Superstrings and Related Matters : Proceedings of the ICTP Spring Workshop, Trieste, Italy 27 March - 4 April 2000

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Proceedings of the Trieste 2000 Spring Workshop on

UPERSTRINGS AND RELATED MATTERS

Editors

C. Bachas J. Maldacena K. S. Narain S. Randjbar-Daemi

World Scientific

s

UPERSTRINGS AND RELATED MATTERS

the

abdus salam international centre for theoretical physics

united nations educational, scientific and cultural organization

international atomic J o 3 ? S energy agency S k y " ,
s

Proceedings of the Trieste 2000 Spring Workshop on

UPERSTRINGS AND RELATED MATTERS

ICTP, Trieste, Italy 27 March - 4 April 2000

Editors

C. Bachas Ecole Normale Superieure, France

J. Maldacena Harvard University, USA

K. S. Narain and S. Randjbar-Daemi W Italy

V f e World Scientific » •

Singapore • Hong Kong Sinqapore • New Jersey 'London •L

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

SUPERSTR1NGS AND RELATED MATTERS Proceedings of the ICTP Spring Workshop Copyright © 2001 by The Abdus Salam International Centre for Theoretical Physics

ISBN

981-02-4525-4

Printed in Singapore by World Scientific Printers

V

PREFACE

The Abdus Salam ICTP's Spring Workshop, saw a variety of currently interesting topics in string theory. These proceedings contain four sets of lectures presented at this workshop. GaberdiePs lectures contain a comprehensive introduction to the boundary state approach to the D-branes and, in particular, non-BPS D-branes. Examples are given that arise in the context of dualities between string theories. Kachru's lectures deal with warped compactification in 5d theories and the brane world scenarios that emerge. Gauge heirarchy and cosmological constant problems in this context are discussed in detail. Finally these lectures also include string theory constructions that involve D-branes wrapped on various cycles of Calabi-Yau spaces and the corresponding superpotential computations in the world volume gauge theory are described. This is important for both understanding supersymmetry breaking phenomenon as well as understanding the quantum geometry of the Calabi-Yau spaces when the branes are considered as probes. Maldacena's lectures include a description of the holographic correspondence between field theories and string M theory. In particular, the correspondence between string/M theory compactification on AdS spaces and conformal field theories living in the boundary, is discussed in detail. The evidences for this correspondence are described in the context of maximally supersymmetric gauge theories in 4d. It is known that the effective world volume theory on D-branes in the presence of NS-NS antisymmetric tensor field is a non-commutative gauge theory. Nekrasov's lectures give a pedagogical introduction to this subject and, in particular, focus on the non-perturbative solutions such as monopoles and instantons in the non-commutative gauge theories.

vii CONTENTS Preface

v

Lectures on Non-BPS Dirichlet Branes M. R. Gaberdiel

1

Lectures on Warped Compactifications and Stringy Brane Constructions S. Kachru

43

Large N Field Theories, String Theory and Gravity J. Maldacena

74

Trieste Lectures on Solitons in Noncommutative Gauge Theories N. A. Nekrasov

141

1

LECTURES ON N O N - B P S DIRICHLET B R A N E S MR. GABERDIEL Department of Mathematics King's College London Strand, London WC2R 2LS, U. K. A comprehensive introduction to the boundary state approach to Dirichlet branes is given. Various examples of BPS and non-BPS Dirichlet branes are discussed. In particular, the non-BPS states in the duality of Type IIA on K3 and the heterotic string on T 4 are analysed in detail.

1

Introduction

The past few years have seen a tremendous increase in our understanding of the dynamics of superstring theory. In particular it has become apparent that the five ten-dimensional theories, together with an eleven-dimensional theory (M-theory), are different limits in moduli space of some unifying description. A crucial ingredient in understanding the relation between the different perturbative descriptions has been the realisation that the solitonic objects that define the relevant degrees of freedom at strong coupling are Dirichlet-branes that have an alternative description in terms of open string theory. x ' 2 ' 3 The D-branes that were first analysed were BPS states that break half the (spacetime) supersymmetry. It has now been realised, however, that because of their description in terms of open strings, D-branes can be constructed and analysed in much more general situations. In fact, D-branes are essentially described by a boundary conformal field theory 4 ' 5 ' 6 ' 7,8 ' 9 ' 10 ' 11 (see also Refs. 1217 for earlier work in this direction), the consistency conditions of which are not related to spacetime supersymmetry 18 ' 19 ' 20 (for an earlier non-supersymmetric orientifold construction see also Ref. 16). In an independent development, Dbranes that break supersymmetry have been constructed in terms of bound states of branes and anti-branes by Sen 2 1 ' 2 2 ' 2 3 ' 2 4 ' 2 5 ' 2 6 (see also Ref. 27 for a good review). This beautiful construction has been interpreted in terms of K-theory by Witten, 2 8 and this has opened the way for a more mathematical treatment of D-branes. 29 ' 30 ' 31 It has also led the way to new insights into the nature of the instability that is described by the open string tachyon. 32 The motivation for studying D-branes that do not preserve spacetime supersymmetry (and that are therefore sometimes called non-BPS D-branes) is at least four-fold. First, in order to understand the strong/weak coupling dualities of supersymmetric string theories in more detail, it is important to

2

analyse how these dualities act on states that are not BPS saturated. After all, the behaviour of the BPS states at arbitrary coupling is essentially determined by spacetime supersymmetry (provided that it remains unbroken for all values of the coupling constant), and thus one is not really probing the underlying string theory unless one also understands how non-BPS states behave at strong coupling. The dualities typically map perturbative states to non-perturbative (D-brane type) states, and thus one will naturally encounter non-BPS D-branes in these considerations. The second motivation is related to the question of whether string duality should intrinsically only apply to supersymmetric string theories, or whether also non-supersymmetric theories should be related by duality. This is certainly, a priori, an open question" it is conceivable that spacetime supersymmetry is a crucial ingredient without which there is no reason to believe that these dualities should exist, but it is also conceivable that spacetime supersymmetry is just a convenient tool that allows one to use sophisticated arguments and techniques to verify conjectures that are otherwise difficult to check. Dirichlet branes play a central role in the understanding of string dualities, and if one wants to make progress on this question, it is important to develop techniques to analyse and describe Dirichlet branes without reference to spacetime supersymmetry. Thirdly, one of the interesting implications of the Maldacena conjecture 39 is that one can obtain non-trivial predictions about field theory from string theory. In the original formulation this was applied to supersymmetric string and field theories, but it is very tempting to believe that similar insights may be gained for non-supersymmetric theories. This line of thought has been developed recently, starting with a series of papers by Klebanov & Tseytlin. 20 Finally, non-BPS D-branes offer the intriguing possibility of string compactifications in which supersymmetry is preserved in the bulk but broken on the brane. Various orientifold models involving branes and anti-branes have been constructed recently, 40,41 ' 42,43 but it is presumably also possible to construct interesting models involving non-BPS D-branes. (Non-BPS -branes in Type II theories have recently been considered in Refs. 44 and 45.) The fact that at specific points in the moduli space their spectrum is Bose-Fermi degenerate may be of significance in this context. 46 The main aim of these lectures is to explain the boundary state approach to D-branes, and to give some applications of it, in particular to the construction of non-BPS D-branes. The structure of the lectures is as follows. In section 2 we explain carefully the underpinnings of the boundary state approach and "Recently, some suggestive proposals have however been made. 1 8 ''

3

apply it to the simplest case, Type IIA/IIB and Type OA/OB. In section 3 we use the techniques that we have developed to construct one of the simplest non-BPS D-branes — the non-BPS D-particle of the orbifold of Type IIB by ( — \)FLI4 — in detail. If we compactify this theory on a 4-torus, it is T-dual to Type IIA at the orbifold point of K3, which in turn is S-dual to the heterotic string on T 4 . This connection (and in particular the various non-BPS states in this duality) are analysed in detail in section 4. The present paper is a shortened version of the lecture notes of Ref. 47. 2

The boundary state approach

Suppose we are given a closed string theory. We can ask the question whether it is possible to add to this theory additional open string sectors in such a way that the resulting open- and closed theory is consistent. The different open string sectors that we can add are characterised by the boundary conditions that we impose on the end-points of the open strings. Conventional open strings have Neumann boundary conditions at either end; if we denote by X^(t, s) the coordinate field, where t G IR and s G [0, n] are the time and space coordinates on the world-sheet of the open string, then this is the condition that dsX»{t,0)=dsX»(t,x)

=0.

(1)

We can also consider open strings whose boundary condition at one or both ends is of Dirichlet type, i.e. X"{t,Q)=av,

(2)

where av is a constant. Finally, we can consider open strings that satisfy Neumann boundary conditions for some of the X M , and Dirichlet boundary conditions for the others

a,x"(t,o) = 0 X"(t,0)

=

av

/x = o,...,p v=p

+ l,...,9.

{>

The endpoint of such an open string is then constrained to lie on a submanifold (a hyperplane of dimension p + 1), whose position in the ambient space is described by a1'; this submanifold is then called the Dirichlet p-brane or Dp-brane for short. The different boundary conditions of the open string are in one-toone correspondence with the different D-branes. We can therefore rephrase the above question as the question of which D-branes can be consistently defined in a given closed string theory.

4

The idea of the boundary state approach to D-branes is to represent a D-brane as a coherent (boundary) state of the underlying closed string theory. The key ingredient in this approach is world-sheet duality that allows one to rewrite the above conditions (that are defined in terms of the coordinate function of the open string) in terms of the coordinate function of the closed string. At first, the coordinate functions of the open and the closed string theory are not related at all: the world-sheet of the open string is an infinite strip, whilst the world-sheet of the closed string has the topology of a cylinder. For definiteness, let us parametrise the closed string world-sheet by r and a, where r G IR is the time variable, and a is a •periodic space-variable a € [0, 2ir] (where a = 0 is identified with a = 2ir). Suppose now that we consider an open string that has definite boundary conditions at either end (and can therefore be thought of as stretching between two not necessarily different D-branes). If we determine the 1-loop partition function of this open string, we have to identify the time coordinate periodically (and integrate over all periodicities). The open-string world-sheet has then the topology of a cylinder, where the periodic variable is t, and s takes values in a finite interval (from s = 0 to s — 7r). Because of world-sheet duality, we can re-interpret this world-sheet as being a closed string world-sheet if we identify t with a (up to normalisation) and s with r. From the point of view of the closed string, the diagram then corresponds to a tree-diagram between two external states; this describes the processes, where closed string states are emitted by one external state and absorbed by the other.

_1_J T

> Figure 1: World-sheet duality

The boundary condition on the ends of the open string become now conditions that must be satisfied by the external states; since we exchange (t, s) with (
= =

0 av\Dp)

fi = 0,...,p v = p+l,...,9,

.v K>

5

and a similar relation for s = T = IT. Here we have assumed that the boundary condition at s = T = 0 corresponds to those of a Dp-bv&ne. It is useful to rewrite these conditions in terms of the modes of the closed string theory. To this end, let us recall that the coordinate field in the closed string theory can be written as X"{T,a)=X£(T

+ a)+X£(T~a),

(5)

where in terms of modes,

n

= ^" + W

XZ

=

e in(T+
1 M+ 1 ,(r _ 2 2 v

a) +

'

*^ 2^

\<e-Mr-a) n

(6) (?)

These modes satisfy the commutation relations

K,<]

= "iJ""^

[<,5^]

=

[aJt.,5-]

=

0

(8) mS^Sm,-n.

In terms of modes the conditions (4) then become p»\Dp) = 0 /x = 0 , . . . , p (9) « + 5^)|£p) = 0 /x = 0 , . . . , p ( < - a l „ ) |£>p) = 0 i/ = p + l , . . . , 9 xv\Dp) = av\Dp) i/ = p + l , . . . , 9 . The boundary conditions for the fermions are more difficult to establish. Ultimately they are determined by the condition that the closed string tree diagram reproduces, upon world-sheet duality, the open string loop diagram (see (31) - (34) below). In order to formulate the appropriate condition, it is necessary to introduce an additional parameter r\ (that corresponds to the different spin structures), and the relevant equations are then (^+i^r)|Bp,r?) > _ < (l#-M^-r)|£P.»7>

=

0

=

0

M

= 0,...,p 10

I/=p+l,...,9.

The actual D-brane state \Dp) is then a linear combination of the boundary states \Bp, rj) with r\ = ± . It is also worth pointing out that the equations can be solved separately for the different closed string sectors of the theory (i.e.

6

the NS-NS and the R-R sector, as well as the corresponding twisted sectors if we are dealing with an orbifold theory). We shall therefore, in the following, usually denote by \Bp, 77) the solution in a specific sector of the theory; the D-brane state is then a certain linear combination of the boundary states in the different sectors and with 77 = ± . In the following we shall always work in the NS-R formalism; we shall also, for simplicity, work in light-cone gauge, and we shall always choose the two light-cone directions to be fx — 0, l.6 The boundary conditions in both light-cone directions will be taken to be Dirichlet; the boundary states we describe are therefore really D-instantons (i.e. they satisfy a Dirichlet boundary condition in the time direction). However, by means of a double Wick rotation, these states can be transformed into states whose boundary conditions are specified as above. 7 In these conventions we necessarily restrict ourselves to D-branes with at least two Dirichlet directions; thus we can only describe Dpbranes with —1 < p < 7. Also, since the two light-cone directions are always Dirichlet, only p — 2 of the transverse directions satisfy a Dirichlet boundary condition in order for the state to describe the Wick rotate of a Dp-br≠ thus in these conventions the boundary states that combine to define a Dp-brane are characterised by the following conditions p*\Bp,ri) ( < + 2£ n )|Bp,T,> {avn-av_n)\Bp,rj) _ x"\Bp,ri)

= = = =

0 0 0 a"\Bp,ri)

/i n i/ v

= 2,...,p + 2 = 2,...,p+2 = p + 3,...,9 = 0, l,p+ 3 , . . . ,9

(W+irrtt^Bp^)

=

0

n =

[tf - irnli'Lr) \Bp,r,)

=

0

v = p + 3,... ,9.

(11)

2,...,p+2

It is actually not difficult to describe the solution to these equations. In each (left-right-symmetric) sector of the theory, and for each choice of 77, the unique solution is of the form \Bp,a,r,)=Af

f J

]J

dk»eik"a"\B^r)),

(12)

i/=0,l,p+3,...,9

where M is a normalisation constant that will be determined further below, and \Bp, k,rj) is the coherent state

—

f I f,

\n>0 6

r 1 rf

L

fl

p+2 p —

M=2

1

9

(13)

77.

v=p+Z

For a good introduction to the covariant approach see the lecture notes by Di Vecchia and Liccardo. 4 8

7 p+2

|Bp,M) (0)

+lTlY, r>0

The ground state is a momentum eigenstate with eigenvalue k, where &;M = 0 for JJL = 2 , . . . ,p + 2; in the NS-NS sector, it is the unique tachyonic ground state, whereas in the R-R sector, it is determined by the condition (10) with r = 0, i.e. ^+iV^)\Bp,k,v)nl

= 0

/i = 2 , . . . , p + 2

(^-ii7^)|Bp,kJJ7)^

= 0

i/=p + 3,...,9.

If the theory under consideration is an orbifold theory (such as the theory we shall discuss later), there are also similar boundary states in the corresponding twisted sectors. The actual D-brane state is then a certain linear combination of these states in the different sectors of the theory and for both values of 77; it is characterised by three properties: 18 ' 34 (i) The boundary state only couples to the physical sector of the closed string theory, i.e. it is GSO-invariant, and invariant under orbifold and orientifold projections where appropriate. (ii) The open string amplitude obtained by world-sheet duality from the closed string exchange between any two boundary states constitutes an open string partition function, i.e. it corresponds to a trace over a set of open string states of the open string time-evolution operator. (iii) The open strings that are introduced in this way have consistent string field interactions with the original closed strings. One is usually also interested in D-branes that are stable; a necessary condition for this is that the spectrum of open strings that begin and end on the same D-brane is free of tachyons. If the underlying theory is supersymmetric, one may sometimes also want to impose the condition that the D-branes preserve some part of the supersymmetry, and that they are therefore BPS saturated; this requires that the spectrum of open strings beginning and ending on the D-brane is supersymmetric. However, there exist interesting D-branes in supersymmetric theories that are stable but not BPS; 22,23,19,28,49,50,27,51,52,53 some examples of these will be described later. The first condition is usually relatively easy to check, although it requires care in all sectors that have fermionic zero modes. (We shall describe the relevant subtleties in some detail for the case of Type IIA and Type IIB in

8

subsection 2.2.) The second condition is in essence equivalent to the statement that world-sheet duality holds. It is a very powerful constraint that determines the normalisations of the different boundary states (as we shall show in the next subsection). This condition can be formulated in terms of the conformal field theory and is sometimes referred to as Cardy's condition (see also subsection 2.4). The third condition is very difficult to check in detail; as far as I am aware, there is only one example (namely the two Type 0 theories) for which it seems to imply constraints that go beyond (i) and (ii). The set of boundary states which satisfies these conditions forms a lattice. This follows from the fact that if the set S = {|-D)i, \D)2,. • •} satisfies these conditions, then so will the set of boundary states that contains in addition to the elements of S any integer-valued linear combination of \D)\, \D)2,.... When we talk about the D-branes of a theory, what we really mean are the basis vectors of this lattice, from which every D-brane of the theory can be obtained as an integer-valued linear combination; this is what we shall determine in the following. In general, a given theory can have different lattices of mutually consistent boundary states that are not consistent relative to each other. In this case, condition (iii) presumably selects the correct lattice of boundary states. (This is at least what happens in the case of the Type 0 theories.) Finally, it should be stressed that the above conditions are intrinsic consistency conditions of an interacting string (field) theory; in particular, they are more fundamental than spacetime supersymmetry, and also apply in cases where spacetime supersymmetry is broken or absent. 2.1

World-sheet duality

Before describing some examples in detail, it is useful to illustrate the condition of world-sheet duality more quantitatively (since the same calculation will be needed for essentially all models). The closed string tree diagram that is represented in Figure 1 is described by /*oo

dl (Dq\e-lH< \Dp),

/

(15)

Jo where Hc is the closed string Hamiltonian in light cone gauge, Hc

=

7rk2 + 2TT

^ li=2,...,9

+2irCc.

oo

r>0

(16)

9

The constant Cc takes the value - 1 in the NS-NS, and 0 in the R-R sector. Under the substitution t = 1/2/, this integral should become the open string one-loop amplitude that is given by |°°^Tr(e-2^P),

(17)

where V is an appropriate projection operator, and H0 is the open string Hamiltonian given as 1 H0 = nf + ^w2+n

Y, fi=2,...,9

°° [Y,a-n< + Y.r^rm n=l

+ *C0.

(18)

r>0

Here p denotes the open string momentum along the directions for which the string has Neumann (N) boundary conditions at both ends, w is the difference between the two end-points of the open string, and a£ and ijj^ are the bosonic and fermionic open string oscillators, respectively; they satisfy the commutation relations K , < ] = mJ'"'Jm,_n

W,rs}

= ^vSTt.B.

(19)

For coordinates satisfying the same boundary condition at both ends of the open string (i.e. both Neumann (N) or both Dirichlet (D)) n always takes integer values, whereas r takes integer (integer + | ) values in the R (NS) sector. On the other hand, for coordinates satisfying different boundary conditions at the two ends of the open string (one D and one N) n takes integer+| values and r takes integer + | (integer) values in the R (NS) sector. The normal ordering constant C0 vanishes in the R-sector and is equal to — | + | in the NS sector (in a' = 1 units) where s denotes the number of coordinates satisfying D-N boundary conditions. The trace, denoted by Tr, is taken over the full Fock space of the open string, and also includes an integral over the various momenta. The calculation (15) can be performed separately for the different boundary states in the different components since the overlap between states from different sectors vanishes. For definiteness let us consider one specific example in some detail, the tree exchange between two Dp-bi&ne boundary states in the NS-NS sector. (The result for the other sectors will be given below.) Thus we want to consider the amplitude

/

dl (Bp, a x , n\e-lH'

\Bp, a2,77>NS,NS,

(20)

10

where \Bp, a,rj)NS.NS is the coherent state in the NS-NS sector given in (12). The momentum integral gives a Gaussian integral that can be performed, and the amplitude becomes0 J C N S / t f l r ^ e — = 1 ^ a - (Bp,0,»7|e-' if «|Bp,0 ) »j) NS . NS . (21) Jo In order to determine the overlap between the two coherent states, we observe that the states of the form

n^(^-n.^,)''|0),

(22)

where n; > Jii+i and if n* = n^+i then fii < fii+i, form an orthonormal basis for the space generated by the modes a£5£ and similarly for

n^(^~V,^V,)''|0).

(23)

i

Here we have used that the bilinear inner product is defined by the relation

W4>\x) = -<0K„x>

(M\x) = »<0|#t„x>,

(24)

and similarly for a^ and ^ together with the normalisation |0)||0)) = 1.

(25)

It is then easy to see that the above amplitude becomes r->

^s-NSy

f°°

9-r

(»l-»2) 2 / 1 ( g )

dll—re

^Jf(jfy

,

,

(26)

where q — e~27r/, and the functions fi are defined as in Ref. 4 oo

/i(9) = g - n ^ - ^ " ) n=l

/2fo) = V2q^f[(l

+ q2n)

n=l oo

/3(g) = s-^n^+s2""1) n=l OO

/4(g) = f ^ n t 1 - ^ 1 ) '

(2?)

c T h e amplitude is bilinear in the external states, and the prefactor is therefore A/"2 rather than A/OV. On momentum eigenstates the amplitude satisfies (/cilfo) = S(ki + £2).

11

Next we substitute t = 1/21, and using the transformation properties of the fi functions,

Me-*'*)

= Me-"*)

ftie-"'*)

=

(28)

h ^ ) ,

the above integral becomes 2 Af2C , N S ^p 2 ^ f°° £litt - *1411 F V * i= a4 ^? ^t ±/|(g) |W NS NS " ' 2t /f(g)

(29)

where q = e "*. This is to be compared with the open string one-loop amplitude

f°° dt Jo ^

Ns(e

2tH

_ vP+1 r°° *

iz±a w , mi 8

{t)

A ®'

*

^WY^h

(m

(30)

where Vp+i is the world-volume of the brane, which together with the factor of (2t) 2~~ comes from the momentum integration. Thus we find that J dl (Bp, q| e-'*« |5p, v),s-NS = AAN2,NS ^

^

/ | TrNS [e"'*-] .

(31)

Similarly we have Jdl(Bp,V\e-,H'

\Bp, -r,U-»s

=^

2 N

,

N S

^ ^

j |

TrR [ e ^ ] , (32)

F

„~tH0

(33) and

=

0.

(34)

We learn from this that we can satisfy world-sheet duality provided we include appropriate combinations of boundary states and choose their normalisations correctly. We have now assembled the necessary ingredients to work out some examples in detail.

12

2.2

A first example: Type IIA and IIB

Let us first consider the familiar case of the Type IIA and Type IIB theories. The spectra of these theories is given by IIA : IIB:

(NS+, NS+) © (R+, R - ) © (NS+, R - ) © (R+, NS+) (NS+,NS+)©(R+,R+)ffl(NS+,R+)ffi(R+,NS+),

where the signs refer to the eigenvalues of ( — 1) F and ( — 1) F , respectively. In particular, the NS-NS sector is the same for the two theories, and consists of those states for which both (—1)F and ( - 1 ) F have eigenvalue + 1 . Given that the tachyonic ground state has eigenvalue —1 under both (—1)F and (—1)F, the boundary state given by (12) and (13) transforms as (-l)F\Bp,a.,r])NS.NS F

(-l) \Bp,a,r))NS^s

=

-\Bp,a,-7j)NS_NS

=

-\Bp,a,-r))NS_NS.

Thus \Bp, a) NS . NS = (\Bp, a, +) N S .NS - \Bp, a, -) NS . N s)

(36)

is a GSO-invariant state for all p. It follows from (31) and (32) that this state does not describe a stable D-brane by itself since the open string that begins and ends on \Bp, a) NS . NS consists of an unprojected NS and R sector, and therefore contains a tachyon in its spectrum. In fact (36) with ^ S - K S ( ^ ) = l4J^T

(37)

describes the unstable Z)p-brane for p odd (even) in Type IIA (IIB) that was considered by Sen in his construction of non-BPS D-branes; 26 ' 27 the unstable Z)9-brane of Type IIA was also used by Hof ava in his discussion of the K-theory of Type IIA. 29 In order to obtain a stable D-brane, we have to add to (36) a boundary state in the R-R sector; since the R-R sector involves fermionic zero modes, the discussion of GSO-invariance is somewhat delicate, and we need to introduce a little bit of notation. Let us define the modes

^

= -L(<±

?

^),

(38)

which satisfy the anti-commutation relations

{V£,^} = 0

{^,^}

= 6^,

(39)

13

as follows from the fact that both the left- and right-moving fermion modes satisfy the Clifford algebras,

{w,r.}

= o

(40)

In terms of ip±, (14) can be rewritten as rl)$\Bp,k,Ti)(J! ^%|Bp,M>&

= =

0 0

/i = 2 , . . . , p + 2 !/ = p + 3 , . . . , 9 .

(41)

Because of the anti-commutation relations (39) we can define p+2

9

\Bp,k,+)£l=l[r+ II C l ^ k , - ) ^ , /t=2

(42)

i/=p+3

and then it follows that p+2

| B p , k , - ) i ° U n ^ I I r+\BP,k,+){°l.

(43)

On the ground states the GSO-operators take the form 9

(-if=n

9

(V2^)=n (^+r-), fj=2

/i=2

9

9

(44)

and

?

(-i) =n(v^^)=n^+-^-)fj=2

( 4s )

^=2

Taking these equations together we then find that (-l)^|£p,M>& F

{-l) \Bp,-k,r))
=

\Bp,k,-r,)£l

(46)

=

p+1

(47)

(-l)

|Bp,k,-»,>W.

The action on the non-zero modes is as before, and therefore the action of the GSO-operators on the whole boundary states is given by (-l) F |£p,a,7?) R . R F

(-l) |Bp,a,j 7 > R . R

=

|Bp, a , - J J V R

(48)

=

( - l ^ + ^ a , - ^ .

(49)

14

It follows from the first equation that the only potentially GSO-invariant boundary state is of the form \Bp, a) R . R = (\Bp, a, +) R . R + \Bp, a, - ) n . R ) ,

(50)

and the second equation implies that it is actually GSO-invariant if p is even (odd) in the case of Type IIA (IIB). In this case we can find a GSO-invariant boundary state |r>p,a) = |£p,a> NS-NS + |£?p,a) R . R . (51) This state satisfies world-sheet duality provided we choose

This gives rise to an open string consisting of a GSO-projected NS and R sector; in particular, the GSO-projection removes the open string tachyon from the NS sector, and the D-brane is stable. The D-brane is also BPS since the open string spectrum is supersymmetric. Actually the condition of world-sheet duality does not specify the relative sign between the NS-NS and the R-R component in (51) since only the square of the normalisation constant enters the calculation:d The opposite choice of the sign defines the anti-brane that is also BPS by itself; however, the combination of a brane and an anti-brane breaks supersymmetry since the two states preserve disjoint sets of supercharges. This can also be seen from the present point of view: the open string that stretches between a brane and an anti-brane consists of a NS and a R-sector whose GSO-projection is opposite to that of the brane-brane or anti-brane-anti-brane open string. 54 In particular, the open string tachyon from the NS sector survives the projection; the system is therefore unstable, and certainly does not preserve supersymmetry. It is also possible to analyse the action of the supercharges on the boundary states directly. 6 We have seen so far that the Type IIA (IIB) has stable BPS D-branes for p even (odd); we have also seen that the theory has unstable D-branes for all values of p. However, these unstable £>p-branes are only independent states if p is odd in IIA (and p is even in IIB). In order to see this, we observe that the normalisation of the NS-NS boundary state in (37) is only correct if p is odd (even) in IIA (IIB). Indeed, if (37) also held for p even (odd) in IIA (IIB), the tree-diagram involving the unstable JDp-brane and the BPS £>p-brane would d

I t also, obviously, does not specify the overall sign, but this is just the familiar ambiguity in the definition of quantum mechanical states.

15

give rise to an open string that consists of - ^ (NS - R)

(53)

and therefore violates (ii) above. The actual normalisation of (37) for p even (odd) in IIA (IIB) is therefore

^«s{Dp) = ±$fa-

(54)

This implies that the boundary state of the unstable Dp-brane is the sum of the boundary states of the BPS Dp-brane and the BPS anti-Dp-brane; it therefore does not define an additional basis vector of the lattice of D-brane states. Finally, we should like to stress that the above analysis shows not only that Type IIA and Type IIB has BPS Dp-branes for the appropriate values of p, but also that these are the only stable D-branes of Type IIA and Type IIB. This is not necessarily the case — as we shall see below, some theories possess stable D-branes that are not BPS. 2.3

A second example: Type OA and OB

As a second example let us examine the D-brane spectrum of Type OA and Type OB. 18 ' 20 ' 55 - 34 These theories can be obtained from Type IIA and IIB as an orbifold by ( —1)F", where Fs is the spacetime fermion number. The effect of (—1)F in the untwisted sector is to retain the bosons (i.e. the states in the NS-NS and R-R sectors) and to remove the fermions (i.e. the states in the NS-R and R-NS sectors). In the two remaining sectors, the GSO projection acts in the usual way 1

+ ( " ! ) F ) f1 + ( - l / ) ~

NS-NS:

PGSO,U = H

R-R:

PGso,c/ = H 1 + ( - 1 ) F ) ( 1 ± ( - 1 ) F ) .

(55)

where the + sign corresponds to Type IIB, and the — sign to Type IIA. In the twisted sector, the effect of ( — 1)F' is to reverse the GSO projection for both left and right-moving sectors. In addition only the states invariant under ( - 1 ) ^ (i.e. the bosons) are retained. Thus the states in the twisted sector are again in the NS-NS and the R-R sector, but their GSO projection is now = H

1

" ( - ! ) F ) f1 " ( - ! ) F ) ) ~(

NS-NS:

PGSO,T

R-R:

PGSO,T=\{1-(-1)F){1*(-1)F)

(56) ,

16

where now the - sign corresponds to Type IIB, and the + sign to Type IIA. Taking (55) and (56) together, we can describe the spectrum of Type OA and Type OB more compactly as the subspaces of the NS-NS and R-R sectors that are invariant under the GSO-projection NS-NS: R-R:

PGSO = \ (1 + ( - 1 ) F + F ) ~ PGso = $(l±{-l)F+F).

(57)

The resulting spectrum is thus given by OA : (NS+,NS+) © ( N S - . N S - ) © ( R + . R - ) © ( R - , R + ) OB:

(NS+,NS+)ffi(NS-,NS-)ffi(R+,R+)ffi(R-,R-).

The NS-NS sector is the same for the two theories: in particular, the low lying states consist of the ground state tachyon (that is invariant under (57) since it is invariant under (56)), and the bosonic part of the supergravity multiplet, i.e. the graviton, Kalb-Ramond 2-form, and dilaton. On the other hand, the R-R sector is different for the two theories (as is familiar from Type IIA and Type IIB). There are no tachyonic states, and the massless states transform as OA :

(8 S ® 8C) © (8 C 8S) = 2 • 8 V + 2 • 56

OB :

(8 S ® 8S) © (8 C ® 8C) = 2 • 1 + 2 • 28 + 70 .

