4

0(5,5;

d

=

5

d

=

6

d

=

3

d

=

2

0(5)

S1(5, Z)\Sl(5)/SO(5) S1(3, Z) x S1(2, Z)\Sl(3)/SO(3)

T 4, is

again

R-R sector

the

happens

what

now see

us

instance, H4

x

E6,(6)(Z)\E6,(6)/Sp(4) E7,(7) (Z)\E7,(7)/SU(8)

Let

now,

Z)\0(5,5)/0(5)

115

and Branes

the

type

1113

case.

piece coming by the cohomology the Hodge diamond for

determined

is

From

The moduli

on,

for

from the NS-NS sector;

the 16 dimensional

(3.101)).

(see equation

in

Sl(2_)ISO(2)

2

H',

groups T4

H2 and

2

4

(3-114)

1

2

we

is

a

IIA

get 8

extra

general

and type

exactly

modulis, for

result

any

string However,

1113

the

same

number

Tdcompactification. theories

are,

after

type IIA

in the

as

The

for this

reason

compactification,

toroidal

case.

This

is that

type

related

with 11, as K3, 0, the moduli for from direct derived be as can inspection different, drastically 1113 we for Therefore, type of the K3 Hodge diamond (see equation (3.39)). from 1 and H from 22 from H', I 2, R-R the HO, from coming sector, get,

by T-duality. IIA

and IIB

which

manifold

a

=

are

sums

up

NS-NS sector.

a

Then, including

the natural

modulis

of 24 extra

total

dim

Therefore,

on

the

IIB(K3) guess for

0(5,21;

=

to

be added to the

58 + 22 of the

dilaton, 22 + 58 + 24 + 1

the moduli

Z)\0(5,21)/0(5)

=

105.

(3.115)

is x

0(21).

(3.116)

taking place. As we can see from (3.111), of find we do not on K3, when type IIA is compactified any appearance IIB the in case type By contrast, words, S-duality. U-duality or, in other the dilaton and, therefore, a modular we find group 0 (5, 2; Z), that contains of the S-duality This is what can be called transformation. the S-duality from observed be which can equation already theory [87], type 1113 string (3.101). In fact, the R-R and NS-NS sectors both contain scalar fields and tensor. the antisymmetric Here, something

quite

surprising

is

Usar

116

String..

Heterotic

3.2.2

and

by

G6mez and Rafael

ideas

productive

HernAndez

of

The idea

in the recent

history

"heterosis", of string

one

of the most beatiful

theory

[88]

was

motivated

all, the need to find a natural way to define non abelian in string theory, without entering the use of Changauge theories Paton factors, the sharpness of the gap in string and, secondly, theory beand right tween left moving degrees of freedom. Here, we will concentrate on to the construction of heterosis. some of the ideas leading In the toroidal comof the bosonic string that the momenta on V, we have found pactification live in a rd,d lattice. This is also true for the NS sector of the superstring. The lattice rdd, where the momenta live, is even and self dual. Taking into between left and right sectors, account the independence we can think on the the left and right to compactify possibility components on different tori, Tdrand T dR, and consider as the corresponding moduli the manifold two basic

facts.

First

0 (dL,

of

dR; Z) \ 0 (dL, dR) / 0 (dL)

x

0 (dR)

(3.117)

-

of this let us try to trying to find out the consistency picture, of moduli The dimension of this moduli simple interpretation get (3-117). is dL x dR, and we can separate it into dL x dL + dL x (dR dL). Let us the first the x standard moduli for compactifications as dL interpret dL, part, T dL; then, the second piece can be interpreted on a torus moduli of as the Before a

-

Wilson

for

lines

gauge group

a

U(I)dR-dL. With

this

when

working

interpretation,

simple

gauge group with

with

a

(3-118),

(3.11.8)

already

the interplay in heterosis the non abelian, potentially left and right When we were parts. and considered toroidal compactifications,

we

gauge group that and differentiating

working

notice

can

be

type II string theory, of the Wilson lines adding, to the moduli space, the contribution for the RR gauge field, A. (in case we are in type IIA). However, in the case of type IIA on T 4, taking into account the Wilson lines did not introduce any 4 heterosis asymmetry in the moduli of the kind (3.117). However, T is not the consider K3 surfaces. we can also only Ricci flat four dimensional manifold; It looks like if T4, K3, and its orbifold surface in between, T 4/ Z2, saturate all compactification manifolds that can be thought in four dimensions. In the of type IIA string case of K3, the moduli (see equation (3.111)) really looks like the heterotic moduli, of the kind (3.117), we are looking for. Moreover, in this case, and based on the knowledge of the lattice of the second cohomology we were

also

of K3

group

(see equation

(3.44)), E8

we can

lines

interpret E8.

of the

a following is interpreting

very

the 16

=

dR

I

-

E8

I

dL units

U I UI as

(3-119)

U,

corresponding

precisely

to Wilson

In other words, and E8 gauge group appearing in (3.119). distant path form the historical one, what we are suggesting

x

moduli

(3-111),

of type

IIA

on

K3,

as

some

sort

of

heterosis,

Strings

Fields,

and Branes

117

magic of numbers is in fact playing in our dL and dR strongly suggest a left part, of critical the critical dimension dimension 10, and a right part, of precisely 26. This was, in fact, the original of the bosonic string, idea hidden under in its left heterosis: as working out a string theory looking, components, and in its right the standard superstring, components as the 26 dimensional bosonic string. However, we are still missing something in the "heterotic" of (3.111), which is the visualization, from K3 geometry, of the interpretation material introduced some of the geometrical gauge group. In order to see this, in subsection 3.1.5 will be needed; in terms of the concepts there introduced, r4,20. We that the (PL, PR) momentum is living in the lattice we would claim dL

with

team,

=

as

4 and

dR

=

numbers

the

The

20.

we

get for

that PL is in the space-like think 4-plane where the holomorphic Recall that they define a S?, and the Kdhler class J, are included. to this 3-plane. Now, momentum vectors, orthogonal space-like 4-plane, can be considered; they are of the type then

can

top form

(0, PR) 2 -2, this Now, whenever p R volume (in fact, with vanishing

p2R (3.26)

=

we

particles.

bosonic

string

M2

that

=

cohomology lattice, we E8. Therefore, related

=

4(p

(16))2

to rational

in

the E8 lattice

observe curves

defined

massless

in K3 of

vanishing

massless

a

26

(3.26), (3,121) appears

used for

minus the Cartan

which allows

here

the second

algebra string

bosons in heterotic

volume,

vector

PR of

the

sign difference

by

vector

.

1),

-

The

2.

for

from

get,

the K3 construction

was

that

==

=

of heterosis, we

+ 8 (.IV

R

condition

spirit

R

3.1.5)

subsection

the

is

the

in

0, for N=O, if (P (16))2

(recall

because

-2

=

x

*

(P(16), P(10)), R

into

M2 so

of E8

easily observe that p2R if we separate, In fact,

dimensional

inside a rational curve K3, 0) The points given by PR J E8. Now, from the mass formulas

the volume is

lattice

be at the root

-2 will

define

will

vector

=

(3.120)

-

of are

to consider

of symmetries when moving in moduli space [81, 89, 90]. Some rational curves can be blown up, which would be the geometrical

enhancement of these

blown down, getting of the Higgs mechanism, or either extra massless fibered K3 surfaces, the different Kodaira Moreover, for elliptically in its Dynkin diagram, the kind of gauge symmetry to reflect, singularities

analog stuff.

be found.

The above theorem

on

discussion

string

summarizes

Quasi-Theorem I Type IIA string on T4

erotic

Previous

equivalences

what

can

be called

the

first

quasi-

[78, 81],

equivalence

string

on

K3 is

equivalent

to

E8

x

E8 het-

.

arguments

by direct

were

so

inspection

general

that

of the different

we can

probably

K3 moduli

obtain

spaces that

extra

have

C6sar G6mez and Rafael

118

been discussed in represented, generated by a

moduli

subsection

3.1.5.

In

for

an

let us consider the modparticular, fibered K3 surface, a fact elliptically terms of the Picard lattice, claming that it is of F"' type, and with the fiber relations section, satisfying (3.28). This in

complex

space of

uli

HernAndez

structures

is

Z)\0(2,18)/0(2)

0(2,18; where

we

lattice

is

have used

(3.62),

equation

(2,18). From interpret (3.122) as

of type

0(18),

x

and the

fact

(3.122)

that

the

transcendental

heterosis

the

point of view, it would be reasonable to heterotic E8 x E8 string, on a compactified T 2. In fact, have 4 real moduli, we will the to Kahler 2-torus, corresponding of T 2, and 16 extra complex moduli associated class and complex structure lines. to the Wilson of (3.122) is However, now the type II interpretation far from being clear, is just the part of the moduli space that is as (3.122) fibration. the elliptic Now, in order to answer how (3.122) can be preserving understood II compactification as a type a similar problem appears as we try of the type IIB moduli on K3, given to work out an heterotic interpretation A simple way to try to interpret in (3.116). (3.122), as some kind of type II is of course thinking of an elliptically fibered compactification, K3, where the volume of the fiber is fixed to be equal zero; generically, J-F= where

F indicates

compactifying this

does

H'

in

not

and

have the

we

a

the

lead

H3,

class

of the

string

II

type

(3-122)

to

will

which

NS field

on

for

vanish.

0, and

the

(3-123)

0,

fiber. the

the

Now,

we can

think

that

we

are

base space of the bundle. However, are type IIA case, as the RR fields

But

what

R field

about and

X,

type

we

IIB?

should

fix

In

this

the

case,

moduli

of these fields base space of the elliptic on the configurations in moduli Here, type IIB S-duality, already implicit (3.116), can fibered help enormously, mainly because we are dealing with an ellipticaly K3 manifold [91, 92, 93]. To proceed, let us organize the fields 0 and X into the complex of

possible

fibration.

X+

and

identify

this

with

the moduli

ie-0,

of the

(3.124)

elliptic

fiber.

Then, the 18 complex complex structures of the elliptic and therefore the moduli of -r field fibration, on configurations R--+b are the base space (provided from the type IIB point -r and equivalent c-r+d of view). These moduli parametrize then the type IIB compactification on the base space B (it is Fl; recall that in deriving (3.122) we have used a base B B -2). There is still one moduli missing: the size space B such that of the base space B, that we can identify with the heterotic string coupling constant. Thus, we arrive to the following quasi-theorem, moduli

dimension

T

of

(3.122)

-

parametrizes

the moduli

of

=

Quasi-Theorem 2 Heterotic theory on the base space of

an

string on elliptically

T 2is

equivalent

fibered

K3.

to

type IIB

string

Strings

Fields, discussion

The previous name of

is

known,

in

the

physics

literature,

119

under

the

[94, 95, 96].

F-theory

generic

and Branes

Wehave been

until now, type II strings on K3, and compared considering, To find torus. what the out on is a string expected moduli for the heterotic the on we trick: can if heterotic use K3, string following string is type IIB on the base space of an elliptically on T' fibered K3, by quasitheorem 2 heterotic fibered K3 should correspond string on an elliptically to type IIB on the base space of an elliptically fibered Calabi-Yau manifold. More precisely, should be compactified basis of an on the type IIB string which is now four dimensional, and that can be represented elliptic fibration, of a lP1 space over another 1P1. This type of fibrations as a fibration are known them to heteotic

in

the literature

determined manifold. another

as

Hirzebruch

through

heterotic

The moduli

interesting

of these small

topic:

spaces,

data,

F,,.

Hirzebruch

given by

bundles

on

the

K3 will

E8

spaces x

put

can

E8 bundle us

in contact

simply on

be

the K3 with

yet

instantons.

to Four Dimensions.. Before conCompactifications let summarize the different us sidering examples, simply superto three dimensions, symmetries we can get when compactifying depending In order to do that, on the need holonomy of the target manifold. we will the results in subsection the number maximum of on 3.2.1, suPersymmetries allowed for a given spacetime dimension.

3.2.3

Heterotic some

Type of String

definite

Target

1I

K3

Heterotic

T

x

Manifold T

2

6

H

Calabi-Yau

Heterotic

K3

H

Bsu(4)

Heterotic

Calabi-Yau

1I

K3

Heterotic

T

x

x

T

T

2

2

6

1I

Calabi-Yau

Heterotic

K3

H

BsU(4)

Heterotic

Calabi-Yau

x

T

2

Holonoiny

Supersymmetry

SU(2)

N=4

Trivial

N=4

SU(3) SU(2)

N=2

SU(4) SU(3)

N=1

N=2

N=1

SU(2)

N=4

Trivial

N=4

SU(3) SU(2)

N=2

SU(4) SU(3)

N=1

N=2

N=1

C6sar G6mez and Rafael

120

In the

1113 16

above

table

The first

.

will

down to four

between

to corresponding dimensional spacetime,

in four

the

introduce

to

use

differentiated

have not

we

lines,

two

supersymmetry

Herndndez

of dual

concept

will

type IIA

with

cases

N

=

and type 2

4 and N

=

be the basic of

pairs

examples compactifications

string

we

dimensions.

of this table, need on the ingredients we yet entering a discussion the holonomy of the moduli space. This holonomy will of course to consider and the type (real, depend on the number of supersymmetries complex or of the representation. Hence, from subsection quaternionic) 3.2.1, we can complete the table below. Before

Supersymmetries

Type

d

=

6

N=2

H

d

=

4

N=4

c2

d

=

4

N=2

c2

Spacetime

Dimension

Using this results, according to

moduli

Let

group.

we can

in the

concentrate

us

U(4) dimensions should

on

which

6

ED

Sp(l)

vectors

respect

U(4),

to

we

the

get

(3-125)

scalars

have

m of

the

to

holonomy

SO(6).

(D

(real)

we

with For

case.

SO(6) part

the

each,

i.

e-,

the number of

these matter

multiplets,

holonomy

is

of the

group

the

acting

be

0 (6, The

U(1) part

just

from

of

(3.125)

will

act

holonomy arguments,

Now,

we

need to compute

and the total of type

IIA.

From the

compactification moduli

This

m.

of

dimension

on

x

the

(3.127)

will

we see

Let

us

x

multiplet

so we

expect,

S1(2)1U(1).

string, be 134.

that

then

(3-126)

of type

0(m)

x

0 (m).

supergravity

moduli

a

For heterotic

table,

manifold.

0 (6)

m) /

0(6,m)/0(6)

first

we

the

Let

(3.127)

answer us

should

now

is clear:

consider

consider

compute

the

K3

M=

the x

dimension

22,

case

T2

as

of the

space:

will

thinking a

4

U(1)

--

we

of the moduli

rules

=

will contain multiplets compactify. Then, if

The matter

part

d

Sp(l) U(4) U(2)

decompose the tangent

now

transformation

its

Holonomy

4

be relevant in the

Calabi-Yau

volume

fiber,

spirit fourfold

when

discussing

of the discussion of

is used for

the

third

in the last

SU(4) holonomy, compactification.

line

part

elliptically

where, by BsU(4)) of previous section, and with fibered,

we are

where a zero

Strings

Fields, Moduli

of metrics of metrics

Moduli

and B fields

K3

on

and B fields

2

which

obtained of IIA

3-form four

T

=

4

bj(K3

x

T 2)

=

2

b3(K3

x

T 2)

=

44

=

2

=

2

Now,

we

N = 4 for

into

in R to

2

-

forms

(3-128)

one

Now, the dual of

in R4 is

2-form

a

scalar,

so we

get the last

moduli.

two extra

in

4

Notice

from

can

dimensions.

80

121

that the 44 in b3 (K3 x T 2) is coming from the S' of T2, and the 22 elements in H2(K3; Z). The S' cycles of T 2 to give 2-forms in be compactified on the

134.

up to

sums

3-cycles

=

on

Axion-Dilaton Duals

and Branes

need to compare the two moduli spaces. If the moduli, the heterotic compactification, the

account

0(6,22;

Z) T-duality,

0(6,22;

Z)\0(6,22)/0(6)

x

will

look

we

S-duality

expect

once

have taken

we

like

S1(2, Z)\Sl(2)/U(1).

0(22)

(3-129)

as the second term in (3-129), Now, we have a piece in IIA looking naturally namely the moduli of the u-model on T 2, where S1(2, Z) will simply be part of the T-duality. Thus, it is natural to relate the moduli of IIA on the torus with the part of the moduli in (3.127) multiplet. coming form the supergravity There is dual pairs in the second line of our table. Let us now consider the conditions Calabi-Yau what under on visualize to a simple general way In fact, with SU(3) holonomy such dual pairs can exist. manifold imagine what we get is a fibration in K3 x T 2 ;then, fibered that K3 is ellipticaly to type IIA is equivalent on of the T 4 tori. on T' on IP1 Now, heterotic

K3,

so

IP',

and that

we

to

expect

that

expect

the

manifold

Calabi-Yau

Therefore, duality works fiberwise. II dual pairs with get heterotic-type

[97, 93].

should

be

from

general

=

2 if

order

to

N

a

K3 fibration

arguments,

we use

on we

Calabi-Yau

precise holonomy, which is U(2) in this case. The and hypermultiplets. vector 2 we have two types of multiplets, In N four real and the hypermultiplet two real contains vector scalars, multiplet scalars. Sp(l), and the moduli into Then, we decompose U(2) into U(1) and hypermultiplet vector part. The manifold. Calabi-Yau consider on the Let us first type IIA string h',' deformations of B and J, h 2,1 complex deformations moduli will contain with a as we are working and V RR deformations (bl does not contribute, The total Calabi-Yau number, in real dimension, is manifold). manifolds

picture,

which

we

need

are

again

K3 fibrations

In

get

a more

the

to work out

=

2h',' where

we

conclude

have used that that

we

b3

have h1,1

==

2(h

vector

+ 2,1

4(h

2,1

+1),

+

(3-130)

1),

in real

multiplets,

dimension.

and h 2,1 + 1

From

(3.130)

hypermultiplets.

we

122

C6sar G6mez and Rafael

Notice

that

4 (h2,1 + 1) is

type

II

so, for

Now, let now consider, of

T',

we

of E8

T',

or

coming from the dilaton and the axion and axion into an hypermutiplet. string on K3 x T 2. The moduli we must

the 2

counting

dilaton

have combined

consider

us

HernAndez

that

heterotic

E8 bundles

x

we

K3,

on

have worked

is much

more

elaborated

difficulty

of the

Part

out.

that

than

from

comes

However, we know, accordding to Mukai's theorem, that anomaly conditions. i. e., hyperkahler, bundles on K3 is quaternionic, the moduli of holomorphic moduli

the

and that the moduli

therefore

on a

of the

T 2 , that

good

a-model

will

be

candidate

for

80. We have yet

K3 is of dimension

on

manifold

a

of

0(2, m)/0(2) the

representing

x

O(m) type,

multiplet.

vector

and

Thus,

we

get

Type IIA hypermultiplets

multiplets

Vector From'our

related

to

p the

with

in the

pair

h"

sense

order

(3.131)

of

control

to

need to watch

we

the

need

out

value

possible

for

the gauge group has been fixed the logic for the identification to

the

I term

contributing

space of the K3-fibration. As can be observed

class

or

++

2

(3-131)

.

m in the

the

complex

in

from

structure

Then,

order

in

heterotic

get

to

a

of m, from the lines that

heterotic can

the

e.,

if

we

the

2-cycle

do not

of view, we T 2 after

point

be defined

on

from the K3 piece. From (3.132) dilaton-axion (3.133)), the heterotic

(3.133), of T',

dual

(3-133)

Wilson

i.

we

statisfy

to

P.

(3.132),

are

(3-132)

I+ p,

K3 manifold.

number of the

Picard

=

M=

In

K3 Heterotic, T

in type IIA previous discussion we know that vector multiplets, manifold Calabi-Yau fibered K3 fiberwise a on h',. Working

h1,1,

for

get,

++

(and by

defined

freeze

either for

minimum value

this

was

is related

p is

the

base

Kahler

the

This

2.

is

the contribution Dynkin diagram of type A2, i. e., the moduli work of line A opens here, in order to identify possible SU(3). the with IIA theories for moduli, of vector quantum type multiplets spaces to

defined

according

to

the

Picard

Seiberg

lattice

4.1

Chapter M-Theory

a

for

and Witten, rank

4.

of

G

=

gauge

theories,

with

(3.134)

P.

IV

Compactifications.

used to say that "meaning is use". Wittgenstein ophycal slogan able to make unhappy the platonic

This

is

the

kind

mathematician,

of

philos-

but

it

is

Strings

Fields,

and Branes

123

of game we are going to play in order to begin the study essence the type without start we will saying M-theory [98, 78, 80, 81, 861. More precisely, what M-theory is from a microscopical point of view, giving instead a precise meaning to M-theory compactifications. with the idea of M-theory Recall that our first contact was in connection of the moduli of type IIA string with the interpretation theory on T'. In that after including RR fields, was of the type case the moduli,

in

of

Z)\0(5,5)/0(5)

0(5,5; to

(4.1)

of moduli

M-theory interpretation equivalence

The

0(5).

x

(4.1)

be

can

summarized

according

the

compactified

M-theory therefore,

and

us

now

compactified

(4.3)

put rule

in

(4.4)

we see

equivalently, This is pactified.

on

B

x

in the

that

M-theory

B or,

a

X

S'

x

fact,

In

on

++

T 4,

on

IIA

on

particular

one

manifold

compactified

M-theory

From

IIA

++

(4.2)

of type

case

B

x

(4.3)

X.

S'

of x

(4.3)

will

S1. Then,

get

can

IIA

on

work.

into

M-theory

considering we using T-duality, consist

5

generically,

more

M-theory Let

T

on

on

S'(R) R on

IIB

++

-+

B

oo

x

B

limit,

S1,

since

S'

x

x

S' (R)

x

S'(-).

a/

on

B

we

get. type IIB

(4.4)

R

theory

string

the second S' becomes

on

uncom-

example to the ones described in previous F-theory compactifications. Namely, the of type in (4.4) can be interpreted R -+ oo limit a compactification as defining B x S' x S1, in fibration IIB string theory on the base space B of an elliptic fiber becomes zero. Following that the limit where the volume of the elliptic between M-theory x S', on B x S' equivalence path, we get an interesting fiber goes in which the volume of the elliptic in the limit as elliptic fibration, sections,

zero,

sult

under

and type derived

in fact

the

IIB

a

generic

on

from the

very

close

name

B. This

of

stands

compactification

as a

rule

when

surprise,

(4.3).

compared

fact, if B is, the compactification. In

for

to the

re-

instance,

of an eleven d, then we should expect that lead 2 S' should d B 11 x x to on S', theory, as M-theory, would when which is ten dimensions. dimensional, lead, However, type IIB, d dimensional on B, to a 10 theory, so that one dimension is compactified requires knowledge of the microGetting rid off this contradiction missing. first be The to of nature required on M-theory is of thing M-theory. scopic There eleven dimensional to have, as low energy limit, course supergravity. and eleven dimensional between type IIA string is a connection sutheory dimensional Kaluza-Klein reduction the on an as corresponding pergravity, of dimension dimensional

-

-

-

C6sar G6mez and Rafael

124

HernAndez

internal of the string S, which allows an identification theory spectrum with In particular, the RR field in ten dimensions the comes from supergravity. component of the metric, while the dilaton is obtained from g11,11. The gll,,, in what is known as the string precise relation, frame, iS17

e-20 0 the type IIA

with

field.

dilaton

get

we

e

of the radius

S1,

R of the

20/3

(4-6)

.

relation

a

coupling

(4.5)

7

In terms

R=

Using now equation (3-21), S', and the string manifold,

e-3-y

=

between

the

R of the

of type

IIA

string

constant

internal

theory,

R = g 2/3 From

(4.7)

it

that,

obvious

is

when g is

region theory. Historically, by Witten [81]. It

large,

R

-+

oo,

astonishing

to make at least

that,

the M-theory enter properly strong coupling regime of string

we

the

in

this is

as

e., working beatiful simple

i.

(4.7)

argument

with

all

the

was

pieces

put

forward

1995

in

nobody

around,

was

relating with the string and to derive from it such a coupling constant, supergravity is that as it described are strongly coupled IIA strings conjecture striking In fact, there are good reasons for such by eleven dimensional supergravity. in the whole community: obstacle first of all, a mental nobody did worry about type IIA dynamics, with only uninteresting as it was a theory pure abelian modes coming from the Secondly, the Kaluza-Klein gauge physics. which have a mass of the order 1, are charged with on S', compactification R But respect to the U(1) gauge field defined by the gll,, piece of the metric. in ten dimensional this A,, field type IIA string is of RR type, so before the discovery of D-branes, there was no candidate in the string spectrum to be with these Kaluza-Klein put in correspondence modes, which can now be able before

identified

with

the comment

the R of eleven

dimensional

D-Obranes.

approach to M-theory can be the conceptual key to solve the problem concerning the missing dimension: in fact, something in the spectrum is becoming massless as the volume of the elliptic fiber, in the case of B x S1 x S', is sent to zero. Moreover, the object becoming massless can be, as mode of an as a Kaluza-Klein suggested by Sethi and Susskind, interpreted fiber goes zero. To understand opening dimension as the volume of the elliptic of this object the nature look more carefully This we should at M-theory. two dimensional theory is expected to contain a fundamental membrane; if Witten's

this

membrane wraps

volume of the fiber

the

standard

the

goes

2-torus

zero.

Kaluza-Klein

Then,

formula

to "'

Wehave identified

g11,11

=

e2-y.

S' all

for

x

S1,

its

mass

what is left

becomes

is to relate

compactifications

on

zero

the

S1,

as

area

which

the with

leads

Strings

Fields,

and Branes

(4.8)

LjL2

R

125

of (4.4). adequate interpretation on a concrete example of (4.4): we will choose fourfold of SU(4) holonomy. X B x S' x SR as representing a Calabi-Yau three After compactification, dimensional a SU(4) holonomy implies theory 2 supersymmetry with N should be expected. Moreover, sending R -+ 00 N In order to work out the spectrum of I theory. leads to a four dimensional standard Kaluza-Klein the three dimensional techniques can be used. theory, 2 of of the the H 3-form C,,,p of eleven on Compactification 2-cycles (X; Z) leads three in dimensional dimensions. to vector a supergravity Moreover, from each 2-cycle. the Kdhler class can also be used to generate real scalars, dimH 2 (X; Z); then, the previous procedure produces Thus, let us assume r scalars In order to define r N 2 vector multiplets and r vector fields. r real with these vector another in three dimensions, set of r scalars is yet fields, These extra r scalars needed, in order to build the complex fields. can, as with the duals, in three dimensions, of the 1-form vector usual, be identified fields: the three dimensional dual photon. be reproducing, the well known inOur next task will using M-theory, 2 supersymmetric theories in in N effects three dimensions. stanton gauge

solving

problem

our

Let

us

the

on

concentrate

now

=

=

=

=

=

=

M-Theory

4.2

order

In

Instantons.

define

to

wrapped using 6-cycles

instantons

6-cycles

the

on

is understood

obtained

from

Calabi-Yau

a as

dimensions,

three

in

of

we

fourfold

X

will

[99].

use

The

5-branes

reason

for

follows:

the gauge bosons in three dimensions of the 3-form C,,p over 2-cycles. Thus, in

the

integration photon, we should consider the dual, in the Calabiof which are 6-cycles. Yau fourfold X, 2-cycles, However, not any 6-cycle with topological can be interpreted as an instanton charge equal one, and therefore will contribute dimensional to the three no 6-cycle superpotential. If we interpret a 5-brane wrapped on a 6-cycle D of X as an instanton, of the type we can expect a superpotential are

order

to

define

the dual

W=: with

e-(VD+i-OD)

(4.9)

VD the volume of D measured in units of the 5-brane photon field, associated with the cycle D. In order

and OD tension, get, associated

the dual to

i)

D,

a

superpotential

To define

fermions

ii)

a

U(1)

are

To associate

like

(4.9),

we

transformation

need

with

respect

which

to

iii)]Po

prove

interpret

that

three

dimensional

charged. with

the

6-cycle

D

a

violation

of U(1)

charge,

amount.

iv To

to

this

OD as

U(1) symmetry the

corresponding

is not

anomalous.

Goldstone

boson.

in the adecuate

C6sar G6mez and Rafael

126

Following

defining

start

fourfold a

these

steps,

dimensional

of three

the

U(1)

and let

X,

we

N

will

extend

transformation.

by

the instanton

dynamics

chapter

1. We will

the

Calabi-Yau

described

Let

D be

N the normal

canonical

its

M-theory

to

2 gauge theories

denote

us

manifold,

Calabi-Yau

=

Herndndez

bundle

is

in

6-cycle

in

bundle

of D in X. Since

X is

trivial,

and therefore

get

we

(4.10)

KD f -- N, KD the canonical

with

of the

space

direction,

bundle

normal

U(1)

the

Locally, Denoting by of D.

bundle.

transformation

-+

z

The

U(1)

transformation

coordinate

X

as

in

the

the total normal

as

eiOz.

(4.11)

by (4.11)

defined

the

be defined

can

interpret

we can z

likely

anomalous, since theory; thus, it for the U(1) symmetry we are looking is a good candidate for. Next, we need dimensional fermions. to get the U(1) charge of the three However, before doing so, we will review some well known facts concerning fermions and Dirac it

of the

is part

operators

on

diffeomorphisms

Kdhler

consider

Wewill

of the

is very

elevean

not

dimensional

manifolds. a

Kdhler

manifold

of

complex dimension

N. In holomor-

phic coordinates, gab

coordinates,

In -these

the

algebra

The

SO(2N) spinorial

-Y

n-particle

states

are

defined

defined

V)(Zl 10 the

O(z,. )

field

by

the Kdhler

can

is defined

by

O(Z' Os?

S+ S-

=

stan-

(4.14)

by

Ilyn I f2 manifold

>

(4.15)

.

takes

>

+Od(Z, f)-yals?

>

values

on

+Oab(z" )-Ya-Y6jS?

f?O,q of (0, q)-forms, generated by cohomology of the Kdhler manifold. chirality spinor bundles are

=

in the

0,

>=

The spaces Dolbeaut

two different

be obtained

condition

the

spinor

bundle

Fock representation:

this

=

on

(4.13)

(4.13)

of

state

^/dly6... A spinor

0,

2gab.

,Yalfl and

becomes

matrices

76

bj

a vacuum

(4.12)

0-

:--:

of Dirac

representations

approach:

dard Fock

gab

b

j,a' 0"

-:::

(K1

(K'/2

/2 & (g

flo,o) f20,1)

(K1 ED (K1 ED

/2 (g /2 (D

S?0,2) DO,3)

ED

0)

the

Dirac

Using

(K1 /2 (K1 /2

0

(g

+

(4.16)

operator,

define

>

this

flO,4) f2O,5)

notation,

the

e

(D

(4.17)

Strings

Fields,

X)

will

(the

change of chirality given by the

and the

be

for

index

aritmetic

the Dirac

and Branes

operator

on

127

the manifold

genus, N

X

where

h,,

=

The

D in

normal

canonical

a

budle

canonical

trivial

with

e.,

the

comments

divisor

the

account

i.

E(-I)nhn,

(4.18)

dim S?O,n.

previous

mensional

=

bundle

on

be

can

applied

readily fourfold

Calabi-Yau

X.

the

to

Now,

we

case

of

should

a

six

take

diinto

N, to D, in X. Using the fact that X is Calabi-Yau, bundle, we conclude that N is isomorphic to KD, D. The spinor bundle on N will be defined by W1/2 D

1/2

K

0

(4.19)

complex dimension of N is one, and the vacum and filled states have, respectively, U(1) charges 12 and 21. On the other hand, defined D will be the spinor budle on by (4.17) ,with K KD. Thus, spinors and on D are, part, taking values in the positive up to the SO(3) spacetime bundles negative chirality In

fact,

this

in

case

the

-

=

(K1 (K1

/2 E) K/2 E) K-

1/2) 1/2)

(9 0

[(K

1/2 &

[(K1

/2 0

j?0,0) 00,1)

6)

(K1/2& f?0,2)], (K1/2 & j?0,3)].

(4.20)

in a change of U(1) charge, with the U(1) charge Now, we are interested .1 and -.1 the charges of the spinor bundle (4.19) on N. by 2 2 the change of U(1) charge is given by of a given chirality, For spinors

defined

dim

Using

(K

now

(9

S?0,0)

Serre's

+ dim

get that

the number of

holomorphic zero

sections

modes with

of fermionic

zero

have used the

index

for

the

0

f?0,2)

-

dim

(00,0)

-

dim

('00,2).

(4.21)

duality, dim

we

(K

in

(K

(9

00,3-n)

=

(0, k)-forms

holomorphic

K0

S?0,3-k

,

('00,n),

dim

and therefore

(4.22)

is

equal

to the

number of

the number of fermionic

U(1) charge equal I2 is given by h3 + hi, and the number modes with U(1) charge, is given by ho + h2 (here we

Dirac

twisted

-

operator

spin

0*,

a +

bundle

with

(4.20)

is

0* the adjoint of 0. Thus, the the Euler holomorphic given by

characteristic,

X(D)

=

ho

-

h,

+

h2

-

(4.23)

h3-

with zero modes is doubled once we tensor Now, each of these fermionic with vertex spinors in R3. In summary, for each 6-cycle D we get an effective a net change of U(1) charge equal to X(D). in a three dimensional in order to get the three dimensional Therefore, look for with 2 theory, need N to we 6-cycles D, 1, as the net X(D) =

=

C6sar G6mez and Rafael

128

change of U(1)

charge

number of fermionic

by

a

6-cycle

D,

and with

a

D-brane

a

flat

in

three

as

be I.

to

we

did

2

dimensional

(4.19),

in

precisely,

More

the

defined

instanton,

massless D-brane.

on

N

p, in flat

the =

dimensional

ten

worldvolume.

D-brane

defines

The a

Minkowski

of

quantization field

low energy

the-

with U(1) supersymmetric Yang-Mills, of this theory to p + 1 dimensions of the p on the worldvolume propagating 1

reduction

excitations We will

lagrangian

The worldvolume

Space.

Flat

of dimension

The dimensional

the

describes dimensional

a

p + 1 dimensional

space, the open superstring ending ory, which is ten dimensional gauge group.

provided,

one,

fermions

modes for

zero

Configurations

consider

Wewill

is

case

of the

2X(D).

is

D-Brane

4.3

that

in

U(1) charge

the

normalize

we

HernAndez

use

will

worldvolume

as

contain

a

U(1)

coordinates

x0, x1....

gauge field

massless

I

XP.

Ai (x,),

fields 9, transp, and a set of scalar p + 1, Oj (x,), j We can geometrically the set adjoint representation. interpret the "location" in of flat the D-brane of fields transas representing (x,) Oj of the previous verse generalization picture corresponds space. The simplest of k > I parallel to configurations D-pbranes. In this case we have, in addiexcitations tion to the massless excitations, a set of k massive corresponding D-branes. to open strings ending on different of this of D-branes would The field theory interpretation configuration broken to be that of a gauge thory with U(k) gauge group, spontaneously between different D-branes representing U(I)k, with the strings stretching we can start charged massive vector bosons. To get such an interpretation, in ten dimensions, I U(k) supersymmetric with N and perform Yang-Mills reduction down to p + 1 dimensions. In this case, we will again dimensional Xj (x,), with j 9, which are now k x k get a set of scalar fields, p + 1, in the adjoint of U(k). Moreover, the kinetic term in transforming matrices, of the form ten dimensions produces a potential

with

i,

s

=

forming

0,

.

.

.

=

,

.

.

.,

in the

=

=

V

9 =

7

T

.

tr[X',

..

,

Xj]2.

(4.24)

i,j=p+l

As

we

possesses

directions

already observed in many examples before, this potential These flat to classical vacumm states. directions, correspoding X' defined by diagonal matrices,

have

flat are

A?, Xi

(4.25)

Ak On each of these vacua, to

U(j)k;

thus,

we can

use

the

U(k)

these

gauge

vacuum

symmetry is spontaneously

configurations

to

describe

broken sets

of k

Strings

Fields,

and Branes

129

D-branes. In fact, for the simpler as we observe parallel p dimensional case of the set of scalars one D-brane, dimensional reduction has the by appearing of the position of the D-brane. In the case (4.25), interpretation geometrical fact consider we can in of the 1"-brane. A,' as defining the ith -coordinate This is consistent with the idea of interpreting the strings between stretching different D-branes as massive vector bosons. In fact, the mass of this string would be

states

gJA1

M

-

(1, m) string. charged boson.

for

classical

cides

with

This is, in

transversal

its

dimensional

theory, described

by

a

this

simple

of k

set

the

fact,

A,'

where

we now

allow

of flat

parallel

D-branes

X',

but

with

meaning

we can

interpretation.

dimensions,

branes,

parallel

matrix

about In

geometrical

down to p + I

complete

the

A nice way to think abelian projection.

of the

i. e.,

reduction,

describes

the

that only space. It is important realizing the moduli space of the worldvolume

of the potential (4.24), is the one possessing the

(4-26)

fact, Higgs mass corresponding massive In summary, merging the previous comments into a lemma: moduli space of the worldvolume lagrangian of a D-brane coin-

a

the

A,J,

-

non

In

of N

I

=

the minima

lagrangian, particular,

U(k)

gauge

fledged dynamics is vanishing off diagonal terms. full

its

(4-25) is again in terms of 't Hooft's (4.25) as a unitary gauge fixing,

of

of

think

The case depend on the worldvolume coordinates. corresponds to a Higgs phase, with Aj' constant functions worldvolume. on the Moreover, we can even consider the existence of which will be points where two eigenvalues singularities, coincide, to

Az1

At1+1'

=

Vi.

(4.27)

obvious realizing that (4.27) so we eximposes three constraints, that on 3-dimensional a p p-dimensional pect, D-branes, region of the two consecutive D-branes can overlap. The p 3 region in the worldvolume, It

is

quite for

-

-

p + 1 dimensional

worldvolume

of view of p + I dynamics, a in 't Hooft's abelian projection. Next to

[117]).

allowed

(p, 1'), on

we

will

In

The

type

In

(some

order

vertices

for

corresponding

the D-brane

IIB.

consider

theory

string

IIB

fact,

to

of the D-brane

monopole,

define

same sense

as

is the

case

brane configurations for type IIA and type refences are those from [100] widely increasing these configurations first work o ut the we will branes.

Dirichlet

a

worldvolume.

the RR fields

from the point

represent,

in the very

some

of the

intersecting to

will

IIA

In type

for

Let

p-brane,

type IIA

us

and

a

p should

and type

IIA

Au Ativp

IIB

X

Btiv

start

with

a

be even, IIB string

of type

vertex

fundamental

string

ending

and odd for

theory

type

are

I

Attvpa.

corresponding strength tensors are, respectively, for type IIB. IIA, and one, three and five-forms

(4-28) two and four-forms

Thus,

the

sources

for are

C6sar G6mez and Rafael

130

of dimensions

D-branes

for

type

which

six

and four

are

D-branes

dual).

Besides,

and its

dual

Let

us

for for

is

a

then

transform

this

IIB.

addition,

In

D-branes

the X field D-7brane.

the

a

Z) duality (p, 1) vertex,

string

threebrane

the

is

source

(p, IF)

of type

vertex

a

theory, and one (Hodge) magnetic duals, and five and string theory,

IIA

type that

S1 (2,

into

vertex

for

IIA

type

have the

type IIB,

in

with the

use

for we

(notice

IIB

type

start

We can

p odd.

and two,

zero

and three

three

HernAndez

in

a

IIB, and

a

self

is

object,

i.

IIB

type

D-pbrane

IIB

type

extended

type

of

symmetry between

in -I

a

with

e.,

to strings D-lbrane,

transformations By performing on the j T-duality spacetime to the worldvolume of the D-brane and the D-string, orthogonal form (p, IF) to a vertex (p + j, IF + j) of two D-branes, sharing j

D-string.

or

directions pass

we

worldvolume

common

coordinates.

If

j

is

even,

we

end up with

a

in

vertex

and if j is odd with a vertex in type IIA. Namely, through a Twe pass from type IIB duality transformation string theory to type IIA. As an example, we will consider the vertex (3, IF) in type IIB string theory. After

IIB,

type

a

S-duality

transformation

T-duality type IIB,

are

in

the

vertex

(5",

3),

S1(2, Z) duality

in the

transformations, we can perform

and two

between

get the

we

duality

a

the

group

(5,3)

vertex

of type IIB for branes.

transformation

solitonic

Neveau-Schwarz

on

it

to

fivebrane

strings, As

we

generate and

a

D-3brane. Let

(5, 3)

us now

and

consider

(5NS, 3)

in

fivebranes,

solitonic

located

at

some

type

with

definite

brane

build up using the vertices configurations consider we will theory [100]. In particular,

IIB

worldvolume

values

of

6

xO, xi, x2, x3, x4

coordinates 7

8

x9, It is convenient

and

and

x5,

organize of the fivebrane the coordinates w) where w as (x6, (x7, x8, x9). By conthe D-3brane will s are two worldvolume struction of the vertex, coordinates, with the fivebrane. in addition to time, Thus, we can consider D-3branes with coordinates worldvolume XO, X1, X2 and X6. If we put a D-3brane in between two solitonic and X61 positions at A in the X6 coordinate, then the fivebranes, 2 of the D-3brane will be finite worldvolume in the X6 direction (see Figure 1). some

x

,

X

X

,

to

=

the macroscopic Therefore, physics, be described can effectively by a 2 +

I

ravel

theory,

brane

what kind

of 2 + 1 dimensional

configuration,

we

must first

by the fivebrane boundary lagrangian for a D-3brane in

between

tions,

in the

This

means

two

solitonic

x1 direction, in particular

gauge

e.,

for

scales

dimensional we are

work out

larger than JX62 X61, 1 theory. In order to unobtaining through this -

the

type of constraint impossed fact, the worldvolume low energy is a U(1) gauge theory. Once we put the D-3brane fivebranes Neumann boundary condiwe imposse conditions.

for that

the fields

for

In

living

on

fields

scalar

a60 and, for

i.

=

the D-3brane

we

worldvolume.

imposse

(4.29)

0

fields,

F,,6

=

0)

p

=

0, 1,

2.

(4.30)

Strings

Fields,

X

and Branes

X6-coordinate

6 X2

6

131

D-5brane

D-5brane

D-3branes

Fig.

4.1.

Solitonic

fivebranes

three

dimensional

the

Thus,

which

dimensional

theory

as

consider

6,

then

we are

(x 6, W2),

and

D-3brane

the

These

(4.29),

is

we

x

the

D-3brane three

multiplet three

w,

reduced

in three

dimensional

the

on

What this

in

with

dimensions.

theory

fields

,

the

4 X

and

U(1)

be at

case,

the

x

5

Therefore, n parallel

A,,,

2

motion

X3'X4

fields. to

we

and for

and x'.

By condition be constant

on

the two ends of the of

Now, if we

get

conclude

D-3branes

x

where the D-3brane

coordinates.

we can

the If

(X6,W I)

positions allowed

scalar

is that

gauge field

and there-

xO, xi,

coordinates

un-

three

configuration.

be constrained

means

is

effective

need to discover

to

three

0, 1, 2,

=

coordinates

worldvolume

can

practice 3

we

them.

an

that

suspended

we

combine

N = 4 vector our

effective

between

two

1) U(n) gauge group, gauge theory 4 3 Denoting by v the vector (x X XI), the Coulomb branch of this theory is parametrized of the n D-3branes by the vi positions In addition, each brane). in chapter we have, as discussed (with i labelling II, the dual photons for each U(1) factor. In this way, we get the hyperkdhler of the Coulomb branch of the moduli. structure Hence, a direct way to get is as follows. The supreserved by the brane configuration supersymmetry persymmetry charges are defined as fivebranes

solitonic

N

=

4

(Figure

for

space

along

D-3brane,

brane

with

the D-3brane

on

scalar

fields

particular

the fivebrane

same x

Next, the

fivebranes R3,

p

the

one

worldvolume

W2- In this

of -these

have the

scalar

with

with

interpret

for

by

unbroken

solitonic

to

A,,,,

gauge field, that we can

gauge theory n D-3branes.

for

the =

have defined

direction.

the

these

forcing

with

the values 6

theory

the coordinates

are

Thus,

ends.

gauge

U(1)

a

as

left of supersymmetry Dirichlet threebranes,

amount

X

U(n)

a

means

stretching

threebranes

Dirichlet

n

U(1)

already

constrained

fore

with

is

a

supersymmetry.

and

,

CLQL + 'ERQR, where

with

,

(4-31)

QL and QRare the supercharges generated by the left and right-moving degrees of freedom, and EL and ER are ten dimensional spinors.

worldsheet

G6mez and Rafael

Usar

132

pbrane,

Each solitonic

Hern6ndez

along xO, xl,...

extending

worldvolume

with

I

XP,

im-

posses the conditions 6L

in terms

rpEL,

...

of the ten dimensional with

D-pbranes,

the

rO

:'--

ER

Dirac

worldvolumes

-ro

::--::

rpER,

...

(4.32)

Fi; on along xO, x1,

gammamatrices,

extending

hand, xP, imply the

the other

constraint 6L

Thus,

fivebrane,

NS solitonic

that

see

we

xO, xi, x2, x3, x4

x5,

and

and

equal

with

values

along xO, xi, x2 and x6, or, equivalently, dimensional theory. described allows array just

of

worldvolume

with

effective

three

The brane

coupling

of

constant

Kaluza-Klein

reduction

x'

(compactified)

on

the

direction

lagrangian,

dimensional

effective

the

N

a

three

reduce

the gauge

eight

=

simple

JX26

on

on

the

of the gauge computation theory: by standard after integrating the over

constant

to

is

an

effective

three

given by

X11 6

_

(4.34)

2

93

at

supersymmetries

4 supersymmetry

lagrangian

the

coupling

2

located

threebranes

dimensional

x6 direction,

finite

to

worldvolume and Dirichlet

w,

preserve

worldvolume

D-3brane

the

(4-33)

-Vp'ER-

rOF1

:::::::

94

constant. Naturally, gauge coupling (4.34) is taking into account the effect on the fivebrane at x6 of the D-3brane In fact, we can position ending on its worldvolume. the dependence of x6 on the coordinate consider of v, normal to the position The dynamics of the fivebranes the D-3brane. should then be recovered when

in terms a

of the four

dimensional that

expression

classical

the Nambu-Goto action

influence

x3, x4

of the

x'),

and

is not

equation, with

fivebrane

of the solitonic

is minimized.

Far from the

are located points where the fivebranes (at large values of of motion is simply three dimensional the equation Laplace's

V2X6 (X3, X4,

X5)

=

(4.35)

0,

solution X

where k and is the

there

a

spherical is

defined

a

well

constant,

are

at

defined 02

limit -

al,

k

the

point

as r

in

the

-+ r

(4-36)

+ a,

=

r

depending

constants

radius

(r)

on

the

(X3, X4, X5). oo; -+

threebrane From

(4.36),

hence, the difference oo

tensions, is

it

A2

-

and

Al

is

r

that

clear a

well

limit.

is that it allows to obtain beauty of brane technology very brane manipulations. Wewill by simply performing geometrical one example, concerning our previous model. If we consider the from the point of view of the fivebrane, brane configuration the n suspended look like n magnetic will threebranes monopoles. This is really suggesting

Part

of the

strong results now present

Strings

Fields,

since,

described

as

chapter

in

II,

analogy can be put above, be transformed view of the threebrane,

more

know that

133

Coulomb branch

the

moduli

SU(n)

space of N = 4 supersymmetric moduli space of BPS monopole

This

we

and Branes

is isomorphic gauge theories with magnetic charge configurations,

(3, 1)

into

'a

we

have

(5 NS 3)

the vertex

precisely:

this

In

can,

to

equal

the n.

described

as

from the point of theory with SU(2) and n magnetic monopoles. Notice that gauge group broken down to U(1), the from build to that by passing configuration (5 NS 3) vertices, up ussing build

the

up with

Next,

(5,3)

we

will

(3, 1) vertex, work out

example now

be

volume

comes

vertex.

four

the

the

made out of two Dirichlet

a

dimensional

Coulomb moduli

same

branes.

configuration,

for

coordinates

D-5branes

the

x0, xi, of x3,

x2

and

with

(4.30),

Wewill

same.

the vertex

with

now

The main difference

boundary conditions (4.29) Dirichlet by boundary conditions.

the

remains

but

from the

replaced

case,

gauge

the

choose

x7, x8and

previous

which should

9,

as

world-

they before, let us denote this positions by (m, x 6), where now m (X3, X4, x1). An equivalent the studied above will be now a set of two D-5branes, to one at configuration of the x 6 coordinate, that we will again call x 6Iand X26 ,subject some points to between them along the x 6 coordinate, MI M2 with D-3branes stretching with worldvolume x2and x 6 x 0, xi, extending again along the coordinates will

be located

at

some

definite

values

x

,

4, x5

and x1.

x

so

that

As

=

=

,

(Figure

2).

Our task

now

will

X

X62

D-5brane

X6-coordinate

D-5brane

D-3branes

Fig.

Dirichlet

4.2.

dashed

lines).

threebranes

extending

between

a

pair

of Dirichlet

fivebranes

(in

of the effective three dimensional description theory on these threethe of D-3branes worlvolumes on the fivebrane will points 7 8 x'. of values This and now be parametrized x x that have means by we in the effective three scalar fields three dimensional theory. The scalar fields 3 x4 x 5 and x 6 of the threebranes to the coordinates x are corresponding forzen values where the fivebranes to the constant located. are Next, we be the

branes.

The end

134

C6sar G6mez and Rafael

should

consider

volume.

happens

what

Impossing

Hern6ndez to the

U(1)

boundary

Dirichlet

gauge field

conditions

on

for

the D-3brane

this

field

is

world-

equivalent

to

F4, i.

e.,

there

is

theory.

=

0,

electromagnetic

no

/_t,

v

effective

the

in

tensor

(4.37)

0, 1, 2,

=

three

dimensional

going on, it would be convenient summarizing the rules we have used to Consider impose the different a Dboundary conditions. and B aM the boundary pbrane, and let Mbe its worldvolume manifold, of M. Neumann and Dirichlet for the gauge field on the boundary conditions worldvolume defined are D-pbrane respectively by field

Before

=

where y and v ordinates to B.

N

--+

F,,p

=

0,

D

---+

Fj "V

=

0,

directions

are

If

B is

(4.38)

of tangency to B, worldvolume

and p are the of a solitonic

of the

part

and if imposse Neumann conditions, Dirichlet brane, we will imposse Dirichlet will

it

is

of the

part

normal

brane,

worldvolume

cowe

of

a

Returning to (4.37), dimensional effective we see that on the three theory, the only non vanishing component of the four dimensional strenght tensor is FA =_ a,,b. Therefore, all together scalar fields in three dimensions we have four or, equivalently, 4 supersymmetry. with N a multiplet Thus, the theory defined by the n suspended D-3branes in between a pair of D-5branes, is a theory of n N 4 massless hypermultiplets. There exits a different the theory, way to interpret namely as a magnetic In fact, dual gauge theory. if we perform a duality transformation in the four dimensional U(1) gauge theory, and use magnetic variables *F, instead of the field F, what we get in three dimensions, electric after impossing D-boundary is a dual photon, or a magnetic conditions, U(1) gauge theory. The configuration chosen for the worldvolume of the Dirichlet and solitonic fivebranes with D-3branes suspended yet allows a different configuration conditions.

=

=

between

D-5brane

a

we

NS-5brane.

a

(4.32)

requirements

persymmetry brane

and

The solitonic

fivebrane

==

=

10

-

-

are

for

the

with

Dirichlet

suspended threebranes

easily

seen

D-3brane

is frozen.

5brane is

equal

In

to the

FOr1F2-V7-V8F96R-

-QCL

-

the

su-

five-

(4-39)

=

the

position

=

(4.40)

F56R)

-FO

-1'O-V1-V2-V66R,

to be consistent.

fact,

6R

imply

IEL

which

consistent

Namely,

imposses

CL

the

(4.33).

have CL

while

is in fact

This

and

The

position mof

the

(X3,

(4.41)

problem

now

4,X 5)

X

D-5brane,

is that

the suspended of the NS-

of the end point and the position

(X7,X8,X9)

Strings

Fields, of the end

point

on

the

D-5brane

The fact

NS-5brane-

the

the

that

is forced

to

D-3brane

i. e., posseses a has no moduli, on it between branes we Using the vertices brane configurations. build quite complicated and left of a fivebrane, to the right are placed

defined

threebranes

at

different

of the fivebrane.

sides

equal

be

is frozen

and Branes to

the

mass

gap.

have

described

W of

position

that

means

135

so

When Dirichlet

the

far

theory we

can

threebranes

can open strings They will represent k2 the number of

connect

the

hypermul-

threebranes as (ki, k2), with k, and transforming is In case the fivebrane of the fivebrane. and right, respectively, to an electric with respect the hypermultiplets are charged group, solitonic, while in case it is a D-5brane, charged. Another possithey are magnetically between with D-3branes extending is that with a pair of NS-5branes, bility A solitonic fivebranes. them, and also a D-5brane located between the two of the whenever will now appear massless hypermultiplet (x 3,X4,X 5) position of the D-5brane. with the m (x 3,X 4, x') position the D-3brane coincides

tiplets to

the

left

=

different for representing configurations gauge two different we have considered types configurations For the examples described of moduli. above, these two types of moduli are as three dimensional follows: the moduli of the effective theory, corresponding and where the suspended D-3branes can be located, to the different positions which of the fivebranes, locations to the different the moduli corresponding from the This second type of moduli specifies, used as boundaries. are being coupling constants; theory, different point of view of the three dimensional the changes and follow of the fivebranes, hence, we can move the location three dimensional theory. Let us then consider taking place in the effective fivebrane two solitonic placed between a case with branes, and a Dirichlet So far

we

have used brane

In these

theories.

them.

Let

right.

In

us

doing

brane

now move

so,

there

NS-5brane

the is

a

on

the

left

moment when both

of the fivebranes

D-5brane

meet,

to

the

sharing

of the hypermultiplet of x'. If the interpretation we have a common value what happens to the hypermulwe must discover presented above is correct, tiplet after this exchange of branes has been performed. In order to maintain after the exchange, should be created the hypermultiplet, a new D-3brane To fivebrane. the Dirichlet fivebrane solitonic to extending from the right Let work. start D-brane at need us will considering dynamics prove this we closed loops, C and C', and suppose electrically two interpenetrating charged move in C'. in C, while magnetically are moving charged particles particles Wilson and number L(C, C') can be defined using the standard The linking flux electric the measure 't Hooft loops. we can passing through Namely, flux passing the or measure C' or, equivalently, magnetic compute B(C), is what In both line the Wilson are doing we i. A(C). cases, through C, e., the field created the dual the C to and C' over by particle moving integrating to the case Let us now extend this simple result in C and C', respectively. and its dual is is a source of 7-form tensor field, A fivebrane of fivebranes. and HD for Wewill call this 3-form. HNSfor NS-5branes, therefore a 3-form.

C6sar G6mez and Rafael

136

D-5branes.

Now, let

HernAndez

consider

us

the worldvolume 3

R

3

R We can of

define

now

linking

the

YNS7

X

YD

number

particle:

a

X

L(YNs, YD)

(4.42)

-

-

before,

did

we

as

fYN SH fy" D

=

fivebranes,

of the two

in the

simpler

HNS

=

case

(4.43)

.

HNS is

locally dBNS globally;

dBNS- Since we have no sources for HNS we can however, this requires B to be globally defined, or invariant. In IIB type string theory, B is not gauge invariant; gauge however, the combination on a D-brane we can define BNS FD, which is invariant, with FD the two form for the U(1) gauge field on the D-brane. Now, when the D-5brane and the NS-5brane do not intersect, the linking number is obviously When they intersect, this linking zero. number changes, which means that (4.43) should, in that case, be non vanishing. Writing The 3-form

use

HNS =

,

-

fy" that

we

observe

for

FD. These

3branes the

with

required

FD

worldvolume

R3

In the

Description

example Now, let

previous

branes

that

fyi'

dBNs

C,

with

C

ending

on

Seiberg-Witten

consider

be used to

sources

YD,

which

the D-

precisely

is

D-3branes.

have considered

us

(4.44)

dFD,

,

x

of

we

-

are

of extra

appearance

and fivebranes.

=

numbers would be adding get linking like on YD and are therefore point

way to

for

sources

D-Brane

4.4

only

the

NS

H

type

define,

Solution.

type IIB IIA

by analogy

theory

string

strings,

where

with

and three

we

have four-

the

previous picture, again be the use of solitonic in between. The only difference fivebranes, the fivebrane does not create a RRfield in type IIA string now is that theory the physics of the two parallel solitonic fivebranes does not and, therefore, have the interpretation of a gauge theory, the type IIB as was the case for N

=

can

configuration Let

us

volume and at

finite

X6,

configurations

some

fixed

x

,

X

value

1 ,

on

X

will,

field as

theory,

in the

X

,

3

4 X

,

x

with

6

type

and

solitonic

x',

coordinate.

worldvolume

will

fivebranes,

located In

at

with IIB

X7

addition,

world-

with

=

let

X8 us

=:

X9

==

are a

finite

and

in the

macroscopic

X6

four

N = 2 supersymmetry. This four dimensional in previous considered section, be defined

case

0

introduce

XO,Xl,X2,X3

coordinates

fivebranes; thus, they we can define worldvolume,

the solitonic

On the fourbrane

dimensional

of infinite

2

of the

fourbranes,

Dirichlet

which terminate

theory

0

The idea

[103].

above described consider

coordinates

direction.

[103]. gauge theories with sets of fourbranes

dimensional

2 four

dimensional

Kaluza-Klein

by standard ory defined of the four

dimensional

will

theory

JX26

as

dimensional

coupling

parameters

of the effective

the points

locate

on

field

the coordinates

x

where the

worldvolume

the fivebrane

Moreover, we can interpret theory on the dimension-

constant.

of the fourbrane

worldvolume

(4.45)

95

moduli

reduced

ally

-6

2

of the five

classical

the-

constant

X1 I

-

-

94 in terms

Then, the bare coupling

be

1 2

dimensional

of the five

reduction

worldvolume.

the D-4brane

on

137

and Branes

Strings

Fields,

4

and

x

D-4branes

5 ,

which termi-

nate.

addition

In

yet include on

the

the

break

imposes the projections 6L

while

we can

of supersymmetry To prove this, we notice

any further

of the fourbranes.

in the worldvolume

theory

fivebranes,

and solitonic

fourbranes

without

sixbranes,

each NS-5brane

that

Dirichlet

the

to

Dirichlet

D-4branes,

-

rO

-`

r5CL,

...

ER

:--:

...

(4.46)

TWER,

localized

worldvolume

with

--VO

at

x0,

x

1

x2

x

3

and

x

6

imply EL

(4.46)

Conditions

(4.47)

and

symmetry

we

break

fivebranes

of the

again half

breaks

which

percharges, As

leads

discuss

will

In

whenever

can

be added with

half

of the

additional

no

super-

the

IIA

string

macroscopic hypermultiplets

four

of type

to the effective

hypermultiplets the mass of these particular, D-4brane

meets

a

One of the main achievements

the Dreal

su-

=

sixbranes

the

on,

while

supersymmetries,

symmetry, leaving eight N 2 supersymmetry.

remaining dimensional

to four

later

be used to add

theory.