(59)

In the case of Type OA, the theory has two 1-forms and two 3-forms in the R-R sector, whereas Type OB has two scalars, two 2-forms, and a 4-form (with an unrestricted 5-form field strength). The states in the R-R sector of Type OA and Type OB are therefore doubled compared to those in Type IIA and Type IIB. One may therefore expect that the D-brane spectrum of these theories is also doubled. In the NS-NS sector, each boundary state \Bp, a, r/) is by itself GSOinvariant; the most general GSO-invariant boundary state in the NS-NS sector is therefore \Bp, a)NS-Ns = a+1Bp, a, +}NS-NS + a_ \Bp, a, -) N S _ N S •

(60)

If a+a- 7^ 0, it follows from (32) that the open string that begins and ends on the same boundary state contains spacetime fermions. Since the closed string sector only consists of bosons, this presumably means that the openclosed vertex of the string field theory cannot be consistently defined; thus

17

condition (iii) suggests that we have to have a+a- = 0.e Thus there are two consistent NS-NS boundary states, and they are given by \Bp, a, +)NS_NS and \Bp, a, -) N S ^NS- As before, neither of them is stable since the open string that begins and ends on this state has a tachyon from the unprojected open string NS sector. In oder to stabilise the brane, we have to add a boundary state in the R-R sector, but as before, these are only GSO-invariant provided that p is even (odd) for Type OA (OB). For each such p we are then left with four different D-brane states (together with their anti-branes) \Dp, a, r], r]') = \Bp, a, 7?}NS.NS + \Bp, a, r/) R . R ,

(61)

where

However not all of these branes are mutually consistent: the open string between the boundary state \Dp, a, +, +) and \Dp, b , —, +) consists of a R-sector together with a NS-sector with a (—1)F insertion, and likewise for \Dp, a, —, —) and \Dp,b, +, —). Thus there are only two mutually consistent D-brane states for each allowed value of p which we can take to be |£>p,a,+,+>

and

\Dp,a,-,-).

(63)

These D-branes have played an important role in recent attempts to extend the Maldacena conjecture 39 to certain backgrounds of Type OB string theory. 20 3

The non-BPS D-particle in

IIB/(-1)FL14

Up to now we have described a general construction of D-branes that does not rely on spacetime supersymmetry. We want to apply this technique now to the construction of stable non-BPS D-branes. From the point of view that is presented in these lectures, the simplest stable non-BPS D-brane is presumably the D-particle of a certain orbifold of Type IIB 1 9 , 2 2 (see also Ref. 72); this shall be the topic of this section. As was pointed out by Sen some time ago, 21 duality symmetries in string theory sometimes predict the existence of solitonic states which are not BPS, e

If the theory actually possesses one brane with a+a- ^ 0, so that the open string is NS-R with the GSO-projection | ( 1 + (—1) F ), the (mutually consistent) lattice of boundary states containing this boundary state has only one stable brane (and anti-brane) for each allowed value of p; this would also seem to be in conflict with the doubled R-R spectrum of the theory.

18

but are stable due to the fact that they are the lightest states carrying a given set of charge quantum numbers. One example Sen considered concerns the orientifold 67-17 of Type IIB by fil4 where I 4 is the inversion of four coordinates. This theory is dual to the orbifold of Type IIB by ( - 1 ) F L I 4 , where FL is the left-moving spacetime fermion number. As we shall explain below, the spectrum of the orbifold contains in the twisted sector a massless vector multiplet of M = (1,1) supersymmetry in D = 6, and this implies that the orbifold fixed-plane corresponds, in the dual orientifold theory, to a (mirror) pair of D5-branes on top of an orientifold 5-plane. 21 Because of the orientifold projection, the massless states of the string stretching between the two D5branes is removed, and the gauge group is reduced from U{2) to SO(2). The lightest state that is charged under the SO (2) is then the first excited open string state of the string stretching between the two D5-branes: in the open string NS-sector the first excited states are

Q

^-3/210) -1^-1/21°)

^ states 8-8 = 64 states

^-1/2^-1/2^-1/21°)

(s) =

(64)

56 states

and in the R-sector the relevant states are 0^183) ^-i|8c)

8-8 = 64 states 8 • 8 = 64 states.

. » [

}

Thus there are altogether 128 bosons and 128 fermions which form a long (non-BPS) multiplet of the M = (1,1) D = 6 supersymmetry algebra. Since these states are stable, one should expect that the dual (orbifold) theory also contains a stable multiplet of states that is charged under this 50(2). However, these states are not BPS, and the corresponding states in the dual theory therefore cannot be BPS D-branes; in fact, as we shall show below, the orbifold theory possesses a stable non-BPS D-particle that is stuck to the orbifold fixed plane and that has all the above properties. 3.1

The spectrum of the orbifold

Let us first describe the orbifold of the Type IIB theory in some detail. For simplicity we shall consider the uncompactified theory, i.e. the orbifold of jf?9,1/( — l) F z -I 4 . Let us choose the convention that I 4 inverts the four spatial coordinates, x6,...,x9. The fixed points under Z4 form a 5-plane at x6 = x7 = x8 = x9 = 0, which extends along the coordinates x2,... ,x5, as well as the light-cone coordinates i ° , x1. In light-cone gauge, type IIB string theory

19

has 16 dynamical supersymmetries and 16 kinematical supersymmetries. The former transform under the transverse 50(8) as Q ~ 8S

Q ~ 8S.

(66)

The orbifold breaks the transverse 50(8) into SO(4)s x S0(4)R, where the 50(4)5 factor corresponds to rotations of (x2,..., x5), and the 5 0 ( 4 ) ^ factor to rotations of (x6,..., x9). The above supercharges therefore decompose as 8 s —> ((2,1), ( 2 , 1 ) ) + ((1,2), (1,2)),

(67)

where we have written the representations of 50(4) in terms of 50(4) ~ 5/7(2) x SU(2). The operator I4 reverses the sign of the vector representation of S0(4)R (the (2,2)), and we therefore choose its action on the 5 0 ( 4 ) # spinors as (2.1) -+ - ( 2 , 1 ) (1.2) -+ (1,2). <™ The action of ( —1) F L is simply ( - 1 ) ^ : Q^-Q

Q->Q,

(69)

and the surviving supersymmetries thus transform as Q ~ ((2,1), (2,1))

Q ~ ((1,2), (1,2)).

(70)

From the point of view of the 5-plane world-volume this is (dynamical, lightcone) Af = (1,1) supersymmetry/ In addition to the untwisted sectors, the theory also contains a twisted sector that is localised at the 5-plane. In the twisted sector the various oscillators are moded as

^-{Li/J;:S:::::S-{z+1/2^::::S Z /i = 2 , . . . , 5 ^- -{L1/2 ;:i;:J - { Z + l / 2 /i = 6 , . . . , 9 . (71) The ground state energy vanishes in both the R and NS sectors, and they both contain four fermionic zero modes that transform in the vector representation of SO (4) s and SO (4) R, respectively. Consequently the twisted NS-NS and R-R ground states transform as ((2,1) + (1,2))® ((2,1) + ( 1 , 2 ) ) , -^The same orbifold of type IIA would yield Af = (2,0) supersymmetry.

(72)

20

where the charges correspond to SO (4) s (SO(4)R) in the twisted R-R (NS-NS) sector. The unique massless representation of D — 6 TV = (1,1) supersymmetry (other than the gravity multiplet) is the vector multiplet ((2, 2), (1,1)) + ((1,1), (2, 2)) + fermions .

(73)

In order to preserve supersymmetry, we therefore have to choose the GSOprojections in all twisted sectors to be of the form Paso,T = \(l-(-l)F)(l

+ (-lf).

(74)

This agrees with what we would have expected from standard orbifold techniques, namely that the effect of ( — 1)FL is to change the left-GSO projection in the twisted sector. In addition, the spectrum of the twisted sector must be projected onto a subspace with either ( — 1)FLX4 = + 1 or { — V)FLX\ = — 1 (in the untwisted sector only + 1 is allowed). Since twisted NS-NS (R-R) states are even (odd) under (—1) FL , and I4 reverses the sign of the vector of 5 0 ( 4 ) a (and leaves the vector of SO(4)s invariant), we conclude that in the present case the twisted sector states are odd under (—1) F L Z4. 3.2

The non-BPS DO-brane

Having described the spectrum and the GSO projections of the various sectors in some detail, we can now analyse whether a D-brane boundary state with the appropriate properties exists. Since the non-BPS state in the orientifold theory is localised at the orientifold plane, one would expect that the corresponding non-BPS D-brane should be a .DO-brane that is stuck to the orbifold fixed plane; we shall therefore analyse in the following whether such a D-brane state exists. For definiteness we shall assume that the DO-brane is oriented in such a way that it satisfies a Neumann boundary condition along the x2 direction. In the (untwisted) NS-NS sector the action of ( — 1)FL is trivial, and I4 acts on the boundary state given in (36) as li\Bp,a)NS_NS

= |Bp,I 4 a) N S . N S ,

(75)

since I4 acts in the same way on left- and right-movers. If a = ao lies on the fixed plane, Xiao = ao, and the boundary state is invariant. Thus we have a physical p = 0 NS-NS boundary state |[/0,ao) = |B0,ao>

(76)

21

On the other hand the p = 0 R-R boundary state is not physical since, as we saw in section 2.2, it is not invariant under the GSO-projection.9 In the twisted sector, the boundary state is of the same form as described before, except that now the moding of the different fields is as described in (71). Since there are only bosonic zero modes for /i = 0,1, 2,3,4, 5, and since x2 is a Neumann direction, the position of the .DO-brane boundary state is described by a 5-dimensional vector b that can be identified with ao- Both the twisted NS-NS and the twisted R-R sector contain fermionic zero modes, and the ground state of the .DO-brane boundary state therefore has to satisfy xl>»\B0, ao, -V)(°LS,T

= 0

for v = 6,7,8,9,

(77)

in the twisted NS-NS sector, and

%\B0,-T))%)K.T

=

0

for i/ = 3,4,5,

in the twisted R-R sector. On the ground states, the GSO operators act as twisted NS-NS : twisted R - R :

(-I)* 1 = Ul=6(V2^) (-l)f = n

U

«

(-if

=

( - l )

Y\l=^<) ?

= n U ^ « (79)

Using the same arguments as before in section 2.2 we then find (-1) |-BO,a0,77)NS_NS,T

=

|B0,ao,-f?)Ns-Ns,T

(-l) F |jB0,ao,Jj) NS . NS , T

=

+|-B0,a o ,-r?) N S . N S ] T ,

(-1)^150, a0,jy)R.R,T

=

|-B0,ao,-7?)R.R,T

=

-|B0,ao,-?7)R.R,T.

(80)

and

F

(-l) |B0,a o ,7 ? ) R . R , T

(81)

Because of (74) it then follows that only the combination |T0, a 0 ) = (|S0, a 0 , +) R . R , T - \B0, a 0 , -) R . R , T )

(82)

in the twisted R-R sector survives the GSO-projection, and that no combination of twisted NS-NS sector boundary states is GSO invariant. In addition, 9 T h i s boundary state is actually also not invariant under ( — 1 ) ^ 1 4 , as follows from the analysis of Ref. 22.

22

the ground states of the twisted R-R sector boundary state are odd under (—1)FLI4, as they are precisely the vector states of SO(4)s that arise in the twisted sector. We therefore have one further physical boundary state, and the total D-particle state is of the form l-DO, ao> - \U0, a 0 ) + |T0, a 0 >.

(83)

We can then determine the cylinder diagram for a closed string that begins and ends on the D-particle, and we find that

L

{D0,ao\e~1Hc\D0,ao)

dl -

rjLl9*/2sf2 3/2

~h

t \

NSNS

/ K g ) " /1(g) , Q1/2V-2 fi(q)fl(q)\ + T

/f®

™' fi(
,M, m

where /; is defined as in (27). Thus if we choose

^-^'-iaisj

A

«"'fi6>—5(^-

<85)

we obtain (compare Ref. 22) j dl 0 b , a o | e - ^ p O , a o ) = ^ j ^ Tr N S . R [i(l + ( - l ) F I 4 ) e ^ 2 t ^ ] - (86) The open string spectrum thus consists of a NS and a R sector, and both are projected by | ( 1 + ( — l)FXi). The tachyon of the NS sector is even under Zi but odd under ( — 1)F, and is therefore removed from the spectrum. This indicates that the D-particle is stable. In addition, 4 massless states are removed from the NS sector, leaving 4 massless bosons, and the R sector contains 8 massless fermions. Including the zero modes in the light-cone directions,'1 this gives the D-particle 5 bosonic zero modes and 16 fermionic zero modes. The former reflect the fact that the D-particle is restricted to move within the 5-plane, and the latter give rise to a long (28 = 256-dimensional) representation of the six-dimensional M = (1,1) supersymmetry algebra. Finally, the D-particle is charged under the vector field in the twisted R-R sector. We have therefore managed to construct a boundary state that possesses all the properties that we expected to find from the S-dual description. ''When counting the zero modes of a D-brane one must include the light-cone directions as well as the physical (transverse) massless states of the open string. See for example Ref. 69 for a discussion of the type IIB D-string.

23

Sen has proposed a different realisation for this state as the ground state of a D-string anti-D-string system. 22 In order to describe this construction, it is useful to consider the theory where at least one of the four circles that are inverted by the action of I4 is compact. (This will serve as a preparation for the following section where we consider the T-dual of the configuration where all four circles are compactified.) Let us then consider a Dl-brane anti-Dl-brane pair that wraps around this compact circle, x6, say. In the reduced space, the branes stretch from the fixed point at x6 = 0 to the fixed point at xe = nR6. As we have seen before, the ground state of the open string between the brane and the anti-brane is a tachyon; this indicates that the system is unstable to decay into the vacuum. However, we can consider the configuration where we switch on a 7Z.2 Wilson line on either the brane or the anti-brane. This implies that the tachyon changes sign as we go around the circle, and thus the ground state energy of the open string is given by 9

m

1

1

2 + if

(87)

In particular, the ground state of the open string is only tachyonic if i?6 > N/2; on the other hand, for Re < V% the ground state of the open string is massive, and the brane anti-brane system is stable. As we shall see in the next section, the non-trivial 2 2 Wilson line implies that the twisted R-R charge at the endpoints of the Dl-brane have opposite sign. Thus the combined system of the brane anti-brane pair only carries twisted R-R charge at one end (but not the other); it also does not carry any untwisted R-R charge, and therefore has the same charges as the non-BPS Dparticle that we have just discussed (see Figure 2). This suggests that the brane

Figure 2: Dl-brane anti-brane pair with a relative Z2 Wilson line and the DO-brane.

anti-brane pair decays into the D-particle if R6 > \/2. This interpretation is further supported by the fact that for R6 < -y/2, the open string that begins and ends on the D-particle contains a tachyon, and thus indicates that the Dparticle is not the stable state. Indeed, the projection | ( 1 + ( - 1 ) F I 4 ) removes

24

the tachyon with winding number 0, but the anti-symmetric combination of winding number w = ±1 is contained in the spectrum; this state has mass

and thus becomes tachyonic if RQ < \/2. One can also compare the mass and the R-R charge of the two states, and they agree indeed (for Re = \/2). 4

N o n - B P S states in Heterotic - Type II duality

If we consider the compactification of the above IIB orbifold on a 4-torus (on which I 4 acts) then the theory is T-dual to IIB on

TA/{-\)FL14

<-^-» IIA on T 4 /X 4 .

(89)

The orbifold of T 4 /X 4 describes a special point in the moduli space of K3 surfaces, the so-called orbifold point. On the other hand, Type IIA on K3 is known to be S-dual to the heterotic string on T4. 75 Under T-duality, the stable non-BPS DO-brane of the Type IIB orbifold becomes a stable non-BPS Z)l-brane of Type IIA on K3; it is then natural to ask whether one can identify the corresponding non-BPS state in the heterotic theory. This is actually an interesting problem in its own right since both theories of the dual pair are quantitatively under control, and one can make detailed comparisons. The following discussion follows closely Ref. 40 (see also Ref. 70). 4-1

The setup

Let us first explain the conventions of the orbifold of the Type IIA theory. In the untwisted sector, the GSO-projections are given as in (35). If we denote the compact coordinates along which I 4 acts by x6,..., x9, the moding of the fields in the twisted sectors is as in (71). Furthermore, the GSO-projections in the relevant twisted sectors are given by twisted NS-NS twisted R-R

\ (l - (-1)^) (l - (-1)?)

(90)

\ (l - (-1)F) (l + (-1)^) •

(91)

Since the theory has D = 6 N = (1,1) supersymmetry, the states in the massless R-R sector must form a vector, and thus the GSO-projection must

25

be the same as for the T-dual I I B / ( - l ) F i I 4 orbifold. Consistency with the operator product expansion, in particular the OPE R-R x R-R,T = NS-NS,T

(92)

then determines the GSO-projection in the twisted NS-NS sector; in fact, since the GSO-projection of Type IIA and Type IIB are opposite in the untwisted R-R sector, the same must hold in the twisted NS-NS sector.' Next let us recall the precise relation between type IIA at the orbifold point of K3 and the heterotic string on T 4 ; the following discussion follows closely Ref. 73. Let us denote the radii of the compactified coordinates by R-Ai and Rhi for Type IIA and the heterotic string, respectively. The sequence of dualities relating the two theories is given by het T

4

- A l T

4

X IIB T 4 / 2 2 A

IIB T 4 / Z 2 ' A

IIA T 4 / Z 2 ,

(93)

where the various Z 2 groups are V2 = (l,m4)

Z£ = (l, ( - 1 ) ^ 2 4 )

Z 2 = (1,I4).

(94)

Here I4 reflects all four compact directions, Q. reverses world-sheet parity, and FL is the left-moving part of the spacetime fermion number. The first step is ten-dimensional S-duality between the (SO(32)) heterotic string and the type I string, 74 which relates the (ten-dimensional) couplings and radii aJ gi ixg^1

Rij oc g^l/2Rhj.

(95)

The second step consists of four T-duality transformations on the four circles, resulting in the new parameters g>

=

vr'gi

oc

V-igh

D/

_

R-i

^

i / 2

r

i

(9b)

where Vi = J\j Rij and Vh = J\j Rhj denote the volumes (divided by (27r)4) of the T 4 in the type I and heterotic strings, respectively. This theory has 16 orientifold fixed points. In order for the dilaton to be a constant, the R-R charges have to be canceled locally, i.e. one pair of D5-branes has to be situated at each orientifold 5-plane. In terms of the original heterotic theory, this means 'For a recent discussion of the subtleties associated with the choice of the GSO-projections in the twisted sectors see Ref. 71. •'Numerical factors are omitted until the last step.

26

that suitable Wilson lines must be switched on to break 50(32) (or E% x Eg) to U(l)16; this will be further discussed below. The third step is S-duality of type IIB. The new parameters are given by 9"

9'-1

=

«

R'l Vj = 9g'-^K >-^R'3

Vh9(97)

oc V^Rex Vl'\,1 h]

Finally, the fourth step is T-duality along one of the compact directions, say a;6. The resulting theory is type IIA on a K3 in the orbifold limit. The coupling constants and radii are given by gA RAj RA6

= =

g"{K)-1 R13'

= =

=

1

=

K')"

g^RMVlh'2 l l2 2Vhh R-hu]hj 1 1/2

fori ^ 6

(98)

2- V~ R

where we have now included the numerical factors (that will be shown below to reproduce the correct masses for the BPS-states) .fc In addition, the metrics in the low energy effective theories are related as 7S Gt

= VhgH2G*v .

(99)

The corresponding point in the moduli space of the heterotic theory has B = 0 and Wilson lines that can be determined in analogy with the duality between the heterotic string on S1 and type IIA on S1 /Q.l\ (type IA). A constant dilaton background for the latter requires the Wilson line A = ( ( | ) 8 , 0 8 ) in the former, 76 ' 77 ' 78 resulting in the gauge group 50(16) x 50(16). The sixteen entries in the Wilson line describe the positions of the D8-branes along the interval in type IA. This suggests that the four Wilson lines in our case should be A»

*-iii)Wi)WOWf (i.o, |,o, l.o, l.o, i.o, i,o, 1,0,i,o), ^In our conventions a'h = 1/2, a'A = 1.

(ioo)

27

so that there is precisely one pair of D-branes at each of the sixteen orientifold planes. Indeed, this configuration of Wilson lines breaks the gauge group 50(32) to 5 0 ( 2 ) 1 6 ~ t/(l) 1 6 , and there are no other massless gauge particles that are charged under the Cartan subalgebra of 50(32). To see this, recall that the momenta of the compactified heterotic string are given as 79 PL

=

(PL,PL)

=

[VK+A^VH,

•g^+w'Ri

(101) PR

=

PR

=

| ^ - - w'R,)

,

where p1 is the physical momentum in the compact directions p% = nl + Bl3Wj - VKAiK - ^A^A^Wj,

(102)

•Wi,rii £ TL are elements of the compactification lattice T 4 ' 4 , and VK is an element of the internal lattice T 16 . For a given momentum ( P L , PR), a physical state can exist provided the level matching condition lpl

+ NL-l

= ±P2R + NR-cR

(103)

is satisfied, where NL and NR are the left- and right-moving excitation numbers, and CR = 1/2 (CR = 0) for the right-moving NS (R) sector. The state is BPS if NR = CRS3, and its mass is given by \m2h = Q P | +NL-lj

+ Q P 2 R + NR-

CR}

= P2fi + 2(NR - cR).

(104)

The massless states of the gravity multiplet and the Cartan subalgebra have NL = 1 and P/, = P ^ = 0. Additional massless gauge bosons would have to have NL = 0, and therefore PL = 2 . If Wi = 0 for all i, this requires V2 — 2 and pi = 0. The possible choices for V are then simply the roots of 50(32), and it is easy to see that for each root at least one of the inner products VK AlK is half-integer; thus pl G 2 + 1/2 cannot vanish, and the state is massive. On the other hand, if Wi ^ 0 for at least one i, the above requires (V + Aw)2 < 2, and it follows that V + Aw = 0, i.e. that the massless gauge particle is not charged under the Cartan subalgebra of 50(32). 4-2

BPS states

In order to test the above identification further, it is useful to relate some of the perturbative BPS states of the heterotic string to D-brane states in

28

IIA on T 4 / 2 2 , and to compare their masses. Let us start with the simplest case - a bulk D-particle. This state is charged only under the bulk U(l) corresponding to the ten-dimensional R-R one-form CR_R. It can be described by the boundary state |D0;a,b,ei)

=

(| J B0;a,b) N S . N S +£ 1 |B0;a,b) R . R ) + (\B0; a, -b) N S . N S +

£l

\B0; a, - b ) R . R ) ,

(105)

where a denotes the position along the uncompactified directions for which the D-brane has Dirichlet boundary conditions, i.e. x°, x1, a;3, x4, x5, and b denotes the position along the compacitified directions, x6,... ,x9. Since the directions x6,... ,x9 are compact, the corresponding momenta are quantised, kl = rrii/RAi, and the momentum integrals are replaced by sums; thus the boundary state becomes

|£0, a, b, r,) = N J

[] i/=0,l,3,...,5

dkve^a"

(f[

£

J™*'/** j \BQ, k^n, r,),

\i=6m,GZ

/

(106) where |B0, k, m, 77} is given by the same formula as in (13). The GSO-invariant boundary state |.B0;a,b) is then again given as in (36) and (50). (Since we are dealing with the untwisted sector of a Type IIA orbifold, the R-R sector boundary state with p even is GSO invariant.) The state in (105) is manifestly also invariant under the orbifold operator I4 since it is the symmetric combination of a DO-brane state together with its image under I4. In order to determine the correct normalisation of the different boundary states we have to perform a similar calculation as before in the case of the uncompactified Type IIA and Type IIB theory. There are, however, two minor modifications. Firstly, since the momenta along the four compact directions are quantised, one cannot simply do the Gaussian integral; instead, one is left with a momentum sum that can be transformed, using the Poisson resummation,

J2 e-'(W«)2 = * J2 e^(nfl)2 ;

(107)

into a winding sum which in turn appears in the open string trace.' Secondly, the open string that one obtains from (105) will have four sectors (depending on whether each end of the string is at (a,b) or at (a, —b)), each of which More details on this can be found in Refs. 22 and 46.

29 consists of [NS-R]i(l + (-l)F).

(108)

However, under the action of Z 4 , the four sectors are pairwise identified, and therefore only half as many open string states survive. Taking this into account, the normalisation of the boundary states in (105) t u r n out to be

RA6RA7RASRA9^S-NS RA6RA7RA8RA9-^R.R(D0)

(-DO)

1 1 Vi 2128(27) 1 1 Vi ~28(27) '

(109) (110)

As before, e\ = ± 1 differentiates a D-particle from an anti-D-particle. T h e mass of the D-particle in the T y p e IIA theory is given by m,A(D0) = 1/gA- Using (98) and (99), the mass of the corresponding state in the heterotic theory is therefore mh(D0) = V*g-lmA{DQ) = v}g^l±9A

= -±-

.

(Ill)

-Kh6

This implies t h a t the corresponding heterotic s t a t e has trivial winding (IUJ = 0) and m o m e n t u m (V = 0, pl = 0), except for pe = t\. Level matching then requires t h a t NL = 1, and therefore the state is a Kaluza-Klein excitation of either the gravity multiplet or one of the vector multiplets in the C a r t a n subalgebra. Next we consider the D-particle t h a t is stuck at one of the fixed planes (which we may assume to be the fixed plane with b = 0). T h e mass and the bulk R-R charge of this D-particle is precisely half of t h a t of the bulk Dparticle t h a t we discussed above; it is therefore sometimes called a 'fractional' D-particle. 8 0 It also carries unit charge with respect to the twisted R-R U(l) at the fixed plane. The corresponding b o u n d a r y state is then \D0f, a; ei, e 2 ) = \B0; a) NS _ NS + ej. \B0; a) n _ R + e21J50; a) N S . N S , T + exe2\BQ; a) R . R , T . (112) As we have seen above, the boundary states in the untwisted sector are GSOand orbifold-invariant. As regards the b o u n d a r y states in the twisted sectors, the analysis is completely analogous to the analysis of the previous section, t h e only difference being t h a t the GSO-projection in the twisted NS-NS sector is now opposite t o what it was there; as a consequence the DO-brane b o u n d a r y state is also GSO-invariant in t h a t sector.

30

The normalisation of the boundary states in the untwisted sector is as for the case of the bulk DO-brane above, RA6RA7RASRA<>Ks-.s(D0f)

=

\ ^ j ^

(113)

RA6RA7RAsRA9^lR(D0f)

=

- \ \ ^ y

(114)

and in the twisted sectors it is •A&.NS.T^O/)

= \ ^

^lR,r(D0f)

= - \ ^

•

(115)

With these normalisations, the open string between two such D-particles is given by [NS - R] 1 ( l + e i e i ( - l ) F ) ( l + ^ 1 , ) .

(116)

If we consider the limit of the bulk DO-brane state as b -» 0, i.e. as the bulk Dparticle approaches the fixed plane, the normalisation of the boundary states of the bulk brane is indeed twice that of the corresponding components of the fractional brane. This demonstrates explicitly that the mass and the untwisted R-R charge of the bulk brane is indeed twice that of the fractional brane. As before, €\ = ±1 and eiC2 = ± 1 determine the sign of the bulk and the twisted charges of the fractional brane, respectively. In the blow up of the orbifold to a smooth K3, the fractional D-particle corresponds to a D2-brane which wraps a supersymmetric cycle. 81 In the orbifold limit the area of this cycle vanishes, but the corresponding state is not massless, since the two-form field B^ has a non-vanishing integral around the cycle. 82 In fact B = 1/2, and the resulting state carries one unit of twisted charge coming from the membrane itself, and one half unit of bulk charge coming from the D2-brane world-volume action term JdzaCp]_R A (F^2^ + B^). At each fixed point there are four such states, corresponding to the two possible orientations of the membrane, and the possibility of having F = 0 or F = ±1 (as F must be integral, the state always has a non-vanishing bulk charge). These are the four possible fractional D-particles of (112). Since there are sixteen orbifold fixed planes, there are a total of 64 such states. In the heterotic string these correspond to states with internal weight vectors of the form y = el£2(02",l,±l,014-2")

(n = 0 , . . . , 7 ) ,

(117)

and vanishing winding and internal momentum, except for pe = ei/2. The sixteen twisted U(l) charges in the IIA picture correspond to symmetric and

31 anti-symmetric combinations of the (2n 4- l ) ' s t and (2n + 2)'nd C a r t a n U(l) charges in the heterotic picture. It follows from the heterotic mass formula (104) t h a t the mass of these states is ™h{D0f)

=-±-.

(118)

As before, this corresponds to the mass mA(D0f) = V-1/2ghmh(DQf) = ~ , (119) ^9 A in the orbifold of type IIA, and is thus in complete agreement with the mass of a fractional D-particle. Additional B P S states are obtained by wrapping D2-branes around nonvanishing supersymmetric 2-cycles, and b y wrapping D4-branes around the entire compact space. One can compute t h e mass of each of these states, a n d t h u s find t h e corresponding s t a t e in t h e heterotic string. Let us briefly summarise the results: (i) A D2-brane t h a t wraps the cycle {x%,x^) where i ^ j and i,j 6 { 7 , 8 , 9 } has mass TIIA = RAiRAj/{%9A)', in heterotic units this corresponds t o rrih = 2Rhk, where k £ { 7 , 8 , 9 } is not equal t o either i or j . T h e corresponding heterotic state has IOJ. = ± 1 , pl = 0 , (V ± Ak)2 = 2, and iVL=0. (ii) A D2-brane t h a t wraps the cycle (x\x6), where i is either 7,8 or 9, has mass m,A = RAiRA&l{^9A)\ in heterotic units this corresponds to rrih = \/{2Rhi). T h e corresponding heterotic s t a t e therefore has pl = ± 1 / 2 , 2 wi = 0, V = 2, and NL = 0. (iii) A D4-brane t h a t wraps the entire compact space has a mass VHA — Yli^-Ai/{^9A)] in heterotic units this corresponds t o rrih = 2Rh6- T h e corresponding heterotic state therefore has WQ = ± 1 , pl — 0, (V ± AA)2 = 2, and NL - 0. 4.3

Non-BPS

states

T h e heterotic string also contains non-BPS states t h a t are stable in certain domains of the moduli space. One should therefore expect t h a t these states can also be seen in the dual type IIA theory, and t h a t they correspond to non-BPS branes. Of course, since non-BPS states are not protected by supersymmetry against q u a n t u m corrections t o their mass, the analysis below will only hold for j k C l and gA
32

The simplest examples of this kind are the heterotic states with vanishing winding and momenta (lo, = Pi = 0), and weight vectors given by V

v

=

(0m,±2,015~m)

= (o2m,±i,±i,o2r\±i,±i,o12-2"-2m)

(120)

The results of the previous section indicate that these states are charged under precisely two f ( l ) ' s associated with two fixed points in IIA, and are uncharged with respect to any of the other U(l)'s. There are four states for each pair of [/(l)'s, carrying ±1 charges with respect to the two £/(l)'s. In all cases V2 = 4, and we must choose NR = CR + 1 to satisfy level-matching. These states are therefore not BPS, and transform in long multiplets of the D = 6 N — (1,1) supersymmetry algebra. Their mass is given by mh = 2V2,

(121)

as follows from (104); in particular, the mass is independent of the radii. On the other hand, these states carry the same charges as two BPS states of the form discussed in the previous section (where the charge with respect to the spacetime £/(l)'s is chosen to be opposite for the two states), and they might therefore decay into them. Whether or not the decay is kinematically possible depends on the values of the radii (since the masses of the BPS states are radius-dependent). In particular, the first state in (120) carries the same charges as the two BPS states with p% = ± 1 / 2 , and weight vectors of the form Vi V2

= =

(02n,l,l,014-2n) (02",l,-l,014-2n),

(122)

where we have assumed that m is even and written m = 2n; if m is odd, the two weight vectors are Vi V2

= =

(02n,l,l,014-2n) -(02n,l,-l,014-2n),

(123)

where m = 2n + 1. The mass of each of these states is l/(2Rhe), and the decay is therefore kinematically forbidden when Rh6<^=2.