(4.48)

breaking.

The solitonic

6brane

into

r0r1r2r3-1'7T8r91ER, sixbranes

certainly

shows that

be recombined

can

EL

which

(4.47)

r0r1T2r3r6'FR-

::_-

will

theory

can

dimensional become

zero

D-6brane. of the

brane

representations the

different

of supersym-

moduli

spaces, ability represent gauge theories namely the Coulomb and Higgs branches, in terms of the brane motions left fivebranes of k fourbranes free. For a configuration connecting two solitonic Coulomb the described have the x1 we one as above, direction, along the is branch of the moduli space of the four dimensional by parametrized theory fivebranes. When the transversal fourbranes the of on the different positions is the

metric

Nf

Dirichlet

ing

is

the

sixbranes

Nf

added to this of

a

four

configuration,

dimensional

field

what

theory

we are

with

describ-

SU(N,)

with N, is the number of D-4branes we are considering), of branch the brane In this representation, Higgs hypermultiplets.

gauge group

flavor

are

Coulomb branch

to

(in

case

C6sar G6mez and Rafael

138

the

theory

ing

on

is obtained

D-6branes on

the

As gauge

effects.

determine

study of we

the locations

Higgs

the

theories,

two

However,

for

several

we

living mostly

will

pieces endbetween

two

concentrate

gauge theories. solution of N = 2 supersymmetric of the theory is corrected by quantum

moduli of effects

types

(determined

beta function

into

pure

Seiberg-Witten

classical

the are

is broken

of the D-4branes

branch.

Coulomb branch

the

know from the

There

totically

when each fourbrane

sixbranes:

different

Herndndez

at

that

loop)

one

the

enter

implies

game:

existence, the infinity point

free

vanishing

a non

the

in the

assymp-

at in moduli space, regime, of a singularity and strong coupling which imply the existence of extra singularities, effects, where some magnetically become massless. The problem we charged particles such a complete characterization are facing now is how to derive of the quantum moduli dimensional N 2 supersymmetric field space of four theory from the dynamics governing the brane configuration. The approach directly =

to

be used

completely

is

different

a type string theory employed in the description N 4 supersymmetric field the standard pling through

IIB

to

from

brane

a

configuration. of the preceding we can theories, S1(2, Z) duality

brane

=

in the

in type IIA type IIB case,

of three

dimensional

construction

In

fact,

section pass from

weak to strong

of type IIB strings; need is to know how brane configurations

cou-

hence,

the

we transform ingredient under this duality symmetry. In the case of type IIA string theory, the situation is more complicated, as the theory is not S1(2, Z) self dual. However, the strong coupling limit we know that of type IIA dynamics is described by the eleven dimensional we should M-theory; therefore, expect to recover the 2 supersymmetric N strong coupling dynamics of four dimensional gauge theories the of M-theory description using strongly coupled type IIA strings. Let us first start weak coupling effects. The first by considering thing to be noticed, the above described of Nc Dirichlet concerning configuration fourbranes between two solitonic extending along the x1 direction fivebranes, where only a rigid motion of the transversal fourbranes is allowed, is that this In fact, in this simple image is missing the classical dynamics of the fivebranes.

essential

=

on the fivebrane worldvolume picture we are assuming that the x 6 coordinate which is in fact a very bad approximation. is constant, Of course, one physical we should as we did in the case requirement impose to a brane configuration, of the type IIB configurations of the previous that of minimizing is section, the total worldvolume action. More precisely, what we have interpreted as Coulomb or Higgs branches in term of free motions of some branes entering the configuration, should correspond to zero modes of the brane configuration, i.

e.,

to

changes

worldvolume do not

in

the

(in

action

constitute

an

configuration other

energy "normal"

only depend on the into the complex coordinate

preserving changes in

words,

expense).

=

X4

+ ix

x

4

5,

condition

x1

The coordinate

coordinates

V

the

the brane

and

x

5 ,

of minimum

configuration can

which

that

be assumed to

can

be combined

(4.49)

Strings

Fields, normal

the

representing

of the

the

to

position fourbranes,

of the away from the position dimensional a two laplacian,

139

fourbranes.

for

x

6

reduces

Far now

(4.50)

0,

=

to

solution

with

X6 (V) for

transversal

the equation

V2X6(V)

and Branes

As

tensions.

(4.51)

+ a,

depend

k and a, that will we can see from

constants

some

brane

JvJ

kln

=

(4.51),

on

the

the

and Dirichlet

solitonic

of x6 will

value

diverge

at

with the type 1113 case, a first as a difference problem constitutes, of equation for the interpretation (4.45). In fact, in deriving (4.45) we have used a standard Kaluza-Klein coupling argument, where the four dimensional the x 6 is defined by the volume of the internal constant space (in this ocasion,

infinity.

This

interval

between the two solitonic

will

deform

would

space at

v

four

equal

to

brane

is

effective

and

since

these

(4.51),

already with

get,

6

by

coordinate

case

four

define

of the

coordinate

disturbing in

1113

case.

can

not

divergent.

effect

x

6

of the

definition

the

of the

However, equations right picture,

be the Let

From

branes

internal

the

us

consider

then

(4.45)

equations

and

v,

2kN, ln(v)

have differentiated

Equation (4.52) loop renormalization

(4-52)

2

94 we

this

are

2

where

the

fourbranes.

I

fivebrane.

values

the

was

that

us

way to

where the

in the type

N, transversal

large

for

x

Since the Dirichlet

natural

region as

indicate

of the

values

the

is

vanishing, coupling

likely

dimensional

(4.51)

the

defined

interval

which

very

configuration we

the

as

infinity,

three

(4.50) a

be

fivebranes).

fivebrane,

solitonic

the

95

the

direction

have

in

which

the

if

the

pull

fourbranes

as meaning very for effective the conequation coupling group let us first this interpretation, In order to justify stant. analyze the physical meaning of the parameter k. From equation (4.51), we notice that if we move is located in v around a value where a fourbrane (that we are assuming is v 0), we get the monodromy transformation

the

one

can

nice

a

we

interpret

it

=

X6 This

_+ X

6+

(4.53)

27rik.

where we add an equation can be easily understood in M-theory, x'O, that we use to define the complex coordinate dimension,

extra

eleventh

X

Now, using radius

R

of view, now

in

the

we

we

terms

fact

have of

that

from

can,

a a

the

extra

6

k identify interpre 'tation

similar in

the

(4-54)

coordinate

(4.53),

change

10

+iX

theta

with

is

compactified

R. Rom a field

of the

parameter.

on

a

circle

theory

monodromy of (4.52), Let

us

then

consider

of

point but the

C6sar G6mez and Rafael

140

loop

one

renormalization

without

gauge theories

Hernindez

SU(N,)

for

equation

group

47r

924 (U)

2N

2

In

4,7r

90

G) A

(4.55)

dynamically generated scale, and go the constant can be absorved coupling through from A scale A, we get to a new going

A the

4,7r-

_2

Thus,

once

we

coupling

fix

a

-

n

reference

scale

constant

2N,

( A'

u

In

-

4,7r

94

bare

2N,

4,7r _

2

-

Ao,

4-7r

the

In

coupling constant. change in A; in fact,

bare

The bare when

supersymmetric

hypermultiplets, 47r

with

N= 2

a

( A')

dependence

(4-56)

A

on

the scale

A of the

given by

is

2N,

T

( X0_ A

In

(4-57)

the dependence on A of the bare coupling to distinguish In the brane dependence on u of the effective coupling. approach, the coupling constant defined by (4.52) is the bare configuration brane configcoupling constant of the theory, as determined by the definite uration. Hence, it is (4.57) that we should compare with (4.52); naturally, units and scales. Once we interpret some care is needed concerning k as the S' of M-theory we can, in order to make contact with radius of the internal g25 with the radius of S', which in M-theory units is given by (4.57), identify It

is

important

constant,

and the

R= with

g the

(4.45)

string

should

coupling

and

constant,

be modified

(4-58)

gl, 1, the string

length,

IT. Therefore, Ci

to

X62 2

-

X6 1-

gl,

94

-

-2N, In(v),

(4.59)

v in Then, we should interpret (4.59) as a with R as -E, or, equivalently, playing the role of natR !ural unit of the theory. becomes the Then, comparing (4.57) and (4.59), R A in the formula for the bare coupling scale constant. In summary, v fixes Ao the scale of the theory. From the previous interan equivalent discussion, where fixes R and therefore in the scale are follows, pretation Ao, changes to changes in the radius of the internal S1. equivalent Defining now an adimensional complex variable,

which

should

dimensionless

be dimensionless.

variable

8

and

a

complexified

coupling

( C6

constant,

+

ix1o)1R,

(4-60)

Strings

Fields, 0

4ri

7 we can

(4.59)

generalize

and Branes

+

141

(4-61)

27r'

to

i7a(V)

=

S2(V)

SI(V))

-

(4-62)

of branes defining a pure gauge theory. simple configuration Now, we how the notice clearly monodromy, as we move around v 0, means a change 0 -4 +21rN,. Let us now come back, for a moment, to the bad behaviour of x'(v) values of v. A possible at large to this solve is problem way modifying the of of with a fourbranes configuration single pair N, fivebranes, extending between them, to consider set of solitonic fivebranes. a larger this Labelling fivebranes with the a coordinate will by a, 0, corresponding n, x. depend on v as follows: for

the

can

=

=

qL

X6(v),

=

qR

REInIv

-ail

REln1v

-

i=1

-bjl,

(4-63)

j=1

where qL and qR represent, the number of D-4branes to the left respectively, and right As is clear from (4-63), of the a" fivebrane. at a good behaviour

large

only

possible if the numbers of fourbranes to the right and left equal, qL qR, which somehow mounts to compensating the perturbation created by the fourbranes at the sides of a fivebrane. The four dimensional field brane array will have now by this theory represented a gauge group fourbranes fl, U(k,), where k,, is the number of transversal 1 and ath solitonic between the a fivebranes. of the Now, minimization worldvolume action will require into account the dependence not only taking Of X6 on v, but also the fourbrane on the positions NS-5brane, represented four dimensional on the worldvolume coordinates by ai and bj in (4.63), XO, X1, X2 and x'. Using (4.63), and the Nambu-Goto action for the solitonic of

a

v

will

fivebrane

be

are

=:

-

fivebrane,

f

we

for

get,

the kinetic

energy,

3

E,91,X6(v,

d4xd 2V

bj(X1i)),9j'X6(v,

ai(x '),

Convergence

of the

v

integration

bj(x")).

(4-64)

implies

a,, or,

ai(x"),

jL=0

ai

bj)

-

0

(4-65)

constant.

(4-66)

=

equivalently, ai

This

position

"constant between

of motion" left

and

right

-

is

bj showing

fourbranes

how the

average

must be hold

of the

constant.

relative Since

the

C6sar G6mez and Rafael

142

configurations

the

dimension

of all

rl,,

we

consider

they

direction

reduce

the group

x

4

SU(k,)

in this

different

sides

to the

and arises.

at both

of

sides

U(1),

x

gauge of a

left

hypermultiplet

the gauge groups as

same

and

that

the

fourbranes

to

force

the

theory

result

right

on

can

of

center

right,

the of

center

mass

describing

we are

be derived

of the first

D-

the

to

if

we

soli-

and last

infinitely

are

rl,,

to

of the fourbranes massless

The

to the left

discussion

and x1 directions.

Hypermultiplets

a

U(k,).

implies

field

the

reduce

massive, we can assume that they do An important will difference appear of of periodic configurations fivebranes, upon compactification in this case, constraint to a circle: (4.66) is now only able to as

the x'

on

that

differ-

will

be associated

can

will

constraint

means

with

(4.66)

general

our

semi-infinite

no

this

now,

which

fourbranes

the

in

fourbranes

With

insteadofH,,,

fivebranes: move

know from

we

be associated

constraint

gauge group Constraint (4.66)

mass.

0;

=

semi-infinite

tonic not

ai

vanish,

to

SU(k,,),

include

if

F_j

sectors

of

will

U(k,)

of the

each sector.

in

have that

we

As

space.

center

is frozen

mass

is

of this

of the

motion

theory fourbranes,

gauge

of the transversal

U(1) part

the

branes,

H,,, U(k,,)

of the

Coulomb branch ent

Hernindez

leaving

alive

a

U(1)

factor.

as strings theory connecting whenever the positions fivebrane; therefore, brane become coincident, right of a solitonic As the hypermultiplets under are charged will a certain transform a+ I fivebrane, they

understood

are

(k, k,,,+,). However,

brane

varies

the

as

as

a

of

can

only

hypermultiplet

fined

variation

rates

01, (Ei ai,,,) naturally

=

on

of

position

function both

x

sides

01, (Ej aj,,+,).

from constraint

=

X

,

be

1

X

,

2

and

x

the

both

on

3 ,

accomplished

the

as

follows bare

to

of

sides

existence

thanks

same, The definition of the are

of

the

again massses

a

fact from

a

five-

well

de-

that

its

(4.65):

comes

then

(4.66): 1

m,,

fourbranes

the

0

1

1:

-

k,,

ai,,,,

-

aj,,,+,.

-

k,+,

(4.67)

the constraint interpretation, (4.66) becomes very natural from view: it of that the of the hypermultiplets states do masses physical point not depend on the spacetime position. of the previous The consistency of hypermultiplets definition be can checked using the previous of the one-loop construction beta function. In fact, from equation (4.62), we get, for large values of v, With this

a

-

The number k,,,

N,.

ir,

(v)

of branes

Comparing theory with Nf flavors,

gauge

with

the

=

(2k,,

in the

-

all

function

beta

we

Nf

=

k,_1 is,

k,_1

+

k,,+,)

as we

for

conclude

-

N

Inv.

(4.68)

know, the number of colours, 2 supersymmetric SU(N,)

=

that

k,,+,,

(4.69)

Strings

Fields, so

that

the number of fourbranes

fivebranes, k,+, pair Notice, from (4.67), of

fourbranes

with

implies

a

at

global

us now come

(hypemultiplets) Nf, becomes a

back to equation

both

at

hypermultiplets

the

fivebrane

global explains

flavor

(4.59).

of

sides

a

143

certain

the number of flavors.

solitonic

This This

sector.

of all

mass

of

sides

symmetry.

symmetry of the adjacent Let

=_

the

that

both

flavor

k,,-l

+

and Branes

the

What

symmetry

same.

is the

This gauge

meaning of (4.67).

physical we

associated

the

are

need in order

to unravel

four dimensional coupling dynamics of our effective gauge theory is the u dependence of the effective dependence that will coupling constant, It is from this dependence effects due to instantons. contain non perturbative the Seiberg-Witten that we read geometry of the quantum moduli space. Strong coupling effects correspond to u in the infrared region, i. e., small u or, large A. From our previous discussion of (4.59), we conclude that equivalently, R -4 0, to the type IIA string the weak coupling limit, regime corresponds and the strong coupling regime to the M-theory reime, at large values of that R (recall theory correspond changes of scale in the four dimensional of the internal to changes of the radius S'). This explains our hopes that M-theory could describe the strong coupling regime of the four dimensional is effectively We will then see now how M-theory working. theory).

the

strong

From the M-theory point of Coupling.. in a can be interpreted are considering four the define the D-4branes to In different are we using particular, way. dimensional macroscopic gauge theory can be considered as fivebraries wrapS'. Moreover, the trick we have used to make ping the eleven dimensional if we conobtained in the x' direction these fourbranes finite can be directly R4 x X, where R' is parametrized with worldvolume sider fivebranes by the 2 0 1 and embedded in the and x 4 and Z is two dimensional, coordinates X X x four dimesional x4, x5, x6 and x1o. If we think in purely space of coordinates with the topolclassical terms, the natural guess for Z would be a cylinder k of D-4branes for a extending along the x' configuration ogy S1 X [X6,2 X6], 1 4.4.1

M-Theory

view,

the

brane

and

we

,

,

,

between two solitonic

direction

because there

pactification, around

Strong

configuration

this

surface

will

fivebranes. is

no reason

produce,

on

the

This

is however

to believe

four

that

dimensional

a

a

very

fivebrane

naive

com-

wrapped

worldvolume

R4,

fact, any gauge field on R4 should tensor field the chiral come from integrating P of the M-theory antisymmetric in of Z. If we wnat to reproduce, fivebrane on some one-cycle worldvolume, better of U(k) or SU(k) gauge theory, four dimensions, we should some kind first Z with a richer consider a surface homology group. However, we can the explicit dependence of the X6 try to do something better when including coordinate on v. In this get a picture that is closer to the right case, we will far away from the true solution. Including the v dependence answer, but still leads to a family of surfaces, of the X6 coordinate parametrized by v, Z, deabout this picture is that v, which fined by S' X [X6,2 A]1 (v). The nice feature of Z in the space Q, defined by the coordinates coordinate is the transverse any form

of

non

abelian

gauge group.

In

C6sar G6mez and Rafael

144

x

4, x5,

x1o,

x6and

is that

similar

now

problem of the of the

have yet the

homology

becomes

Herndndez

of Z. The

group

trying

we are

genus or, for following

reason

keep alive

to

the

moduli

the

to

in

the

of

previous

interpretation

however,

Z,;

general

more

terms, line

of the

v

of

we

the first

thought,

coordinate

as

of the Coulomb branch. This is, in fact, the reason moduli, or coordinate with the genus, as we are using just one complex giving rise to the difficulties of the rank of the gauge group, something we are coordinate, independently forced to do because of the divergences in equation (4.51). The right In fact, to we must try M-theory approach is quite different. from the particular brane configuration we are working get Z directly with, and define the Colomb branch of the theory by the moduli space of brane Let us then define the single valued coordinate configurations. t, t

and define

the surface

Z

=-

looking

we are

for

F(t, v) From the classical totic

for

large

very

=

small

very

(4.72)

Conditions

assymptotic function

for

two

(4-72) k

(4-73)

imply

different

roots

(4-73)

F(t, v)

that

fixed

for

have, for fixed

will It

v.

must

values

be stressed

that

of

the

(4.72) and (4.73) corresponds to the one loop beta theory with gauge group SU(k), and without hypermulthe previous conditions will be of the generic satisfying

behaviour

field

a

A function

tiplets.

know the assymp-

t,

and

while

we

k

t-v-

t, k roots,

(4-71)

0.

t, t-V

and for

through

of the fivebrane

of motion

equations

behaviour

(4.70)

exp, -s,

type

F(t,v) with

A,

B and C

(4.74)

function

polynomials

t to terms

these

one

undetermined

t/constant.

v)

=

constant.

The

of of the

equations

+

B(v)t

degree

in

v

of

t2

+

B(v)t

one can

the

In order

of this

meaning loop beta function, be read

+

k.

C(v), From

(4.74)

(4.72)

and

(4.73),

the

rescaling

+

to kill

rescaling written

(4-75)

constant, this

constant,

we can

can

be

understood

as

(4.72)

easily and

(4.73).

rescale In

in

fact,

as

s

and therefore

A(v)t2

becomes F (t,

with

=

=

-kln

of R goes like

(V)

(4.76)

Strings

Fields, R'

v

(4-77)

equivalently,

or,

t( ) R'

t -4

and based

Thus, observe N

of the

2 pure

=

the

on

constant,

theories,

gauge

B(v) we

(4.75)

in

=

k

v

k-2

of the

SU(k).

k-3

+ U3V

If

scale,

of the

theory. Seiberg-Witten

get the

gauge group

+ U2V

definition

the

the scale

B(v)

this

solution

for

is chosen to be

(4.79)

+ Uk)

+

we

With

get the Riemann surface

finally

P Riemann surface

group.

(4.78)

.

on

defines

we can

with

k

R

above discussion

the constant

that

interpretation

a

145

(R' R)

-k In

s -+

and Branes

Moreover,

of genus we can

k

+ I

=

1, which

-

try

now

of the fivebrane

worldvolume

B(v)t

+

(4-80)

0,

is

describing

fact

in

visualize

to

this

original

our

the

rank

of the

Riemann surface brane

gauge

the

as

configuration:

each

fourbranes and the transversal to P', cna be v-plane can be compactified with k 1 a surface as gluing tubes, which clearly interpreted represents This image corresponds handles. with k disjoint to gluing two copies of P', 2k branch points. cuts on each copy or, equivalently, Thus, as can be observed D-4brane there correspond two branch points from (4.80), to each transversal -

and

one

If we

cut

should

P'.

on

we are

in SU(k) gauge theories with hypermultiplets, relations, (4.72) and (4.73) by the corresponding

interested

first

replace

t

(4-81)

V

-

then

and t

for

large

t

and

curve

respectively.

small,

the beta functions

for

these

with

(4.82)

These are, in If we take

theories.

fact, k,,

the relations =

we

0, and Nf

=

get from

k,+,,

the

becomes t

the

V- k-k.+l

-

C(v) a polynomial hypermultiplets,

in

2+ B(v)t v,

of

+

C(v)

degree Nf,

=

(4-83)

0,

parametrized

by

the

masses

of

Nf

C(V)

f

11 (V

-

Mj),

(4-84)

j=1

f a complex constant. Summarizing, we have been able to find a moduli N 2 supersymmetric four dimensional reproducing with

=

of brane

configurations

SU(k)

gauge theories.

C6sar G6mez and Rafael

146

HernAndez

solution is obtained of the worldSeiberg-Witten by reduction dynamics on the surface Z,.,, defined at (4.80) and (4.82). 4 reducing the fivebrane dynamics to R on Z,, leads to an effective

The exact

volume fivebrane

Obviously, coupling

finishing

Before

this

the brane

construction.

the

Z,

curve

compactified

R4,

in

constant

in terms x

4

k

the

-

section,

it

First

all,

of

I

should

it

to stress

some

that

be noticed

configuration, This

-r(u)

1 Riemann matrix

-

important

is

of the brane

and x1 directions.

k

x

requires of the brane

is part

of Z,

of

peculiarities

the definition

working philosophy,

with

of un-

where

A different in flat particular configuration spacetime. approach will consist in directly working with a spacetime Q x R1, with Q and consider worldvolume Z x R4, a fivebrane some Calabi-Yau manifold, with R4 C R', and Z a lagrangian submanifold of Q. Again, by Mc Lean's theorem, the N 2 theory defined on R' will have a Coulomb branch with Betti number of Z, and these deformations dimension equal to the first of Z in the four dimensional scalar fields in Q will represent theory. Moreover, the holomorphic top form 0 of Q will define the meromorphic A of the SeibergIf we start with some Calabi-Yau solution. Witten manifold Q, we should provide some data to determine Z (this is what we did in the brane case, with Q non compact and flat. Z directly we want to select If, on the contrary, from Q, we can only do it in some definite to cases, which are those related the geometric mirror construction [118, 119]. Let us then recall some facts The data are mirror. about the geometric with

start

must

we

a

=

-

-

-

The Calabi-Yau A

lagrangian

A

U(1)

flat

is

the

manifold

theorem

condition

Secondly, mension

of

on

9.

This

of data

family with

D-2brane.

of the In

moduli

points

we

of abelian N

=

this

2

fiber

family

space

the

jacobian

varieties

particular cases, of Q. This will mirror

this

with

is

we

--+

frist

use

Mc

Q, preserving

of dimension

is

a

varieties

above,

moduli

Z

bl(Z).

of Z, which is of dithe quantum moduli of

defines

of abelian

points

of

This

supersymmetry,

second and third

some

Namely,

solution.

of deformations

submanifold.

lagrangian

each of these

gauge

set

Seiberg-Witten

of the

get the

to

theory, equal bl(Z). Moreover,

a

Z -4Q.

Z.

on

crucial

a

of abelian Lean's

bundle

is equivalent Z as a D-brane in Q. to interpreting requirement data, in order to get from the above points the structure

The third

This

Q.

manifold

submanifold

i.

e.,

gauge group of rank of the is the moduli the

Q itself

moduli or,

of Z

as

properly, equal one,

a

more

be the case for Z of genus i. geometric In the this the in characterization of Z case. is Q simple SU(2) e., cases, fibration. The relation to describing between geoQ as an elliptic equivalent and T-duality metric mirror produces a completely different physical picture. In fact, when Z is a torus, consider in type IIB a threebrane with we can, classical moduli given by Q. After T-duality or mirror, we get the type IIA In summary, it is an important in terms of a fivebrane. description problem the

for

Strings

Fields, understand

to

string strings.

In order

study

the

in

theories

with

IIA

type

description

coupling

Dimensional

147

and type IIA

of type

Field

Theories.

N = I supersymmetry, the first thing recall the way R-symmetries us then

Let

dimensional

of four

case

between

mirror

strong

R-symmetry.

be

will

defined

were

M-theory

of N = 1 Four

field

to consider

will

we

of quantum

relation and the

Description

Brane

4.5

the

theory,

IIB

and Branes

N

2

=

and

supersymmetry,

through compactification theories. The U(I)R in four supersymmetric gauge field the euclidean or SO(3)R in three are simply dimensions, dimensions, group of rotations in two and three dimensions, respectively. Now, we have a four dimensional t and v, and a Riemann by coordinates space Q, parametrized surface To characterize Z, embedded in Q by equations of the type (4.74). transformations transform we can consider on Q which non R-symmetries, its holomorphic will then trivially top form Q. The unbroken. R-symmetries the Riemann surface defined by the brane conin Q preserving be rotations If we consider behaviour of type (4.72), or only the assymptotic figuration. (4.81), we get U(1)R symmetries of type three

N

dimensional

dimensional'N

4 supersymmetry,

=

of six

I

=

t

Akt,

v

Av.

(4-85)

U(1) symmetry is clearly broken by the curve (4.80). of the U(I)R symmetry is well understood in

This

This

breakdown

the

containing invariant

under

Let

us

now

dimensional locate

effect.

induced

instanton

x

volume 6

suspend the positions now

1

2

set

v

=

and X3

,X

is

macroscopically

,x

,X

1

,X

6

,

4

+iX

5,

and

The effective a

four

now we

be defined

on

x

8,

and

w

x

=

this

on

7

+

have

only

N

x

4

=

iX8,

gauge

2

the

N

=

2

as

I

is

curve

=

=

At

=

X

5

time

=

They on

definite

some

x'

before, parametrized

be

are,

in

as

by

theory,

with

the

previous

worldvolume

by

cases,

of fourbranes

set

coupling

constant

6 0

supersymmetry,

we

fivebranes.

the two solitonic

defined

X6,

As

0.

=

will

Of

value

worldvolume

with

(4-86)

g1s

dimensional

an

space

I four reproduces N IIA type string theory, and x' 0 with, worldas usual,

which

D-4branes

theory

field

take

in

but this

in between.

dimensional

the four

that

=

fivebrane,

9

can

X8

=

X

Moreover,

we see

theory the larger

space.

and x1.

4

X,x

of k D-4branes x

X7

=

3

and

coordinates .

x

2

solitonic

X1,X 2,X3,X7

The worldvolume 0

at

another

a

X

0

we

configuration Wewill again start

[117]. x

9coordinates, 7 (x ,X 8 ,x9)

Q,

brane

a

fivebrane

coordinates

x

considering

of

in the

consider

theories

say xO ,we locate coordinates x 0,

x8 and

7

rotations

solitonic

a

If instead

spontaneous

field

as

(x

0

no

,X

1

massless ,X

2 ,

X3).

bosons In

fact,

Usar

148 at

v

x'

the line

and x1

G6mez and Rafael

0,

=

0 and x'

=

Notice

that

would be v, since

the

same

the

in

argument,

case

hand,

other

we

0

=

by

that

only possible massless scalar project out ? and w. On the we have projected and, therefore,

0 the

=

so

Hernindez

all

out

at

X60

w

we

0

scalars.

massless

of two solitonic

=

have

fivebranes

X8 x9 but at x1 0, we have one 2 complex massless scalar that is not projected out, which leads to N The previous discussion means that w supersymmetry in four dimensions. v, and x9 are projected scalar fields; out as four dimensional however, w and v moduli parameters classical of the brane configuration. are still each of Now, we return to a comment already done in previous section: in between the solitonic the fourbranes fivebranes we are suspending can bd as a fivebrane interpreted wrapped around a surface defined by the eleven S' of M-theory, dimensional multiplied by the segment [0,A].0 Classically, the four dimensional reduction of theory can be defined through dimensional worldvolume the fivebrane Z. The coupling surface will on the be constant given by the moduli T of this surface, located

different

at

values

of

x

6

=

=

=

=

27rR

I 2

9

length

S the

with

field

symmetric case,

we

of the

have not

a

[0, X6], 0

interval

theories,

the

on

classical

(4.87)

S

M-theory

in

and, therefore,

moduli

units.

of what takes

contrary

In N

place

we can

=

define

not

I super-

the

in

N a

=

2

wilso-

constant coupling depending on some mass scale fixed by a vacuum value. This fact can produce some problems, into once we take expectation 6 account the classical this dependependence of x on v and w. In principle, dence should be the same as that in the case studied in previous section,

nian

*6 *6

Using

for

defined

the t coordinate

large

we can

and small use

these

kInw.

-

(4.70),

in

t

k W

become

(4.89)

equivalently, Taking into

or,

(4.88)

k

V

(4.86).

in

(4.88)

equations

t

t, respectively, relations

kInv,

-

t

-

v

account

k'

t-1 the

_

Wk Now, .

units,

we can

write 9

with

k

with

the

_=

Yang-Mills

N,.

As

loop theory,

one

we

did

beta

2

in

-

the

function

A

V

NJIn N

for

-

R

+ In

W

-],

2 case,

=

N

=

I

we can

ftexp

try

to

supersymmetric

87r2 =

(4.90)

R

-

3N,g(1t)2

compare

SU(N,)

(4.90) pure

(4.91)

Strings

Fields, In order

to

get the scale from (4.90)

we

V

with

(

some

with

constant

I

N, In

-

In order

to

make contact

(4.91)

with

(2=

we

1

have used

associated

to four

R

in order

to

(4.92) Using (4.92)

we

impose

(AR)3,

t

V

(kt-1

(4.94)

A. Using (4.92), theory,

measure

get

(4.93)

must

N = I field

dimensional

(4.90)

and

R2

we

R where

1

(length)2.

of

units

149

impose

(W-

=

and Branes

we

get the

curve

k

k W

(4.95)

(W-

V

set of brane only depend on (k The different k are given by values of (, with fixed ( These N, roots the N, different vacua parametrize predicted by tr (_I)F It is important the coupling to observe that constant arguments. jIT we are We can interpret it as a complex defining is the so called wilsonian coupling. 0 number with Im -.L Hence, the value Of IM (k fixes the 0 parameter 87r2 of the four dimensional theory. defines For a given value of C, (4.95) of genus zero, a Riemann surface

The

by (4.95) compatible

defined

curve

configurations

will

.

(4.95)

with

.

=

.

-

L e.,

a

rational

curve.

We will

coordinates.

This next

is

curve

observe

now

that

embedded in the

these

curves,

space

(4.95),

are

of

(t,v,w)

the

result

corre"rotating" solution, [106] the rational curves in the Seiberg-Witten sponding to the singular points. However, before doing that let us comment As mentioned above, in order to define an R-symmetry on U(I)R symmetries. we need a transformation on variables (t, v, w) not preserving the holomorphic top form,

of

S?

A rotation

in the

defining

and

an

--

dv A dw A

w-plane, compatible R-symmetry, is V

t W

Now,

(4.95). the

it

is

clear

More

curve

that

interesting

(4.95):

this

symmetry is

an

exact

with

dt t

(4.96)

R.

the

assymptotic

conditions

(4.89),

V,

t,

e27ri/k is

(4.97)

W .

broken

spontaneously

U(1) symmetry,

that

can

by the

curve

be defined

for

G6mez and Rafael

Usar

150

HernAndez

e'J

V

t

e

w

As

be

can

seen

from

(4.96),

with

respect

charged

Fields

momentum in the

(i.

interval

charges, SQCDdegrees fields

out

in

all

fields

Rotation

4.5.1

any of these

the

A different

of Branes..

angular

SQCDdo not be decoupled

=

1

should

on

carry

dimension

equivalent

is

S? is invariant.

since

should

momentum in the eleventh

of N

U(1) charge This

R-symmetry,

an

linear

previous discussion field theory.

dimensional

four

or

(4.98)

w.

U(1) symmetry

this

The fields

with

of freedom.

the

i6kt,

-'d

is not

to

plane,

w

branes)

zero

e.,

so

v or

this

e

v,

the

to

definition

from

the

N

=

1

have projected way of the effective N 1 we

=

present

way to

carry

the above construc-

of branes. Wewill a rotation now concentrate on this by performing of NS-5branes with worldvolumes The classical exconfiguration procedure. 3 2 0 1 4 5 tending along x x x x x and x can be modified to a configuration 4 5 fivebranes from the v has been rotated, where one of the solitonic x + ix in contained the be also it to so that, (x 7, x')-plane, by moving direction, in the (x 4 X5 X7 X') space. Using the same a finite angle y, it is localized the brane configuration, where a fivebrane notation as in previous section, the rotation is has been moved to give rise to an angle [i in the (v, w)-plane, to impossing equivalent tion

is

,

,

,

,

,

=

,

,

,

(4.99)

/-tV.

W=

the brane configuration we obtain, points on the rotated in the (x 4,X 5,X 7,X 8) by the (v, w) coordinates parametrized conditions therefore imposse the following assymptotic [116]: In

t t

respectively

(X 6,X10,X i.

e.,

for

large

describes

uration

for

V

=

V-

=

and small

k

t.

Let

surface,

W=

k

are

We can

pv'

W=

,

fivebrane space.

(4.100)

0,

us now assume

that

this

',

brane

config-

embedded in the space 4,X 5, X7,X 8) , and let us denote by Z the surface in the N = 2 case, the graph of the function is simply jL = 0. In these conditions, a

Riemann

(4.99) as telling us that w on Z posseses a simthe rest of the Riemann over extending holomorphically If we imposse this condition, the projected surface. surface we get that Z, i. of N 2 the In is theory, fact, it is a well genus zero. e., the one describing in the theory of Riemann surfaces that the order of the pole known result at infinity depends on the genus of the surface in such a way that for genus be forced to replace than we will 4va for some larger (4.99) by w zero, A is there the in trying to on no problem priori, genus. power a depending of modification instead the of rotate w pole using, type Mv, some higher from W pva, for a > 1. This would provide Z surfaces with genus different w on

Z.

ple pole

We can

at

interpret

infinity,

=

=

=

=:

Fields,

Strings

and Branes

151

immediately find problems with equation (4.90), and constant all dependence of the coupling on v and w. be rotated the that that conclude to produce curves can we only Therefore, N I theory are those with zero genus. This is in perfect dimensional a four solution. agreement with the physical picture we get from the Seiberg-Witten of add soft the the term a once we /_ttr02' type Namely, only points breaking the the moduli of real the in as vacua are theory singular remaining space where the Seiberg-Witten curve points, degenerates. however,

zero; we

will

we

would to kill

be unable

=

and Scales.. In all our previous discussion we have QCDStrings enough in separating arguments related to complex or holoand those related The M-theory deto Kdhler structure. morphic structure, however relevant information For instance, contains on both aspects. scription of curves, we were mostly interested derivation in our previous in reproducing of the Seiberg-Witten for instance, the complex structure the as is, solution, wilsomoduli dependence on vacuum expectation values, i. e., the effective nian coupling constant. However, we can also ask ourselves on BPS masses embedding of Z in the ambient and, in that case, we will need the definite top form defined on Q. As is clear from the space Q, and the holomorphic the holomorphic fact that we are working in M-theory, top form on Q will and we will on R, i. depend explicitely coupling constant, e., on the string that will depend explicitely find BPS mass formulas therefore on R. We, will in the case of N 1. supersymdiscuss this type of dependence on R first field theory we have described I four dimensional contains, metry. The N One is the constant in equation in principle, two parameters. C introduced (4.92) which, as we have already mentioned, is, because of (4-90), intimately S'. Our first connected with A, and the radius R of the eleven dimensional is task would be to see what kind of four dimensional dynamics dependent value of R, and in what way. The best example we can on the particular of course use is the computation of gaugino-gaugino condensates. In order four should minimize dimensional for to do that, to a we try suerpotential this will define I theory. the N W we Following Witten, superpotential function of Z, and with critical the when as an holomorphic points precisely Z is a holomorphic in Q. The space Q now is the one with surface curve 4.5.2

not

been careful

=

=

=

coordinates

x

4 ,

X

5 ,

X

6 ,

7 X

,

X

8

and x'O

used to prove that rotated Moreover, we need to work with the

persymmetry.

superpotential: a functional

a

that

this

second

condition

was

of genus equal zero) necesarily I subecause of N curve holomorphic are

there are two different about priori, ways we can think maybe the simplest one, from a physical point of view, defined volume is given by on the volume of Z, where this

J the Kdhler

class

=

of Q. The other

W (Z)

-

=

A

Vol(Z) with

(notice

curves

one

=

fB

J?,

as

(4.101)

J.Z'

posibility

this is

is

defining

(4.102)

C6sar G6mez and Rafael

152

with

B

such that

3-surface

a

Definition when Z is

Hern6ndez

Z

OB,

=

(4.102) automatically curve a holomorphic

that

terms,

H1,1 (Q)

in

that

the

Z is

an

H2 (Q).

n

top form in

of

being stationary, holomorphy condition

condition

the

Q. Notice

in

Z means, in mathematical of Q, i. e., an element lattice

on

holomorphic

and S? the

satisfies

element This

of the what

is

Picard

allows

us

temporarily abandoning (4.101), to Wis being stationary What we require the approach based on (4.101). for arbitrary be defined in principle, but it should, for holomorphic curves, surfaces Z, even those which are not part of the Picard group. Equation i. e., if the homology class of (4.102) is only well defined if Z is contractible, to

be

W(Z) where

OB

now

physical

From

=

multiplied Z.

into

(4.95),

Zo

0',

and

if

W(Z)

==

we

simplicity,

fB

S?. Let

t

r

V

r,

we

ePe io ,we

r

=

W _+

I

the

perform

for

e27ri/k

can

and

v

-+

ZO

0.

number of transverwe can

write

complex plane us first map the complex plane complex plane, Z,- as given by take

B

the

as

(4.104)

k

r

V

=

f (p)r,

W

=

(P-P)r-

,

=

a

=

superpo-

as

=

0 for

(4.105) p < 1.

symmetry of

of the

reparametrization

k the

H3 (Q; Z) of the

zeroes

k

t

v, is

assume

of

(r-1.

2, and f(p)

p > w

define

will

be Zk invariant,

then

us

=

f(p) t'

with

ZO needs

(4-103)

set

to

I [0, 1], and let by an interval on this Denoting r the coordinate is defined by

Writing

t _4

we

the

ZO

choose

surface

reference

a

fB 9,

=

know that

w

with

case,

by Zk symmetry,

Therefore,

for

reason

W(ZO)

-

For

-

we

be related

fourbranes.

W(Zo)

Z U

=

arguments

should

tential sal

the

is

If that is not the H2 (Q; Z) is trivial. defined, and (4.102) is modified to

Z in to

and this

however,

use

The Zk transformation

(4.105)

if,

at the

same

time,

r-plane

P _* P,

0

with

b(p)

entering

=

the

0 for

p

definition

!

-4

0 +

- -' for 1, and b(p) k of B, is given by =

(4.106)

b(p), p

:5 -1.

Thus, the 3-manifold

Fields, t

g (P,

such that

for

0

a

have g

we

0')

153

r,

U)r-1,

(g(-P,

W

and Branes

k

r

V

Strings

1, and for

o,

=

(4.107) 1,

we

g(p)

get

f (p).

Now,

with 0

get

we

W(Z)

fB

kR

The

dependence on R is already clear pendence on C we need to use (4-107),

W(Z) for

g

=

kR(

=

g(p, o-).

f

Thus

dt

Rdv A dw A

dv A dw A

dr

(4.109)

r

(4.109).

from

( 9g+

dadOdp

(4.108)

t

9gOP

OU

In

9g+ 9gap 9a

order

)

get the de-

to

(4.110)

I

get

we

W(Z) Notice that the superpotential (4.111) sponds to the volume of a 3-manifold. gaugino-gaugino condensate, we need this multiplying by -17,,; thus, we get R <

AA >-

kR ,

-

(4.111)

given

is

In

to obtain

kR(

in

order

R6

units to

(length)',

as

make contact

(length)

-3

units.

corre-

with

the

We can do

A',

(4.112)

equation (4.108). A different way to connect C with the QCDstring and computing M-theory context, its of C independent Witten, we will then try an interpretation of (4.90), by computing in terms of C the tension of the QCDstring. Wewill then, to define the tension, consider the QCDstring as a membrane, product of a string in R4, and a string living in Q. Let us then denote by C a curve in Q, and assume that C ends on Z in such a way that a membrane wrapped on C defines in R4 ". Moreover, a string we can simply think of C as a closed in Q, going around the eleven dimensional curve S', where

we

have

used

in defining, tension. Following

A is

the

t

ilk

to

V

(V

W

This it

will 18

curve

is

trivial

a non

produce

Notice

that

option

to

an

if

wrap

element

ordinary

we were a

type

working

threebrane

exp(-27rior),

to

in

IIA in

around

(4.113) H, (Q; Z), and a membrane wrapped on string; however, we can not think that

type 1IB string theory, Z, in order to define

we a

would

string

on

have the 4

R

Herndndez

C6sar G6mez and Rafael

154

R'

S',

H, (Q; Z)

Z, and 1-cycles in only candidates for non trivial homology, Q. However, we can define QCDstrings using cycles in the relative non trivial cycles ending on the surface Z. To H, (Q1 Z; Z), i. e., considering compute H, (Q1 Z; Z), we can use the exact sequence is

QCDstring

the

of type

curves

a

type IIA string.

(4.113)

implies

The map

determined

is

t

very

by the

map

defining

Z)

be defined

can

tIlk0

one

to,

V

=

to

e

(v-

I

(4.115)

-

v

=

k)

,

and thus

we can

=

minimum is obtained

(4.116)

-

27rio-/k

(4.117)

(4.117),

of

Using the

( should

Ilk

The tension

of the k roots.

pendent is given by

string

(4.114)

follows:

as

=

of R, because t is fixed.

and its

(t

Z

Zk

t

W

with

=

likely,

HI (Q1 Z; Z)

in

curve

then

(Z; Z)

HI (Q; Z) ItH,

=

HI (QIZ; A

x

HI (Q; Z) -4 H, (Q1 Z; Z),

-+

H, (Q/ Z; Z)

that,

conclude

=

be the

will

HI (Z; Z) which

Q

If

metric

on

be

(4.117) (4.118)

I

n2

when

of

1/2

t2/n n2

is inde-

by construction, Q, the length

t2/n

Thus, the length

QCD

of the

1(11/2

(4.119)

n

which the

has the

tension

this

identify

right

length

units,

as

C behaves units,

need to go to (length)-' with A, we get tension

we

A

equivalently,

2

QCDresults news, as they imply that the theory QCD, posseses O-brane modes, with we have not decoupled the M-theory

Thus,

consistency

Next, the

ones

before with

we

obtained

doing that the description

using we

will

to

1 WE .

to define

Then,

we are masses

A of the

of models with

These

R

of

order

are

in order

with,

working

we

not to

good match

A, and therefore

modes.

superpotential techniques

instanton

conclude

if

(4.120)

R2

requires

compare the

standard

In order

3

with

would like

(length)2. again using

1(11/2 -

n

or,

as

this

brief

review

described

M-theory.

in on

brane

N = 4 supersymmetry.

above with

However,

configurations

Strings

Fields,

and Branes

155

fourbranes

Beta Function.. with Let us come Vanishing with n + 1 solitonic with k, Dirichlet fivebranes, configurations th extending between the a pair of NS-5branes. The beta function,

derived

(4.68),

N

4.5.3

2 Models

=

back to brane in

is

2k,

-

SU(k,)

each

for in

all

to

implies

of the x6 direction

pactification

of the

mass

k,,

all

that

does not allow

we

compactify

will

the

are

same.

the

U(1)

factors

all

to eliminate

to vanish

Now,

com-

in

removed, so that the gauge group U(1) x SU(k)'. Moreover, using the definition the hypermultiplets for periodic we get, configurations,

the gauge group: one of them reduced from fl,,=, U(k,) to

(4.67)

section,

the beta function

circle

a

immediately

sectors

(4.122)

k,,-,,

+

in the gauge group. In this of radius L. Impossing

factor

the x6 direction

ka+l

+

of

can

be

not

EM,

(4.123)

0.

=

is

a

hypermultiplets

The

of

consists

Let

us

are

in

now

copy of the adjoint consider the simplest

of type

representations representation,

a

of N

case,

2

=

k 0

and

a

neutral

SU(2)

x

U(1)

k,

and therefore

singlet. four

dimensional

adjoint [103]. The corand two solitonic a single fivebrane, The mass of the hypermultiplet is clearly fourbranes. Dirichlet zero, and the A geofour dimensional theory has vanishing beta function. corresponding of the is a fibering metric procedure to define masses for the hypermultiplets in a non trivial that the fourbrane so v-plane on the x1 S' direction, way, modulo a shift in v, are identified positions with

theory, responding

one

brane

hypermultiplet configuration

in the

contains

X6

6 X

V

so

now, the

that

(4.124),

as

E,,,

From the

compactified of

S'

value

and,

mass

m,,

=

on a

circle,

S1. This space of x" is changed

x

in

addition,

x10

O-angle

can

(4-124)

+ M,

V

hypermultiplet,

of the

of

_+

now can as

is the constant

M-theory,

of radius

be made

rn

appearing

in

be defined

the

R. The

x10 coordinate

(x 6, if,

trivial

non

has also

been

x1o) space has the topology when going around x', the

follows:

X6

X6

X10

x10

X"+27rR.

of genus one, and moduli (4.125) can be understood the

27rL,

+

m.

of view

point

representation

+

+

Relations

depending as the O-angle

27rL,

(4.125)

OR,

(4-125)

define

L and 0 for

on

of the four

fixed

a

Riemann surface

values

dimensional

of R. 0 in

field

theory:

as

X10 __I

-

_

R

X10 _2

(4.126)

C6sar G6mez and Rafael

156

x1O

with

of

1226).

(4.

and x1O 1

x1O(27rL),

=

x1O(O). Using (4.125),

=

of the four

O-angle

bare

is the

This

HernAndez

92

the

massless

by

and y restricted

Y2 -r

we

have

degree

a

(x

coupling

the bare

with

=

-

el

(-r)) (x

k in v,

The moduli

z)

of

=

e-2

-

we

Vk

_

of Z are,

parameters

=

now

try

to

solve

will

we

need is

defining

Z

through

(4.128)

0,

E,

(T)) (X

-

e3

(4.129)

(7)),

by (4.126) and (4.127) [120]. In require F to be a polynomial

defined

fourbranes,

F (x, y,

z)

y,

equation

constant

of k

collection

the

us

theory

of the

constant

L and 0. Let

C, Thus, all what

F(x, x

coupling

the bare

be given by a Riemann where E is the Riemann surface defined

x

by (4.125), and C is the v-plane. of the type an equation with

(4.127)

The solution

case.

in the space E

Z, living

surface

coupling

of the bare

R

R, and changing

of

the value

as

theory.

=:

we can move

the value

fixed

model for

this

that

clear

It is therefore

get 0

27rL

I

keeping

the value

inmediately appears concerning right answer should be

A question the

constant:

we

dimensional

case

of

will

f, (X,

Y)Vk-i+ point,

this

at

(4.130) hidden

in

the

functions

point (x, y) (4.130) (x, y) spectral curve defining a, branched covering of curve in the sense of Hitchin's as a spectral E, i. e., (4.130) can be interpreted at has If some a pole point (x, y), then the same fi system [121]. integrable of These to root poles have the interpretation infinity. vi(x, y) should go In the are we fivebranes. case solitonic the of simple the position locating the Coulomb branch of the theory will with a single fivebrane, considering, a simple on E with pole at one be parametrized by meromorphic functions functions k have As fivebrane. we of the is entering the position point, which the is which will be branch the Coulomb of dimension right k, (4.130), the

fi (x, y)

in

(4.130).

for

one

a

theory

Now, after introduce will E

x

C,

but

Let

with

this

the

us

denote vi

(4.130)

that

in E. Notice

the

is

U(1)

discussion mass.

non

trivial

a

x

SU(k) gauge group. of the model with massless

The space where defined fibration

X6

X6

V

flat

equivalently, this

bundle

+

now we

hypermultiplets,

need to

define

we

Z is

not

through 21rL,

X10 +OR,

X10

or,

at the

of

the roots

-4

(4-131)

V+M

the space obtained by fibering all E, with the exception over

C non trivially of one point

on

po.

E. We can

Away

from

Fields, this

the solution

point,

given by (4.130).

is

If

Strings

(4.130)

write

we

and Branes

in

157

factorized

a

form, k

F (x, y,

z)

fj

=

(v

(x, y)),

vi

-

(4.132)

i=1

f,

write

we can

(4.130)

in

as

the

sum

k

(X, Y);

Vi

(4.133)

i=1

therefore, f, will the hypermultiplet with

this

w

have

that

this

In N

we

for

return

a

the

R7

eleventh constant.

The brane

X

fivebrane, Q, defined by

solitonic

particle

to

If

Z is

a

lagrangian

(4.135)

of !P in Lean's

theorem,

bi (Z),

in

"

Recall

four

M-theory Q

with

=

S'.

x

we

a

manifold

in R'

with

lagrangian

V such that

holomorphic

of

f1w, is zero,

we

have

a

the

the

V (w)

top form of

we

of

x

Z

Z

:

Q,

x

of N

will

string

1.25].

Let

2 four

=

the

2 and

theory then

us

dimensional

M-theory

consider for

=

on

flat

(compactified)

the

string coupling Q turns out to be equivalent to a R4, where Z is a complex curve to

0.

=

(4.134)

Q.

-4

then

N

=

condition

2

(4.135)

we can

existing

manifold

is

f

-

(where

of this

relation defined

(S2)

interpret

the

moduli

space

theory as the space of deformations of lagrangian submanifold". BY Mc

the dimension

0

124,

N

in

embedding

an

dimensional know that

123,

The S1 stands

worldvolume

f, with

po,

of

obtained

R proportional

radius

configuration

defining

[122,

approach,

R1

the

preserving

agreement that

mass

residues

with

description

representation

di

of the effective

of the

identify

that

with

limit

the brane

with

with

equivalent

is

sum

we

M-theory

the

F(t, v) This

the

that

theories,

gauge

point

In the

Q, dimension,

in

y

The

of the differential

Theory.

compare

moment to

gauge theories.

spacetime,

will

dimensional

performing

upon

dx*As

infinity,

at

String

and

section

I four

=

point

=

fivebrane.

the residue

with

w

of the

positions

m.

M-Theory

4.6

the

at

pole with residue

the

at

be identified

differential

abelian

the

means

poles

will

=

w

by

Vol

is

space of deformations

between the

is

the genus of Z and

condition"that

(Z),

the

Kdhler

class

of

Q),

and S2 the

C6sax G6mez and Rafael

158

the

Hernindez

the

gauge group in keeping in mind that

of the

rank

effective

theory. It is approach two ingredients the holomorphic top form dimensional

four

in the M-theory important and used: the curve defined by (4.134), are being Q of Q, which explicitely depends on the radius R of the eleventh dimenof the noticed in the discussion as already sion. This will be very important, because an explicit dependence on the string coupling I superpotentials, N =

will

be induced

A different

approach

constant

[1271.

engineering

in the

In this

BPS mass formulas.

(4.134)

to

case,

(4.135)

and

procedure

the

is

based

that

is based

on

on geometric set of following

the

steps: 1.

is

number of vector

ate

2. A

threefold on a Calabi-Yau X, with the apropicompactified in four dimensions. multiplets enhancement of gauge symmetry in the to classical corresponding

theory

String point

moduli

space of the

threefold

Calabi-Yau

must be localized.

by performing

3. A

defined

4.

rigid Calabi-Yau threefold is manifold Calabi-Yau The rigid

is used to define

5.

Going

the

limit. particle surface Seiberg-Witten a

point

Z.

form

to

corresponding

uration

type

IIB

type

into

a

worldvolume

set

Z

6. The BPS states

from the the

type IIA

to

an

of fivebranes x

theory

string

represents

ALE space with singularity that can be interpreted

a

of as

a

brane some

fivebrane

configDynkin with

W.

are

through holomorphic

defined

Calabi-Yau

meromorphic one-form A, derived top form, in the rigid point particle

the

limit. show previous set of steps, that we will explicitely both between approaches example, the main difference at work in one definite theory. 'There is also is at the level of the meromorphic form in Seiberg-Witten related to the implicit in the underlying difference philosophy, an important of the above heterotic-type steps, use in the string approach, described in the manifold. Calabi-Yau choice of a particular us to the 11 dual pairs, driving of instead a certain The most elaborated geometric engineering approach uses, determined local of by the II dual pair, a set geometrical data, heterotic-type mirror and interested maps to generalizes on, type of gauge symmetry we are field we dimensional four the all these In theory this set of local data. cases, the as extra coupling will not on string parameters, depend are going to obtain approach, where field theories On the other hand, the M-theory constant. the might be on string coupling constant, depending explicitely are obtained of direct to a phenomena rich as explanation provide enough dynamically of the point context in the more restricted that can not be easily understood of limit string theory. particle example [124]. In Next, we will follow steps I to 6 through an explicit with order to obtain a field theory with gauge group SU(n) we should start of a K3the structure with h2,1 manifold n, and admiting a Calabi-Yau We and additional details). threefold fibered (see chapter II for definitions,

As

we can

see

from the

=

Strings

Fields,

In

24)+

24

(X 1

+X2

order

I

X12 3

12

(4.136)

that

I

b +

which

-OOXIX2X3X4X5-

5

2

(4.136)

as

1(XIX2X3 6

1

)6_

K3-fibration

a

manifold

we

12

(XIX2)

159

whose

12

(4.136) perform

will

0.

=

the

of variables XI

T4 (

IX2

+

visualize

clearly

to

change

so

Calabi-Yau

the

is the

mirror I

to a SU(3) case, corresponding ]p24 weighted projective space 1,1,2,8,12)

consider

will

and Branes

be rewriten

can

1/24,

b-

XoiII12,

2

X1

z.

a

surface,

Parameters a

b

The parameter

I

1

K3

can

fibered

(4.138)

in

_06/0, 0

=

be

Next, we should look for The discriminant singular.

the

0-2, 2

b

=

a

XOX3X4X5 :--

parametrized

space to those

=

in

(4.136)

02 /021.

IP1

in

by the through

(4.139)

]P':

(]P').

be written

0,

(4.138)

IP'

C

Vol

points

can

( 7C

the volume of

as

log

a

related

are

interpreted -

over

b=

,

(4.137)

form

in the

-2)x'2 0 +12 X312 + 3X43 + IX25 + (XOX3)6+ 2 WC

represents

coordinate

il/12

= -

1

1

-

IX2

(4.140) which

over

the

K3 surface

is

as

2

-/-A

11 (i

K3

-

e (a,

b, 6)

e,-. (a, b, 6),

(4.141)

i=O

where eo 1

-

VJ-C)2-

C

I

C

a)2

e2 The Calabi-Yau

bC2

-

el

V/((l -

c

-

-

a)2

-

C)2

-

bC2

(4.142)

C

manifold

will

be

singular

,"Calabi-Yau

==

whenever

11(ei

_

two roots

ei

ej)2.

coalesce,

as

(4.143)

i<j

We will

consider

SU(3)

symmetry.

through

the

singular

Around

this

point

point

in

the we

moduli will

space

introduce

corresponding coordinates,

new

to

C6sar G6mez and Rafael

160

=

-2

b

=

a'A6,

C

=

I

0,

in

e.

=

e1

=

2U3/2

e 2

=

-2U3/2

There

parameters

u

and

exits

get

(2U3/2

3V3v

a

set

(4.144) of roots

3.\,F3V)2

+

-

(u,

A6)

v;

on

A6,

(4-145)

definition

the

of

Riemann surface

a

singular SU(3) point, picture geometrical

natural

(4.144),

which

for

definition

the

is

and in

Z, defined point par-

the

understanding of the

of lp24 of the space of complex structures 1,1,2,8,121 of the and From this u v in view, parameters point,

the moduli

point.

ej

3v/3v

+

at the

in

v

+

we

3vf3v).

+

oo,

=

as

a

(4.143),

in

13/2Z

a

eo

(-2U3/2

a/3/2

_

limit

Now, we can use (4-145) by the Calabi-Yau data limit.

(Ce/U)3/2,

a

Going now to the a' -4 0 with z defined a z-plane,

ticle

HernAndez

blow

the up,

in

SU(3) singular (4.144) will be

associated with a vanishing two-cycles of the case we type A,-, singularity, singularity (in e., n are considering, 3). These vanishing cycles, as is the case with rational with Dynkin diagrams of non affine are associated type. The singularities, define the the branch points curve on z-plane (4.145) related

the

to

of the

volume

rattonal

i.

of

set

orbifold

an

=

y'

=

II(x

-

6)),

ei(u,

(4.146)

v; A

i

which

can

0, with

F

also be represented given by [129, 130]

as

F(x, z) where

B(x)

theories, This

has

configurations, is

that

and Z

now

as

space in

a

it,

but

rigid

limit F (x,

which

defines

a

threefold

of

(4.148)

to

get the meromorphic

map from

A =

z

locus

of

a

F(x, z)

polynomial.

+ B (x),

+ z

(4-147)

a

embedded in

Calabi-Yau

vanishing

polynomial in x of degree three; in the general case of SU(n) polynomial will be of degree n. exactly the same look as what we have obtained using brane with the space Q replaced by the (x, z) space. The difference the (x, z) space as a part of spacetime, we are not considering

is

the

the

as a

the

Calabi-Yau third

we use

Z

defined

as

in

(4.147)

define

to

z)

+

Y2

+

W2

=

(4.148)

0,

(x, y, z, w) space. And, in addition, of the point particle limit. representation one-form A, and the BPS states, we need in the

homology

a

by the equation

group,

H3(CY),

of the

Calabi-Yau

we

think

In order to

define

manifold,

a

Strig,,,and

Fields,

Branes

161

in H3(CY) of three-cycles of S1, K3, correspond to vanishing cycle general type The three-cycles with the topology S' circles of S3 can be in the z-plane. from of the the north south with a to a as S3, pole interpreted path starting and another of K3. This at ending vanishing two-cycle vanishing two-cycle, from the to in Once have to we z-plane, paths going corresponds, ei+ e,-.

HI(Z).

into

This

defined

this

be done

can

S'

the

follows.

as

SI

with

x

The

a

map,

f

:

H3 (CY)

H, (Z),

-+

(4.149)

define

we

(C))

A (f

(4.150)

Q(C),

=

top form. can be done for

holomorphic analysis computing the mass of BPS states, the In fact, in A and the meromorphic one-form brane framework. we can that such in C consider a two-cycle Q 0 the

with

A similar

(4.151)

ac C Z'

words, C E H2 (QIZ;

in other

or,

Z).