(124)

More generally, the above non-BPS state has the same charges as two BPS states with Wi = 0, and internal weight vectors Vi

=

±(0m)l,0*,l,014_m_*)

V2

=

±(0m,l,0fc,-l,014-m-fc) ,

(125)

33

where again the non-vanishing internal momenta are chosen to be opposite for the two states. The lightest states of this form have a single non-vanishing momentum, pt — ±1/2 for one of i = 6,7,8,9, and their mass is l/(2Rhi). Provided that R

"<-^m

* = 6,7,8,9,

(126)

the non-BPS state cannot decay into any of these pairs of BPS states, and it should therefore be stable. Similar statements also hold for the non-BPS states of the second kind in (120). We should therefore expect that the IIA theory possesses a non-BPS Dbrane that has the appropriate charges and multiplicities. This state is easily constructed: it is a non-BPS D-string that stretches between the two fixed planes into whose fractional D-particles it can potentially decay. Let us for simplicity consider the state that stretches along x6 from the origin to the fixed plane with coordinates (TTRA6,0, 0, 0), and let us denote the transverse position 5 by c (where c has non-trivial coordinates along ) . Then the boundary state is given as \Dl,c;6,e)

=

|Bl,c;0) N S . N S (127) ie +e (|B1, c; 0) R . R , T + e |B1, c; (TTRA6, 0, 0,0)) R . R , T ) ,

where 0 denotes a Wilson line which originates from the fact that the x6 direction is compact so that the NS-NS vacuum can be characterised by a winding number u>6.m In fact, the boundary state |i?l,c;0) NS _ NS is defined by \B1, c; 0)NS.NS = J2

e%eW6

\B1, c; ™6)NS-NS ,

(128)

where | B l , c ; w 6 ) NS . NS is given by (36), (106) and (13) except that |J31,k,77)(°) in (13) is now replaced by \Bl,-k,w6,Ti)W. (129) This tachyonic ground state has winding number w§ along the x6 direction, and momentum equal to kl for i ^ 6. Because it describes a I)l-brane with a Neumann direction along x2, we also have that k2 — 0; furthermore the momenta k% for i = 7,8,9 are again quantised. This boundary state is (as before) obviously invariant under the GSO-projection; invariance under the ""The relevant closed string Hamiltonian contains then also an additional term where v is the winding length.

v2/(4TT),

34

orbifold projection requires that 6 = 0 or 6 — w (since I 4 maps w6 i-» — we-) The correct normalisation will turn out to be RA7RASRA^S.NS(D1)

= ^ j

~

,

(130)

where V2 — KRASVI-, with V\ being the volume along the a;2-direction along which the Dl-brane has a Neumann boundary condition. The two boundary states in the twisted R-R sector are localised at different fixed planes, and are otherwise standard boundary states. Since the twisted RR sector does not have any fermionic zero modes in the x6 direction, the ground state satisfies the same zero mode condition as the DO-brane boundary state discussed above; this also implies that it is GSO-invariant. The parameter e takes the values ± 1 , and determines the sign of the twisted R-R charge at one end of the .Dl-brane. The correct normalisation will turn out to be A/-R2.R,T0Di) = - I J L ,

(131)

where V\ is the world-volume along the x2 direction. In order to describe the corresponding open string it is convenient to use a different description for the orbifold. 22 Let us denote, as before, by J4 the reflection of the four coordinates d let I4 be defined by , 24

J xl M- -x{ • \ x6 - • 2irRA0

Hi ^6 - x6 .

,

s

{i6Z)

Let us consider the compactification where initially the radius of the sixth circle is 2RA6- The 7Z.2 x Z2 orbifold of this theory that is generated by I4 and I 4 is then equivalent to the above orbifold. In order to see this we observe that I 4 and I | commute with each other, and that both are of order two. The 2Z2 x 2Z2 orbifold can therefore equivalently be described as the I 4 -orbifold of the I 4 I 4 -orbifold; however, 141'4 is the translation x6 t-t x6 + 2TTRA6, and its effect is simply to reduce the radius from 2RA6 to RAGFor the above choice of normalisation constants, the spectrum of open strings that begin and end on the above D-string is then given as [NS - R ] i (1 + ( - l ) ^ ) (1 + (-1) F X 4 ) ,

(133)

where the terms that involve I 4 come from the twisted R-R sector localised at 0, the terms involving Z4 come from the twisted R-R.sector localised at (TTRA6, 0,0,0), and the remaining terms arise from the untwisted NS-NS sector.

35

(More specifically, the term with the unit operator corresponds to the contribution where w6 is even, whereas the term l / 4 ( - l ) F I 4 ( - l ) F I ' 4 = l/4(x 6 >->• x6 + 2TTRA6) comes from the terms with WQ odd.) Since 9 and e can only take two different values each, there are four different D-strings for each pair of orbifold points. These four D-strings are only charged under the two twisted sector U(l)s associated to the two fixed planes, and the four different configurations correspond to the four different sign combinations for the two charges. These charges are of the same magnitude as those of the fractional D-particles, since the ground state of twisted R-R sector contribution satisfies the same zero-mode condition, and has the same normalisation (compare (115) and (131)). Furthermore, it follows from (133) that the Dstrings have sixteen (rather than eight) fermionic zero modes, and therefore transform in long multiplets of the D = 6, J\f = (1,1) supersymmetry algebra. These states therefore have exactly the correct properties to correspond to the above non-BPS states of the heterotic theory. The open string NS sector in (133) contains a tachyon. However, since the tachyon is (—l) F -odd, and since I4 reverses the sign of the momentum along the D-string, the zero-momentum component of the tachyon field on the D-string is projected out. Furthermore, since Z4I4 acts as x6 —>• x6 + 2TTRAG, the half-odd-integer momentum components are also removed, leaving a lowest mode of unit momentum. As a consequence, the mass of the tachyon is shifted to 2

K

A6

For RAG < \/2 the tachyon is actually massive, and thus the non-BPS .Dl-brane is stable. On the other hand, for i?^ 6 > y/2 the configuration is unstable and decays into the configuration of two D-particles that sit at either end of the interval. These D-particles carry opposite untwisted R-R charge, and their twisted R-R charge is determined in terms of the twisted R-R charge of the D-string at either end.

Figure 3: The non-BPS Dl-brane and the two fractional DO-branes into which it can decay.

36

One can also understand this instability from the point of view of the two fractional BPS D-particles. Since they carry opposite untwisted R-R charge, the open string between them consists of [NS-R]i(l-(-l)F)(l±I4).

(135)

The ground state of the open string NS sector therefore has a mass m 2 = - 1 + (TritUeTo)2 = - i + (^f^

,

(136)

and so becomes tachyonic for RAG < \/2, indicating an instability to decay into the non-BPS D-string. The D-string can therefore be thought of as a bound state of two fractional BPS D-particles located at different fixed planes. This is also confirmed by the fact that the classical mass of the D-string (141) is smaller than that of two fractional D-particles (119) when RA6
(137)

and thus the D-string is stable against decay into two fractional D-particles in this regime (see Figure 3). In terms of the heterotic string, this decay channel corresponds to (122). Given the duality relation (98), the domain of stability of the non-BPS D-string (137) becomes in terms of the heterotic moduli V~1/2Rh6

< 2>/2.

(138)

Thus the D-string is stable provided that Rh6 is sufficiently small; this agrees qualitatively with the regime of stability in the heterotic theory (124). (Since we are dealing with non-BPS states, one should not expect that these regimes of stability match precisely.) Other decay channels become available to the D-string when the other distances R&i (i = 7,8,9) become small. In particular, the D-string along x6 can decay into a pair of D2-branes carrying opposite bulk charges, i.e. a D2-brane and an anti-D2-brane, that wrap the (a;2,a;6) cycle. Since the mass of each D2-brane in the orbifold metric is RAiRAe/{^9A), this suggests that the D-string is stable in this channel when R

*i>7M

(* = 7 > 8 > 9 )-

( 13Q )

This is further confirmed by analysing whether the different configurations carry tachyons on their worldvolume: on the D-string the tachyonic ground

37

<

>

++

++

Figure 4: A D2-brane anti-brane pair and the non-BPS Dl-brane into which it can decay. The twisted R-R charge of each D2-brane at each of the four corners is one half of that of the non-BPS Dl-brane.

state of the open string carrying one unit of winding in the xl direction is not projected out, and this state becomes actually tachyonic if RA% < l / v 2 ; 25 on the other hand, there is a tachyon in the open string between the two D2branes when RAi > 1/A/2. In terms of the heterotic string, the decay channel into D2-brane pairs are described by (125). Using the duality relation (98) as before, (139) then becomes

K

-1/2

Rhj < 2\/2

for j # 6 .

(140)

Thus the D-string is stable against this decay provided that Rhj is sufficiently small, and this agrees again qualitatively with the heterotic domain of stability (126). A similar analysis can also be performed for D-strings that stretch between any two fixed points. One can also compare the mass of the non-BPS -Dl-brane with that of the dual heterotic state. As we mentioned before, one should not expect that these are related exactly by the duality map since for non-BPS states the masses are not protected from quantum corrections. Indeed, the classical mass of the above non-BPS Dl-brane is given by mA{Dl)

RAG

V2gA

(141)

where the factor of \/2 comes from the fact that the normalisation of the untwisted NS-NS component (130) is by a factor of \/2 larger than that of the standard BPS D-brane of Type II (52). In heterotic units, this mass is oc 1/V/,, and therefore does not agree with (121). In the blow up of the orbifold to a smooth K3, the non-BPS D-strings correspond to membranes wrapping pairs of shrinking 2-cycles. Since such curves do not have holomorphic representatives, the states are non-BPS. For each pair of 2-cycles there are four states, associated with the different orientations of

38

the membrane; the membrane can wrap both cycles with the same orientation, or with opposite orientation. In either case the net bulk charge due to B = 1/2 can be made to vanish by turning on an appropriate world-volume gauge field strength (F = ± 1 in the first case, and F = 0 in the second). The decay of the non-BPS D-string into a pair of fractional BPS D-particles is described in this picture as the decay of this membrane into two separate membranes, that wrap individually around the two 2-cycles. It would be interesting to understand in more detail how non-BPS branes behave away from the orbifold point; first steps in this direction have recently been taken in Ref. 84. The entire discussion also has a parallel in the T-dual theory that we analysed in the previous section. The non-BPS D-string that stretches along x6 is mapped under T-duality to the non-BPS D-particle of the IIB orbifold. (The two different values 6 = 0,TT correspond to the two possible positions, and e to the sign of the charge of the D-particle.) The non-BPS D-string can be formed as a bound state of a fractional D-particle and a fractional anti-Dparticle (see Figure 3). Under T-duality, the D-particle anti-D-particle pair becomes a pair of a BPS D-string and an anti-D-string of the IIB theory that stretch along the a;6 direction; since the D-particles sit on different fixed planes, the BPS D-strings have a relative Wilson line. Thus the non-BPS D-particle can be understood as the bound state of a Dl-brane anti-Dl-brane pair with a relative Wilson line; this reproduces precisely the construction of Sen 22 (see Figure 2). By T-duality it follows that the D-particle is stable provided thatf1 Ri > 4 = v2

i = 6,7,8,9.

(142)

Similarly the other decay channels can also be related to decay channels considered by Sen. Finally, it may be worth mentioning that in addition to the non-BPS .Dlbrane, the IIA theory also has a non-BPS Z?3-brane for which an analogous analysis applies (see Ref. 47 for more details). Acknowledgements I thank Oren Bergman and Ashoke Sen for many useful conversations about issues that are covered in these lectures. I also thank Eduardo Eyras for comments on a first version of these notes. "In the previous section we considered the uncompactified theory where all radii are infinite; in this regime the D-particle is stable.

39 These lectures were given at the 'Spring workshop on Superstrings and related matters', Trieste, 27 March - 4 April 2000. I thank the organisers for giving me the opportunity to present these lectures, and for organising a very successful and enjoyable meeting. I also thank the participants for asking many useful questions that have helped to improve the presentation of these notes. I am grateful to the Royal Society for a University Research Fellowship. The work was also partially supported by the PPARC SPG programme "String Theory and Realistic Field Theory", PPA/G/S/1998/00613. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

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40

20. I.R. Klebanov, A.A. Tseytlin, Nucl. Phys. B546, 155 (1999), hepth/9811035; Nucl. Phys. B547, 143 (1999), hep-th/9812089; JHEP 9903, 015 (1999), hep-th/9901101. 21. A. Sen, JHEP 9806, 007 (1998), hep-th/9803194. 22. A. Sen, JHEP 9808, 010 (1998), hep-th/9805019. 23. A. Sen, JHEP 9808, 012 (1998), hep-th/9805170. 24. A. Sen, JHEP 9809, 023 (1998), hep-th/9808141. 25. A. Sen, JHEP 9810, 021 (1998), hep-th/9809111. 26. A. Sen, JHEP 9812, 021 (1998), hep-th/9812031. 27. A. Sen, Non-BPS States and Brants in String Theory, hep-th/9904207. 28. E. Witten, JHEP 9812, 019 (1998), hep-th/9810188. 29. P. Hofava, Adv. Theor. Math. Phys. 2, 1373 (1998), hep-th/9812135. 30. S. Gukov, Commun. Math. Phys. 210, 621 (2000), hep-th/9901042. 31. O. Bergman, E.G. Gimon, P. Hofava, JHEP 9904, 010 (1999), hepth/9902160. 32. A. Sen, JHEP 9912, 027 (1999), hep-th/9911116. A. Sen, B. Zwiebach, JHEP 0003, 002 (2000), hep-th/9912249. N. Berkovits, A. Sen, B. Zwiebach, Tachyon condensation in superstring field theory, hep-th/0002211. 33. J.D. Blum, K.R. Dienes, Phys. Lett. B414, 260 (1997), hep-th/9707148; Nucl. Phys. B516, 83 (1998), hep-th/9707160. 34. O. Bergman, M.R. Gaberdiel, JHEP 9907, 022 (1999), hep-th/9906055. 35. R. Blumenhagen, A. Kumar, Phys. Lett. B464, 46 (1999), hepth/9906234. 36. S. Kachru, J. Kumar, E. Silverstein, Phys. Rev. D59, 106004 (1999), hep-th/9807076. 37. S. Kachru, E. Silverstein, JHEP 9811, 001 (1998), hep-th/9808056; JHEP 9901, 004 (1999), hep-th/9810129. 38. J.A. Harvey, Phys. Rev. D59, 026002 (1999), hep-th/9807213. 39. J.M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998) and Int. J. Theor. Phys. 38, 1113 (1999), hep-th/9711200. 40. I. Antoniadis, E. Dudas, A. Sagnotti, Phys. Lett. B464, 38 (1999), hep-th/9908023. 41. G. Aldazabal, A.M. Uranga, JHEP 9910, 024 (1999), hep-th/9908072. 42. G. Aldazabal, L.E. Ibanez, F. Quevedo, JHEP 0001, 031 (2000), hepth/9909172. 43. C. Angelantonj, I. Antoniadis, G. D'Appollonio, E. Dudas, A. Sagnotti, Nucl. Phys. B572, 36 (2000), hep-th/9911081. 44. S. Mukhi, N.V. Suryanarayana, D. Tong, JHEP 0003, 015 (2000), hepth/0001066.

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43

LECTURES O N W A R P E D COMPACTIFICATIONS A N D STRINGY BRANE CONSTRUCTIONS SHAMIT KACHRU Department of Physics and SLAC, Stanford University, Stanford, CA 94305/94309, USA E-mail: [email protected] In these lectures, two different aspects of brane world scenarios in 5d gravity or string theory are discussed. In the first two lectures, work on how warped compactifications of 5d gravity theories can change the guise of the gauge hierarchy problem and the cosmological constant problem is reviewed, and a discussion of several issues which remain unclear in this context is provided. In the next two lectures, microscopic constructions in string theory which involve D-branes wrapped on cycles of Calabi-Yau manifolds are described. The focus is on computing the superpotential in the brane worldvolume field theory. Such calculations may be a necessary step towards understanding e.g. supersymmetry breaking and moduli stabilization in stringy realizations of such scenarios, and are of intrinsic interest as probes of the quantum geometry of the Calabi-Yau space.

1

Introduction

Scenarios for an underlying string theory description of nature have been considerably enriched following the duality revolution of the mid 1990s. Perhaps the most striking qualitative new feature is the emergence of scenarios in which standard model gauge fields are confined to some submanifold of a larger bulk spacetime, while of course gravity propagates in the bulk. For instance, such models are natural in the Horava-Witten extension of the E8 x Eg heterotic string theory, where finite string coupling opens up an additional dimension with the geometry of an interval, and the Es x Es gauge fields live on the boundaries *. More generally, after the realization of the important role played by D-branes in string duality 2 , it was found that the world-volume quantum field theory on coincident D-branes enjoys a non-Abelian gauge symmetry 3 . This makes it natural to construct type II or type I string models where the standard model gauge fields are confined to stacks of D-branes (see e.g. 4 for a discussion of some such attempts). String constructions of this sort have also motivated new ideas in long wavelength effective field theory for reformulating the gauge hierarchy problem 5 ' 6 and the cosmological constant problem 7 ' 8 ' 9 in terms of brane world constructions. The translation of these problems to brane world language does not solve them, but certainly provides a different way of thinking about

44

them, and opens up exciting new possibilities for phenomenology. In the following lectures, we will first review some of the new ideas for reformulating the hierarchy problems of fundamental physics in the language appropriate to such "brane world" scenarios. We will then switch tracks and talk about the detailed investigation of one class of microscopic brane constructions that exist in string theory. These latter lectures will start with a telegraphic review of some aspects of closed string Calabi-Yau compactifications. They will then focus on superpotential computations in models with 4d M — 1 supersymmetry, since these are quite relevant to issues of physical interest like supersymmetry breaking and stabilization of moduli. 2 2.1

Trapped Gravity and the Gauge Hierarchy Trapping Gravity

Our world might actually be contained on a 3+1 dimensional defect, e.g. a domain wall, in a higher dimensional spacetime 10 . Why would we see 4d gravity in such a model? Suppose the extra spatial dimension, parametrized by X5, is a circle of radius r, and we are localized around some point on this circle. The global 5d metric looks like the metric on R3'1 times a circle. The 5d Einstein action is: S5 =

f d5x v ^ G R Ml

(1)

where M5 is the 5d Planck scale. Integrating out the "extra" x 5 dimension gives rise to a 4d effective action with an effective Planck scale

Ml ~ r Ml

(2)

Hence, at length scales larger than r, gravity will appear to be fourdimensional with a Newton's constant determined by (2). For sufficiently small r, this of course would reproduce everything we know about gravity from present day experiments. However, there is a more general possibility. The metric can be warped. For instance, consider pure 5d gravity with a cosmological constant, and a source term for a domain wall located at x$ = 0:

S =

J d5x y/=G(R-A)

+ f d4x ^j(-Vbrane)

(3)

where 9liv = 5ff6?GMN(x5 = 0), /i, v = 1, • • • ,4, and M, N = 1, • • • 5. Following Randall and Sundrum n , we will find solutions of (3) which give rise to 4d gravity and in which non-trivial warping plays an essential role. If we want

45

the 4d world to look flat, we should look for solutions of the equantions of motion following from (3) which exhibit an 50(3,1) symmetry (the Poincare invariance of our world). The most general such ansatz for the 5d metric is: ds2 = e2A(x5)'qllvdx^dxv

+ dx\

(4)

With this ansatz, Einstein's equations just become equations for the warp factor A (primes denote differentiation with respect to X5): 6(A')2 + ^A = 0

3A" + ^V6(x5)

(5)

= 0

(6)

Choosing A < 0, a negative 5d cosmological constant, one can quickly solve (5) to find

A = ±kx5, k=^J-—

(7)

Integrating (6) from x$ — — e to a;5 = e to pick up the delta function contribution, one finds: 3A(A') = - \ v

(8)

where A denotes the discontinuity across x$ = 0. So to solve the equations with the ansatz (4, we must take A = — kx$ for x5 > 0, A = kx5 for X5 < 0. Furthermore, we must tune the brane tension V in terms of the bulk cosmological constant A so that

V = 12k=12]J-±

(9)

This yields a solution where ds2 = e~2k^riiivdx>1dxv

+ dx\

(10)

The warp factor is sharply peaked at £5 = 0, where the domain wall, which we will call the "Planck brane," is located. This fact leads to the existence of localized gravity at the Planck brane n . Namely, doing the naive 5d to 4d reduction by simply integrating over the x$ direction, one finds M( = Ml f dx5e-2k^1

(11)

46

This is finite despite the existence of an infinite 5th dimension, so the 4d Newton's constant on the Planck brane is finite, and an observer there would see effective four-dimensional gravity. There is a natural concern that arises in this case, that does not arise in the case of a 5d theory compactified on a circle of radius r. In the latter case, the lightest 5d Kaluza-Klein (KK) modes have masses which go like -. For r small enough to avoid experimental detection, this leads to a gap in the KK spectrum, and the low energy theory is clearly just 4d general relativity coupled to the brane worldvolume fields. In the warped case, however, the infinite extent of the X5 dimension means that there is no gap in the spectrum of bulk modes! So, one should seriously worry that they will appear as particles with a continuum of masses in 4d, and ruin 4d effective field theory. It has been argued in e.g. n and 12 that despite the gapless KK spectrum, a physicist on the Planck brane would still see effectively four-dimensional physics. This is because, although the KK spectrum is gapless, most of the bulk KK modes have wavefunctions with support far from x5 = 0 where the brane fields are localized. Due to this very small overlap of wavefunctions in the x5 direction, the brane fields and localized graviton couple only very weakly to the bulk continuum. So for instance the Newtonian form of the gravitational potential V(r) ~ receives only small ^5- corrections (curiously, as if there were two extra flat dimensions) n > 1 2 . 2.2

Hierarchies from Multiple Branes

Consider now a case with two branes, located at x5 = 0 and x$ = TT. We will take £5 to live on the interval between 0 and 7r , so the space now has an extra dimension of finite extent. The total action looks like:

S= I d5x V^G(R - A) + SSM + Sn

(12)

where SSM is the action on the "Standard Model brane" located at x$ = IT and Spi is the action on the "Planck brane" located at x$ = 0, i.e. SSM = / d4xy/-gSM(£sM Spi = j ^x^g^iCpi

- VSM) - Vpi)

(13) (14)

Following Randall and Sundrum 6 , we will demonstrate solutions of the theory (12) which give rise to large hierarchies between scales in a somewhat natural way.

47

We again look for a warped metric which maintains the 4d Poincare invariance we desire: ds2 = e-2A{x5)rll,udxtidx,/

+ r2dx\

(15)

It follows from (15) that the size of the x$ interval is nr. The Einstein equations are now: 6 ^

+^

=0

(16)

SA'fr2 + ~^6(x5) + i — < 5 ( x 5 - TT) = 0 (17) 2 r 2 r As before, we can define k = \J-j^ (and take the bulk A to be negative). Then, (16) is solved by taking A(x6)

= kr\x5\

(18)

Notice that (18) is consistent with a Z2 symmetry under which x5 is reflected; our strategy will be to find a Z2 symmetric solution where — TT < x5 < n, and then orbifold by the Zi to get the desired setup. From (18) (and the periodicity of x$, where £5 = ±7r are identified), it follows that A"

= 2kr (6(x5) - 6(x5 - TT))

(19)

Comparing this to (17), we see that we should choose VPi = -VSM

= 12*

(20)

in order to find a Poincare invariant solution. Taking this as our background gravity solution, what is the 4d effective field theory on the "Standard Model" brane that follows from it? First of all notice that it is natural to write the metric ansatz in terms of 4d fields as follows (here x runs over the dimensions other than X5): ds2 = e-2kTW

(77M„ + h^(x))

dx»dx" + T{xfdx2

(21)

There are two dynamical 4d fields in (21); the four-dimensional metric 9iiv{x) = hpV(x) + r]^, and the 4d scalar field T(x) (the so-called "radion"). We recover the desired vacuum solution by choosing the radion to have a constant VEV (T(x)) = r. One can easily compute the 4d effective action for the metric g by starting from the 5d Einstein action; one finds a 4d Einstein term with Mj = Ml{l-e-2kr«)

(22)

48

for the 4d Planck scale. In particular, notice that for r of reasonable magnitude (in Planck units), M\ depends only very weakly on r. This is intuitively because the 4d graviton is largely localized in the vicinity of the Planck brane, which is at x$ = 0 . This leads to an interesting phenomenon. If we compute the metrics gpi and gsM which appear in the source terms for the Planck and Standard Model branes, it follows that 9t = 9^

(23)

while gSM

=

e

-2*r,r^

(24)

This reflects the fact that an object with energy E at the Planck brane would be seen, at the Standard Model brane, as an object with energy Ee~krn; equivalently, length scales at the Standard Model brane are "redshifted" to be longer than the corresponding lengths at the Planck brane. This is a familiar manifestation of scale/radius duality in AdS/CFT 13 . However, here it has the interesting consequence that if one starts with dimensionful parameters in the Standard Model Lagrangian CSM (e.g. a Higgs mass) of order the 4d Planck scale, they can easily be "redshifted" down to energies which are hierarchically smaller, simply by the factors of the metric (24). Hence, to find TeV scale physics on the Standard Model brane, one simply needs to choose r to be of order ten times the fundamental scale. This sounds relatively natural, and provides a candidate solution to the hierarchy problem. Finally, one should ask, how easy is it to accomplish the stabilization of the radion around the required value (that leads to an "explanation" of the hierarchy)? Goldberger and Wise have argued that the presence of a bulk scalar field, with fairly natural bulk and brane couplings, can do the job 14 . 2.3

Some Remarks on Randall-Sundrum

scenarios

In this section, we make some remarks which have relevance to both the naturalness of RS scenarios, and their possible embedding into a fully consistent microscopic theory of gravity (like string theory). There has been a great deal of research on this topic. There are several different things to say about this. One is that, via the AdS/CFT correspondence 13 , the Randall-Sundrum scenario is more or less a strong coupling version of an older idea for solving the hierarchy problem just within quantum field theory. It has long been realized that if one starts with some ultraviolet fixed point CFT around the UV scale (say, just below

49 the Planck scale) and perturbs it by a marginally relevant operator (whose dimension is very close to 4, say 4 — e) then one can naturally generate scales much lower than M p j. Namely, the RG running of the couplings in the perturbed field theory is logarithmic, and therefore the relevant coupling will have significant dynamical effects only after a vast amount of RG running (in energy scale space). Roughly speaking, the scale at which the operator produces significant dynamical effects might be M = e~1/tMpi. A scenario of this sort was advocated recently by Frampton and Vafa 15 . Obviously, for e finely tuned close enough to zero, one can achieve M « Mp[. However, it might be quite a challenge to find a 4d conformal field theory whose most relevant perturbation is of dimension 4 — e with e small; and if one cannot find such a theory, then this mechanism becomes unnatural (because the other, more relevant operators will also be activated along the RG flow). This concern has a direct translation into the Randall-Sundrum scenario, via the AdS/CFT dictionary 16 . a Their scenario requires the existence (between the Planck and the Standard Model branes) of a region many AdS radii in size where the 5d metric is approximately that of AdS space. By introducing a Planck brane, they have also rendered normalizable those modes in AdS space which are normally non-normalizable (due to divergent behavior near the boundary of AdS). These modes will fluctuate. It is difficult to think of concrete scenarios where none of these fluctuating modes in the gravity theory is a "tachyon" (which maps, via the AdS/CFT duality, to a relevant operator). Such tachyons, when they fluctuate and condense, will destroy the AdS form of the metric between the Planck and Standard Model branes. The question of why tachyons (except perhaps those which are very close to being non-tachyonic) should be absent, is the same as the question raised above in the field theory picture. This is not surprising; by AdS/CFT duality, the Randall-Sundrum scenario is exactly the same as the previous one, except in the limit of strong field theory 't Hooft coupling. Of course, such a limit was never tractable before, so interesting new features could emerge there. Without addressing these concerns, one can still ask whether one can plausibly realize the mechanism of §2.2 in some class of string theory backgrounds. A good argument that this is possible has been provided by H. Verlinde 17 . He recalls that in certain compactifications of F-theory on a Calabi-Yau fourfold X4, one can introduce

"This clear relationship and the corresponding concerns were stressed to the author by Maldacena and Witten on separate occasions.

50

space filling D3 branes to satisfy tadpole cancellation conditions (at least if the sign of the Euler characteristic is correct). Since the known list of Calabi-Yau fourfolds includes some with \x\ ~ 200, 000, this can lead to the introduction of large numbers of D3 branes. Of course, with 4d Af = 1 supersymmetry, dynamics will undoubtedly dictate the positions of these D3 branes in the end (there will be a superpotential for the chiral fields which fixes their positions). But it is quite plausible that large numbers of D3 branes will be stacked on top of one another, generating an AdS throat which is "glued" into the CY fourfold asymptotically. Then, the D3 brane field theory will hopefully, in the infrared, manufacture some analogue of the RS Standard Model brane, while the gluing of the throat to the CY fourfold acts effectively as a Planck brane. More explicit models of what the TeV brane might look like have emerged in the recent papers 18 . 3 3.1

Brane Worlds and the Cosmological Constant The Problem

It is an old idea, going back at least to Rubakov and Shaposhnikov 19 , that if the Standard Model were confined to a defect in a higher dimensional space (e.g. a domain wall), this defect might naturally like to be flat. Suitably interpreted, the flatness of the defect could then explain the extreme smallness of the cosmological constant in our 4d world. In this section, we discuss the extent to which this idea seems realizable in the "wall world" scenarios which have become common today. To see that the idea doesn't always fare well, lets begin by reviewing the relevant portions of the Randall-Sundrum scenario. For simplicity, we discuss the scenario of §2.1, but that of §2.2 would differ in no essential way. So, suppose we did live on a domain wall in a 5d gravity theory with bulk A < 0. The key point is to recall that, in searching for a Poincare invariant 4d world, we were forced by the Einstein equations to tune the tension of the brane V in terms of the bulk cosmological term, as expressed in equation (20):

Now, if we imagine the Standard Model degrees of freedom living on the wall at x$ = 0, small changes in the Standard Model parameters (the electron mass, QCD scale, weak scale,...) will result in a renormalization of the brane tension V —> V + AV. Equivalently, quantum loops of Standard Model fields enter in V. But under such a shift, the relation (20) will be violated, and hence one will no longer be able to find a flat solution!

51

This is the manifestation of the cosmological constant problem in such wall world scenarios. One must tune the brane tension V, which depends in a sensitive way on the Standard Model parameters, in terms of other microscopic parameters, or one cannot find a Poincare invariant 4d world. 3.2

Adding Scalars

In most microscopic theories which could be responsible for the 5d bulk action in a wall world scenario, there are degrees of freedom other than the 5d metric. For instance, in string theory generic compactifications result in massless scalar moduli. So, it is natural to consider a theory with additional 5d bulk scalars, and see if the situation of §3.1 improves. In fact, as discussed in 8 ' 9 , it does to some extent. So, take now for the action:

S= Jd'xV^U-liV^A+Jd'x^gi-fm

(26)

In addition to the action for the 5d gravity and scalar field, we have a source term for a domain wall at x$ = 0. In the presence of the scalar <j>, it is natural to take the wall tension to be ) = Ve»t>

(27)

Most of what we say generalizes to far more generic /((/>), as detailed in 8 . To look for Poincare invariant 4d worlds, we again choose the metric ansatz: ds2 = e2A(-x^r]^dx^dxv

+ dx\

(28)

and we take the scalar {x$). The resulting equations are (where again ' denotes differentiation with respect to x^): ^4>" + y ^ V = Weh*6{xs)

(29)

52

6(A') 2 - \{')2 = 0 3A" + ±(')2 = -±eb*V8(x&)

(30) (31)

An important fact which is immediately evident from the equations above is that finding fiat solutions will NOT require any fine tune of the coefficient V in (27) in terms of any microscopic parameters. This is obvious because the only non-derivative coupling of the scalar <> / is through the brane tension term (in /()). So given a solution for one value of V, a shift of V to V + AV can be compensated by an appropriate shift in the zero mode of <j>, leaving the equations of motion unchanged. Why is this significant? The Standard Model physics at x$ = 0 is purely reflected (in this approximation, where the theory is in its ground state) through the wall source term. Now, suppose the Standard Model gauge couplings are independent of . Then varying Standard Model parameters, or summing Standard Model radiative corrections, will shift V in a way that is independent. Hence, one can effectively absorb any cosmological constant generated by Standard Model physics, while still finding a Poincare invariant 4d world 8 - 9 . A picture where is, at leading order, unrelated to Standard Model couplings is not unreasonable. For instance if we are in string theory, we could let the brane at £5 = 0 be a D-brane and 0 be a geometrical modulus for a cycle the brane does not wrap. Alternatively, if we wish to treat as the dilaton, we could imagine the brane at x§ = 0 is a wrapped NS brane whose effective gauge coupling is determined by some geometrical modulus, as in the examples of 20 . 3.3

What about 4d gravity?

To proceed, lets write down the explicit solutions to the Einstein equations. Solving the bulk equations of motion, we find 3 4 4>(x5) = -log\-x5+c\+d A' = ±0'

(32) (33)

Notice that at X5 = — §c, there is a singularity: the scale factor vanishes (A goes to -00), and the |curvature| - • 00. If one momentarily views x5 as a time-like direction, and the slices of constant x$ as 4d spatial slices in a

53 cosmology, then this singularity looks like a big bang or big crunch singularity where the spatial slices collapse to zero size. Next, we need to include the wall source terms. For simplicity, we specialize to the case: /(,/,) = Ve-$*

(34)

However, with one exception (to be mentioned later), basically all of our considerations carry over for much more generic /() 8 . From the form of the bulk solutions, it is clear that the solution with the wall should have the general form: 3 4 <j){xh) = -log\-x5

+ci\ + d i , xh < 0

(35)

3 4 4>{xz) = jlog\-x5

+c2\+d2,

(36)

xa>0

and A' = \''. Imposing the matching conditions at the wall, we find a Z2 invariant solution (symmetric under left/right exchange) if ci = - c 2 = c ,

4

Ac

d1 = d2 = d, e~3d = — — V \c\

(37)

with arbitrary c. Now, suppose we choose c positive. Then, the solution (35) (36) has curvature singularities at x$ = ± | c . Let us suppose the space ends at the singularities, so that the x$ dimension is effectively an interval (with the Standard Model brane in the middle). Then, a simple computation reveals: 2A < oo Mi ~ M? J dx5 e

(38)

so there is indeed four dimensional gravity coupled to the brane field theory at x$ = 0. 3-4 Discussion of Several Important Issues There are several issues that need to be addressed about this framework for discussing the cosmological constant in wall world scenarios. 1) What about bulk quantum corrections? In general, choosing a bulk action with vanishing 5d cosmological term and only kinetic terms for the bulk scalar , as in (26), is only sensible in an

54

approximate sense. We are assuming the bulk is supersymmetric, with the brane breaking supersymmetry. Still, eventually the interaction of bulk and brane fields will transmit the SUSY breaking to the bulk, and there will be subleading results which correct the action (26) and lead to slight curvature of the previously Poincare invariant slices. How do we estimate the size of these effects? It follows from the matching conditions (37) that if one chooses the brane tension Vf((j)(0)) to be roughly a TeV, and one fixes M 4 ~ 1019GeV, then the 5d Planck scale is fixed to be 105TeV (and the size of the x$ interval is about a millimeter). Interactions of bulk and brane fields are suppressed by explicit powers of j ^ - ; therefore, bulk corrections to the 4d effective field theory will arise in a power series in e = (TeV/M$). Hence, in this scenario one can arrange to cancel the leading Standard Model (TeV)4 contribution to the effective 4d cosmological term, but there will be contributions at subleading orders in e. While these are too large to be tolerated given the observed value of the cosmological constant (unless one can somehow cancel the first few terms in the power series), they are nevertheless hierarchically smaller than the expected answer. So, our philosophy should be, that we are looking for a system where the induced cosmological term is hierarchically smaller than what is expected (and we can postpone understanding how to get precisely the right magnitude of the suppression). 2) We have shown there are generically Poincare invariant solutions to the equations, independent of Standard Model parameters. However, are there also other curved solutions, which would be characteristic of a 4d effective field theory with nonzero cosmological term? For generic f(<j>), it turns out that curved solutions with de Sitter or anti de Sitter symmetry do exist 8 . For fine tuned /(), e.g. /() ~ Ve±3
55

were proposed in 22 , but have a very non-generic low energy effective field theory (with Bose-Fermi degeneracy) and have only been studied at low orders in perturbation theory. Indeed it has been advocated by Banks 23 that perhaps M theory does not admit nonsupersymmetric, Poincare invariant solutions. We find it intriguing that these "wall worlds" do admit Poincare invariant solutions, and are quite similar to systems one can realize in string theory with wrapped branes. 3) What about the singularities? Of course, the 5d effective field theory defined by (26) breaks down in regions of large curvature. However, it is often the case that string theory can regulate and provide a definition of singular geometries. So, it is an important problem to find a microphysical realization of these systems, which regulates the singularities or describes the physics occurring there. There are obvious analogies between our x5 interval and the intervals encountered in e.g. Horava-Witten theory or Type I' string theory. Instead of expanding on those here (see e.g. 8 for more discussion), we concentrate instead on the similarity to geometries arising from RG flows in AdS/CFT duality. Polchinski and Strassler, for instance, have studied a class of RG flows from the deformed Af — 4 super Yang-Mills to confining gauge theories with less supersymmetry 24 . Because their geometries involve a 5d gravity theory with small curvature in the UV (near the "Standard Model" brane, in our language) and large curvature in the IR (which corresponds to our singularities), they are quite similar to our setup. As discussed by Bousso and Polchinski 25 , we can then use the results of 24 to infer some important "facts" about the singularities we encounter here. What Polchinski and Strassler find is that the RG flow results in a "discretuum" of possible IR branes - there are roughly e*N possibilities for the IR brane, where TV is a large number in the (super)gravity limit. This translates immediately to the statement that, in our solutions, it is quite likely that the integration constants which arise in {x$) cannot take arbitrary values, but are rather quantized to certain allowed values at the singularities. The question is then, do the allowed values allow for a solution which is closer to Poincare invariant than would be possible with the expected (TeV)4 cosmological constant? The answer seems to be yes. As argued in 25,b the AdS/CFT results strongly suggest that cosmological constants which are suppressed from TeV ''This was independently known and stated by several others including N. Arkani-Hamed, E. Silverstein and the author.

56

scale by powers of e~^N should be achievable. The question of why the allowed singularity with the smallest possible norm of the 4d cosmological constant would be cosmologically preferred is a difficult one. A possible scenario, following earlier work of Brown and Teitelboim 26 , was presented in 25 (in a different context, involving M theory compactifications with four-form fluxes). Several aspects of their work might generalize to the case under discussion. A related approach can be found in 27 , and a critical discussion appears in 2 8 . Another method of dealing with the issue of singularities, is to attempt to find solutions that retain the "self-tuning" property (the existence of a flat solution independent of "Standard Model" parameters), but are either not singular or have singularities which are physically innocuous. One interesting approach of this sort has been detailed in 29 , where they find a self-tuning model which has no naked singularities and is known to arise in exact string theory constructions. While the particular solution they find is not attractive for other reasons (there is a strongly time-dependent 4d Newton's constant), the basic idea seems promising. Another recent self-tuning construction which is free of singularities appears in 30 . There has been some controversy in the literature about the various brane world approaches to studying the cosmological constant problem (regarding issues like physical admissibility of the singularities). While my point of view is well represented here, alternative viewpoints can be found in e.g. 31 - 32 .

4

Calabi-Yau Compactifications and Closed String Mirror Symmetry

In the next two lectures, we will work up to the study of building blocks for microscopic "brane worlds" which clearly are realizable in string theory. These backgrounds involve D-branes in curved geometries, and the open string sectors which live on these branes. Such brane theories exhibit some interesting duality symmetries, which we will also briefly explore. To make the discussion self-contained, we must provide a brief description of the relevant closed string backgrounds first.

4-1

Type II Calabi-Yau Compactifications

Suppose one wants to find a supersymmetry-preserving compactification of the type IIA or type IIB theory, by compactifying on a smooth manifold M

57

of complex dimension d.c One can argue that a necessary condition is Holonomy of M C SU(d)

33

(39)

The possible choices of M become more and more plentiful as d is increased. For d = 1, the only choice is the two-torus T 2 , and the resulting 8d theory has 32 supercharges. For d = 2, one can choose either T4 or K3, which preserve 32 and 16 supercharges respectively. Finally, in the case we will utilize later on, d — 3, there are (at least) thousands of choices (for the earliest large compendium of such spaces that I am aware of, see 3 4 ) . The generic choice of such a complex threefold preserves 8 supercharges, corresponding to 4d M — 2 supersymmetry. The study of such compactifications has been a rich and beautiful subject about which we will necessarily be very brief here: for a much more comprehensive review, see 35 . These so-called Calabi — Yau manifolds are Ricci flat and Kahler. The Ricci-flat Kahler metrics on a Calabi-Yau space M come in a family of dimension /i 1 ' 1 (M) + 2/i 2,1 (M), where h1,1 parametrizes the choice of a Kahler form and 2h2'1 is the dimension of the space of inequivalent complex structures on M. A simple example of such a space is the quintic Calabi-Yau threefold in CP4. CP4 is defined by taking 5 homogeneous coordinates (zi, • • •, Z5), subject to the identification {z\, • • •, z5) ~ (Azi, • • •, Xz5) where A is a nonzero complex number (and with the origin deleted). The quintic is defined by a homogeneous equation of degree 5 in this space, for instance 5

P = J>? = 0

(40)

i=i

The complex structure deformations of this manifold are parametrized simply by monomial deformations of the equation (40), modulo linear changes of variables Zi —> AijZj. In the end, this leads to a 101 possible (complex) deformations of the equation (40). The Kahler deformations are, in this case, simply inherited from those of P4 - there is a single real Kahler parameter, parametrizing the overall volume. 4-2

Spectrum of IIA or IIB String Theory on M

Compactifying either type II string theory on a Calabi-Yau threefold M results in a 4d, W = 2 supersymmetric theory in the remaining R3'1. Such a theory admits two kinds of light supermultiplets: c We want to achieve Poincare supersymmetry in the remaining 10 — 2d dimensional theory, so we will not have to worry about e.g. the Freund-Rubin ansatz and AdS solutions.

58

• The vector multiplet, which consists of a complex scalar field, a vector field, and fermions, all in the adjoint representation of the gauge group Q. • The hypermultiplet, which consists of 2 complex scalars and fermi superpartners, in any representation of the gauge group Q. The moduli of the Ricci-fiat metric on M show up as scalars in such multiplets in the compactified IIA/B theory. However, the correspondence between the geometry of M and the type of multiplet is different for the two theories. Type IIA on M: The IIA theory in ten dimensions has a metric, an NS _BM„ field, a dilaton (f>, and 1 and 3 form RR gauge fields. On dimensional reduction on M, this gives rise to hl'l{M) abelian vector multiplets (where the scalars come from the real Kahler moduli of the metric plus the B^v field, and the vector comes from Cfivp). On the other hand, the complex structure moduli of the metric together with the scalars coming from absorbing 3-forms on M with CM„P give rise to (the scalar components of) h2,1{M) hypermultiplets. It turns out that the dilaton in the IIA theory is also part of a hyper, yielding a total of /i 2 l l (M) + 1 hypers. Type IIB on M: In the IIB theory, in addition to the metric, NS B field and dilaton, there are 0,2 and 4 form RR gauge fields. These give rise to h2'x(M) abelian vector multiplets in the low energy theory (with the scalars coming from complex structure moduli, and the vectors coming from C^px). On the other hand, the Kahler moduli, the Z?M„ and CM„ fields (NS and RR two forms), and the RR 4-form give rise to /i1,:L(M) hyper multiplets coming from the (1,1) forms on M. Including the dilaton, which again transforms as part of a hyper, this yields a total of /i 1 ' 1 (M) + 1 hypers. By J\f = 2 supersymmetry, there are several simplifications in the low energy effective action for these theories. First of all, with this much supersymmetry, there is no potential generated for "flat directions" which are present in the tree-level theory. Hence, there are moduli spaces M of exactly degenerate supersymmetric vacua (the physical reflection of the moduli space of Ricci-fiat metrics on the Calabi-Yau M, if you will). Furthermore, because of the extended supersymmetry, the moduli space M takes the form of a product of vector and hypermultiplet moduli spaces: M = Mv x Mh

(41)

where the metric on Mv,h is independent of VEVs of scalars in the "other" kind of multiplet.

59

4-3

Quantum Corrections

String theory on M comes with two natural perturbative expansions: an expansion in string loops, controlled by the dilaton gs = e~^', and an expansion in sigma model perturbation theory (or curvatures), which is roughly controlled by ^r where R is some characteristic "size" of M (controlled by the Kahler moduli). Large gs corresponds to strong string coupling, while large sigma model coupling means that classical geometry is not necessarily a good approximation, and the "stringy" phenomena characteristic of quantum geometry can occur 3 5 . One can then consider corrections to the tree-level picture in both of these expansions. To be concrete, let's consider the geometry (metric) of the vector multiplet moduli space Mv. It is controlled, as is familiar from SeibergWitten theory 36 , by a holomorphic prepotential F(i) where fa are the scalar moduli in the vector multiplets (F also determines the kinetic terms of the gauge fields, the so-called "gauge coupling functions"). • In the IIB theory, Mv is independent of the Kahler moduli and gs, because both of them are in hypermultiplets and the geometry of the vector moduli space is independent of the VEVs of hypers. Therefore, it is exactly determined at both string and sigma model tree level: it is computable in terms of classical geometry. To be slightly more precise, each Calabi-Yau manifold M is characterized by a holomorphic (3,0) form fi, which is unique up to scale. If we let i,j,k index directions in the moduli space of complex structures, then d3F C «• - , » , ~ / ilAdidjdkQ

d fad fad fa JM

(42)

A detailed explanation of this formula can be found, for instance, in 3 7 . • In the IIA theory, there is a more complicated story. Because the Kahler moduli are in vector multiplets now, there are quantum corrections to the prepotential F controlled by the sigma model coupling. However the dilaton is still in a hypermultiplet, so there are no gs corrections - F is computable at string tree level. Considerations of holomorphy dictate that the form of F is such that <93F will contain contributions which are either tree-level or non-perturbative in ^ - (i.e. going like e - v ) . This is because the Kahler parameter R is real, and its scalar partner (which arises from the dimensional reduction of the NS B field and complexifies it) is an axion a 3 8 . Although the continuous shift symmetry for the axion can be broken non-perturbatively, there is a discrete symmetry under which a shifts by 2TT (in a natural normalization). This forbids any corrections to the prepotential in perturbation theory, but is consistent with nonperturbative corrections.

60

What is the source of the non-perturbative corrections to sigma model perturbation theory? At tree level in the gs expansion, the string worldsheet -R2

is a sphere, and "instanton" corrections suppressed by e~^~~ can arise when the worldsheet wraps a holomorphic sphere (of radius R) embedded in the Calabi-Yau space M 39 . Denote by Hi a basis for the homology 4-cycles in M, and by b{ a dual set of 2-forms (i = 1, • • •, b2(M)). The Hi are in 1-1 correspondence with the scalars fa in the N = 2 vector multiplets. At large radius, when non-perturbative corrections to the sigma model are irrelevant, there is an elegant formula expressing the prepotential F in terms of the intersection numbers of M (see e.g. 37 ) fl

d3F f , a , » , ~ / hAbjA

d(t>idjd
As argued in

bk

(43)

JM

39

, this f is corrected by instantons to an expression of the form ^ V f f f -Area(C) /f biAbjAbk /f bj /f b, k e - A r?e a ( C ) (44) 83F , , + vy^ ~^ f h , , dfadfijdcpk JM

c

^C

JC

^C

where the sum runs over holomorphic spheres C passing through all three of the cycles i?ij,fc4-4

Mirror Symmetry

Mirror symmetry basically is the statement that Calabi-Yau manifolds naturally come in pairs M and W such that the type IIA theory on M is exactly equivalent to the type IIB theory on W (see 40 and references therein for the development of this idea). A moment's thought reflects that this is an extremely nontrivial statement about the geometry of Calabi-Yau spaces, and a powerful computational tool for physics. For instance, the roles of the Kahler and complex structure moduli of M and W will be interchanged by the symmetry, since their physical role in the low energy effective M = 2 gauge theory that arises from the string compactification is interchanged in the IIA and IIB theories. For a quick indication of the mathematical power of this statement, recall that the intricate prepotential (44) in the IIA theory will be computable in tree-level of both gs and a' perturbation theory in the IIB theory, by a formula of the form (42). Equating the two makes highly nontrivial predictions about e.g. the multiplicity of holomorphic curves in M; this has been exploited to great effect beginning with the work 41 . A heuristic proof of this statement (for some classes of Calabi-Yau manifolds) was provided by Strominger, Yau and Zaslow 42 . Their reasoning goes

61 roughly as follows. Suppose the IIA theory on M is really equivalent to the IIB theory on W. Then the full theories, including all BPS states and their detailed properties, should match. Now, what are the SUSY brane configurations allowed by Calabi-Yau geometry, that will give rise to BPS states in the two cases? There are basically two sets of possibilities for CY threefolds 43 : • One can wrap D2, D4 or D6 branes on holomorphic 2,4 or 6 cycles. • One can wrap D3 branes on "special Lagragian" three-cycles; by definition, a special Lagrangian three-cycle E is a submanifold of M such that the Kahler form u) restricts to zero on E, and Im(il)\s = 0 as well. Since the IIA theory only has supersymmetric Dp branes for even p, while the IIB theory only has supersymmetric Dp branes for odd p, the holomorphic cycles are relevant for IIA while the special Lagrangian cycles are relevant for IIB (as long as one is focusing only on point particles in the transverse R3'1). It is a common abuse of terminology to simply call both special Lagrangian cycles and holomorphic cycles "supersymmetric cycles," for obvious reasons. So, lets start by considering the simplest possible case, the DO brane in type IIA on M. The worldvolume theory is a supersymmetric quantum mechanics with 4 supercharges, and its moduli space is intuitively just given by the manifold M itself. Hence, if IIB on W is exactly equivalent to IIA on M, it must contain a "mirror" supersymmetric brane whose moduli space is also M\ As discussed above, it must be a D3 brane wrapping a SUSY 3-cycle S C W. What are the properties of E? In particular, one needs the complex dimension of the moduli space of the wrapped D3 brane to be 3. By McLean's theorem 44 , E itself has &i(E) moduli as a supersymmetric cycle in W. In addition, Wilson lines of the U(l) gauge field on the wrapped D3 brane provide another &i(E) moduli. Thus, we learn that we must have &i(E) = 3 to match the expected dimension of the moduli space. Furthermore, if we fix a point in the moduli space of the special Lagrangian cycle and simply look at the Wilson lines, they give rise to a T3 factor in the moduli space of the wrapped brane. Hence, we learn that in some sense, M must be a T3 fibration! Now obviously, switching the role of M and W would yield an argument that W must also be a T3 fibration. Hence, an elegant conjecture is that both M and W are fibered by special Lagrangian T 3 s, and in particular the mirror of the DO brane on M is a D3 brane wrapping the supersymmetric T3onW. This is intuitively sensible: T-dualizing on the 3 circles of the T 3 would turn the IIB theory into the IIA theory, and change the D3 brane into a DO brane. This chain of arguments indicates that all Calabi-Yau manifolds with

62

mirrors are T3 fibrations; it is furthermore a constructive argument, since one can in principle explicitly construct the mirror manifold by T-dualizing the supersymmetric T3s. In practice this is of course very difficult. Indeed, simply demonstrating the supersymmetric T 3 fibration is out of reach except in very special cases; weaker results about Lagrangian fibrations do exist. For a recent review, see 45 . 4.5

An Example

Since the previous subsection was fairly abstract, we close this section with a (trivial) example. Consider IIA on a T 2 which is a product of two circles, T2 — S^ x 5^ 2 where Ri^ are the radii. We can clearly view this as a T 1 (i.e., 5 1 ) fibration over S1. From standard results in elementary geometry, the complex structure of this torus is parametrized by

(45)

r ~ ifwhile its Kahler structure (or volume) goes like p ~ iRiR2

(46)

The i appears in (46) because the string theory modulus p satisfies p = B + iJ, where B is the NS B field and J is the geometrical Kahler form. T-dualizing along the S^ circle has the following effect. Define R[ = ^ -

(47)

Hi

(we are setting the string scale to unity for simplicity in this subsection). Then R2 Tnew = i-jZT = 1R2R1 = Pold

R[

(48)

and Pnew = iR[R2

.R2

- i-£~ = T0id

(49)

And of course, T-dualizing along one circle exchanges the IIA and IIB theories. We have succeeded in taking IIA on a torus with (complex,Kahler) moduli (r, p) to IIB on a torus with moduli (p, r ) . This is mirror symmetry for T 2 , and it has precisely arisen here as T-duality on the S1 "fibration." One can do the slightly less trivial case of KZ with as much success, by viewing K2> as a T 2 fibration42.

63 5

Open Strings and Mirror Symmetry

In the previous lecture, we saw that Calabi-Yau threefolds come in pairs M, W such that the IIA theory on M is equivalent to the IIB theory on W. This yields a powerful tool for the study of 4d J\f = 2 supersymmetric string vacua. By a slightly more elaborate construction, we can also manufacture 4d M = 1 models starting with type II strings on Calabi-Yau spaces. Namely, we should compactify the type II theory on a Calabi-Yau, and then introduce additional (space-filling) D(p + 3) branes wrapping supersymmetric p cycles.d It is natural to ask: what does mirror symmetry do for us in this context? Let's begin the discussion in type IIA string theory. If we wish to make a "brane world" in type IIA string theory by compactifying on a Calabi-Yau M and then wrapping D{p + 3) branes on p cycles in M, and we also want to preserve 4d M = 1 supersymmetry, then the only possibility is to wrap D6 brane(s) on supersymmetric (special Lagrangian) three-cycles. Recall that a 3-cycle E C M is called special Lagrangian iff • W|E = 0 • /m(0)|E = 0 where to is the Kahler form of M, and fi is the holomorphic (3,0) form. Such cycles are volume minimizing in their homology class. 5.1

How to produce examples o / E

Although quite generally it is difficult to produce examples of special Lagrangian 3-cycles in compact Calabi-Yau manifolds, there is a rather special construction that can be used to give a simple class of examples. Suppose we have local complex coordinates 21,2,3 on M, chosen so that: u> ~ 2_. dzi A dzi

(50)

i

Q ~ dz\ A dzi A dzs

(51)

Furthermore, suppose that M comes equipped with a so-called real involution I, which acts at X:

Zi^Zi

(52)

Consider now the fixed point locus of 1 in M, i.e. the locus of points where Zi = z;. Let us call this E^. It is clear from (50) and (51) that I acts "In the full construction, one will also have to introduce orientifolds to cancel the RR tadpoles.

64

on the Kahler form and the holomorphic three-form as X : w ->• - w , f l 4 ( i

(53)

So in particular, we read off from (53) that on E j : w|Ex=0,

Jm(fi)|El=0

(54)

Hence, the fixed point locus S j of a real involution 1 acting on M is always a special Lagrangian cycle. Let's be very concrete by working out an example. Consider the CalabiYau hypersurface in WP^ 2 2 2 defined by the equation: p = z\+z\+z\

+ z\ + z\-2(\ + e)z\z\ = 0

(55)

Notice that p = dp = 0 is soluble when e —> 0, indicating that the hypersurface (55) becomes singular at that point in moduli space. We will consider the region of small positive e. Now, consider the real involution 1 : Zi —>• Zj acting on (55). The fixed point locus is obviously the locus where all of the z^ are real. What is its topology? Let us define u = z\, and work (without loss of generality) in the 02 = 1 patch. Then p = 0 implies u2 - 2(1 + e)u + 1 + Q = 0

(56)

Q = zl + zi + z45

(57)

u± = 1 + e ± \ / e 2 + 2e - Q

(58)

where

Solving (56) we find

What is the point of this? The solutions (58) go imaginary for large Q, so Q is bounded to lie in some domain of size basically 2e (for small e). The locus Q < 2e intersects the fixed point locus of X in a 3-ball B3, and has boundary Q = 2e which is (up to finite covering) an S 2 . The two different branches of solutions for u in (58) are glued together along this boundary 5 2 ; so altogether E j consists of two B$s glued together on an S2. But of course this is nothing but an 5 3 . It follows from these manipulations that the size of the S 3 goes to zero as e —> 0; the singular point in moduli space is related to the existence of this collapsing 3-cycle.

65

5.2

D6 Branes wrapping special Lagrangian cycles

Now that we have gotten some feeling for very simple examples of special Lagrangian cycles, lets start to consider the physical theory living on a D6 brane which wraps such a cycle E. Since the brane breaks half of the supersymmetry, the low energy theory on the brane (living in the noncompact i? 3 ' 1 ) will be a 4d J\f = 1 field theory. The D6 brane gauge field will descend to yield a C/(l) gauge supermultiplet in 4d. The other kind of light multiplet in N — 1 theories is the chiral multiplet; how many of these will be present? It follows from the work of McLean 44 that the "geometrical" moduli space of E has (unobstructed) real dimension 6i(E). String theory complexifies this with Wilson lines of the U{\) gauge field, yielding a moduli space of vacua with foi(E) complex dimensions for the brane worldvolume field theory. What are good coordinates on this moduli space? Choose a basis jj for fli(E), and choose discs Dj C M such that dDj = "/j. Define LOj =

/ JDj

LO

(59)

which is the area that a holomorphic disc in DjS relative homology class would have (if it existed). To complexify this, consider also the &i(E) Wilson lines

a, =

f A

(60)

where A is the U(l) gauge field on the D6 brane. Together, (59) and (60) yield the scalar components of &i(E) chiral multiplets 4>j which live in the 4d M = 1 theory on the wrapped brane 46>47: 4>j = - 4 + *a7 H J

a!

J

(61)

Now, in any M = 1 supersymmetric field theory, a quantity of great interest which governs the vacuum structure and tends to be exactly computable is the superpotential W{<j>) as a function of the chiral fields . How do we compute W(j) in the theories at hand? First of all, there is no superpotential for the (j>j to all orders in a!. The proof of this statement is quite analogous to the one used in discussions of heterotic string compactifications 39 . Because the Wilson line a,j has a shift symmetry under large gauge transformations, W((f>j) must not have any polynomial dependence on a,j. But since a,j appears in the chiral field 4>j as in (61), and I f is a holomorphic function of the chiral fields, this implies that there can be no polynomial terms in W(j) at all. This is consistent with

66

McLean's result in pure mathematics, which roughly speaking sees a' perturbation theory. On the other hand, terms of the form g-K-Ha,-) = e-4>i (Q2) are consistent with shifts a,- —> a,j + 2TT which occur under large gauge transformations, and hence such terms in the superpotential cannot be ruled out. What would the source of such terms be? Just as closed string theories have worldsheet instantons, D brane theories on Calabi-Yau spaces can have "disc instantons." At tree level, the open string worldsheet is a disc D. One can consider holomorphic maps D —> M such that dD = jj C E, and such that the normal derivative to the map at the boundary is in the pullback of the normal bundle to E in the Calabi-Yau. The claim is then that the superpotential in these D6 brane theories is entirely generated by such disc instanton effects. For instance, one can formally compute couplings like the Fifyjfa coupling that would arise between two scalars and the auxiliary field F in chiral multiplets if there is a nontrivial superpotential. This is discussed at length in 47 (and is very closely related to the discussion in 4 6 ) . The upshot is that one can give a formula for this three-point function on the string worldsheet in terms of an infinite sum over disc instantons. If we call the vertex operators for the spacetime fields appearing in this coupling VF,Vl,V£, then the three-point function (VpVlV^)

has the following expression. Denote by

<4^j")(*, j , k) the number of holomorphic maps from the disc D to M with the following properties: i) [3D] = VJj mat ii) yi>i>k are mapped in cyclic order to the intersection of 3D with the 2-cycles in E dual to "/i,j,kiii) [dD — YJj miD{], which by i) is a closed 2-cycle in M, satisfies

[dD-Y^miDt] l

= Y,n«K"

(63)

a

where the Ka are a basis for H2{M). Then one can derive the statement: 61(E)

hx'x(M)

,_, 1=1

= 1l a„=

e~ „m,,n . .a>0 .n

JdD

JOU

au JdD •'

J 3D

JOU

(64) where 6l is the harmonic one-form associated to 7$, and ta = JK UJ is the area of Ka- This formula is, in some natural sense, the open string analogue of the

67

instanton sum formula (44) for the prepotential in closed string Calabi-Yau compactifications. In some very special examples, such disc instanton sums have proven directly computable 4 8 . Closed String Parameters How do the closed string moduli of the Calabi-Yau M play a role in the brane theory? From (64) above, it is clear that the superpotential W of the brane theory really depends on the closed string Kahler moduli; they enter as parameters ta, so we should really denote W as W((f>j;ta) to indicate the relevance of the closed string background. In fact, it was argued rather generally in 49 that in this class of theories (so-called "A-type" branes), the Kahler (complex structure) moduli of M will only enter in the superpotential (FI D-terms) of the wrapped brane theory. This is in accord with (64), where the Kahler dependence is manifest and there is no explicit complex structure dependence. The dependence of the FI D-terms on the complex structure of M has been explored, for instance, in 50>51. 5.3

Type IIB "Mirror" Brane Worlds

Thus far, we have been focusing our attention on the brane worlds we can construct in the IIA theory, but of course it is possible to make analogous type IIB constructions by wrapping 5,7 or 9 branes on holomorphic 2,4 or 6 cycles (or indeed, by having D3 branes transverse to the Calabi-Yau). In §4.4, we saw that mirror symmetry was of great use in "solving" the AT = 2 theories that come out of string theory, by making the prepotential, which receives an infinite series of quantum corrections in the IIA picture (44), explicitly computable at tree level in the IIB picture via (42). Mirror symmetry should be a similarly powerful tool in studying brane worlds of the type under discussion. Computations of superpotentials, which by (64) are dauntingly difficult in the IIA picture, are much simpler in the IIB picture. In fact, it was argued in 49 that in the case of D(p + 3) branes wrapping holomorphic p cycles, the superpotential is exact at tree level (receives no a' corrections whatsoever). Hence, in the IIB theory, superpotentials are effectively as computable as e.g. prepotentials in the closed string case. The challenge, then, is to find a mirror IIB brane configuration for a IIA configuration consisting of a D6 brane wrapping a special Lagrangian three-cycle E d . Clearly, the IIB theory will be compactified on the mirror manifold W; the question is, what is the mirror brane setup? To find concrete examples it is most convenient to focus on cases where the mirror IIB setup turns out to be a D5 brane wrapping a rational curve C C W. The generalities of this kind of correspondence were discussed in 47

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and very concrete examples, where a disc instanton generated superpotential in the IIA theory maps to a tree level superpotential in the IIB theory, were presented in 52 . So, what is the physics of a IIB D5 brane wrapping CI As always, there is a U(l) gauge supermultiplet. The number of massless chiral multiplets is given by the number of small deformations of the curve, parametrized by H°(C,Nc), where Nc is the normal bundle of C C M.e However, by classical deformation theory, deformations of C C W can be obstructed; this corresponds precisely to a massless chiral multiplet which has a nontrivial higher order superpotential! If h°{C,Nc) = 1, and we call the chiral multiplet <j>, then an iVth order obstruction is reflected in a superpotential on the brane which looks like W ~ 4>N+2

(65)

As in the IIA D6 brane theory, closed string moduli enter in the IIB D5 brane theory as parameters in the Lagrangian. From 4 9 , the superpotential depends only on the complex structure of W, while the FI D-terms for the U(l) gauge field can depend on the Kahler structure of W. In concrete examples, the moduli space of C can exhibit very intricate behavior as one varies the complex structure parameters ipa of W. So we should write the superpotential as W(i;ipa) where a runs over the complex structure deformations of W, and the 4>i parametrize the deformations of C. Therefore, if one finds a mirror pair consisting of a D6 brane wrapping a special Lagrangian cycle S c M and a D5 brane wrapping C C W, then the IIA disc instanton sum (64) should basically map to purely classical geometrical data in the IIB theory. The superpotential W(<j>i;ta) in IIA will encode the same data as W (fa; ipa)- The map between the parameters ta and tpa will of course be the mirror map between the closed string moduli spaces. On the other hand, working out the map between the open string fields <j> and is a complicated problem about which little is known at this point 52 . In practice, how does one go about constructing such mirror pairs? The strategy followed in 52 was to wrap D5 branes on curves C C W which collapse to zero volume at some particular point in the Kahler moduli space of W. Then the 3-cycle S that the mirror D6 brane wraps must collapse at the mirror point in the complex structure moduli space of MJ In examples where e

If one were to wrap a higher genus curve, there would of course also be Wilson lines; but flat bundles on P1 do not lead to any additional degrees of freedom. •^A simple argument for this is that in the closed string theory, there will be light nonperturbative states associated with the collapsing curve; to replicate this phenomenon, there must also be a vanishing cycle on the mirror manifold.