The

by Q= R the BPS mass will

and thus

M-R

v(t)

with

dt

A

t

be

given by

fc

dt

A dv

t

holomorphic

=

top form

on

Q is given

(4.152)

dv,

R

fac

dt t

V(t),

(4.153)

given by

F(t, v)

=

(4.154)

0,

corresponding Seiberg-Witten curve, Z. Notice that the same analysis, using (4-150) and the holomorphic top form for (4.148) will give, by contrast to the brane case, a BPS mass formula independent of R. and geometric Next, we will compare the brane construction engineering 1 [126, 128]. in the more complicated case of N for the

=

4.7

Local

Let

V be

Models an

for

Elliptic

Fibrations.

fibration,

elliptic

p

zA

with curve.

is

a

an

Let

singular

algebraic us

curve,

denote fiber.

lap}

Each

and

the finite

singular

Cp

:

(4.155)

V

V'

(a), set

fiber

of

Cp nip

with

points can

(9ip,

in any point in A such that

a

be written

A,

an

(P-'(ap)

elliptic =

Cp

as

(4.156)

C6sar G6mez and Rafael

162

Oip

where

are non

the intersection

satisfy

rational

singular

Different

numbers.

Hern6ndez

types

(i9ip.0jp).

matrix

0?

with

curves,

singularities

of

All

-2, and nip

integer by (4.156) and types of Kodaira singularities ==

'P

are

characterized

are

different

the relation

C2

=

P

r(u)

Let

elliptic

be the

path

For each

a

S,

transformation

S

with

=

S1(2, Z), acting

in

S,,

-r

fiber

on

-r(u)

c,,,-r(u)

as

+

b,

+

d,

define

point a

u

E ' A.

monodromy

follows:

(4.158) by

is characterized

the

at

we can

-

=a,,r(u)

(u)

singularity

of Kodaira

Each type

of the elliptic A f ap},

modulus

H, ( A%

in

(4.157)

0.

a

particular

monodromy

matrix.

fibration In order to define an elliptic point will be (45], the starting to be of genus zero, curve A, that we will take, for simplicity, algebraic and a meromorphic function J(u) on A. Let us assume J(u) 4 0, 1, oo on A function ,A' holomorphic T(u), japl. Then, there exists multivalued with j the elliptic modular with Im r(u) > 0, satisfying J(u) &(u)), we define on the upper half plane. As above, for each a E H, j-function a monodromy matrix S,, acting on T(u) in the form defined by (4.158). define an elliptic Associated to these data we will fibration, (4.155). In order of S, and let, us the universal define let us first to do that, covering of Z ' over A', with the elements in transformations the covering identify H, (zA'). Denoting by fi a point in Z, we define, for each a E H, (zl% the ii -+ aii, by covering transformation an

=

-

=

-r(afi) words, Using (4.158),

in other

we we

considerr

as a

define

we

and ni,n2

the

integers,

g(a,nj,n2) by 9

Denoting

the

(4.159)

S,,,-r(fi); holomorphic

valued

single

function

on

define

f,(fi) Next,

=

A'x

product the :

=

(c,,,,r(fi)

+

d,)-l.

C and, for each

(4.160) (a,

ni,

n2)

,

with

a

E

171 (A'),

automorphism

015A)

group

of

-+

(afi,

f,,(fi)(A

+

automorphisms

ni-r(fi)

(4.161),

+

n2)).

we

define

(4.161) the

quotient

space

B'

=-

(A'

x

C)

(4.162)

since g, as defined has no fixed non singular surface, by (4.161), A'. F) om (4.16 1) and (4.162), it is clear that B' is an elliptic fibration of elliptic with fiber elliptic curves on A', Thus, by the previous modulusr(u). the elliptic fibration we have defined construction, away from the singular points ap,

This

points

is

a

in

Strings

Fields,

Ep

neighbourhood

local

and Branes

163

point ap, with local comonodromy associated t, Sp t(ap) around ap. By Up we will denote the universal with a small circle covering of E' defined with coordinate by Ep ap, p P Let

denote

us

a

and such that

ordinate

=

analog

(4.159)

of

will

log

(4.163)

t.

be

7-(P If

be the

-

P

The

of the

Let

0.

=

+

Sp-r(p)-

1)

(4.164)

points ap, k times, we should act with SP'; hence we by the winding number k. The group of automorphisms small closed paths around ap, becomes

go around the each path

we

parametrize

(4-161),

reduced

g(k, Denoting

to

n2) (p, A)

ni,

by 9p the

=

group

(p

+

k, fk (p) [A

(4.165),

define

we

ni-r(p)

+

+

n2l)

elliptic

the

(4.165) around

fibration.

ap

as

(up Next,

4.8

Let

we

extend

will

two different

consider

us

assume

of

Sp

fibration elliptic depending on

the

cases,

Singularities

C)/9P.

X

new

case,

is of finite

we can

variable

us

or

point

ap. Wecan order of Sp.

infinite

(4.166)

extend

(4.167)

Id-

=

to the

singular

simply

points,

defining

a

a as

denote

and define

singular

order,

0" Let

to the

the finite

Type D4: Z2 Orbifolds.

(sp)m In this

(4.166)

D

local

a

the group

g(ni,

(4.168)

t.

=

neighbourhood GD of automorphisms

in

n2)

:

(a, A)

=

(or,

a-plane

the

A +

njT(a)

+

of the

point

a

=

0,

(4.169)

n2),

and the space F

=

(D

x

(4.170)

C)IGD-

fibration F defines an elliptic Obviously, E D, an elliptic curve of modulus -r(a).

over

D,

From

a

with

(4.167)

fiber and

F,

at

(4.160),

each point it follows

that

A (U) with

k

=

0(m).

of transformations

Thus,

we can

(4.165):

define

=--

a

(4.171)

17

normal

subgroup X of 9p

as

the set

HernAndez

C6sar G6mez and Rafael

164

n2)

g (k, ni,

Comparing

now

(4.169) (Up

Using (4.172)

and

(D'

=

cyclic

C the

(4.174)

From

:

we

FIC Now, FIC

The

a

a

=-

t.

The

cyclic

(Up

can

since

from fixed

(4-177)

=

(4.160)

with four

a, b

=

irreducible

by

C

((9o, (9')

external

points

to

ap,

namely

(4.176) Sp

case

points

order, regularize.

of finite

is

that

we can

is

( 01

01

(4.177)

the order

case,

in this

(-

-4

'2-r(O)

is

m=

simply

2, and

we

(4.178) At the point

-1.

=

define

becomes

-A),

9,

get f, Z2 orbifold

case

a

=

0

we

have

points, b

+

(4.179)

-), 2

of these four singular points will produce 0, 1. The resolution have the irreducible In we addition, (91'...'194. components,

component 00, defined -64 cycle, we get the

with

in

ap,

we

(0,

(4.175)

to

(4.175)

the standard

points,

to

singular

In this

group

and

by

Q19p U FoIC.

x

have

(01, A)

four

(4.173)

FO.

(4.174)

extension

=

transformation.

parity

by a'

-

get the desired

SP L e.,

F

-=

A (9) A)

-+

by FIC. simplest example corresponds

defined

(4.172)

n2)

gljv,

=

extended

fibration

elliptic

the

Thus,

+

get

C)IGD

x

nir(p)

(e 27rik/mor,

(a, A)

(4-173),

and

A+

m, defined

of order

9k

k,

get

we

group

+

we

C with

(p

-+

(4.172),

C)IJV

x

(4.165)

and

(p, A)

:

points

=

of

of the torus.

((90,(92)

=

the

curve

2(90 =

D-diagrams

+

el

((q()'(93) can

itself

+

at

e2 =

+

a

=

e3

((90,(94)

be associated

0.

+

Using

the relation

04, =

with

a

2

=

t'

(4.180) 1. In

the

general, four

the

four

Z2 orbifold

Fields,

Singularities

4.9

Wewill

now

of

the

consider

is of infinite

and Branes

165

Type case

n

SP which

Strings

0

A local

order.

(4.181)

1

model for

monodromy

this

be defined

can

by t

-r

Using the phisms,

p defined

variable

g(k, and the local

n2)

ni,

model for

(4.163),

in

(p, A)

:

(up i.

e.,

fibers

way to think

[63].

erings order

Let

of

n,

of the type of about these us

a curve

elliptic elliptic

recall

that

Thus, for

g

of

=

C,

1,

we case

out

+

of the

gp

of automor-

n2),

(4.183)

singular

point,

(4.184) elliptic

is in terms

cyclic

C of genus g, is

in

A + ninp

with

curves,

a

the group

C)/9P'

curves

=

modulus

X

for

get,

k,

+

(4.182)

t.

fibration,

elliptic

the

we

(p

-+

log

n

7r,

unramified

a curve

ng + I

-

0

modulus

cyclic covering, of

np.

A

Moreover,

the basis

of genus

(4.185)

n.

n. 1, for arbitrary Denoting by -r the elliptic modulus of 0 is given by 1, the elliptic

get g

(4.186)

of C

H, (C; Z)

of

& and

generators a,

cov-

C, of

H

nT.

homology

simple

unramified

are

given

in

terms

of the

as

H&

H4

=

a,

=

nO,

(4.187)

H and (4.183), -+ C. From (4.186) we can interprojection fibration fibers elliptic (4.184) as one with elliptic given by-n-cyclic unramified modulus p or, equivalently, coverings of a curve C with elliptic 1 logt. There exits a simple way to define a family of elliptic curves, with modulus given by elliptic ' logt, which is the plumbing fixture construc27ri tion. Let Do be the unit disc around t 0, and let CO be the Riemann sphere. Define two local coordinates, z,, : U,, -+ Do, Zb : Ub -+ Do, in disjoint neigbourhoods U, Ub, of two points P,, and Pb of Co. Let us then define

with

H the

pret

the

=

W

=

J(p, t)It

E

Do,p

E

p

Co

-

EUb,

U,

-

with

Ub,

or

JZb(P)J

p E >

U,,,

ItIli

with

Jz,,(p)J

>

ItI,

or

(4.188)

166

C6sar G6mez and Rafael

and let

S be the surface

S We define

of

family

the

(p, t)

Ixy

=

t; (x,y,t)

=

x

Do

E

through

curves

w n u,,

E

HernAndez

For each t

we

the

get

a

genus

non zero

t

and at

curve,

one

t

homology cycles.

t

==

,t)

(Pb) t)

(Pb),Zb

Zb

7

0

we

pinching

The

(4.189)

identifications

(z,, (p,,),

Do

Dol.

x

following

the

(Pb)t)Ewnu,xDo

pinching by

Oo

x

E

S,

(4.190)

E S.

get

a

region

nodal

curve

(4.191)

XY

defines

which

a

singularity -r

for

some

constants

C,

t, such that

Ao,

I

(t)

27ri

the

now

result

unramified

CIt

+

+

=

(4.186) (4.182),

and

(4.192)

and the

covering

is

(

=

we

)

i

I

0

1

get,

for

(4.183).

group

of the

(4.192) choice a

of coordinate

of type

singularity

(4.193)

.

the

The

is

curves

C27

=

SP Using

109t

=

modulus

C, and C2. Wecan use an appropiate 0. The singularity at t 0 is C2 to classification, corresponding

=

in Kodaira's

Ao. The elliptic

of type

given by

by

is characterized

cyclic covering pinching region

order

of

of the

n,

cyclic

given by XY

=

(4.194)

tn'

i. e., for the surface C2 /Zndefining the An-i singularity, of the singularity at t 0. The proceed to the resolution of the singularity 1 exceptional resolution (4.194) requires n divisors, rational we have the curve On- I In addition, (9o, defined by the ei, 0, complement of the node. Thus, we get, at t

of

instead

Now,

we

(4.191),

can

=

-

-

-

-

,

-

=

190

C

with

(190, 01)

=

Dynkin diagram. cyclic unramified is given by

(00, (9n-1) The group covering is

+

*

-

1, and (ei, of

covering

Zn, and

(4.195)

19n-1;

+

Oj+j)

=

transformations

the action

(9i

--+

(9i+j,

On-i

--+

eo-

over

1, which

is

of the

theAn-1 nth

the components

order

(4.195)

(4.196)

Strings

Fields,

Singularities

4.10

This

is

case

reasoning

SP

-1

-n

0

-1

by

g(k, Using 1907

n2)

ni,

192n) with

...

fixed

Z2 orbifold

the

with

case

Defing for

all

characterized

The

=

2(9o

+

-

+

-

affine which

-2,

framework

general

N

=

I

+

is

(91

a

+

affine

(C)2

Decompactification

mensional

-+

-6n+4

of the

non

=

get

Oi

2e,,,

get

by C2

we

+ ninp

of irreducible

set

93

+

It

is

W,

(4.199) to

easy

=:

Notice

2g

-

2,

we

for

that

conclude

rational

fiber

of Kodaira

Vol

in

limit

(. )

which

working

we are

-+

AbE type.

then 0.

g

1,

=

singularities, get self

we

order

in

we

get

to

=

N

=

know,

for

to

Vol

the

(E)

-+

locally

oo,

2 super-

and the four

around

Calabi-Yau

H

C in B, where the fiber is singular, four. Let us now see what

on

E the

(4.201) work

will

di-

by changing

1

we

four

compactifications 0, with (E)

=

corresponds

Now,

As

(E)

E-+X the locus

in

and Affinization.

Vol

to

that

Dynkin diagrams of ADE type [44], corresponds to genus equal zero.

of M-theory is that gauge theories Calabi-Yau the limit in Vol fourfolds,

dimensional

that

see

(4.200)

=

dimensional

given by

is then

0.

=

type.

+

diagram.

by C'

fiber

e2

components get the four

we

fibered elliptically As described between fiber. above, we can interpolate elliptic N and 1 in in four three dimensions, dimensions, symmetry the radius R through

The three

same

(4.198)

n2))

+

192n-i- In addition, above. The singular fiber

described

points

also

the

(4.197)

(_ 1)k (A

+ p,

what

the genus of the singular of Kodaira singularities

intersection

4.11

u 2= t,

intersections we

(k

-+

the identifications

C

this

(p, A)

:

variable

a new

Through

examples.

the group

above,

as

of the two previous Gp is given, for

combination

a

167

-bn+4-

Type

of

and Branes

a

fourfold

singular X,

(4.202)

+B, is of codimension

one in B, i. e., of singular fiber in the In this case, we have Vol (E) limit. three dimensional oo. A possible way to the point at infinity. In the represent this phenomenon is by simply extracting in previous as described case of An-1 singularities, subsection, taking out the the irreducible corresponds to decompactifying point at infinity component

real

dimension

happens

to the =

C6sar G6mez and Rafael

168

00,

that

as

fibration with

intersecting

(E)

Vol

basis

the

M-Theory

to the

considering

what the

doing,

irreducible

at

in this

A,,-,.

case,

More

we

then

generally,

a global we can section, as the one decompactify When we decompactify, the level is of the fiber,

leads

which

component,

Holornorphic

and

Instantons

to

fibration.

elliptic

extra

affine,

possesses

we are

clear

was

non

going

we are

of the

limit,

0

=

As

itself

curve

A,,-,,

component

compactifying precisely affine Dynkin diagram. 4.12

the

we are

irreducible

the

the

with

diagram,

the affine

elliptic

the

select in

associated

was

pass from

Hermindez

to

the

Characteristic.

Euler

in a Calabi-Yau instanton fourUsing the results of reference [991 a vertical D defined will of be divisor that of such the a by X, fold, H(D) type (4.202), is of codimension one in B, and with holomorphic Euler characteristic

X(D, OD) It

is

in

can

bundle

U(1)

(4.203)

case

define

that

contribution

to D in

locally

is

a

to

[99],

modes

zero

associated

on

and

we

the normal

N, D, we define

D. For

complex line bundle

the

transformation

eiat,

t -4

with

(4.203)

1.

have two fermionic

we

superpotential X, which

a

=

t

a

of the

coordinate

U(1) charge equal

of N. The two fermionic

fiber

Associated

half.

(4.204) zero

modes have

divisor

D, we can define a and real scalar field OD that, together with Vol (D) defines the imaginary Under U(1) rotations (4.204), OD transforms as parts of a chiral superfield. one

OD + X(D)a.

OD In

three

field

fect

sense,

as

Goldstone vertical

for

(4.205)

the transformation of the dual phoprecisely [24]. However, transformation (4.205) has perin the four dimensional instantons, decompactification

this

dimensions,

ton

the

to

is

boson

limit. Let

gular

us

assume

rical

now

consider

ofA,,-,

fiber that

the

engineering

elliptically

an

type,

singular spirit,

over

a

fiber

is

will

we

Calabi-Yau

C of codimension

constant

over

fourfold, one

C. Moreover,

in

with

sin-

B. We will

in the

geomet-

impose

hi,o(C) Hi(C)

fibered

locus

=

h2,0(C)

=

0

(4.206)

This prevents transforus from having non trivial by going, on C, around closed loops, since all closed In addition, that C we will loops are contractible. assume, based on (4.206), After impossing these assumptions, surface. is an Enriques consider we will divisors n over C, in a trivial Di, with i 0, 1, defined by the fibering

and, thus, mations

on

the

=

0.

fiber

=

-

.

.

.

,

irreducible

of the

way,

of the

C

for

get,

=

F4

C1

X(Dj) Interpreting

now

X

N of D in

Zn

ODj

:

now

(4.205),

we

(4.209)

and

ODj

:

(4.211)

we

OD5 with

j

=

Let

0,

C

fiber

gular careful sum

This

n

.

f

C,

1, and

(4.211)

n

modulo 27r,

27rj

(4.212)

+ C'

=

n

independent of j. by fibering

D obtained

divisor

defined

(4.195).

in

In

this

over

C the

we

need to

case

sin-

be

to

(Enj=O (9j) as

interested

we are

is

by C2

defined

2(1 =

_

2g

any other

for

in.

Di

each divisor modes.

In

the

the

graph,

2, with

Kodaira

heuristic

as

have, of the

The

soaking

interpretation

a

as

A,,-,

shown in

each node,

where from

mode lines.

we

case

In

fact,

g(rn-1 9,)), _j=o -

the result Wecan try to intepret of each component ej, and the

an

get,

27r*

+

=

which,

is

(4.210)

the topological consider If we naively Compute (4.207). n. we will of components (9j in (4.207), get the wrong result X(D) Euler but not for the holomorphic would be correct result topologically,

order

in

1

inside

(4.209)

by

ODi

---

constant

c a

the

En-1 'j=O 19j,

=

characteristic

as

-

consider

now

us

divisors.

get

Zn

Combining

these

ODi+,)

-+

the

under

law, to

.21ri/nt.

t _,

Using

ODj associated

fields

of Kodaira

description

transformation

the

derive

bundle

normal

of the

fiber

the

previous

our

being defined

Zn transformation

the

(4.207)

(C)

(4.208)

on

used in

t variable

we can of type A,,-,, singularities Z,, subgroup of U(1), of the scalar Namely, from (4-196) we get

with

C2

Ussing

[137],

1.

=

(4.204)

t variable

the the

as

(19i)

[126].

singularity class

surface,

Enriques

an

fDi

1

X(D) we

ej of the A,,-, holomorphic Euler

components

representation

the Todd

169

and Branes

Strings

Fields,

what

C2 the self

singularity,

X(D) (4.213)

of the result

of the

Thus,

we

get

g

=

1, and

(4.213)

of the

singularity, Figure 1,

up of fermionic

in

cycle

0.

consequence

representing

(4.207) for (4.195), cycle (4.195)

write

genus of the

zero.

in terms

topology

should

intrsection is

=

we

g the

with

of

of the fermionic

cycle.

(4.208),

we can

one zero

ei,

we

modes

(4.213).

In

fact, two

zero

fermionic

soak up all

zero

have two fermionic

represented

modes

associated

in the

to zero

modes

zero

figure

Usar

170

G6mez and Rafael

HernAndez ............

Feromonic

..........

Fig.

O-Parameter

4.13

We will, with

this

in

both

In

cases,

X(Dj)

=

order

of the

Soaking

4.3.

and

up of

zero

mode

modes for

zero

Gaugino Condensates.

section,

only consider

and for

each irreducible

1. Associated

this

to

[99]

f

singularities component

divisor,

A,,-,

of

we can

Oj, get

we a

and

get

a

b"+4 divisor

superpotential

d2Oe- (V(Di))+i95Di

type.

Di, term

(4.214)

where V(Dj) means the volume of the divisor Di. As explained above, we are instanton divisors using vertical of (9i over Di, defined by a trivial fibering the singular locus C c B, satisfying conditions In order to get the (4-206). 1 -+ 0. Since four dimensional N 1 limit, take the limit we will Vol (E) R the singular fibers are, topologically, the union of irreducible components (see =

(4.195)

and

(4.199)),

=

we can

write

Vol with

Cox the Coxeter

the total

number of the

number of irreducible

1

(00

=

(4.215)

RCox'

which equals corresponding singularity, Therefore, we will define Vol (Dj)

components.

as

Vol If

we

first

consider

(Dj)

=

lim

(C)

Vol

R-+oo

1

the N = 2 supersymmetric three M-theory on the Calabi-Yau

by compactifying

tained

(4.216)

RCox

dimensional fourfold

theory

ob-

X,

L e., in the case, is decom-

only the divisor (90, for the A,,-, an elliptic diagram describing to singularity the non Artin a rational, describing like, singularity [44]. In that case, the volumes of the Oi components, for i 0 0, are free parameters, to the Coulomb branch of the N 2 three dimensional corresponding theory. In the three dimensional theory, the factor Vol (C) corresponds to the bare coupling constant in three dimensions, limit

R -+

pactified,

0,

we

know that

passing from affine diagram

the affine

=

Vol

(C)

2)

93

(4.217)

Strings

Fields,

(Dj)

and Vol

for

0, with

becomes

(Dj)

Vol Let we

us

now

concentrate

lim

-

R-*oo

(94n 1

exp

Let

us

fix

now

the

these

get,

we

g23

+i

( +C))

with

order

In

that,

do

to

dimensional

four

to the

respect

ODj

E ODi

-4

U(I)

R

we

+

0

if

fact, we

(4.220) 0

define

is

direct

a

U(I)R

under

rule

of

the

N

0

anomaly

SU(n)

for

is

assigning

(4.219)

is

with

the

anomaly equation:

axial

F.P,

(4.222)

n

1,

=

5

167r2

F.P.

(4.222) from (4.223), 2, differing UMRcharge I2 to the fermionic

parameter

1 0-

given by

a1_1 j The factor

U(1)

of the

consequence

as

=

(4-221)

0 +,na.

-4

327r2 the

use

(4.220)

na.

parameter, In

will

point of view, symmetry. From

i=O

transformation

the

(4.219)

n-1

_7 precisely

j

e"t,

t -+

i=O

is

Using (4.214)

n.

=

Dj,

each divisor

n-1

This

where Cox

case,

that

(4.218)

g24 COX

for

From the

rules

(4.216),

use

1

1

RCox

(4.219).

(4.205).

under

that

must

n

in

c

the transformation

are

(4.205)

2

constant

rules

transformation

the

-

we

171

Coulomb

dimensional

three

case,

I

theA,-,

on

superpotential

following

get the

the

Xi

dimensional

In the four

coordinates.

branch

:

i

3

and Branes

topological

sum

En-1 i=O

(4.223) reflects

ODj

the

modes.

zero we

get that

fact

that

we are

Identifying the

0-

the

constant

c

in

simply 0 C

(4.224)

=

n

so

that

exp

with

j Let

gularities.

=

finally

then

we

0,

.

.

.

us now

-

,

n

try

Defining

(94 -

1 2

+i n

('

1, which to extend

again

superpotential

the

obtain

+

is the

the

the four

'))

A3 e 27rii/neiO/n,

(4.225)

n

n

correct

previous

value

for

argument

dimensional

the

gaugino

to the

O-parameter

bn+4 as

the

condensate.

type of sin-

topological

172

C6sar G6mez and Rafael

SUMOf

0Dj

(4.199),

for

HernAndez

the whole set of irreducible

the transformation

0 where

Coxeter

the

now

groups, 27r we

with

N

=

-6,-,+4

for

2n+8,

we

-6n+4

for

get that

components

get

k

singularities

However,

N-2.

of k to each irreducible a

of N

set

(

k 1, particular =

a

.

.

.

,

N

-

value

as

cycle

0 (N)

Of

0Dj

for

7

gauge modulo

any irreducible

0

of k to components of diagrams of type

N

(9i

do not know how to associate

for

1

D prevents

2))

0 +

N

-

(4.228)

'

do not know how to associate will

be

associating intersection; however,

"vanishing

non

diagram. Using (4.227), gaugino condensate for O(N)

the

N-2

However, we still possibility

value

a

-6n+4

27rk

+

of k. A with

(4.227)

2'

-

of the

values

2 94(N-2)

2.

value

+

2

-

now we

2 different

exp

(9i

-6n+4

Since 0 is defined

N-2.

=

the

component

-

groups:

with

the

(4.226)

Interpreting

is 2n + 6.

COX(-6n+4) 27rk

get

for

0 + Cox.a

-

Y

we

get,

is

component,

with

we

rule

to each

consecutive

values

topology Notice that the doing that globally. puzzle we find for O(N) gauge groups, the

from

us

problem we have is the same sort of concerning the number of values for < AA >, and the value of the Witten terms is simply the number of nodes of the index, which in diagramatic diagram. In order to unravel this puzzle, let us consider more closely the way fermionic

(4.199); we

modes

zero

for

the

get divisors

soaked up

are

components

(9'

with

Now,

X

=

1.

from the Todd representation

f)n+4

on a

04

to

for

of the X

,

diagram.

associated

the

Wewill the

components

holomorphic =

to

Euler

Use

Z2 orbifold

2190,...'219,, characteristic,

4.

the

cycle points, get,

we

(4.229)

2(9, with 192 -2, has self intersection Euler characteristic of the holomorphic when fibering over C any of the cycles n. 2(9j, with i 0, Equation (4.229) implies 8 fermionic zero modes, with the topology of the soaking up of zero modes of the -6n+4 diagram, as represented in Figure 2. Notice that the contribution to X of 2(9 is different form that Of (191 + (92) with ((91-192) 2 for the second. 0; namely, for the first case X 4, and X For the bn+4 diagram, we can define: i) The Witten index tr (-I)', as the number of nodes, i. e., 5 + n; ii) The Coxeter number, which is the number of The

reason

for

this

Of course, divisor obtained -8.

is that

(4.229)

the

refers

cycle

to

=

the

=

.

=

irreducible

components,

=

i.

e.,

2n + 6 and

iii)

.

.

,

=

The number of intersections

as

by the dashed lines in Figure 4, i. e., 8+4n. From the point of view of the Cartan algebra, used to define the vacuum configurations in [22], we the number of nodes. The 0-parameter can only feel is able to feel the Coxeter represented

Fields,

.

.

.......

Fig.

173

line

Feromonic

Soaking

4.4.

and Branes

........

Dynkin ...............

Strings

zero

up of

mode

zero

modes for

f)n+4.

of to the intersections related number; however, we now find a new structure to cycles 2(9j, the graph. In the Witten index case, the nodes corresponding with one, in the number of < AA > values with with i n contribute 0, =

two,

.

.

.

,

and in the

number of intersections

with

four.

This

value

four

calls

for

of the definition of these nodes. The topological interpretation two cycles, into implies the split of this orientifold implicitely 0-parameter the F-theory description [138] of the Seiberg-Witten a phenomena recalling the only possible of the orientifold, [35]. Assuming this splitting splitting topology for the soaking up of zero modes is the one represented in Figure to four zero inside the box is associated orientifold" 3, where the "splitted 02 for a cycle 01 + 192, with 191-192 to X modes, corresponding an

orientifold

Feromonic

...............

Fig.

4.5.

Orientifold

zero

mode

splitting.

hand, each node surrounded by a circle in Figure 5 reprerational thus, we sum of two non singular sents curves; with the intersecfour rational mode "orientifold" each by curves, represent lines inside the box of Figure 5. When we forget about internal tions depicted It of structure '--ZN-2 in Figure 5, we get the cyclic (4.228). equation Z2n+6 is clear that much more is necessary in order to reach a complete description On the other itself

of the

the disconnected

O(N)

vacuum

structure.

Hernindez

C6sar G6mez and Rafael

174

Domain Walls

4.14

and Intersections.

section already raises the problem known as discussing again only the SU(n) case, the transforfo the 0-angle mation law (4.221) as the together with the very definition 1 that would 0 the scalar is field sum imply O'D of the Enj= 0 0Dj topological I (9j. On the basis of (4.205), 6-cycle associated to the An-1 cycle, C F_j=Q =0 of zero. This is, in be equivalent this will to saying that X(D) n, instead the mathematical The mathematical solution comes from 0-puzzle. terms, this result, relate the fact that X(D) 0. In this section we will on the value Euler chareacteristic, of the holomorphic of dornain walls to the appearance To let consider C start us a cycle el + 192, with with, [132, 133, 134]. be 1. The intersection self can expressed as (01 e2)

The discussion

0-puzzle.

in the

fact,

In

previous

and

n

=

=:

=

=

=

-

(C.C) where the the

contributions

-2

intersection

fibered

an

on

=

come

-2

cycle

six

be written

can

now

section

term

charge oposite

of chiral the

cycle

up two

of the

singularities, modes, leading to

equation

represent

(4.230)

is

we

term

A,-,

C of

zero

and the

two contributions

get

(9i

we

(4.230).

most

vacua,

we

components. that

X(C)

=

natural or

to

X(C)

answer,

values

change

and net

When we do this

each intersection

Figure

in

the inter-

sense,

modes,

0. A

is

for

soaking

graphical

way to

4.

term

i+l-vacua 4.6.

wonder about

will

of the

and

In this

zero

get that

the result

presented

Fig.

leading

from

trivially

of one, coming a, contribu-

independently,

+2 in

intersection

Now,

comes

(4.231)

i-vacua

terms

+2

(_C2).

to two fermionic

that

to

2

e2, considered

be associated

can

=

(4.230)

decomposition

the

the components (91 and of -1 from the intersection

tion

,

as

1 X

Using

1921 and (922

from

between (91 and e2. As usual ,we can consider C Euler chracteristic Enriques surface. The holomorphic

corresponding

from

(4.230)

2,

2 +

-

=

is

0 for

certainly

of < AA >.

the all

Intersection

physical Kodaira domain

term.

interpretation

singularities. walls extending

of the intersection

The

simplest,

between

and

different

Strings

Fields, of view

From the

point effectively

behaves these

fermionic

Oj

of

Thus, the

modes,

zero

Oj+l

with

of let

3

(4.232), interpreted fact is the surprising simply a point; considering

In result most

the

V)j+,,j

is associated

with

the intersection

e

as

to the

intersection

19j

of

volume as

we can

order

of the intersection

wrapped

on

of the intersection

way

same

terms

we

as

six-cycles

the

the

term,

geometrically of the gaugino

now we are

A3 in the computation of the divisor. In the

fivebranes

19j+,.

(4.232)

of A3, since

think

and

components,

.27ri/n).

_

the contribution

the factor

instantons

instanton,

27rij/n(1

term"

modes. One of

zero

bjj+j,

appearance

the

from

comes

M-theory

define

two fermionic

the

A

to

say

175

"intersection

the

counting,

with

done for irreducible computation of the black box in Figure 4 should be of the

naively

contribution

condensate

us

and the other

extending

interpret

mode

zero

anti-instanton

as an

and Branes

used

fivebranes

as

cycle C x f (i9j-(9j+j) I, i. e., the product of the singular locus C and the intersection point. The fivebrane wrapped on this cycle defines, in let us say interwining between the vacua i, four dimensions, a domain wall, where the coordinate at X3 X3 is +oo, and the vacua i + 1, at X3 -oo, It is in this sense that we should use with the unwrapped direction. identified (4.232) to define the energy density, or tension, of the domain wall. In the four 1 the local Vol (E) goes zero as R dimensional limit, engineering ; moreover, of the limit where volume the singular locus C is very approach works in the terms behave like large, so that we can very likely assume that intersection dimensional limit. in A four of theA,,-, the 3, but only Cyclicity (4.232), with 1 the in different from the two j to j + vacua diagram allows us to pass ways: should define the I steps, or a single one. The sum of both contributions n physical domain wall; thus the energy density will behave as wrapped

the

on

=

=

-

nA The extension

certainly ence

of the

involved,

more

of orientifolds.

It

Finally,

we

of

geometry

le 27rij/n

previous

(1

will

say

and Y

details).

Z is =

S1

would

some

QCDstrings

rational

a

x

The

e27ri/n) 1.

argument

is

curve

case

of

O(N)

b diagram, topology be certainly interesting studying

words

on

intimately

in

the

this

Z)

=

to

to

the

In reference

topological

the to

the

that

(4.234) configuration Z is embedded

a

[117], fact

Z,

space where

is then associated

groups is and the presthe interplay

case.

QCDstring.

related

associated

R1 is the ambient

QCDstring

the

to

(4.233)

of the

H, (Y/Z; where

-

due to the

and domain walls

orientifolds

between

3

partially

of

fourbranes,

(see [117]

for

wrapped membrane H, (Y1 Z; Z) is defined

of H, (Y1 Z; Z). Recall that was done boundary on Z. The previous discussion de9cribed in for SU(N) gauge groups. Using our model of A,,-, singularities, Then section 2, the analog in our framework of (4.234) is equation (4.187). on

a non

trivial

by one-cycles

element

in

Y,

with

C6sar G6mez and Rafael

176

we

the

in

can,

paths going

spirit

same

HernAndez

[117],

in reference

as

from Pk to Pk+1

where Pk

,

are

1909k-l Geometrically, root

is

clear

domain

wall

it

of the

here ends

suggesting as

To end up, let suggested in

of

<

that

It

AA > does not

some

QCDstring is the the QCDstring

of this

comments

known that

is

coincide

with

the

to

(4.235)

By construction, L e., on intersection domain walls,

on

[135].

tension

QCDstring points,

the

Pk-

tension.

include

us

the

::--

associate

the intersection

on

the

points.

the existence

of extra

vacua,

coupling

strong

weak

square we are

coupling

computation more computation;

precisely[136], < AA >sc<

In

of

framework

the

will

tors

depend

in

M-theory particular

fourfold.

Calabi-Yau

In the

< AA

>wc

(4.236)

-

the numerical faccomputations, of complex structures of the coupling regime we must consider struc-

instanton on

the

strong

moduli

and the Picard lattice. In the fibration structure preserving the elliptic is performed in the Higgs weak coupling regime, where the compuatation to the value contributing phase, the amount of allowed complex structures the previous of < AA > is presumably Obviously, larger. argument is only suggesting a possible way out of the puzzle (4.2M). level, the extra vacua, with no chiral symEqually, at a very speculative the singular to the cycle D defining could be associated fiber, metry breaking, and therefore does not produce any gaugthat leads to X we know a cycle 0, Notice that any other cycle with X :A 0 will lead, if clustering ino condensate. D with X 0 is used, to some non vanishing so that, gaugino condensates, the in candidate like looks to extra vacua, suggested a,possible [135]. If this tures

==

=

argument L

is correct

this

gluodynamics. theq' mass

for of

-L, N

N

=

which

In

the

will

appears

dimensional

gauge

extra

vacua

e. in any ADE N = 1 four to stress It is important

N

=

that 0

case

for

theory.

the

0-puzzle

the

Witten-Veneziano

is

not

dependence of the vacuum of entangled "vacuum" states. entanglement is due to the fact

also indicates

a

singularity

any Kodaira

exclusive

of

energy

N

[139,

formula on

=

1

140]

0 in terms

to our approach that X 0 for the origin 0 means that the set of divisors Di, plus the cycle. In fact, X(D) singular the i. domain invariant under are walls, intersections, U(1), as implied by e., in N think of something similar If we naively 0 and we equation (4.205). in intersections of vacuum entanglement look for the origin we maybe should of intersections the topology into topological think in translating properties of abelian proyection gauges [58].

1 the

means a

set

of this

In

=

=

=

Strings

Fields,

M(atrix)

A.

The

holomorphic

topology

Principle.

Holographic

The

work first

was

dimensional

sphere in R', S(2) contain:

of two In

suggested by 't Hooft in [141]. Let originally with the be a surface Let S(') spacetime. and let us wonder about how many orthogonal

principle four

in

quantum states of states.

we

can

order

this,

do

to

we

find

will

will

an

the

use

upper bound for

hole. Let area of the black the entropy and the horizon A( the number of states inside S(2); the entropy can be defined exp, S

S2 is the horizon

of

a

hole,

black

S_

number relation

us

call

then

as

(A.1)

M.

=

we

this

Bekenstein-Hawking

between

If

177

Theory.

A.1

us

and Branes

[142]

have

1 A

A.2)

4121 P

A the

with

all

states

of

q-bits

we

contained

information

defined

as

length

in Planck

of the horizon,

area

physical q-bits,

the

inside

quantum systems of

units.

the

S(2)

Now, let surface,

two states.

us

in

translate terms

The number

of

(A.3)

N = 2'.

(A.3)

to

of

given by

need is

Using (A.1)

n

we

then

get A

1 n

(AA)

41n2 121 P

which

is

the number of

essentially

cells,

of

area

12,P covering

the surface

S(2)

all three dimensional physics inside S(2) surface q-bits, living on the two dimensional S(2). Wewill call these q-bits the holographic degrees of freedom. What we need now is the two dimensional dynamics governing these two dimensional the three in holographic projection, degrees of freedom, able to reproduce, of S(2), instead dimensional physics taking place S(2). We can even consider, The exteninto two regions. of dimension two, dividing an hyperplane space tell that will of the us holomorphic principle, sion, to this extreme situation, for the 2 I of in described terms be some + the 3 + I dynamics can dynamics the of freedom hypersurface. living on holomorphic degrees allows to introduce of the holographic This picture M(atrix) principle In this case, we will of the M-theory. projection holographic theory [143] as to ten dimensional physics. What we will need, pass from eleven dimensional will be in order to formulate theory, M(atrix) What can

we

from

learn

be described

i) An explicit ii)Identifying

using

definition the

this

is

states

of the

holomorphic

that

of

holographic degrees of

projection. freedom.

C6sar G6mez and Rafael

178

iii)Providing

ten

a

conjectured

The

Hernindez

dimensional

dynamics for these degrees of freedom.

[143]

in

answer

i), ii)

to

momentum frame. i) The infinite as degrees of freedom. ii)D-Obranes iii)The dynamics is implemented through

iii)

and

are

the worldvolume

of the

lagrangian

of D-Obranes. These set the infinite

of conjectures

M(atrix)

define

momentum frame

boosting

is

theory

at

The idea

present.

eleventh

in the

direction

of

eleven

in

in such a way that p1l, dimensional the eleventh spacetime, component of the momentum, becomes larger than any scale in the problem. In this frame, with an eleven dimensional massles system of momentum p we associate,

(p,

1,

p

i

),

a

galilean

dimensional

ten

E

If

we

on a

introduce

an

S1,

circle

mass

pi 1, and energy

P2I

=

2p11

(A.5)

.

cut

off,

by compactifying

R, the

pl,

is measured in units

infrared

of radius

system with

the eleventh

dimension

1. Then, R

of

n

P11

=

(A-6)

_.

R

We will

number of partons n as the which are necessary to interpret -value with of These the system given by (A.6). pl, partons are dimensional degrees of freedom we are going to consider as holographic

describe ten

a

variables.

Using (A.4), sional n

=

we can

particle,

massless

p11R

means

is the

define in

the

number of

dimensions, holographic degrees

that

r9

TP9 and the radius

objects

for

Natural

r,

characterizing

in ten dimensions

candidates

are

dimensional

ten

eleven

the with

-

some

of freedom

eleven

an

dimen-

given p1l. fact, which, by (A.4)7 In

p11R,

size, mass

D-Obranes.

of

size

with

will

equal this,

From

(A.7) (p, 1 R) 1/9 1p Now,

be

.

to the

it

mass

seems

the worldvolume dynamics of D-Obranes will of M-theory. description holographic M(atrix) theory would hence simply be defined As for any other type of D-branes, of D-Obranes. that

be

of

we can

look

parton, T'R to conjecture L e.,

a

natural

good candidate

a

for

the

is

defined

dimensional

as

the

dimensional

Yang-Mills

with

reduction N

=

as

down to

I supersymmetry.

the worldvolume

this

theory theory

worldvolume

0 + I

dimensions

If

consider

we

of a

set

ten

of

matrices D-Obranes, we have to introduce 9. As usual, X, with i 1, the diagonal in terms of the classical part of this matrices can be interpreted of the N D-Obranes, and the off diagonal terms as representing positions the exchange of open strings. for Thus, the worldvolume lagrangian we get N

=

.

.

-,

Strings

Fields, N D-Obranes

a'

U(N) Yang-Mills

is

1, the bosonic

=

part L

in units

where

1p

I

lagrangian .

[

=

2g

1, and with

=

we

.

.

-

2

g the

theory.

in

named

(A.8)

in

are

fact

IIA

type

string modes

Kaluza-Klein

In the

M-theory.

defini-

In this

constant.

dimensional

ten

a

D-Obranes

a

.2

coupling

string

which

in

X3]

tr[X,

-

units

179

simply

is 1

D-Obranes

consider

have

.

trXX'

simply However, we know that of an eleven dimensional theory spacetime, the D-Obranes have

tion

Using

quantum mechanics.

of this

and Branes

eleven

dimensional

given by

momentum pil

I P11

with

R the radius of the

physics that in

the kinetic

(A.9).

In

of the eleventh

defined

partons

(A.9)

R' The way to relate momentum frame

dimension.

in

(A.8)

in

term

=

infinite

the

coincides

with

the

(A-5)

equation

(A.8) is

for

to

the

observing given pil

using the relation

fact,

R

(A.10)

gl,

=

the galilean 1, we notice that (A.8) is precisely choosing units where 1, the worldvolume of mass -1. Thus, we will interpret for particles lagrangian g infinite momentum frame of -the Mas the dynamics of D-Obranes, (A.8), D-Obranes. theory the brane spectrum directly in deriving consist Our main task now will the different branes as colfrom the M(atrix) (A.8), interpreting lagrangian be necessary it will of D-Obranes. In order to achive this, excitations lective the relation to work in the N -+ oo of (A-8). Using

and

=

1' between

the

by defining

string Y

=

length,

9 13

-

where Dt Mills

=

theory.

at

+

iA,

In order

and the

tr

with to

Plank

variables,

In Y

L=

g-1131

=

scale,

and with

[2R DtY'DtY' 1

A

equal

get (A.8),

(A.11)

P)

-

pass to

we can

1p

1R[Y',

4

1,

we

yj]2

1

=

Plank

units

get

(A.12)

1

A0 piece of the ten dimensional going to the temporal gauge Ao

Yang-

the

=

0 is all

what is needed.

Now, some of the ingredients namely, the matrices P and Q matrices,

any matrix

Z

can

introduced defined

be written

in

in

(1.87).

chapter In

I

terms

will

be

of this

needed; basis

of

as

N

z

=

E n,m=l

Z'r1'mpnQm.

(A.13)

180

Usar

Taking

into

G6mez and Rafael

HernAndez

that

account

=

QPe21ri/N

e'f,

Q

PQ

(A.14)

define

we can

P

=

=

e'4,

(A.15)

with 27ri

(A.16)

N

(A.15)

Replacing

(A.13)

in

we

z

get

=

E Zn,me inp eimd

(A.17)

n,m

looks

like

the Fourier

is that

this

function

which ence

and

operations

the

e.,

(A. 12); L

P11

we

then

dpdqk'(p,

=

2

f

I -+

-

N

becomes the

conmutator

what

In the N -+

N

---

q)

Z (p,

function

only differby P this interpret

The

.

quantum phase space defined oo

[P, 4]

limit

limit,

we can

Thus,

0.

=

Z (p,

replaced by functions become, in this limit,

[X, Y]

in

a

a

be

can

trZ

i.

of on

in this

since

as

the matrices

The matrix

(A.16).

satisfying classical,

4 variables

quantum space

limit,

transform is defined

q),

in the N

--+

00

by (A. 17).

defined

as

q) dp4q,

Z (p,

[aqX19py

Poisson

-

aqyapXl

(A.18)

Now,

bracket.

we can

use

(A.18)

get is

&'(p,

q)

f

-

Pi

I

dpdq[i9qY'49pYj

-

aqYjapY']

,

(A.19) where pil The interest of (A.19) is that this result R eleven diemensional lagrangian for the eleven dimensional

light interpreted the

toroidal

A.2

Toroidal

simple. to

Wewill

i in

(A.19)

directions

goes from

to the

up the %

is

with

the

supermembrane in 9, which can be

1 to

supermembrane

worldvolume.

of M(atrix) consistency Next, we will try to define

of the

Compactifications. of toroidal consider

dimensional

ten

clear

write

that

alraedy a good indication of M-theory. as a microscopic description of (A.8). compactifications result

The definition with

Notice

the transversal

as

The previous

theory

frame.

cone

coincides

N

procedure,

[145]

compactifications the worldvolume =

1

of

supersymmetric

we

will

keep

all

of

M(atrix)

lagrangian Yang-Mills

indices

for

of

theory is quite D-Obranes, starting

in a

R'

while,

x so

S'. that

In order we

will

Strings

Fields,

181

and Branes

(A.20)

X?l

k,1

for

the

X'.

matrix

Now, if

we

live

to

in

R'

different

hence label

k and 1 will

The indices

D-Obranes

force

S,

x

D-Obranes.

S'

interpret

and

as

(A.21)

R/F,

21ffl, we can think by a vector e of copies of each D-Obrane, parametrized by integers n, depending on the cell should be changed to of R/_V where they are. Then, (A-20)

with

F

lattice

dimensional

a one

defined

=

X71

k,m;l,m

We can

,C

now

integers.

are

forget

about

the indices

The

lagrangian

(A.8)

Now,

we

should

imposse

Xnm,

write

to

where

q

symmetry

n

and

M

with

with

respect

to

the

(A.23)

r

of

action

F,

implies

XMn -XMn

X,',,n The

1,

becomes

_X3mqX'n)(X',X3m-Xn,X'm)jr n

2g

which

meaning of (A-24) in X1 for

difference of the

by

k and

then

-[trMin-k an+-tr(Xm'qXq'n 2

=

(A.22)

-

one

compactified

Xoi,n

index,

'C

1

tr''

2g

n

that

n-1

XM-1

n-1

2-xR1

=

is

Xm'-1

the

+ 1

trSn (Sn )+

-

-

n

On,

(A.24)

n-1,

X1 is periodic, D-Obranes, is simply

matrices

can

(A.23) 1 2

be

so

that

the

simply

the

length labelled

becomes

trT

-

j=2

=1

m

1i

coordinate

and

n D-Obranes, Using (A.24), Xn, and the lagrangian

n

'

+.Xm'm--1

direction. =

i

jk(Tik)+],

n

n

(A.25)

j,k=2

where

Y

Snj

QXq,I Xnj -qD

-

27rRnXnj,

q

Tnj

k

=

E[xi,

q

xkn-ql-

(A.26)

q

Once

we

(A.21),

get lagrangian

we can

compare it

with

the worldvolume

fact, for D-lbranes the worldvolume lagrangian is 1 + I dimensional theory, with gauge fields A' and A', super Yang-Mills We Yj (with j in the adjoint and matter fields representation. 2,...,9) A' 0. On the other hand, work in the temporal can then gauge, fixing takes form D-Obranes to D-lbranes. on S' Hence, on T-duality performing the dual S' the worldvolume lagrangian for D-lbranes should coincide with

lagrangian

for

D-lbranes.

In

=

=

C6sar

182

G6mez and Rafael

of D-Obranes

that

worldvolume

'C

R9

in

lagrangian

Hernindez

S',

x

i.

in the

f

dx

_7rRl 2g [

can

be

compared with (A.26)

dt

I

k'k'+

tr

S',

=

[y2,y3]2_

tr

2

R'

radius

with

A' A'+ I

tr

(A.25).

lagrangian

with

e,

dual

tr

The D-lbrane 1

2-7rR

[01Y -i[A

1, yii?, (A.27)

of

Y'(x),

X,1,

and

as

if

the Fourier

we

just

n

as

the Fourier

modes

A'(x):

modes of

A'(x)

X'

interpret

E einx/R' Xnl,

=

n

Y'

(X)

E

=

e

inx/R'

Xni-

(A.28)

n

Hence, ified

readily

we can

following

the

induce

Td is equivalent

on

M(atrix)

result:

theory compactdual on the Yang-Mills

d + I

supersymmetric the time direction,

to

j d x R, with R standing for and the supersymmetric reduction from N I ten diYang-Mills theory defined through dimensional mensional Yang-Mills theory. This is a surprising result, connecting M(atrix) with Yang-Mills with far reaching a relation compactifications contheories, =

sequences,

some

M(atrix)

A.3

of which

Theory

simple teresting

cases.

Let sides

of

and

defined

wrapped We will

case

consider

on

first

concerning

Li be the lengths

on

we

the

,

.

.

M(atrix)

compactify the U-duality of T'.

length.

on

The dual

torus

will

be

t

will

then

an

be defined

12 =

-

8_'

in-

13P

with

(A.29)

Li

of the eleven

In terms

have

T 4, which

symmetry.

length

1, the string

what follows.

on T d, as supersymmettheory, compactified tdXR i. e., as the worldy'olume lagranglan td Let us then work out some dual torus,

zi with

in

Quantum Directions.

M(atrix)

Wecan then represent ric Yang-Mills theory of d D-branes

will

we

dimensional

Planck

scale,

0S,

1p,

(A.30)

and therefore

zi Let one

unit

us

now

of p1l,

consider and

one

the infinite unit

13 =

P

LiR

(A.31)

'

momentum frame

of momentum in E

R

pi=

-

2P11

2L?

71

some

energy internal

of

a

state

direction,

with

Li,

(A.32)

Strings

Fields, This

corresponds,

state

with

tion

ification

the

A(Ci)

line

means a

(recall

through

flux

that

X' behave

components

line

Yang-Mills,

supersymmetric

Wilson

compactified

Wilson

trivial

non

in

trivial

a non

and Branes

configura-

to a gauge in the toroidal

Yang-Mills

as

Ci. This

energy

183

compactfields.

is

This

given by

2YMZ2 9S i

(A-33)

ZI Z2 Z3 Z4

(A.32)

Identiying

(A.33)

and

we

92SYM Using (A-31)

we

get

Z2 Z3 Z4

R Z,

2Li'

Z2i

get 16

R3 El E2 E3 Z4

2

9 sym,

P

that

2

(A-35)

2LlL2L3L4R'

2 16

P

which

(A-34)

-

expected

in 4 + 1 dimensions, has units of length. that M(atrix) we expect compactifications, 4 4 will reproduce type IIA string on T theory on T that, has been derived in under the U-duality chapter III, is invariant group, S1(5, Z). Thus, our task is this U-duality to unravel invariance, supersymmetric considering Yang-Mills x R. From (A-35), on j 4 we observe a clear S1(4, Z) invariance of the gauge means

g

,

as

From the definition

theory.

M(atrix)

exchange

These transformations In

ant.

of

order

extend

to

needs to be defined. constant

itself

in

has dimensions

described

by

of a

of dimensions

this

symmetry

A way to do this is 4 + I directions that,

length.

In

this

5 + I dimensional

Zi, with

way,

all

leaving

Zi, S1 (5,

to

Z),

usingas as

theory,

with

an

product

extra

such direction

clearly

be

can

we can

their

think

that

seen

the

Z5

coupling

from

M(atrix)

space dimensions

invari-

dimension

(A.35), on

a

torus

T 4 is

T',

1

This is exactly the 1, 2, 3, 4, and Z5 LIL2L3L4R* understood same picture we have in M-theory, limit of as the strong coupled type IIA string theory. There, we associated the RR D-Obranes with KaluzaKlein modes of the extra dimension. In the gauge theory context we should look for objects in 4 + 1 dimensions, that can be interpreted as Kaluza-Klein modes of the extra dimension As candidates to these required by U-duality. Instantons with the H3 homowe can use instantons. are associated states, topy group of the gauge group so that, in 4 + I dimensions, they look like with the gauge coupling conparticles. Moreover, their mass is given by stant

(recall

Therefore,

that

using

the Kaluza-Klein We can,

F1

i

is

(A.35),

=

=

the we

for

action

get the desired

modes of the extra

fact, try using string

the

instanton

result,

in

namely

3 + 1

that

dimensions). instantons

ar

dimension.

understand

of dynamics is playing supersymmetric Yang-Mills as the worldvolume theory on T 4, with gauge group U(N), can be interpreted for N fourbranes of type IIA, lagrangian wrapped around T 4. In M-theory, the role

here,

in

to

theory

what

language.

kind

The

C6sar G6mez and Rafael

184

eleventh

the internal

and

is the correct

instanton

and the

eleventh

dimension.

we

picture, expected

fivebranes

as

dimension.

direction,

the extra If this

fourbranes

this

interpret

we can

Herndndez

When we

effectively we can mass

get

a

to

wrapped strong coupling,

5 + I dimensional

around

in

we

gauge

by comparison of the 4 wrapped around T and

check it

of the

of the fivebrane

The energy E

partially

move

would then

mass

of the

the internal

be

LIL2L3L4R =

open

theory.

(A.36)

16

P

which

exactly

is

the

mass

of the instanton, I 9

In order

gl,

and

=

the scales

R

we

described

theory.

the

dimension R3-d

92SYM It

is

equal a

zero

barrier

strong

(A.38)

form

clear

gives

a

appears

coupling

coupled

d

in

in the

for

that

weak

=

4.

field

above, it would be convenient to Using relations (A.32) to (A-35), d, 3d-6

is

(A-38)

Lqgd-3

d < 3 the

limit

supersymmetric In

fact,

theory.

for

of string

coupling theory. Yang-Mills

d > 4 the

One of these

of the quantum dimension

generation

(A-37)

P

entering generic

for

get,

LlL2L3L4R 16

the effect

understand

to

briefly

discuss

I 2

needed for

limit

g

copling U-duality.

strong

constant

However, 0 leads

-+

effects

to

is the

Acknowledgments work is partially supported by European Community grant RXCT960012, and by grant AEN-97-1711. This

ERBFM-

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q-Hypergeometric Representation Vitaly

Functions

and

Theory

Tarasov

St.Petersburg

Branch

of Steklov

Mathematical

Institute

Introduction Multidimensional

hypergeometric theory

functions

well

known to be

closely realgebras and quantum groups, see [27], [35]. The basic way to connect these subjects goes via integral for solutions of the Knizhnik-Zamolodchikov representation (KZ) equations. There are three the most essential points on this way. First, one consideres the twisted de Rharn complex associated with a one-dimensional local system and its cohomology and homology groups, the pairing between the twisted differential forms and cycles producing multidimensional funchypergeometric The hypergeometric tions. functions obey a system of differential equations which describes of the Gauss-Manin connection sections periodic associated lated

to the

with

the local The

representation

and it turns

functions KZ

The reduces

geometric a result,

groups with coefficients in terms of the representation

are

in the local

theory

the system of difference equations by the Gauss-Manin connection can be identified in the

appearing

homology in

groups

coefficients

in the local

one

get

can

a

geometric

proof

of the KZ equation

case

associated

with

quantized

of vertex

of the

form factors

in massive

qKZ equations solutions

operators

qKZ equation were

models, of the

cf.

of quantum affine had been considered

integrable

derived

[15]

as

a

a

the

natural

theorem

sernisimple

Lie

the quantum group is

closely

algebra

Uq (g),

a

algebras. earlier

see

difference

An important in

[26]

as

special equations for

models of quantum field theory. for correlation functions

are

that

the KZ equation in a suitable in [9] as equations for matrix

equations

and references

qKZ equation

with

(qKZ) equation

Knizhnik-Zamolodchikov

of the KZ equation and it turns into The qKZ equations had been introduced

integrable

system have

of the Kohno-Drinfeld

associated

analogue elements

with

of the

[35].

limit.

nat-

theory.

representation

with

a

representation theory of quantum groups. This the problem of calculating monodromies of the KZ equation to a between twisted problem of computing certain relations cycles. As terms

given by the R-matrices

The

system admit

Kac-Moody algebras, for the hypergeometric

of

out that

induced

the monodromies g

Lie

system.

equation

description

are

Kac-Moody

cohomology

description

ural

of

therein. related

Later in

Integral representation to diagonalization

the

lattice

for of the

K. Behrend, C. G´omez, V. Tarasov, G Tian: LNM 1776, P. de Bartolomeis et al. (Eds.), pp. 193–267, 2002. c Springer-Verlag Berlin Heidelberg 2002


Vitaly

194

transfer-matrix Bethe ansatz

Tarasov

of the

method

corresponding

integrable

lattice

model

by

the

algebraic

[32].

in of the qKZ equation for solutions are studied representation Integral only in the simplest sf, case. But the form of the results suggests that extended to the general case. it is plausible they can be naturally It is shown in [30], [31] that the geometric picture for the KZ equation multidimensional connects can be naturally quantized and its quantization with the representation functions theory of affine quantum q-hypergeometric i.e. Yangians, quantum groups. quantum affine algebras and elliptic groups, with a de Rham complex associated difference One can define the twisted local discrete one-dimensional system, and the corresponding (difference) cohomology and homology groups, the pairing between them producing qThere is a difference functions. analogue of the Gauss-Manin hypergeometric functions the q-hypergeometric obeying a system of difference connection, equations which describes periodic sections of the discrete Gauss-Manin con-

detail

associated

nection

Both

the

with

the discrete

cohomology

local

system.

homology groups in natural description and

with

coefficients

in

a

dis-

terms of the representation system have a the system of diftheory of affine quantum groups. This allows to identify functions induced by the discrete for the q-hypergeometric ference equations and to express the transiwith the qKZ equation, Gauss-Manin connection solutions of the between asymptotic tion functions qKZ equation via suitable crete

local

analogous It odromies of the KZ equation. and the cohomology homology R-matrices,

which

is

to the

Kohno-Drinfeld

is remarkable

that

theorem

the

on

in the difference

case

mon-

both

represented spaces quantization brings up more forms" and "cycles". symmetry between "differential of the above mentioned geometric There are three versions construction, For the rational and elliptic case considered the rational, one. trigonometric functions in [31], are given by multidimensional integrals q-hypergeometric in terms of the rational is written the of Mellin-Barnes qKZ equation type, and the tranof modules over the Yangian Y(51,), intertwiners R-matrices, intertwiners functions sition are computed via the trigonometric R-matrices, the of modules over quantum loop algebra Uq( [:). in [30]. In this case the qKZ equations is The trigonometric case is studied functions the transition R-matrices and of the in terms written trigonometric of modules intertwiners via the dynamical R-matrices, are expressed elliptic the over quantum group Ep,.y(s[,). elliptic In the elliptic system of difference [11], the corresponding case, see [10], the of the modification is some quantized KnizhnikqKZ equation, equations Its solutions Zamolodchikov-Bernard are given by elliptic (qKZB) equation. functions. the For of elliptic qKZB equaq-hypergeometric generalizations for the hypergeometric solutions tion one consideres a monodromy problem functions between asymptotic the transition instead of calculating solutions; of functions

very

similar

to

groups each other.

can

be

So the

as

certain

q-Hypergeometric

Functions

and

Representation

Theory

195

for

the qKZB equation with the elliptic modulus p and the step a the monmodulus a odromy problem produces the qKZB equation with the elliptic and the step p. In a sense, this in the elliptic means that case we reach complete symmetry between cohomology and homology spaces. In these lectures the relation of the q-hypergeometric we review functions and the representation theory of quantum groups via the qKZ equation in the trigonometric To avoid introducing of notions case. and notations a lot in advance we start the exposition from relatively elementary constructions in Section3. satisfied a remarkable formulating identity by multidimensional the hypergeometric Riemann identity q-hypergeometric functions, [28], see Theorem 3.1. In Section 4. we outline the proof of the hypergeometric RieIn particular, down a system of difference mann identity. we write equations for q-hypergeometric functions and give asymptotics of the q-hypergeometric functions in a suitable In the asymptotic zone. sequel we explain the geometof the system of difference ric origin section equations as a periodic equation for a discrete Gauss-Manin connection, the relation as well with as expose the representation and the elliptic theory of quantum loop algebra Uq( [,)

Ep,.,(s[,).

quantum group

Technically,

with

the

representation theory is described via version of the tensor cotrigonometric ordinates transforms the difference associated with the discrete equations Gauss-Manin connection while the elliptic to the qKZ equation, of version the tensor coordinates is responsible for expressing the transition functions between asymptotic of the qKZ equation via the dynamical elliptic solutions

the

the

so-called

tensor

relation

coordinates.