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6i(E) > 0 but the curve C has less than &i(S) unobstructed deformations, the D6 theory must "lose" some its tree-level moduli to an instanton generated potential. Examples of this sort were produced in 52 , which is strong evidence for the presence of the disc instanton effects (64). It would be extremely interesting to actually find an open string analogue of the mirror map, which lets one directly map the IIA superpotential to the IIB superpotential. As a byproduct, one might obtain nice counting formulas for holomorphic discs with boundaries in a special Lagrangian cycle 4 6 . Acknowledgements These lecture notes are being submitted to the proceedings of both TASI 1999: Strings, Branes and Gravity, and the 2000 Trieste Spring Workshop on Superstrings and Related Matters. I would like to thank the local organizers of TASI 1999 for providing such a wonderful atmosphere for the school, and the students for their enthusiastic participation. I would also like to thank the organizers of the 2000 Trieste Spring Workshop on Superstrings and Related Matters for providing a very stimulating environment in which to deliver these lectures. My thinking about the subjects in the first two lectures was developed in collaborations with M. Schulz and E. Silverstein, while for the latter two it was developed in collaborations with S. Katz, A. Lawrence and J. McGreevy. In addition, I would like to acknowledge discussions with T. Banks, S. Dimopoulos, N. Kaloper, J. Maldacena, S. Shenker, R. Sundrum, L. Susskind, S. Thomas, H.Verlinde and E. Witten which greatly influenced my thinking on some of these subjects. Some of the research described in these lectures occurred while the author was enjoying the hospitality of the Aspen Center for Physics and the Institute for Advanced Study in Princeton. This work was supported in part by an A.P. Sloan Foundation Fellowship, a DOE OJI Award, and the DOE under contract DE-AC03-76SF00515. References 1. P. Horava and E. Witten, "Heterotic and Type I String Dynamics from Eleven Dimensions," Nucl. Phys. B460 (1996) 506, hep-th/9510209. 2. J. Polchinski, "Dirichlet Branes and Ramond-Ramond Charges," Phys. Rev. Lett. 75 (1995) 4724, hep-th/9510017. 3. E. Witten, "Bound States of Strings and P-branes," Nucl. Phys. B460 (1996) 335, hep-th/9510135. 4. G. Aldazabal, L. Ibanez, F. Quevedo and A. Uranga, "D-branes at Singularities: A Bottom-up Approach to the String Embedding of the Standard

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Model," JHEP 0008 (2000) 002, hep-th/0005067. 5. N. Arkani-Hamed, S. Dimopoulos and G. Dvali, "The Hierarchy Problem and New Dimensions at a Millimeter," Phys. Lett. B429 (1998) 263, hep-th/9803315. 6. L. Randall and R. Sundrum, "A Large Mass Hierarchy from a Small Extra Dimension," Phys. Rev. Lett. 83 (1999) 3370, hep-th/9905221. 7. E. Verlinde and H. Verlinde, "RG Flow, Gravity and the Cosmological Constant," JHEP 0005 (2000) 034, hep-th/9912018. 8. S. Kachru, M. Schulz and E. Silverstein, "Self-tuning Flat Domain Walls in 5d Gravity and String Theory," Phys. Rev. D62 (2000) 045021, hepth/0001206; S. Kachru, M. Schulz and E. Silverstein, "Bounds on Curved Domain Walls in 5d Gravity," Phys. Rev. D62 (2000) 085003, hep-th/0002121. 9. N. Arkani-Hamed, S. Dimopoulos, N. Kaloper and R. Sundrum, "A Small Cosmological Constant from a Large Extra Dimension," Phys. Lett. B480 (2000) 193, hep-th/0001197. 10. V. Rubakov and M. Shaposhnikov, "Do We Live Inside a Domain Wall?," Phys. Lett. B125 (1983) 136. 11. L. Randall and R. Sundrum, "An Alternative to Compactification," Phys. Rev. Lett. 83 (1999) 4690, hep-th/9906064. 12. S. Giddings, E. Katz and L. Randall, "Linearized Gravity in Brane Backgrounds," JHEP 0003 (2000) 023, hep-th/0002091. 13. J. Maldacena, "The Large N Limit of Superconformal Field Theories and Supergravity," Adv. Theor. Math. Phys. 2 (1998) 231, hep-th/9711200; S. Gubser, I. Klebanov and A. Polyakov, "Gauge Theory Correlators from Noncritical String Theory," hep-th/9802109; E. Witten, "Anti-de Sitter Space and Holography," Adv. Theor. Math. Phys. 2 (1998) 253, hep-th/9802150. 14. W. Goldberger and M. Wise, "Modulus Stabilization with Bulk Fields," Phys. Rev. Lett. 83 (1999) 4922, hep-th/9907447. 15. P. Frampton and C. Vafa, "Conformal Approach to Particle Phenomenology," hep-th/9903226. 16. J. Maldacena, unpublished; E. Witten, unpublished; S. Gubser, "AdS/CFT and Gravity," hep-th/9912001. 17. H. Verlinde, "Holography and Compactification," Nucl. Phys. B580 (2000) 264, hep-th/9906182. 18. I. Klebanov and M. Strassler, "Supergravity and a Confining Gauge Theory: Duality Cascades and \SB Resolution of Naked Singularities," JHEP 0008 (2000) 052, hep-th/0007191;

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J. Maldacena and C. Nunez, "Towards the Large N Limit of Pure M = 1 Super Yang-Mills," hep-th/0008001. V. Rubakov and M. Shaposhnikov, "Extra Space-Time Dimensions: Towards a Solution to the Cosmological Constant Problem," Phys. Lett. B125 (1983) 139. S. Kachru, N. Seiberg and E. Silverstein, "SUSY Gauge Dynamics and Singularities of 4d M = 1 String Vacua," Nucl. Phys. B480 (1996) 170, hep-th/9605036; S. Kachru and E. Silverstein, "Singularities, Gauge Dynamics, and Nonperturbative Superpotentials in String Theory," Nucl. Phys. B482 (1996) 92, hep-th/9608194. R. Myers, "New Dimensions for Old Strings," Phys. Lett. B199 (1987) 371. S. Kachru, J. Kumar and E. Silverstein, "Vacuum Energy Cancellation in a Nonsupersymmetric String," Phys. Rev. D59 (1999) 106004, hepth/9807076. T. Banks, "Cosmological Breaking of Supersymmetry? Or Little Lambda Goes Back to the Future 2," hep-th/0007146. J. Polchinski and M. Strassler, "The String Dual of a Confining FourDimensional Gauge Theory," hep-th/0003136. R. Bousso and J. Polchinski, "Quantization of Four-Form Fluxes and Dynamical Neutralization of the Cosmological Constant," JHEP 0006 (2000) 006, hep-th/0004134. J. Brown and C. Teitelboim, "Dynamical Neutralization of the Cosmological Constant," Phys. Lett. B195 (1987) 177; J. Brown and C. Teitelboim, "Neutralization of the Cosmological Constant by Membrane Creation," Nucl. Phys. B297 (1988) 787. J. Feng, J. March-Russell, S. Sethi and F. Wilczek, "Saltatory Relaxation of the Cosmological Constant," hep-th/0005276. T. Banks, M. Dine and L. Motl, "On Anthropic Solutions of the Cosmological Constant Problem," hep-th/0007206. G. Horowitz, I. Low and A. Zee, "Selftuning in an Outgoing Brane Wave Model," hep-th/0004206. Z. Kakushadze, "Self-tuning and Conformality," hep-th/0009199. E. Witten, "The Cosmological Constant from the Viewpoint of String Theory," hep-ph/0002297. S. Forste, Z. Lalak, S. Lavignac and H. Nilles, "The Cosmological Constant Problem from a Brane World Perspective," JHEP 0009 (2000) 034, hep-th/0006139. M. Green, J. Schwarz and E. Witten, "Superstring Theory: Volume II,"

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Cambridge University Press, 1987. 34. P. Candelas, M. Lynker and R. Schimmrigk, "Calabi-Yau Manifolds in Weighted P(4)," Nucl. Phys. B341 (1990) 383. 35. B. Greene, "String Theory on Calabi-Yau Manifolds," hep-th/9702155. 36. N. Seiberg and E. Witten, "Electric-Magnetic Duality, Monopole Condensation, and Confinement in J\f = 2 Supersymmetric Yang-Mills Theory," Nucl. Phys. B426 (1994) 19, hep-th/9407087. 37. P. Candelas and X. de la Ossa, "Moduli Space of Calabi-Yau Manifolds," Nucl. Phys. B355 (1991) 455. 38. X.G. Wen and E. Witten, "World Sheet Instantons and the Peccei-Quinn Symmetry," Phys. Lett. B166 (1986) 397. 39. M. Dine, N. Seiberg, X.G. Wen and E. Witten, "Nonperturbative Effects on the String World Sheet," Nucl. Phys. B278 (1986) 769. 40. D. Cox and S. Katz, Mirror Symmetry and Algebraic Geometry, Mathematical Surveys and Monographs No. 68, American Mathematical Society, 1999. 41. P. Candelas, X. de la Ossa, P. Green and L. Parkes, "A Pair of Calabi-Yau Manifolds as an Exactly Soluble Superconformal Theory," Nucl. Phys. B359 (1991) 21. 42. A. Strominger, S.T. Yau and E. Zaslow, "Mirror Symmetry is T Duality," Nucl. Phys. B479 (1996) 243, hep-th/9606040. 43. H. Ooguri, Y. Oz and Z. Yin, "D-Branes on Calabi-Yau Spaces and their Mirrors," Nucl. Phys. B477 (1996) 407, hep-th/9606112. 44. R. McLean, "Deformations of Calibrated Submanifolds," Duke University PhD thesis, Duke preprint 96-01: see www.math.duke.edu/ preprints/1996.html. 45. D. Morrison, "Geometric Aspects of Mirror Symmetry," math.ag/0007090. 46. C. Vafa, "Extending Mirror Conjecture to Calabi-Yau with Bundles," hep-th/9804131. 47. S. Kachru, S. Katz, A. Lawrence and J. McGreevy, "Open String Instantons and Superpotentials," Phys. Rev. D62 (2000) 026001, hepth/9912151. 48. H. Ooguri and C. Vafa, "Knot Invariants and Topological Strings," Nucl. Phys. B577 (2000) 419, hep-th/9912123. 49. I. Brunner, M. Douglas, A. Lawrence and C. Romelsberger, "D-branes on the Quintic," JHEP 0008 (2000) 015, hep-th/9906200. 50. S. Kachru and J. McGreevy, "Supersymmetric Three Cycles and Supersymmetry Breaking," Phys. Rev. D61 (2000) 026001, hep-th/9908135. 51. M. Douglas, "Topics in D-Geometry," Class. Quant. Grav. 17 (2000)

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1057, hep-th/9910170. 52. S. Kachru, S. Katz, A. Lawrence and J. McGreevy, "Mirror Symmetry for Open Strings," hep-th/0006047.

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LARGE N FIELD THEORIES, STRING THEORY A N D GRAVITY"

Lyman

JUAN MALDACENA Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA E-mail: [email protected]

We describe the holographic correspondence between field theories and string/M theory, focusing on the relation between compactifications of string/M theory on Anti-de Sitter spaces and conformal field theories. We review the background for this correspondence and discuss its motivations and the evidence for its correctness. We describe the main results that have been derived from the correspondence in the regime that the field theory is approximated by classical or semiclassical gravity. We focus on the case of the M = 4 supersymmetric gauge theory in four dimensions.

1

General Introduction

These lecture notes are taken out of the review : . A more complete set of references is given there. Even though though string theory is normally used as a theory of quantum gravity, it is not how string theory was originally discovered. String theory was discovered in an attempt to describe the large number of mesons and hadrons that were experimentally discovered in the 1960's. The idea was to view all these particles as different oscillation modes of a string. The string idea described well some features of the hadron spectrum. For example, the mass of the lightest hadron with a given spin obeys a relation like m2 ~ TJ2 + const. This is explained simply by assuming that the mass and angular momentum come from a rotating, relativistic string of tension T. It was later discovered that hadrons and mesons are actually made of quarks and that they are described by QCD. QCD is a gauge theory based on the group SU(3). This is sometimes stated by saying that quarks have three colors. QCD is asymptotically free, meaning that the effective coupling constant decreases as the energy increases. At low energies QCD becomes strongly coupled and it is not easy to perform calculations. One possible approach is to use numerical simulations on the lattice. This is at present the best available tool to do calculations in QCD at low energies. It was suggested by 't Hooft that the theory might simplify when the number of colors TV is large 2 . The hope was that one could solve exactly "These lecture notes are based on the Review written by O. Aharony, S. Gubser, J. Maldacena, H. Ooguri and Y. Oz, hep-th/9905111.

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the theory with N = oo, and then one could do an expansion in 1/N = 1/3. Furthermore, as explained in the next section, the diagrammatic expansion of the field theory suggests that the large N theory is a free string theory and that the string coupling constant is 1/N. If the case with N = 3 is similar to the case with TV = oo then this explains why the string model gave the correct relation between the mass and the angular momentum. In this way the large N limit connects gauge theories with string theories. The 't Hooft argument, reviewed below, is very general, so it suggests that different kinds of gauge theories will correspond to different string theories. In this review we will study this correspondence between string theories and the large N limit of field theories. We will see that the strings arising in the large N limit of field theories are the same as the strings describing quantum gravity. Namely, string theory in some backgrounds, including quantum gravity, is equivalent (dual) to a field theory. Strings are not consistent in four flat dimensions. Indeed, if one wants to quantize a four dimensional string theory an anomaly appears that forces the introduction of an extra field, sometimes called the "Liouville" field3. This field on the string worldsheet may be interpreted as an extra dimension, so that the strings effectively move in five dimensions. One might qualitatively think of this new field as the "thickness" of the string. If this is the case, why do we say that the string moves in five dimensions? The reason is that, like any string theory, this theory will contain gravity, and the gravitational theory will live in as many dimensions as the number of fields we have on the string. It is crucial then that the five dimensional geometry is curved, so that it can correspond to a four dimensional field theory, as described in detail below. The argument that gauge theories are related to string theories in the large N limit is very general and is valid for basically any gauge theory. In particular we could consider a gauge theory where the coupling does not run (as a function of the energy scale). Then, the theory is conformally invariant. It is quite hard to find quantum field theories that are conformally invariant. In supersymmetric theories it is sometimes possible to prove exact conformal invariance. A simple example, which will be the main example in this review, is the supersymmetric SU(N) (or U(N)) gauge theory in four dimensions with four spinor supercharges (Af = 4). Four is the maximal possible number of supercharges for a field theory in four dimensions. Besides the gauge fields (gluons) this theory contains also four fermions and six scalar fields in the adjoint representation of the gauge group. The Lagrangian of such theories is completely determined by supersymmetry. There is a global S£/(4) i?-symmetry that rotates the six scalar fields and the four fermions. The conformal group in four dimensions is 5 0 ( 4 , 2), including the usual Poincare transformations as well as

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scale transformations and special conformal transformations (which include the inversion symmetry x>* —> x^/x2). These symmetries of the field theory should be reflected in the dual string theory. The simplest way for this to happen is if the five dimensional geometry has these symmetries. Locally there is only one space with 50(4,2) isometries: five dimensional Anti-de-Sitter space, or AdS$. Anti-de Sitter space is the maximally symmetric solution of Einstein's equations with a negative cosmological constant. In this supersymmetric case we expect the strings to also be supersymmetric. We said that superstrings move in ten dimensions. Now that we have added one more dimension it is not surprising any more to add five more to get to a ten dimensional space. Since the gauge theory has an SU(A) ~ 50(6) global symmetry it is rather natural that the extra five dimensional space should be a five sphere, 5 5 . So, we conclude that N = 4 U(N) Yang-Mills theory could be the same as ten dimensional superstring theory on AdS$ x 5 5 4 . Here we have presented a very heuristic argument for this equivalence; later we will be more precise and give more evidence for this correspondence. The relationship we described between gauge theories and string theory on Anti-de-Sitter spaces was motivated by studies of D-branes and black holes in strings theory. D-branes are solitons in string theory 5 . They come in various dimensionalities. If they have zero spatial dimensions they are like ordinary localized, particle-type soliton solutions, analogous to the 't Hooft-Polyakov6,7 monopole in gauge theories. These are called D-zero-branes. If they have one extended dimension they are called D-one-branes or D-strings. They are much heavier than ordinary fundamental strings when the string coupling is small. In fact, the tension of all D-branes is proportional to l/gs, where gs is the string coupling constant. D-branes are defined in string perturbation theory in a very simple way: they are surfaces where open strings can end. These open strings have some massless modes, which describe the oscillations of the branes, a gauge field living on the brane, and their fermionic partners. If we have N coincident branes the open strings can start and end on different branes, so they carry two indices that run from one to N. This in turn implies that the low energy dynamics is described by a U(N) gauge theory. D-p-branes are charged under p + 1-form gauge potentials, in the same way that a O-brane (particle) can be charged under a one-form gauge potential (as in electromagnetism). These p + 1-form gauge potentials have p + 2-form field strengths, and they are part of the massless closed string modes, which belong to the supergravity (SUGRA) multiplet containing the massless fields in flat space string theory (before we put in any D-branes). If we now add D-branes they generate a flux of the corresponding field strength, and this flux in turn contributes to the stress energy tensor so the geometry becomes curved. Indeed it is possible to

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find solutions of the supergravity equations carrying these fluxes. Supergravity is the low-energy limit of string theory, and it is believed that these solutions may be extended to solutions of the full string theory. These solutions are very similar to extremal charged black hole solutions in general relativity, except that in this case they are black branes with p extended spatial dimensions. Like black holes they contain event horizons. If we consider a set of iV coincident D-3-branes the near horizon geometry turns out to be AdS$ x 5 5 . On the other hand, the low energy dynamics on their worldvolume is governed by a U(N) gauge theory with M = 4 supersymmetry 8 . These two pictures of D-branes are perturbatively valid for different regimes in the space of possible coupling constants. Perturbative field theory is valid when gsN is small, while the low-energy gravitational description is perturbatively valid when the radius of curvature is much larger than the string scale, which turns out to imply that gsN should be very large. As an object is brought closer and closer to the black brane horizon its energy measured by an outside observer is redshifted, due to the large gravitational potential, and the energy seems to be very small. On the other hand low energy excitations on the branes are governed by the Yang-Mills theory. So, it becomes natural to conjecture that Yang-Mills theory at strong coupling is describing the near horizon region of the black brane, whose geometry is AdS$ x S 5 . The first indications that this is the case came from calculations of low energy graviton absorption cross sections 9 ' 10 ' 11 . It was noticed there that the calculation done using gravity and the calculation done using super Yang-Mills theory agreed. These calculations, in turn, were inspired by similar calculations for coincident D1-D5 branes. In this case the near horizon geometry involves AdSz x S 3 and the low energy field theory living on the D-branes is a 1 + 1 dimensional conformal field theory. In this D1-D5 case there were numerous calculations that agreed between the field theory and gravity. First black hole entropy for extremal black holes was calculated in terms of the field theory in 12 , and then agreement was shown for near extremal black holes 13,14 and for absorption cross sections 15>16.17. More generally, we will see that correlation functions in the gauge theory can be calculated using the string theory (or gravity for large gsN) description, by considering the propagation of particles between different points in the boundary of AdS, the points where operators are inserted 18,19 . Supergravities on AdS spaces were studied very extensively, see 20 ' 21 for reviews. See also 2 2 ' 2 3 for earlier hints of the correspondence. One of the main points of these lectures will be that the strings coming from gauge theories are very much like the ordinary superstrings that have been studied during the last 20 years. The only particular feature is that they are moving on a curved geometry (anti-de Sitter space) which has a

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boundary at spatial infinity. The boundary is at an infinite spatial distance, but a light ray can go to the boundary and come back in finite time. Massive particles can never get to the boundary. The radius of curvature of Antide Sitter space depends on N so that large N corresponds to a large radius of curvature. Thus, by taking N to be large we can make the curvature as small as we want. The theory in AdS includes gravity, since any string theory includes gravity. So in the end we claim that there is an equivalence between a gravitational theory and a field theory. However, the mapping between the gravitational and field theory degrees of freedom is quite non-trivial since the field theory lives in a lower dimension. In some sense the field theory (or at least the set of local observables in the field theory) lives on the boundary of spacetime. One could argue that in general any quantum gravity theory in AdS defines a conformal field theory (CFT) "on the boundary". In some sense the situation is similar to the correspondence between three dimensional ChernSimons theory and a WZW model on the boundary 2 4 . This is a topological theory in three dimensions that induces a normal (non-topological) field theory on the boundary. A theory which includes gravity is in some sense topological since one is integrating over all metrics and therefore the theory does not depend on the metric. Similarly, in a quantum gravity theory we do not have any local observables. Notice that when we say that the theory includes "gravity on AdS" we are considering any finite energy excitation, even black holes in AdS. So this is really a sum over all spacetimes that are asymptotic to AdS at the boundary. This is analogous to the usual flat space discussion of quantum gravity, where asymptotic flatness is required, but the spacetime could have any topology as long as it is asymptotically flat. The asymptotically AdS case as well as the asymptotically flat cases are special in the sense that one can choose a natural time and an associated Hamiltonian to define the quantum theory. Since black holes might be present this time coordinate is not necessarily globally well-defined, but it is certainly well-defined at infinity. If we assume that the conjecture we made above is valid, then the U(N) YangMills theory gives a non-perturbative definition of string theory on AdS. And, by taking the limit N —> oo, we can extract the (ten dimensional string theory) flat space physics, a procedure which is in principle (but not in detail) similar to the one used in matrix theory 25 . The fact that the field theory lives in a lower dimensional space blends in perfectly with some previous speculations about quantum gravity. It was suggested 26,27 that quantum gravity theories should be holographic, in the sense that physics in some region can be described by a theory at the boundary with no more than one degree of freedom per Planck area. This "holographic" principle comes from thinking about the Bekenstein bound which states that

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the maximum amount of entropy in some region is given by the area of the region in Planck units 28 . The reason for this bound is that otherwise black hole formation could violate the second law of thermodynamics. We will see that the correspondence between field theories and string theory on AdS space (including gravity) is a concrete realization of this holographic principle. Other reviews of this subject are 2 9 ' 3 0 ' 3 1 , 3 2 ' 1 . 2

The Correspondence

In this section we will present an argument connecting type IIB string theory compactified on AdS5 x S5 to Af = 4 super-Yang-Mills theory 4 . Let us start with type IIB string theory in flat, ten dimensional Minkowski space. Consider N parallel D3 branes that are sitting together or very close to each other (the precise meaning of "very close" will be defined below). The D3 branes are extended along a (3 + 1) dimensional plane in (9 + 1) dimensional spacetime. String theory on this background contains two kinds of perturbative excitations, closed strings and open strings. The closed strings are the excitations of empty space and the open strings end on the D-branes and describe excitations of the D-branes. If we consider the system at low energies, energies lower than the string scale l/ls, then only the massless string states can be excited, and we can write an effective Lagrangian describing their interactions. The closed string massless states give a gravity supermultiplet in ten dimensions, and their low-energy effective Lagrangian is that of type IIB supergravity. The open string massless states give an M = 4 vector supermultiplet in (3 + 1) dimensions, and their low-energy effective Lagrangian is that of M = 4 U(N) super-Yang-Mills theory 8 , 3 3 . The complete effective action of the massless modes will have the form S = Obulk + S'brane + ^int •

(1)

S^uik i s the action of ten dimensional supergravity, plus some higher derivative corrections. Note that the Lagrangian (1) involves only the massless fields but it takes into account the effects of integrating out the massive fields. It is not renormalizable (even for the fields on the brane), and it should only be understood as an effective description in the Wilsonian sense, i.e. we integrate out all massive degrees of freedom but we do not integrate out the massless ones. The brane action S'brane is defined on the (3 + 1) dimensional brane worldvolume, and it contains the TV = 4 super-Yang-Mills Lagrangian plus some higher derivative corrections, for example terms of the form a' 2 Tr(F 4 ). Finally, Slnt describes the interactions between the brane modes and the bulk modes. The leading terms in this interaction Lagrangian can be obtained by

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covariantizing the brane action, introducing the background metric for the brane 3 4 . We can expand the bulk action as a free quadratic part describing the propagation of free massless modes (including the graviton), plus some interactions which are proportional to positive powers of the square root of the Newton constant. Schematically we have Stuik ~ ^ j

f y/gK ~ Udhf

+ K{dh)2h + •••,

(2)

where we have written the metric as g = r\ 4- K,h. We indicate explicitly the dependence on the graviton, but the other terms in the Lagrangian, involving other fields, can be expanded in a similar way. Similarly, the interaction Lagrangian Sint is proportional to positive powers of K. If we take the low energy limit, all interaction terms proportional to K drop out. This is the well known fact that gravity becomes free at long distances (low energies). In order to see more clearly what happens in this low energy limit it is convenient to keep the energy fixed and send / s —> 0 (a' —> 0) keeping all the dimensionless parameters fixed, including the string coupling constant and N. In this limit the coupling K ~ gsa'2 —> 0, so that the interaction Lagrangian relating the bulk and the brane vanishes. In addition all the higher derivative terms in the brane action vanish, leaving just the pure Af = 4 U(N) gauge theory in 3 + 1 dimensions, which is known to be a conformal field theory. And, the supergravity theory in the bulk becomes free. So, in this low energy limit we have two decoupled systems. On the one hand we have free gravity in the bulk and on the other hand we have the four dimensional gauge theory. Next, we consider the same system from a different point of view. D-branes are massive charged objects which act as a source for the various supergravity fields. We can find a D3 brane solution 35 of supergravity, of the form ds2 = rl/2{~dt2

+ dx\ + dx\ + dx2) + fl'2{dr2

^5 = (1 + *)dtdxidx2(ixsdf~1 / = 1+ ^

,

,

4

+ r2dQ2) , (3)

l2

R = A-K9sa N .

Note that since gtt is non-constant, the energy Ep of an object as measured by an observer at a constant position r and the energy E measured by an observer at infinity are related by the redshift factor E = f~l,iEp

.

(4)

This means that the same object brought closer and closer to r = 0 would appear to have lower and lower energy for the observer at infinity. Now we

81

take the low energy limit in the background described by equation (3). There are two kinds of low energy excitations (from the point of view of an observer at infinity). We can have massless particles propagating in the bulk region with wavelengths that becomes very large, or we can have any kind of excitation that we bring closer and closer to r = 0. In the low energy limit these two types of excitations decouple from each other. The bulk massless particles decouple from the near horizon region (around r = 0) because the low energy absorption cross section goes like a ~ ui3R8 9 , 1 ° , where u is the energy. This can be understood from the fact that in this limit the wavelength of the particle becomes much bigger than the typical gravitational size of the brane (which is of order R). Similarly, the excitations that live very close to r = 0 find it harder and harder to climb the gravitational potential and escape to the asymptotic region. In conclusion, the low energy theory consists of two decoupled pieces, one is free bulk supergravity and the second is the near horizon region of the geometry. In the near horizon region, r
dr + dx\ + dx\ + dx\) + R2— + R2dtt\,

(5)

which is the geometry of AdS5 x S5. We see that both from the point of view of a field theory of open strings living on the brane, and from the point of view of the supergravity description, we have two decoupled theories in the low-energy limit. In both cases one of the decoupled systems is supergravity in flat space. So, it is natural to identify the second system which appears in both descriptions. Thus, we are led to the conjecture that M = 4 U(N) super-Yang-Mills theory in 3 + 1 dimensions is the same as (or dual to) type IIB superstring theory on AdS$ x S5 4 . We could be a bit more precise about the near horizon limit and how it is being taken. Suppose that we take a' —> 0, as we did when we discussed the field theory living on the brane. We want to keep fixed the energies of the objects in the throat (the near-horizon region) in string units, so that we can consider arbitrary excited string states there. This implies that \fa'Ep ~ fixed. For small a' (4) reduces to E ~ Eprj\fo!:. Since we want to keep fixed the energy measured from infinity, which is the way energies are measured in the field theory, we need to take r -t 0 keeping r/a' fixed. It is then convenient to define a new variable U = r/a', so that the metric becomes ds2

,

J4TT

T

o

N^~dl2

, 9

+

^

, o

+ dX2

,

+ dx

^

Ov

+

f~,

z~z CLLJ

^4n9sN~[p~

+

V^9sNdft25 (6)

82

This can also be seen by considering a D3 brane sitting at r. This corresponds to giving a vacuum expectation value to one of the scalars in the Yang-Mills theory. When we take the a1 —» 0 limit we want to keep the mass of the 'W-boson" fixed. This mass, which is the mass of the string stretching between the branes sitting at r = 0 and the one at f, is proportional to U = r/a', so this quantity should remain fixed in the decoupling limit. A U(N) gauge theory is essentially equivalent to a free U(l) vector multiplet times an SU(N) gauge theory, up to some ZJV identifications (which affect only global issues). In the dual string theory all modes interact with gravity, so there are no decoupled modes. Therefore, the bulk AdS theory is describing the SU(N) part of the gauge theory. In fact we were not precise when we said that there were two sets of excitations at low energies, the excitations in the asymptotic flat space and the excitations in the near horizon region. There are also some zero modes which live in the region connecting the "throat" (the near horizon region) with the bulk, which correspond to the 1/(1) degrees of freedom mentioned above. The U{\) vector supermultiplet includes six scalars which are related to the center of mass motion of all the branes 3 6 . From the AdS point of view these zero modes live at the boundary, and it looks like we might or might not decide to include them in the AdS theory. Depending on this choice we could have a correspondence to an SU(N) or a U(N) theory. The 1/(1) center of mass degree of freedom is related to the topological theory of Bfields on AdS37; if one imposes local boundary conditions for these B-fields at the boundary of AdS one finds a C/(l) gauge field living at the boundary 38 , as is familiar in Chern-Simons theories 24 ' 39 . These modes living at the boundary are sometimes called singletons (or doubletons) 4 0 ' 4 1 ' 4 2 ' 4 3 , 4 4 ' 4 5 ' 4 6 ' 4 7 , 4 8 . Anti-de-Sitter space has a large group of isometries, which is SO(4,2) for the case at hand. This is the same group as the conformal group in 3 + 1 dimensions. Thus, the fact that the low-energy field theory on the brane is conformal is reflected in the fact that the near horizon geometry is Anti-de-Sitter space. We also have some supersymmetries. The number of supersymmetries is twice that of the full solution (3) containing the asymptotic region 36 . This doubling of supersymmetries is viewed in the field theory as a consequence of superconformal invariance, since the superconformal algebra has twice as many fermionic generators as the corresponding Poincare superalgebra. We also have an 50(6) symmetry which rotates the S 5 . This can be identified with the SU{A)R R-symmetry group of the field theory. In fact, the whole supergroup is the same for the M = 4 field theory and the AdS5 x S5 geometry, so both sides of the conjecture have the same spacetime symmetries. We will discuss in more detail the matching between the two sides of the correspondence in section 3.

83

In the above derivation the field theory is naturally defined on R ' , but we could also think of the conformal field theory as defined on S 3 x R by redefining the Hamiltonian. Since the isometries of AdS are in one to one correspondence with the generators of the conformal group of the field theory, we can conclude that this new Hamiltonian \{PQ + KQ) can be associated on AdS to the generator of translations in global time. This formulation of the conjecture is more useful since in the global coordinates there is no horizon. When we put the field theory on S3 the Coulomb branch is lifted and there is a unique ground state. This is due to the fact that the scalars 7 in the field theory are conformally coupled, so there is a term of the form J dixTi(cj)2)TZ in the Lagrangian, where TZ is the curvature of the four-dimensional space on which the theory is defined. Due to the positive curvature of S3 this leads to a mass term for the scalars 19 , lifting the moduli space. The parameter N appears on the string theory side as the flux of the five-form Ramond-Ramond field strength on the 5 5 , F5 = N. (7) / .s 5 From the physics of D-branes we know that the Yang-Mills coupling is related to the string coupling through 5 , 4 9 T

= -r~ + IT 2?r

=

— + ^r -

(8)

9YM 9s 2TT where we have also included the relationship of the 9 angle to the expectation value of the RR scalar %• We have written the couplings in this fashion because both the gauge theory and the string theory have an SL(2, Z) self-duality symmetry under which r -4 (ar + b)/(cr + d) (where a, b, c, d are integers with ad — be = 1). In fact, SX(2,Z) is a conjectured strong-weak coupling duality symmetry of type IIB string theory in flat space 5 0 , and it should also be a symmetry in the present context since all the fields that are being turned on in the AdS$ x S5 background (the metric and the five form field strength) are invariant under this symmetry. The connection between the SL(2, Z) duality symmetries of type IIB string theory and M = 4 SYM was noted in 51>52>53. The string theory seems to have a parameter that does not appear in the gauge theory, namely a', which sets the string tension and all other scales in the string theory. However, this is not really a parameter in the theory if we do not compare it to other scales in the theory, since only relative scales are meaningful. In fact, only the ratio of the radius of curvature to a' is a parameter, but not a' and the radius of curvature independently. Thus, a' will disappear from any final physical quantity we compute in this theory. It is

84

sometimes convenient, especially when one is doing gravity calculations, to set the radius of curvature to one. This can be achieved by writing the metric as ds2 = R2ds2, and rewriting everything in terms of g. With these conventions GN ~ 1/./V2 and a' ~ l/\/gsN. This implies that any quantity calculated purely in terms of the gravity solution, without including stringy effects, will be independent of gsN and will depend only on N. a' corrections to the gravity results give corrections which are proportional to powers of 1/' \fgsN. Now, let us address the question of the validity of various approximations. The analysis of loop diagrams in the field theory shows that we can trust the perturbative analysis in the Yang-Mills theory when g2YMN ~ gsN ~ ^ - « 1.

(9)

Note that we need gYMN to be small and not just g\M- On the other hand, the classical gravity description becomes reliable when the radius of curvature R of AdS and of S5 becomes large compared to the string length, R4 -^ ~ gsN ~ g2YMN » 1.

(10)

We see that the gravity regime (10) and the perturbative field theory regime (9) are perfectly incompatible. In this fashion we avoid any obvious contradiction due to the fact that the two theories look very different. This is the reason that this correspondence is called a "duality". The two theories are conjectured to be exactly the same, but when one side is weakly coupled the other is strongly coupled and vice versa. This makes the correspondence both hard to prove and useful, as we can solve a strongly coupled gauge theory via classical supergravity. Notice that in (9)(10) we implicitly assumed that gs < 1. If gs > 1 we can perform an SL(2, Z) duality transformation and get conditions similar to (9)(10) but with gs —)• l/gs- So, we cannot get into the gravity regime (10) by taking N small (N = 1,2,..) and gs very large, since in that case the D-string becomes light and renders the gravity approximation invalid. Another way to see this is to note that the radius of curvature in Planck units is R4 jl4 ~ N. So, it is always necessary, but not sufficient, to have large N in order to have a weakly coupled supergravity description. One might wonder why the above argument was not a proof rather than a conjecture. It is not a proof because we did not treat the string theory nonperturbatively (not even non-perturbatively in a'). We could also consider different forms of the conjecture. In its weakest form the gravity description would be valid for large gsN, but the full string theory on AdS might not agree

85

with the field theory. A not so weak form would say that the conjecture is valid even for finite gaN, but only in the N -» oo limit (so that the a' corrections would agree with the field theory, but the gs corrections may not). The strong form of the conjecture, which is the most interesting one and which we will assume here, is that the two theories are exactly the same for all values of gs and N. In this conjecture the spacetime is only required to be asymptotic to AdS$ x S5 as we approach the boundary. In the interior we can have all kinds of processes; gravitons, highly excited fundamental string states, D-branes, black holes, etc. Even the topology of spacetime can change in the interior. The Yang-Mills theory is supposed to effectively sum over all spacetimes which are asymptotic to AdS$ x S5. This is completely analogous to the usual conditions of asymptotic flatness. We can have black holes and all kinds of topology changing processes, as long as spacetime is asymptotically flat. In this case asymptotic flatness is replaced by the asymptotic AdS behavior. 2.1

Brane Probes and Multicenter Solutions

The moduli space of vacua of the M = 4 U{N) gauge theory is (R )N/SN, parametrizing the positions of the N branes in the six dimensional transverse space. In the supergravity solution one can replace N

N

i=l '

1 '

and still have a solution to the supergravity equations. We see that if |r| S> [r^j then the two solutions are basically the same, while when we go to r ~ T{ the solution starts looking like the solution of a single brane. Of course, we cannot trust the supergravity solution for a single brane (since the curvature in Planck units is proportional to a negative power of N). What we can do is separate the N branes into groups of Nt branes with gsNi 3> 1 for all i. Then we can trust the gravity solution everywhere. Another possibility is to separate just one brane (or a small number of branes) from a group of N branes. Then we can view this brane as a D3-brane in the AdS$ background which is generated by the other branes (as described above). A string stretching between the brane probe and the N branes appears in the gravity description as a string stretching between the D3-brane and the horizon of AdS. This seems a bit surprising at first since the proper distance to the horizon is infinite. However, we get a finite result for the energy of this state once we remember to include the redshift factor. The D3-branes in AdS (like any D3-branes in string theory) are described at low energies by the Born-Infeld action, which is the Yang-Mills action plus some higher derivative

86

corrections. This seems to contradict, at first sight, the fact that the dual field theory (coming from the original branes) is just the pure Yang-Mills theory. In order to understand this point more precisely let us write explicitly the bosonic part of the Born-Infeld action for a D-3 brane in AdS 34 ,

S = ~ {2ir) ,_ s*g a12,2 J Id4xf-'[ s

\j-

det(7/a/3 + fdardpr

+ r2fgijdaPdpQJ

+ 2TTQ'y/fFaf,)

- 1

A-wgsal2N (12) where 6l are angular coordinates on the 5-sphere. We can easily check that if we define a new coordinate U = r/a', then all the a' dependence drops out of this action. Since U (which has dimensions of energy) corresponds to the mass of the W bosons in this configuration, it is the natural way to express the Higgs expectation value that breaks U(N + 1) to U(N)xU(l). In fact, the action (12) is precisely the low-energy effective action in the field theory for the massless U(l) degrees of freedom, that we obtain after integrating out the massive degrees of freedom (W bosons). We can expand (12) in powers of dU and we see that the quadratic term does not have any correction, which is consistent with the non-renormalization theorem for Af = 4 super-Yang-Mills 5 4 . The (dU)4 term has only a one-loop correction, and this is also consistent with another non-renormalization theorem 5 5 . This one-loop correction can be evaluated explicitly in the gauge theory and the result agrees with the supergravity result56. It is possible to argue, using broken conformal invariance, that all terms in (12) are determined by the (dU)4 term 4 . Since the massive degrees of freedom that we are integrating out have a mass proportional to U, the action (12) makes sense as long as the energies involved are much smaller than U. In particular, we need dU/U < U. Since (12) has the form C{gsN{dU)2/U4), the higher order terms in (12) could become important in the supergravity regime, when gsN 3> 1. The Born Infeld action (12), as always, makes sense only when the curvature of the brane is small, but the deviations from a straight flat brane could be large. In this regime we can keep the non-linear terms in (12) while we still neglect the massive string modes and similar effects. Further gauge theory calculations for effective actions of D-brane probes include 57 ' 58 ' 59 . 2.2

The Field -H- Operator Correspondence

A conformal field theory does not have asymptotic states or an S-matrix, so the natural objects to consider are operators. For example, in Af = 4 super-

87

Yang-Mills we have a deformation by a marginal operator which changes the value of the coupling constant. Changing the coupling constant in the field theory is related by (8) to changing the coupling constant in the string theory, which is then related to the expectation value of the dilaton. The expectation value of the dilaton is set by the boundary condition for the dilaton at infinity. So, changing the gauge theory coupling constant corresponds to changing the boundary value of the dilaton. More precisely, let us denote by O the corresponding operator. We can consider adding the term J dixo{x)0(x) to the Lagrangian (for simplicity we assume that such a term was not present in the original Lagrangian, otherwise we consider (j>o(x) to be the total coefficient of 0(x) in the Lagrangian). According to the discussion above, it is natural to assume that this will change the boundary condition of the dilaton at the boundary of AdS to
-dt2 + dx\ H

h dx\ + dz2

More precisely, as argued in 1 8 ' 1 9 , it is natural to propose that gJ

d4x0{£)0(£)

)CFT

—

Zstring

4>(x,z)

= cp0(x)

(13)

where the left hand side is the generating function of correlation functions in the field theory, i.e. 0 is an arbitrary function and we can calculate correlation functions of O by taking functional derivatives with respect to (f>o and then setting 0O = 0. The right hand side is the full partition function of string theory with the boundary condition that the field <> / has the value cf>o on the boundary of AdS. Notice that o is a function of the four variables parametrizing the boundary of AdS^. A formula like (13) is valid in general, for any field (p. Therefore, each field propagating on AdS space is in a one to one correspondence with an operator in the field theory. There is a relation between the mass of the field
=2

+

d2

+ R2m2

(14)

Therefore, in order to get consistent behavior for a massive field, the boundary condition on the field in the right hand side of (13) should in general be changed

88

to
(15)

and eventually we would take the limit where e —>• 0. Since <j> is dimensionless, we see that fo has dimensions of [length] A ~ d which implies, through the left hand side of (13), that the associated operator O has dimension A (14). A more detailed derivation of this relation will be given in section 4, where we will verify that the two-point correlation function of the operator O behaves as that of an operator of dimension A 18>19. A similar relation between fields on AdS and operators in the field theory exists also for non-scalar fields, including fermions and tensors on AdS space. Correlation functions in the gauge theory can be computed from (13) by differentiating with respect to 0. Each differentiation brings down an insertion O, which sends a particle (a closed string state) into the bulk. Feynman diagrams can be used to compute the interactions of particles in the bulk. In the limit where classical supergravity is applicable, the only diagrams that contribute are the tree-level diagrams of the gravity theory (see for instance figure 1).

o-e-o

Figure 1: Correlation functions can be calculated (in the large gsN limit) in terms of supergravity Feynman diagrams. Here we see the leading contribution coming from a disconnected diagram plus connected pieces involving interactions of the supergravity fields in the bulk of AdS. At tree level, these diagrams and those related to them by crossing are the only ones that contribute to the four-point function.

This method of defining the correlation functions of a field theory which is dual to a gravity theory in the bulk of AdS space is quite general, and it applies in principle to any theory of gravity 19 . Any local field theory contains the stress tensor as an operator. Since the correspondence described above matches the stress-energy tensor with the graviton, this implies that the AdS theory includes gravity. It should be a well defined quantum theory of gravity since we should be able to compute loop diagrams. String theory provides such a theory. But if a new way of defining quantum gravity theories comes along we could consider those gravity theories in AdS, and they should correspond to some conformal field theory "on the boundary". In particular, we could

89 consider backgrounds of string theory of the form AdSs x M5 where M 5 is any Einstein manifold 6 0 ' 6 1 ' 6 2 . Depending on the choice of M5 we get different dual conformal field theories. Similarly, this discussion can be extended to any AdSd+i space, corresponding to a conformal field theory in d spacetime dimensions (for d > 1). 2.3

Holography

In this section we will describe how the AdS/CFT correspondence gives a holographic description of physics in AdS spaces. Let us start by explaining the Bekenstein bound, which states that the maximum entropy in a region of space is Smax — Area/AGN 2 8 , where the area is that of the boundary of the region. Suppose that we had a state with more entropy than Smax, then we show that we could violate the second law of thermodynamics. We can throw in some extra matter such that we form a black hole. The entropy should not decrease. But if a black hole forms inside the region its entropy is just the area of its horizon, which is smaller than the area of the boundary of the region (which by our assumption is smaller than the initial entropy). So, the second law has been violated. Note that this bound implies that the number of degrees of freedom inside some region grows as the area of the boundary of a region and not like the volume of the region. In standard quantum field theories this is certainly not possible. Attempting to understand this behavior leads to the "holographic principle", which states that in a quantum gravity theory all physics within some volume can be described in terms of some theory on the boundary which has less than one degree of freedom per Planck area 2 6 ' 2 7 (so that its entropy satisfies the Bekenstein bound). In the AdS/CFT correspondence we are describing physics in the bulk of AdS space by a field theory of one less dimension (which can be thought of as living on the boundary), so it looks like holography. However, it is hard to check what the number of degrees of freedom per Planck area is, since the theory, being conformal, has an infinite number of degrees of freedom, and the area of the boundary of AdS space is also infinite. Thus, in order to compare things properly we should introduce a cutoff on the number of degrees of freedom in the field theory and see what it corresponds to in the gravity theory. For this purpose let us write the metric of AdS as 2

ds2 = R2

2

(^) ^ 2 +(T^^ 2 +^ 2 )

(16)

In these coordinates the boundary of AdS is at r = 1. We saw above that when

90 we calculate correlation functions we have to specify boundary conditions at r = 1 — S and then take the limit of 5 —> 0. It is clear by studying the action of the conformal group on Poincare coordinates that the radial position plays the role of some energy scale, since we approach the boundary when we do a conformal transformation that localizes objects in the CFT. So, the limit <5 —> 0 corresponds to going to the UV of the field theory. When we are close to the boundary we could also use the Poincare coordinates , , -dt2 + dx2 + dz2 5—: , (17) zz in which the boundary is at z = 0. If we consider a particle or wave propagating in (17) or (16) we see that its motion is independent of R in the supergravity approximation. Furthermore, if we are in Euclidean space and we have a wave that has some spatial extent A in the x directions, it will also have an extent A in the z direction. This can be seen from (17) by eliminating A through the change of variables x —> Xx, z —> Xz. This implies that a cutoff at ds'=Rz

z~ 6

(18)

corresponds to a UV cutoff in the field theory at distances 5, with no factors of R (5 here is dimensionless, in the field theory it is measured in terms of the radius of the S 4 or S 3 that the theory lives on). Equation (18) is called the UV-IR relation 63 . Consider the case of Af = 4 SYM on a three-sphere of radius one. We can estimate the number of degrees of freedom in the field theory with a UV cutoff 5. We get S ~ N2S~3, (19) since the number of cells into which we divide the three-sphere is of order 1/63. In the gravity solution (16) the area in Planck units of the surface at r = 1 — 5, for S -C 1, is Area VS*R38-3 2 3

4G^ = ^ ^ r ~

N S

•

(20)

Thus, we see that the AdS/CFT correspondence saturates the holographic bound. 63 . One could be a little suspicious of the statement that gravity in AdS is holographic, since it does not seem to be saying much because in AdS space the volume and the boundary area of a given region scale in the same fashion as we increase the size of the region. In fact, any field theory in AdS would be holographic in the sense that the number of degrees of freedom within some (large enough) volume is proportional to the area (and also to the volume).

91

What makes this case different is that we have the additional parameter R, and then we can take AdS spaces of different radii (corresponding to different values of N in the SYM theory), and then we can ask whether the number of degrees of freedom goes like the volume or the area, since these have a different dependence on R. One might get confused by the fact that the surface r = 1 - S is really nine dimensional as opposed to four dimensional. From the form of the full metric on AdS§ x S5 we see that as we take 8 —> 0 the physical size of four of the dimensions of this nine dimensional space grow, while the other five, the S5, remain constant. So, we see that the theory on this nine dimensional surface becomes effectively four dimensional, since we need to multiply the metric by a factor that goes to zero as we approach the boundary in order to define a finite metric for the four dimensional gauge theory. 3

Tests of the A d S / C F T Correspondence

In this section we review the direct tests of the AdS/CFT correspondence. In section 2 we saw how string theory on AdS defines a partition function which can be used to define a field theory. Here we will review the evidence showing that this field theory is indeed the same as the conjectured dual field theory. We will focus here only on tests of the correspondence between the N = 4 SU(N) SYM theory and the type IIB string theory compactified on AdS$ x S 5 ; most of the tests described here can be generalized also to cases in other dimensions and/or with less supersymmetry, which will be described below. As described in section 2, the AdS/CFT correspondence is a strong/weak coupling duality. In the 't Hooft large N limit, it relates the region of weak field theory coupling A = 9yMN in the SYM theory to the region of high curvature (in string units) in the string theory, and vice versa. Thus, a direct comparison of correlation functions is generally not possible, since (with our current knowledge) we can only compute most of them perturbatively in A on the field theory side and perturbatively in 1/y/X on the string theory side. For example, as described below, we can compute the equation of state of the SYM theory and also the quark-anti-quark potential both for small A and for large A, and we obtain different answers, which we do not know how to compare since we can only compute them perturbatively on both sides. A similar situation arises also in many field theory dualities that were analyzed in the last few years (such as the electric/magnetic SL(2,Z) duality of the N = 4 SYM theory itself), and it was realized that there are several properties of these theories which do not depend on the coupling, so they can be compared

92 to test the duality. These are: • The global symmetries of the theory, which cannot change as we change the coupling (except for extreme values of the coupling). As discussed in section 2, in the case of the AdS/CFT correspondence we have the same supergroup SU(2,2\4) (whose bosonic subgroup is 50(4,2) x SU(4)) as the global symmetry of both theories. Also, both theories are believed to have a non-perturbative SL(2,Z) duality symmetry acting on their coupling constant r. These are the only symmetries of the theory on M. . Additional ZJV symmetries arise when the theories are compactified on non-simply-connected manifolds, and these were also successfully matched in 6 4 ' 3 7 6 . • Some correlation functions, which are usually related to anomalies, are protected from any quantum corrections and do not depend on A. The matching of these correlation functions will be described in section 3.2 below. • The spectrum of chiral operators does not change as the coupling varies, and it will be compared in section 3.1 below. • The moduli space of the theory also does not depend on the coupling. In the SU(N) field theory the moduli space is R /Sjv, parametrized by the eigenvalues of six commuting traceless N x N matrices. On the AdS side it is not clear exactly how to define the moduli space. As described in section 2.1, there is a background of string theory corresponding to any point in the field theory moduli space, but it is not clear how to see that this is the exact moduli space on the string theory side (especially since high curvatures arise for generic points in the moduli space). • The qualitative behavior of the theory upon deformations by relevant or marginal operators also does not depend on the coupling (at least for chiral operators whose dimension does not depend on the coupling, and in the absence of phase transitions). There are many more qualitative tests of the correspondence, such as the existence of confinement for the finite temperature theory 65 , which we will not discuss in this section. We will also not discuss here tests involving the behavior of the theory on its moduli space 5 7 , 6 6 ' 5 8 . b

Unlike most of the other tests described here, this test actually tests the finite N duality and not just the large N limit.

93 3.1

The Spectrum of Chiral Primary Operators

The Field Theory Spectrum The TV = 4 supersymmetry algebra in d = 4 has four generators Q-£ (and their complex conjugates QaA), where a is a Weyl-spinor index (in the 2 of the 50(3,1) Lorentz group) and A is an index in the 4 of the SU(4:)R R-symmetry group (lower indices A will be taken to transform in the 4 representation). They obey the algebra

{Qa,Qf} = {Q&A,Qf,B} = o, where az (i = 1,2,3) are the Pauli matrices and (a°)aa = — o~aa (we use the conventions of Wess and Bagger 67 ). TV = 4 supersymmetry in four dimensions has a unique multiplet which does not include spins greater than one, which is the vector multiplet. It includes a vector field AM (/i is a vector index of the 50(3,1) Lorentz group), four complex Weyl fermions XaA (in the 4 of SU(4)R), and six real scalars cf)1 (where i" is an index in the 6 of SU(4)R). The classical action of the supersymmetry generators on these fields is schematically given (for on-shell fields) by

{Qt \0B) ~ (<7n« A {Qtrxf}-^)^*1,

+

^WA\ (22)

with similar expressions for the action of the Q's, where a^" are the generators of the Lorentz group in the spinor representation, X>M is the covariant derivative, the field strength F^v = [Z>M,X>„], and we have suppressed the 517(4) ClebschGordan coefficients corresponding to the products 4 x 6 - > 4 , 4 x 4 — » 1 + 15 and 4 x 4 ->• 6 in the first three lines of (22). An yV = 4 supersymmetric field theory is uniquely determined by specifying the gauge group, and its field content is a vector multiplet in the adjoint of the gauge group. Such a field theory is equivalent to an TV = 2 theory with one hypermultiplet in the adjoint representation, or to an TV = 1 theory with three chiral multiplets $ 2 in the adjoint representation (in the 3 2 / 3 of the 5C/(3) x V{\)R C 5£/(4) fl which is left unbroken by the choice of a single TV = 1 SUSY generator) and a superpotential of the form W oc

94 CijfcTr($'$-7$fc). The interactions of the theory include a scalar potential proportional to ^2JJ Tr([(/>7,4>J)2), such that the moduli space of the theory is the space of commuting matrices (j)1 (I = 1, • • •, 6). The spectrum of operators in this theory includes all the gauge invariant quantities that can be formed from the fields described above. In this section we will focus on local operators which involve fields taken at the same point in space-time. For the SU(N) theory described above, properties of the adjoint representation of SU(N) determine that such operators necessarily involve a product of traces of products of fields (or the sum of such products). It is natural to divide the operators into single-trace operators and multipletrace operators. In the 't Hooft large N limit correlation functions involving multiple-trace operators are suppressed by powers of N compared to those of single-trace operators involving the same fields. We will discuss here in detail only the single-trace operators; the multiple-trace operators appear in operator product expansions of products of single-trace operators. It is natural to classify the operators in a conformal theory into primary operators and their descendants. In a superconformal theory it is also natural to distinguish between chiral primary operators, which are in short representations of the superconformal algebra and are annihilated by some of the supercharges, and non-chiral primary operators. Representations of the superconformal algebra are formed by starting with some state of lowest dimension, which is annihilated by the operators S and K^, and acting on it with the operators Q and PM. The Af = 4 supersymmetry algebra involves 16 real supercharges. A generic primary representation of the superconformal algebra will thus include 2 16 primaries of the conformal algebra, generated by acting on the lowest state with products of different supercharges; acting with additional supercharges always leads to descendants of the conformal algebra (i.e. derivatives). Since the supercharges have helicities ±1/2, the primary fields in such representations will have a range of helicities between A — 4 (if the lowest dimension operator ip has helicity A) and A + 4 (acting with more than 8 supercharges of the same helicity either annihilates the state or leads to a conformal descendant). In non-generic representations of the superconformal algebra a product of less than 16 different Q's annihilates the lowest dimension operator, and the range of helicities appearing is smaller. In particular, in the small representations of the AT = 4 superconformal algebra only up to 4 Q's of the same helicity acting on the lowest dimension operator give a non-zero result, and the range of helicities is between A — 2 and A + 2. For the N = 4 supersymmetry algebra (not including the conformal algebra) it is known that medium representations, whose range of helicities is 6, can also exist (they arise, for instance, on the moduli space of the SU(N) Af = 4 SYM

95 theory 68,69,70,71,72,73,74,75^. j t j g n o t c j e a r jf g^,^ m e c j i u m representations of the superconformal algebra 76 can appear in physical theories or not (there are no known examples). More details on the structure of representations of the M = 4 superconformal algebra may be found in 77 . 78 . 79 . 80 . 81 . 82 . 76 and references therein. In the U(l) M = 4 SYM theory (which is a free theory), the only gaugeinvariant "single trace" operators are the fields of the vector multiplet itself (which are (p1, XA, A 4 and FM„ = d^A^). These operators form an ultra-short representation of the M = 4 algebra whose range of helicities is from (—1) to 1 (acting with more than two supercharges of the same helicity on any of these states gives either zero or derivatives, which are descendants of the conformal algebra). All other local gauge invariant operators in the theory involve derivatives or products of these operators. This representation is usually called the doubleton representation, and it does not appear in the SU(N) SYM theory (though the representations which do appear can all be formed by tensor products of the doubleton representation). In the context of AdS space one can think of this multiplet as living purely on the boundary of the space83,84,85,86,87,43,88,89;9o,9i,92] a s e x pected for the 17(1) part of the original U(N) gauge group of the D3-branes (see the discussion in section 2). There is no known simple systematic way to compute the full spectrum of chiral primary operators of the TV = 4 SU(N) SYM theory, so we will settle for presenting the known chiral primary operators. The lowest component of a superconformal-primary multiplet is characterized by the fact that it cannot be written as a supercharge Q acting on any other operator. Looking at the action of the supersymmetry charges (22) suggests that generally operators built from the fermions and the gauge fields will be descendants (given by Q acting on some other fields), so one would expect the lowest components of the chiral primary representations to be built only from the scalar fields, and this turns out to be correct. Let us analyze the behavior of operators of the form Qhh—in = T r ^ 1 ^ 2 • • -(f>In). First we can ask if this operator can be written as {Q,ip} for any field ip. In the SUSY algebra (22) only commutators of 0 7 's appear on the right hand side, so we see that if some of the indices are antisymmetric the field will be a descendant. Thus, only symmetric combinations of the indices will be lowest components of primary multiplets. Next, we should ask if the multiplet built on such an operator is a (short) chiral primary multiplet or not. There are several different ways to answer this question. One possibility is to use the relation between the dimension of chiral primary operators and their R-symmetry representation 9 3 , 9 4 ' 9 5 , 9 6 , 9 7 , and to check if this relation is obeyed in the free field theory, where [ 0 / l / 2 " ' / n ] = n. In this way

96 we find that the representation is chiral primary if and only if the indices form a symmetric traceless product of n 6's (traceless representations are defined as those who give zero when any two indices are contracted). This is a representation of weight (0, n, 0) of SU(4)R; in this section we will refer to SU(4)R representations either by their dimensions in boldface or by their weights. Another way to check this is to see if by acting with Q's on these operators we get the most general possible states or not, namely if the representation contains "null vectors" or not (it turns out that in all the relevant cases "null vectors" appear already at the first level by acting with a single Q, though in principle there could be representations where "null vectors" appear only at higher levels). Using the SUSY algebra (22) it is easy to see that for symmetric traceless representations we get "null vectors" while for other representations we do not. For instance, let us analyze in detail the case n = 2. The symmetric product of two 6's is given by 6 x 6 —> 1 + 20'. The field in the 1 representation is Tr(V), for which [ Q ^ , T r ( 0 V ) ] ~ CAJBTr(\aBJ) where CAIB is a Clebsch-Gordan coefficient for 4 x 6 —> 4. The right-hand side is in the 4 representation, which is the most general representation that can appear in the product 4 x 1, so we find no null vectors at this level. On the other hand, if we look at the symmetric traceless product Tr({J(/>J}) = Tr(^7^>J) - \8IJ1r{(j)K~K) in the 20' representation, we find that {QA,Tv((f>V<j>J})} ~ Tr(A a B 0 K ) with the right-hand side being in the 20 representation (appearing in 4 x 6 —> 4 + 20), while the left-hand side could in principle be in the 4 x 20' —> 20 + 60. Since the 60 does not appear on the right-hand side (it is a "null vector") we identify that the representation built on the 20' is a short representation of the SUSY algebra. By similar manipulations (see 19.98>78>8i for more details) one can verify that chiral primary representations correspond exactly to symmetric traceless products of 6's. It is possible to analyze the chiral primary spectrum also by using M — 1 subalgebras of the H = 4 algebra. If we use an M = 1 subalgebra of the H — 4 algebra, as described above, the operators On include the chiral operators of the form T r ( $ n $' 2 • • • $*") (in a representation of SU(3) which is a symmetric product of 3's), but for a particular choice of the M = 1 subalgebra not all the operators On appear to be chiral (a short multiplet of the TV = 4 algebra includes both short and long multiplets of the M = 1 subalgebra). The last issue we should discuss is what is the range of values of n. The product of more than N commuting 0 N x N matrices can always be written as a sum of products of traces of less than N of the matrices, so it does not c

We can limit the discussion to commuting matrices since, as discussed above, commutators always lead to descendants, and we can write any product of matrices as a product of commuting matrices plus terms with commutators.

97 form an independent operator. This means that for n > N we can express the operator 0Ill2"In in terms of other operators, up to operators including commutators which (as explained above) are descendants of the SUSY algebra. Thus, we find that the short chiral primary representations are built on the operators On = 0^hl2"In^ with n = 2, 3, • • •, N, for which the indices are in the symmetric traceless product of n 6's (in a U(N) theory we would find the same spectrum with the additional representation corresponding t o n = 1). The superconformal algebra determines the dimension of these fields to be [On] = n, which is the same as their value in the free field theory. We argued above that these are the only short chiral primary representations in the SU(N) gauge theory, but we will not attempt to rigorously prove this here. The full chiral primary representations are obtained by acting on the fields On by the generators Q and P of the supersymmetry algebra. The representation built on On contains a total of 256 x j^n2(n2 — 1) primary states, of which half are bosonic and half are fermionic. Since these multiplets are built on a field of helicity zero, they will contain primary fields of helicities between (—2) and 2. The highest dimension primary field in the multiplet is (generically) of the form QiQ4On, and its dimension is n + 4. There is an elegant way to write these multiplets as traces of products of "twisted chiral Af = 4 superfields" 98>78; see a l s o " which checks some components of these superfields against the couplings to supergravity modes predicted on the basis of the DBI action for D3-branes in anti-de Sitter space 1 0 °. It is easy to find the form of all the fields in such a multiplet by using the algebra (22). For example, let us analyze here in detail the bosonic primary fields of dimension n + 1 in the multiplet. To get a field of dimension n + 1 we need to act on On with two supercharges (recall that [Q] = | ) . If we act with two supercharges Q^ of the same chirality, their Lorentz indices can be either antisymmetrized or symmetrized. In the first case we get a Lorentz scalar field in the (2, n — 2,0) representation of SU(4)R, which is of the schematic form ea0{Qa,

[Qp, On]} ~ e^TrikcAXpB^1

• • • 4> J "- 2 )+Tr([<^, ^

• • • 4>J-i) + Tr{\aA\pBKi

•••^ - ' ) . (24) Both of these fields are complex, with the complex conjugate fields given by the action of two Q's. Acting with one Q and one Q on the state On gives a

98 (real) Lorentz-vector field in the (1, n - 2,1) representation of form

SU(4)R,

of the

• • • ^ - * ) + K) a d Tr((I? M 0 J )
{Q Q , [Qa, On]} ~ Tr(A ayl Af ^

• A complex scalar field in the (0, n — 2,0) representation, given by Q 4 C„, of the form Tr(F^<(>h • • • 0 1 "- 2 ) + • • -. • A real scalar field in the (2,n - 4,2) representation, given by of the form e ^ e ^ T r ( A ^ 1 A^ 2 Xf> A| 2 0 7 ^ • • • 0 / —') + • •..