The

R-matrices. In

addition

qKZ equation qKZ equation taking the

to

of the

solutions

we

consider

values

in

a

the

dual

tensor

qKZ equation, product of Uq(,S[.,.)-

of the dual qKZ equation modules, and solutions taking values in the dual the of the the spaces qKZ equation we identify geometric picture space. Using of solutions of the qKZ and the dual qKZ equations with the tensor product of the corresponding modules over the elliptic and its quantum group Ep,.,(sf,) dual space, respectively. In this context the hypergeometric Riemann identity the hypergeometric solutions of the qKZ and dual qKZ equations means that transform the natural of the to the natural pairing spaces of solutions pairing of the target solutions T1 and TI* of the qKZ namely, given respective spaces; and dual qKZ equations we have that

(ValueT/*, In

particular,

deformation cf.

ValueT/) one

of

can

target say that

Gaudin-Korepin's

spaces

the

spaces

hypergeometric

formula

for

of solutions

Riemann

norms

identity

is

a

[17],

of the Bethe vectors

[32]. We preface

dimensional structive

the

discussion

examples, both for understanding

of the the

the

multidimensional

differential main

ideas

and difference in

a

simpler

case

one.

with

the

They

context.

In

one-

are

in-

these

Vitaly

196

Tarasov

also pay attention to another appearance of the which is ignored in the multidimensional

examples

we

Riemann

identity

we

show that

relation

one-dimensional of

local

homology

for

classes

identity

Riemann

(co)homology

the

of intersection differential

in the

is

an

with

groups

system and the dual local

analogues

difference

obtain

hypergeometric

the

Riemann bilinear

cohomology

of

The deformation

case.

coefficients

In other

system.

forms

hypergeometric case. Namely, analogue of the in

words, classes

a

we

and

of the Riemann

Riemann surfaces was obtained in [25]. hyperelliptic the the is main glance, hypergeometric Though 4iemann identity in it fact of of the role which be stem a can lectures, plays a depicted topic knowledge, and which allows using only things of a common mathematical themes in the process of its proof. At the more sophisticated us to introduce Riemann identity includes moment the proof of the hypergeometric virtually all essential results concerning the geometric picture of the qKZ equation, so it serves well as an entering point to the subject. the paper were obtained in The results of the joint as a part presented A. the author with Varchenko of together project developed by /, University and G. Felder, ETH Mrich. I am very grateful North Carolina at Chapel Hill, collaboration and valuable I thank organizers discussions. to them for fruitful

bilinear

for

relation

the

at first

CIME

of the

school

summer

and all

the

lectures,

1.

One-dimensional

In this

section

period example

we

.

.

differential

considered

was

Let zi,

participants

consider

.

,

be

z,,

for

opportunity

their

to

present

interest.

example

example of

an

in

the kind

for

of the school

of the twisted

matrices

Cetraro

at

Riemann bilinear

the

for

relation

de Rham (co)homologies.

one-dimensional

This

[4]. complex numbers.

distinct

pairwise

Fix

noninteger

com-

n

plex

numbers

A,,

.

.

.

,

A,,

such that

0

A,,,

Z. Consider

a

one-dimensional

M,

local

system

n

4

(t;

zi,

-

-

-

,

Zn)

=

11

(t

-

Zm) I-

-

m=1

The local

system operator

boundary

determines

the

differential

twisted

Op Namely, for

a

f !P-1

d4i

.

(t)

functionf

dp we

and the

twisted

have

n

d,p f

=

d

f

+

=

d

f

+

f

1:

Amwm

m=1

where

dt Wm

and for

function

a

contour

g(t)

-y the

t

-

twisted

Z"'

'

boundary

m=

1,

..

.,n,

ap -y is such that

for

any rational

q-Hypergeometric

Functions

ly ) ,i

Let in

C\ Izi

simple

d,5

!P

g

be the space of rational functions and let be the J ,j C Tl z,, I ....

poles and vanishing at infinity. the de holomorphic

Izi....

z,,}.

cohomology The following

For

is

statement

dp )

in

197

.

one

variable

subspace

Rharn

(.) ,j)

f2l

E

w

t

Theory

by Lwj

are

with

regular at most

(J ,j),

(fl*

complex

denote

which

of functions

E

p

dp ) on H1 (S?*, d,5 ) its

straightforward.

quite

1.1.

dimH1(f2*,dq5

that

the

n

Moreover,

1.

-

.

is clear

It

form

a

Representation

class.

Proposition H1 (f2*,

g 1,9 ,

=

Consider C\

and

differential

forms

Wn form

wi,

a

basis

S? 1

in

(Yj)

and n

E Amwm.

4 1

m=1

Set pm =wm-wm+ll

Corollary

).

A contour

-y

defines

S?1 (J ,I)

forms

The

1.1.

H1 (S?*, d,,,

m=

1,---'n-1.

cohomology

is

cycle,

twisted

the

on

form

a

basis

in

of differential

space

rule:

(-Y, (,)) If

Lpn-jj p

(-y, .),p

functional

linear

a

by the

Lpjjp,...'

classes

9,p

i.e.

=ly

,,

'fi

W

0, then the functional can be (-Y, -)"' functionals on space H, (Q*, dp ) of linear the cohomology space H1 (f?*, dp ). Below we give examples of twisted cycles; twisted namely, we will describe certain important cycles 71, -,Yn-lTo simplify from that zi, now on we assume notations, zn are real and -y

a

considered

as

an

element

y

=

of the

-

-

zi

<,

< zn-

a

small

positive

=

exp(27riA,,,),

number

A, oriented oriented

-.

=

from z,,, + s to zm+I circles:

starting

zm

E,

,ym

=

Am+

Aj,

Let

[Z"'

I

"In-1

+ E, Z,,,+,

ft I It

respectively.

1'...

M=

Z,4

and let

Z at

-,

Wedenote

m Fix

-

-

.....

-

In.

be the

following

intervals:

6]

Z

be the counterclockwise

Z-1

Set

1

-

m+i

(1.2)

Vitaly

198

that

Tarasov

is >

W+

f

> +1

and

we

I arg(t

<

-r

arg(t 17r

+

arg(t

It

is

to

smooth deformations we assume

z+1

points to

that ....

ending a

-y and the

Z,

27r,

t E

Z+ k

t E

Zk

<

Ir

(71, -) (71, -) (-yn-1, 'An- 1, Zi A,, -

C\ Izi

,

Z

....

is

k

respectively.

abusing

In what follows

differential

form

corresponding

w

and its

the dual

local

0, depend

7m

Tn

=

on

pairwise distinct going from zj counterclockwise starting

there

each A,,,

notations

,

if it

71,

are

is

zm

an arc

class

1

well

as

n.

distinguish as a cycle

confusion.

no

form

rn

do not

Lwjp

causes

7n-

zn 4- e,

=

usually

we

cohomology

cycles

i94

.....

(-y, -),p

functional

1,...,n.

=

cycles:

loop going around z,,, we have Originally,

a

1. 2. The twisted Prop osition H, (S?*, d4i ). mology space

Consider

k=1,--.,n,

,

e. (-yn-1, .).p do not under coherent change -),p Zn:: in C \ Iz,.... zn; namely,

such that

zn

n,

=

do not

...,

-

-

1

1<1
twisted

are

7j,...,7n_j

....

of

E

at zm,

Remark 1. 1. between

t E

at any moment of the deformation

and each

z,,-,+,

and

z

k,

1
functionals

the functionals

Moreover,

prescriptions:

I

7r/2,

<

that

see

and the

1,

-

<

Zk)

-

Al

t E

1

following

the

tEZI

Zk)

-

0

ir/2,

Zk) I

-

arg(t

<

easy

n

<

arg(t

0 < -ir

Zk) 1 :5

-

Zk) I

-

by

of the function P

branches

fix

basis

a

in

the

ho-

system n

V(t;

Zn)

Z1,

=

11

(t

obtained twisted

via

replacing

cycles

-yj

f

(-YM' using

the

same

way

[20]; they

=

are

Am

VW

-

'a-

intervals

The intersection

-An- Wedefine

-

accordingly

the

-yn-1:

.....

1YM/

An by -Al,

A,,.

(1.3)

Zn)

zi,

M=1

6M+1

6M+1 6M+1 J -

An-,

A1 ......

numbers -yj given below:

o

-ym'

0;+1

(

+

6M

Vw +

6M

+1

and circles can

_TM+

Z:':,1

be calculated

(1.4)

1

L Z'

+

n

in

a

VW, as

simple

(1.5)

before.

geometric

q-Hypergeometric -YM

7

0

^ M_j -YM -Y,

completely

71

M

determine

1,

g

H, (f2*,

:

there

is

form.

by requirement 01

extend

f

=

it

1.6

-+

pairing

)

to

a

C,

-+

pairing (C

d,,5,

*

01 (Xi)

(9

(1.9)

Formulae

g

C

=

(1.12) in [4]

0.

was

explicitly

forms

described

pi,

pn-

were

I

cal-

therein: PM *

PM-1

0

pig

PM

P.

W,n

more

pl,

numbers for

the intersection

see a

one

can

Tm

Al <,

-

-

-,

-

.

-

,

Pn-

1

Can

the

in the difference

the intersection

m=

numbers

-

-,

+An'

formulate

the

Riemann

bilinear

(1.16)

1,...,n,

1,...,n,

-

Al <, can

find

+An

1

we

feature

similar

.

transparent.

-

Now

(1-15)

> 1.

Wn:

Wn

W1 0 WM

MI

(1.14)

1,

from

and 1.13

1.12

-

numbers of the forms

Wewill

even

1.1,

-

(1.13)

n,

...,

1'...,n

11

=

degeneration

7j',...,-yn_j.

wi,

0

,

Am 1,...,n,

l'M

1,

=

m=

-

-

the intersection

be

Tn

,

AMAM+1

PM* PM-1

rational

relations

of the forms

-

0,

7j,...,7n_j, where it will

Using

"M

that

by

be obtained

=

=

is remarkable

case,

nondegenerate

numbers of the differential

and the intersection

cycles

(.FI)

> 1.

(1.10)

H1 (Q*, d,,,,

0

us

g

MI

nondegenerate.

is

S?1

e

(1.8)

H, (S?*, d,,,,

t2i

d,p f :

0

natural

Let

:

0

It

dp )

a

11

-

-

form

H1 (S?*, dip )

:

the intersection

9

(yj oym)n",=1 1,

=

1,

n

n,

M=

the intersection

e

culated

(1.6)

1,...,n,

M=

M

the matrix

The restriction

199

(1.7)

0,

=

is known that

called

Theory

_YM1

o

It

W -+i

-

-

_YM1_1

0

0

0

that

Notice

(1

1

Representation

and

Functions

relation

154

m.

(1.17)

for

the

twisted

de Rham (co)homologies. Theorem 1. 1. relation

[4]

For any

differential

forms

w, w'

E

S?1

the

following

hold: n-1

27ri(w

0

W1)

=

E 1,M=1

(9-

I

)IM

I

I

W

(1-18)

Vitaly

200

Twasov

difference

One-dimensional

2.

C'

Let

10}.

C\

=

example IpI

Fix p E Cx such that

pZ

Set

< 1.

fps I

=

s

E Z

I.

Let

00

(U).

(U; P).

=

][1

=

(1

P'U)

-

s=O

0(u)

and let

(u),,o(pu).(p).

=

For any vector V.

on

Fix

denote

we

complex numbers

nonzero

lowing

be the Jacobi

V

space

notations

compact

x

by V*

space of linear Yn- We will

Xn, Y1,

xl,.

(Xi,

=

theta-function. the

Xn),

(Y1,

Y

Yn)-

-

functionals use

the fol-

We assume

that

for X1,

1,

any -

-

-

/Y.

=

1,

m

n)

,

X1

Y1,

simple

poles

-

-

Y[x; y]

Let

X1/X- Vp" .

.

and any It

Fix

at

Let

zero.

-

-

-,

dim.F[x]

that E

a

generic

P-S-1

CP1 \ IX1,

is clear

and third

functions

in

and say that

cases,

hold. variable

one

with

at most

points

at

pole

in

second

if the above conditions

be the space of rational

psXM,

regular

the

m in

are

Yn

,

-

1

n,

,

.

/Y. V pZ'

Yi

C'.

.97[x] Xnj =

For any

Ym,

,

Y[x]

and

dim P [x]

a] f (t)

W(t;

X; Y;

: [x; y]

C

vanishing =

(t)

functionf

D[x; y;

E

m

zero

or

subspaces of functions infinity, respectively.

at

n.

set

o

=

be the

at

Z>.v,

(t;

x; y;

a) f (pt)

-

f (t)

(2.1)

where n

a)

=

H

a

M=1

We call It

easy

operator

D[x;

tional

space

to

y;

paper

we

a].

) [x; y]

top cohomology

isomorphic

see

that

So

to

the the

we

quotient equality

the definition

y;

a

a] f F[x; y]

discrete

y;

a]

space

a]

=

:

the is

twisted

local

system

o(-; local

J [x; y] ID [x; J [x; y] ID[x;

T[x; y]

a], -+

(2.2)

y;

a].) [x;

with on

x; y;

y], y]

and denote

by

H'[x;

y;

a]

respect to the C' with the func-

a),

see

[30].

The

system is canonically

a]J [x;

y;

differences.

total

invariant

of the discrete

of the space H1 [x; y;

L.]

-

coefficient

connection

H'[x;

H1 [x; for

space

have

and the

space

take

the

tlym t1XM

-

1

of the form D[x; y;

functions

is

I

see

[30].

In

this

q-Hypergeometric

201

Theory

Representation

and

Functions

projection.

the canonical

n

that

Say

if

generic

is

a

V pz

a

and

H X./Y.

a

V P.

M=1

Proposition

Let

2.1.

over,

[.F[x]] The

be viewed

described

integral

geometric

z

i4 z'.

:A lp'xn I,

r

I

spanned by

y]

T, [x;

T,

I Itl

m=

I

r

=

y]

U [x;

Let

=

For any r, r'

Lemma2. 1.

y]

in U [x;

integral

I

p'y,,,

It

1

=1 M

0(tlx.)

is

a

see

4i(t;

that

if

-

Set

s

E Z

(2-3)

I. are

homologous

x;

y)

set

f (t)

dt t

=

holomorphic

=

Int

=

[x; yJ (f ) f (t) such

function a

on

right hand is functionf,

in the

if

a

(2.5)

.

that

the

C' and

f (t)

say by Laurent function:

n,

(2.4)

)

same reason

[x; y] (fv)

dim.F,11 [x; a] following

be the

y]

fj,[x;yJ

F,11 [x; a] be the space of functions

can

circle

Y_ I
is easy to see that the integral due to Lemma 2.1. Moreover,

f (pt)

Let

sEZ, IpI

n,

in U [x;

27ri

=

Int

One

E Z

C[PSYM].

T, [x; y] and TrI [X; y]

=

depend on r f (p t), then for the

does not

f (t)rjn

s

1:

m

the contours

[x; y] (f )

converges.

given by fV (t)

Let

can

each other

1,...,n,

m=l,

functionf

Int

side

a]

small

oriented outside

are

-

>r

straightforward. (t) holomorphic

is

For any

if the

y;

annihilating using the hyper-

-

proof

The

\ lp'xm,

C'

on

counterclockwise.

oriented

C[P'Xn]

sEZ, Jp4lx7n I

m=l,

HI [x;

More-

n

1:

+

n.

such that

r

54 JP'y"'J'

t E C

C[z'l

and

n =

=

functionals

such functionals

C[z]

the circles

r

T, be the circle

Let

a]

a].

y;

by C[z] the counterclockwise

number

positive

a

y;

functionals

of linear *

H'[x;

below.

For any z E Cx denote around z. We assume that

Take

a]

We construct

differences.

total

y;

H'[x;

=

(J [x; y])

of

subspace

a

as

the twisted

L.F'[x]J

=

proof is straightforward. homology space HI [x;

The

Then dim

be generic.

a, x, y

series.

product

Vitaly

202

Tarasov

(t;

!D

(t.lx.).

Y)

X;

R (t.1y.).'

-=

(2.6)

M=1

It

solves

a

difference

equation n

(p t;

fi

Y)

X;

(t;

p

=

X;

Hi

Y)

1

-

-

M=1

cf.

(2.2).

di(t;

We call

WE F,11

[x; a]

y)

x;

the

tlym t/xM

phase function.

it

For

(2-7) any

w

cz

) [x; y]

and

set

[x;

I

y;

a] (w, W)

[x; y] (w

Int

=

WP(-;

x;

y))

(2.8)

form the

integraL Notice that for hypergeometric number finite of small circles in a only given of of the the contour the definition t, a pole integration containing (2.3) to the hypercontributing integrand 4i(.; x; y) w Winside and, therefore, Int [x; y] (4 (-; x; y) w W). geometric integral defines the hypergeometric The hypergeometric integral pairing We call

the

integral

functions

I

[x;

We will

y;

also

of this

and Wthere

w

a]

P[x; y]

:

consider

it

by the

Proposition

2.2.

denoting

.

li(t)

For any

[x;

follows

an

F,11 [x; a]

E

J [x; y]

=

[30]

4i(pt)

isomorphism

For

generic

i [X;

Y;

of

-+

I

-+

[x;

a] (f, g)

y;

.

(2.9)

map

(.) [x; y]) *,

vector

:

I

and WE 411

a]

w,

w(pt)

[x;

spaces.

0

=

W(pt)

and

y;

a]

[x; a]

-

we

have

.

-+

di(t)

w(t) W(t).

(2.5).

maps the

a, x, y the

-' eli

W)

[x; a]

imply that

(2.8)

from

a]

y;

(2.7)

and

means that proposition space H, [x; y; a].

Theorem 2 A.

linear

:

a] (D [x;

(2.2)

Hence the statement

is

w

y;

W(t)

homology

0 g

letter.

D[a]w(t)

The last

a]

y;

same

(2.1),

Formulae

C, f

-+

corresponding

the

I

Proof

F,11 [x; a]

0

i [x;

is

space

map

H, [x;

y;

a]

F,11 [x; a] into

the

q-Hypergeometric

To obtain

parameters

a

xj,

Define

.

and

Corollary

local

system

and

replace

.

.

with

x,

,

yi,

Shapovalov

the

S [X; It

from Proposition [2.2] picture of the dual discrete

follows

The theorem

Y] (f, g)

.

.

y,,

,

S[x; y]

pairing

[X; Y] (fg)

Int

=

.

F[x]

E

[A.1] we 1

a-

0 P [y]

f

,

by

a

F[x]

:

Theory

Representation

and

Functions

f

=

1.

the

interchange definitions.

in all

C by the rule:

-+

,

for

203

g E

F'[y]

-

is easy to check that n

E Res (f

S[x; y] (f, g)

(t) g(t) t-1 dt) lt=x_

(2.10)

M=1 n

E Res (f

(t)

g (t)

1

t-

(2.11)

dt) I t=ym

M=1

Similarly,

we

define

S,-ll [x;

a]

y;

Shapovalov

elliptic

the

pairing

a-']

Yell [x; a] OXll [y;

:

-+

(2.12)

C,

n

Sell [x;

a] (f

y;

g)=1:

,

Res

t-1 dt)lt=x_

(f (t) g(t)

(2.13)

M=1

f

f (p t)

Since

g (p

E

f (t)

t)

[x; a],

Xll

g (t),

a-']

Xell [y;

g E

(2.14)

check that

one can n

Sell [x;

y;

E

a] (fg)

Res

t-1 dt)lt=ym

(f (t) g(t)

(2.15)

M=1

the

also consider

Wewill

S[x; y]

Shapovalov pairings

maps

S[X; y] S,11[x;y;a] Proposition

S[x;y] Sell [x; Proof. pairing

is y;

2.3.

[30]

r[y]

x,y

for

We prove

S is similar.

and functions

Consider

gn'

g,',...,

gm(t)

E

functions

gm(t)

=

Xll [y;

a-']:

I

=

Then

generic

of the

nondegeneracy

the

91,

a]

a

pairing -

-

-

,

gn G

the the

R 1=1, IzAm

linear

(2.17) Shapovalov Shapovalov

Sell. The proof Yell [x; a]:

0(tlxm)

n

as

(2.16)

0(a-lt/xm)

0(& tlxm) OWYM)

y;

(.F,11[x;a])*.

-

be generic.

Moreover, nondegenerate. a] is nondegenerate.

Sll [x;

F[X] *,

-4

Yell[y;a-1]

:

Let

:

and

0(tlxl) 0(t/Yl)

pairing pairing

for

the

204

Vitaly

Taxasov n

where a

H x,,,

a

=

and

/y,,,.

It

is easy to

8,11 (gi, gm)

that

see

=

0 unless

S'al (gm, gm ')

0/(1)

where

=

0(a-1 ) OR

-

0'(1)

_!jO(U)ju=1 du

=

n

n

11

Consider

[x;

y;

the

a]

[x;

1=1, 10M

(P)3

-

O(Xm/Yk)

k=1

The last

.

formulae

denoting

them

Theorem

identity.

It

is

by

[2.2]

clearly

imply

pairing

f:

an

Theorem 2.2.

I'[x;

an

[y] (&.Fqj [y;

y;

a]

:

1]

a-

-4

C,

[y; a-']

Jl ,jj

abstract

form of the

hypergeometric

of Theorem 1.1 in the differential

following

The

_7"

:

letters.

same

describes

analogue

a]

y;

the

maps

(.F[x])

-+

the

hypergeometric

1'[x;

C,

-+

linear

Fe,11 [x; a]

:

of the

restrictions

corresponding

a]

y;

following

T[x] O.Fjj [x; a]

:

and the I

m

=

I

11

O(xm/x,)

claim.

I

1

M=1

diagramm

a]

I

is

Riemann

case.

commutative:

[x;y;a]

(S [X;yl) ply]

(I'[x;y;a])* As

we

geometric independent Let

a

will

multidimensional

identity

does not

case,

involve

the

proof

Theorem 2.1.

of

So,

the one

hypergets

an

proof of Theorem 2.1 using Propositions 2.3 and Theorem 2.2. be generic. Then by Proposition Theorem 2.1 we have isomorphisms

H'[x;

.F[x] .F,11[x;a] which

the

in

see

Riemann

y;

HI[x;y;a],

allow

a],

P [y]

; ,,

.17,11[y;a-1]

H1 [y; --

x;

a]

,

HI[y;x;a],

the pairing us to consider S[x; y] as a nondegenerate pairing of cohomology spaces and the pairing S,11 [x; y] as a nondegenerate pairing of the homology spaces, see Proposition 2.3. Moreover, Theorem 2.2 means that these pairings are difference forms (1.11) analogues of the intersection and (1.10), respectively. To obtain an analytic form of the hypergeometric Riemann identity similar to the formula (1. 18) we have to pick up bases of the vector spaces mentioned in the commutative diagram. For any m the following n introduce functions: 1, the

=

.

.

.

,

q-Hypergeometric

(t;

W'.

(t; WM

Wm (t;

W,',,(t; where a,,,

t

YM

1<1<

0(aM't/xM) =

x; y;

II

0(tlx,,,) a)

E

Y' [y]

0(t/yi) 0(tlxl)

F

Let x, y be

,

Then

generic.

.w,, x; y) form a basis in functions w, x; y), .wn' x; y) form a basis in functions w', x; y), The functions W,, (.; x; y; a) form a W, (.; x; y; a), n F,ll [x; a] provided arl 1 <1 <mxl /yl 0 p''M

the

.

.

the

.

.

.

give

Wewill

proof

a

c).

of claim

The

-

proofs

F[x]; P[y];

the space the space basis

,

.

.

=

Proof.

a]

a-11,

F,11[,y;

(=-

(2.18)

Xll [x;

E

-

a-')

y; x;

Y[X]

YJ

-

I<1 <"'

W(t;

=

X1

-

t

E

XI

-

t

H""

YJ

-

t

1<1<m

Ym -

t

H

X"'

-

205

aflj<j<mxl/yl-

=

Lemma2.2.

a) b) c)

t

Y)

a)

x; y;

t

Y)

X;

Theory

Representation

and

Panctions

the

in

space

1.

a)

of claims

and

b)

are

similar. Since

dim Fell

=

n

suffices

it

to

Observe Wm(yl) independent. if Hn tions Wj,..., Wn are linear independent =1 provided M xilyl 0 pz, arjj
=

Wn are WI, the funcHence, holds which W", (Ym) 0 0,

the functions

show that

that

0 for M

1 <

=

0 for

=1 M

c)

is

(2.19)

or

(2.20)

m.

Wm (x,,)

W1,...' Hence, the functions 54 0, which holds provided

pz,

ajjj
I >

res

M=

0,

-

-

-

,

n

under the

0 and

Sll (Wi, W,n)

1

-

.

,

Wn are

(2.20)

.

of the

assumptions

holds

.

(2.19)

n.

..

res Wm (xi) Similarly, if rIn linearindependent

.

m.

lemma, claim

proved.

S(wl,

Lemma2.3.

w'M

where

am

=

a

111<1<m xilyl

and

I

=

m.

Ym' 0(am-') 0(am x,,,Iy,,) 0'(1) O(X./Y.) M

-

0 unless

YM

S(wm, W, Se 11 (WI, W,n)

=

0'(1)

XM

-

d

=

du0(u)IU=1

=

-

(p),

00.

Moreover,

Taxasov

206

Vitaly

Proof.

Weprove the statement

S,11.

pairing

the

(WI (t) W.' (t)) I t=x,,0

Res

hence, Sll (WI, W'm)

=

0 unless

proof

The

Sll (WI, W,n)

=

1,

Tn

for

< k <

the

pairing

unless

1 < k <

(2.15).

This

proves we

I < m,

Res

=

of the statement

see

1,

Similarly,

from the above consideration

Sell (Wm, W,n) and the second part

(2.12).

see

0

=

0 unless

Moreover,

of the statement.

unless

m<

(Wl (t) W.' (t)) I t=y ,

Res

hence,

for

Observe that

S is similar.

the first

also

part

have that

(W,,, (t) W,' (t)) I t=x_ ,

follows

from the

straightforward

calcu-

lation.

Set

Nm

-P )3 O(Xm/YM) 0(am-') 0(am xm/y.) 00

--

where

am

equivalent

to each of the

S(f, g)

1:

Theorem 2.2 is

afjj<j<mxl/yj.

=

following

formulae:

n =

W,,,) V(g, Wm)

NmI(f,

m=1

for

f

any

and g E P [y],

F[x]

E

and

n

Sell (f, g)

=E (1

X.

-

I(Wm' f) 1, (Wm' g)

/Y.)

m=I

for

f

any

Xll [x; a] and

E

g E

a-'].

Fll [y;

Set n

n

X (t,

(I-Xm/Ym)Wm(t)W'

U)

(U), X, 11(t,u)

NmWn(t)Wm(u).

m

m=1

m=1

(2.21) Then

we

have

g)

S (f, for

f

any

E

T[x],

P[y],

g E

Sell (f, 9) for

any

f

integration There

E

Xell [x; a], variables

are

explicit

g E

for

It

=

=

Fll [y;

the

&

IU

(f Wg (U),

U))

and

It

0

a-'].

Iu

(X (t, U), Here

hypergeometric

formulae

Xal (t,

for

f W9(u))

subscripts integrals.

the functions

X (t,

in

u)

It

0

Iu indicate

and Xll

(t, u).

the

q-Hypergeometric

Theory

Representation

and

Functions

207

Lemma2.4. n

t

(t, U)

x

=

t

,

U)

J1

_

U

-

M=1

(

W'. W(- t-lU)

-

(t (t

Y

U

XM) YM)

-

X"') (U

-

0(&-'

0(a-1 t1u)

n

t/U)

0(a-1)

OWY-) O(u/x.) O(u/y,,,) 0(tlx,,,)

M=1

n

where ii

a

=

11

XM/YM.

M=1

Proof

formula

second

the

We prove

proof

The

lemma.

of the

of the

first

is similar.

formula

follows 1. The general case easily for n to with induction n. respect by particular 1 and denote by j (t, u) the right hand side of the second formula. Let n functionk (-, u) belongs to J7,11 [x; a]. So the u) is proportional For any u coefficient on u: the Lemma depending cf. to W1, proportionality 2.2, suffices

It

to

the formula

prove

from this

=

case

=

x

Substituting lemma is Define

for

a

f

t

pairing

f

any

pairing

y;

a]

:

S Y,

9)

=

g E

) [y; x]. y;

coincides

[x; y]

can

J [x; y] It

0

Then

a] f, g)

: [x; y],

E

0 P [y]

.F[x]

[x;

a] (D [x;

S [x; y;

for

equality

gets f (u)

one

N, W1, (u).

=

The

proved.

) [x; y],

E

into

W1Wf (U)

=

above

the

yj

=

(t, U)

'P[y; x]

(9

(f (t)

Iu we

by

the rule:

U))

Xell (t,

9 (U),

have that

=

a] (f, D[x;

a-] g)

S[x;

y;

g E

J [y; x],

and

the

Shapovalov

with

C

-+

be considered

as a

y;

the

=

0

of

restriciton

S[x; y].

pairing

analogue of

difference

[x;

y;

a]

on

Therefore,

the

the intersection

form

S2,

of the The

that

the spaces

space X,11 [x; a]

same

degeneration

and w' M as well

as

J71

( kl)

-+

(C

forms.

of the differential Notice

o

their

as

F[x] p -+

and P [x]

can

0 and then

a

connects

0 and

a

-+

as

oo,

degenerations respectively.

with the functions w,,, functionW,,, This fact generalizes a similar pairings.

the the

Shapovalov

be considered -+

Vitaly

208

Tarasov

observation

for the intersections

numbers -yj

o

and pt

-y,,,

*

in the differential

p,,,

case.

In this

section

differential a

we are

suitable

modifications

taking

situation

a

for absolute

striving examples.

not

and difference

In

of words.

special

value

fact,

One

also

can

of the

similarity similarity

the

consider

parameter

H xn/yn.

=

But

we

will

not

do it

in these

in the

be improved

by

sophisticated for instance, 1 a

a,

more

or

=

n

a

of formulae can

lectures.

M=1

3. In

hypergeornetric

The the

of the

rest

Unlike

paper

the one-dimensional

identity

Riernann

we

consider

the

will

multidimensional

difference

case.

with

certain finiteintroducing dimensional called the hypergeometric spaces of functions spaces with pairings between them being given by the q-hypergeometric and forintegrals, mulate the hypergeometric Riemann identity. Then we will describe relaof the hypergeometric tions to the spaces and q-hypergeometric integrals

(co)homology

groups

with

of the discrete

sections

the relation

case

we

coefficients

start

in

Gauss-Manin

a

discrete

local Further

connection.

system and periodic we

will

concentrate

theq-hypergeometric integrals ory of the quantum loop algebra quantum group Uq(gt,) and the elliptic the relation Ep,.,(s[,), skipped from discussion in the one-dimensional difference example. will be basically of the one-dimenThough the exposition independent sional examples considered in the previous two sections we are going to use sometime the one-dimensional difference of example as a helpful illustration the general story. Weomit almost all the proofs for the multidimensional case to [30]. But to show the main technical a reader referring ideas we will give for the one-dimensional some proofs It also seems to be instructive case. as an introduction to [30], since the one-dimensional example is not specially on

considered

of the

Here them

we

Let

fps I

S

Let

to

C'

3. and 4.

3.1, 4.1,

are

the

concerning

hypergeometric

Riemann

[28].

due to

notations

describe

were

in order

of Sections Theorems

see

Basic

3.1

the representation

there.

The results

identity,

with

basic

introduced

make the rest =

X1. (u),,,

C\

101.

which

notations

already

are

in Section

of the paper

used all

2., but

we

over

will

the paper. Some of mention them again,

selfcontained.

Fix p E C' such that

IpI

< 1.

For any X C Z set

pX

E

the Jacobi

=

(u; p),,.

theta-function.

=

r1:0 (I S=

-

psu)

and let

0 (u)

=

(u),,,,

(p u),,,, (p),,,,

be

Fix

r

f

0'...'

=

77

called

V pZ1

r+I

Say

parameters.

k,

1 and any

-

f. Take

integer

nonnegative

a

y,,

yi,

77

the paper

over

77

X,

if for

any

have

r

V PZ

Yk 1Y.

for

:A

k

m,

Xk/Ym VPZ-

r

that

we assume

209

xj,...'

parameters

we

n,

m=

Iq

All

the

XklXm V PZi

r

complex numbersq, are generic

nonzero

that

Theory

Representation

and

Functions

q-Hypergeometric

the parameters

unless

generic,

are

other-

wise stated.

(ti,

functionf

For any

.

.

tj)

,

.

permutation

and any

[f I,

t1b

Ua > Ub

77

1
f

MCI

=

I

*

.

11

tol )

,

.

Ua > Ub

For any I

([-,,

=

-

defines

an

In)

Zn.O

-,

-

IM

1,

For any

mE

I < defines

which

a

If

Remark 3. 1.

Zn,

m,

partial

E

identify

C n and I E

X >

1

<

M,

for

any

on

the set

Zp.

a

(tj,

functionf

residue

Res

Res We often t

=

.

.

,

(f (t) (dtlt)'

(f (t) (dtlt)t

Res

.

(

...

use

(tj'...'tt)'

tj)

t

=

1....

I

U

(3-5)

1

-

In <, > 1-11 then dominance ordering

the inverse

we

t'

define 77

NI)

Z_

the 1,

point

X1....

and

) lt=t.

a

point

by

B

t*

77

1

(tj,

[,q ]

E

771-[2

X1,

)

=

I

x >

.

.

.

,

X2)

I-In

t;)

C"

in the

for

Xn) we

follows:

as

I

...

(3.6)

X2,

Xn)

....

define

a

-

multiple

) lt=t. Res

the

of f.

...

For

(3.4)

U.

partition

a

with

coincides

Zt'

(771-

[77

Sj.

group

M=

I E Z n 0 with

ordering on Zin partitions corresponding x

(3-3)

*

say that

we

ordering

introduced

For any

It,. ) tl /t1a )

0 (77

set

we

+IM,

11

if

we

(3.2)

t'.

I E Zn> 0 I

M

tO'a

-

t"

symmetric

of the

action

1, <'...'

=

Zn

Let

the

E

set

0(,q-,t",

77

I
Each of the formulae

S1

E

or

(f (ti, paper

.

.

.

,

the X

=

tj) (dtj 14)) 1 t,

-

-

-

(3-7)

(dt, It,))

I t,

=q

.

compact notations:

following

(Xi'

=t;

,

Xn)

,

Y

=

(Y1,

-

-

-

,

Yn)

-

Vitaly

210

Tarasov

For any vector

linear

operator this

In

space V

A

we

paper

X1,

Xn,

Z1,

Zn in

Y1,

-

-

,

[30]

The

Let

(P(t;

this

Yn in

77)

MZM'

(t;

X; Y;

following

be the

1

(Xm /ta)

Inj

Int

Ix,,,l

> 1 and

X1,

-

-

-

,

Xn,

-

-

-

,

Cli

in

1, jyml

(27HY

1,

>

fv

V

V

and

for

6,

n

(3.8)

n

1

(77-1ta/tb)oo define

we

1,

m=

f

'

below

(t)! (t)

the

we

(dtlt)'

(C

I Iti I of

continuation

Then

...,n.

values

t E

arbitrary

by the analytic

Yn

M

I

a,b=la:Ab

-

a

Y1,

The parameters

the parameters

(ta /Ym) ooH

I =

=

B We define

oo

<

rji=l dta/ta Int x;y;,q](fP)

(dtlt)

a

! ).

(f

Int[x;y;,q](f4 ) where

to

1

holomorphic

integral

Assume that

and for

space,

function:

functionf(tj,...'tj)

any

C2 ZM

1

11 H

77)

hypergeometric

[30].

from

correspond

YM

m=1 a=1

For

vector

integral

n

A

the dual

operator.

results

use

paper

hypergeornetric

x; y;

by V*

A* the dual

follows:

as

XM -=

3.2

denote

by extensively

we -

we

denote

with

=

set

(3-9) Itil

1,

the

=

I

parameters

77,

the

pa-

(3. 1),

the

respect

to

rameters.

Proposition

3. 1.

For

hypergeometric integral Int [x; function of the parameters. Proof. For generic integrand f (t)! (t) ta a, b

=

1,

=

.

intersections

.

.

0, ,

f,

values are

ta a

:A b,

of the

=

at

] (f 4;)

y;,q

of the

1,

ta

,

.

.

hyperplanes)

.

the parameters is

well

,

n,

s

of

PsYm

E Z >o.

x, y,,q,

and is

see a

x, y, 77,

following =

defined

holomorphic

the singularities hyperplanes:

parameters

most at the

P'xm

m=

of

values

generic

,

ta

=

P-s?7 tb

The number of

configuration

(3.10)

,

of the

(3.10)

edges (nonempty

and dimensions

of

edges are always the same for nonzero generic values of the parameters. the topology of the complement in C of the union of the hyperTherefore, does not planes (3.10) change if the parameters are nonzero generic. The rest of the proof is similar to the proof of Theorem 5.7 in [31]. the

q-Hypergeometric It

from

clear

is

as

of

Proposition

integral

hypergeometric

the

parameters

proof

the

Functions

Representation

and

[3.1] Int [x;

that

for

y;,q

(f

f (t)

4;(t;

x; y;

of the

torus

Theory

211

values

of the

generic

4;)

represented

be

can

integral

an

[x;

Int

y; 71

')

] (f

ti[x;y;,q]

where

depend It

is

fj,[x;Y;nj

V

suitable

a

see

for

that

deformation

77) (dtlt)'

(3.11)

V which does

not

f

1 the

=

(f !P)

of Int

definition

present

equiv-

is

(2.4).

the definition

to

(27ri)i

f

on

is easy to

alent

=

follows we are integrals using the hypergeometric form. f which have a certain particular symmetric functions coincide with the symmetric A-type In this case the hypergeometric integrals cf. Appendix B.. Jackson integrals,

Remark 3.2.

3.3

The

Let

F[x;,q;

what

In

(f fl only

Int

for

hypergeornetric

]

f

be the

of rational

space

hypergeometric

and the

spaces

f (ti,

functions

pairing

tj)

such that

the

product

H H (t

ri t-1 .=,

is

tj,

symmetric

a .

.

.

,

(3.2)

.T,[X;,q;

I

=

f

f (ti'.

There

Remark 3.3.

paper.

degree

.

are

I

tj)

called a

tj

tj f

...

the

are

few motivations functions

trigonometric origin of the

name

with

variables to the

respect

-1

;

f(3-13)

hypergeometric

spaces.

tj)

(tj,

(3.12)

tb

-

each of the

in

n

invariant

E

trigonometric

no

the

To describe

,

quite

are

though

.

than

less

tb

-

ta

1
a=1

T[x;,q ; f ] group St. Set

symmetric

The spaces Y and P

trigonometric,

of

77 ta

H

of the space

of the f

n=1

polynomial

Elements

ti.

action

i

n

i

4)

f (ti,

T[x;

call

to

71

the

spaces

will

appear let trigonometric

Y and P

in the actually us change the

variables: ,q

=

exp

Then elements

(c h)

ta

,

of the spaces

functions. trigonometric geometric spaces degenerate such that the product i.e.

W(U1

....

I

=

'F

exp(cua)) and

T" are

In the limit into

x,,,

rational

c -+

functions

Ua

(Ua M=1 a=1

of exponentials, hyper'trigonometric

functions

0 both the

the space of rational

ut)

(3.14)

exp(c .,,,).

=

1
-

Ua

O (ul,

Ub + h -

Ub

.

.

.

,

uj)

Vitaly

212

is

a

Taxasov

polynomial

symmetric

The last

Ul'...'uj.

of

space

degree

less

the

called

is

than

each of the

in

n

hypergeometric

rational

variables see

space,

[31]. Fixa

E

Let.Fll[x;,q;a;f]

C'.

g(tj,...,tj)

be the space of functions

such

that n

g(ti....

1
m=1 a=1

is

a

holomorphic,

symmetric

g(ti, Elements

(3.3)

-

symmetric

Remark 3.4. a

=

r,77

j_jrjn

are

]

in Cx t and

tj

,

a,2-2a g(t,'...

=

; a; t

]

77; a; f

=

called

The parameter =g; 1' cf. (3.8).

are

invariant

)'ell

[x; 77-1; a-';

tt)

,

with

(3-16)

-

respect

to the

action

here

a

elliptic

the

hypergeometric

related

is

the

to

spaces.

parameter

r,

in

[30]:

M

Proposition

dim.F[x;

and

tj)

-,

-

.

(3-15)

St. Set

group

T"11 [x; The spaces Xell

-

[x;,q

of the space Xell

of the

of ti,

function

-'Pta,

-

0(7? ta/tb) 0(ta/tb)

0(ta/xm)

tj)

I

[30]

3.2.

77 ; f

]

dim.77'[x;

=

77 ; f

]

q, xi,

=

xn

dim Xell

[x;,q

have that

we

a; t

)

n

+ f

-

in

1

-

from Lemma [A.2].

follows

The statement

For any a,

Proof. In what follows we do not indicate hypergeometric maps if spaces and related pressed arguments are supposed to be the

explicitly it

all

causes

same

for

no

all

for

arguments

the

confusion.

The supthe spaces and maps

involved. Remark 3.5.

degenerations In

this

limit

The

trigonometric hypergeometric spaces elliptic hypergeometric spaces as p

of the

functionO(v)

the

.Fell [a]

degenerate into we use correspondence hypergeometric spaces. Another degeneration metric

one occurs

p

=

Then in the limit the space

Fll [a]

after

exp

c -4

two

slightly

,

different

elliptic change of

0 the into

a

-

=

v)

and the

versions

hypergeometric variables exp

functionO(e")

(p)

0 and then

(3.14)

,

is to be

the space of functions

Because

of the

as

a -+

Xell [a]

spaces

P, respectively.

F and

spaces

of the the

(E p)

turns

the

(1

into

turns

be considered

can

-+

0.

and

of this

trigonometric

space to the

supplemented

trigonoby

0 < Im p < 27r.

replaced b (ul, .

by sin(7ru/p) .

.

,

uj)

such that

and

q-Hypergeometric

Functions

and

Theory

Representation

213

i

(u

uj)

1,

((7ri

exp

n

E u, 1p)

p)

-

x

a=1

sin(7r(Ua sin(7r(U,,

n

1111

x

(7r(ua

sin

,,,)Ip)

-

m=1 a=1

is

a

polynomial

symmetric

e2-xiul

...,e

space,

see

[31].

4i(t;

x;

Let

1
27riul .

y;,q)

degree

of

is another

This

(ta /xm) (ta /YM)

I, I,

X; Y; 77 )

m=1 a=1

P(t;

We call

x;

y;,q)

phase function.

the

Ct;

X;,q-1)

Y;

I

[x;

f

0 g

y; 77 ;

f We also

consider

Int

-+

I'[x;

:

y; 77;

a]

these

(Ct;

=

X; Y;

induces

Y[X; 771 [x;

y; 77

:

-97[y;

&

77

pairings

]

77)) hypergeometric

the

Yell [X;

77;

x;

a]

pairings

71;

y; x;

maps

(3.18)

C)

-+

y;,q)) [y;

0

(fg!P(.;

linear

as

(3.17)

spaces:

] (f g!P(.;

[y; x;,q-1]

0 g F4 Int

hypergeometric

that

hypergeometric

a]

variables

(77 taltb) (77-lta/tb)

1
Notice

(3.11)

hypergeometric integral and elliptic trigonometric

The of the

each of the

in

n

function:

n

(t;

than

-

of the trigonometric

version

following

be the

less

h)lp) Ub)/P)

Ub +

-

,

a]

-+

C,

77-1)).

denoting

by

them

the

same

letters:

Remark 3.6. ditional

I[X;Y;77;al

-Fell[X;77;a]

1'[x;y;,q;a]

Y.111[y;77;a]

In this 1

factor

-2-7riy

Proposition

3.3.

paper

V

we

multiply

compared

[[30]]

Let

the

with

(.F[x;77])*,

(3.19)

hypergeometric

the

the parameters

pairings

hypergeometric x, y,,q

by

pairing

in

ad-

an

[[30]].

Assume that

be generic.

n

a

0 psnr

,

a

772-21

11

xM/YM :A P-,-177-"

r

=

0,

-

..

,

f

-

M=1

Then the

hypergeometric

The statement

follows

pairing from

I

[x;

Corollary

y;,q;

[A.1].

a]

is

nondegenerate/.

I

,

s

E

Z>o

-

Vitaly

214

Corollary

Taxasov

Let the parameters

3.1.

Assume that

be generic.

x, y,,q

n

a

54 P-s-177r'

11

772-21

a

x,,,/ym

i4 ps?7-,r

0,---'f-

=

1,s

E

tj f

(tj'...

Z>0.

M=1

hypergeometric

Then the

Proof

Let

1'[x;

pairing

following map: ir commutative is diagram

be the

7r

Then the next

f (ti,

:

nondegenerate/.

is

.

.

.

tj)

,

I[Y;X;n-,;P-,.-,]

77-1; p-la-1]

Tell [Y;

a]

y;,q;

77 ;

and the vertical

3.4

The

(ti,

Res[x;71](f)

tj)

1

[x;

is

E

F[x;,q;

=

Res

t E

Wedefine

geometric

is

the

spaces

equivalent

as

0 g

Sell [X;

f

(3.21) and the

by (3.6)

defined

are

f

E

0 g

and g E P [y; 77; t

(jg)

[x;,q

Xell [x;

71; a; t

multiple

we

have

(jg)

.

[[30]]. Fell [y;,q;

and g E

a; f

]

we

have

(-1)'Res[y;,q-1](jg).

=

to

]

]

(- 1)'Res [y;,q-']

=

LemmaC.8 in

to

equivalent Shapovalov

LemmaC.3 in

pairings

of the

[[30]].

trigonometric

and

elliptic

hyper-

follows:

S [X; Y; n

f

ZjI,

t

Res[x;,q](jg) The statement

the statement.

Zj"

Cx',

f

For any

Lemma3.2.

proves

(dt1t)')jt=x,.[,q]

(f(tj,---'tj)

Res

E

[17 ]

] (fg)

y; 77

The statement

which

by (3.7).

For any

Lemma3. 1. Int

x >

(3.20)

set

E

=

ME

where the points residue is defined

invertible,

are

tj).

pairings

Shapovalov

functionf

For any

arrows

I

I--

[X;Y;77; a]

a]

...

(Tly;n-'])*

-

7r

Tell 1y;

tj

-4

I

F-+

Y; 77; -+

:

T[X;

Res

a] Res

[x; :

77; f 77

] (DY' [Y;

] (jg)

Xell [X;

[x;,q ] (jg)

77;

(3.22)

tC'

,

77; a; -

1y;

77; a; t-4

C,

q-Hypergeometric We also

consider

these

and

Functions

pairings

linear

as

Representation

denoting

maps,

Theory

them

215

by the

same

letters:

S

[X;

S,11[x;y;77;a] Proposition

:

For

3.4.

Y'

Y;,q

T,',I[y;,q;a]

generic

(.F[x;

[Y;,q

(3.23)

77

-+

x, y, 71

Shapovalov

the

S [x; y;,q

pairing

]

is

nondegenerate/. follows

The statement

Proposition

Lemma [4.1]

from either

Let the parameters

3.5.

or

Assume that

be generic.

x, y,,q

[A.3].

Proposition

n

71' V p,

a

77

a

r+2-21

11

pl,

X.1y.

r

=

0,...,f

-

1.

M=1

Shapovalov

Then the

The statement

3.5

Now

we

formulate

identity

the

this

)7eli [X; I I

[X;

Y; 77;

a])

-

('7:1 ell 1y; theorem

section

the

in the next

nondegenerate/.

[A.3]. identity

of this

both the

hypergeometric Shapovalov

the

section,

hypergeometric

and

Rie-

pair-

section.

be generic.

x, y,,q

n ;

we

weight

a])

(V [X; Y; 77; a])

*

Then the

functions

an

Theorem

[2.2]

hypergeometric corresponding

important

(4.1)

(.F[x;

-

(4.3).

in the spaces,

77

I

*

>

for the

describe

[Y;X;77; a] _

I

generalizes

relations

I

a]

71 ;

Given bases of the bilinear

is

following

di-

commutative:

agram is

This

main result

result

a]

y;,q;

Proposition

Let the parameters

Theorem 3.1.

(S-

8,11 [x;

Riemann

which involves

We prove

ings.

from

follows

hypergeometric

The

mann

pairing

-T'

])

*

(-W

(S[X;Y;nl YI

[Y; 771

one-dimensional

[3.1]

Theorem

hypergeometric

example of the bases

integrals. -

case.

translates

into

In the next

the bases

given by

Vitaly

216

4.

coordinates

Tensor

this

section

we

hypergeometric

and

spaces

maps

give

a

bases of

the

We introduce

the

on

hypergeometric

the In

Tarasov

Riemann identity. proof of the hypergeometric and ellipthe in functions trigonometric weight

the tensor and using these bases define coordihypergeometric spaces, introduce we see on the Furthermore, (4.7). hypergeometric spaces, form of and formulate the hypergeometric an equivalent maps, see (4.11), To prove the idenRiemann identity, the hypergeometric see Theorem [4.1]. certain systems maps satisfy tity we use the fact that the hypergeometric cf. (4.19), of difference equations, (4.20), and then study asymptotics of the zone. asymptotic hypergeometric maps in a suitable

tic

nates

4.1

hypergeornetric

Bases of the

For any ( E

define

Zj1

spaces

the functions

W( by the formulae:

wr and

wt n

n

]1 H

1

m=1 S=1

-

E I H 1,

V

OESt

aEFm

m=1

t'

( ta

-

1<1<m

Wt(t;x;y;n;a)

ta

11 J1 m=1 S=1

n

H H(

X

o,ESt

where

0(77

m-1taIX.) 0(ta/xm)

2a-2

m=1 aEF_

fm}

1 + Im .......

F,,,

a

and

am

ITJL 1<1<m

=

Y1

-

-

1.

n =

77)

x; y;

ta

J1

Xm

(t;

X1

)1

0 (77)

'

01

(4.2)

x

0078)

(4. 1)

=

0(talYI) 0(talXl)

arjl<,<mxl/yl,.

1,...,n.

Tn

Set n

,

W I

(t;

X; Y;

n)

=

H Yt. .

m=1

Wf'(t;x;y;77;a) The functions

wr,

w, and Wr,

respectively. functions, are given by (2.18).

weight case

Proposition form a basis

4.1. in

The statement

Corollary basis

in

[30]

W(

WI(t;

Y; X;

77-')

(4.3)

a=1 =

The

1

are

WI(t;y;x;,q-';a-1). called

weight

the

functions

in

the

For

from

Proposition

elliptic

one-dimensional

generic x, y,,q the functions I wt (t; hypergeometric trigonometric space F[x;,q].

follows

and

trigonometric

x;

y;,q)

[A.1].

For generic I w, (t; x, y, 77 the functions hypergeometric trigonometric space P[y;,q].

4. 1.

the

the

11 ta

x;

y;,q) I

(EZ.

form

a

q-Hypergeometric

Proposition a,q

-r

[30]

4.2.

Then the

functions

geometric

space

from

jWj'(t;x;y;,q;a)}tc,z,-

X,'ll [x;

basis

a

I

The bases

biorthogonal for

the next

}

wr

I w'r }

and

with

respect the bases I Wt

a

basis

hyper-

elliptic

the

in

of Proposition [4.2] the functions the elliptic hypergeometric space

in

hypergeometric trigonometric space are Shapovalov pairing (3.22), and the same hypergeometric spaces, see Wl' I of the elliptic

of the the

to

[30] 1

n

S(W('w')

M

H H

61.

=

M=1 8=0

at,m

Proof.

One

Res

=

arl,

can

(w ( (t;

see

/yj

xj

and

from the definition

(dtlt))

y;,q)

x;

-2[j

j<.n

.c'

(1 -,qs+,)

01( 1) 0(778+1) 0(77-sxm/ym)

M=1 8=0

where

n),qs Y. ,(4-5) (x- -wym)

I.-i

n

Sell (WI, W

0'(1)

of the

I t=x,,[,7]

W((t;X;Y;?7)It=y>n[n-11

d

=

TUo(U)lu=l

=

weight

functions

_

(P )3

00.

that

=

0

unless

I <

n

orf

=

0

unless

I >

n or

(4.6)

n,

n,

similarly,

Res

(w' (t; M

x;

y;,q)

W'M (t;

(dtlt))

X; Y;

by (3.22)

Therefore,

using m, while unless f > m or I

(4.4)

lemma.

Lemma4.1.

and

Assume that

generic.

[A.2].

assumptions

form

217

r=0,...'2t-2.

form

IEZ,'

Theory

a].

77 ;

holds

a) I

y;,q;

Proposition

the

Under

4.2.

be

x, y,,q

M=1'...,n-1,

I Wt (t; x; Fll [x;,q; a].

follows

The statement

Corollary

Let the parameters

Opz,

jj,
Representation

and

Functions

=

71)

and the

formula

(3.21)

and

m.

Moreover, =

Res

we

(4.5).

unless

m< n

or

m=

unless

m> n

or

m= n.

S(wj, we

proof

w'M

0 unless

=

obtain

( <

S(wr,w'

that

n,

M

m or

0

get

(wt (t) w'( (t) (dtlt)')

calculation The

have

we

Lemma [3.1]

addition

straightforward

0

0

in

S(wt, w'[)

of the first

=

of the residue

lt=x>f yields

of the second formula

[n] the

I

right

is similar.

hand side

218

Vitaly

4.2

Tensor

Let

V

Tarasov

coordinates

=ED

C v,,

and let

They

are

B [x; y; 77 V*M

B'[x;

the

-+

V*

:

W"'

*

-+

M

(vt*, vm)

tensor

coordinates linear

(t;

B, [x;

77)

hypergeometric

the

y;,q

V*M

B,,, [x;

F'[y;,q

W,M(t; X; Y;

m.

on

77)

X; Y; -+

Jr

=

-+

y; 77;

v*M

,

;

a]

V*

:

W. (t;

a]

X; Y; 77 ;

V*

:

paper

(' ) [X; (u,v)

=

y;,q

spaces

]

:

used in this

,(4.7)

a]

the tensor

differ

coordinates

from the tensor

factors.

Shapovalov

and the

coordinates

the

a)

W,,,(t;x;y;?I;a).

-+

Remark 4. 1. The tensor coordinates in [30] by normalization coordinates

on

a],

;

Fell [y;,q;

-+

[4.2]

The tensor

[30].

cf.

spaces,

Tell [x; 77

-+

of Propositions Under the assumptions [4.1] and of the respective vector are isomorphisms spaces.

forms

by

Denote

space.

maps:

F[x;,q

-+

V*

y; 77 V

pairing:

following

the

dual

MEZe"

canonical

the

maps

C v*M be the

V*

MEZj" Introduce

hypergeometric

and the

(3.23)

pairings

bilinear

induce

V and V*-

V0 V

((, )) [x; ((u, v))

C,

-+

(B-'S-'B*-lu,v),

y;,q;

a]

V* 0 V*

:

1) (u, (B,

C,

-+

Sll B.

V)

11

(4.8) the

Weomit these

common

pairings

in the second line.

arguments

(m-i

n

(VI, VM)

=

H 11

Jr.

M=1

[,,,

((v*,1

V*M

Jf

Mn HH -i

al,m

the next

=

for

afjj<j<m-n

-2[

(I

-

S =0

n"') (X. (1 -,q) ns

nsy.)

-

(4.9)

Y.

0( qs+,) 0('q--qXM1YM)

'Xjlyj,

cf.

Lemma [4.1].

(4.10)

(p),00

These formulae

imply

proposition.

Proposition is

formulae

7780(n) 0(,qsa-')I'm 0(?j1-s-(-at,mxm/ym)

M=1 3=0

where

explicit

The

are:

4.3.

nondegenerate/.

Let The

the parameters form )) is

x, y,,q

be generic.

Then the

nondegenerateprovided

form

that

n

a

77-'

0 pZ'

a

?7r+2-2i

]I

Xrn/ym 0 pZ'

r

=

0,...,

M=1

and a

?I-jjj
OpZ,

m=1,.-.,n-1,

r=O,...,2f-2.

t

-

1,

q-Hypergeometric Remark 4.2.

Let the parameters 77

a

-47ri,y

q

the

Uq(sf,,)

of

[8.]

weight elliptic

subspace

((

Notice

that

here.

with

Consider I

following

the

[x;

y;,q;

a]

1

B* I

=

V*

:

We call

[3.1]

Theorem

Theorem 4.1. metric

Let

a].

y;,q;

That

Difference

Let

L,,, [x;

y;n

The operators

y;,q; =

a] (B')

omit

following

to the

: *

the

from the

notations

vector

spaces

any u,

=

for

and L' M[x; y; 71

v

],

E V*

=

V,

-+

(4.11)

,

arguments

common

in

maps.

Then the

hypergeo-

(,)[x;y;,q],

forms

have

(1[a]u,P[a]v). hypergeometric

the

k

we

V*

P B',,,

statement.

be generic. the respect

x, y,n

n, be linear

1,

hypergeometric spaces T[x; are defined by their actions

trigonometric weight

we

P[x;y;77;a]

equations

]

and

parameters

for

subspace.

types.

1 and P the hypergeometric

the

weight

slightly there

of different

P

((u,v))[a] 4.3

differ

_P[x;

equivalent

is

space to the

a

the

pairing.

[8.]

V,

1[x;y;n;a],

maps

[x;

is

either over

maps:

given by (3.19)

are

the second line.

to the canonical

of the space V with Verma modules

B ,,,

where I and P

subspace in weight

space to the

transforms

,

distinguish

coordinates

-+

weight

a

space.