Q2Q2On,

• A complex vector field in the (1, n—4,1) representation, given by Q3QOn, of the form Tr^F^V(j)J4>h • • • 0 7 "- 2 ) + • • •. • An complex anti-symmetric 2-form field in the (2, n — 3,0) representation, given by Q2Q2On, of the form Tr(i^„[0 J l , 4>J*]4>h • • • 0 1 - 2 ) + • • -. • A symmetric tensor field in the (0,n — 2,0) representation, given by Q2Q2On,

of the form Tr(V{ll4>JVv}K11 • • • ^--=>) + • • •.

The spectrum of primary fields at dimension n + 3 is similar to that of dimension n + 1 (the same fields appear but in smaller SU(4)R representations), and at dimension n + 4 there is a single primary field, which is a real scalar in the (0,n — 4,0) representation, given by Q4QiOn, of the form Ti^F^,)// 1 • • •
YM

M

the Lagrangian density coupling to 6 (of the form Tr(F A F)). For later use we note that the chiral primary multiplets which contain scalars of dimension A < 4 are the n = 2 multiplet (which has a scalar in the 20' of dimension 2, a complex scalar in the 10 of dimension 3, and a complex scalar in the 1

99

of dimension 4), the n = 3 multiplet (which contains a scalar in the 50 of dimension 3 and a complex scalar in the 45 of dimension 4), and the n = 4 multiplet which contains a scalar in the 105 of dimension 4. The String Theory Spectrum and the Matching As discussed in section 2.2, fields on AdSf, are in a one-to-one correspondence with operators in the dual conformal field theory. Thus, the spectrum of operators described in section 3.1 should agree with the spectrum of fields of type IIB string theory on AdS$ x S 5 . Fields on AdS naturally lie in the same multiplets of the conformal group as primary operators; the second Casimir of these representations is Ci — A(A — 4) for a primary scalar field of dimension A in the field theory, and C 2 = m2R2 for a field of mass m on an AdS5 space with a radius of curvature R. Single-trace operators in the field theory may be identified with single-particle states in AdS$, while multiple-trace operators correspond to multi-particle states. Unfortunately, it is not known how to compute the full spectrum of type IIB string theory on AdS$ x S5. In fact, the only known states are the states which arise from the dimensional reduction of the ten-dimensional type IIB supergravity multiplet. These fields all have helicities between (—2) and 2, so it is clear that they all lie in small multiplets of the superconformal algebra, and we will describe below how they match with the small multiplets of the field theory described above. String theory on AdS$ x S5 is expected to have many additional states, with masses of the order of the string scale l/ls or of the Planck scale l/lp. Such states would correspond (using the mass/dimension relation described above) to operators in the field theory with dimensions of order A ~ (Ss-ZV)1/4 or A ~ A?"1/4 for large N,gsN. Presumably none of these states are in small multiplets of the superconformal algebra (at least, this would be the prediction of the AdS/CFT correspondence). The spectrum of type IIB supergravity compactified on AdS$ x S5 was computed in 101 . The computation involves expanding the ten dimensional fields in appropriate spherical harmonics on S5, plugging them into the supergravity equations of motion, linearized around the AdS5 x S 5 background, and diagonalizing the equations to give equations of motion for free (massless or massive) fields^. For example, the ten dimensional dilaton field r may be expanded as r(x,y) = YlT=o Tk{x)Yk(y) where x is a coordinate on AdS$, y is a coordinate on S5, and the Yk are the scalar spherical harmonics on S5. These spherical harmonics are in representations corresponding to symmetric traced

T h e fields arising from different spherical harmonics are related by a "spectrum generating algebra", see 1 0 2 .

100

less products of 6's of SU(4)R; they may be written as Yk(y) ~ y^y*2 • • -yIk where the y1, for / = 1,2, •••,6 and with ^2I=zl{y1)2 = 1, are coordinates on S5. Thus, we find a field rh(x) on AdS$ in each such (0, k, 0) representation of SU(4)fi, and the equations of motion determine the mass of this field to be mj? = k(k + 4)/R2. A similar expansion may be performed for all other fields. If we organize the results of101 into representations of the superconformal algebra 77 , we find representations of the form described in the previous section, which are built on a lowest dimension field which is a scalar in the (0,n,0) representation of SU(4)R for n = 2,3, • • •, oo. The lowest dimension scalar field in each representation turns out to arise from a linear combination of spherical harmonic modes of the S5 components of the graviton /i° (expanded around the AdS$ x S5 vacuum) and the 4-form field Dabcd, where a,b,c,d are indices on S5. The scalar fields of dimension n + 1 correspond to 2-form fields Bab with indices in the S5. The symmetric tensor fields arise from the expansion of the ^WSs-components of the graviton. The dilaton fields described above are the complex scalar fields arising with dimension n + 2 in the multiplet (as described in the previous subsection). In particular, the n = 2 representation is called the supergraviton representation, and it includes the field content of d = 5,7V = 8 gauged supergravity. The field/operator correspondence matches this representation to the representation including the superconformal currents in the field theory. It includes a massless graviton field, which (as expected) corresponds to the energy-momentum tensor in the field theory, and massless SU{4)R gauge fields which correspond to (or couple to) the global SU(4)R currents in the field theory. In the naive dimensional reduction of the type IIB supergravity fields, the n = 1 doubleton representation, corresponding to a free U(l) vector multiplet in the dual theory, also appears. However, the modes of this multiplet are all pure gauge modes in the bulk of AdSs, and they may be set to zero there. This is one of the reasons why it seems more natural to view the corresponding gauge theory as an SU(N) gauge theory and not a U(N) theory. It may be possible (and perhaps even natural) to add the doubleton representation to the theory (even though it does not include modes which propagate in the bulk of AdS$, but instead it is equivalent to a topological theory in the bulk) to obtain a theory which is dual to the U(N) gauge theory, but this will not affect most of our discussion in this review so we will ignore this possibility here. Comparing the results described above with the results of section 3.1, we see that we find the same spectrum of chiral primary operators for n = 2,3, • • • ,N. The supergravity results cannot be trusted for masses above the order of the string scale (which corresponds to n ~ (gsN)1/4) or the Planck

101

scale (which corresponds to n ~ TV1/4), so the results agree within their range of validity. The field theory results suggest that the exact spectrum of chiral representations in type IIB string theory on AdSs x S5 actually matches the naive supergravity spectrum up to a mass scale m 2 ~ N2 /R2 ~ N3/2M2 which is much higher than the string scale and the Planck scale, and that there are no chiral fields above this scale. It is not known how to check this prediction; tree-level string theory is certainly not enough for this since when gs = 0 we must take N = oo to obtain a finite value of gsN. Thus, with our current knowledge the matching of chiral primaries of the N = 4 SYM theory with those of string theory on AdS$ x S 5 tests the duality only in the large N limit. In some generalizations of the AdS/CFT correspondence the string coupling goes to zero at the boundary even for finite N, and then classical string theory should lead to exactly the same spectrum of chiral operators as the field theory. This happens in particular for the near-horizon limit of NS5-branes, in which case the exact spectrum was successfully compared in 1 0 3 . In other instances of the AdS/CFT correspondence (such as the ones discussed in io 4 . 105 . 106 ) there exist also additional chiral primary multiplets with n of order N, and these have been successfully matched with wrapped branes on the string theory side. The fact that there seem to be no non-chiral fields on AdSs with a mass below the string scale suggests that for large N and large gsN, the dimension of all non-chiral operators in the field theory, such as Tr(0 J tfi1'), grows at least as (gsN)llA ~ (Sy^JV) 1 ^ 4 . The reason for this behavior on the field theory side is not clear; it is a prediction of the AdS/CFT correspondence.

3.2

Matching of Correlation Functions and Anomalies

The classical JV = 4 theory has a scale invariance symmetry and an SU(4)R R-symmetry, and (unlike many other theories) these symmetries are exact also in the full quantum theory. However, when the theory is coupled to external gravitational or SU(4)R gauge fields, these symmetries are broken by quantum effects. In field theory this breaking comes from one-loop diagrams and does not receive any further corrections; thus it can be computed also in the strong coupling regime and compared with the results from string theory on AdS space. We will begin by discussing the anomaly associated with the SU(4)R global currents. These currents are chiral since the fermions XaA are in the 4 representation while the fermions of the opposite chirality A^ are in the 4 representation. Thus, if we gauge the SU(4)n global symmetry, we will find an Adler-Bell-Jackiw anomaly from the triangle diagram of three SU(4)R currents, which is proportional to the number of charged fermions. In the SU(N)

102

gauge theory this number is N2 — 1. The anomaly can be expressed either in terms of the 3-point function of the SU(4)R global currents,

(26) where dabc — 2Tr(Ta{Tb, Tc}) and we take only the negative parity component of the correlator, or in terms of the non-conservation of the SU{A)R current when the theory is coupled to external SU(4)H gauge fields F^„, N2 - 1 How can we see this effect in string theory on AdS§ x S5 ? One way to see it is, of course, to use the general prescription of section 4 to compute the 3-point function (26), and indeed one finds 107 ' 108 the correct answer to leading order in the large N limit (namely, one recovers the term proportional to N2). It is more illuminating, however, to consider directly the meaning of the anomaly (27) from the point of view of the AdS theory 19 . In the AdS theory we have gauge fields Aa which couple, as explained above, to the SU(4)u global currents J ° of the gauge theory, but the anomaly means that when we turn on non-zero field strengths for these fields the theory should no longer be gauge invariant. This effect is precisely reproduced by a Chern-Simons term which exists in the low-energy supergravity theory arising from the compactification of type IIB supergravity on AdS$ x S 5 , which is of the form i N2 r — -2 / 96TT

JAdS5

d5x{dabce»vXp°Ald„A\dpAca

+ •••)•

(28)

This term is gauge invariant up to total derivatives, which means that if we take a gauge transformation A^ —> A^ + (VfiA)a for which A does not vanish on the boundary of ^cLS^, the action will change by a boundary term of the form 2 ?N

- ^ ^

384TT

f

2

/

JdAdS5

d'xc^d^^FlF^.

(29)

From this we can read off the anomaly in (V^J^) since, when we have a coupling of the form / d4xA% J°, the change in the action under a gauge transformation is given by f d4x(VJ-A)aJ° = - J d4xAa(Vfl J£), and we find exact agreement with (27) for large N. The other anomaly in the M = 4 SYM theory is the conformal (or Weyl) anomaly (see 109,110 and references therein), indicating the breakdown of conformal invariance when the theory is coupled to a curved external metric (there

103

is a similar breakdown of conformal invariance when the theory is coupled to external SU(4)R gauge fields, which we will not discuss here). The conformal anomaly is related to the 2-point and 3-point functions of the energymomentum tensor 1 1 1 , 1 1 2 ' 1 1 3 , 1 1 4 . In four dimensions, the general form of the conformal anomaly is WT^)

=-aE* - cl4,

(30)

where 167T /4 =

-16^(^^-27e-

( 3 1 )

+

3^2)'

where 72M„pCT is the curvature tensor, 72M„ = TZ^pv is the Riemann tensor, and 72 = 72^ is the scalar curvature. A free field computation in the SU(N) M = 4 SYM theory leads to a = c = (N2 - l)/4. In supersymmetric theories the supersymmetry algebra relates g^T^ to derivatives of the R-symmetry current, so it is protected from any quantum corrections. Thus, the same result should be obtained in type IIB string theory on AdS5 x S 5 , and to leading order in the large N limit it should be obtained from type IIB supergravity on AdS5 x S 5 . This was indeed found to be true in H5,ii6,ii7,ii8e; w h e r e t h e conformal anomaly was shown to arise from subtleties in the regularization of the (divergent) supergravity action on AdS space. The result of i15>n6,117,118 implies that a computation using gravity on AdS^ always gives rise to theories with a — c, so generalizations of the AdS/CFT correspondence which have (for large N) a supergravity approximation are limited to conformal theories which have a = c in the large N limit. Of course, if we do not require the string theory to have a supergravity approximation then there is no such restriction. For both of the anomalies we described the field theory and string theory computations agree for the leading terms, which are of order A^2. Thus, they are successful tests of the duality in the large N limit. For other instances of the AdS/CFT correspondence there are corrections to anomalies at order \/N ~ gs(a'/R2)2; such corrections were discussed in 12° and successfully 121 122 123 compared i n . . /. It would be interesting to compare other corrections to the large N result. e

A generalization with more varying fields may be found in 1 1 9 . 'Computing such corrections tests the conjecture that the correspondence holds order by order in 1/iV; however, this is weaker than the statement that the correspondence holds for finite N, since the 1/JV expansion is not expected to converge.

104

4

Correlation Functions

A useful statement of the AdS/CFT correspondence is that the partition function of string theory on AdSs x S5 should coincide with the partition function of M = 4 super-Yang-Mills theory "on the boundary" of AdS5 18 ' 19 . The basic idea was explained in section 2.2, but before summarizing the actual calculations of Green's functions, it seems worthwhile to motivate the methodology from a somewhat different perspective. Throughout this section, we approximate the string theory partition function by e~IsuGRA, where ISUGRA is the supergravity action evaluated on AdSs x S5 (or on small deformations of this space). This approximation amounts to ignoring all the stringy a' corrections that cure the divergences of supergravity, and also all the loop corrections, which are controlled essentially by the gravitational coupling K ~ gsta'2 • On the gauge theory side, as explained in section 2.2, this approximation amounts to taking both N and gyMN large, and the basic relation becomes p-IsvORA

°

~ 7 —7 — p-W — -^string — -^gauge — c

>

(on) \°^v

where W is the generating functional for connected Green's functions in the gauge theory. At finite temperature, W — (5F where 0 is the inverse temperature and F is the free energy of the gauge theory. When we apply this relation to a Schwarzschild black hole in AdS$, which is thought to be reflected in the gauge theory by a thermal state at the Hawking temperature of the black hole, we arrive at the relation ISUGRA — 0F. Calculating the free energy of a black hole from the Euclidean supergravity action has a long tradition in the supergravity literature m , so the main claim that is being made here is that the dual gauge theory provides a description of the state of the black hole which is physically equivalent to the one in string theory. We will discuss the finite temperature case further in section 6, and devote the rest of this section to the partition function of the field theory on R . The main technical idea behind the bulk-boundary correspondence is that the boundary values of string theory fields (in particular, supergravity fields) act as sources for gauge-invariant operators in the field theory. From a D-brane perspective, we think of closed string states in the bulk as sourcing gauge singlet operators on the brane which originate as composite operators built from open strings. We will write the bulk fields generically as <j>(x, z) (in the coordinate system (17)), with value (p0(x) for z — e. The true boundary of anti-de Sitter space is z = 0, and e # 0 serves as a cutoff which will eventually be removed. In the supergravity approximation, we think of choosing the values ] m the region z > e subject to these

105

boundary conditions. In short, we solve the equations of motion in the bulk subject to Dirichlet boundary conditions on the boundary, and evaluate the action on the solution. If there is more than one solution, then we have more than one saddle point contributing to the string theory partition function, and we must determine which is most important. In this section, multiple saddle points will not be a problem. So, we can write Wgauge[4>0] = -\og(efdixMx)0(x))

^

~ extiemmaISUGRA[4>]

•

(33)

0| _ =00

That is, the generator of connected Green's functions in the gauge theory, in the large N,gyMN limit, is the on-shell supergravity action. Note that in (33) we have not attempted to be prescient about inserting factors of e. Instead our strategy will be to use (33) without modification to compute two-point functions of O, and then perform a wave-function renormalization on either O or so that the final answer is independent of the cutoff. This approach should be workable even in a space (with boundary) which is not asymptotically anti-de Sitter, corresponding to a field theory which does not have a conformal fixed point in the ultraviolet. A remark is in order regarding the relation of (33) to the old approach of extracting Green's functions from an absorption cross-section11. In absorption calculations one is keeping the whole D3-brane geometry, not just the nearhorizon AdSs x S5 throat. The usual treatment is to split the space into a near region (the throat) and a far region. The incoming wave from asymptotically flat infinity can be regarded as fixing the value of a supergravity field at the outer boundary of the near region. As usual, the supergravity description is valid at large N and large 't Hooft coupling. At small 't Hooft coupling, there is a different description of the process: a cluster of D3-branes sits at some location in flat ten-dimensional space, and the incoming wave impinges upon it. In the low-energy limit, the value of the supergravity field which the D3-branes feel is the same as the value in the curved space description at the boundary of the near horizon region. Equation (33) is just a mathematical expression of the fact that the throat geometry should respond identically to the perturbed supergravity fields as the low-energy theory on the D3-branes. Following 18 - 19 , a number of papers—notably 125,126,107,127,108, 128-139_ have undertaken the program of extracting explicit n-point correlation functions of gauge singlet operators by developing both sides of (33) in a power series in 4>Q. Because the right hand side is the extremization of a classical action, the power series has a graphical representation in terms of tree-level Feynman graphs for fields in the supergravity. There is one difference: in ordinary Feynman graphs one assigns the wavefunctions of asymptotic states to

106

the external legs of the graph, but in the present case the external leg factors reflect the boundary values 0. They are special limits of the usual gravity propagators in the bulk, and are called bulk-to-boundary propagators. We will encounter their explicit form in the next two sections. 4-1

Two-point Functions

For two-point functions, only the part of the action which is quadratic in the relevant field perturbation is needed. For massive scalar fields in AdSs, this has the generic form

S = riJ^xy/g[^d(l>)2

+ ^m24>2],

(34)

where n is some normalization which in principle follows from the ten-dimensional origin of the action. The bulk-to-boundary propagator is a particular solution of the equation of motion, ( • —m2)<j) = 0, which has special asymptotic properties. We will start by considering the momentum space propagator, which is useful for computing the two-point function and also in situations where the bulk geometry loses conformal invariance; then, we will discuss the position space propagator, which has proven more convenient for the study of higher point correlators in the conformal case. We will always work in Euclidean space 3 . A coordinate system in the bulk of AdS$ such that ds2 = % {dx2 + dz2)

(35)

provides manifest Euclidean symmetry on the directions parametrized by x. To avoid divergences associated with the small z region of integration in (34), we will employ an explicit cutoff, z > e. A complete set of solutions for the linearized equation of motion, ( • — m2)4> = 0, is given by = el^'xZ(pz), where the function Z(u) satisfies the radial equation u5du—d u u3

-u2

-m2R2 Z{u) = 0 .

There are two independent solutions to (36), namely Z(u) = u2I\-2{u) Z(u) = U2KA~2(U), where Iv and Kv are Bessel functions and A = 2 + A/4 + m2R2 . 9

(36) and

(37)

The results may be analytically continued to give the correlation functions of the field theory on Minkowskian K , which corresponds to the Poincare coordinates of AdS space.

107

The second solution is selected by the requirement of regularity in the interior: IA-2(U) increases exponentially as u -* oo and does not lead to a finite action configuration. Imposing the boundary condition <j>(x,z) = <po(x) = elpx at z = e, we find the bulk-to-boundary propagator 4>(x,z) = Kp-(x,z)=

{pe)2KAMpe)er

.

(38)

To compute a two-point function of the operator O for which 0 is a source, we write d2W \4>0 = \ieipx

+

\2ei*x] A1=A2=0

= (leading analytic terms in (ep)2)

2

4

+

2A-4

^ *- >fifcij^
<39)

4-h (higher order terms in (ep)2), ^ 2 A _ 8 2 A - 4 T(A + 1) 1 =^ ' 2 A 7r r(A - 2) |f - y| 2 A

Several explanatory remarks are in order: • To establish the second equality in (39) we have used (38), substituted in (34), performed the integral and expanded in e. The leading analytic terms give rise to contact terms in position space, and the higher order terms are unimportant in the limit where we remove the cutoff. Only the leading nonanalytic term is essential. We have given the expression for generic real values of A. Expanding around integer A > 2 one obtains finite expressions involving log ep. • The Fourier transforms used to obtain the last line are singular, but they can be defined for generic complex A by analytic continuation and for positive integer A by expanding around a pole and dropping divergent terms, in the spirit of differential regularization 140 . The result is a pure power law dependence on the separation \x — y\, as required by conformal invariance. • We have assumed a coupling J d4x(x,z = e)0(x) to compute the Green's functions. The explicit powers of the cutoff in the final position space answer can be eliminated by absorbing a factor of eA~"4 into

108

the definition of O. From here on we will take that convention, which amounts to inserting a factor of e 4 ~ A on the right hand side of (38). In fact, precise matchings between the normalizations in field theory and in string theory for all the chiral primary operators have not been worked out. In part this is due to the difficulty of determining the coupling of bulk fields to field theory operators (or in stringy terms, the coupling of closed string states to composite open string operators on the brane). See 10 for an early approach to this problem. For the dilaton, the graviton, and their superpartners (including gauge fields in AdS*,), the couplings can be worked out explicitly. In some of these cases all normalizations have been worked out unambiguously and checked against field theory predictions (see for example 18 ' 107 ' 132 ). • The mass-dimension relation (37) holds even for string states that are not included in the Kaluza-Klein supergravity reduction: the mass and the dimension are just different expressions of the second Casimir of 50(4,2). For instance, excited string states, with m ~ l/\/o7, are expected to correspond to operators with dimension A ~ (^y M A r ) 1 / 4 . The remarkable fact is that all the string theory modes with m ~ 1/R (which is to say, all closed string states which arise from massless ten dimensional fields) fall in short multiplets of the supergroup SU(2, 2|4). All other states have a much larger mass. The operators in short multiplets have algebraically protected dimensions. The obvious conclusion is that all operators whose dimensions are not algebraically protected have large dimension in the strong 't Hooft coupling, large N limit to which supergravity applies. This is no longer true for theories of reduced supersymmetry: the supergroup gets smaller, but the Kaluza-Klein states are roughly as numerous as before, and some of them escape the short multiplets and live in long multiplets of the smaller supergroups. They still have a mass on the order of 1/R, and typically correspond to dimensions which are finite (in the large gyMN limit) but irrational. Correlation functions of non-scalar operators have been widely studied following19; the literature jndudes 1 4 1 ' 1 4 2 ' 1 4 3 ' 1 4 4 > 1 4 5 ' 1 4 6 ' 1 4 7 ' 1 4 8 ' 1 4 9 - 1 5 0 ' 1 5 1 . For M = 4 super-Yang-Mills theory, all correlation functions of fields in chiral multiplets should follow by application of supersymmetries once those of the chiral primary fields are known, so in this case it should be enough to study the scalars. It is worthwhile to note however that the mass-dimension formula changes for particles with spin. In fact the definition of mass has some convention-dependence. Conventions seem fairly uniform in the literature, and a table of mass-dimension relations in AdSd+i with unit radius was made in

109 152

from the various sources cited above (see also 98 ) y/d2 + 4m 2 ),

scalars:

A ± = \{d±

spinors:

A = | ( d + 2|m|),

vectors:

A± = \(d ± y/(d - 2) 2 + 4m 2 ),

• p-forms:

A = \{d ± y/(d-2p)2

• first-order (d/2)-forms (d even): • spin-3/2:

+ 4m 2 ), A = | ( d + 2|m|),

A = | ( d + 2|m|),

• massless spin-2:

A = d.

In the case of fields with second order lagrangians, we have not attempted to pick which of A± is the physical dimension. Usually the choice A = A + is clear from the unitarity bound, but in some cases (notably m 2 = 15/4 in AdS$) there is a genuine ambiguity. In practice this ambiguity is usually resolved by appealing to some special algebraic property of the relevant fields, such as transformation under supersymmetry or a global bosonic symmetry. For brevity we will omit a further discussion of higher spins, and instead refer the reader to the (extensive) literature. 4-2

Three-point Functions

Working with bulk-to-boundary propagators in the momentum representation is convenient for two-point functions, but for higher point functions position space is preferred because the full conformal invariance is more obvious. (However, for non-conformal examples of the bulk-boundary correspondence, the momentum representation seems uniformly more convenient). The boundary behavior of position space bulk-to-boundary propagators is specified in a slightly more subtle way: following107 we require KA{x,z;y)

-> z4-A54(x-y)

as

z-+0

.

(40)

Here y is the point on the boundary where we insert the operator, and (x,z) is a point in the bulk. The unique regular K& solving the equation of motion and satisfying (40) is

K

^*

= 1^=2) {z>H*-vvY •

(41)

110

At a fixed cutoff, z = e, the bulk-to-boundary propagator KA(x,e;y) is a continuous function which approximates e4_A<54(ir — y) better and better as e —¥ 0. Thus at any finite e, the Fourier transform of (41) only approximately coincides with (38) (modified by the factor of e 4 _ A as explained after (39)). This apparently innocuous subtlety turned out to be important for two-point functions, as discovered in 107 . A correct prescription is to specify boundary conditions at finite z = e, cut off all bulk integrals at that boundary, and only afterwards take e —> 0. That is what we have done in (39). Calculating twopoint functions directly using the position-space propagators (40), but cutting the bulk integrals off again at e, and finally taking the same e -> 0 answer, one arrives at a different answer. This is not surprising since the z = e boundary conditions were not used consistently. The authors of107 checked that using the cutoff consistently (i.e. with the momentum space propagators) gave two-point functions (0(xi)0(x2)) a normalization such that Ward identities involving the three-point function {0(xi)0(x2)Jp{xz)), where JM is a conserved current, were obeyed. Two-point functions are uniquely difficult because of the poor convergence properties of the integrals over z. The integrals involved in threepoint functions are sufficiently benign that one can ignore the issue of how to impose the cutoff. If one has a Euclidean bulk action for three scalar fields fa, fa, and fa, of the form = / d5x ^ 5 Y,\{dfa)2

+ \m2i2i+Mi2fa\ ,

(42)

and if the fa couple to operators in the field theory by interaction terms J d4xfaOi, then the calculation of (O1O2O3) reduces, via (33), to the evaluation of the graph shown in figure 2. That is, (01{xi)02(x2)03(x3)) =

= -A /

d5xy/gKAl(x;x1)KA2(x;x2)KA3{x;x3)

|fl - f 2 | A l + A 2 ~ A 3 | ^ l - ^ 3 | A l + A 3 - A 2 | ^ 2 - f 3 | A 2 +A 3 " A l '

(43) for some constant a\. The dependence on the X{ is dictated by the conformal invariance, but the only way to compute a\ is by performing the integral over x. The result 107 is r [|(Ax + A2 - A 3 )] r [|(A! + A 3 - A 2 )] r [|(A2 + A 3 - AQ] ai 2niT(A1 - 2)r(A 2 - 2)r(A 3 - 2) r [ i ( A 1 + A 2 + A3)-2] . (44)

111

Figure 2: The Feynman graph for the three-point function as computed in supergravity. The legs correspond to factors of K& , and the cubic vertex to a factor of A. The position of the vertex is integrated over AdSs.

In principle one could also have couplings of the form (fridfadfo. only to a modification of the constant a,\.

This leads

The main technical difficulty with three-point functions is that one must figure out the cubic couplings of supergravity fields. Because of the difficulties in writing down a covariant action for type IIB supergravity in ten dimensions (see however 153 ' 154,155 ), it is most straightforward to read off these "cubic couplings" from quadratic terms in the equations of motion. In flat ten-dimensional space these terms can be read off directly from the original type IIB supergravity papers 1 5 6 ' 1 5 7 . For AdSs x S5, one must instead expand in fluctuations around the background metric and five-form field strength. The old literature 101 only dealt with the linearized equations of motion; for 3-point functions it is necessary to go to one higher order of perturbation theory. This was done for a restricted set of fields in 130 . The fields considered were those dual to operators of the form Tr(Jl 0 J 2 . . .
O / =C5 1 ... J t Tr0( J l ...^)

(45)

112

follow from an action of the form

5

= Wf I d5xV~g{ ? ^r-h(VS/)2"l(l ~4)s1

Y^ Ghi2i3whwhwh ^ 3 h,h,h Derivative couplings of the form sdsds are expected a priori to enter into (46), but an appropriate field redefinition eliminates them. The notation in (45) and (46) requires some explanation. / is an index which runs over the weight vectors of all possible representations constructed as symmetric traceless products of the 6 of SU(4)R. These are the representations whose Young diagrams are • , m , r r n , • • •• Cjt Jt is a basis transformation matrix, chosen so that C^ JeCji Je = 5IJ. As commented in the previous section, there is generally a normalization ambiguity on how supergravity fields couple to operators in the gauge theory. We have taken the coupling to be J d4x sjO1, and the normalization ambiguity is represented by the "leg factors" w1. It is the combination s1 = w's1 rather than s1 itself which has a definite relation to supergravity fields. We refer the reader to 1 3 0 for explicit expressions for Ai and the symmetric tensor Qi1 I2J3 . To get rid of factors of w1, we introduce operators O1 — w101. One can choose w1 so that a two-point function computation along the lines of section 4.1 leads to (OI^x)O,^0))=5^-i.

(47)

With this choice, the three-point function, as calculated using (43), is {&1{$)0I*{X'2)0I*{X3)) VA 1 A 2 A 3 (C / l C / 2 C / 3 )

_ 1_ ~ N |fi - f 2 |

Al+A2

~

A3

(48)

|^i - f3|Al+A3~A2|^2 - £3|A2+A3~Al '

where we have defined inhnhphx {U

C L

— nh

rh

)—^j1...jiKl-KjcJi--JiL1-LkL'K1-KjL1-Lk-

plz

(AQ\

\*y)

Remarkably, (48) is the same result one obtains from free field theory by Wick contracting all the (f>J fields in the three operators. This suggests that there is a non-renormalization theorem for this correlation function, but such a theorem has not yet been proven (see however comments at the end of section 3.2). It

113

is worth emphasizing that the normalization ambiguity in the bulk-boundary coupling is circumvented essentially by considering invariant ratios of threepoint functions and two-point functions, into which the "leg factors" w1 do not enter. This is the same strategy as was pursued in comparing matrix models of quantum gravity to Liouville theory. 4-3

Four-point Functions

The calculation of four-point functions is difficult because there are several graphs which contribute, and some of them inevitably involve bulk-to-bulk propagators of fields with spin. The computation of four-point functions of the operators O^ and Oc dual to the dilaton and the axion was completed in 158 . See also 126 - 131 ' 133 ' 134 - 159 - 160 ' 137 ' 135 - 161 ' 162 for earlier contributions. One of the main technical results, further developed in 1 6 3 , is that diagrams involving an internal propagator can be reduced by integration over one of the bulk vertices to a sum of quartic graphs expressible in terms of the functions £,A1A2A3A4(^1^2,^3,^4)

4

r /

=

d^Xy/g^KAi(x,z;Xi), 1=1

(50)

A

KA(x,z;y)=^2

(l_yy

+

The integration is over the bulk point (x, z). There are two independent conformally invariant combinations of the Xi: -1

_

-*9 -»9

1

-t'13-t'24 x

x

J

. _ u

x

-O

J/

*• 12 34 ~ '14 '23

-*9

12 x 34 x

12 34

-*9 -*9

x

14 x 23

J

J

/n\

' '14 '23

One can write the connected four-point function as 16:?