[7.],

explicitly

we

linear

( )

the dual

or

219

27rip

of evaluation

Sections

in

the tensor

dual

the canonical

into

notations

either

the form its

e

=

the dual

identifications

Ep,.,(sI,),

turns

particular,

In

associated

))

,

p

I

or

Theory

by

2.7ri-/

modules,

product

tensor

a

quantum group

Then the form used

in

Representation

V with

space

Verma

do similar

we

e

=

subspace. Under these identifications pairing of the weight subspace and In Section

and

q, -y, p be defined -

identify

we

product

tensor

e

=

[7.]

Section

In

Functions

maps

operators

y;n ] and P [x; y;n on the bases of the

acting in the ], respectively. trigonometric

functions: n

Lk [X;

Y; 77;

a]

Wf

(*;

X; Y;

77)

=

(a '01-1 11

X",

/Y",) 'Wk

f

(_;

k

k

X;

Y;

77), (4.12)

y;

n),

M=1 n

L'[x;y;77;a k

W,r(.;x;y;,O)

H Xm/ym)

(an'-'

=

W,,(.; k

k

X;

k

M=1

where

kf kX

=

(Xk+1 -,Xn,X1....

7

Xk)

k 7

y

=

In, 13.1

(Yk+1

...

Yn,Vl,---,Vk)-

It),

(4.13)

220

Vitaly

Recall

that

Using 1,

M=

Tarasov

[I

.

.

by

n,

,

+ (,,,,

f, <,

=

n,

particular K,,,,, KMrI operators and in

P ---

=

f.

End

((B[x;y;,q])-'Lm[x;y;,q;a]B[x;y;,q])*, ((B'[x;y;,q])-'L' [x;y;,q;a]B'[x;y;,q])*.

Km'[x;y;?I;a] define

we

1,

introduce

(V),

the formulae:

Km[x;y;,q;a]

We also

m=

coordinates

the tensor

(4.14) (4.15)

M

Mm[x;

operators

y; 71;

a]

(V*),

E End

n:

m

(aq'-"

M.[x;y;77;a]v,*

xjlyj)-",

11 1<j<m

(4.16)

Qh

Qh, 1

Let X1,

n

shift

multiplicative

be the

Qh mf(XI, f(hxl,...,

hx,,,,

QP

m

Theorem 4.2.

[30]

Qm

Set

=

M,

[x;

y; 77;

xn; Yi, xn;

xm+,,

functions

on

of

(4.17)

Yn)

...,

hyl,...,

hym,

Yn).

ym+,,.

(4.18)

n.

The

system of difference

lowing

acting

operators

Yn:

Xn, Y1,

a]

=

hypergeometric equations:

K,,, [x;

a] f [x;

y;,q;

map

1 [x;

y;,q;

a] Mm[x;

P [x;

y;,q;

y;,q;

a] satisfies a],

y; 71;

the

fol-

(4.19)

n.

Tn

The hypergeometric 4.3. Corollary equations: system of difference

Q,nP[x; m=

y;,q;

a]

=

K,n[x;

y; 77;

map

a] P[x;

a] satisfies

a] (Mm[x;

y;,q;

y; 71;

the

a])

following

(4.20)

1,...,n

The last

claim

results

from the commutativity

of

diagram (3.20)

and formulae

(4-3). Remark 4.3. of the

elliptic

The numbers pf,m weight functions:

are

related

to

the transformation

H (Xjlyj)

QMWI

i

properties

WE

(4.21)

WE'

(4.22)

1<j<m

QMW(,

'M

11 (Xjlyj)I<j<m

q-Hypergeometric Consider

Remark 4-4.

QmT/(x; y) Its

the system

that

T1

Tn

I,-,

=

any

.

[5.]

In Section

(4.23)

solves

the

Gauss-Manin described

,

Mn are invariant

h,

so

see

via

[5.2],

[7.]

In Section in

(4.23)

[5.1].

be viewed

can

bundle

Solutions

invariant

(7.12).

(7-11),

see

the dual

space of the

A be

Let Y1,

=

-

the

K,,,(x;y)Tf'(x;y),

system (4.23)

of the

following

elliptic

hyper-

for

functionT/'

a

asymptotic

lx,,,/x,,,+Jl 1xM/YM1(x; y)

tends

and the ratios

xn

functionf (x; y) HMAf. Notice that a

is invariant

with

/y,,,

has

--

zone

<

m

1,

rn

to limit

XM/XM+1

h.

the

into

same

as

(4.24)

n,

taking

values

in the

dual

maps

of

the

parameters

X1,

Xn,

'Yn:

We say that

it

I,-,

m=

hypergeometric

of the

A

If

the pe-

(5.24).

qKZ equation (7.3) weight subspace.

Asymptotics

4.4

an

the system (4.23) with the qKZ equation with identify subspace of a tensor product of Uq(sf,) Verma modules, We also identify the system of difference equations

Q,,,Tf'(x;y) with

of

the discrete

with

of the

sections

as

equa-

we

weight

a

(5.23),

formulae

see

shift

to the

respect

system of difference The factor Y plays the role

cohomological

Theorem

[5.2]

with

the last

the system

certain

a

Theorem

in

.

the last construction results geometric bundle. Slightly modified, be to hypergeometric map, so Theorem [4.2] nearly the appears

values

system

1'...'n.

M=

coefficients.

show that

we

equation for connection,

section

.

nonzero

has constant effectively in adjusting [30]. map

Theorem

221

n,

(V*)

Y where Y E End

Mi,

the operators for Qh n

tions

are

Theory

equations

M'; 'Y*Y)'

=

Qh'...' l

operators

riodic

of difference

Representation

equations

Qmy(x;y) Notice

and

Km(x; y) TI(x; y),

=

have the form

solutions

of difference

Functions

a

in A and write

0,

HMAf to the

1 n

(x; y) 3

M=

and yn Ix,,, finite limit

the limit

respect

-

1,...,n I,-,

remain as can

shift

n

depend

operators

on

1,

-

we

x1,

Q,

.....

-

(4.25)

-

A if

bounded for

(x; y) 3 A,

I

-

any

denote -,

n

for

-

.

limit

this

Xn, Y1,

Qn'

1,

M=

-

-,

any

.

.

,

n.

by

Yn, but nonzero

Vitaly

222

Taxasov

(x; y) 3 A,

(x; y)

.

I >

(x; y)

..

I

I

[x;

y;

a] (wj

x;

y), W.

x; y;

(x; y)

=

V[x;

y;

a] (w ,

x;

y), W.'

x; y;

=

m

(,,-1

(?7-1),,,,(7j1a-1)oc) I'm

77'

H

rl

=

that

ar,m

=

(,q-s-,).

=

_77-sar,

H 11

M

A.

E

[6.]

Section

(p).

hypergeometric

It'.

limA

(,qs-1+(ma-1

(pi7-3at,.). (,qs+,).

(x; y)

It

integral 0 unless

=

m

I <

m or

YM/XM)OO

WOO"M

(778y./Xm).

Propositions [4.4] and [4.5] in the one-dimensional proof Proposition [4-4] in the general case Theorem 6.2 in [30]. Proposition [4.5] can be proved in a

we

prove

to show the main idea of the

equivalent

similar

to

way.

Corollary

1'[x;

the

Zj'

(77)

M

0 unless

.

Moreover

M=1 S=O

case

integral =

(pn'-s-'-a(,mx./y.). (n--9xM/yM)-

I<j<mq-2(jXj/yj

all

[.-I

n

is

a)) a))

a

For any 1, 4.5. Proposition 3 limit has a finite as (x; y) and (3.5), m, cf.

In

I,.

=

M=1 8=0

HMAI(If

and

(x; y)

n

liMA It( Recall

in the asymptotic matrices. by triangular (x; y) by the formulae:

4.4.

has

m or

If

limits

represented

be

can

[30] Yor any 1, m E Zj' the hypergeometric _3 A. Moreover liMA It limit as (x; y) finite (3.5), and m, cf.

Proposition Ir

and the limits

the functions

Namely, define

I and P have finite

pairings

hypergeometric

The zone,

y; 71;

For

4.4.

a]

any

finite

have

HMA1 Vf*

hypergeometric maps Y [x; _3 A. Moreover, (x; y)

the

a,,q

limits

as

V1

HMAIt

I

+

E V..

HMA1,

y;,q;

a]

and

1

M>1

and

HMArV(

V1

HMA11'f

+

E Vm liMA

I'm

f

-

M
Kn, KI, Kn, are respectively and ordering with respect to the basis I V1 }(Ez,of the limits HMAK; ' and HMAKm' are equal to

Remark 4.5. It is easy to check that the operators KI, I A, and the limits limits cf. (4.14), have finite as (x; y) .

lower

and upper

(3.5).

The

Mm*,

cf.

triangular

diagonal

parts

.

(4.16): HMAKmvf

p-1 I'm

v(

+

E Vn (vn*

HMA&,vc)

7

n>1

liMA Kmvf

pt,m vr

+

E Vn (vn* n
7

HMAK,,,vr).

.

,

.

.

q-Hypergeometric

Furthermore, liMA I Vj* is a with eigenvalues liMA

the

Ki,...' Proof

4.5

Now we

hypergeometric

of the

are

going to equivalent

grals.

Two

[4.1].

Wewill

I

any

operators

limA

Similarly,

(4.20)

common

Zj'

E

223

the

vector

liMA K,, K,.... implies that for

of the operators eigenvector respectively.

Riemann

identity

the bilinear

prove

forms

of the

for the hypergeometric identity are given identity by Theorems [3.1]

theorem.

the latter

prove

for

that

eigenvector I respectively. A-

Theory

Representation

and

of the

vector liMA I1V(* is a liMA K,, with eigenvalues

Z,1

I C-

any

(4.19)

from

follows

it

common

Functions

Proof

[4.1].

of Theorem

inte-

and

Consider

the

functions

G,

..

(x;

a)

y;,q;

We have to prove

for

that

both

sides

that

assume

(x;y;,q;a)

of the

above

Let

Lemma4.2.

such that

Mr,f

a

jz.,e

of the

In the rest

be generic.

for

proof

any k we

a] vj*, 1 [x;

ME

3j"

=

]

y;,q

M

.

((vf*,vm*))[x;y;,q;a].

be

will

do not write

functions

of a,

following

the

use

by (4.16).

defined then

1,...'n,

(4.26)

analytic

are we

Let pr,# =

a] v,*, ) [x;

y; 77;

I

=

explicitly

we can

statement.

If 1,

ME

'Zil

are

m.

arguments

and 'q for

a

all

functions.

the involved

Proof.

=

1,

y; 77;

equality particular,

In

generic.

is

a

any

..

Gr Since

(P [x;

=

By the definition

of the

Shapovalov

S[x; y]

pairing

it

is

easy to

see

that

S[X;Y](W.[(.;IX;ky),Wlk.(_;kX;ky)) (4.13),

cf.

that

Therefore,

for

S (Lk wf, L, w,,,)M any u, v E V we have is

(K,,, [x; y] Since the

u,

K'

[x; y] v) [x; y]

,

hypergeometric

equations (4.19) system of difference

ence a

Qk Gj

..

(x;y)

maps

and

finite

that

limit

as

(4.20),

=

1[x; y]

=

S[X;Y](Wf(.;X;Y),W

S (wr,

w,' ),

(u, v) [x; y] and P [x;

respectively,

k

=

I,-,

,M

y] satisfy

X;

M

=

1,

-

(4.12).

cf.

n,

-

n

-,

Y))

-

the systems of differ(x; y) satisfies

the functionG

equations =

where pf,f are given by On the other hand, a

=

fil,ep-1 M't Gt,,,(x;y),

k

=

1,

...,

(4.27)

n,

(4.16).

Corollary [4.4] shows (x; y) -3 A. Taking the limit

that in

the functionGj equations (4.27)

(x; y)

m

we

has

obtain

Vitaly

224

Taxasov

RMAGt,,, RMAGt

Therefore, Observe

0 for

=

Gr

that

now

is m

I

k

by Lemma[4-2]. functionof analytic x,

i4

an

m

y,,q,

a.

So it

suffices

to

n, and

by

the claim

prove

Gl,,(x;y)

[:A

for

0

M

the assumptions

under

Jal (4.27)

we

IM

1,

=

1X""/Y"'J

I p t,m 1

assumptions

Under these

..

IG(m(x;y)l (x; y) the

Therefore, functionG

(x;y)l

any

Zj'

and

I liMA Grml system (4.27) reads

1,...,n.

m=

1,

liMA Gr

r.

with

In

Gim(x;y)

k

the

to

For

0.

=

the

1,...,n,

=

this

particular,

respect

1,...,n.

shows that

shift

the func-

all

Qh' ..,Qh

operators

n

h.

nonzero

Obviously,

=

=

i.e.

0,

=

Gff(x;y),

=

invariant

are

k

=

gives Gt f (x; y)

Grr(x;y)

tions

any f E

lGr.(x;y)i,

=

QkG,I(x;y) which

1 for

m=

have

JQkGr

for

EMA Gt.,

N't ILM't

=

right

the

hand side

(4.26)

of formula

enjoys

the

same

proper-

ties:

((V

((v*,f v*))f [x; y]

and

Hence,

M

with

verify

to

(4.8)

(4.7),

and

local

Discrete

I

"

Corollary

[4.5]

[4.4],

Propositions

5.

to

:A

m,

Qh for

Qh'I

n

any

h.

nonzero

=

((vr*,vr*)),

Zn'

words,

in other

RMAIt cf.

respect

I

that

limA Git or,

for

0

X; Y

is invariant

remains

it

V'*"

liMA

I(I r'(Vf;

[4.4].

This

and formulae

systems

Vr) is

a

=

((Vr*

I

Vf*)),

straightforward

(4-10).

(4.9),

and the

calculation

Theorem

discrete

[4.1]

is

using proved.

Gauss-Manin

connection In

this

section

we

explain

the

(4.19), (4.20) equations systems are essentially

satisfied

these

Gauss-Manin notions were

of

a

connection discrete

introduced

in

local

[30],

geometric origin of the systems of difference by the hypergeometric maps. Weshow that the periodic section equations for the discrete

assosiated

system

[31].

with

a

and the

suitable discrete

discrete

local

Gauss-Manin

system.

The

connection

q-Hypergeometric Discrete

5.1

Consider

by

B C C'

with

(3 ...... be

and discrete

called

3.)

the

311

subset

connection

base space.

225

systems

The lattice

ZI acts

the

on

B if for

ZM)

any

Say

that

E B there

z

if the

V(zl,...,

-*

Ak (ZI....

there are

a

is

a

vector

subbundle

discrete

A,,,

=

1 ....

I

of

M.

commute:

(5-1)

)PZki

...

if

3 is given subspaces is

in

family

and the

k

=

zm) Aj (zi,

)

Z"')' A,,

isomorphisms

Aj(zl,...,zm)Ak(Zl,---,PZj,...,Zm) =

1 E Zn'.

base space.

of the

)

flat

is called

The connection

P1. 3.),

I

...

over

V(Zli---,PZk....

:

Say that a distinguished

local

Theory

isomorphisms

and linear

Ak(Zl,---,Zm)

(P 1.,

-+

invariant

an

discrete

a

V(z)

space

Representation

dilations:

1:

bundle

' space C

vector

a

base space

Let

connections

flat

and

Functions

ZM)

....

-

subspace

a

in

with

invariant

every

fiber

respect

to

is

the

connection.

A section

Ak (Z1, A

-

-

(zl,

functionf f (Zl

Periodic

-

.

.

.

PZk....

...

,

zm)

-,PZk2

-

bundle

7

is called

Zm)

....

form

sections

The dual

periodic

called

is

if

values

its

with

invariant

are

connection:

ZM) 8 (ZI)

,

-

s(z)

s : z t-+

to the

respect

a

with

a

module

-

-

-

the

-

7

zM),k

-,

-

if

quasiconstant

over

dual

s(zi,

=

f (ZI,

::--

the

ZM)

ZM) 7k

ring of quasiconstants.

V*(z)

has fibers

connection

and isomor-

phisms A*k Let

(zi,

.

s 1...

,

zm)

.

.

)

SN

:

be

V* a

(zi,

basis

.

.

.

,

zM)

V*

-+

of the

of sections

morphisms Ak of the connection

are

(zi....

given by

)Pzk7

...

7

initial

bundle.

matrices

A(k):

(5.2)

ZM). Then the

iso-

N

Ak(Zli---,Zm)Sa(Zl,---,PZk,...,Zm)

=

1:

(k) Aab

(Z1....

)

Zm)

Sb

(Z1

ZM)

b=1

For

a

section

coordinate

z F-+

0 (z)

of the

bundle

dual

denote

by

!P

-+

: z

Tf

(z)

its

vector,

IPa(z) The section

V)

is

system of difference

=

(O(Z),Sa(z))-

if

and only periodic equations

Tf(zl,...,PZk,---,zm)

F=

A(k)

if

its

(Z1'...'

coordinate

ZM)

Tf

vector

(Z,,...,

satisfies

zM)

,

the

k

Taxasov

Vitaly

226

1, equation. =

.

.

Say

system of difference

This

m.

,

.

functions

that

(ZI

Vj

Zm)

1

I...

the

periodic

form

zl,...)zm

section

system

a

of

if

coefficients

connection

variables

in

pj,...'(p,,,

is called

equations

(ZI

Ok

Wk(ZI,

=

-

7

...

-

-,

7pzj,---,Zm) --M)

Oj

=

(zi....

PZk....

7

I

zM)

flat connection define a discrete on m. These functions 17. vector secbundle, cf. (5.1). A periodic complex one-dimensional a phase Anction tion ! of the dual bundle is called of system of connection coefficients:

j, k

for

all

the

trivial

P(zj....

)

=

-,

Pzk....

Z1,

.

.

.

Zkf (Zli k

I,-,

=

Zm)

Ok

=

,

(Z1

Zm acting 1

...

Zm)

Wk(ZI

=

oj,

i

...

.

.

Zk the

We say that

one-dimensional local

:

F

local

k

Wmdefines

,

.

(zi,

.

.

ZM) f (Z15

PZki

....

1,...,M.

twisted

the

zm) by

,

.

=

shift

the rule:

7zm))

...

of zj,

.

k

=

.

,

zm

such that

1,.

the operators

m.

..,

coefficients

connection

system.

.

of F:

F,

-+

space F and the

discrete

I

.

ZM)

I

...

,

Zm commute with each other.

Zj,...'

Let F be a vector space of functions Zm induce linear isomorphisms Z1, , .

.

functionf

on a

The operators

m.

Zm) ! (Z1

....

i

coefficients

of connection

The system

operators

i

F is called

the

W1,

.

.

.

functional

,

Wmform

a

space of the

system. The de Rham complex,

coefficients the top

F/DF

the

cohomology

cohomology and homology

local

in the discrete

system

group HI is

are

defined

in

isomorphic

canonically

of Cxrn with particular, [31]. the quotient space

groups

[30], to

In

where M

DF

=

E(Zkfk k=1

-

A) I fk

E

F,

k

=

}

M

.

-

total DF are called the twisted differences. of the Hm equality FIDF for the definition The dual space Hm (Hm)* is the top homology group. It can space H1 functionals of the space F* spanned by linear be considered as a subspace differences. total the twisted annihilating

The elements In this

of the

paper

we

subspace take

the

=

=

q-Hypergeometric Discrete

5.2

There a

is

Let

7r

on

.

C'+m

:

is called

total on

.

Let fer

C m be

the

tj coordinates , the total space. .

F,

functional T1) Ji For

Z11 point I

a

the fiber

system

be

over

227

discrete

t1,

.

[30],

[31].

base with

fiber

see

the

onto

.

ti,

,

.

zi,

.

.

zn

,

.

the

on

base,

coordinates

are

system on Cx(+m). We resystem, so that F is the total

local

local

corresponding

the

connections,

construction,

zm be coordinates

,

.

flat

twisted

shift

by

operators

zM.

I f

I f

and

Hj I

z

a

system F J., V, (.; z),

local

the functions

all

restricting

H'Jz

.

.

a

total

Cx m we define

E

z

=

by

Wedenote

7

...

z

Flz

the

as

We denote

space.

...

fiber,

z1,

that

so

with

projection

Let

space.

the

Theory

Representation

connection

affine

an

xl,...,xm

local

this

to

of bundles

construction

-+

and

connection

of the Gauss-Manin

version

C'. Ci+"n tJ'

Gauss-Manin

geometric

a

discrete

Functions

to the

},

E F

('; Z)

(Pa

the top

cohomology

.

.

.

Z)

,

on

fiber: Wa1-7r

=

and

-

1

(z)

homology

-

spaces of the

fiber.

This

provides a bundle isomorphisms

construction

Flz

fibers

with

tion

Zkl(,,,...,z_)

FI( ,,...'PZ

:

by

On the

other

Flz

fibers

the

Z1,

operators

hand,

and their

the

.

.

.

f ,

on

.

Z-)

.....

Wk(*;

z--

Z,,,,

acting T1,...,Tj

operators

action

FJ(z

-4

z_)

Zk I (zl,...,z-) induced

C' with

over

a

discrete

flat

connec-

and

the fibers

Z) f

on

k

I

,

functional

the total

naturally

be

can

m,(5-3)

1,

=

with

is consistent

space F.

restricted

to

isomorphisms

the

(5.3):

Zkl(zl,...,z-)

Ta1(z,,...'z_) which

Zk I (z denoted over

from the commutativity space. Hence the

follows

the total

functional

.

....

H1 I (,,,, ---,PZkt---,'-)

:

z_)

by the

Cx In with

Gauss-Manin

same

flat

connection

Zkk I (zi, In what follows twisted

that

(5.3)

H%z,...,._)

--+

the

vector

Ta and Zk acting induce

,k

on

isomorphisms 1....

=

H J;'

spaces

This connection

connection.

connection on

*

the

so

flat

of the operators

isomorphisms

7

form

IM, a

bundle

is called

the discrete

bundle

induces

connection.

The Gauss-Manin

dual

letters,

discrete

a

TaJ(z1,_..,pzk,-..'Z_)

zkl(zl,...,z_)

=

shift

....

the

z_)

:

Ht I (zl,

_

usually

do not

acting

duced isomorphisms causes no confusion.

cohomological

the

bundle:

operators

we

on

homological ....

Z-)

Hi1(Z1'---'Pzk'---'ZM)

distinguish

of the functional

on

the or

the

explicitly total

functional

cohomological.

in notations space

spaces

of

between

and the

fibers,

in-

if it

Vitaly

228

Tarasov

system associated

local

Discrete

5.3

with

hypergeometric

the

integrals In this

section

describe

we

discrete

certain

a

related

closely,

B

on we

deal

clearly

base, ti,

the Y1,

-

-

,

-

affine

an

.

-

tj

function

rational

I

with

projection Ci. fiber

on

ir

xi, on

the

by

dilations

the

to

[30].

in

B C (C

Let

2n

be

obey (3.1)

Yn

-

system

From

p.

now

B.

Ct+2n

:

Let

on

the total

respect

over

coordinates

-

Xn, Yi,

Vp

ql+'

q such that

X1'.

bundles

coordinates

are

Yn

I

-

2n

invariant

base with

the

E C

discrete

with

Consider onto

f (x; y)

=

B is

The set

local system. The local type local system studied

to the

trigonometric Take a nonzero complex number the subset of generic points: is

.

.

.

fiber,

the

total

space (Ct+2n be coordinates on Yn

of the total

-

-

-

,

that

so

ti,

Consider

space.

space with

2n

Y1,

Xn,

,

(C

_+

.

-

-

the

simple poles

at most

1

4,

X1,

space

::

-

I

Xn,

F[,q ]

at the

following

tb

P871 ta

of

hyperplanes ta

P_8Xm

=

ta

,

8+1

P

=

ta

YM,

=

8+1

77 tb

P

=

I

(5.4) <

a

< b <

f,

M

1,

=

.

I

-

.

s

n,

poles

E Z >0,

and any

0,

Y.

at

the

coordinate

hyperplanes ta a

=

I

following

....

I

t,

M

system

=

=

11

0,

X"'

Let

n.

....

=

(t;

X; Y; 77;

a)

11 a
ntb

77 ta

be the

n

11

711

a

ta

01,- --,On

Xl,---,Xn,

11

Xm/y,"

tb

-

Pta

11

774

-

P 71 ta

I
a

m=1

M=1

X

0,

coefficients:

of connection

n

Pa

=

-

tb

ta

-

-

Ym

X

Xm

(5.5)

a

i

XM(t;

X;

V)

=

11

(ta

-

P Xm)

(5-6)

1

a=1 M

00;

X;

Y)

=

11

(ta

-

n.

PYm)-I,

a=1

Denote

by T1[?7;a],---'Tj[,q;a],

twisted

shift

of the space space

T[,q I

9perators. It F[71 ]. Hence,

Xi,...'XM'

Yl,...,Ym

the

corresponding

is easy to check that they induce linear isomorphism local system with the functional we have a discrete

and the system

of connection

coefficients

(5.5),

(5.6).

q-Hypergeometric A

g (t;

y) !P(t;

x;

i

n

4T,(t;

Y;,q)

X;

(taIXM)O() (ta /Ym)

11 11

=

m=1 a=1

(3.17),

cf.

g(t;

and

y)

x;

is

an

g(t1,---,Pta,---,ti;x;Y) Xt;

X1,

-

-

,

-

ixn;Y)

the

introduced

induced

local

We take sider

the

of the

local

functions

rational

hyperplanes

the a

system

=

fiber

the

W[x;

y;,q;

local

system

on

fiber

a

of the

over

variables

has

(5.7)

,

that

-

-

Y),

4; X;

-,

(5.8) (5-9)

tj,

.

.

local

system and

The functional

(x; y)

point

a

total

the

as

fibers.

on

(5.6)

(-PYrn)_ 9(t;X;YI,---,Yn)-

=

systems

229

(-Pxm)'g(t;xl,---,Xn;Y),

=

9(t;x;yl,..-,Pym,...,Yn)

1
a?72-2a g(ti,

=

Pxm....

(77taftb). (77-ltaltb).

functionsuch

arbitrary

(5.5),

coefficients

11 oo

Theory

Representation

of the system of connection where x; y;,q)

phase function

the form

and

Functions

.

,

with

tj

(C

E

at

2n

the

is

simple

most

con-

F[x;

space

j

y;,q

space

of

poles

at

(5.4)

and any poles at the coordinate hyperplanes t,, = 0, coefficients The system of connection of the local system on

pj

(.;

a]

the

x;

y),

.

.

.

,

Wj (.; x;

corresponding

W[x;

a]

y;,q;

y)

=

F[x;

y;,q

(5.5).

by formulae

given

is

top cohomology

by

We denote

space:

] ID.F[x;

a]

y;,q;

where

D.77[x;

E(Ta[?7;a]j(x,V)fa

f

-

fa) I

fa

a]

y;,q;

(5.10)

J [x;y;n],a

E

a=1

There

is

an

of the

action

symmetric

Sj

group

the

on

total

functional

space

.F[,q ] given by (3.2): [f1O-(tI,---,4;X;Y)

11

f(tO-1,---,tai;x;Y)

=

0a > Ub

I
and

similar

a

by

note

^

j [,q ]

'H_,[x;y;,q;a] action

Y1,

.

on .

is for

[X-f

.

,

St -action

and the

the total

J , [x;

any

11

f =

c

F[,q]

X.[f

y; 77

subspacein functional

Y,, and naturally

]

1

a

space of

the subspaces

W[x;y;77;a]

of

^

St

E

Sj

we

a

the operators

fiber

-invariant

F[x;

t1b y; 77

the operators

T,

[,q; a],

'

tO'a

-

].

Wede-

functions

by.F,[x;y;,q].

generated

space commutes with

transforms

and

11

the functional

on

t" 77

and

The

XI, Tj [,q; a], that .

.

.

have

[Y-f 11

=

Y-If

11

1

[Taf ]a

by

SjXn,

=

TO'a [ f 10'

Vitaly

230

This

Tarasov

that

means

cohomological

F[,q ]

Let

the

C

with

J , [,q ]

that

degree product

X;

]I ]I

Y) F1 t-1 a

a=1

(ta

f (t;

functions =

f (t;

m=1 a=1

ta

1
y)

x;

x;

77 ta

X.)

-

of the

connection.

f (h t; hx; hy)

zero:

subbundle

discrete

a

n

i

f (t;

form

Gauss-Manin

space of rational

of

the

a]

y;,q;

discrete

the

be the

functions

homogeneous Cx, and such

ft, [x;

spaces

bundle

y)

which

for

are

any h E

tb

-

tb

-

of deqee less than n in each of the variables symmetric polynomial let Y.[,q] be the space of rational functions C Similarly, of degree zero and such that the f (t; x; y) which are homogeneous functions product is

a

tj,...,tj.

n

f (t;

]I J1

Y)

X;

(ta

-

We call

tt. tj, geometric .

polynomial

symmetric

a

,

.

.

(3.12),

see

hypergeometric

ric

the functional

(3.13), fiber

space of the

-T'[X;'q-

X.

a particular hypergeometric subspaces of fi, [x; y;

in

subbundle

discrete

I'H,,[x;y;77;a] Xlyl,

-

-

-,

X.Y.

Fix

nonzero

space of the

Ci

.

.

.

,

xn

ED Cn

the

to

a

total

trigonometproduces

fiber

functional

in

that

space:

77

f

x;

y)

f

E

Y[71

71

f

X;

Y)

f

E

J7.

[q

of y is irrelevant for the above relations. spaces W[x; y; n ; a] and W. [x; y; 77 ;

a]

are

after

suitable

a

of

The collections

}(x;Y)E]s

are

of the

reduction

17i[X;

spaces

with

invariant

connection. Y; n ;

respect

}(X;Y)EB

a]

the

to

and

operators

-

from Lemma [5.1]

complex numbers

vi,

...,

and v,,

[5.2].

Proposition

and consider

the

following

sub-

base space

Cvn X1,

from

.

follows

proposition

The

introduced

spaces

specialization

hyper-

trigonometric

hypergeometric

same

variables

cohomology a] generated

[33]

5.1.

each of the

total

be obtained

tb

by F[x;,q I and X. [x; 77 ], respectively. advance that the hypergeometric cohomology spaces form with respect to the Gauss-Manin connection But they 71;

do form the subbundle

Proposition

[,q ]

tb

-

-

in

n

the

from the total

choice

that

It is not clear a

[X;

than

and F.

can

by the

spaces

The

less

trigonometric

Y[x;

Notice

degree

F[,q ]

spaces

The

spaces.

[3],

Section

the

of

ta

I
m=1 a=1

is

77 ta

H

Xm)

=

I (x; y)

E

being coordinates as

a new

total

C2n I on

space.

YM = VMXM7

Cn. We take In other

words

m=

Cn

as

we

1,

-

..

a new

allow

,

n

}

7

base space and

only simultaneous

q-Hypergeometric coordinates

of the

dilations

Introduce

n.

m

and y, thus local system

x,,,

a

discrete

f I (C i

Y, functional

total

the

as

f

ED C,,n

local

the

of

space

Representation

and

Functions

Theory

231

x"'1y"',

the ratios preserving on C' E) Cn taking

E

and the

system

of

restrictions

coeffi011 Xn On as the system of connection W1, twisted shift the corresponding are T, so that Til operators cients, XnYn. The local systems on fibers are clearly kept intact under the XJYJ,..., bundle of the base space, so the obtained described reduction cohomological Cxn over cohomological Cvn n (Cx2n is a natural reduction of the original bundle over (Cx2n.

functions

.

.

.

Xi

Oj'

,

)

...

*

Corollary

the

Gauss-Manin

cohomological

hypergeometric bundle cohomological The

5.1.

of

bundles

Let WE, I E

is

equivalent

3,",

be the

with Cxn .

over

X;

Y;,O)

wl'(t; for

77) Zj',

x; y;

any I E

I W&; X; of the

bases

are

to

sub-

discrete

to

discrete

the

wl'(t;

=

'

H ta

.

y; x;

(4.1),

see

and let

(t;

77),

X; Y;

WE

a=1

q-'),

(4.3).

cf.

Then wc C-

Y[77 ]

n) I

Zj' }

and w(*

E

and the sets

n) I

Y;

Zj' I

I E

I

,

Y[x;,q],

spaces Let

ro*t (-;

X; 9;

respectively,

I E

see

[4.11

Proposition

[4.11.

Corollary

and

form

respect

f

11 xt.

=

M=1

that

spaces

to Proposition [5.1]. weight functions, trigonometric n

W. (t;

[,q ]

I

connection.

The statement

.F.

*

=

V

so

*

kWr(t;X;Y;71)

W, #;kX;ky;n)'

=

kWo(t;X;y;,O)

Wo,(t;kX;ky;

=

7),

where

k( k k

k

0,

=

.

.

.

,

n,

I kWE (-;

x; y;

Itt

[I,

I

I

Xk)

X

(Xk+l,

Y

(Yk+1,---,YnY1,-.-,Yk),

-

,

Xn,X1,

,

(4.13).

cf.

For

Lemma5.1.

(fe+JL,

any

71) 11

E

k

Zt' I

the

n

bases

are

of

the

I

sets spaces

k

(-;

WE

x; y;

T[x;,q ],

T.

71) 11 [x;,q ],

E

tively. The statement

Set

Zk

=

is

Xlyl

clear, ...

since

XIYk,

T[x;,q

)7[k

k n

Or,e[,q;a]

=

(aql-'

H xM/YM) M=1

n.

Ie

x;

771

Let

and T.

[x;

q

)7,

[k

Zt'

respec-

X;

Vitaly

232

Taxasov

Proposition

The

5.2.

functions

weight

trigonometric

have

following

the

properties: Zkwr

Or,e[7j;a]kW(

=

(T. [,q ; a] fr,.

+

-

fr,.)

(5.11)

91,a)

(5.12)

a=1

Zk Wo E

Ot'r[77

=

k

tj

;

W I

(Ta [,q; a]

+

91,a

-

a=1

k

fr,,

where

n,

gi,

t,

a

a,

functions

certain

are

belonging

to

X177 I. equality

The first

is

show the idea of the

proved proof

Proof

of

that

ke(M) =C(M-t)

k <

n.

Proposition

Recall

proof

The

consider

[5.2]

Proof. Notice m<

[321.

in we

for for

of the second

is similar.

one

below the one-dimensional t

1. Let

=

e

(m)

=

(o,

ke(M)

1
.

.

lm-th,

,

.

-

-

-

0)

,

e(m-f+n)

=

for

that

W"(M)

(t;

X;

Y)

t

=

t_YI

fli<,<.

t-x_

t-XI

I

n

n

Ta aj f (t;

x;

y)

=

a

f (p t;

x;

t

H XM/YM 11

y)

M=1

t

M=1

Y"'

-

XM

-

Zk f (t;

X;

Y)

=

f(t;pxl,...,Pxm,xm+l,---,Xn;PY1,...,Pym,ym+l,...,Yn)

X

X

t

H

taken

together

-

pXj

t-pyj

I<j
These formulae

To

example.

imply

ZkWe(m)

=

kwe(M)

for

k < m,

for

k >

n

T,,,[a]Zkw,(n,)

=

II

a

k

XM1yM We(nj)

M=1

Since

1 for

(5.11).

gives

Consider

have

The

a

functionf

any

m

proof of (5.12)

function(o,

fl,

-

-

-

,

fj

E

Oe(n,),e

E

- [77]

the section

f)

is

xm/ym

a

for

k > m, this

M=1

bundle, y;,q

0 defines =

identically

The tensor

=

is similar.

functional

(O,f)(X;Y) The

n

and

V) of the homological 0 (x; y) on ) [x;

section

linear

a

k <

m.

that

] vanishing a

function

is for on

any

DJ [x;

(0, f )

on

(x; y)

a].

y;,q; B

E B

we

For

by the rule

(O(X;Y),f(.;X;Y)).

zero

coordinates

if

(T,,[,q; a] on thg'4igonometric f

f,,

-

f,,)

for

some

hypergeometric

q-Hypergeometric

(4.7),

see

vector

space V

(D

=

0

the section

translate

space,

Functions

and

into

a

Theory

Representation

taking

functionTf

233

values

in the

(C v.:

MEZin

(X;

77)

Y;

E (0 (X;

=

The functionT

Recall

is similar

that

Y),

.'

to

coordinate

a

Q1,

the operators

W.

(-;

X; Y;

77))

(5-13)

v

E Z,"

ta

.

Q,,

,

.

.

vector

of the section

act

functions

on

of x, y

(Q.f)(x1'---'X-;y1'---'y.) f(pX11

=

cf.

(4.17),

P xM) XM+1I

I

...

and for

any

f

E

Q,.(O,f) by definition

Zl*,

be

a

-

where

the

=

.

,

.

.

Zn*

,

.

Proof.

K,,,,(x;y)TI(x;y),

Applying

QkT'(X;

by (4-14).

given

are

the section

0

both

to

H

(aql-'

77)

Y;

IEZ,' Here relations

(5.14)

left

hand side.

Taking

see

(4.12)

The last

sides

and

The system

coincides

a

the

we

section

we

Remark 5. 1.

It

ogy space

N(x;

.F[x; 771

contains

sertion

results

and W. [x; y; 77

is

a]

no

from

;a].

(,O(X; y),

of

k

W&; X;

0 have been used

obtain

we

11))

Y;

VE,

the

to transform

K1,

of the operators

system of difference the the

with

construct

.

.

.

,

Kn,

same as

discrete

periodic

the

equations

periodic

Gauss-Manin

sections

of the

by equation

satisfied

section

connection.

homological

In

bundle

integral. in

[30]

T[x;,q]

is

proved

space y; n;

the

nearly

bundle will

(5.11)

equality

the statement.

prove

map is

hypergeometric

hypergeometric

/ym)te

the definition

account

shows that

discrete

certain

..

of the

M=J

into

(4.14),

theorem

X

and the invariance

hypergeometric

the next via

(5-15)

M=11111)n)

n

for

-

(4-23).

with

the

(5.14)

M

with of the homological bundle, invariant Then the satisfies JunctionTf, cf. (5.13), Zn*.

Kl,...,Kn

operators

,

section

respect to the operators Zl*, a system equations of difference

QmT/(x;y)

.

Yn)

P Ym, Ym+l,

....

(Z; ?P'ZMf)'

=

Letio

I

=

have

we

of the operators

Theorem 5.1.

xn; p Yi

I

...

follows:

as

via

that for generic x, y,,q, a the trigonometric cohomolisomorphic to the hypergeometric the canonical In other words, the space projection.

functions

Corollary Therefore,

which

[A. 1]

are .

the total

The

same

twisted is true

differences. for

the spaces

This

as-

F. [x; 77

Vitaly

234

Taxasov

dim W[x; y;,q

Furthermore,

it

is

a]

;

possible

[x;,y;,q

dim 71,,

=

)

n

explicitly

show

to

a]

;

+ f

In

-

I

cohypergeometric implies

the

that

-

that

spaces W[x; y;,q; a] and 'R. [x; y; 77; a] are the same, which in addition to difference the functionT/ equations (5.15) satisfies

ence

equation

homology

with

f (t; and it

bijectively

maps

functional

of the total

isomorphism

linear

to the transformation

respect

X;

Y)

ti

-+

a.

a

differ-

corresponding

The

is

Y)

X;

hypergeometric

trigonometric

the

(t;

4f

p

F[,q ]

space ...

-+

a

space

F,'[x;?jj

to

77 ].

F[X;

implicitly

Aomoto

Moreover,

[1]

showed in

dimfi,[x;y;?7;a]

=

for

that

almost

all

x, y, 77,

a

(n)+t-ln-l.

is a model for Thereby, the trigonometric hypergeometric space F[x;77] and the defined functionT/ section by a periodic W, [x; y; 71 ; a], '0 of the homodifference shift bundle solves twisted the all to equations corresponding logical their not to X,,Yn. X, Yj'...' only Y, products XIYI,..., operators Xj,...' form of the additional difference an explicit equations is not Unfortunately,

known yet.

sections

Periodic

5.4

hypergeometric Consider cf. it

(3.16).

cf.

the

vanish

will

Let

us

J [x;

pairing

D.) [x;

on

extend

y; n;

a]

y; 77 &

]

F,11 [x;

(9

F,11 [x;

Wedescribe

(t;

71;

77 ;

a],

spaceX,11 [x; 77; a] below. explicitly be the phase function (3.17)

hypergeometric

elliptic

bundle

via

the

space F,11 [x; n ; a] introduced the hypergeometric pairing I

hypergeometric

elliptic

the

(3.15), (3.18), to

[3.],

homological

of the

integral

this

a]

-+

C in

such

in Section

[x; a

y;,q;

a],

way that

inducing a map from the homology space j * [x; y;,q; a].

thus

to the

extension

y;,q)

and Int

[x;

]

hyper(3. 11). Take functions f E P[x; y; 77 ] and g E F,11 [x;,q ; a]. geometric integral for substituting into The product as an integrand fg P(; x; y; n) is eligible and the result does not Int [p-'x; p'y; p'77 ] if s is a large positive integer, depend on s. So, for given f, g we define Let !P

x;

I[x;y;,q;a](f,g) taking

a

sufficiently

large

i [x;

y;,q

;

=

:

J [x;

In the one-dimensional

(2.9). with

The restriction

I

[x;

y;,q

;

a]

-

of

]

y;,q

f

0 g

case

definitions

i [x;

y; 77 ;

a]

be the

1nt[p-'x;p-'y;p',q](fg P) the

s, and extend

a]

y; 71

(9 -+

on

hypergeometric

T,11 I

(5.16)

[x;

[X;

71 ; Y; n;

a]

-4

al(f,

pairing

as

follows:

(5.17)

C,

g)

-

(5.16), (5.17) are equivalent J7[x; 77 ] (&Xjj [x; 77 ; a] clearly

to

(2-8),

coincides

q-Hypergeometric

[30] a] (f g)

Proposition

5.3.

have I

[x;

Proof

(idea).

Proof

Denote for

y;,q

;

=

,

while

a

f

For any 0.

the shift

by Ua

UaAt1,---,t1) For any

(t)

functionF

eligible

DJ [x;

E

a]

y;,q;

with

a]

respect

to

ta:

[x;

y;,q

]

we

At1,---'Pta,-,W-

=

integral

hypergeometric

the

F,11 [x;,q;

and g E

operator

235

Theory

Representation

and

Functions

Int

has the

property: Int

[x;

y; n

] (Ua F)

Observe that

(9

Va cf.

(*;

(5.8).

(5-7),

(5.5),

!P

x;

of the local

coefficients

of connection

] (F)

y;,q

g!P(.;

product

the

[x;

Int

=

77))

X; Y;

Hence,

y;,q)

any h E

J [x; [x;

I

y; 77;

implies

which

the required

The last

proposition F,11 [x;,q; a]

space

Notice

that

to the

spect

a] (Ta[71; a]

a

Wa(*; x; y; 71;

a)

(5.16)

formulae h

y;,q)

gx;

h,

-

g)

I

show that

a

(5.10).

cf.

[x;

0,

=

(5.18)

and

a) maps the elliptic homology space fi* [x; y;,q; a]. Int [x; y; 7) ] is integral hypergeometric that

hypergeometric

y;,q;

the

transformations

xm -+

p xm and ym

1nt[x;yj,---,PYrn,---'Yn;?71(F) hypergeometric

the

Consider B.

To have

a

a

functiong(-; orbit of (x; y) a

a

ji

dual

discrete

......

bundle

y)

x;

the

flat

f7m 9(*;

xi.... X; Yi....

integrals bundle

fibers

are

the

connection

i7n

acting

on on

PYM) ....

a]

F,11 [x;,q; that

and

the bundle

for for

equal

(x; y)

point

a

any

(x; y)

E B to

any

point

in

to

-defining

over

X.,11

[x;

71;

the

a].

:---

^

Z2n-

Introduce

commuting operators

Yn)

-=

(-PxmY9(*;X1,---'Xn;Y), (-P Ym)

_'

9 (';

X; Y1,

-

(5.20) -

-

,

Yn)

,

E

have

sections:

Pxm,---,Xn;Y) )

Jnt[x;y;,q](F)

means

same

(5-19)

meaningful.

fiber

a]. Notice,

Y,11 [x;,q;

E

are

a

re-

is

=

=

B with

g of this

section

kn 'ki,

Xrn 9(';

over

with

invariant

p ym, that

-+

Int[xl,...,Px,,,,---,Xn;Y;77](F)

whenever

of the system

function

fiber:

Ua(hg!P(-;x;y;y))

=

statement,

means

into

the

phase

a

(5-18)

=

have

we

]. Therefore,

y; 77

is

system of

(Ta[77;a]h)g,P(-;x;y;7)) for

a

,

Vitaly

236

,m

1,

=

call

.

.

.

,

Taxasov

a

for

operators the discrete

with

elliptic

g of the

section

corresponding

the

these

bundle

the twisted

flat

shift

bundle hypergeometric bundle: homological

of the

section

(i [71 ; a] g) (x; y) hypergeometric elliptic bundle. That homological

bundle

the

a] 5f,,,

hyper-

y;,q

to the

a]

;

y)

discrete

the

dual

g (.; x;

.

Take

functionf

a

by

denote

1

[71; a]

g

.

(5.20)

connection

Gauss-Manin

connection

i [77 ;a] f7,n

f [77 ;a],

on

on

the

M

act on sections where the operators Xl*,. Yl*, , Xn*, , Yn* connection. to the dual Gauss-Manin bundle according .

E

.

) [77 ]

and

.

a

of

.

elliptic

g of the

section

homological

the

hypergeometric

Let

bundle.

F,n (x; y) Denote for

by U.,,

while

a

=

f [x;

To prove the first

)xn)

relation

shift

x;

y), kn with

operator =

1 [x;

=

a] (f (-;

y;,q

respect

;

for

a] (Xm f (.;

x;

y),

Given

a

to

(5-7),

section

with

space

produces

with

elliptic

g of the

a

(x;

functionTf.

bundle

hypergeometric

coordinates

the tensor

y; 77;

x.:

on

a)

the

with

.

.

.

n

,

y)) (5.9)

the map I

trigonometric

values

1,

and

(5.20)

from the invariance

=

bined

.

m=

g (-; x;

(5.6),

Formulae

operator.

any

Un (f k,,,g 45). So the claim follows Xn f g 4i integral. (5.19) of the hypergeometric is similar. The proof of the second relation that

y))

g (-; x;

h(xl,...,Pxm,...,Xn)-

have to show that

we

UmFm(x; y) where Xn is the twisted

y; 77 ;

the shift

U,,, h(xj....

imply

elliptic

is

X,*n f [77 ; a],

=

i [x;

=

i [77; a] transforms

The map

Lemma5.2.

Proof.

We

operators.

the

connection

bundle.

geometric For

taking

and

n,

the obtained

[,q; a]

com-

hypergeometric

in the vector

space V

by

the rule

-[[X;Y;,q;al(w.(.;X;Y;,q);g(.;X;Y))v.,

Tf,(x;y;,q;a) MC

(5.13).

cf.

Theorem 5.2.

the

Let

that

9('; cf. over, way.

X1,

(5.9).

Pxm

...

Then the

for generic

a

(5.21)

ZI section

g be invariant

with

respect

to

the

operators

is

)xn;Yl,---,PYm,---,Yn)

junctionT9 all

solutions

=

(xn7,/Ym) g(';X;Y),

the difference satisfies equations (5.15). of the system (5.15) can be obtained

(5.22) Morein

this

q-Hypergeometric

and

Functions

Theory

Representation

237

The first part of the statement follows from Lemma (sketch). The second part follows from Proposition [5.11. [3.2] and [5.2) of the hypergeometric cf. Proposition the nondegeneracy pairing, [3.3].

Proof.

Proof

and Theorem

W[ (t; weight function hypergeometric elliptic

elliptic

The

WC[71; a]

W1[77 ; a] (x; y) transformation

Its

x; y;,q

of the

differ

properties

a),

;

Wr (-;

=

a

little

(4.2),

defines

natural

way:

cf.

bundle

in

a

x; y; 71 ;

a)

section

.

(5.22),

from

a

containing

one more

multiplier

Q-W[

i

(Xjlyj)

rl

ttl'.

=

Wt,

I<j<m

(an'-"

1.

xjlyj)-

rl

In,

1, <,

-

-

+Im,

-,

I<j<m

cf.

(4.21).

the

functionT/W,[,7;,]

results

This

slight modification compared with (5.15): in

1,

[x;

,

.

a]

y;,q;

system of difference

The last

n.

of difference

system I

.

.

(4.19)

equations

definition

the

for

(5.23)

equations is nothing else but the by the hypergeometric map

satisfied

because

(x; y)

T1w, by

equations

=pr,.Km(x;y)1Pw,[,O;,,,j(x;y),

Q,nTfw,[,?;,](x;y) m=

of the difference

a

of the

[x;

y; n;

coordinates

tensor

on

a] vj*

(5.24) hypergeometric

elliptic

the

space.

Asymptotics

6.

In this to

case

is

given over

.

.

.

in

this

Section

duced in

(j),

we

illustrate

All

e

prove Propositions the idea of the proof.

section

case

,

e

hypergeometric

of the

[4.4]

and

The

maps

[4.5]

the one-dimensional

for

proof for the multidimensional

[30]. section

[2.]

we assume

the

that

t

one-dimensional

=

1.

We use notations

intro-

the labels particular, which should be used for the weight functions according to nogeneral case, are replaced by labels 1, (3.5) n, the ordering

(n),

tations

in the

being

translated

for

In

case.

.

Weconsider

A

as an

follows:

e(f)

asymptotic

lx'lx'+l 1xM/YMI

<

zone

I

e(m)

for

1 >

A of the parameters

<

m=

1,

M=

.

.

,

m.

1,...,n 1'...,n

xi, -

.

.

Xn i Y1,

I I

-

-

Yn:

Vitaly

238

Taxasov

(x; y) -14 A if xm/ym and ym/xm

writing

xm

Ix,,,,+l

11',n(x;y) [4.5]

now

are

Proposition

equivalent For

6.1.

to the

l,m

any

=

HMAI?nn?,

Wegive

I'm (x; y)

a

proof

(2.8)

fi

=

27ri

(2.4).

and

(t

=

0

m, and

(0'-'Y-Ix-)-

(pc'-)-

Ilm (x; y). The proof for the integrals

integrals

given by

is

wt

(t;

y) Wm (t;

x;

a)

x; y;

4

(t;

y)

x;

'[X;Y1

(2.6)

Using formulae

(2.18)

and

we

dt

(6.2)

t

that

obtain

the inte-

-

X0 (P

M-1 t I xm)

Xm/t)

(t/Ym)

.

t

ri

t

1
has

singularities

0,

t

is easy to

=

see

we change same multiple.

if

A the

xj,

integrals

HMAIlm

Moreover

has the form

0 (a

It

hypergeometric

-4-'A.

(x; y)

[4.41

Propositions

(po,-X-/Y-)(X-/Y-)(P)-

-a-

the

for

(x; y) 1

(X; Y)

Ii.

=

as

(6-1)

,

is similar.

The functionllm

t

We

alll
am =

Proof.

it

ratios

(4.25).

cf.

(-;2)-

=

HMAIm/711

so

1 and the

-

1,...,n,

the

1,...,n

=

limits Ilm (x; y) and I,m (x; y) have finite 0 unless 1 > unless 1 < m, limA IIm

grand

any

n

m

are given by (2.18). statement. following

wj,w1,W,,W,,,

functions

where the

cf.

m

I[x;y;a](wl(.;x;y),W.(-;x;y;a)) I[x;y;a](wl(.;x;y),W,,,(-;x;y;a))

Il.(x;y)

Here

any

for

the functions

consider

and

0 for

-+

bounded

remain

yj

replace

that

yj

-

t

the

=

P-8Yk

So without

loss

following

/t)

J1

j :

m<

integral

Int

of t,

x,....

generality

m
we can

k < n,

(6.2)

the

integration

contour

t, [x; y] by

8

fl [x; y] (w, Wm 7xn,

E

that from

as

by the

(x; y) I

zero,

Therefore,

Z>o.

does not

Yl,---,Yn

assume

parameters xm, ym remain bounded and separated 0 for I < j < m and Xk, Yk -+ oo for m < k < n. in

(tlXk)oo (t/Yk)oo

points:

1 <

7

dilations of

Yj

-1-1, (Pxjlt). I<j<,m

hypergeometric

make simultaneous

(P

TT

Xi

-

at most at the

P'xj'

(P)

.

while we can

q-Hypergeometric

t(N) [Xrn,

Y7 nj

r

yr

=

and

Functions

E

+

Representation

239

C[PS YM

PsX,71]

C

Theory

-EZ>-N

-EZ
jpsx >r

jP1

Y-

I
sufficiently large positive integer N, since all other small circles C[p8xi], inside and, appearing in (2.3) contain no poles of the integrand C[p'yi] to the integral. thereby, do not contribute limit have a finite the integrand on t(N) Furthermore, [X 1, Yrn] as xj, yj 0 for I < j < m and Xk) Yk -+ oo for m < k < n, while xn, yn remain fixed.

for

a

r

The limit

equals 0(a;1tIx-) (PX-/t)-

t

(P)-

(t/Y-)-

0(a-"t/x-) *

t

(X-/t)-

(t/Y-)-

(P)-

0

be calculated

fj

-L-

and

27ri

n,Yn]

rating

7.

an

1

=

M,

for

I >

M.

(PX-/t)f'1(t/Ym)O(C1;1t/Xn)

(N)[X .,yJxm/t)-

O(c t)

r

(t/y-)-

anticlockwise

I psa I

The quantum

s

-

t

oriented

Z
E

and

dt

(P)-

t

dt

TpF- T

I p8l

(p blc),,. (ab) c,

around

contour

loop algebra

equation

(ac),,

dt

27rik(a/t)c,.(bt),,,) the sets

for

using the formula I

where C is

I < M,

o(a-'tlx-)

fj'(N)[X

21-7ri

can

for

integrals

remaining

The

77

b

I

s

E

U'(gf,)

the

origin

t

=

0 sepa-

Z>o

and the

qKZ

hypergeometric spaces and theory of the quantum loop and the elliptic the We identify algebra Uq(gf,) quantum group Ep,',(s(,). and the with difference equation, trigonometric qKZ equations (4.23) (5.14) cf. (7.11), (7.12), and the difference equation (4.24) with the dual qKZ equathe hypergeometric tion (7.3). Further, we interpret map as a map from a module over module over the elliptic quantum group to the corresponding the quantum loop algebra.

In

Sections

the tensor

7. and 8.

coordinates

we

show in what way the in the representation

arise

240

Vitaly

Taxasov

7.1

Highest

weight

Let

q be

a nonzero

the quantum group

U. (s (.,)-modules complex number which

is not

U. (s (,)

E, F, qH

with

generators

qHq-H qH E

qE qH,

=

2H

coproduci A(q

, A(E)

=

H

:

)=

E0 1 +

Uq (s (,) q

H

itq (s (,)

0

be

A(q -H)

A(F)

E,

0

q-'F qH,

-2H

itq (S

-4

=

=

qH,

(9

q2H

defines

coproduct

The

A

=

given by

q-H

=

0

q-ff

(7-1)

F 0 q -2H +10F.

Uq(sl,)-modulestructure

a

Consider

and relations:

1,

=

qH F

[E, F] Let the

q-HqH

=

of unity.

root

a

of

product

tensor

a

on

Uq (s [,)-modules. We take

coproduct

here

the

from the

coproduct

used in

[30].

bedding

Uq(sf,) (7.4)

V,(gf,)

and the evaluation

Remark 7.1.

Uq(sQ, Let

cf.

-+

and

also in

differs

which a

modification

highest

qA

em-

Uq'(B[,)

homomorphism'E:

weight

slightly of the

(7.5).

Uq(M,)-modulewith

V be the

U,,(sf,)

for

This results

"0

Let

V

ED VA-1

=

be

1=0

its

weight

operator

For decomposition. uA-H E End (V) by

UA-H V

7.2

The

=

any

1V

for

U

any

v

E

to

&0) 2-3

relations

loop algebra Uq ( F,) is a unital associative 70) < j :5 i < 2, and L !), 1, 2, s i, j -3 V r

)

=

Z

(7.2),

-

algebra

-

eij be the 2 x 2 matrix of the -i-th row and 'j-th

(uq

R(u)

with

+

in the

the

u(q

generating

-

only Set

1) (ell

-

q-1)

series

quantum loop algebra

the

column.

q-1) (ell

-

(u

+

Introduce

=

with

an

1, 2,-

..,

genera-

subject

(7.3).

Let tion

define

VA-1

The quantum tors

u we

U,(gf.)

algebra

quantum loop

complex number

nonzero

(2) ell

L: (u)

Uq(j1,)

entry

+ e22 (9

e22)

+

ell)

+

(2) P-22 + e22 0

e12 0 e2l

=

L

1 at the intersec-

nonzero

+

0)

(q

q-1)

-

00

+

EL

have the form

e2l

& e12

8)u :'.

The relations

q-Hypergeometric

Li( O)Li,70)

L_ (1_)

where

[7].

R(xly)

L+(j(1)

=

L 2)(2 (y) L+(,)(1 (x) R(xly),

R(xly)

L ,)(1 (x) L 2)(2 (y)

L 2)(2 (y)

(x)

eij 0 10

L: (u)

=

=

1

(7.2)

(02)

UIqI

Uq' W2

:

L-4-)(2

PX

is

L(I-lO)q

There is also

V+

2

restricted

=

obtained

V(x)

u

V_

1,

E

-+

the

0

=

L:: (u).

to

For any is

are

1

2

2

[24],

central

the

corresponding

a

Hopf algebra

(u)

kj

is

(7.3)

1,

1

2

ik

L: (u)

Uq' 012 )

(u)

algebra

quotient with

a

coproduct

.

family

(

of

automorphisms

is called

the

Lt(u/x), U,(s[2-)

contains

homomorphism

0 and

_

-

we

arrange

the

as a

H

C :

qH q-Hu_1 q-H E (q q-1) U-1

-+

L(120)1(q-q_1)-

UqI OfFz) 1

-H

generating

-

subalgebra the

module

evaluation

the

(7.4)

Uq(-5[z):

--+

F qH (q q

Hopf subalgebra,

q7-1) qHU_ I -

series

j Lt(u)

(7.5) into

automorphisms p,, and the homomorphism Uq(.slz) are the identity maps.

Both the

Uq(sf,)-module/ from

-+

-L2(+10)1(q-q-')Fq

evaluation

-T

t-4

2 matrix

x

-H

an

L(u) where

see

U,'(0[2):

loop algebra embedding being given by -*

1

one-parametric

The quantum

H

L2( OW1TO)

L(,TO)L2( O)

1

(u),

13

Uq W2

px

q

10 eij & Lt

(u)

I1

important

an

Uq'(01,

:

R(xly),

1L2(+20)L(lo) L2( O) L,(TO) 1,

(u) There

(7-2)

1,2,

relations:

Uq'(j [)

loop algebra

241

L 2)(2 (y) L ,)(1 (x) R(xly),

and denote

Uq(gl,). The quantum

and

following

-

to relations

=

L2(+20)L(1+10)

L1(+10)L2(+20),

(1 +10) L 2(+2 0) L(, O)L2(20)

A

=

L+(,)(x) L+(2)(y) (1 (2

L

by

i

1,

=

Weimpose the

in addition

=

U

L+(2)(y) L+(,)(x) (2 (1

(u)

Uq (g [,).

ii

Theory

Representation

R(xly)

The elements

in

Li(,-O)L(ii+o)

1,

=

ii

ii

and

Punctions

V denote

V via module.

the

by V(x) the ^Uq'(j [_,)homomorphism E o

a E

module which p,

The module

Taxasov

Vitaly

242

V1, V2 be Verma modules

Let

Uq (st, .)

over

with

generating

vectors

V1 i V2

generic complex numbers x, y the Uq(g[2)- modules V, (x) 0 The modules are generated V2 (y) and V2 (y) 0 V, (x) are isomorphic. by the is unique up to normalization The intertwiner vectors V1 0 V2 and V2 0 vi. and maps the subspace C (vi 0 V2) to C (V2 0 vj). The normalized intertwiner where Pyly, has the form Pvv2 Rvv, (xly) : V, 0 V2 -+ V2 0 V, is the pervalues in End (VI 0 V2) having mutation map and Ry, v, (z) is a functionwith the properties: For

respectively.

Ry, v2(z)

(F (F

Rv, v2(z)

^

0

q-2H

0 1 +

Such

a

functionRv,

R-matrix.

also

(z)

V

the

(E

Rv, y2(z) particular,

In

(z

and is

following

q2H

Rvy2(z)

+

Oq

=

OE)

=

IOE) the

respects

-2H

+

V1 0 V2

=

0 z

F)

10

R*V,

(7.6)

v2(z),

F) Rvv2 (z)

.

i

It is called the uniquely determined. product V, 0 V2. The trigonometric

relations: H

v2

q-2H

(FOI+

0 V2

vi

the tensor

2H 0 1 + q

E0

(z)

for

,(z)qH

Ry,

Rv, y2(z)

v2

exists

R-matrix

satifies

=

q-2H OF) =-(FOq

z

Ry, sf, trigonometric

1OF)

+

=

q

H

0

qHRy,

(EOq2H

y2

(z) E) Rvv2 (z)

+ 10

(zEol+

q

2H

(7.7)

0

decomposition

weight

,

E) Rvv2 (z) of V, 0 V2

with highest over U,(s[,) V2)j be'the weight subspace of weight is a rational qA., +A2 - . Then the restriction function v2 (z) to (Vi 0 V2)j f- 1 Moreover, r 0, of z which is regular for z 0 q2A,.+2A2-2r Ry, y2 (z) 1. f is nondegenerate/ 0, for z 0 q 2r-2AI -2A2, r

7.1. Let Vj,V2 Proposition weights qA1, qA2, respectively.

Verma modules

be

(Vi of Ry,

Let

(9

,

=:

=

The

R-matrix

trigonometric

Py For

U,(sf,)

the

Yang-Baxter

Rv,

v,

All

these

[29],

[3]).

v2

Rv,,v2(z)

Verma modules

(x1y) Rv, properties

satisfies

-,

relation

the inversion

=

-

-

.

(Rv2v, (z-1))

-1

Pyj

V1, V2, V3 the corresponding

(7-8)

v2

R-matrices

satisfy

equation: v,

(x) Rv2v, (y) of

trigonometric

=

Rv2 v3 (y) Ry, R-matrix

are

y,

(x) Rv,

well

v2

known

(xly). (cf.

(7.9)

[51, 114],

q-Hypergeometric 7.3

The

Let

Vj,...,

A,

q ric

Vn be Verma modules

...,q ^

respectively.

,

R-mairices.

(u)

be the

(VI

End

E

Uq(st,)

over

Rvivj

Let

Rij (x)

Let

243

qKZ equation

trigonometric A,,.

Theory

Representation

and

Functions

with highest weights corresponding trigonomet-

Vn)

(9

be defined

in

standard

a

way: i-th

i-th

Rij (u) provided

id 0

Rvi

that

(u)

yj

%r(u)

...

(u)

r

& r(u)

E End

0

%r(u)

(Vi

0 Vj).

0

0 id

...

For any X E

Uq (S 1z)

set m-th

Xm Let p,

r,

ide

=

complex numbers.

be

A. (zl,.

Zn)

-,

.

[9]

Theorem 7.1.

For any

0

1,

m=

-

The linear

I

set

Rn,l(pzm/zi)

(7-10)

x

Rn,m+l(zm/zm+l)

...

A, (z)....

maps

n

-,

-

...

-H-Rm,n(zm/zn)

A-

(9 id.

...

Rn,m-,(pzm/zm-,)

=

X r.

0% X

...

An (Z) obey the flatness

con-

ditions

A, (zi,

pzm

Zn)

zn) Am(zi,

....

Am(zl,...,pzl,---,Zn)Al(zl,---,Zn),

=

n.

M

Yang-Baxter

from the

follows

The statement

(7.9)

equation

and the inversion

(7.8).

relation

The maps A, (z), over (Cxn with

An (z) define a discrete o Vn- This V, (9

.

fiber

bundle

...

flat

connection

on

is called

connection

the trivial

qKZ

the

connection.

By (7.7) in

V1 0

the 0

...

A, (z),

operators

and, therefore,

=

&

1,

.

.

.

,

n,

of (Vi

isomorphisms

0

0

qKZ equation system of following T (Zi

I

.

,

An (Z)

the

commute with

....

Pzm,

=

E

Vn)i

M=

decomposition

action

of q H

-

-

-,

of V, 0

.,n, 0

...

Vn-

zllz,,,

f

-

1,

define

-

(7.10) and Proposition zn) functionT/(zl,. difference equations [9]: .

=

-

weight subspace of weight q 0,..., 0 p' for all r An (z) Z, the linear maps A, (z), -2r

a

Zn)

1'.

be the

2AI +2A_

.9

0,

from for

The

q

m,

...

follows

The statement is the

54

1

qH ]

Vn)j

0

...

(Cxn such that

E

,

weight

the

(VI

Let

Zn)

-,

respect

Lemma7. 1. z

.

Vn:

[Am (zl,..

For any and 1, m,

.

An(zl,...,

-,

[7-1]. with

zn) TI(zj,...'

values

zn),

in

V,

0

Vn

(7-11)

244

Vitaly

.

m=

1,

tions

with

els,

.

.

Tarasov

The qKZ equation is a remarkable system of difference in representation applications theory and quantum integrable ,

.

n.

[15],

[9], [26],

see

To make easier the

following

comparing of the qKZ equation and the differthe qKZ equation hypergeometric in maps we rewrite

the

way:

T/(PZI,...,pz,,,,z,,,,+,,...,Zn) ,rn

M

AfmI(zj,_,Zn) R,,, (z:L,

.

.

.

,

z,,,)

(7.12)

V]L

0

0

...

V.

V.+j

Zn),

,

R2,m+I(Z2/Zm+l)

X

x

Rm,m+l (zm/zm+,)

...

OV.&V-,(&

...

(7.14)

x

........................

0

(7.13)

Ri,m+,.(zl/zm+,) ...

Rm,n(zm/zn)

x

]R"rxl(zl,...

...

R2,n(Z2/zn) x

:

KAI-HI

Ri,n(zl/zn) x

Let P.

AI'I(zj,.--'Zn)!P(Z1,---'Zn),

=

where

1,...,n,

=

mod-

[19].

[21],

the future

for

equations

ence

equa-

...

(&V,,,,

be the rotation

map:

P.

V(I)

:

Then the tensor

V(.)

0

P.R. (z:L,

product

products

of

Uq(g[,)

P nR.(zj

Pm,Rm(zi,

-

-

-,

Zn)

:

-

.

,

-

Zn)

.....

VI (9

(9

:

...

...

is

the

is the

TI,

.

.

following

(P

Z,....

=

&

V(I)

VI(Zi) 0

Vn (Zn) -+

0

0

...

-

-

-

i

V(-)

(&

...

intertwiner

0

Vn(zn)

VI (ZO

VM+1 0

...

0

of the

Zn)

(Alml(zl,..-,Zn))*-IT"(Zl,---,;n)ra=l,...,n.

(7.15)

-

0

...

VM(ZM)

0 Vn 0 V1 0

generating vectors of the modules V1, with values equation for a functionT/' system of difference equations: P Zm, Zm+l,

&

normalized

Here vj, are , v,, The dual qKZ .

V(n)

(&

modules:

(9 Vn

...

(0

Zn)

evaluation

VM+1(ZM+1)

-+

V(-+I)

'4

.

.

.

in

,

...

-

OVM-

V, respectively.

(Vj

0

Vn)* (7.16)

q-Hypergeometric 7.4

coordinates

Tensor

trigonometric

the

on

Theory

Representation

and

Functions

245

hypergeometric

spaces

be

Let

Vl,...,V,,

A,

A

q

...,q

and

modules

Verma

generating

vectors

Uq(.sf,)

over

v,

,

....

with

Vn,

respectively.

0

Vn)k

weights

highest Let

00

Vi

(

Vn

0

(Vi

k=O

(V1

0

Vn)

0

...

functional

on

having weight

k

V,

0

...

0

0

E

Vn)*,

0

...

weight

the

k. Denote by (vi

-

0

Vn)

0

...

subspace *

a

linear

V1 0

Vn)

0

...

00

Vn) *, V)

0

...

q

product,

tensor

A-

Vn such that

(9

((V1 ((V1

of the

decomposition

weight

be the

for

0

=

any

v

E

(D (Vj

(9

0

...

Vn)

k

-

k=1

Fix

f and consider

integer

nonnegative

a

the

vector

=ED

V

spaces

(C vm

mEZ."

C v' m like

and V'

in Section

Assume that

4.-

MEZ" q

4A_

-2r

:-

the space V with the dual space

We identify

1,

,

7

r

weight

the

(VI

space V' with

0

0

...

f

=

subspace (Vi Vn)j* by the rule:

rIl<j
I

:--

1.

-

Vn

...

0

Vn)j

,

(7.17)

-

v,

I

=

...

...

0

Vn)*

and the

-

Then n

ri

V-)

I.-i

q2s+2) (q4A--2s

11

1

M=1 3=0

-

(7.18)

q2

which are related to the hyperthe parameters we identify Furthermore, of the the related with to representations parameters geometric integrals follows: and the as qKZ equation quantum loop algebra P x.

Consider

=

=

q

77

P,

2A-

yn

zM'

dual

the

q2,

=

a

=

q

(D

V*

spaces

-2A-

=

n

q21-2-2 E A-, M=

zM'

C v*m and V'*

dual

(D

=

mEZj'

(7.19)

1'...'n.

mE

C v*m with

the

Ze"

bases:

V.) Recall

that

spaces

T[x;

the 71 ]

tensor

and

J7'[y;,q

=

(V'[*,

61

coordinates

]

are

the

on

the

following

V,m

Jim.

trigonometric linear

maps:

hypergeometric

Vitaly

246

Tarasov

B [x; y; 71

V*M

cf. and

(4.7), (4.3).

]

V*

:

-+

Comparing

(

,

of V and V'

(VI

subspace

V'*

:

0

actually 0

...

S [x; y;,q

*

])

y;,q

]

]

T'[y;,q

[17],

Theorem 7.2.

and its

B'[x;

[30]

y; 71

(.F[x;

-+

hypergeometric

trigonometric

coincides

Vn)j

bilinear

find

with

the

dual

space

(VI

form

that

canonical 0

vector

particular,

In

types. a

(4.1)

cf. the

the

on

of the

pairing 0

...

V.

pairing

Vn)j*.

The

words is

S [x; y; 71 ]

:

of different

V''not immediately

we

(7.20)

weight functions, here distinguish

of V and

(4.9)

and

P [y;,q

-+

W,M(t;x;y;,q),

-+

coordinates

pairing

a

in different

statement

(B [x; Here

(4.8), is (7.18)

see

formulae

) [x; y;,q]

same

]

y;,q

v1*M

,

the tensor

with

(, ) [x; y;,q],

weight

Y;,q)

X;

B'[x;

where wn, and w' are the trigonometric Notice that, unlike in Section [4.], we

spaces associated now

(t;

W..

T[x;,q

-+

]

id E End

=

0

0

...

Shapovalov

the

is

71

((Vi

Vn)j )

.

pairing

of the

by formulae

(7 17).

-1(1-1)/2

*7.21)

spaces.

Let

vf',

vj,

I

3j',

E

be

given

Then

LT2 12 (ti) X

LT2(tj) 12

...

rjt.,

rIn

a

(1

,

M

(9

vi

...

(9 vn

=

(q

Xm/ta) r11
-

r1i=1

X

Proof.

(t;

L j(tj))*(v-j

In

=1 M

a

Proof

...

(idea). y;,q).

(1

-

(9

0

...

-tb ta-tb

Vn)*

E

(t;

wM

(7-21)

Y;,q)

X;

E A_

qf(t-1)/2-i

=

E

ta-tb

the formula

A-

V.

MEZ'-

71-Ita-tb

Ym/ta) f11
Consider

q'E

q-1)'

77 ta

-

-

(L j (ti)

-

(7.22)

X

, (t; X; Y; WM

77) vM'

MEZt" as a

definition

of the func-

using the coproduct and the commutation relations for the quantum loop algebra one can show that these functions belong to the hypergeometric trigonometric space F[x; 71 ] and have the double triangularity properties (4.6). Hence, by Lemma [4.1] they are linear combinations of the trigonometric which also have the double triangularity weight functions properties (4.6). This means that the connection matrix is diagonal, and one has only to show that its diagonal entries equal one, which can be done by comparing either residues at points x t> m[lq I or values at points y > The proof of formula (7.22) is similar. tions

w.

Formulae

x;

(7.21)

Then

and

(7.22)

are

hypergeometric

spaces

of the quantum

loop algebra.

Corollary we

have

7.1.

[30],

and the

[31]

the

key points

tensor

For any

v

E

in

connecting

coordinates

(Vi

0

...

(&Vn)j

with

the

the

trigonometric representations

and V* E

(V10

...

OVn)j*

q-Hypergeometric

(v* (q

x

LT2(ti) 12

,

LT2(tj) 12

...

(9

vi

Ovn)

...

L j21 (ti)

&

Vn)

x

qf(t-1)/2-i

Theory

(1

x

71 t

11

Xm/ta)

-

(B'[x;

v)

y; n

II

A-

tj)

a=1

The statement

of

tification

Kl,...,Kn,

from

(9

Vn)

Let

and formulae

qKZ

g'

(Alm (4.23)

taking

F,',I[y;?I;a] of the

be sections

(7.23)

Km.

'='

q

nearly coincides with the weight subspace (Vj 0 (4.24) nearly coincides with

in the

values

the system of difference equations taking equation (7.3), the functionV and

EAj

V

Km,

equations

the functionT/

F,11[x;,q;a]

g and

EAj '=1

q

=

(7.12),

while

j,

the dual Let

(7.20)

system of difference

the

qKZ equation ...

tb

-

.....

-21

is

1
m=1

tb

-

Corollary

and

Alml That

ta

x

(7.21), (7.22). [7.1] imply that after a suitable idensee A#I,...,A#n, notations, (7.17), (7.19), the operators (A#n)*-' given by (7.13) almost coincide with the operators K,, given by (4.14): K, follows

(7.15)

Formulae

-1ta

Y-1t.)

(I

tb

-

] v) (ti,

n

I:

tb

-

a

ta

I
m=1

LY, 21 (tj)

...

247

(B[x;y;,q]v*)(tj,...'tj)

=

11 11 a=1

((VI

Representation

n

q-1)' qE An-i(i-I)/2

-

and

Functions

values

(Vi

in

0

...

Vn) j*,

0

elliptic hypergeometric spaces. bundles with fibers hypergeometric that is for any point (x; y), functions [y;,q; a] are given. Let I [x; y; 77; a] pairings (3.18). The sections g and g'

be the

elliptic

a], respectively, a] and.F,',I[y;,q; .F,11[x;,q; g(-; x; y) E-Fell [x; 71; a] and g'(-; x; y) E and I[x; y;,q; a] be the hypergeometric define functions T1, and P,', taking values

(Vi

in

0

...

OVn)j

and

(Vio

...

0 Vn)

J*

I

respectively:

Tfg(x;y;77;a)

=

E I[x;y;,q;a](w ME

Tf,',(x;y;,q;a)

=

..

(.;

x; y;

77);

g (.;

x;

y))

v.

Z,"

E 1'[x;y;?I;al(wm(.;x;y;77);g'(.;x;y))vm7

I

MEZI' cf.

(5.2 1).

Recall are

that

related

the via

Theorem 7.3.

parameters

xi,

Xn,

Y1,

Yn and zi,

zn,

A,,

.

(7.19). Assume that

the section

9(,;Xl,..-iPxm,---,Xn;Yl,---,PYm,---,Yn)

g has the

property =

q

VA_

g (.; X;

Y)

,

.

.

,

An

Vitaly

248

1,

m=

-

Proof.

-

n

,

.

Taxasov

X1,

1,

=

-

-,Pxm7

-

[5.21

Pym,

....

functionT/g',

(7.3).

g'

is

(7.23).

and formulae

has the property -

-

-,

Yn)

=

q-2fA_ gi (.;

Y)

X;

,

of the dual qKZ equation

solution

a

(7.12).

of the qKZ equation

solution

a

the section

Xn; Yl)

i

...

Then the

n.

is

from Theorem

Assume that

Theorem 7.4.

m

follows

The claim

9'(-;

JunctionT/9

Then the

-

proof is similar to the proof of Theorem [7.3]. of solutions of the qKZ and dual qKZ equations is pairing functionof The hypera quasiconstant. clearly a -p -periodic Zn i.e. zi, this quasiconstant in terms of the elgeometric Riemann identity expresses Shapovalov pairing of the sections g and g. liptic

The

The

.

-

-

9

,

Theorem 7.5.

(Tf,', (x; The

8.

In this

Fix =

we

the definitions

elliptic

dynamical

see

.

y), g'(.;

x;

y))

x;

77 =

Ep,.y(.sf,) Ep,.,(sf,)

the

concerning R-matrices

elliptic

quantum group

associated

detailed

a more

such that

p,,y

OWO(Ah) X) 0 (A)

0 (,q

R(x,.\) a(X /\-l)

Remark 8. 1.

with

exposition

tensor

the

on

Set p

Im. p > 0.

=

[12]

the

the

with

ell

0 ell

prodsubject

=

e27riP

only

nonzero

-_

0(,q) O(XA) 0 (77 x) 0 (A) 1 at the intersection

entry

+ e22 0 e22 + e12 0 e2l

+

a(x, A)

ell

P(X, A-1)

quantum group

Ep,., (sf,)

1/2)2p

(m

0 e22 e2l

+

0 e12

is described

-

in terms

theta-function 00 =

E exp(7ri(m

-

+

+ 27ri

+

1/2) (u

+

1/2))

M=-00

which

and

Set

P(X,,\)

elliptic

'3(X, A)

,

column.

e22 & e-11 +

In

additive

e(u)

.

Let

Let eij be the 2 x 2 matrix of the ^i-th row and ^j-th

of the

a] (g (.;

;

[8], [12].

a(x, A)

+

For

complex numbers

two

e-47ri-y

modules.

y;,q

quantum group

recall

and the

proofs

and

S,11 [x;

=

elliptic

the

^Ep,.y(-sl,)-

of

ucts

a))

y; j7 ;

quantum group over

section

Ep,7 (s (,)

77

elliptic

Modules

8.1

a), T. (x;

y; 77 ;

is related

to

the

e(u)

multiplicative =

i

exp(7rip/4

theta-functionO(x) -

7riu) O(e 27riu)

by the equality .

q-Hypergeometric be the

Let

one-dimensional

Say that eigenspaces

module.

an

dimensional

v

Panctions

Lie

algebra diagonalizable

V is

Representation

with

the

if

Theory H. Let

generator

V is

a

249

direct

V be

of finite-

sum

of H:

ED vt'

=

and

Hv

=

fo

liv

VI,

E

v

r

.

A

For

a

any

v

functionX(p) E V/,. V1,

Let

.

.

.

taking

id 0

=

(2)

V,

(2)

...

0

diagonalizable tions Tij (u, A), i, j relations following

module

a

[Tij(u,A),H] 0

=

values any

ule

in

C2

E

v

i

as

(VI

(Vj) ,j

0

0 ...

V,,)

(9

...

0

(Vn)/In

ij

n

102H(&l

Tij (u, A),

A) R(xly,

=

A).

E id

T(2) (u, A)

o eij

o

(ell

-

e22)/2.

Hv [k]

be

a nonzero

=

a

diagonalizable

T12 (U, A)

(U 'X) ,

V

[k]

=

(A

-

k) V[k].

complex number. Set

Til(u,A)vlkl

T21

Tij (u, A),

Tj

Example. Fix a complex number A. Consider VA (D (CV[k] such that

z

set

1,2,

=

kEZ>o

Let

we -

the

over

ij

and H acts

decomposition

T(1) (X, /\) T(2) (Y In2HO101 /\)

0 id 0

for

K)

in End

(j-i)H,

10102H

E eij

v

^

T(2) (Y A)T(1) (X,

-

T(l) (u, A)

o

...

X(P)

=

elliptic quantum group Ep,',(5[,) End (V)-valued functogether with four in u, A E C' and obey the are meromorphic

1, 2, which

R(xly,

v

Wehave the

-

module V

a

Here

(D

X(H)

set

id.

...

pn) taking p.) v for

a

=

is

(V,)Al

m-th

functionX(pl,...' X(H,,..., H,,) v X(pl,...' see [12], By definition, For

we

modules

=

0% H

...

(V)

End

in

V,, be diagonalizable

,

V,

Set Hm

values

V

0(?,A-kU/Z) 0(77-k A) 0 (77 AU/Z) 0 (A)

=

[k]

=0(71

0(nk-l-AA-lU/Z)

A-k-IA

U/Z) 0(77). O(n AU/Z) O(A)

o(n2A-k+l)

t,(77 AU/Z) t,(A-1)

0(,Ok) -

0(,q)

77 kV[k]

71 kV[k+l]

'I

1-k

v[k-1],

mod-

Vitaly

250

Tarasov

T22 (U) A) These formulae

VA

make

evaluation

the

V

into

=O(nk-A

U/Z) O(n 2A-k A) V[k] 0(nAU/Z) O(A)

[k]

-Ep,7(s(,)-

an

VA(Z)

module

highest

Verma module with

is called

which

A and evaluation

weight

point

z

[12]. For any complex vector space V denote by Fbn(V) on Cx. The space V is naturally meromorphic functions of constant functions. as the subspace Let V1, V2 be complex vector spaces. Any ftmction o induces

linear

a

'n.

an(Vi)

-+

^Ep,.y(s(2)-

For any

module

Pin(V).

space

Fun,

Fbn(V2), V

define

we

the

f (A)

:

_+

=

f (A)

Til (u)

Ti2 (U)

such

operator

associated

operator

generated

is

f (A)

algebra

meromor-

u

of the

operator-valued

mero-

below:

Ti2 (u, A) f

-+

by

Til (u, A) f (77 A)

F-+

obeyed by the generators

(77-1 A)

of the operator

algebra

are

described

[12].

in

V1, V2 be the

that

W(A)f(A).

W(A,,qH) f (A),

with respect and by values and residues to defined i, 1, morphic functions j 2, Tij (u),

Let

(Hom (VI, V2))

_4

A,,qH acting pointwise,

in

The relations

f(A)

:

algebra

The operator

o(A, qH)

in detail

E Fbn

map

:

acting on the phic functions

the space of ^V-valued embedded in Fbn(V)

^

Ep,,y (sf,)-

induced

An element the

intertwines

W E Fbn

actions

(Hom (Vi, V2))

of the

respective morphism of Ep,,t (z I,)- modules V1, V2. A morisomorphism if the linear map p(A) is nondegenerate/ map

is called

algebras

modules.

Ful p

phism W is called an generic A. Example. Evaluation

^

a

for

if,qA

=

The

M with 77

elliptic

the

tautological Aell:

Tij (u, A)

F-+

VM(x)

are

isomorphic

isomorphism.

quantum group

Ae":

VA(x) and

Verma modules

Ep,.y(s[2)

H

E (1

-+

&

has the

H 0 1 + 10

Tik (U)

77

A111:

coproduct H,

A)) (T k j (u, A)

2HO1

k

The

precise

meaning of

module structure If

V1, V2, V3

V,

0

(V2

0

on ^

are

V3)

are

coproduct is that it defines an ^Ep,.Y(5[2)product V, 0 V2 of Ep,,y (s [,) modules V1, V2. modules, then the modules (Vj 0 V2) 0 V3 and isomorphic [8]. the

the tensor

Ep,,y (zG)naturally

^

-

q-Hypergeometric

for

A,

any

Mand

Let

VA(X), VM(y)

generic

x, y there

modules VA(X) & V M(y), of ^Ep,,y(.s(,)viol V[01 Oviol. has the form Moreover,

Verma modules.

be evaluation

is

V

251

Uq'(j1').

loop algebra

the quantum

[10]

[12],

The

Theory

Representation

Aell which is opposite to the yll is in a sence opposite to

coproduct coproduct

take the

[10].

for

A taken

Theorem 8.1. Then

we

[8], [12],

used in

coproduct

the

that

Notice

Remark 8.2.

coproduct,

and

Functions

a

M(y)

(&

F?

isomorphism

unique

VA(X)

s.

F?

t.

(__,y)

(x,y)

(' X)VIO]

0

=

(A)

R (x,y)

VA

where

VAVM

C'

ell

map and R

permutation values

with

Corollary

VA

in End

ell

VAVM

The

functionRell

the tensor

(x, A)

Re" VA

[10]

[12], n

'I""

ell

coordinates

Let

Vje(zj),...,Ve,(z,,) weights VI, be

highest Ve, 1

....

A,, the

called

is

meromorphic

A) satisfies

the

of

function

U, A E

relation

inversion

elliptic

dynamical

the st,

complex dynamical

ell

VAVN(xl

A)

VMVN(y,

Tensor

a

(R ellVMVA(X- I ,A)) -1P VAVM

=

For any the

R

8.2

is

VM(x,

satisfy

R-matrices

VM(xly,

A)

R-matrix

for

VA (g VM.

product

ing elliptic

VA

VAV

VA VM

Theorem 8.2.

VM(u,

M(x, A)

R

P

R

VM).

(&

junctionRell

The

8.1.

(VA

A)

VM VAVM(xly,

VA 0 VM_+ VM0 VA

P is the

ell

P

=

A)

R

A)

"ll

R A'

on

VN(X

the

A,, corresponding .

.

1(&2HOI

A)

ell

RVAVM(xly,

hypergeometric

Verma modules

and evaluation

,

772HO101 A)

11

VMVN(y,

Re

elliptic

be evaluation .

17

,

A, M, N the correspondYang-Baxter equation:

numbers

points

modules.

zi,

.

.

over .

,

z,

A).

spaces

Ep,,(S(,) respectively.

with Let

Let

00

Ve & I

...

(Ve1

(g Ve

&

...

(g

Ve)t

f=0

weight decomposition

be the ...

has

0

V,,e)j

a

(0110

basis ...

of the tensor

product,

the

weight subspace (VIe

0

i with respect to . The weight subspace having weight E A,,, We denote by 1 E Zj'. 0 0-1, given by the monornials vII-I 0 0 V"'); 0 v I'd) *, I E Zin, the dual basis of (Ve 0 -

...

...

*

Vitaly

252

Taxasov

Recall

that

given

by

are

Yn and zi,

Y1....

Xm

The next

elliptic

the

i

...

(4.3).

A,,

zn,

.

=qA-Zm,

theorem

x; y;,q

We relate

An

,

.

elliptic

an

[10],

Theorem 8.3.

.

the

a), W.' (t;

;

parameters

x; y; 77 ;

a)

xl,...,Xn,

follows:

as

77-A-

Ym =

is

W. (t;

weight functions

(4.2),

formulae

ZM I

analogue

(8.1)

m

of Theorem

[7.2].

[30] T;2(tl, 77

=

[It-'

f(f-1)/2

E

X

VIO]

A)**

W., (t;

0(7) 0(,7-"\)

-0

X; Y; n;

(9

0

11

&

...

(77

-

1

V

t

a

/

t

(8.2)

=

b

)

O(ta /tb)"

-a
VIM.]

n-"\)

[01

VIM.],

MEZjn

(T _, (ti, 71"A)

0(?,1-s-i+2EA_A)

i

S=O

n

]111

0(77)

a=1

*

m=1

0(ta/Xm) 0(ta/Ym)

W.(t;x;y;,q;A)N

x

A)) (v 1010

T _, (4,

...

1
(A) (VIM-]

..

...

0

vIO)*

0(77 ta/tb) O(taftb)

'=

(8-3)

X

VIM-)

0

MC_zn

where n

Nt (A)

1111

=

M=1 8=0

At,m

and The

proof

=

to the

the vector

dual

proof of

Theorem

(C V M,

=

ED ME

...

(8.4)

[7.2].

velf

=

(

C V M,

MEZe'

=

v,,,,*

c V m*,

=

Z,"

it (V [ *, Vm) (8.3) we identify the 0 Vn)j by the rule

that

M)

spaces

v,*11

and

-

?780(,qsX_1)O(,q1-s-f ,+zAmAr, I'm

MEZn t

so

-s)

spaces

V0,11 and their

2A_

\Fj1<j<mq2Aj-2[j.

is similar

Consider

0(778+1) 0(,R

ED

c

vm'*,

mEZn

m,

(V C*, Vm')

spaces

V,',,

=

and

it

V,*,,

m

-

by formulae (8.2) weight subspace, (Vle (D

Motivated

with

the

q-Hypergeometric V

f,

that

the

OW

.F,ll

[x;,q

;

A]

[y;

and

B

A]

77 ;

[x;

,,

as

the

y;,q;

A]

VM*

[x;

B'

14

A]

y; 17;

cf.

(4.7),

(7.20),

(4. 10)

of different

we

that

obtain

the canonical

with

vector

V,*,,

of

V,,')

spaces

V,',, *,

and

(Vl'

with

associated

the

(8.4), (8.5) with (4.8), coincides In other 0 V,,e);.

formulae

and

1

(8-6)

A],

j7;

Comparing

(Vle

of

A]

A),

77;

the

types.

pairing

the

pairing

Y,',, [y;

W.' (t;

spaces

A),

0; 71;

V,11

:

depends on A. hypergeometric

maps:

411 [x;,q;

V,*,,

again distinguishing

coordinates

linear

0. (t;

VM1*

tensor

(8-5)

elliptic

the

on

following :

253

V11-1

of the second formula

coordinates

the tensor

Theory

V[1.1

0

...

V

-p) 3_- -N(()T)

hand side

right

We consider

(3

i

V1* Notice

V11-1

=

Representation

and

Functions

cf.

0...

words

(-l)'(B,,,[x;y;71;A])*S,11[x;y;77;A]B',,,[x;y;,q;A] ((Ve

id E End where

Sell [x; y;,q;.\]

Corollary

:

Fe'l [V;,q; A]

hypergeometric

of the elliptic

For any

8.2.

v

e)*)

V

(g *

[x;,q;

A])

Vne)j

and v* E

Shapovalov

is the

pairing

spaces. E

(Vj'

0

0

...

(Ve

0

...

0

Vne);

we

have

(V*

i

T12 (h

i

A)

771 -1 A)

TI 2 (ti;

...

V

[010

...

0

V[01 ) =(8.7)

(B',,,[x;y;77;,q-'A]v*)(tl,...,ti) x

((V[01

&

77

rji-l

t(t-l)/2

...

0

V

[01

-0

"(") 0(,I-,.\)

T21 (tl

)

rl,.,,

771 - A)

(B, [x;

0(771-s-f+2EA-A)

x S=O

The statement

(P),00 follows

n

11 a=1

from

(8.6)

in=1

x

y;,q;

0(ta/xm) 0(ta/Ym)

and formulae

O(n-"t _a
...

A]

O(ta ,)'

T21 (ti,

v) (t,....

/t

A) v) I

Itb) 0(ta/tb)

0 (77 ta 1
(8.2),

(8.3).

tj)

(8.8) x

Vitaly

254

hypergeometric

The

8.3

[4]

Section

In

Taxasov

[X;

I

Taking

we

introduced

A]

y; 77;

7

into

V,*,,

:

following

linear

[8.]

hypergeometric maps 1 and P, see (4.11). linear they are the following maps

[x;

V,

-+

of vector

A]

y; 77;

V1.

V, 11

:

of the parameters,

correspondence

the

account

and the identification

the

[7.],

of Sections

In notations

maps

(8-1),

(7.19),

cf.

hypergeometric

the

spaces,

the

maps induce

maps:

I[z;A] A]

[z;

:Vj'O

(Vl'

:

(9

V.')

0

...

V",

0

...

*

V,

-+

-+

(Vi

0

(&

0

...

V,

0

...

V.) *,

with Verma modules over Uq(-sf,) are weights highest below properties of the maps f and We describe respectively. of the hypergeometric P reformulating the corresponding properties maps.

Vl,...,V,,

where q

A,

A,,

...,q

,

Proposition

For generic

8.1.

f [z; A]

and A the maps

z

and

P[z; A]

non-

are

degenerate/. Introduce

Propositions [3.3], [4.2] Alml (z,,. Zn; A), M

operators

n,

[4-2].

[3.1],

and Corollaries

from

follows

The statement

acting

V,

in

0

V,,:

Al'1(zj,..-,zn;

(711+HA)Aj-H

A)

(8.9)

Zn)

Rm(zi,

j

j=1

where H = Hi

also

<,

.

.

.

,

operators

+Hn and Rm(z:L, U#n(A) acting

z,,,)

Uf-l(,X) A M

=

A,

where 1,...,n, +A,,,.

difference

[P

['(zAl'I Im

I'

-

=

+

-fE

i)

f [z; A]

and

V11.1

(D

V[[.]

At

t

<,...,+(m,

I-,

0

...

(g

0 (D

Introduce

Vn

e

VIE-]

(8-10)

V

I-,

=

P[z; A]

...

and

obey the following

Alml

systems

of

equations ZZ1

....

zi)

....

(8-11)

PzM,zM+1,---,ZZn;A1 =

P[P

77

The maps

Theorem 8.4.

I

r.

given by (7.14).

is

Vj'

in

Al'I

(zi,

-

Pzm,zm+l,---,-n;Al =

-

-

,

Zn;

A) f [zi,

-

-

-

,

Zn;

=

(Al'l(zl,.--,Zn;A))*-lP[zl,---,Zn;A](Ulml(A))*-

AI Ulm' (A) (8-12)

q-Hypergeometric

Proof. lae

follows

The statement

Functions

and

[4.2],

from Theorem

[4.3]

Corollary

255

and formu-

(7.23).

Remark 8.3.

The next

(8.11)

The system

(8.12)

the system

while

is

is

a

For any

(v*, v)

Vn'

0

...

(P [z; A]

=

qKZ equation (7.12) qKZ equation (7.3).

of the

of the dual

[4.1].

Theorem

to

VI'O

E

v

modification

modification

a

equivalent

is

statement

Theorem 8.5.

(Vl'

and v* E

v*, i [z; A]

v)

that

for

k

for

1,

Z n>

mE

0

E mj

j=I

all

n

1,

-

any A the

maps

limiVIE',

i [z;

z,,,Iz,,,+,

A ] and

0

*

we

have

and

k

e

EIi

<E Mj

j=1

j=I

j=I

(3.5).

see

Assume that

Theorem 8.6.

Vn)

0

...

n

E Ij

( < mif

write

we

0

.

n

Recall

for

Theory

Representation

I'[z;

V[En]

0

A ] have

FE'vi

=

1:

+

0

-+

for all m limits. finite

0

Fmlvl

0

Ftn

n

Moreover,

vn lim

F'"Vn

(9

Then

1.

-

it 1.

lim

+

I

E

M>1

(VIt',

limp

0

0

0")*

(FEv,

=

E (Fm,v,

+

(9

(2) F

...

Fm,

0

EI

Vn)

Vn)

*

*

lim

iml

linI

li't

+

r

M<[

where lim

im

1' EE

lim

At,m

where

M *

=

E

(pns-2A-) (p)OO (77s-I+ .-zAnA-1 I,M)O.

(pns+l)oo

00

follows

A]

from to with

[4.4],

Propositions

values

(Ve

in End

[4.5]. Vne) satisfying

0

0...

the differ-

equations ...

1,...,n,

(VII

0

is

values

=

in

8.3. Corollary i) The JunctionTf,

)zn;Al

P ZM) ZM+1....

7

called

and

an

1 [z; A]

Y [z;

A]

(Vi

0...

Let A

(z; A)

0

=

is

nq a

v

and

t-l-E

map

(VI

A_

of

0...

=

0

E

v

Y[z; A],

(z; A)

.

solution

Given

map.

,TI,'.

Vn)j

U17nI(A)Y[z,,---,Zn;A1,

=

adjusting adjusting

an

VnI);

0

...

TIv (z; A)

taking

2s-2A_

AE,. (p?7)oo (pn-8)q,.)oo

M=1 5=0

functionY[z; Y[pz1)

v

-77

1111

of A. In particular,

AJJJ<j
=

The statement A

(m-1

n =

functions

suitable

are

I

1

limfu

ence

im'

lim

c,

(Ve

define

i'[z;

Vn);,

A] (Y [z; A]) *v respectively.

Then the

Vne),

functions

qKZ equation

(7.12).

Vitaly

256

ii)

The

Tarasov

[7. 11

Corollaries

is

[8.21

and

Vle (zi)

and we

V,,(z,,).

0

[z; A]

the maps I

(Vle

E

0

and the

(zi)

modules V, v

A]

I'[z;

and

Uq'(j[, :)

algebra

let

instance,

For

i(I-i)/2-tE

q =

X

(V[ol

f

0

...

(D

Vne),

(9

...

n

X a=1

where the

integration

U,,' (gf,)-

L21,21 (u)

0

vi

...

12

(tj)

771 _tA)

T21 (tl

0

(8.13)

x

E A /\)

vi

(9 Vn

X

X) V)

X

...

(ta/tb)oo

77_lta/tb)oo

a,b=1a:$-b

t[x;

y; 71

]

by

(9 Vn

0

-

-

,

-

loop algebra

V, (zi)

the vector

=

xj,

0 vi

(D

0

...

...

_

(dtlt)'

by (3. 11)

is defined

The parameters

parameters

T21 (tt

...

i

(Py./t.). (X'lt')'

generated

module

(ti)

via the quantum

of the

values

[0)

(8. 1).

be discovered

can

generic

V

contour

(7.19),

mind relations

00

qql-s-i+2

8=0

12

3

(P)

[X;Y;771

(9

(g

ra=l

A-

(q-q-I)IfI

(27ri) x

^

[8.5].

[8-4],

to describe

have

I[z;A]v

turn,

us

(7.3).

qKZ equation

dual

the

of the quantum loop in the corresponding

quantum group Ep,.y (s [,)

Then

for

allow

of the actions

in terms

of

solution

a

from Theorems

follows

The statement

entirely elliptic V,, (z,,)

(z; A)

junctionTf,'.

and

Y1,

Xn,

action

Vn(zn)

-

as

is

0 Vn which

-

-

,

we

in

their

Namely,

well. an

keep

Yn in

irreducible

obey relations

1

n

(u)

v,

(9

v,,

=

q-

A-

11

(1

-

U/Ym)

Vi 0

Vn,

M=1 n

L+ 22 (u)

vi

0

Vn

q

=

A-

U/Xrn)

Vn,

V,

M=1 n

22 (U)

vi

0

Vn

=

A q- E

XM/u)

Vi

Vn

M=1

Moreover, ties

the

Uq'

module is

isomorphism. for generic Similarly, up to

irreducible

obey

^

Ep,,y (sf,)-

v

1010

determined

of the parameters Vle (zi) module generated by the vector values

relations

T21 (u, A)

uniquely

viol

=

0,

0 v

by these

proper-

Vn (zn) e

v

I']

is

an

which

q-Hypergeometric T11 (u, A) v101 T22 (U) A) and the

V

o

V

[010

V

=

V[01

Ep,,y (sl,)-

^

[01

Functions

1010

0

...

and

-

0(u/x,,,) O(U/Y.)

11

O(A)

M=1

uniquely

module is

Theory

257

V[01'

O(Arjx./y,.)

=

Representation

determined

V101,

V101

by these properties

up to

11 stands for fj" rn=1 1 [z; A] naturally Therefore, appears as a map from the module over the module over the quanelliptic quantum group Ep,,y (s (,) to the corresponding tum loop algebra (i Uq (z): isomorphism.

Here

I

[Z;

I

Ve (Zl)

P[z; A]

Similarly,

[Z;

claims

The modules

E.g.

for

There to

One of the

odromy This ries

a

fact

of

of its

group is

product described

establishes

simple

The

important

tensor

Lie

[35]. transition

solutions

VI! (Z'n')

[8.4]

a

for

[8-6]

for

being

-r

of the

of representations in terms of the and their

map 0

...

these the

(&

VIn (Zln)

maps,

identity

cf.

[30].

In

permutation

qKZ equation

characteristics For

a

qKZ equation.

of the

remarkable

algebras

and

Vn (Zn))

pairings. by isomorphic modules.

V" (Z")

_+

of the

solutions.

(8-14)

-

the canonical

consider

construction

solutions

most

group

this

(Zn)

0

replaced

be

can

one can

of Theorems

asymptotic

monodromy in

-r, -r'

(V 1 (Zl)

maps respect

(8.14)

V.

modules:

n

we use

Asymptotic

values

these in

0

describe

9.

right

1

section

(Zj)

V1

0 Ve (Z.))

0...

that

analogues

are

map of the

involved

I,,,,[z;A]

(g Ve (Z,,)

...

permutation

any

the next

a

(Vi' (Zi)

A

[8.5]

Theorem

is

(g

connection

a

differential

differential of

equation is the equation with

KZ

simple Lie algebra its moncorresponding quantum group. a

between

quantum groups,

representation see

[16],

[6), [18],

analogue of the monodromy group for difference equations functions between asymptotic solutions. For a difference

theo-

[27],

is the set of

equation zones in the domain of the definition of the asymptotic equation and then an asymptotic solution for every zone. Thus, for every pair of asymptotic zones one gets a transition function between the corresponding cf. [30], asymptotic solutions, [31]. In this section we describe and asymptotic solutions, zones, asymptotic their transition for the qKZ equation with values in a tensor product functions one

defines

suitable

of

Taxasov

Vitaly

258

Uq (s [,)-modules

fact

A remarkable

-

that

is

the

functions

transition

are

R-matrices elliptic acting in the tensor product of This fact establishes the corresponding a modules, cf. (9.3)-(9.5). ^Ep,.y(s[,)theories of the quantum loop algebra correspondence between representation and the elliptic quantum group [30]. be Verma modules over Uq(.sf,) with Let Vl,...,V,, weights highest the for functionT/ in Consider with values a qKZ equation qAl,...,qA-. described

V')t

(Vi

:

......

T (Zi

1, asymptotic

m=

that

IZE

tends

z

for

all

an

Say that asymptotic

a

-

Zn)

-,

(7. 11).

limit

For every

in the

in

hN (z)

0

...

A,.

aji,...,

Each

tween

any

in the

equation the

consider

we

an

m=

zone,

1,...,n

I A, if

z

1

-

}.

z,_/zm+l

0

-+

(9. 1)

form

vector

hj (z) (vj

V1,

-

o(l)

-

-

-

-

,

-,

,

functions

Zn) to

A,,...,

vj

o(l))

=

ajm

such that

hj(zl,...,

VN are constant

tends

0

An

their

is

+

as z

Zn) which form

vectors

a

basis

_3 A,.

have finite

common

limits

the

in

asymptotic eigenvalues

with

eigenvector

respectively.

ajn,

Given for

asymptotic

=

-'PZM,

The operators

Remark 9.1. zone

1,

<

meromorphic

are

numbers aj,,,, 0 Vn) i , and

suitable

(VI

S,,,

E

r

in the

hj(zl,.. for

(9-1)

Zn),

of the qKZ equation TIN of solutions if zone A, asymptotic

!P1,

solution

(z),

permutation

IZT_+l I

ZT_

Tj (z) where h,

Zn) Tf(zi,

1.

-

basis

Am(zi,

=

given by

I I

(Cxn

to

1,...,n

m=

-

in (Cxn

A,

zone

=

pZM,

(7-10),

cf.

n

,

A,

Say

of the

in terms

permutation asymptotic

asymptotic

solution asymptotic T111, TIk of the qKZ transition functions one A,, gets C,,,, be-

an

-r

.

zone

.

.

,

solution: N

7"

TIiI (Z) (C',"

Tfk7

(Z%k

(9.2)

j=1

variables

zi,

-

-

-,

i.e.

Zn,

C,,, (z) Example. and let the

Y[z; A]

monomial

=

Let be

r

an

basis

(z)

C,,,,

functions

The transition

functions of each of the ^p, -periodic and they have the properties

are

quasiconstants,

C7,17" (2) C'r'

id,

be the

identity

adjusting

f 0-10

...

17""

permutation.

map which is I E 0 V IW

I

(Z)

=

Let

C""' A

=

nondegenerate/ _7j' 1 of (Ve 0...

(Z) n

-

q

and 0

U-2-2

E A-

diagonal

Vne) j

.

Set

in

q-Hypergeometric

Tff(Z) I Tir I I asymptotic

Then the

Z11 1,

E

is

of the qKZ equation (9.1) asymptotic solution Corollary [8.3], Theorem [8.61 and Proposition

an

A id,

zone

259

VIE-]

[z; A] Y[z; A] V[1-1

I

=

Theory

Representation

and

Functions

in

see

[8-1]. asymptotic permutation

[z; A]

1r

analogous right

i [z; A],

to

any

V,,e) by

(Vle

End

for

(Ve,.

E

0

Define

.

A, for Ve (Z.) "

.

(of,)

map

a

(g V

...

0 Ve

...

(8-13).

V &

_+

zone

V11 (zi),

quantum group Ep,,,

Consider

Ve

v

asymptotic

way. Let

elliptic

the

over

Ve

:

hand side of formula

to the

similar

a

A, respectively.

A,,...,

weights

highest

in

Verma modules

be the evaluation with

be described

T can

in the

qKZ equation

of the

solution

An any

set

i,[z;

A] (A), Ul'

operators

v

be

to

equal

U'T' (A)

E

the rule

U'T"(A) V11-1 + zA

2A k

A(,,,

& V11-1

V[17*nl

V[frl]

It-zltf

< O'l, < or_

where

t

11 <,

and

+

a

r-'.

=

Yr [z; A] of the difference

A solution

equations

Y,[zi,.-.,Pzm,...,Zn;,Xl corresponding

in End

values

with

n,

M=

permutation

to the

Um(/\)Y-r[zl,---,Zn;Al,

=

(Vle

Vne)

0

is called

adjusting

an

map

-r.

U-2-2 E A_ and let Y, [z; A] be an adjusting [30] Let A n q the monomial basis I VIE-] 0 in and which is diagonal nondegenerate/ map Set Ve I 0 E 0 VI"I Of } I n)j Zin (Ve,

Theorem 9.1.

=

.

(Z)

1

Then the

I TVE' I

=

I E Zn

asymptotic

1,

the

P,,,,Re,,,(z;

Ve

of

'1"r

(Z 71 )0

P'r"'rW11,r,,,r(z; Here

P,,,,

is the

...

(D

solution

asymptotic

an

V[ Tnl, of

I E

the

1

i, [z; A] i,, [z; A] equals modules: the 'Ep,.y(s[,)-

ratio

A)

p',"Reii -4

is

(g

Zin

qKZ equation

(9. 1)

in

A,.

zone

By construction, twiner

[Z; -X] Y' [Z; Ä] VIEr.)

I,

(Z; A) Ve

rn

:

(Zr-)

A) vlol

permutation

Ve

(Z', )

(g

...

the

(& Ve Irl

n

')

normalized

_+

I

(&...

map

(9

vIO1

=

vlolg...'O

V[01.

inter-

(9-3)

260

Vitaly

Tarasov

p,,,,

Ve

PT',T and

Re7111-r (z;

For

example,

A)

V

is

I[,, 10...

0

'.

a

if -r'

suitable 7-

-

Ve & 71

0 Ve

ri

1

V

product

(m,

(D Ve

...

m+

1)

of the

dynamical

(m,

where

m+

1)

is

a

R-matrices. elliptic then transposition,

ell

,.(m,m+,),,(z;,\)

R ell

R

V

T_

"\n2H&

(&

...

n2H

=

(9.4)

id(D (n-m+l)

M-1

By Theorem [9.1]

for

permutation

have an asymptotic soluzone AT. The asymptotic transition functions corresponding C,,,, between the asymptotic solutions, cf. (9.2), of the respective tensor products equal the intertwiners of evaluation Verma modules over the elliptic to the weight quantum group restricted subspace and twisted by the adjusting maps: tion

I Tfrr I

I E

C"', (Z)

Zj' }

:

any

of the

(Ve

&

C,,,, where A

=

n

q

U-2-2

E A-.

qKZ equation

...

(z)

&

=

V" ), Y, [z;,\]

_+

T

in

we

the

(Ve -'R'

Ve

'n

!,l (z; IT

A) Y,, [z; A]

(9.5)

q-Hypergeometric

Six

A.

Let A

aqI-ijj,'

=

by

say

Let

(n

=

Laurent w

=

formulae

Representation

determinants

pairing

S[A] be the A(-u) -nf (U). It is Let

x,,,. =

Theory

261

Shapovalov

of the

pair-

1.

holomorphic

space of

easy to

that

see

functions

dim

E[A]

=

n,

series.

exp(27ri/n).

-A-1.

for

hypergeometric

,=,

f (p u)

C' such that

and

formulae

determinant

S Wepresent here explicit ings S and S,11, and the on

Dinctions

complex

Fix

and

numbers

such that

n

=

p and

Set n

191 (u)

Ul

=

111 O(_( I-IW'u)'

-

.'n.

M=1

Notice, that 6 and C.

the functions

Clearly,

Proof.

V1

V,, do

1,

functions

The

LemmaA.I.

id

E S

[A]

i9j,

for

....

any I

=

191(wu) that

is the functions

with

distinct

For any I E

'01, eigenvalues.

Zj'

-

gf(t;

let

-

=

a

1,

Moreover,

.

n.

,

and

G[(t;

XM

choice

a)

x;,q;

(t; ta

11

77 ta

1
X;

be the

following

-

functions:

7))

X

n

tb

-

operator

independent.

11 11

tb

ESi

m=1

tM "

n

1111

0(ta/xm)

m=1 a=1

I
rn-I

Here

F,,

I +

E

The

hypergeometric

metric

form

a

basis

in

the

X

n

0(ta/tb) 0(71 taltb)

T_ ]I aESt

]I

m=1

'dM(tO'a)

aEI'-

M

E 0 1,

I

k=1

LemmaA.2.

'

aE.P_

GI(t;x;,q;a)

X

of

S[A].

the space

in

of the translation

,

x;,q)

ta

m=1 a=1

.

basis

particular

WI-1191(u),

n

11 ri

.

?9n are eigenfunctions Hence, they are linearly -

on a

On form

9f

X

depend

not

M=

1'...'n.

f=1

functions space

elliptic

gr(t; x;,q), F[x; 27 ; i ].

hypergeometric

I E

Zin, form

The

functions

space

F,11 [x;

a

basis

Gf (t;

in x; 77 ;

77 ; a; t

].

the trigonoa), f E Zn,

Vitaly

262

Proof.

We prove

elliptic

The

the

to

in

9 [A]

for

the second claim.

proof of the first one is similar. Y,11 [x;,q ;a; f ] is naturally space isomorphic of the space E[A] the space of symmetric

hypergeometric symmetric power

'f-th

functions to

Taxasov

tj,

.

any

,

.

.

a

The

-

tj which considered

1,

=

.

.

t.

,

.

as

isomorphism

The

f (ti,

-

-

4)

,

f (ti,

-+

-

-

-

m=1 a=1

f

E

Let

Xii [x;

Wt,

wr,

a]

77 ;

Q(x; y;,q)

I

.

Now the statement

Zj',

E

y;,q

;

a) by

W((t;x;Y;,q)

variable

0 (77 ta

(4.1),

belong

Itb)

0 (taltb)

from Lemma[A.

functions

t,,

follows:

1
follows

weight

be the

Q111 (x;

and

one

as

]1 H O(ta /X",)H

4)

,

of

reads

1

n

-

functions

(4.2).

'

1]. Define

matrices

the rule:

E Qf.(X;Y;77)g.(t;x;,q),

=

MEZI,

Wr (t;

x; y; n;

a)

E QeI1 IM (x;

=

ME

a)

y;,q;

G..

(t;

x;,q;

a).

Zj"

Set

d(n,

f, s)

m,

(m)-l+im-1 ij !O

(n)-m-l+jn-m-l.

i+j
(A.1) Proposition

[29],

A. 1.

[30,

7]

Theorem A.

i-I

det

Q(x;

Proposition

77)

y;

=

[30,

A.2.

11

11

s=O

1<1<m
=

n-I

II

F

n

11

/Xj)d(n,m4,s)

II

S=1-t

x

x,,,)(n)+i-s-2n-1.

-

B.8]

Theorem f-1

detQell(x;y;,O;a)

(77,y,

x

M=1

i-I

(m-n)(n)+t-1n

0(?jsy1/xm)(n)+f-s-2n-I

YM

M=1

s=O 1<1<m
where n-1 2

(p).I-n

(

H M=1

Consider ric

I E

a

basis

gf(t)

hypergeometric Zj", of the elliptic

=

space

tj-1

...

ti

F'[y;?7;f]

hypergeometric

1

Opm) W-

gr(t; and

-

y; a

space

I

)

n-m

77-1),

]

1 E

(n)+i-in

Zj',

G,(t) Ye'll [y;,q ; a;

basis

of the =

f

].

(A.2) trigonomet-

Gr(t;y;?7-';a-1),

q-Hypergeometric

Proposition

(_l)n(n-l)/2.(n)+t-lnnn(n+l)/2-(n)+f-ln+i

=

IMEZZ'

1-1

(1

H

x

n

)n

-

II

-,qs+lTn

8=0

I'M=1

(n)+f-s-2n-1

1 xi

-

X

77SYM

[S.

det

=-2

263

Theory

Representation

A.3.

[S(g(, gm)]

det

and

Functions

n

(-l)n(n-l)/2.(n)+t-ln,,n(3-n)/2.(n)+I-ln+1 t-1

0(,,)no (nsa-1) 01(1)nO(,qs+l)n

S=:11OI

x

Here

is

0

x

M

(n)+i-s-2n-l

(,qs+2-21 aflxm/ym)

rj

X(n-1)(n)+t-1n

M=1

0(n-SXI/YM)

rIrI

by (A.2),

given

11

for rIn

stands

=1 M

and

rlrj

stands

for

11n1= 1 11nM=1* Weprove the second formula.

Proof.

Lemma [4.1]

det

we

[Sell (GE, G,.,,)]

HH H answer

we use

order

of the

the

, ') 0(,q'-s-'-

is similar.

By

a[,.

x./ym)

0(,q-sxmlym)

01( 1) 0(,qs+,)

m=1 s=O

IEZ,"

changing

one

detQ(y;x;77-1;a-1))-1x

77s 0(,q) 0(,qsaj-

n

uct

proof of the first

(detQ(x;y;77;a)

1,-E Z"e

x

To get the final

The

have that

Proposition [A.2] and simplify the triple prodand applying Lemma [A.3] several products

times.

The

LemmaA.3.

following

holds:

identity

1

E (j)+aj

(1)+m-am

(j)+k+ak

=

(j)+kk

(j)+k+l+m+lj+k+m+l.

a=1

The statement Let

I

=

Proposition

I

can

[x;

be

y; 77;

A.4.

a]

with respect proved by induction be the hypergeometric pairing.

[30,

Theorem det i-I

,-n(n)+i-In+1

[I(wr,

W.)]

f,mEZ,'

H 11 0(,q,+j-jCe-yjj.,: l 8=1-t

m.

5.9]

n-1

M=1

to 1 and

=

<, y,/Xl)d(n,m4,s)

x

Vitaly

264

Taxasov

(n-1)n (?78a-1)c)O (p ?7s+2-2'a rl xm/ym)()o (77-8-I)n (P )2n-I rj(n-,xm/ym).

X 8=0

X

00

00

(n)+i-s-2n-l.

(77'yI/xM)OO (77 -9X'/Ym)'

X

'

1<1<m
11

Here

Corollary

[I(gc,

i-1

Here

35-

f, s)

by (A. 1).

given

are

The statement

X

(n)+i-s-2n-I

IIrI(?7-Sx1/y-).O

00

rl

follows

The Jackson

B.

m,

-1)00 (P 77 s+2-2ta 11 x1n/ym)OO

by (A.2),

rIn1= 1 rInM=1* Proof.

a

(P)2n-l+n(n-l)/2

00

given

is

8

17

cx)

(,q-s-l)n

S=O

d(n,

exponents

=-1,0-n(n+1)/2-(n)+t-1n+1

G

(,,-I)n

I,

and the

=, M

[30]

A.1.

det

X

for rIn

stands

rIn

for

stands

[A.1],

from Propositions

integrals

the

via

rIn

and

=, M

[A.2]

for

stands

and

[A.4].

hypergeometric

integrals Consider

the

functionf

(ti,

hypergeometric

tj)

Int[x;y;?7](f! ),

integral

(3.11),

cf.

for

a

of the form

P(tI'...'t,)e(t1)

11

)tt)

...

(ta/tb)oo

(B.1)

1
where P (tj, the variables on

.

.

tj)

,

.

ti,

.

is .

,

.

(ti,

and (9

tj

C't such that

19(h) for

some

inition and

...

1

Pta)

A. The

constant

E (C

x

n

f E

.

tj)

,

.

tj)

=

is

A

a

pairings, x,

Zn,

,

s

...

E

)

degree

of

-n

(9(tl,

integrals

(3-18),

see

Xn, YI,

V let

-

-

x >

,

-,

at most Min each of

holomorphic

symmetric

(-ta)

hypergeometric

hypergeometric A determined by a,,q, any

.

...

of the

For

polynomial

symmetric

a

t,)

...

which fit

this

function

(B.2) in the

appear case

for

M=

defn

-

Yn-

(1, s) [77 ]

t be the E CI

following

point: x >

Sf,.

P

For

(1, S) [771

+"+'..+sE"

instance,

+E'77 if

+...+St.

(P"'

s

-EzX2,. =

(0,

.

71

I-(.,.X,,

PsE'+I'X2, .

.

,

0),

then

x >

Ps2+...+s(.,972-[.,.X,'...' -

-

-

P'

qt -r"

(1, s) [,q ]

=

+'+s'e?7 x >

[

PS(-'XI 1-(

[,q ],

Xn,

cf.

(3.6).

P9" Xn)

I

q-Hypergeometric

Proposition f (ti, tt)

B.1.

Let

parameters

form (B. 1),

have the

.

Flinctions

be

x, y, 71

(B. 2).

Theory

Representation

and

Let

generic.

a

265

function

A ssume that

n

II xm'

pn A

I

(1, Iq 1

min

<

1

M=1

Men the

[x;

Int

sum

below is

] (f 4;)

y;,q

and

convergent

Y

=

Res

(f (t) 4 (t;

x;

(dtlt)tI

y;,q)

77

mEZI ,e sEZ'>>0

if I p' A rIn

Similarly, vergent

=,Y_1 I>

M

max

M

(1, 177 1'-1),

then

[X;

IrIt

E

E

Res

Y; 77

(f (t) !i(t;

1 (f 4 )

sition

[2].

sum

below is

con-

x; y;

=

77) (dtlt)i

) jt=y,(m,s)

MEZIn SEZ'
the

and

proof

[B.1]

is similar

coincide

to the

proof of

the symmetric

Theorem FA in

A-type

Jackson

[30].

The

integrals,

[n- i]

sums see

for

in

Propoexample

Vitaly

266

Tarasov

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Series

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to

the

171

,

Knizhnik-Zamolodchikov 1995

,

99-137

equation

Constructing Gang

symplectic

invariants

Tian

Department

of

Mathematics,

Massachusetts

Technology

of

Institute

MA02139

Cambridge,

Introduction These lectures Their

have been written

to

first

review

problems

in

symplectic

aim is

moduli

Then

notes

"Quantum Cohomology",

session

will

we

symplectic

review

manifolds

and

maps

where A E H2 (X,

Z)

give

:

a

30-July

given at the C.I.M.E. 8, 1997, at Cetraro, Italy.

course

of virtual

fundamental

The construction

geoemtry.

of Gromov-Witten

some

H*(X; (Q)xk

applications. a symplectic x

*

H

ffig,

k;

Q)

classes

here follows for

invariants

for

invariants

*XA,g,k

for

construction

construction

The Gromov-Witten linear

June

manifold

X

[LT3]. general

are

multi-

(0-1)

Qi

--+

for

intek, g are two non-negative any homology class, of 9A.,e, the space Deligne-Mumford compactification of smooth k-pointed invariants and deThey are symplectic genus g curves. fined by enumerating into X. In holomorphic maps from Riemann surfaces [Gr], Gromov first studied symplectic manifolds by exploring invariant propof the moduli space of pseudo-holomorphic erties maps from the Riemann sphere. Subsequent progress was made by Ruan [Ru]. He introduced certain invariants of genus zero and used them to distinguish symplectic symplectic gers,

ffg,k

and

manifolds.

ants,

It is

refered

The first

to

is

is the

now as

known that

Ruan's invariants

Gromov-Witten

theory

mathematical

are

part

(abbriviated

invariants

of GW-invariants

and its

of extensive as

invari-

GW-invariants).

applications

to quan-

in late 1993 (cf. cohomology by [RT1], not constructed the GW-invariants of any genus for semionly [RT2]). They manifolds positive but by using inhomogeneous Cauchy-Riemann equations, tum

was

established

also proved

fundamental

Calabi-Yau

manifolds

axioms for

Ruan and Tian

these

invariants.

All

Fano-manifolds

and

examples mansymplectic manifold of complex dimension less than 4 is symplectic In 1995, the semi-positivity was removed by Li and Tian for semi-positive. virtual moduli cycles. An alternative algebraic manifolds by constructing conof virtual struction moduli cycles was given later Behrend and Fentachi. by More recently, the construction of virtual moduli cycles was extended to general symplectic manifolds first by Fukaya and Ono, Li and Tian, followed by B. Siebert (cf. [FO], [LT2], [Si]). Another approach was proposed by Ruan [Ruj, making use of a result in [Si]. ifolds.

Also

are

special

of semi-positive

any

K. Behrend, C. G´omez, V. Tarasov, G Tian: LNM 1776, P. de Bartolomeis et al. (Eds.), pp. 269–311, 2002. c Springer-Verlag Berlin Heidelberg 2002


Gang

270

Tian

I conclude

this

giving

the

for

me

work

This

Euler

1.

introduction

short

my thanks

with

of

oppotunity presenting supported by partially

was

weakly

of

class

de Bartolomeis

to Paolo

my lectures NSF grants.

Cetraro.

at

V-bundles

Fredholm

of smoothly stratified the notions introduce orfor Fredholm or V-bundles, simply weakly short, bispaces, pseudocycles Then we will construct orbispaces. WFV-bundles, over any smooth stratified This construction is due to the Euler class for weakly Fredholm V-bundles. the one in [LT2]. J. Li and myself [LT3] and refines which of orbifolds, A smoothly stratified orbispace X is the generalization with finite are those only by quotient singularities topological groups. spaces For of X is where X can not be made smooth locally. As usual, a singularity the case to consider the GW-invariants, the purpose of defining we suffice which are essentially that X has only singularities by finite groups. quotients to the case of more complicated It is possible to extend this theory singularsuch as quotient by compact Lie groups. For example, in singularities ities, one the case of the Donaldson type invariants connections, using Yang-Mills the Euler class of more general bundles involving needs to construct quotient by compact Lie groups. singularities will be 'assumed to be Hausdorff. All topological spaces in this section In this

section,

we

will

first

and

In

orbispaces

Smooth stratified

1.1

this

subsection,

we

stratified

certain

Definition

X which subsets

1.1.

admits

Note that structure

a

=

definition

our

topological

of

f f

a

then

X X

stratified stratification

1

spaces

which

admit

any

=

space

closed

U,,cj

smooth Banach mani fo ld.

a

knowledge

of the normal

cone

X,. Y between

--+

-+

X is

topological

Xc, by locally

we mean a

space,

each X,

doe not require

each stratum

along

is

class

a

such that

strata,

A morphism continuous map

f I x,,,, f, manifold,

By a smoothly locally finite

called

Xj,

introduce

structures.

two

Y such that

for

smoothly

stratified

each stratum

X, the

spaces

is

stratum smooth map into a certain Y,3 of Y. If Y is a smooth a morphism. f,, is f above simply means that each restriction

smooth. In the

following,

we

will

call

smoothly

will occur. space if no confusion the notion Next let us introduce

local

charts.

uniformization

stratification

By this,

a

restriction

X

=

of

Let X be

U,,,c =j X, by locally

we mean

for

each

x

E X the

stratified

space

simply

by stratified

orbispaces and their smoothly stratified finite topological space with a locally closed subsets X, called strata.

a

set

ja

E I

I

a

E

X,,}

is finite.

Constructing Definition

of

sists

stratified

a

U

7ru(t)

=

is

(3)

induces

iru

If there

a

is

no

confusion,

it

denote

is

a

Gfj

--+

n

in

and in

and

Definition

J(Ui, Gfj,)}I
n

equivariant

V)

---

7r

of

spaces

the

x

can

1. 2 E

quotient

and

(V, G- ,)

if there

covering

is

map

the

and commutes with n

stratified

V)

-4

U n V.

orbispaces.

if it is orbispace smoothly stratified X stratification U,,,,, X,' by lobe covered by quotients of local uniformizasuch that for any finite of charts -collection

niLIU,, V

local

For sim-

V)

n

7rul (U

smoothly

a new

than the former

U1 (U

fj"

quotient

V.

to

(U, Gfj)

continuous

stratified

U n V C X and

the notion

X,

subsets as

an

between

-4

space

a

charts

V)

and

defines

(t, Gfj)

Now let

is finer

by

X such that

and U the

GO) of

We say that X is a with a locally finite

1.3.

topological cally closed

the later

con-

fJ

on

and each stratum

chart

a

(7rU-'(V), -

-+

X

U.

onto

restriction

by (U, Gfj) I v

7rvl (U

:

morphism

Now we introduce

Definition

it

of X. We say that

7rV1 (U

projections

the

effectively

fJ

:

X,)

n

fJlGfj

then

subset,

of X, called

WfjV

is

map,7ru

of

chart

acting

(t, Gfj)

call

will

G,[j

271

invariants

uniformization

ru-1 (U

=

we

open

an

homomorphism G, ,

tion

of X; by fJc,

Gf.;

often

will

we

such that

continuous

homeomorphism from

chart

be two charts a

group

given

are

V C U be

uniformization

plicity,

finite

a

(t, Gfj).

of the chart Let

A local

open subset

an

of fJ

under

is invariant

above.

G-Cj-invariant

a

The strata

as

fJ,

space

and

morphisms

(1) (2)

Let X be

1.2.

symplectic

of

there

=

is

(V, GV)

chart

a

and

(V, GV)

(V, GV)

is

such

finer

that

than

x

all

(Ui, Gi-j,). orbispace. orbispace is smooth if all its charts morphisms desmooth dimensional finite In smooth. a smoothly fined in (1.1) are particular, Also if X is a smoothly stratified stratified orbispace, orbispace is an orbifold. of open that is, they are the quotients its strata X,, are Banach orbifolds, finite Banach manifolds by groups. If X and Y are two stratified by a map from X into Y, we orbispaces, for Y X such that -+ : a continuous any x E X, there is a chart map f mean Y of and chart of a a homomorphism Gij- F-+ GV and X, G. ,) (U, Gfj) and f descends V such that U V E x E U, f (x) F-+ map f : Gij-equivariant call

We will

each stratified

Clearly,

f IU f

to

that

f

:

X

smoothly

:

is

-+

stratified

orbispace

simply

stratified

A stratified orbispace. space is a stratified 1.3 and corresponding in the Definition

V c Y. Wewill call such I a local representative are morphisms. morphism if all its representatives if it has an inverse morphism f Y is an isomorphism

U

of

-+

A

a

-1

:

f We say morphism .

Y

F4

X.

Gang

272

Tian

Weakly pseudocycles

1.2

subsection,

In this

we

topological

two

work with

will

spaces

continuous

a

All

X and Y.

f

map

:

homology theories

X

--+

Y between

rational

with

are

coefficients.2 First

recall

we

dimension

open subsets

from k

a

such that

an

open subset

d and

XI

*

*

;

i

the transition

boundary

of such

a

with

(1)

for

in

L,,

space which each a, there is

Ix,

=

coordinates

0,3

function manifold

0),

! 1

is defined

*

admits

Mof

covering by homeomorphism 0,

a

>

*, xk

Rd; (2)

of

Oa lo,,

0

A manifold

corner.

topological

are

Xd

*

U,3 54 0,

n

The

<

is

corner

JU,,J

onto

k (a)

=

U,

U,,

of manifolds

definition

the

d with

a

0}

Rd,

C

where

For any a and 0 with nuq) is a diffeomorphism.

(u

to be

OM=U U Oa1(L,nfxi=o})_ Clearly,

is W

a

manifold

Now we define d in

sion

X is

d with

dimension

S

manifold

(p, M), two

define

(p, M),

(p, M), set3.

that

plu,

=_

there p

o.

o

M is

the

reversed

and

(p', N)

ni

E

say that it is 0 if there the complement of their

such that

-

we

pseudomanifolds (p, M) the sum E niSi, where we

d

is

an

1 with

S of dimenpseudomanifold smooth manifold of oriented, continuous For a pseudomap.

an a

(plam, aM)

to

(p

be

.

We define

We define

orientation. U p',

-S to be the

MU N)..We

of

sum

then

can

Z, of pseudomanifolds are

two

union

For any inductively. subsets and U, U2 of disjoint open in Mis at most a d I dimensional -

reversing pseudomanifold

orientation

A rational

corner.

An oriented

and p: M-+ X is define OS to be

corner

where M is M with

Mso that

so

==

of dimension

pseudomanifolds. pair (p, M), where

a

(1.2)

1
a

isomorphism is

a

formal

W: sum

U,

-+

U2

of oriented

with rational A rational coefficients. pseudomanifolds pseudomanifold of it is integral if an integer and is zero. multiple Let f : X -+ Y be a continuous map between topological spaces. define weakly f-pseudocycles.

is

zero

We now

A rational

is given by weakly d-dimensional f -pseudocycle is a rational Id in X, K is a pseudomani 2' such that closed subset in Y of homological dimension d no more than the closure of M p-1 f -1 (K) is compact in M and the rational (d 1) dimensional pseudomanifold (plam, o9M p-'f -1 (K)) is 0. Two such cycles (pl, M1, K1) and (P2, M2, K2) are quasi-cobordant if there is a (d + 1) -dimensional pseudomanifold (p, M) in X, a closed subset K C 1 Yt'P of homological dimension 1 such that M p- 1 f no more than d (K) is compact in M, K, and K2 are contained in K and (pi, Mi) (P2) M2)

Definition

triple

a

1.4.

(p, M, K),

where

(p, M)

-

-

-

-

-

-

-

-

'

In

3

By By

4

fact, this this

hold for integers. following image of finitely many d- I dimensional maxJk; Hk (K, Q) :A 01 < d 2.

many discussions we mean

that

we mean

in the

it is the

-

manifolds.

-

Constructing

(p, (9M).

symplectic

invariants

if there is a cycles S and S' are equivalent S that is S1 so quasi-cobordant So, Si So,

Two such

f -pseudocycles St. St

273

of weakly

chain

Si+J

to

=

and

=

of this

The usefulness

by the following

is illustrated

definition

Proposition.

Let (p, M, K) be a weakly d-dimensional 1.1. f -pseudocycle. Proposition element in Hd (Yt'P, Then f o p: M -+ Y defines a unique Hd (YtOP). K) element in Hd(Y). such cycles define two equivalent an identical Further, =

the

First,

Proof.

homology

two

homological

has

dimension

d

<

-

groups are canonically 2. Now we show that

isomorphic f o p defines

K

since a

cycle

is Hd (YtOP, K). Since M is smooth and the closure of M p-'f -'(K) of the M that in a can so we f op(M) boundary triangulation M, give compact is contained in K, thus it defines an element in Hd (Y, K). For the same reason, then they give if two weakly f -pseudocycles S, and S2 are quasi-cobordant,

in

-

rise

the

to

homological Hd (VOP). elements

over

a

this

On the

other

natural

the

-

Hd(Y). V-bundles

Fredholm

the we introduce subsection, orbispace smoothly stratified

this

In

It in

Weakly

1.3

in

< d

dimension

Hd(Yt'P,

into

Hd(YtOP, K).

hand, since K is of homomorphism from Hd(Y"P) 1, define the same homology class in and is so S, S2 injective, K) define identical that equivalent follows weakly f-pseudocycles class

same

and the next

U,,,Cj

X,.

Fredholm

X0

is open and dense in

UaEIta

Also, if tJ is a chart of X, then t has a stratification U n X,. that U, corresponds to the stratum X.

Definition is a

Let X be

1.5.

a

pair (E, P), (of stratified projection

X is

where E is

a

a

so

indexed

A V-bundle smoothly stratified orbispace. a smoothly orbispace and P: E stratified

orbispaces) (resp.

(ti,

of charts

collection

V-bundles

throughout

We assume

0 E I and the stratum

that

section

=

weakly

of

notion X

X

such that

(Et,,

GCj,)

Gfj,))

(resp. for

E)

which

over

X

-4

by

covered

is

following

the

holds:

(1) if Ui there

(2) is

is

of the

(3)

of ti

of

and

finer

both

-equivariant

x

is

of Ui,

restricting

t, with given by E, ,, 1,Cj.,

bundle are

then over

and P-'(Ui) P from P-'(Ui) onto Ui;

chart

a

to a

over

U,

Gl[j,

which

the

bundle

-linear are

EfJ, lfj , Fur-

action.

simply

pull-backs

by Pfj,.

For any two charts

than

Efj,

Etj,

fJi representing

P1f,,:ECj,

the strata

strata

C X then

Ui

over

Banach vector

spectively)

Gfj,

chart

be any stratum

smooth

thermore,

a

projection

U,

Let a

a

is

E

(ti,

Ui

(ti, n

Gfj,)

Gfj,)

Uj,

there

and

isomorphisms

(tj,

and

(fJj, Gfj,)

is

local

a

G.Cj

of

X

(over Ui

uniformization such

that

X

chart E

and

Uj

re-

(Uk Gfjk) i

Uk and there

are

Tian

Gang

274

Eij

I U,,,

,

the map given in (1. be the (4) let

OEcj,

E be

Now let E

-->

We say that will

We fix local

its

and

subbundle;

(3)

U

finite is

U, is

a

(5)

:=

is

a

Flu

4i: X

X is

-->

is

an

compact.

strati-

the

morphism isomorphism. For simplicity, a

P with

section

bundle

compact with

We

support. in terms

section

t,

fJ

(with

is

t

C

C

is

P-'

E

U=

F

subspace of E&

Flfj,:,

Efjlfj

C

U,,,EIU,).

is

smoothly

Further

we

(0)

n

of

the :

of

Efj

d

such that:

restricting

stratified UO

,

chart

a

smooth

require

complement has codimension bundle, and its restriction where OF and the map OF I U,,

fio,

and

a

vector

space =

at least

of

U n UO 2 in

U;

to each stratum

vector

bundle

of -*

t

over

U and its

smooth vector

w

stratified

equi-dimens%onal

an

strata

bundle

vector

inclusion

the

continuous

a

of (E, -P) consists approximation a E, representative of 4it : U GCj)

dOF(W) is

i

X under

to

of E is

A section o

of this

Gt-linearized

smooth section;

At each

P

smoothly

a

is

4 _1(OE)

a

smoothly

Gt-equivariant

open and dense in

F

(Efj,

equi-rank

rank

dimension

(4)

is

Ofj kij

-

structure

finite

chart

P-'(F)

:=:

they define isomorphic

it

X.

that if

X with

over

A local a

each stratum

to

4 _I

for

Fredholm

1.6.

a

where

approximations.

of X,

F is

orbispaces

that

over so

(OE)

the

finite

a

(1)

bundle

4 -'(O)

finite

(U, Gfj)

V-fiber

V-bundle

a

Definition

a

support

describe

now

(Efjj)

Afjkfij

each stratum,

then

of Et,,

We require

-P has compact

write

along

structures

sections

E.

stratified

as

Eij JujnUk

and

1). zero

OE of

fied suborbispace P. projection

we

(Eij,)

fJi

the Banach bundle

preserving

(1>: X

AfJk

-_

nuk

Pt I u:

=

U

--+

F,

differential

T,,to

--+

Et,,lw/Flw

surJective. Such

a

local

finite

An orientation line

bundle

given by UO

a

of

will

approximation

(t, Gfj, Efj, F)

A"P(TU) Gt-invariant

0

A"P(F)-'

is

over

nonvanishing

be denoted

a

by

Gfj-invariant

UO. Note that

section

(0, Gfj, Et, F). of the real

orientation

of AIIP

such

(TU)

an

0

orientation

AtOP (V)

-

1

is over

-

of

The index

(t, Gfj, Efj, F) is defined (t', Gfj,, Efj,, F') is

Now assume that

of the

same

Definition

if

we

index 1.7.

have the

(1) (U', Gfj,)

over

U'

to

be rank F

another

-

dim U.

local

finite

finer

than

approximation

C X.

Wesay that

(fY, G&, E&, F')

following: than (U, Gfj); is finer

is

(fJ, Gt, Efj, F)

Constructing

(2)

Let

7ru-,' (u,

W:

n

u-1 (u

u)

v)

n

symplectic

(which

is

invariants

(1. 1))

in

wuu,

275

be the

is a continuous subbundle W*F C V*ECj _= ECj,lu,nu of V and also a smoothly stratified subspace; 1. 6 for fj and (3) Let (U, F) and (U', F') be those defined in the Definition U' respectively, then for any w in U01 n W-1 (UO), the natural homomorphism

covering

then

map,

T,,,,U'IT,(,,)U is

isomorphism;

an

(t, Gfj, Efj, F)

(fY, Gfj,, Efj,, F)

(4)

In

also

require that the orientations of (t, isomorphic through the isomorphism

are

(F'IW*F)I,,

--+

both

case

AtOP induced

(TwU')

by

At"P(F'I,,)-l

0

Note that

(5)

from

smooth submanifold

phism (3)

%

subsets

9A

w

!P-'

is

point

then

U01

is

(1.3)

W-1 (UO).

n

locally

a

rank F. Hence the

-

we

(fY, Gfj,, Efj,, F')

in

o-' (UO)

n

oriented,

are

and

AtOP(Fjw(w))-1,

0

any

U01

1.6, rank F

closed

homomor-

1.7 is well-defined. 7

of

covers

Definition

of the Definition

of codimension

of X such that

say that

where

I (Uj, OU, EU,, Fi)}ic-r, In the following, (X, E,

=

approximations

Gfj, Efj, F)

AtOP(T W(w)U)

(3),

in

Definition

in the

Now let

-=

isomorphism

the

and

Ai

Gjj,)

(Uj,

(0)

if !V

An index

1

(0)

is

a

be

collection

a we

of oriented

denote

will

uniformization

is contained

by

chart

in the

union

finite

Uj the open We over Uj.

of Uj in X.

weakly smooth structure of (X, E, fl is I (Uj, Ou,, Eu,, Fj)} jEK of index r oriented smooth approximations such that 9A covers 0-1 (0) and that for any (0j, Gfj,, Efj,, Fi) and (Uj, Gfjj, Efjj Fj) in Qt with x E Uj n Uj, there is a local finite approximation (U, Gt, Et, F) E 2( such that x E U and (U, Gfj, Efj, F) is finer than both (Uj, Gfj,, Efj,, Fj) and (Uj, Gfjj, Efjj, Fj) a

1.8.

collection

9A

oriented

r

=

7

It

follows

finite

from

easily

f (ti,

approximations

definition

this

Gi Ej, Fi)}

that

I
7

(t, Gfj Efj, F)

approximation

which

7

for

in 9A and

any number x

E

n =j

contains

x

U,,

of local

there

finite

is

and is finer

a

local

than

all

(Uj, Gi, Ej, Fj). Let 21' be another Wesay that p E U n (V

and

1

(U, Gfj,,

index

2V is finer

than

(0), there Efj,, F)

is

smooth structures

a

oriented

r

%if for

(t',

any

Efj,, F) E 9A' over U' such (U, Gfj, Efj, F). Wesay that

Gfj,,

is finer

than

9J, and %,

are

weakly smooth structure (U, Gjj, Ej-j, F) E 9A over

equivalent

if

21,. is finer

than

of

(X, E, 0).

U C X and p C: U,

that

weakly

two

9A, and vice

verse.

Definition

holm is

A V-vector

1.9.

V-bundle

compact and

bundle

if

it

admits

an

is

contained

in

with

oriented

finitely

(X, E, 0)

section

weakly

smooth

many strata

of

is

structure

X.

a

weakly and

Fred-

0-1(0)

Gang

276

Tian

Definition

weakly Fredholm V-bundles (X, E, 4i) and (X, E', V) Fredholm V-bundle a weakly (X x [0, 1], R, TI) such (X, E, (P) over X x f 01 and to (X', E', V) over X x I I}. Two

1.10.

if there

homotopic

are

is

that

it

1.4

Construction

Let

X be any stratified dimensional a finite

restricts

to

of the

Euler

class

orbispace

with

the

smooth orbifold

Y be

stratification

X

f : Xt'P

and

let

U,C,,X,,

=

Yt'P

-4

be

a

smooth

map.

following

The

is the

Theorem 1. 1.

main result

(X, E, fl

Let

be

the Euler

constructing

on

class.

WFV-bundle

index r. Let 9A of relative 1. 8. We (X, E, fl given Definition in 21 and U U,C,U, the Go, ECj, Ffj) a

pre-weakly that for any (t, further the stratification, representative ffj,,,: U,, -+ Y,,, is a submersion. Then we can to class e(X, E, 0) of weakly f -pseudocycles, an equivalence assign (X, E, 45) called the Euler class of (X, E, 0). It depends only on the homotopy class of (X, E, fl. Further, the image of this class under f,, is a well-defined homology be the

Fredholm

structure

assume

class

on

=

H, (Y, Q)

in

The class

e

-

(X, E, 0)

image under f.

its

or

the

is

cycle

virtual

we

set

to

construct.

the

In

remainder of the

struction

The

complete

[LT2]

in

of this

required appear

(U, Gfj, Efj, F)

model

consisting

stratified

a new

as

possible.

the

topological

:

section

'

r

dimensional

5

we

By this, can

is

be

the

general.

will

the

to

locus

zero

tions,

so

of

cycle.

r

-

zero

is

r,

In the

choose these

we mean a

Given

section

the

in

steps

of

con-

(X, E, fl.

construction

stratified U and

over

is

with

is that

this

of

transversality

2 set K C U

so

that

section

of F away from K will

case

we

work with

perturbations

so

of S*E with

locally represented by a finite symmetric fiber product of E.

set

that

certain

only b

a smoothly 0: U -+ F. to perturb 0

like

be

a

We find

0

Since

the

to

the

so

finite

a

that

dimension

of

construction

vanishing

matching

multi-valued

solution:

many local

their

along, not points

near

perturb

K.

of F

section

zero

smooth

to

our

affect

of unordered

Note that

we

to the

we can

not

U,

is ill-defined

the

set

dimensional

section

class,

F is

approxima-

finite

a

smooth

transversal

as

it

orbispace

a

of the Euler

that

finite

local

a

to

Moreover, we have to allow Here is only an orbispace.

X is

dimension

transversal

is

F

difficulty

smooth.

because

homological of the

F

construction

-+

main

associate

we can

dimensional

of U. Thus the notion

where F is not section

U

follows:

as

X,

V-bundle

Here the

strata

near,

finite

a

rank

Following to

of

finite

is

UC

over

outline

[LT3].

in

The idea of the construction tion

will

we

by using the WFVstructure of the approach is a modification class

of this

account

and will

section,

Euler

an

approximalocus

patch

and which properties of E, where S*E

sections

S*E is not

a

vector

bundle

in

Constructing

together of

to

form

(X, E, fl

weakly f-pseudocycle

well-defined

is the

class

equivalent description, that these perturbed at a time, one chart

As is clear to make

a

it

invariants

277

X. The Euler

in

class

represents.

from this

sure

symplectic

the main

vanishing

of this construction difficulty Wewill patch together.

is

locus

do

and in the mean time make sure that the perturbation chart chosen in the the current on perturbation agrees with the perturbations of charts are arises that in the intersection One charts. difficulty preceding chart is finer than all the preceding not open. However if the current charts, be an open subset in the preceding will then the intersection charts and a chart. first task the in is subset closed Hence current to our pick a good locally Let % set of charts. J(Ui, 0j, Ei, Fi)JiEK be the weakly smooth structure denote the projection of (X, E,fl. For simplicity, -+ Ui : fji we will irui and denote denote the denote restriction by 7ri, Filui by Fi 4ii '(Fi) by Ui, we can assume 4 i 1 ui : Ui -+ ti I ui by Oi : Ui -- Fi. Without loss of generality, that for any approximation (Ui, Gi, Ei, Fi) cz 9A over Ui and any U' C Ui, the restriction (ti, Gi, Ei, Fi) I u, is also a member in 9A. The following lemma enables us to pick a good set of charts. =

There

Lemma 1. 1.

which

(1) (2)

The set

V'

For any

pair

the

approximation

Proof.

is

a

use

collection

C C IC and

a

of 'C of

ordering

total

Uf 7i (Ui) I i E C}; j E L, approximation (Gj, Gj, Ej, Fj) (ti, Gi, Ei, Fi).

(0)

is

contained

X, (a c- 10) be all the I0 so that for each a we assume lo f 1, =

-

-

-,

To prove the lemma, and 0 < 1 < 1, such that

is

of

structure

of X that

strata cz

which

proof,

the

stratified

the

the union

in

the

i <

We now outline

We will

finite

holds:

following

the

10,

1

V

(0)

n

k}. we

A,: For any distinct i, j E A2: For each a E 10 and I

suffice

C', , <

1,,

X,3

,,,

to construct

Ui

n

Uj

=

than

elementary.

(X, E,!P)

as

we

4i'(0).

intersect

U,6

finer

is

is compact.

C', ,

subsets

did For

C

[LT21.

in

Wegive

an

Let

order

to

convenience,

IC, where

a

C-

I0

0;

the set

U

U

OJ) !(a,l)

kECjo

z',=4i-'(o)nx,-

Uk

of dimension less than in the image of finitely many submanifolds and 1 < j; or a X, where ( , j) > (a, 1) means that either a < < C", C' of distinct with x A3: For any pair (i, j) E (a', P), the chart (a, 1) finer than is (Ui, Gi, Ei, Fi) (Uj, Gj, Ej, Fj).

is contained

1 in

=

a

-

In

particular,

We will that

CI,,

all

construct

may be

Ui with

i E

U,3 1.,,, U0<1<1,,,C1

CI, inductively,

empty.

0z

We now

assume

0

cover!V'(0)

from the

starting that

for

n

largest

some

a

a

E

X,. E

10,

10. Note we

have

Gang

278

Ll,,

constructed

V

finite

Tian

Lla+2

+17

a

for

i

all

jUili

c IC such that

L,,,.

1. We now construct

Vj

c

!1-1(0)

covers

n

compact. Let 1, be the maximum of jdim(Uj 1, > 0. Since tj is a stratified space with assumption, V such that subsets find C Ui we can open it

since

(1)

let

7r-I(U

U

fil

n

(2)

Let

X,. n

X,) I

is

(X,,,)

in

R in

(3) f U I

i E

L'-

to

71

define

a

ir,7 n

ui

ui

n

71

submanifold

then 1

V1

(X,),

still

Ll-

Clearly,

A,

We want

-

to

n

X,.

same

(in Uj

less

than

After

fixing

an

(Uj

the

U'j (X,))

of

7ri1,;

i E 1

7ri

as

I

n

is

n

X"'), of

closure

X;

in is

smooth

a

ordering

on

L',

we

of charts i E L'.

JU Uj,i

Gi, Ej, Fj)

satisfies

a

constructed.

(0)

strata

closure

the

R

-

dimension

V'

collection

(ti,

R

the

is

denotes

then

7r

covers

be the

U_

where

(X,,), 17, of (X,) 7rin

7ri-I (x,)

V n

pick a possible L'j. By

We first This

A3. Now we assume that L". For each

construct

x

a

(1.4)

Ll a...... E Z'

Ll

a

have been

find

we can

a ,

a

is finer such that it contains x and it (fJj,Gj,Ej,Fj) approximation We denote this collection in Ul,>IL". than all approximations by L' again. still denote by L, Since Zl,, is compact, we can pick a finite subcollection, Z' such that jUili covers E L'} By our choice of the charts, Zl,' is contained less than 1. in X,, of dimension in the image of finitely many submanifolds i E L', still cover Next we choose U'j C Uj such that (1) 7r,71 Z,,,; (2) let (Uli), i

finite

a

-

.

1

7r,-.

in

7r-.l (V,)

=

finite

then

(3)

(X,);

If

7r-1 (Z' ) R 7r,-.

n

let

we

=

is the I

Zl,,

n

of

the closure

same as

(Uli)

of smooth submanifolds

union

a

2

z

.

1(E

T?

then

less than

of dimensions

7r-1 (Z'

contained

R is

-

n

1

-

in

a,

1. Now fixing

of charts ordering on L' as before, we define L` to be the collection 1 obtain the for this defined by (1.4). 2, 0, we procedure By continuing of the Lemma. the This L' 1 for 0 : , desired proof completes : lo,. an

-

,

We now

briefly

sketch

(Uj,Gj,Ej,Fj)j(=-,c

the

be the

of

strategy

by Oj: Uj -+ Fj the UaEl. U,"" where 10 is

Lemma 1. 1. We denote

stratified

Ui

set,

7ri is precompact

f I

:

As 7

Oi.

this

17Vi in

-+

Ui .6 By a

define

we mean

that

need to

wherever

general.

both

compact,

Uf-xj(fV)

finite

i

E

assumption,

our

find

we can

I

Lj we

tv-i

open

covers

C

(V'(0)

and

for

each

have that

tj fVj

with n E

a

Uj 10,

submersion.

before,

approximations,

finite

is

such that

Y,, is

first

Wedo not (Pki

=

mentioned

We will

6By 7

Ui,,,, we

V'(O)

Since

of Lemma 1.1.

1(7r#Vj))

=

proof of Theorem 1.1. Let approximations given by finite models. As a corresponding the ordered index set in the proof the

of local

collection

we

need to

multi-valued its

closure

we can

sides

axe

sections

associated

to

the

to

perturb

collection

of

is compact.

multi-valued

use

multi-valued

use

sections

arrange

sections (pji

well-defined.

if for

defined But

the in

the

given

(1.1) latter

collection

such that may not

of local (Pkj

0

(Pji

be true

=

in

Constructing local

finite

k-th

symmetric

sum

of ki

8

SkjljF

in

of sections fiber

Skili

C

j

abbreviate

multi-valued

a

U

-+

S'iFi, it

we perturb perturbation

be

a

Fi,

any

x

p,- jijl

(2)

E U

(Uj)

for

for

if

Oj,

sections

no

279

j,

we

(P generates

of Gi.

the

Oil'

have

ki li

a

Oj1j

=

k 3 1j,

=

-

multi-valued

a

following,

In the

identify

and

we

the

locally

is

1 with

confusion

with

i, V)i

any

li, Ij

need to

we

We begin

for

any i >

some

the section

generate

generic.

is

(1)

such that

Clearly,

may

the

of

where ni is the order sections as sections

multi-valued sections

V be

(1.1).

in

Before that

near

Fj along

invariants

Gi, Ej, Fi),c=L. By a multi-valued section, Ski _+ : U, Oi Fi, where Sk Fi denotes the

product

of Fj

sections

where Wji is defined section O'i : Uj t-+ will

(fJi,

approximations collection

mean a

symplectic

we

Oi with

any

occur.

clarify by which general situation.

we mean

Let

smooth map between two open smooth manifolds. Let subset and s : U F-+ S'E be a smooth multi-valued pre-compact

p:

K C U

a

section

by FU (S' E) the set of all smooth 1-valued sections of E over U. As we said before, FU(SIE) may not have an additive so we can not simply structure, apply the transversality sections. theory to multi-valued However, we can still add sections of E to if h is any section of E over more precisely, section, any given multi-valued If ti, are any U, we can get a new section s + hl E Fu(S'E) as follows: -, tj local representations of s, then tj + h, We s + hl locally. tj + h represent t of s relative s + t' E Fu (S' E) is a p-generic to K perturbation say that with compact support if the following holds: (1) The support of h as a section of E is pre-compact in U and (2) there is a precompact open neighborhood Knbfd of R C U so that the graph of t is transversal to the zero section of E in Knbhd and p: Knbhd n t-'(O) immersion." -+ V is a strong Here, we mean that by a strong immersion p: A -+ B between open manifolds, p is an immersion and further, for any x, y E A with p(x) p(y), p.T,,A is 9

of

smooth vector

a

bundle

E

denote

U. We will

over

-

-

-

-

-

,

=

=

transversal

p.T.A.

to

Let

Lemma1.2.

submersion be the

space B

=

jh

of

E is

of

with

section

with

such to

+

s

V is

-+

For we

that

dense

is

respect

U

:

dim V.

Fu(S'E),

E

support Then B,

K.

to

compact}

h is

s

compact

f

that

-

relative

s

as before. Suppose bigger than dim U

multi-valued

ru(E)

Fu(E) I

E

define hl

B,

is

and

ft'

t, (m)

By this,

}

=

t,'

respectively,

there

we mean

that

its

local

we mean

that

for

any local

is

a

permutation

representatives

are

-

the

C'-topology.

the

by k

''

b and V)',

of

M

f

a

and open in

bundle F, if a section 0 in Sk F is locally represented any vector sections of t 1, tk of F, then V51 is locally t1, represented by 1-multiple ) Two sections V) and 0' of Sk F are equal if for any local representations

that

a

any pre-

For

*

9

h E

perturbation

generic 8

of

set

be

situation

rank

C U and any

compact K to

the

and the

of

o-

11,

-

-

-

,

k}

made of smooth

tk

-

It.1 such

sections

of

E. 10

By this,

open subset

in

WC

nbhd

WnK

denote

by

t

-

1

U, the

%raph n

nb

and p: K (0) the union

d

ti-

representation

of each tj 1

of the

(0)

-+

zero

t 1,

is transversal

V is loci

a

strong

of any local

tj

of t

to the

immersion.

any small of E section

over

zero

Note that

representatives

ti

we

of t.

Gang

280

Proof

B,

f

since

is

Since t-1

a

(0)

in

is

,

The lemma is

may

V is

an

be

that

assume

We begin

the

with

ordinary

use

first

by U,. Wefirst the proof of Lemma I f : U, --- Y is a submersion

arguments and t-1 (0) is also transverin a neighborhood of k.

that

immersion

(0)

-+

V is

a

of

Oi

one

define

simply

we can

immersion.

strong

-

at e

a

We

time.

(X, E, fl

to

smooth

a

1

W)

U,,

C

U,,

U

locus

zero

rank

UO."

M,

in

n

of F is greater

perturbation

small the

It follows on

from that

U,, with

property

Since

not

u,

to

only we

dense in X

:

-,

U. Recall

that

Y. Wethen

pick

n Wof


find

Y. Let

element First

M, so

neighborhood

closed

a

homological

dimension

=

f

locus

-1

extend of

0,,

<

T

L," -

C

Y,

2 and its

(L,).

U,3 4 0.

that

we can

vanishing

the

Wewant to extend

0. in

to

U,,

neighborhood W_ is contained

a

n

of 0, has its 0, then U,3 is not dense in U and the dim Y. Hence, by last lemma, we can find than dim U of 01 u,,, uu,, extending 0, so that it is an f -generic per-

U,

(U,

u

we

U,3).

may

assume

0

>

(Uc,

u

If

that

this

extension

-

F is

a

because

many smooth manifolds

to

Iu,,

U is

can a

of finite

UO) n (fV \M,,)

uuO (F).

continuous

compact support

holds

has

it

uu,, relative be 00 csection

new

we

follow,

compact support, dense in U, by the assumption, it

is not

we can

that

01 u,,.,

of

turbation

case,

easy to

F, W, C tj by 17V and Ul,, U, = 0, where 10 appeared

morphism f

relative

a

orbifold,

such

of M,, n

in the interior

is

go-'(o)nw-)

f (Oa (0) in boundary is a smooth orbifold Let 0 E 10 is the next largest to UO in an appropriate way. n

this

1. In

is

2. Thus

dim Y is

it

the given

for

.

at least

has codimension

of

which

Assume that

1.

of perturbation f-generic denoted by 0, : U, -+ Y, Since U,,,

Since

f.,

of F. To make the notation

small

Let

in U

less than dim V

transversality

perturbations

member in

sections

in

a

smooth submanifold

of dimension

model 01 : U, -+ F, by 0: U --+ look at the largest a E 10 so that

the finite

denote

of

a

zero.

need to

a

is

assume

1

of k.

neighborhood

a

(x)

t-

:

Y, otherwise,

dim

<

r

f

the

Wenow show how to construct may

1

-

we

h such that

perturb proved.

f

the standard

a

We may further

dim V in

of submanifolds

union

a

theodense.

is

it

=

each

submersion,

V). neighborhood of K, by using if necessary, h slightly perturbing of f so f : t-1 (0) sal to fibers (x

E

show that

to

theorem, we can find a small h E Fu(E) is transversal to s + h' graph of t of R. It follows that t-'(O) is a union of

than

less

transversality

ordinary

remains

the

neighborhood

a

of the

It

open.

such that

of dimension

submanifolds

Next,

in

C B is

transversality

support

section

zero

application

easy

an

the standard

compact

the

is

that

clear

is

by

First, with

proof

The It

rem.

Tian

Then

bundle

be extended

Hausdorff

over

f

is

with a

strong

compact support. immersion

U and any continuous

from

function

to U. This extention continuously by finitely topological space stratified

dimension.

Constructing a

neighborhood further

can

in the

sense

manifold <

r

f (0 Clearly,

<

<

-

f (o) n u, M, is Clearly,

n

W)

graph of

the

in

the

L,

(0)

n

Welet

and for of Li.

interior

of G1. Let

by using section

Un)

M6

C U

of

open

an

10,

> 0 in

f

-

U, be

=

(L

dimen-

M,

standard

Y.

in -I

f

any -y

Put

U,1

the

zero

n

closed

homological

of

L, is the closure

assume

Uy

L,, transversely

immersed,

an

we

smooth orbifold

a

C Y

may

we

to the

( -'

f

is

orbifold

action

generality,

homological dimension neighborhood L,3

well.

as

subset

a

smooth

a

the

is transversal

For any -y > 0 in Io,

2

-

of of

281

retraction

L,3

of

r

is

is of

U L,,

0 of 0, such that is

slightly,

Un\Ml;

(2)

find

of U n W. Then

perturbing

and

<

loss

(0)) \L,,,

strong

boundary

can

under

invariant

compact neighborhood theorem

the

contained

is

(00 (0))

closed

a

dimension

procedure, we 2 and a perturbation Y whose boundary

in

1

f

find

1

invariants

boundary

the

(0

f

that

2. So

-

this

r

subset

(1)

r

(0)) U L,,, such that L,3 has homological

Continue

of

follows

easily

one can

Y. Without

n

It

-

of dimension

of

sion

that

of orbifold

2. Then

-

W, \m, into ' f (00 (0)) intersects

(o)

of

assume

symplectic

1

(Li).

a

pre-

transverality

that of F

is contained

neighborhood

a

over

in

the

of

interior

Lj;

(3)

The

is transversal

(0),

to

where UO denotes

of U. of F may not be Gi-invariant. section j in 1,U(Sn'F) by

The section

nl-valued

invariant

Uln \ M,

of UO n

boundary

the main stratum

However, we can get a Gjputting together all sections

a*0 (a E GI), where nj is the order of G, (I > 1). a subset C Y of homological Suppose that we have constructed Lj_j < r 2 and a Gj -invariant dimension perturbation j of 03jmj in Fu (Smi Fj) for each j < i divisible 1, where Tnj are some integers by all m, for 1 < j -

-

-

fll<,

and

(1) Lj__j orbifold

(2)

For each

j

j

such that

Ujn

closure

of

open subset

in

pre-compact

neighborhood

an

boundary

Y whose

is

a

smooth

Y;

in

Mi-1,

-

such that

ni,

is the

<

i, there

is

a

is transversal

the

to

where Mi-I

-1

f

of

section

zero

Fj

Ujn

over

c

Uj

of

Uj

n

fV_j

neighborhood

a

of

(Li-1);

k

(3) of

oj*k

For any j < k < i,

Fk,

(4)

where

For any j

submanifolds

Ujk <

in

-1

is

equal

j Wjk i, the

(Uj) and cPjk boundary of

;

n

=

1

(0)

UJI1,

where

section 01 be the ml-valued that it satisfies the above (1)

Let clear

Now we want to continue where Wji submanifold

are

defined of Uj.

in

By

this

(1. 1). our

Ujn\Mi-l

n

Uj,o

denotes

is

(4).

Then each

So

we

to

Mk-valued

transversal

the main stratum

a*(O)

made of those -

as

may

for

assume

Uj. Let Uji be

(p,-.jiil (Uj

assumptions,

sections

(1.1);

in

Uj,O

procedure

induction

& over Ujk

to

is defined

n

Ujn for

-

a

that

E

to

those

of

Uj.

G1. It

is

i > 2.

W37jji ' (Uj)

U

<

01

Mi- 1) is a smooth any j < k < i, we

Gang

282

Tian

ni

Wj*i(oj 3

have

get

can

0'

section

a

that

with

coincides

it

simplicity,

still

we

Next,

we

before,

)

Uji

on

OTi Omi'

denote

outside

0'

so

that

10 is the

that

that

We can extend

neighborhood by 0'.

a

Wj*i(oj) 3

Note

R of

together, this 0' 0' to Uj

such

i.12

For

Uj
we

is

C U

a

extension

perturb

want to

we assume

these

Uj.

C

over

over

this

Gluing

Uki.

n

Uj
of Smi Fj

of

perturbation

small

As

mi

W*kj(0 1k k

j

satisfies

it

set

(1)

the above

of indices

such that

a

(4)

-

j.

for

corresponding

10 such that U)3 0. Since f : Ui,, -+ Y,, is a submersion, the largest a E 10. with boundary in Ui,,,. By last lemma, Mi-1 n Ui,,, is a smooth manifold to Uj a f-generic Mi-I such we can find perturbation 0. of 0' relative

Ui,,,

stratum

is

We also

nonempty.

for

is compact

Uj,O

and

any

is

these

arrange

E

a

open and dense in

-

that

it

is

unchanged

transversal

homological

near

boundary

to

of Li-1.

dimension

f (01,,,

(0)

Wj)

R. We may also

<

r

follows

It

2. Thus

-

assume

that

we can

f (0

find

n

U Lj-

1

such that

-1 C'

is

its

(0)

Wj)

n

n

17Vi)

Lio,

neighborhood

smooth orbifold

procedure

in

as we

(1)-(4)

properties

is

is of

U Li-i

for

Y did

Lj

Oj.

and

We let this m E L. procedure until we reach the largest of homological closed neighborhood dimension corresponding

Keep doing C Y be the

L,,, <

2.

-

Let

Then

Ai (1/mj)[7rj( ,--1(O))nUjn] for by our construction,

pseudomanifolds.

i E

each

i, j

Ll

covers

!V'(0),

if

=

we

E

It is

a

pseudomanifold.

rational

C,

Z1j1uinuj-f-'(L-)

patch together of these Aj. j sufficiently perturbations be a weakly f-pseudocycle.

zA be the

Welet

m).

(i::

=

Z-Ailuinui-f-I(L-) as

a

this

-

(0)

closed

a

boundary < r 2. Continuing dimension and is of homological before for 01 on U1, we will get Li and Oi satisfying

of

f (of

that

Since

choose the

Ifv-i

close

I to

This cycle (zA, Lm) will (X, E,fl. Let (X x [0, 1], U, T-1) be a homopoty between two WFV-bundles (X, E, and (X, E', V), Suppose that the above arguments have already given us two e(X, E, (P) and e(X, E, V), (ZA, L) and (,A', L') representing f-pseudocycles

Oi for

all

i, then the pair of

is the Euler

class

respectively. pseudocycle

(2i, 1)

Then

one

can

show

O(zA, L) implies

The theorem

12

above

arguments

that

there

is

a

f-

=

(A, L)

-

L').

e(X, E, fl depends only on the homotopy class of (X, E, proved. true from the arguments is often useful and clearly The following corollary the above proof of Theorem 1.1

This

in

by

such that

that is

Wemay need to raise

0'

to

certain

power

and then

get its

extension.

Constructing

symplectic

283

invariants

be a WFV-bundle of relative Let (X,E,!P) index and f : Y of finite dimension. Let morphism into a smooth orbifold X, be the strata of X (a E 1) such that X0 is the main stratum of X. We P- I (o) n xo is a smooth submanifold in X0 of dimension r and assume that in Xi of for any a > 0, !P-1 (o) n x,, is contained in a smooth submanifold

Corollary

X

1.1.

Y be

i-+

dimension e

a

<

(X, E, fl

E

r

H,, (Y, Q)

Sometimes

include

to

One interesting arguments as in the proof

topological

space.

identical

is

a

f -pseudocycle

in

Y

representing

-

desirable

is

it

f ( V'(O))

Then

2.

-

e(X, E, fl

the is

case

case

where Y is

when X

=

a more

f

Y and

of Theorem 1.1, one class in H* (X, Q) for

general By

idX.

=

the

construct

can

homology any WFV-bundle It admits a pre-weakly Fredholm structure 9A such that (X, E,!P) satisfying: in % and t the stratification, for any (t, Gfj, Et, F,[j) if L C U,,,G-rU, each such L subset closed that either is smooth is n a U,,, empty or a U0 then it has a closed neighborhood M in fJ such that it is a submanifold, of L and for any 0 c I, aMnU,3 is a smooth hypersurface deformation retract for those WFV-bundles involved is actually in U,3. This condition true in defining the GW-invariants. Euler

class

as a

=

GW-invariants

2.

In this

for

2.1

Stable

First

g,

the

Definition is

xi,

.

.

,

xk)

the

set

points

minJ3

a

map such

(1) the.composite

least

is

f

-

o

k

RC

'

2g(R), 01

of distinguished

stable

-pointed

smooth

symplectic

Z)

and let

we mean a connected prestable curve, and only normal crossing singularity.

map is

connected

C2 -smooth,

satisfying the

points

and the nodal points

a

A E H2 (X,

a

2.1].

Definition

(f, Z,

collection

prestable

curve

and

x,

f

...

:

Z

7

Xk)i

-+

X

that

is

7r

w,

X be

and let

C2-maps [LT2,

of stable

A k-pointed .

of Z; (2) Any component at

invari-

all.

k-pointed points

a

maps. Let

form

k marked

the notion

continuous

a

by

with

2.1.

(Z,

and for

once

following, curve

of stable

symplectic

given

a

fixed

We recall

where

the definition

with

complex

Gromov-Witten

constructing

to

maps

E Z be

In

we

review

we

manifold n

apply Theorem 1.1 manifolds. symplectic

section, general

ants

of

where

(f o,7r). ([R])

on

=

0 E

the normalization

H2 (X, Z)

must have

where on R, points of distinguished Zt consists of all preimages of the marked

number Z.

Z is

7r:

Gang

284

Tian

will abbreviate (f, Z, x,.... Xk) to (Xi)) or will We call arise. Z the dof if no confusion Two stable maps (f Z, (xi)) understood. main of f with marked points and (f ', Z, (xi')) are said to be equivalent if there is an isomorphism p: Z -+ Z' ' When (f Z) -= (f ', Z), such a p such that f and x'i f o p p (xi). of is called an automorphism Z). We will denote by Autf the group of automorphisms of (f Z). X be the space of equivalence classes [f, Z] of C'-stable Welet BA'g' maps k Z) with k-marked points such that the arithmetic genus of Z is g and A E H2 (X; Z). With X, A, g and k understood, we will simply ([Z]) write BAX,g,k as B. There is a natural topology on B, which we will define in Section 3.3. However, B is not a smooth Banach manifold It as one hopes.

convenience,

For

(f Z)

to

we

simply

or

,

)

to

,

,

,

=

,

,

=

does admit

a

f

a

stratification

smooth strata.

with

Given any almost complex structure generalized bundle E over B as follows. X be the

-*

:

A011

with

in

of section

of T Zto

complex stable

canonical

Aut f acts of the to be

TX

f

=

-

Ao,,'f

(f Z)

equivalence

v

-

j,

where

(f ', Z')

that

Ao" /Aut f

quotient

(0, 1)-form are

we mean a

j

denotes

two

the

equivalent

Z', then there is a automorphism group

map p: Z -+

follows

It

(0, I)-forms

of

sections

-

and

,

conformal

Ao". f

the

in the

representative f.

Z, Cl-smooth

-+

Here, by a f*TX-valued J v over Et such that

associated

the

isomorphism on A011 and

A" /Aut f

7

7r:

Zt. Assume that

of

maps with

with

define

be any stable map and let where is the normalization

f*TX.

f*

structure

f

one can

w,

(f, Z)

Let

be the space of all

to

f

values

v

of

composite

of Z. We define

with

compatible

J

independent

is

f

(f, Z).

of

class

the

So

of the

we can

choice

define

A011

Put

U

E

A0,1

[f]EB It

has

infinite

an

obvious

projection linear

dimensional

There is

a

natural

Pi defined

as

P: E

we

conformal

have

finite

quotients

of

spaces.

:

B

satisfying

E

---

Piffl Obviously,

are

map

For any stable

follows:

B whose fibers

-+

fii (f ) 4ii

map p. Thus

=

=

map

df

+ J

(Pi (f ')

-

descends

P

f, -

we

define

df J

E

p if to

Pi

o

a

f

=

Id13)

Aof,'

and

map B

f

'

-+

equivalent through a E, which we still denote

are

by -fij. 1

Mxa'

It is the moduli space of J-holomorphic (0) by g, t class with stable maps A, having k-marked points and whose dohomology Note that a stable map [f] is J-holomorphic mains have arithmetic genus g.

Wewill

if

Pi(f)

=

denote

0.

4i

*

symplectic

Constructing the

Stratifying

2.2

In this

basic

section, properties.

Given

we

stable

a

space

give

a

of stable

(f Z),

map

maps

stratification

natural

of B

associate

we can

,

285

invariants

to

=

BAX,g,n

it

a

dual

study

and

its

graph rf

as

of Z

corresponds to a vertex v, in rf together with a marking (g, A,), where g,,, is the geometric genus of Z, in X; For each marked point xi of Z, and A, is the homology class f,([Z,]) each For intersection a leg to v,; we attach components Z, point of distinct and and Z,3 we attach an edge joining v, vp and for each self-intersection denote we will point of Z, we attach a loop to the vertex v,. In the following, F. the of all of F aiAd set vertices of of all the set edges Ed(.V) by Ver(T) in the of dual is the representatives independent If graph Ff Clearly, denote the dual graph of If] by _V[f ] so we can simply Ff. Moreover, the Each irreducible

follows:

Z,

component

=

genus g of If] of g, for all

homology homology Given let

B (r)

r[f]

==

is the

Ver(F)

E

a

class

A of

class

of the

Clearly,

The main result

Proposition nach orbifold Further

(f, Z, (xi))

prove to

each dual restriction

that

to

B(r) the

construct

and let

If'

Zt,

Recall

that

the

is

the

normalization

and Z.2. To avoid

homology class

If ] B(F). following.

graph F, the of E to B(F)

B(F) is

a

U f Zt,

is

legs,

we

B with

a

is

a

B(r) smooth

orbifold

smooth

is

a

smooth Ba-

orbifold

bundle.

section.

Banach orbifold.

I

a

of the

points

maps in

the

be the

to

A and k

of stable

space

neighborhoods

points of Z. Note that Z, thus has two distinguished

and the nodal node in

sum

of all

of

of

If ]

in

B (F)

.

Let

f

of Z. Then

be the normalization

distinguished

A,

is defined

which

classes

is the

section

=

where

g and

genus

of!Pi

restriction

suffices

It

For

2.1.

Wefirst

with

of this

and the

the

Proof.

r

of all

sum

space of all equivalence B is a disjoint union

be the

1".

f,, ([Z]) graph F[f].

is the

graph

any

the genus of r[f 1, which is defined to be the and the number of holes in the graph.V[f]. Also,

same as

G

Ver

(.V)

1,

(2.1)

to a. component Z, corresponding of marked points are preimages

edge e E Ed(F) corresponds to a points in ". We denote them by Zel

each

real 2-dimenwe denote by Zt'P the underlying is given by an almost on `P complex structure with j2 which is a homomorphism j : T`P " Tt'P complex structure -Id. Two almost complex structures j and j' give rise to the same complex if and only if there is a diffeomorphism structure 0 of `P such that j' such that jo is of almost complex structures Let it be a family do J (do)-'. . is the of derivative Then the given complex structure v(jo):= (dldtjt)t=o almost T-t-valued of structures If is another family a complex j' t (0,1)-form. any

manifold

sional

confusion,

of

'.

A

=

=

-

Gang

286

such that

Tian

corresponding This u of T-t. by

an

the

(0,1)-form

v(jo)

shows that

local

Thus 7

where -, i

Ef,

of

is the

are

the

then

some

section

parameterized

0 of

family

universal

'

E

does,

+ N for

preimage

of xi

marked E Z and

form

JjPi

QX

IlPel

X

i

Q, Pi, Pel and P,2

where

local

a

it

as

v(jo)

as

complex deformations

7

is of the

structure

be written

can

H!,(TZ).

of

complex

same

(', (JV-0, (Zel Ze2))

near

Ed(F),

E

induces

open subset

curves e

j' t

each

neighborhoods of .7ci, set of the k-marked

neighborhoods

small

are

(2.2)

Pe2)

X

e

Ze2)

Hl,(TZ-)

of 0 in

', respectively.

ze, and Ze2 in points and (Zel,

Here,

denotes

2

and small

(.`Cij)

usual,

as

#Ed(F)-marked

-

is the

points,

ordering of Ed(-V). 0 we have H! (TZ) The latter is the By duality, H', , (T *' 02) Z of differential forms. holomorphic quadratic space Let V be a small neighborhood of 0 in the space of PTX-valued vector ' C2 fields in the Here X. For each : -+ f v E V, we can along -topology. C2 -map expi(v) associate from 7 into X, where exp is the to it a unique

fixing

after

an

the

Serre

map of

exponential

fixed

a

metric

X.

on

by Gf the automorphism

We denote

naturally

acts

=

on

H!

(TZ)

of

Z,

Z

This

-

of

group

because

is

f.

each

or

-

automorphism of generality, we an

in

1

(q, (Xil))

(Zell

any

a

E

5

Next

F, Xel He X

we X

we

(Zell

7

Hi Pi

and

-action

on

E

Gf

HI

on

x

V

and

group

naturally

lifts

to

-

I

-

ft,

X

0

finite

a

TZ).

Pel'

Without

Pe2

X

Gf

are

-

Any point

follows:

as

loss

a

product it

to

1

E

(ZII

Pi)

7

e

Z12) e

Pel

C

(or. (q), WXD); Wzeli) manifold

two

define

Xe2- We can define

7r(q, (4)

=

7

We also

-

Q, x

Pe2 and

X

v

E

V7

define

Ze12) V)

associate

Xe2

q E

7

construct

we

of

Xel

V, Q

Gf

natural

a

Ze12) v),

Gf,

a(q, (x'),i e

that

assume

action

an

is

V is of the form

x

given

can

Now we define

invariant.

induces

it

so

It

a

XF

copies Ajr

=

as

of

X, He Ae,

smooth evaluation

(eXPIM

I V) (Zel)I Ze2) i

1

17(42))

follows:

for

7

V

each

Xj,

a-1)

*

edge

or

loop

Xe2. X.V where Ae is the diagonal of

say

map

(Zel)

i

Define

7F:I

x

V

-+

=

Xr by

eXPI(V) (Ze2)).

e

One

can

easily

The finite

f

in

B(F).

show that

group

This

Gf

7r

acts

is transversal

on

to

Wsmoothly,

the first

of this

7r_1

Ar. S9 thus

WlGf

is

a

(,Ar)

is smooth.

neighborhood

of

proposition. Wenow prove the remaining Note that each point part of the proposition. in VV is represented by a stable map Z, (xi)) and (', (Xi), (Zel, Ze2)) E Let Ef be the set of all C'-smooth, Put over . f *TX-valued (0,1)-forms proves

part

Constructing

symplectic

287

invariants

U fE*

kW is

Clearly, to

linear

a

a

action

kW

on

Elw

Moreover,

Gf-equivariant.

such

that

so

which

acts

E is

an

on

kW

projection

natural

the

kWlGf,

=

Gf,

and

bundle,

Banach

smooth

orbifold

W, lifts VV is

-4

bundle

over

B.

lifts

4ii

the section

Finally,

,Pw(f) Obviously,

is time

Now it a

stable

in

B.

Let

two

k,

E,

a

< 3

F be

its

Ver,,(-r),

E

the

df -j_r

E

*

defined

by

Ef.

proposition.

of stable

maps

describe

.

edges

As before

normalization.

and g,,,

proves

-

over

=

number of

edges.

+ J

space

!w

section

of B. Let [f ] E B be represented the topology by (f, Z, (xi)). Wewant to construct the neighborhoods of [f ] For each a E Ver(P), we define the dual graph of f k,, to

to

map f

be the and

of the

a

df

=

This

is smooth.

Topology

'2.3

as

it

to

=

0}

then

legs attached to it. Here we count denote by Z, the corresponding still

and we

loop

to

a

component

to be Ja E Ver(r) c Ver(F) I Ver,,(I) When Ver,(F) the complement of Ver,,(-r).

We define

and define

Z,

a

contains

one

or

two

distinguished

points.

We add

to whether to Z, according Lt. contains one one marked point(s) point. We also require that the curve Z is smooth and distinguished Note that this is always at these added points. the differential df is injective of f Wedenote by (yj) 1 <j<1 the 4 0, by the stability possible since f. ([ZJ) stable is a Deligne-Mumford to Z. Then (Z, (xi, yj)) set of all added points with k + 1 marked points. curve of local deformations k + 1 marked points, with As a stable curve Z. differentials admissible on quadratic by (Z, (xi, yj)) are parameterized differendifferential An admissible quadratic quadratic q is a meromorphic nodes and double at at tial with at most simple poles satisfying: poles xi or yj 0 of Z near a node, i.e., Z is defined by W1W2 If W1 W2 are local coordinates in C2 near such a node, then

two or

or

two

.

=

i

lim W1-*0,W2=0

2 W I

q

dW2 1

-

lim W2_.

O'wl=o

2 W 2

q

dW2 2

Let as follows: Neighborhoods of (Z, (xi, yj)) in Ng,k+l can be constructed of U Gf be the automorphism group of (Z, (xi, yj)), then a neighborhood (Z, (xi, yj)) in Ng,k+l is of the form Z 10f, where 1 is a small neighborhood in the space of admissible of the origin quadratic differentials. two submanifold For each yi, we choose a codimension Hj C X such that We orient at and intersects f transversely (yj). uniquely Hj so that f (Z) Hj with f (Z). intersection it has positive

Gang

288

We fix

(xi, yj). curve

0

Kx

map

compact

a

We may into

CU such that

K

on

We then

0

collection

fibers

U

a

=

a

and restricts

points

universal

neighborhood to the identity

of

J > 0.

K, J and 0 given U(U, Hj, K, J, 0)

Hj,

marked

CU be the

from

0

over

fix

Wealso

neighborhood

a

diffeomorphism

a

preserves

of U,

all

K is

choose

f (Z, (xi, yj)) 1.

x

it

to

that

assume

Z.

Z\Sing(Z) containing Let Gf -invariant

K C

set

over

To each ciate

Tian

above,

as

we can

follows:

as

asso-

'(J

Define

Hj, K, J, 0) to be the set of all tuples (f ', Z', (x'j, yj))3 satisfying: (Z, (x, , y )) is in 0; E Hj; f is a continuous map from Z' into X with f'(yj) X; f' lifts to a C2-map Ilff '0 flIC2(K) < < J, where dx denotes the distance function of dx (f (Z), f '(Z'))

U(U,

(1) (2) (3) (4) (5)

-

Riemannian

metric

any

topology of Z' tuple (f ', Z, (x ,,

One

can

Note that Given scendant. B.

in

from

the

Let

large.

(f u(xi).

Wedefine

,

yj'))

is

U, then

E

each

yj"

j,

(f ')

The E Hj. f (a-' (yj")) assumption that Hj

yj'. f

Clearly, I =

-r(xi')

induces as

=

a

x'j'.

above.

we assume

reduced

that

of Z.

(f ', Z', (x'i))

f

de-

its

2

to

this

yj"

U

on

1,

-

a

and

==

to

to

?1

u(Z')

the

one

map

show that

can

is the

identity

open subset

-r(yj), case.

so

yj"))

sufficiently denoted by a,

r

is

a

the

by

Note that

be

[f']

It

may not

Conversely,

B.

in

Z" such that we

may

(Z, (xi)).

of

as a

-r

local

map.

same

[f ']

f

f

The

-r 'r

acts

on

-(J

a

U is of the

uniformization

to the

7-(yj)

in o

that

Hence,

is close

that

if

assume

This

f/1.13

(f

follows

identity

yj"

descend to the -+

map. Then

of Z.

to

yj such that of such yj" are assured

small,

serve

and xi

near

Z'

are

=

3

identical

-r

and f/

Gf a(Z') E

Yj,,)),

(a (X'j)'

(f ", Z", (x ',

a

special

Z'

on

uniqueness to f (Z).

K, 5, 0) can U(U, Hj, K, 6, 0).

that

by sending

in

t(U, Hj, is

a

(Z'),

point

descend

let

Gf

sufficiently

and K is

small

t:

of

transversal

map,

large

on a

sufficiently

U

-

and

Thus

shows that

sufficiently consequently,

'

biholomorphic When J and 0

of B whose quotient

First

acts

a

UlGf.

This

from call

we

stable

are

unique

are

biholomorphic

defined

form

is

a

existence

a(f)

(f " Z" (X ' Yj'))

U, then there and

U,

in

action

(f

=

the

and

0

u

is the

I

i'

is

natural

a

a

where for

'

f

J and

that

assume

Then there

(X

show that

may be different

Yj'3))

map and gives rise to a point [f U be the set of all equivalence classes of stable maps descended in t. Then U is a neighborhood of [f ].

tuples

Now

fixed

a

X.

on

identity

is close

general

chart

to

case

yj'

map

can

and be

symplectic

Constructing

generated by all such neighborhoods U(U, Hj, K, 6, 0). and satisfies topological space can be proven to be Hausdorff

The

in Definition

with

with

orbispace stratified homology class A,

Therefore,

1.3.

Equiped

Theorem 2.1.

a

289

of B is

topology resulting all properties The

Given

invariants

B(F),

where F

runs

B is

a

dual

all

over

smoothly graphs of

legs.

genus g and k

t,

chart

uniformization

local

above,

described

topology

the

strata

have

we

define

we

E7Cj

where

E(p,z,,(,, ,y ))

the

normalizati n`of

the

conditions

of all

C'-smooth,

Z'.

such

Ef.'s

All

in Definition

1.5

of moduli

Compactness

is

of E. One

Therefore,

satisfied.

all

are

(0,1)-forms

f'*TX-valued

form charts

(B, E, Pj)

The above

Theorem 2.2.

2.4

consists

we

over

show that

can

have

proved

V-bundle.

a

spaces

stable theorem for J.-holomorphic we prove a compactness subsection, The compactness theorem of this sort first appeared in the work of Pansu and extended by Parker and Wolfson, studied further Gromov [Gr],

In this

maps.

proof

and Ye. A

following

[SU].

was

However,

the compactness here

uses

Recall

detailed

a

For the

erature.

also

given

proof

theorem

for

of

that

TZX,

The rest

w

assume

available

not

in

lit-

complete proof of maps. Our presentation a

that

J-holomorphic

is

a

is devoted

subsection

J is

an

qtable

maps with

genus g.

subsection.

(X, w)

next

compact manifold

almost

space

9R.Xa,o,e

proof. The by simply taking to

its

complex

structure

with is

a

compatible

compact in B. may

skip

theorem

for

readers this

compatible

i.e.,

metric,

g(u, V) with

of stable

Then the moduli

subsection

and g is the induced

We start

is

case

present

and of arithmetic

points of this

J.

structure

and go to the

proof granted Wealways

the

spaces

will

space of

is the moduli

Assume that

of this

harmonic

[RT].

Here is the main result

complex

moduli

on

general we

maps of any genus, 80's maps in early

holomorphic

for

the

conveniences,

ideas

Theorem 2.3.

for

reader's

homology class A, k-marked

almost

[RT1]

in

work of Sacks and Uhlenbeck

the

=

W(u, Jv),

the monotonicity

formula

u,

for

v

G TX.

holomorphic

maps.

with

Gang

290

Tian

Lemma2.1. in

o9(f (D) radius

X be any

c(X, g)

an e

(

d dr where y E

is

c

f (D),

uniform

respect

depending

constant

metric

W)

f (D)nB,(y)

e"r-2

any

r

denotes

the

to

only

where D is

map,

disk

a

and y E X, the geodesic ball <

E

if of

then

g,

0,

>

(X, w).

on

particular,

In

if

then

f (D)nB,(y)

By choosing

Proof.

contained xi

a

Br(y)

where

at y with

center

J-holomorphic that for

> 0 such

f (D) nMr(Y),

=

and with

r

D

:

is

Br(Y))

n

f

Let

C Then there

in

sufficiently

> 0

c

coordinate

a

2,-,r.

W > 7rr

chart.

small,

we

may

fXi}l
Let

assume

B, (y)

that

be coordinates

such

is

that

0 at y and

=

n

dxi

w

Since

J is

compactible

with

a

a

jY ( It follows

that

Oxi

)

=

w,

09Xn+i

the distance

whose actual

a

19xn+i

)

assume

a =

we

value

from

always denote by

may vary

in

i

axi -,

-

p from y is different

bounded by cp 2. Note that

section,

dxi+n-

may further

we

jY (

,

A

c a

different

,

that

=

1,

yn

uniform

at

-

-

y,

-,

we

have

n.

X2 by

a

constant

function in this

places.

Write n

a

2E(xidxi+n

=

xi+ndxi),

i=1

then w

dO be the

Let

Since

f

is

da

=

induced

J-holomorphic,

and

volume

by

form

"r(1 2 r

This

is

equivalent

2

+ +

-2

e2cr

cp).

+

niMr (y)

we

by the

obtain

W a

cr) Jf(D)n8B, cr) -.4dr

(I

X, f(D)

on

the assumption,

ff(D)nB,(y) ff(D)naB,(y) <

jal :5

(Y)

dO

(ff(D)nB,(y)

to

d

arNow the lemma follows

(r

f (D)nB,(y)

by integrating

this

W) differential

> 0.

inequality.

metric

g.

Constructing

Corollary a

-xe-"

f (M) then f (R) is contained of connected components contained

is

Proof.

If

the

in

f (M))

>

The

is

by

x

one

x

in

R such

dz >

E2.

proved.

almost

and

actually

Let

(X, w)

holds

for

harmonic

We define

for

E D and

x

0, we often write E, (f Denoting by p, 0 the polar =

,

x)

jVf12 (z)dz.

E, (f ).

as

of C centered

( 2

27r

af

27r

I +

19P

is at least

I

jxj,

-

fl' -Xj
=:

f. Z

there

that

< oo,

coordinates

r

follows

< I

r

one

2(r',

af ao

P2

dX (f (y),

'qf 2) 90

r'

E(r/3,r)

0)

dO < Er

consequently, sup

be a symplectic manIf f : D\101 F4 X is then f extends to a smooth J.

structure

f D JVfJ2 (z)dz

with

E, (f ) it

least

(B-5 W)

of Singularity) complex

(Removable

a

write

so

jVfJ2 -1

ff

at

lemma,

Uhlenbeck

K.

find

we can

last

is

Er (f, x) If

f

dz >

due to

compatible ifold a J-holomorphic map map from D into X. Proof.

then

Riemann surfaces.

Theorem 2.4. with

false, Then

corollary

So the

following

maps from

is E.

jVfJ2

JR

A contradiction!

X be any

-+

in M.

statement

dx (f (x),

that

291

with

is

c

R

:

Riemann

1, where

>

invariants

J-holomorphic map, where R is For aR. surface boundary any E > 0 such that each last connected in component of lemma, if given ball of radius a geodesic E in X and f JVfJ2 dz' < 152, R in a geodesic ball of radius 4me, where m is the number

f

Let

2.1.

connected

symplectic

f (y'))

<

at the

origin,

we can

PdpdO,

such that

(f ),

V/2irEr (f

jyj=jy1j=r1 where dv denotes Now take

ri

=

the Ir

31

distance for

all

function

sufficiently lim i

Choose r

z

E

T7-+I

T,

as

above,

00

E, (f )

then

metric by By the assumption,

of the induced

large =

i.

0.

W

and J.

Gang

292

Tian

lim

>

c

0, by

ly I I y I a

<

r}.

the

corollary

Letting

8E.

implies infinity,

i go to

that

that

follows

tended

f (x)

lim,:,o

continuously

Next

we

0.

exists,

is contained

for

that

any x, y E

claim

proved and f

is

can

want to bound the

Vf

derivative

Since

.

f

is

continuous,

we

choose po < 1 so small that f (B2po (0)) is contained in a coordinate 2 of X. For simplicity, U C V1. Fix any y E Bp,, (0). we consider In the polar coordinates (r, 0) centered at y (r = 0 at y), the

:5

c

be any constant

may

chart

fBp (y)I9r

19f

190

0.

then

integrating

by parts,

we

have for

Ep(f)

+

Ep (f )

+

-2

fp0 f

Ep(f) -2

follows

27r

+ 2

fop f

00

2-7r

p

2"

(f

f

2-7r

(f

A,

-

A,

-

A,

-

!X2L)rdrdO a0

9f

fo' L 27r 2 f (f 27r 2 f (f

+ 2

2rdrdO

Of + .1 J r

Ep(f)

(f

Jaafo)(p J 2a o ) (p,

A, Af ao )dO

-

+ 2

J 'f

aO )drdO

A, V _L Or

2r

fol L ( f p

2

-

-

A 22-

27r

fop f

2

(f

2

fp

c

-

c

If

sup aBp (y)

depends only

on

-

the

AI)Ep(f,

+ 2

1001

fP0 f('&07r (f

y)

-

(f -A,

derivative

A

by the Poincar6

inequality

f

I =

2 7r on

A, V

J

'9 ao

Of ao

of J.

I

27r

fdO. 0

the

2 -ir

o

unit

circle,

fo Lf 2

we

27r

_

)drdO

a2f A, J araO )drdO

Now choose

Then

ao

J 1')drdO 9r

r 'rl'"

JO

-

27r

where

(JOf

O)dO

J J'L)drdO '90

V

-

-

O)dO

7

Af ar )drdO,

that

(1

U

Cauchy-

P0,

0

it

af

r

R2',

in

vector

I +

Or Let

ex-

becomes

09f

P

be

D.

to

Riemann equation

in

DTj,

< 8e.

the

so

(DTj\D,i)

f

we see

dv (f (x), f (y)) It

=

converge to a point in X. For any sufficiently is ar such that for any i > j, each connected is contained in a ball of radius E, where DT =

j\DT,))

Then the last

of radius

ball

f (r, 0) above, there

f (,9(DT

of

component

jyj=jy'J=T

that

Now we claim small

f (y'))

dv (f (y),

sup

i"

A12do

< -

00

do

have

)dO,

Constructing

by the'Cauchy-Riemann

So

(I

If

sup aB, (y)

c

-

equation,

Ep(f,y)

7pthe above is the

so

2(1-c f

If-

sup

continuous,

is

27r

IVfI2

dO.

dO,

fdOI)Ep(f,y) choose po

<

p9

5p- Ep(f,y).

small

that

27r

fn

27r

so

fdo I

<

2c*

-

0

(p- 1 Ep (f y))

> 0.

,

19P for

IVfI2

0

0

If

a

that

f27r

P2

21r

we can

Then

follows

f.

-

sup aBp (y)

It

2

same as

OBp(y)

Since

293

27r

fo

p

=

1

:5

But

invaxiants

have

we

AI)Ep(f)

-

symplectic

BpO (0),

any p < po and y G

Ep (f y) :5 2pEj (f ,

lemma, f

By the Morrey's

2 (1

It

follows

So

we

then

-

1/2-1161der

is

cvl-p-) Ep (f y)

Ep (f y) :5 Cp2

that

,

deduces that

The

following

f

< p

,

for

any p

of

have bounded the derivative is smooth in

provides

f

Then

continuous.

:5 -

po,

By

a

5-P Ep and

(f, y).

consequently, I'V f 12 (y) :5 C. theory, one elliptic

the standard

D.

the basic

gradient

for

estimate

pseudo-holomorphic

maps.

Theorem 2.5.

complex for

that

structure any

Let

(X, w)

be

symplectic

a

Then there

J.

J-holomorphic

are

map

fD

f

c

>

D,

i--*

c, :

manifold with a compatible almost 0, depending only on (X, w, J) such X with

IVfI2 (z)dz

< E,

r

then sup

IVfI2

Dz

< -

C

IVfI2 (z)dz,

r2

2

where D, induced

by

=

w

jIzI

<

and J.

rl

and the

norm

I

is

taken

with

respect

to

the

metric

Gang

294

Proof.

This

Tian

So

By scaling,

proved by sketch we just

be

can

theorem.

last

may

we

the that

assume

arguments

same

proof,

its

I 00 12 (ro,

r

out

1. There

=

those

as

pointing is

a

proof changes.

the

in

necessary

(3/4, 1)

E

ro

of

such that

27r

0)

(f 0)

dO < 6E,

< 6E.

,

implies

This

0'))

dx (f (ro, 0), f (ro,

sup

6Tf V67re.

<

0<0,0'<27r

is

sufficiently

p in

X. It

follows

dx (f (x),

f (y))

We may

assume

BV6--;j-, (p)

for

that

some

E

We have shown in

the

4-V6--xE,

:5

proof

for

theorem

of last

for

that

C

D:3.i

x, y E

any

f (D,,,)

3.4,

any y

E D2 and

1

P<

121

2(1

dx (f (z), f (z')))

sup

c

-

Ep (f, y)

lz-yl,lz,-Yl
Combining this

with

the 2 (1

above estimate

any y

on

cv'-E) Ep (f y)

-

:

,

by using the arguments E D1,

Then

we

<

observe:

given

For any

where A denotes

places.

different

p(9

(f, y). i p- Ep

deduce from this

pEp (f y). '9P

7

theorem,

we

have that

for

C.

h

metric

L 1,Vfl2dV h. Note that

<

Theorem 2.3.

Now we prove

First

we

proof of last

in the

lvfl(y)

to

By Corollary

small.

=

Z and

on

[f, Z, (xi)]

E

9A 32cj'O'e,

[w] (A).

the volume form of h and the

is taken

norm

always denote by dv a volume form, which the energy of f depends only on Therefore,

we

with

respect in

may vary

its

homology

A.

class

Next

9RX It

'O't

we

observe:

and S is

an

There is

irreducible

then

It

component

fs follows

from

these

the number of irreducible genus g, the

homology

lVfl2dv

and the

>

stability

components class

number J such that

positive

a

of Z

on

which

there

is

f

if

[f, Z, fxi}]

is not

E

constant,

j > 0. that

of Z. This

A and the target

bound

manifold

a

uniform

bound

depends only

(X, w).

on

on

the

Constructing

Z., (x,j)]

[f,,,,

Let

be

of stable

sequence

a

symplectic

invariants

maps in

295

Because

.

of

by taking a subsequence if necessary, we may assume of a. of Z, is independent that the topology class of metrics consider the following We will part g, on the regular bounded geometry, of Z,. The metrics namely, for each g,, have uniformly chart (U, z) of regular point p of Z, there is a local conformal coordinate with the unit ball D, U is identified I Izi < 1 Z, containing p such that the above observations,

=

in C and

Ou for

some

o(z)

satisfying:

1kPj1Ck(U) where Ck

el'dzd.

=

uniform

are

:

for

Ck,

any k >

of

independent

constants

0,

require that n,) satisfying:

We also

a.

there

Z, (i 1, finitely N,,,i many cylinder-like (1) n, are uniformly bounded independent of a, balls many geodesic (2) The complement Zc, \ Uj N,,i is covered by finitely and < in of where R are bounded; uniformly Z, m, j :5 m, g, BR(Paj) (1 of the form S' x (a, b) (a and b to a cylinder (3) Each N,,,i is diffeomorphic of S' x [0, b) If s, t denote the standard coordinates may be oo) satisfying: then x (a, 0], or S' e"(d '92 + dt2), gajN ,j necks

are

C

=

=

where

p is

We will

Admissible

Clearly,

that

say

if all

admissible

g,

such

to

a

is

g,,

admissible

are

always

metrics

suffices

it

satisfying

smooth function

a

uniform

admissible.

on

metrics

construct

on

we can

easily

down

write

negative, hyperbolic

then the uniformization

obtained

by fatting

admissible

those

Now we fix introduce there

is

follows For Wewill

9'

metric

on

S with

those metrics

a

cusps can

of

sequence

to

regular

on

theorem

in

finitely

can

C

or

seen

Euler

of those

number of S is

Euler

gives

and g,, It

the

either

in

complex analysis

cylinder-like. uniformly.

follows:

as

if

of Z,

C\101,

many cusps,

be

be

uniformly etc..

cl,

component of Z..

part

S. If the

above.

Ig,} R,

each irreducible

metric

a

This

stated

as

call

constants

Z,,.

any

of the Let S be any connected *component S then either is number of S is nonnegative, cases,

We will

uniform

with

exist

bounds

on

a

S is

hard to

is not

unique simply

see

that

be chosen

uniformly

admissible

metrics

g.

on

Z,,.

Wewill

metrics admissible j, on Z,, such that sequence of uniformly of f,. Once it is done, the theorem uniform bound on the gradient a new

a

easily. simplicity, define

If sup_, such that

ldf Ig

we

write

Z for

each

j,, by induction. :5 16, then we simply

given

Z, and

g for

g,

and

f for f,.

=

e

=

ldf Iq(pi)

define

=

sup E

to be g.

ldf Ig

>

16,

Otherwise,

let

pi

E Z

Gang

296

and

Tian

local

be the

z

Write

g

=

coordinate

ewdzd,

as

specified

of Z

above,

define

and

above such that

the

g outside

=

z

region

0 at pi.

=

where

jzj

< 1

e

91

where q

for

t

R

:

[2,

E

77'(t)

0 <

R is

1-4

e

11,

-

a

Clearly, Moreover,

e

=

we

have 1

we

have

respect

1)

(pl,

denotes

the metric

to

the

> e,

t

ji.

t <

may

check that

to

1, q(t)

assume

is

=

t

that

uniformly

ball

of radius

I and centered

at p,

with

from Theorem 3.3 that

ldf 12dv 9

(pl,,)

depneds only on (X, w). < 16, then we sup_, ldf 1,

I for

we

I df 1 1

geodesic

41

=

moreover,

! g. It is easy

follows

It

q(t)

satisfying:

for

SUPB1(pi,ji) where B,

g'

(ejjzF)

function

cut-off

and q (t)

< 1.

admissible.

=

6 > 0,

>

where 5 If

take

otherwise,

ji,

we

choose P2 such

that e-

=

ldf 11 (P2)

ldf 1,

sup

=

16,

>

z

then

E

P2

Z\B2 (PI) ,).

with

struction

g

Now

replaced

by j.

Clearly,

the above get j2 by repeating with 1 on B, (pi, j2 coincides

B, (pi,

2)

B, (pi,

we can

con-

ji),

so

We also

=

have

B, (P2 2)

n

5

and

41 If

df 1 2

sup

process

follows

that

SUPZ ldf bL Now

we

L < 12

< 16.

(pi,j2)

V Id f 12d 9

B, (pi,

> -

we simply put inductively j, -

0

ji)

j > 0,

:5 16,

and construct

41 It

j).

2 -

-

,

1,2.

i

-

Otherwise,

we

continue

the

such that

L

jdfj2dV>6>0' 9 -

(Pi,L)

(A),

Wethen

have construct

therefore, take

to

a new

the process be Lof

sequence

has to stop

uniformly

at

some

admissible

L when metrics

such that sup

I df,,, I ,:,

< 16.

z

Moreover, by scaling j, appropriately, i $ V. By the uniform admissibility

for

we

of

may

,,

assume

when

a

is

d(xCi, sufficiently

that

xci,) large,

we

Constructing that

may have m, 1 and R such

there

symplectic

invariants

many

cylinder-like

finitely

are

297

necks

1, (i -, 1) satisfying: balls complement Z,,, \ Uj N,,i is covered by finitely many geodesic in Z,; BR(Paj ja) (1 < j :, m) (2) The marked points x,j are all contained in the union of those geodesic balls BR(Paj) ice); of the form S1 x (aa,i, b,,j) to a cylinder (a,,i (3) Each N,,,i is diffeomorphic

N,,i (1)

Z,,,

c

=

-

-

The

7

oo).

may be

b,,i

and

We may further

41 where

c

given in taking

is

Now by

j,

(Zp,j)

to

I

a

Ec,,,j

is

a

any

N,,i,

E

x

< e,

if necessary,

we

ZO 00

Riemann surface

'j

assume

may as

pointed

for

that

metric

each

spaces,

is of the form

Ec,o,j \ I where

for

(x,

subsequence

Z,,D,j

a

Jdfajj(,,dv

that

Theorem 2.5. a

converge

such

moreover,

assume

qj7j

qjl,

compact Riemann surface. and on ZO g,,.,j

More

precisely,

there

a

are

nat-

such that

point p,,,,,j ZOO'j, J sufficiently large, there is a diffeomorphism and 0,",, (pc"'j) p"j 0,,, from B, (pc,,,j, goo,j) onto B, (p,,j, i,,) satisfying: the in over to the pull-backs C'-topology uniformly converge go,,,j B, (p,,,,,j, goj). Note that such a convergence of i,, is assured by the uniform admissibility. admissible

ural

metric

any fixed

for

>

r

00

0, when

a

in

a

is

=

Next

follows:

Zo,c,,j large,

we

put

together

all

these

Zo,,,j

form

to

a

connected

curve

ZO.

and Z,,,j,, we identify For any two components Zo,),j punctures yj, for if be to with yj,., E Zooj, equal j) (j may any a and r sufficiently above and B,(p,,,j,,i,,) of B,(p,,j,i,) the boundaries specified

contained

in

a

neck N(a,

cylindrical

stable)

(not necessarily

Zo,,,j,

since

i).

In this

each Z,

way,

we

get

a

connected

as

E

are

curve

is connected.

bounded in terms of j, by taking a of fa are uniformly Since the gradients subsequence if necessary, we may assume that f, converge to a J-holomorphic into X. By the Removable Singularity Theorem, the map foo from Uj Z' j X. Morefrom into to extends a J-holomorphic ZOO smoothly map fo,) map x,,i converge to x,)Oi as a tends over, we may assume that the marked points each xo,)i belongs to the regular to the infinity, part of Ec'), clearly, a stable The tuple map, since there (fo, ZOO, Jxc,,j 1) is not necessarily 00

to a constant map and which components Zj where f, restricts and yj,6 (defined fewer than three of x,: ,,j to CP1 and contains set of Z' 00 ). There are three in the singular possibiliabove and contained

may be

is

conformal

ties

for

such

Zcj's.

If

Z,,OJ

contains

contains component; If Z,,,j this component and identify contract

this

nents

of

ZOO;

If

Z,,j

contains

one

no

no

xOOj

xcoi

but

yj,3 x,,,,i

but

one

two

and yj,3, as and one yj)3,

yj,3

we simply drop yj,3, then we and y,),O',

points then

in other we

compo-

contract

this

Gang

298

Tian

and mark the point Carrying out this process inyyo, as x,, j. obtain a connected curve Z,,,, such that the induced we eventually ductively, is a stable (f, Z,,, jx,,ij) map. this stable map has the same genus as that of Z, and k marked Clearly, points. It remains to show that the homology class of f,, is the same as that of f,. By the convengence, we have

component

f Since

IVf z

12dV

cylindrical S'

then

(a, b),

N(a, i),

be

This

can

N(a, i),

It

(p,j,

j,,)

suffice

to

B,,

fS1

lim a_ 00

00

as

seen

follows:

IVfa 12dv

jB,(p,,,,j, ,) Z, is contained

in

for

show that

each

=

in

the

i,

if

of

union

N(a, i)

0.

(a+r,b-r)

choice

our

of

IVfa 12 dv

<

,,,

we

know that

for

any

C.

-

(p,j.)

Theorem 2.5 that

I Vfa 12

sup N(a,i)

where

c

is the uniform

fc, (S'

x

ja

+

in

particular,

27r-\,/_cE,

fu

IVfa 12dv x

By

41

from

follow

lim a--+Oo

we

lim r

p E

liM r_ 00

=

Uj

of

complement

the

necks

x

00

-

rj)

lb

x

there

are

-

rj)

two

Since

in Theorem 2.5.

given

constant

fa (S'

and

< CC,

are

contained

in

smooth maps

e

geodesic

h,,j

:

such that

D,

small,

is

balls F->

both

of radius

X

(j

=

1, 2)

I

sup D,

IV h,,j I ,x

:5 8

Ivfal

sup

Slx{a+r,b-rl

and

h,,,IaDi The maps map from

faIN(a,j)

and

S2 into

homologous.

fS1

It

fodS1xfa+rj, h,,j

X. Since

=":

fOdS1xfb-rj-

easily put together to form gradient is small everywhere, this

can

its

k

(a+r,b-T) we

ha,21aDi

be

a

continuous map is null

follows

IVfaI2dV x

Therefore,

f.*w x

(a+r,b-r)

=

f

D,

h,*,,wdv

-

f

h,*,,2WT Di

have

fS1 This

`

I Vfa 12 dv x

<

implies the required the stable Therefore,

guments

also

show that

topology

of B defined

IVfa 12.

sup

c

S1

(a+r,b-r)

x

f a+r,b-rl

convergence.

If,,, Z,,,, Za, (x,j)]

map

If,,

in last

section.

(x,,,,i)]

is

in

99ix

to

If,

converge So Theorem 2.3 is

The above

Z, (x,,i)] proved.

in

ar-

the

symplectic

Constructing GW-invariants

Constructing

2.5

of this

The main purpose and GW-invarinats X be

Let

a

299

invariants

subsection

smooth

cycles

moduli

the virtual

is to construct

manifolds.

general symplectic manifold symplectic

for

with

given symplectic

a

form

W

of complex n, and let A E H2 (X, Z). Let 9A.,, t be the empty set if curves moduli space of k-pointed, the and k if 2g + < 3 genus g stable dimension

2g

+ k > 3.

of this

Here is the main theorem

mension

for

Then

n.

(X, w)

Let

Theorem 2.6.

be

r

symplectic As

(A)

2c, (X)

=

(X)

2(n

+

A,

there

g)

-

is

(Og,t

Hr

G

3) (1

-

manifold of complex difundamental class

compact symplectic

a

each g, k and

eA,g,k where

section.

X

+ 2k.

a

virtual

30 Q) this

Moreover,

eA,gk

(X)

is

a

invariant.

application,

an

let

us

GW-invariants

define

the

(93t_,,t,

Q)

fe

(X)1

Let

now.

2g

+ k > 3.

We define

Ox A,g,k to

H*

:

oxA7g,kA017'* 0

E H*

Xe

9X_,,t ak) (0) 01 W). Let (B, E, Pj) be of

tion

x

to

OA g of (X,

k

Q),

ai

its

i-th

by

ev

2g components

before.

2g + k > 3. apply Theorem

(as

in Theorem

For this

We continue

over

fJ. U is

Recall

<

k) are

and 7ri is the projecoften write we will

symplectic

evaluation

where Zred by contracting

of Z

(2.4)

7rk*k+lak

invariants

map

30

x

(f (xi)),

A

...

simplicity,

natural

9N_,,t

is

the all

empty

set

if

non-stable

its

that

t

developed

notations

G, and

let

Et

is of the form

fJ(U,

group

of U and CU be the

a

property

universal

so

class

fundamental

(B, E,!Pj)

admits

for

the

evaluation

far.

Let

fJ

1.1).

to use the

sufficiently

the virtual

constructing

need to show that

the submersion

with

We know that

1.1 to we

purpose,

corresponding

uniformization that

X

-+

reduction

stable

structure

the

Ered

::::::

,

B

a

A

OAX,gk

All

There is

7r2*al 2

< i

For

component.

as

eA,g,k (X). Fredholm

with

(X, Q) (1

7PX A,g,k (0) (ai)).

:

A

if

We will

stated

g, k

as

(f Z, (xi))

+ k < 3 and the

7rj*O

A,

E H*

ev

defined

(2-3)

0,

-+

integrals

be the

where

(30, Q)

S)*

x

be

a

a

weakly map

chart

(0,1)-form

on

of B

chart of E corresponding 0 the local Let be 6, K, 0). Hj, 0. We curve over may assume

be the

small. TX-valued

ev

CU x X is

an

endomorphism.

Gang

300

Tian

TCU -+ TX

v

such that

J

jCU

where

complex

C'-smooth

all

of

is the

Sing(CU).

1.

Given each For each f

denotes

(f Z, (xi, yj))

=

VIf M vIf

Clearly, f

section

this

is

vj,

-

can

over

by

section

section

a

vIf

-+

v.

For

the fiber

on

t.

To avoid

o,

Gf

E

,

the

of

be the space vanish near

E'[J

of

section

a

fibers

the

in

follows:

as

E Z.

X

f

over

introducing pull-back

o-*

In this

.

new

TX)o.

v,

-,

TX)o

X which

If by

v

Efj

x

singularities

associate

define

be any 1 sections in A1,1(CU, the 1 IGf I sections assume that -

of the

set

v(x, f W),

=

CU

on

we can we

way,

notations,

(vi)

is

a

Without

we

section

loss

of

obtain

we

a

denote

still

t.

over

Let

generality,

we

-

fu*(vi) Efj

of

the

fJ,

E

,

AM(CU,

CU. Let

on

(0,1)-forms

AM(CU, TX)o,

c

v

jCU,

-V

=

structure

TX-valued

Sing(CU)

Here

of CU over

v

are

linearly

bundle

in

<

1,u

E

Gf}

We define F independent everywhere. F(vl,... the above I I Gf I sections. F is Efj generated by

and is

a

Gf -equivariant

Suppose f Lf, where Lf

Lemma2.2.

< i

vi)

=

be the subbundle vector

11

subbundle

(f Z, (xi, yj))

of

t

,

a

to

trivial

Eij.

Sp (vi, vj) I f are Cauchy-Riemann equation at f. Wefurther J is sufficiently assume that small and K is sufficiently big in the definition of fJ. Then (-(J, Gf Efj, F) is a local finite approximation of index r, where r is the index of Lf which can be computed in terms of the homology class of f (Z), the genus of Z and the number of marked cl (X), transverse

to

=

E

,

is the

linearization

of

and

the

,

14

points.

This

1,

u

Lf

E

follows

from the

Gf ) generate

for

every

Next

we

f

and

denote

by

L

a

2(AO") f

natural

stratum

by W','(f

(with respect to J). Notice that

Implicit

In

fact,

one

in

Efj

CT

the

to

Uo

and

Theorem because a*

which

orientation

of

*TX)

Sobolev

=

is transverse

the

above

!Pjl(F)

space

of all

n

(vj)

to the

(I

can

show that

:

W1,2 (f *TX)

4ij'(F)

is

a

-+

<

j

cokernel

(U, Gf Efj, F). any f

W1,1-sections

G

of in

f f

of dimension

UO,

*TX *TX

structure

L2(A0,1) f

smooth manifold

r

<

of

Let

,

UO. For

the space of L 2-integrable with values (0,1)-forms the norms induced by w and the almost complex

Lf 14

Function

subbundle

fJ.

in

assign

UO be the main we

a

+ 1.

symplectic

Constructing is

linear

Fredholm

a

det(Lf).

So

operator.

smoothly

varies

It

with

have

we

and

f

well-defined

a

gives

rise

to

line

determinant

bundle

line

determinant

a

301

invariants

Uo. Let FIf be the fiber of the bundle F at f Then it is a finitely 2 decomposition so we have an orthogonal subspace in L (Ao"), f and induced 2-inner w L induced the by J) to product respect

det(L)

over

.

dimensional

(with

L Let

7rf

L2 (Ao'1)

:

FIf

=

P

+

F-L be the orthogonal

F-+

f

2(A0,1) f

Then Tf U is natu-

projection.

Lf

6f

that

such

af

both

Bf vary smoothly det(L) is isomorphic

and

has

On the

canonical

a

Coker(5f),

and

(U, Gf Efj, F) ,

Now

we

A and k

IF,}

such that

Un

to

is

!V

< i

1,)

<

It

still

v1,

-

-

-

,

small

!P-1 (F) It

is

Bf of order 0. Moroever, operator U0. Hence, Lf is homotopic to af,

an

in

since

vi.,

-

vi

-

,

U,

n

-

is

det(09)

bundle

line

with

is

a

B(F)

It

follows

so

the

that

the

evaluation

there

finitely

that that

are

is

to

ev

a

of

stratum

of

fJ corresponding

:

U,

many ui,

E

map

ev

i-+

9R,,t

x

30

A0,1 (CU, TX)o

such that :

I

v,

such that

f

E

!Pi (f )

U,,

the

Since

Sp (vi, sufficiently large,

U, is tedious,

a

--

,

vi)

E

Sp (ui,)

If }

F-+

number of such a's contains

the restriction

all

9Xg, is

Sp (ui,). of

ev

x

t

finite,

30 we can

Then for

to each

choose

6 sufficiently

nonempty

stratum

submersion.

but

rather

straightforward,

to

check

that

those

locally

smooth structure

defined as weakly provide approximations (B, E, Pi). Combining this with the compactness theorem of last section, Fredholm V-bundle. is actually a weakly conclude that (B, E,!Pj) theorem. main of the the This completes proof

nite

each

stratum

graphs of genus g and homology only finitely many dual graphs

by U, there are 0 0. Let U, be the any

of

restriction

where U,,

submersion,

the dual

classified

obvious

submersion.

n

(F)

is

a

and K

1

Given

Un

ev

is

are

legs.

submersion.

a

(1

B(F).

By

decomposition

to the determinant

other

choose

to

t

of

The strata

class

is

f

det(L).

-

want

nonempty stratum

fJ.

+

naturally

is

orient

to

its determin ant J-invariant, Df hand, on Ker(6f) induced by the complex structures orientation canonical so does has a orientation, det(L) consequently,

det(6f).

fibers

Bf

with

consequently,

t9f

=

and

J-invariant

is

It

o

det(L)

that

follows

Lf. rally isomorphic Thus we suffice to At'P(TU) 0 At'P(F)-'. isomorphic find a canonical can we computations, straightforward of 7rf

the kernel

to

above

a

fiof we

Gang

302

Tian

Composition

2.6

laws

subsection,

In last

GW-invariants

for

GW-invariants

have constructed

for general symplectic such as the Puncture properties, the String equation and the Dilaton equation which the generating of GW-invariants The most useful property is the composisatisfy. we

These invariants

manifolds.

equation, function

satisfy

certain

which we will formulate in the following. Wewill GW-invariants, proof. Assume that 2g + k > 4. Given any decomposition g 91 + 92 and S S1 U S2 of f 1; ki, where 2gi + ki > 2, there is k} with I Si a canonical x embedding is 9R,,,e, which assigns stable curves (Zi, xj,..., 1, 2) to their union E1 U Z2 with xkl,+l xk,+,) (i 2 identified to x, and remaining points renumbered by f 1, k} according to law for

tion

drop

its

_':

---::

=

...

7

=

S. There is another

together

the last

One

can

define

ii as

Q)

any 01 through respectively,

For

F_, a2jK2j,

9N_,,tSimilarly,

one

TTg_j_,t+z

9R,,e

-+

E

H*(9R,.,e.+_,,Q)

the Poincare where ali,

define

can

ii

:

H*

Q)

S5*

x

cycles in (resp. the homology class represented in

:

obtained

by gluing

points.

homomorphism

a

them

represent

io

maps

H*

:

follows:

natural

two marked

a

02

and

duality E

a2j

by

-4

,

Q) we

F-i K2j)

cycles

(resp.

il(01A)

by the rational

e

E

rational

Q and Ki.j

then

S5* (9 JZ_,,

is

the

aijKij

integral

are

Poincare

dual

Ejj alia2jis(Kii,

cycle

and of

K2j)

homomorphism

(9n"

e+ Z'

Q)

*

Si

-+

(9no,

e,

by using the

map io. Now we state the

Theorem 2.7. mension

Then

n.

for

Let

Let a,

any

01

composition

(X, w)

,

for

basis

e(S) of

H*

,

consists

compact symplectic

of two formulas.

manifold

of complex

di-

(X, Q) E

we

have

(ai)) (811 (ai)iES,,

ej

X

)O(A2,.92,k2+1)

(82, e*,i (aj)jES2),

i

any

00

E H*

(TZ_q_j,e+z,

oxA,g,k (" (00), (0i)) where

which

Q), 82

E

X 6(S)0A3.,gj,kj+1 and

a

ak be in H,,

0X A,g,k ('1 (01) 02) A=AI+A2

be

law,

is

the

(X, Q)

=

Q),

we

E V)(XA,g-l,k+2)

sign

of permutation

S

and

fe I

basis.

S

is its

have

dual

=

S,

(00, (ai), U

S2 of 11,

ej,

e ),

k},

Jejj

is

a

symplectic

Constructing the readers

refer

We will

subsection,

In this

Kdhler

form

w

spaces

stable

All

J.

standard

space with

complex projective

structure

subsection

this

maps in

assumed to be of genus 0.

are

Then

H'(Z,

bundle

of

f *T','X)

T',OX is f *T','X

that

the standard

Since

restricts

component of Z,

any irreducible

J is

vanishing an integrable

f *T',OX) the Euler

n

class

complex structure,

eA,O,

complex

is the

n

=

Then

line.

each ai

we can

can

be

curves

k :A 3d through 3d

the GW-invariants

3.

=

write

I

-

are

identify

by

-

A

Cauchy-

of the

ev(9JTXM',,',)

where

repre-

equal

is

r

6

particular,

ev*(7r*al2 QL'

o'

we

A

...

have

7*+10k)k

t

A

for

OX A,O,k*

any

point

If in

d[t]

=

Cp2'

OCP d,O,k (1, (ai))

in

'

further

2

general generalization

points the

+ 2n + 2k

Z. We can write

OX d,O,k

Lf

1.1 that

X

I and otherwise

-

can

30, Q),

e x

of X. In

f9x

represented

whenever

direct

one

Corollary

H, (9&,

in

dimension

2, H2 (Cp2, Z)

computations

from

linearization

of the

follows

it

k (X)

OX A,O,k(1) (ai)) If

tangent

a sum

lemma follows

so

cokernel

f. Then

at

to

2c,(X)(A) and

genus 0.

theorem.

the

with

Riemann operator sents

map

and each component of Z is CP'. line bundles on of nonnegative

bundle

positive

a

that

implies

H'(Z,

0,

=

the

X.

Notice This

,

of holomorphic

stable

be any J-holomorphic where T',OX denotes

(f Z, (xi))

Let

Lemma2.3.

or

complex

and

"

proof.

its

projective

X be any

let

we

for

for

GW-invariants

Rational

2.7

[RT2]

to

303

invariants

position. of classical

where f is any the Poincare

then is the

This

complex of

dual

OCp2 d,O,k (1, (ai))

=

0

number of rational

example shows

enumerative

that

invariants.

applications

Some simple

applied to many other branches of mathematmirror quantum cohomology, geometry, algebraic ics, such as enumerative time of Because and topology. symplectic systems symmetry, Hamiltonian will Here we these give two applications. and space, we can not cover all for the general symquantum cohomology briefly: (1) Construct applications there are differential to show that manifolds; (2) Use GW-invariants plectic structures. which admit infinitely symplectic manifolds many different The GW-invariants

15

This ever,

proved in [RT2] arguments can be easily

theorem

the

symplectic

have been

was

manifolds.

for

semi-positive

modified

to

give

symplectic a proof

manifolds. in

case

of

How-

general

Gang

304

Tian

3.1

Quantum cohomology

Let

(X, w)

ring

of X is the

be

symplectic

compact

a

cohomology

manifold.

Here QJH2 (X)} by GW-invariants. appeared in Novikov's study of the Morse

[No]).

(cf.

It

sum

qd+d'

Ei., .

(E di qi).

is

finite

any

in H*

(X, Q,

we

(a

is the

just

is

a

choose

di

q1

qd ,

...

S

a

basis

with

the

ring, that is q qd' grading defined by deg(q d) graded homogeneous ring generated d with ndqd satisfying: nd E Q, all q d

=

=

number of nd with or

H*

on

w(E diqi)

<

c

symplectic

monotone

(X, QJH2 (X) 1).

multiplication

E

a

ring.

group

structure

07,Y)=

q

=

multiplicative

Fano manifold

a

the quantum 0

a

d

natural

a

and the

X is

ring

a

define

monomial

Ed=(dl,,--,d,)

If

> 0.

define

can

has

degree

same c

into

QJH2 (X) I

QJH2(X)}

then

Now we

ring

series

power

for

manifold,

H2 (X)

turns

Then

0 have the

the

a new

[HS], [MS], [RTIJ):

(cf.

follows

as

identify

we

multiplicative

by all formal

:/

Z),

This

diqi.

This

2c, (X) nd

be defined

can

of H2 (X,

q,

cohomology

ring structure denotes the Novikov ring. It first functions theory for multivalued

defined

q1,

The quantum

(X, Qf H2 (X) 1) with

H*

a 9

For any a,

3

0 by

OAX,0,3(070,7)qA

(3-1)

AEH2(X,Q) where -y E H* (X, Q) and of H* (X, Q) with a basis

denotes dual

is

a *

0

basis

in H*

if A

aiqi,

(X, QJH2 (X) 1) a

=

Equivalently,

product.

if

lei}

then

(a' 0, ei)

e*

qA

(3.2)

i

identify

we can

cup

lei* 1,

E E V)AX,0,3

=

A

Note that

the

Eadq

(a,,

A with

be written d ,

a,).

In

general,

any a,

as

)3

=

E#d,

q

d'

d

where ad

7

Od,

are

in

H*

(X, Q. a *

0

We define

=

E ad

0

Pd,

q

d+d'

(3.3)

d,d'

Recall

that

cation

preserves

the

is associative.

degree Of ad qdis deg(a) + deg(qd). It follows that the multiplithe degree. However, it is not clear at all if the multiplication Given a, 0, -y, J in H* (X, Z), we have

a a

(a

0

So the associativity

sign)

18) * 71 6) (0 7)7 6) *

=

:_

means

X (a) EA,B Ei 7PA,0,3 F-A,B Ei OAX,0,3 (a,

that

for

any fixed

0, ei,

X * -y,J) )OB10,3(ei X 6)OB,0,3(01 _Y1 el)

ei

A in H2 (X,

,

Q)

we

have

(up

to

symplectic

Constructing

EAj+A2=AEiOAXj,0,3(a,fl,ej EA,+A2=A Ei V)AX,,0,3(07 But of

by

the

3-1)

are

law of last

composition equal to

X

PD(p) denotes W,,f. Therefore,

p, in

the we

section,

0,

a,

is

tum

ring

The quantum multiplication supercommutative, associative, on H,, (X, Qj H2 (X) 1). structure,

both

sides

of any point

OAX,0,3

fl

consequently,

associative,

is

9

i. e.,

structure,

one literatures, product becomes

the quantum

so

a *

sign,

class

mathematical

and sometimes

e-tw(q,))'

homology

graded ring

an

ph,rsics

In

up to

have

Themem3.1. there

y,

6),

-y,

of the

dual

Poincare

that

we see

305

(e ,

6)OA2,0,3(07

6i,

(PD (p),

A,0,4

where

X

)V)(:,A2,0,3)

invariants

quan-

substitutes

q

ej)e!e-tw(A)

(07 fl,

by

(5-5)

A

In

particular,

0,

then

this

to the classical

converges

a

a

o

product

cup

U0 +

as

t -4

oo.

(X)

If cl

>

OA ei(X)(A)>O

where OA has

degree deg (a

(A).

2c, (X)

U

was Example 3.1. The quantum cohomology of the Grassmannian G(r,n) bundle over G(r, n) of computed in [ST], Wi2]. Let S be the tautological complex k-planes in C'. It is known that H* (G(r, n), Q) is given by

Q[xi, Isn-r+l, where si

are

Segre classes, Sj

In

fact,

corresponds

xi

-Xlsj-l

-

the i-th

to

'

'

*

X'r1 SnI ,

inductively

defined

=

*'

-

-

-

-

-

by

Xj-jSl

Chern class

-

ci

Xj.

(S) (i

=

r).

1,

It

can

be

shown that H*

(G (r, n), Qf H2 (G (r,

In

above

tj

[13e], fact, *

as

Any

w

are

fsn-r+l

examples of computing

More

[GK],

(Q[xl,

n

[CM], [KM], there a

is

special

E H*

rational.

a

[CF],

family

*

i

*

x,,

'

,

Sn-1,

quantum cohomology

q]

Sn +

can

(-I)rq}

be found

in

[Ba],

[Lu]. of

new

quantum multiplications,

containing

the

case.

(X, Q)

can

We define

be written

the quantum

E tiej. Clearly, w E H* (X, Q) if all multiplication 9,, by

as

Gang

306

(6.1)

Tian

EA Ek>O

=

(f ail)

px A,O,k+3 (a 0

k!

7

Theorem 3.2.

Each quantum

solve

the readers

...

ti,

q

[Ti]

or

is

e,,,

for

more

details

associative.

nondeformation equivdistinguish to will we use GW-inNariants subsection, due is to which following stablizing conjecture,

manifolds.

is to

In this

of the

case

==

manifolds

of the GW-invariants

symplectic a special

[RTI]

to

multiplications

symplectic

of

One application alent

)ti

0

We refer

Examples

ei,

is the sign of the induced permutation and e (f ail) this multiplication 0. reduces to at W Obviously, of *.,,, is equivalent to the so called above, the associativity

the

3.2

- ) ei ......

ei.

argued in WDVV equation. we

1

(X, Q,

where a, 0, y E H , on odd dimensional

As

'

Ruan.

Conjecture

homeomorphic symplectic if and only if. the stablized diffeomorphic CP1 with the product symplectic/structures are X and Y

Then X and Y

manifolds

X

deformation It

Suppose that

3. 1.

4-manifolds.

CP1 and Y x

x

are

two

are

equivalent.

follows

from

a

result

of M. Freedman that

two 4-manifolds

X and Y

homeomorphic if and only if X x CP1 and Y x CP1 are diffeomorphic. The stablizing of this between the can be viewed as an analogy conjecture smooth and the symplectic The first of examples pair supporting category. the conjecture were constructed by Ruan in [Rul], wherQ X is the blow-up of CP1 at 8-points and Y is a Barlow surface. Ruan also verified Furthermore, the conjecture for the cases: (1) X is rational, Y is,,irrational; (2) X and Y are irrational but have different In the following, number of (-I) curves. we will and prove the stablizing compute certain genus one GW-invariants for simply connected surfaces K'p,q This is due to Ruan elliptic conjecture and myself in [RT2]. are

.

Let's

generic Then

recall 9

Epnq

the

be obtained

can

smooth fibers if

and hence

n

p, q

is

Theorem 3.3.

plectic

(p', q')

structures

from En

Let

El be the blow _Up Of (Cp2

connected

by logarithmic

p and q.

coprime.

topological

fiber

Moreover,

of

n

copies

transformations

Note that the

sum

Euler

Epnq

is

simply

number

at

of V.

alone two connected

X(Epnq)

=

12n,

number.

Manifolds Ep",q x CP' and Epl,ql ) CP1 with product symare symplectic equivalent if and only if (p, q) deformation n

n

-

Combining

[FM]),

a

are

n

En be the

multiplicity

with

only if

and

and let

points,

Ep",q.

of

construction

we can

with prove

known results

about

the smooth classification

of

Ep",n, q (Cf'

Constructing

Corollary Let

The

stablizing

be two

multiple

3.1.

Fp, Fq

Ap Then

Ap

and F be

[Fp],

:--

[F]

=::

P

,

Aq

`

Aq

=

2)F

(p

+

1)Fp

-

(q

+

Then Theorem 3.3 follows

Proposition Ep"',

q

piece of topological

where

a

is

(1, a)

2)pq

-

+

(p

-

I)p

+

(q

information

1)q)A.

-

(3.4)

proposition.

q(TnA p(mA

m=

2pa(A);

Tn

0;

m

=

Ap), Aq),

=

=

4 Omodp or

cohomology

2-dimensional

a

((n

following

the

( 2qa(A);

I

OmA,1,1

from

=

Let

Wehave

3.1.

X CIP

1)Fq

-

fiber.

[F]*

=

-

general

a

[Fq]-

The primitive class is A [F]Ipq. Another is the canonical class K is Poincare dual to

(n

307

invariants

for Epnq*

holds

conjecture fibers

symplectic

(3-7) and

q

m<

pq,

class.

theory of elliptic proof here. By the deformation that all singular such structure on Jo complex Epq that the complex fibers are nodal elliptic assume we curves. can Furthermore, fibers whose of multiple structures are generic, are neither i.e., j-invariants CP'. Then X 0 nor 1728. Let jo be the standard structure on complex Ep`,q x CP`1 has the product x structure Jo jo. complex describe for rn < pq and rn i4 Omodp or q. For any f E Let's

Proof. surfaces,

We will

outline

we can

its

choose

'r, n

a

n

9AX

its

image Im(f)

is

a

connected

Im(f)

holomorphic

effective

curve.

Write

EaiCi,

=

i

where ai

Ci

> 0 and

are

Since

irreducibe. mA

=

E ai[Ci], i

each Ci is of the form and ti

is

a

point

consequently,

in

Cil

CP'.

we can

x

Jti},

E aiCi'

singular

fibers,

is

a

each

Cl' is

is a holomorphic connected, so all

curve

xi

in

coincide

Ep,q and

write

Im(f ) where

where

However, Im(f)

connected

Cil

is either

=

(E

effective a

aiCil) curve

multi-section

x

in

JxJ,

(3-5)

Epq. By n

or

a

fiber.

our

assumption

A multi-section

on

has

Gang

308

positive a

< pq,

Im(f)

with

intersection

general

m

Tian

fiber.

It

each

Cil

can

Fp

x

is either

general

a

follows

from

only

Jx}

gAX

be

0 for

mA,I,l

[F]

mA

-

Fq

or

fxJ.

m

:

has

fiber.

with

intersection

zero

Cil

each

0 that

=

multiple

a

x

A fiber

fiber.

Because

is

a

Since

fiber.

of connectedness,

particular,

In

0 mod p

or

and

q

m<

pq.

(3-6)


(3-7)

Hence,

*X?nA,1,1(1)0) Now

assume

m00 mod

=0 for

that

Ap,

mA =

so

q,

m=

por

A

and

q

m

straightforward

computation

shows

giff moduli

This

space is

expected

the

f

Let

normal

E

9)TXA

f

should

f

and

'I',

bundle

to

Aut(Fp)

general

in

not

dimension

=

A,,1,1

position

Im(f)

=

Fp

x

is clear

corresponding

be

Jx}-

One =

1

we

obtain

of dimension

4 while

map,

we

define

its

Jx}),

show

NFI, (Ep, q)

&

TX CP

(NFp (Ep, q); Q Tn

an

=

C*

have

the cokernel consequently, Cauchy-Riemann equation

and

is

holomorphic

x

easily

can

H'(Nf) the

it

that H

Hence,

since

(3-8)

be

Nf It

CP1.

be 2.

f *TXIT(Fp where

x

obstruction

of at

bundle

ob

=

TxCP'

Lf is Tx CP1 where Lf f. Putting together all ,

7r*TCP', 2

=

is the linearization

these

where 72 denotes

of

Coker(Lf), the

we

projection

from X onto CP'. One

can

show

image of the Euler

by arguments class

of ob

2 that

section

in

over

9RAXpJ

eAp,I,I(X)

1

=

under

eAp,,,,

(X)

the evaluation

is

the

simply map

ev.

So

2[Fp]

consequently,

and

oxAp The

same

yield

arguments

111(XA The

proposition

is

(1, a)

1,1

proved.

q,

=

2

fFp

a

=

2qa(A).

(3.9)

a).

(3-10)

that ,1)

(1, a)

=

2p(Ao

-

symplectic

Constructing

proof

the

complete

Now we

all, if (p, q) equivalent

of

First

of Theorem 3.3.

309

invariants

=

(p', q'),

as Kdhler complex deformation Ep",,ql p,q It follows transform. the logarithmic where we perform surfaces regardless structures are with product x CP' symplectic that Ep",q x CP1 and E p,q

and

En

n

known to be

were

n

n

,

equivalent.

deformation

Conversely,

equivalent.

deformation

Ep",q

that

suppose

n

Then there F

such that En,

Ep",q n

:

XCp1

,

b F.P("),,, A

X

CP1 and

x

is

n

x

ql

CP1

are

symplectic

diffeomorphism

a

CP1

EP',

Epn,,q,

-+

X

Ep, q XCpl

1

(3-11)

F*

(3-12)

CP1)

(3-13)

CP1)

(3-14)

CP

n

(1, a)

1

bA

=

q

(1,

'l

1

,

and

F*ci(En p

q

F*pl (Ep,

,q'

I

X

CPI)

=

Ci(Epq

X

CPI)

=

PI

n

X

and

Let

I_J2

eo E

((Cpl

,

Z)

be the

positive

0

neo +

=

(Ep"

n

pi

q

for

n

=

X

CPI)

X

CP1

=

we

claim

(3-15)

eo.

0

some

X

First,

generator.

F*(eo) Suppose that F* (eO) Pontrjagan class

(Ep' ,q

E

H'

(EP,

i

q

PI

(Ep'r,q) 0

P1

(Fp ;ql) 54

n

Z).

Note that

the first

0

and P1

Let

(Ep,ql n

4(Epnq, Z)

-y(Ep,q)

'n

'Y(Ep,q)

-y(E , p

q

and pi

)

in

the

(Ep" 'q')

F*7(Epn,,ql) 1

1*

=

multiple

a non-zero

Then,

n

Then pi

way.

same

is

0*

be such that

E H

'Y(Ep,q)[Ep,q] Define

n

=

(Epnq) is a of -Y(Epn,,q,).

7(gq)' Tn'

X (CP U eo)[Ep,,ql (^Y(Ep1,q1) ;n I ('Y (Ep, q' ) U eo) [Ep, q X CP 1 'f (Epq) U (neo + 0) [Epq X (CP

=

F*

=

r n

r n

n.

Hence

n

=

1.

F*

Furthermore,

(eo)

(eo +0)2 Therefore,

2eOO

=

0 and C1

02

(Epq n

=

=

=

0. Then

2eoO

+

02

0

0, consequently,

X

CP1

C1

=

(Epq) n T

0. =

+

0.

2eO.

nonzero

Thus

multiple

of

Gang

310

Tian

(3-14)

By (3.13),

(3.15),

and

F*

classes

primitive

F sends

However,

oxF. where X

q'

<

InAl

p'.

=

En p

q

,

V)

such that

is

and

nonzero

F,, This

implies

=

to

=

(Ap)

=

so

=

E,

Aq(=pA) and

p

are

T

classes,

OnYA,J,l q

are

Ap,

=

(A)

F,,

Suppose

the first and

so

=

A and

(1, a)

(CP1

x

,

Ap,, F,, (Aq)

and the

Aq, Aq'

.

that

q < p and

second class

of

Hence

-

that P=P1, q

We finish

( Epq)* ;

.1

primitive

(1, a)

(nA),I,l

CP1 and Y

x

Ap(= qA)

Then

(c, (Ep,,q,))

proof

the

=

q1.

of Theorem 3.3.

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[Wil]