<0*(fl)0 C (*2)0*(*3)0 C (£4)> = ( 4 ^ 64f24i 1T?2" s

16af| 4 l TW «

24 ( Y ~ 1 ) 46 ~ q ^_^ "

4 4444 — + 7 T ^ 2 — - D 33355 3 5 5 ++ " 5 " ^ — - D 22255 2 5 5 —1 I4D4444

—

40 Q-2

8 -^2244 — 0-6

"

-2 n

^

-D3344

Dii55

(52)

'

-^1144 + 64^24-04455

An interesting limit of (52) is to take two pairs of points close together. Following 158 , let us take the pairs (xi,x3) and (£2,£4) close together while

114

Figure 3: A nearly degenerate quartic graph contributing to the four-point function in the limit | x i 3 | , | x 2 4 | < |xi2|-

holding Xi and £2 a fixed distance apart. expansion implies that {0Al{xl)0A2{x2)0A3(x3)0A4{x4))

Then the existence of an OPE

= 2 ^ ^Al+A3-Am .A2+A4-A„ ' n,m

X

l3

(53)

^24

at least as an asymptotic series, and hopefully even with a finite radius of convergence for X13 and £24. The operators On are the ones that appear in the OPE of 0\ with O3, and the operators Om are the ones that appear in the OPE of O2 with O4. O4, and OQ are descendants of chiral primaries, and so have protected dimensions. The product of descendants of chiral fields is not itself necessarily the descendent of a chiral field: an appropriately normal ordered product : 0<j,0$ : is expected to have an unprotected dimension of the form 8 + 0(l/N2). This is the natural result from the field theory point of view because there are 0(N2) degrees of freedom contributing to each factor, and the commutation relations between them are non-trivial only a fraction l/TV2 of the time. From the supergravity point of view, a composite operator like : 0$0$ : corresponds to a two-particle bulk state, and the 0(l/N2) = 0(K2/RS) correction to the mass is interpreted as the correction to the mass of the twoparticle state from gravitational binding energy. Roughly one is thinking of graviton exchange between the legs of figure 3 that are nearly coincident. If (53) is expanded in inverse powers of N, then the 0(\/N2) correction to A n and A m shows up to leading order as a term proportional to a logarithm of some combination of the separations Xij. Logarithms also appear in the expansion of (52) in the |a?i31, |af241 -C |aTi21 limit in which (53) applies: the

115

leading log in this limit is ,Sl\i6 log ( Xl024 J. This is the correct form to be interpreted in terms of the propagation of a two-particle state dual to an operator whose dimension is slightly different from 8. 5

Wilson Loops

In this section we consider Wilson loop operators in the gauge theory. The Wilson loop operator W(C) = Tr P exp ( i f A

(54)

depends on a loop C embedded in four dimensional space, and it involves the path-ordered integral of the gauge connection along the contour. The trace is taken over some representation of the gauge group; we will discuss here only the case of the fundamental representation (see 164 for a discussion of other representations). From the expectation value of the Wilson loop operator (W(C)) we can calculate the quark-antiquark potential. For this purpose we consider a rectangular loop with sides of length T and L in Euclidean space. Then, viewing T as the time direction, it is clear that for large T the expectation value will behave as e~TE where E is the lowest possible energy of the quark-anti-quark configuration. Thus, we have (W) ~ e~TV^

,

(55)

where V(L) is the quark anti-quark potential. For large JV and large gyMN, the AdS/CFT correspondence maps the computation of (W) in the CFT into a problem of finding a minimum surface in AdS 165>166. 5.1

Wilson Loops and Minimum Surfaces

In QCD, we expect the Wilson loop to be related to the string running from the quark to the antiquark. We expect this string to be analogous to the string in our configuration, which is a superstring which lives in ten dimensions, and which can stretch between two points on the boundary of AdS. In order to motivate this prescription let us consider the following situation. We start with the gauge group U(N + 1), and we break it to U(N) x U{1) by giving an expectation value to one of the scalars. This corresponds, as discussed in section 2, to having a D3 brane sitting at some radial position U in AdS, and at a point on S5. The off-diagonal states, transforming in the N of U(N), get a mass proportional to U,m = U/2TT. SO, from the point of view of the U(N) gauge theory, we can view these states as massive quarks, which act as a source

116

for the various U(N) fields. Since they are charged they will act as a source for the vector fields. In order to get a non-dynamical source (an "external quark" with no fluctuations of its own, which will correspond precisely to the Wilson loop operator) we need to take m -> oo, which means U should also go to infinity. Thus, the string should end on the boundary of AdS space. These stretched strings will also act as a source for the scalar fields. The coupling to the scalar fields can be seen qualitatively by viewing the quarks as strings stretching between the N branes and the single separated brane. These strings will pull the N branes and will cause a deformation of the branes, which is described by the scalar fields. A more formal argument for this coupling is that these states are BPS, and the coupling to the scalar (Higgs) fields is determined by supersymmetry. Finally, one can see this coupling explicitly by writing the full U(N + 1) Lagrangian, putting in the Higgs expectation value and calculating the equation of motion for the massive fields 165 . The precise definition of the Wilson loop operator corresponding to the superstring will actually include also the field theory fermions, which will imply some particular boundary conditions for the worldsheet fermions at the boundary of AdS. However, this will not affect the leading order computations we describe here. So, the final conclusion is that the stretched strings couple to the operator W(C) = Tr

Pexp ( (fiiA^

+ 6I4>IV±^)di

(56)

where x^{r) is any parametrization of the loop and 61 (I = 1, • • • ,6) is a unit vector in R (the point on S 5 where the string is sitting). This is the expression when the signature of K. is Euclidean. In the Minkowski signature case, the phase factor associated to the trajectory of the quark has an extra factor "z" in front of 61 h. Generalizing the prescription of section 4 for computing correlation functions, the discussion above implies that in order to compute the expectation value of the operator (56) in J\f = 4 SYM we should consider the string theory partition function on AdS^ x S 5 , with the condition that we have a string worldsheet ending on the loop C, as in figure 4 166>165. In the supergravity regime, when gsN is large, the leading contribution to this partition function will come from the area of the string worldsheet. This area is measured with the AdS metric, and it is generally not the same as the area enclosed by the loop C in four dimensions. '"The difference in the factor of i between the Euclidean and the Minkowski cases can be traced to the analytic continuation of V±^. A detailed derivation of (56) can be found i n 1 6 7 .

117

Figure 4: The Wilson loop operator creates a string worldsheet ending on the corresponding loop on the boundary of AdS.

The area as defined above is divergent. The divergence arises from the fact that the string worldsheet is going all the way to the boundary of AdS. If we evaluate the area up to some radial distance U = r, we see that for large r it diverges as r\C\, where \C\ is the length of the loop in the field theory 165 ' 166 . On the other hand, the perturbative computation in the field theory shows that (W), for W given by (56), is finite, as it should be since a divergence in the Wilson loop would have implied a mass renormalization of the BPS particle. The apparent discrepancy between the divergence of the area of the minimum surface in AdS and the finiteness of the field theory computation can be reconciled by noting that the appropriate action for the string worldsheet is not the area itself but its Legendre transform with respect to the string coordinates corresponding to 61 and the radial coordinate u 167 . This is because these string coordinates obey the Neumann boundary conditions rather than the Dirichlet conditions. When the loop is smooth, the Legendre transformation simply subtracts the divergent term r\C\, leaving the resulting action finite. As an example let us consider a circular Wilson loop. Take C to be a circle of radius a on the boundary, and let us work in the Poincare coordinates. We could find the surface that minimizes the area by solving the Euler-Lagrange equations. However, in this case it is easier to use conformal invariance. Note that there is a conformal transformation in the field theory that maps a line to a circle. In the case of the line, the minimum area surface is clearly a plane that intersects the boundary and goes all the way to the horizon (which is just a point on the boundary in the Euclidean case). Using the conformal transformation to map the line to a circle we obtain the minimal surface we want. It is, using the coordinates (17) for AdS5, x = \/a2 — z2(ei cos 6 + e^sin^),

(57)

118

where e*i, e2 are two orthonormal vectors in four dimensions (which define the orientation of the circle) and 0 < z < a. We can calculate the area of this surface in AdS, and we get a contribution to the action 1

, R2 f Jn r dza R2 ,a ,, S~-—-A= I d6 I —T- = — ( 1), (58) h y 2ira' 2na' J J€ z2 a1 e ' where we have regularized the area by putting a an IR cutoff at z = e in AdS, which is equivalent to a UV cutoff in the field theory 63 . Subtracting the divergent term we get (W) ~ e~s ~ eR2/a' = e^A*9'N. (59) This is independent of a as required by conformal invariance. We could similarly consider a "magnetic" Wilson loop, which is also called a 't Hooft loop 1 6 8 . This case is related by electric-magnetic duality to the previous case. Since we identify the electric-magnetic duality with the SL(2, Z) duality of type IIB string theory, we should consider in this case a D-string worldsheet instead of a fundamental string worldsheet. We get the same result as in (59) but with gs —> l/gsUsing (55) it is possible to compute the quark-antiquark potential in the supergravity approximation 166 ' 165 . In this case we consider a configuration which is invariant under (Euclidean) time translations. We take both particles to have the same scalar charge, which means that the two ends of the string are at the same point in S 5 (one could consider also the more general case with a string ending at different points on S5 1 6 5 ). We put the quark at x — —L/2 and the anti-quark at x = L/2. Here "quark" means an infinitely massive Wboson connecting the N branes with one brane which is (infinitely) far away. The classical action for a string worldsheet is S = —*— [ dTdoJdet(GMNdaXMdpXN), (60) Zwa J v where GMN is the Euclidean AdS$ x S5 metric. Note that the factors of a' cancel out in (60), as they should. Since we are interested in a static configuration we take r = t, a = x, and then the action becomes

S^mf

,,,^±1.

,61)

We need to solve the Euler-Lagrange equations for this action. Since the action does not depend on x explicitly the solution satisfies

*V(&*)2 + l

constant.

(62)

119 Defining z0 to be the maximum value of z(x), which by symmetry occurs at x = 0, we find that the solution is2 x

= zo

f1

dyy2 u

Jz/zo \ / l - 2 /

4

. 63 . ( )

where ZQ is determined by the condition f1

L _

dyy2

_

V27T3/2

= ^O^TTT^-

(64)

The qualitative form of the solution is shown in figure 5(b). Notice that the string quickly approaches x = L/2 for small z (close to the boundary), |

- a; ~ z 3 ,

z -> 0 .

(65)

Now we compute the total energy of the configuration. We just plug in the solution (63) in (61), subtract the infinity as explained above (which can be interpreted as the energy of two separated massive quarks, as in figure 5(a)), and we find i i

r(

i)4L

> '

We see that the energy goes as l/L, a fact which is determined by conformal invariance. Note that the energy is proportional to (g2 M^)1/2:, as opposed to gYMN which is the perturbative result. This indicates some screening of the charges at strong coupling. The above calculation makes sense for all distances L when gsN is large, independently of the value of gs. Some subleading corrections coming from quantum fluctuations of the worldsheet were calculated m

169,170,171

In a similar fashion we could compute the potential between two magnetic monopoles in terms of a D-string worldsheet, and the result will be the same as (66) but with gyivi —> ^/9YMOne can also calculate the interaction between a magnetic monopole and a quark. In this case the fundamental string (ending on the quark) will attach to the D-string (ending on the monopole), and they will connect to form a (1,1) string which will go into the horizon. The resulting potential is a complicated function of gyM 172 , but in the limit that gyM is small (but still with gYMN large) we get that the monopole-quark potential is just 1/4 of the quark-quark potential. This can be understood from the fact that when g is small the D-string is very rigid and the fundamental string 'All integrals in this section can be calculated in terms of elliptic or Beta functions.

120

U= oo

u=o (a)

(b)

Figure 5: (a) Initial configuration corresponding to two massive quarks before we turn on their coupling to the U(N) gauge theory, (b) Configuration after we consider the coupling to the U(N) gauge theory. This configuration minimizes the action. The quark-antiquark energy is given by the difference of the total length of the strings in (a) and (b).

will end almost perpendicularly on the D-string. Therefore, the solution for the fundamental string will be half of the solution we had above, leading to a factor of 1/4 in the potential. Calculations of Wilson loops in the Higgs phase were done in 1 7 3 . .Another interesting case one can study analytically is a surface near a cusp o n M . In this case, the perturbative computation in the gauge theory shows a logarithmic divergence with a coefficient depending on the angle at the cusp. The area of the minimum surface also contains a logarithmic divergence depending on the angle 167 . Other aspects of the gravity calculation of Wilson loops were discussed in174>175>176.177>178. 5.2

Other Branes Ending on the Boundary

We could also consider other branes that are ending at the boundary 1 7 9 . The simplest example would be a zero-brane (i.e. a particle) of mass m. In Euclidean space a zero-brane describes a one dimensional trajectory in anti-deSitter space which ends at two points on the boundary. Therefore, it is associated with the insertion of two local operators at the two points where the trajectory ends. In the supergravity approximation the zero-brane follows a

121

geodesic. Geodesies in the hyperbolic plane (Euclidean AdS) are semicircles. If we compute the action we get S = m [ds = -2mR [ —^t==, J Je z\Ja2 — z2

(67)

where we took the distance between the two points at the boundary to be L = 2a and regulated the result. We find a logarithmic divergence when e —> 0, proportional to log(e/a). If we subtract the logarithmic divergence we get a residual dependence on a. Naively we might have thought that (as in the previous subsection) the answer had to be independent of a due to conformal invariance. In fact, the dependence on a is very important, since it leads to a result of the form e

c

a2mR'

which is precisely the result we expect for the two-point function of an operator of dimension A = mR. This is precisely the large mR limit of the formula (14), so we reproduce in the supergravity limit the 2-point function described in section 4. In general, this sort of logarithmic divergence arises when the brane worldvolume is odd dimensional 179 , and it implies that the expectation value of the corresponding operator depends on the overall scale. In particular one could consider the "Wilson surfaces" that arise in the six dimensional J\f = (2,0) theory. In that case one has to consider a two-brane, with a three dimensional worldvolume, ending on a two dimensional surface on the boundary of AdSj. Again, one gets a logarithmic term, which is proportional to the rigid string action of the two dimensional surface living on the string in the M = (2,0) field theory 180 - 179 . One can also compute correlation functions involving more than one Wilson loop. To leading order in N this will be just the product of the expectation values of each Wilson loop. On general grounds one expects that the subleading corrections are given by surfaces that end on more than one loop. One limiting case is when the surfaces look similar to the zeroth order surfaces but with additional thin tubes connecting them. These thin tubes are nothing else than massless particles being exchanged between the two string worldsheets 164>180. 6

Theories at Finite Temperature

As discussed in section 3, the quantities that can be most successfully compared between gauge theory and string theory are those with some protection from supersymmetry and/or conformal invariance — for instance, dimensions of chiral primary operators. Finite temperature breaks both supersymmetry and

122

conformal invariance, and the insights we gain from examining the T > 0 physics will be of a more qualitative nature. They are no less interesting for that: we shall see in section 6.1 how the entropy of near-extremal D3-branes comes out identical to the free field theory prediction up to a factor of a power of 4/3; then in section 6.2 we explain how a phase transition studied by Hawking and Page in the context of quantum gravity is mapped into a confinement-deconfinement transition in the gauge theory. 6.1

Construction

The gravity solution describing the gauge theory at finite temperature can be obtained by starting from the general black three-brane solution and taking the decoupling limit of section 2 keeping the energy density above extremality finite. The resulting metric can be written as ds2 = R2 u2(-hdt2

du2 + dx\ + dx\ + dx\) + —^ + dflj (69)

1-4'

«O=TT-

It will often be useful to Wick rotate by setting tg = it, and use the relation between the finite temperature theory and the Euclidean theory with a compact time direction. The first computation which indicated that finite-temperature U(N) YangMills theory might be a good description of the microstates of N coincident D3-branes was the calculation of the entropy 181>182. On the supergravity side, the entropy of near-extremal D3-branes is just the usual BekensteinHawking result, S = A/4GN, and it is expected to be a reliable guide to the entropy of the gauge theory at large N and large g\MN. There is no problem on the gauge theory side in working at large N, but large gyMN at finite temperature is difficult indeed. The analysis of181 was limited to a free field computation in the field theory, but nevertheless the two results for the entropy agreed up to a factor of a power of 4/3. In the canonical ensemble, where temperature and volume are the independent variables, one identifies the field theory volume with the world-volume of the D3-branes, and one sets the field theory temperature equal to the Hawking temperature in supergravity. The result is FSUGRA =

8

~^rN2VT\ (70)

4

FsYM

= -FsUGRA

•

123 The supergravity result is at leading order in ls/R, and it would acquire corrections suppressed by powers of TR if we had considered the full D3-brane metric rather than the near-horizon limit, (69). These corrections do not have an interpretation in the context of CFT because they involve R as an intrinsic scale. Two equivalent methods to evaluate FSUGRA are a) to use F = E — TS together with standard expressions for the Bekenstein-Hawking entropy, the Hawking temperature, and the ADM mass; and b) to consider the gravitational action of the Euclidean solution, with a periodicity in the Euclidean time direction (related to the temperature) which eliminates a conical deficit angle at the horizon.J The 4/3 factor is a long-standing puzzle into which we still have only qualitative insight. The gauge theory computation was performed at zero 't Hooft coupling, whereas the supergravity is supposed to be valid at strong 't Hooft coupling, and unlike in the 1 + 1-dimensional case where the entropy is essentially fixed by the central charge, there is no non-renormalization theorem for the coefficient of T 4 in the free energy. Indeed, it was suggested in 1 8 3 that the leading term in the \/N expansion of F has the form F=-f(gYMN)^-N2VT\

(71)

where figy^N) *s a function which smoothly interpolates between a weak coupling limit of 1 and a strong coupling limit of 3/4. It was pointed out early 184 that the quartic potential c ^ ^ T r ^ 7 , c/)J]2 in the M = 4 Yang-Mills action might be expected to freeze out more and more degrees of freedom as the coupling was increased, which would suggest that f(gYMN) is monotone decreasing. An argument has been given 185 , based on the non-renormalization of the two-point function of the stress tensor, that f(gYMN) should remain finite at strong coupling. The leading corrections to the limiting value of f(gyM^) at strong and weak coupling were computed in 1 8 3 and 1 8 6 , respectively. The results are 3 f(9YMN) = 1 - -J^92YMN + ••• for small gYMN, (72) f(9yMN) = \ + f2 ( g f ^ ) 3 / 2 + • • • for large g\MN. The weak coupling result is a straightforward although somewhat tedious application of the diagrammatic methods of perturbative finite-temperature field •'The result of 1 8 1 , SSYM = ( 4 / 3 ) 1 / 4 5 s f / G i J A , differs superficially from (70), but it is only because the authors worked in the microcanonical ensemble: rather than identifying the Hawking temperature with the field theory temperature, the ADM mass above extremality was identified with the field theory energy.

124

theory. The constant term is from one loop, and the leading correction is from two loops. The strong coupling result follows from considering the leading a' corrections to the supergravity action. The relevant one involves a particular contraction of four powers of the Weyl tensor. It is important now to work with the Euclidean solution, and one restricts attention further to the nearhorizon limit. The Weyl curvature comes from the non-compact part of the metric, which is no longer AdS$ but rather the AdS-Schwarzschild solution which we will discuss in more detail in section 6.2. The action including the Q' corrections no longer has the Einstein-Hilbert form, and correspondingly the Bekenstein-Hawking prescription no longer agrees with the free energy computed as (31 where I is the Euclidean action. In keeping with the basic prescription for computing Green's functions, where a free energy in field theory is equated (in the appropriate limit) with a supergravity action, the relation I = /3F is regarded as the correct one. (See 187 .) It has been conjectured that the interpolating function f{gYM^) *s n o t smooth, but exhibits some phase transition at a finite value of the 't Hooft coupling. We regard this as an unsettled question. The arguments in 188 - 189 seem as yet incomplete. In particular, they rely on analyticity properties of the perturbation expansion which do not seem to be proven for finite temperature field theories.

6.2

Thermal Phase Transition

The holographic prescription of18'19, applied at large N and gyMN where loop and stringy corrections are negligible, involves extremizing the supergravity action subject to particular asymptotic boundary conditions. We can think of this as the saddle point approximation to the path integral over supergravity fields. That path integral is ill-defined because of the non-renormalizable nature of supergravity. String amplitudes (when we can calculate them) render on-shell quantities well-defined. Despite the conceptual difficulties we can use some simple intuition about path integrals to illustrate an important point about the AdS/CFT correspondence: namely, there can be more than one saddle point in the range of integration, and when there is we should sum e~IsuGRA over the classical configurations to obtain the saddle-point approximation to the gauge theory partition function. Multiple classical configurations are possible because of the general feature of boundary value problems in differential equations: there can be multiple solutions to the classical equations satisfying the same asymptotic boundary conditions. The solution which globally minimizes ISUGRA is the one that dominates the path integral. When there are two or more solutions competing to minimize ISUGRA, there can be a phase transition between them. An example of this was studied

125

in 190 long before the AdS/CFT correspondence, and subsequently resurrected, generalized, and reinterpreted in 19 ' 65 as a connnement-deconfmement transition in the gauge theory. Since the qualitative features are independent of the dimension, we will restrict our attention to AdS5. It is worth noting however that if the AdS$ geometry is part of a string compactification, it doesn't matter what the internal manifold is except insofar as it fixes the cosmological constant, or equivalently the radius R of anti-de Sitter space. There is an embedding of the Schwarzschild black hole solution into anti-de Sitter space which extremizes the action 1

k/^^i)-

<73)

167rG5

Explicitly, the metric is ds2 = fdt2 + ~dr2 r2

+r2dfl23, (74)

pi

The radial variable r is restricted to r > r + , where r+ is the largest root of / = 0. The Euclidean time is periodically identified, t ~ t + f3, in order to eliminate the conical singularity at r = r+. This requires 2itR2r+

^HTff-

(75)

Topologically, this space is S 3 x B2, and the boundary is 5 3 x S1 (which is the relevant space for the field theory on S 3 with finite temperature). We will call this space X2. Another space with the same boundary which is also a local extremum of (73) is given by the metric in (74) with \x = 0 and again with periodic time. This space, which we will call X\, is not only metrically distinct from the first (being locally conformally flat), but also topologically B4 x S1 rather than S3 x B2. Because the 5 1 factor is not simply connected, there are two possible spin structures on X\, corresponding to thermal (anti-periodic) or supersymmetric (periodic) boundary conditions on fermions. In contrast, X2 is simply connected and hence admits a unique spin structure, corresponding to thermal boundary conditions. For the purpose of computing the twisted partition function, Tr(—l) F e~ / 3 i f , in a saddle-point approximation, only X\ contributes. But, Xi and X2 make separate saddle-point contributions to the usual thermal partition function, Tre _ / 3 H , and the more important one is the one with the smaller Euclidean action.

126

Actually, both I(X\) and /(X2) are infinite, so to compute /(X 2 ) — I{X\) a regulation scheme must be adopted. The one used in 6 5 ' 1 8 3 is to cut off both Xi and X2 at a definite coordinate radius r = R0. For X2, the elimination of the conical deficit angle at the horizon fixes the period of Euclidean time; but for Xi, the period is arbitrary. In order to make the comparison of I(Xi) and /(X 2 ) meaningful, we fix the period of Euclidean time on X\ so that the proper circumference of the S\ at r = Ro is the same as the proper length on Xi of an orbit of the Killing vector d/dt, also at r = Ro. In the limit Ro —> 00, one finds v ) - ^ t f - r ' ) (76) I(X2)-I(X1)-4G5{2rl+R2) , (76) where again r + is the largest root of / = 0. The fact that (76) (or more precisely its AdS4 analog) can change its sign was interpreted in 19° as indicating a phase transition between a black hole in AdS and a thermal gas of particles in AdS (which is the natural interpretation of the space Xi). The black hole is the thermodynamically favored state when the horizon radius r+ exceeds the radius of curvature R of AdS. In the gauge theory we interpret this transition as a confinement-deconfinement transition. Since the theory is conformally invariant, the transition temperature must be proportional to the inverse radius of the space S3 which the field theory lives on. Similar transitions, and also local thermodynamic instability due to negative specific heats, have been studied in the context of spinning branes and charged black holes jyiost of these works are best understood on the m 191,192,193,194,195,196,197 CFT side as explorations of exotic thermal phenomena in finite-temperature gauge theories. Connections with Higgsed states in gauge theory are clearer in 198,199 T h e r e l e v a n c e t 0 confinement is explored in 1 9 6 . See also200,201,202,203 for other interesting contributions to the finite temperature literature. Deconfinement at high temperature can be characterized by a spontaneous breaking of the center of the gauge group. In our case the gauge group is SU(N) and its center is Zyv- The order parameter for the breaking of the center is the expectation value of the Polyakov (temporal) loop (W(C)). The boundary of the spaces X\,X2 is S3 x S1, and the path C wraps around the circle. An element of the center g e Zjv acts on the Polyakov loop by (W(C)) ->• g(W(C)). The expectation value of the Polyakov loop measures the change of the free energy of the system Fq (T) induced by the presence of the external charge q, (W(C)) ~ exp(-Fq(T)/T). In a confining phase Fq{T) is infinite and therefore (W(C)) = 0. In the deconfined phase Fq(T) is finite and therefore (W{C)) # 0. As discussed in section 5, in order to compute (W(C)) we have to evaluate the partition function of strings with a worldsheet D that is bounded by the I(X)

f f

127

loop C. Consider first the low temperature phase. The relevant space is X\ which, as discussed above, has the topology B4 x 5 1 . The contour C wraps the circle and is not homotopic to zero in X i . Therefore C is not a boundary of any D, which immediately implies that (W(C)) — 0. This is the expected behavior at low temperatures (compared to the inverse radius of the S 3 ), where the center of the gauge group is not broken. For the high temperature phase the relevant space is X2, which has the topology S 3 x B2. The contour C is now a boundary of a string worldsheet D = B2 (times a point in 5 3 ). This seems to be in agreement with the fact that in the high temperature phase (W(C)} ^ 0 and the center of the gauge group is broken. It was pointed out in 6 5 that there is a subtlety with this argument, since the center should not be broken in finite volume (S 3 ), but only in the infinite volume limit (R ). Indeed, the solution X-}, is not unique and we can add to it an expectation value for the integral of the NS-NS 2-form field B on B2, with vanishing field strength. This is an angular parameter tp with period 2TT, which contributes iip to the string worldsheet action. The string theory partition function includes now an integral over all values of ip, making (W(C)) = 0 on S 3 . In contrast, on R one integrates over the local fluctuations of ip but not over its vacuum expectation value. Now (W(C)} ^ 0 and depends on the value of ip € U(l), which may be understood as the dependence on the center Zjv in the large N limit. Explicit computations of Polyakov loops at finite temperature were done in 204 - 205 . In 65 the Euclidean black hole solution (74) was suggested to be holographically dual to a theory related to pure QCD in three dimensions. In the large volume limit the solution corresponds to the M = 4 gauge theory on R x S 1 with thermal boundary conditions, and when the Sl is made small (corresponding to high temperature T) the theory at distances larger than 1/T effectively reduces to pure Yang-Mills on R . Some of the non-trivial successes of this approach to QCD are summarized in 1 .

Acknowledgements These notes were made by selected pieces of1. I thank 0 . Aharony, S. Gubser, H. Ooguri and Y. Oz for collaborating on writing that review. I also thank the organizers for a very nice school. This work was supported by the Sloan, Packard and Mac Arthur foundations and by the DOE grant DE-FGO2-91ER40654.

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TRIESTE LECTURES ON SOLITONS IN N O N C O M M U T A T I V E G A U G E THEORIES Nikita A. NekrasoV1 IHES, Le Bois-Marie, 35 route de Chartres, Bures-sur-Yvette, F-91440, France We present a pedagogical introduction into noncommutative gauge theories, their stringy origin, and non-perturbative effects, including monopole and instantons solutions. 1

Introduction

Recently there has been a revival of interest in noncommutative gauge theories86, They are interesting examples of nonlocal field theories which in the certain limit (of large noncommutativity) become essentially equivalent to the large N ordinary gauge theories 3 7 , 2 7 ; certain supersymmetric versions of noncommutative gauge theories arise as a' -> 0 limit of theories on Dp-branes in the presence of background J3-field24,82; the related theories arise in Matrix compactifications with C-field turned on 1 5 ; finally, noncommutativity is in some sense an intrinsic feature of the open string field theory 9 3 , 5 4 , 8 0 . A lot of progress has been recently achieved in the analysis of the classical solutions of the noncommutative gauge theory on the noncommutative versions of Minkowski or Euclidean spaces. The first explicit solutions and their moduli where analyzed in 74 where instantons in the four dimensional noncommutative gauge theory (with self-dual noncommutativity) were constructed. These instantons play an important role in the construction of the discrete light cone quantization of the M-theory fivebrane 3 , 2 ; and they also gave a hope of giving an interpretation in the physical gauge theory language of the torsion free sheaves which appear in various interpretations of D-brane states 5 9 , 4 3 , in particular those responsible for the enthropy of black holes realized via D5-D1 systems 8 8 , and also enter the S-duality invariant partition functions of J\f = 4 super-Yang-Mills theory 9 2 . One can also relax the selfduality assumption on the noncommutativity. The construction of74 easily generalizes72 to the general case 9^0 (see also 3 1 , 3 2 ). In addition to the instantons (which are particles in 4+1 dimensional theory), which represent the D0-D4 system, the monopole-like solutions were found39 in U(l) gauge theory in 3+1 dimensions. The latter turn out to have a string attached to them. The string with the monopole at its end are the noncommutative field theory "also at Institute for Theoretical and Experimental Physics, 117259 Moscow, Russia

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realization of the D3-D1 system, where Dl string ends on the D3 brane and bends at some specific angle towards the brane. One can also find the solutions describing the string itself 77 , 40 , both the BPS and in the non-BPS states; also the dimensionally reduced solutions in 2+1 dimensions 40 ' 1 , describing the D0-D2 systems; finite length strings, corresponding to U(2) monopoles 41 . The solitonic strings described in these lectures all carry magnetic fluxes. Their S-dual electric flux strings represent fundamental strings located nearby the D-branes with space-time noncommutativity. Also we do not consider theories on the noncommutative compact manifolds, like tori. Some classical and quantum aspects of the Yang-Mills theory on the noncommutative tori are analyzed in 7 8 . These lectures are organized as follows. The section 2 contains a pedagogical introduction into noncommutative gauge theories. Section 3 explains how noncommutative gauge theories arise as a' —> 0 limits of open string theory. The section 4 constructs instantons in noncommutative gauge theory on R 4 for any group U(N). The section 5 presents explicit formulae for the f/(l) gauge group. The section 6 presents monopole solutions in U(l) and U(2) noncommutative gauge theories. The section 7 is devoted to some historic remarks. The format of these lectures does not allow to cover all interesting aspects of the noncommutative gauge theories, both classical and quantum, and their relations to string/M-theory, and to large N ordinary gauge theories. We refer the interested readers to the review23 which will address all these issues in greater detail. •Exercises are printed with the help of small fonts. The symbol i denotes y/— 1, not to be confused with the space-time index i. The letter 9 will be used only for the Poisson tensor Q%\ or its components, while gauge theory theta angle will be denoted by i9. The symbol * denotes Hodge star, while * stands for star-products. 2 2.1

Noncommutative Geometry and Noncommutative Field Theory A brief mathematical

introduction

It has been widely appreciated by the mathematicians (starting with the seminal works of Gelfand, Grothendieck, and von Neumann) that the geometrical properties of a space X are encoded in the properties of the commutative algebra C(X) of the continuous functions / : X —> C with the ordinary rules of point-wise addition and multiplication: (/ + g){x) — f(x) + g{x),f • g(x) = f(x)g{x). More precisely, C(X) knows only about the topology of X, but one can

143 refine the definitions and look at the algebra C°°(A') of the smooth functions or even at the DeRham complex fi'(X) to decipher the geometry of X. The algebra A = C(X) is clearly associative, commutative and has a unit (l(x) = 1). It also has an involution, which maps a function to its complex conjugate: f^(x) = f(x). The points x of X can be viewed in two ways: as maximal ideals of A: f £ lx O f(x) = 0; or as the irreducible (and therefore one-dimensional for A is commutative) representations of A: Rx(f) = f(x), Rx ~ C. The vector bundles over X give rise to projective modules over A. Given a bundle E let us consider the space £ = T(E) of its sections. If / £ A and a £ £ then clearly fa £ £. This makes £ a representation of A, i.e. a module. Not every module over A arises in this way. The vector bundles over topological spaces have the following remarkable property, which is the content of Serre-Swan theorem: for every vector bundle E there exists another bundle E' such that the direct sum E®E' is a trivial bundle X x CN for sufficiently large AT. When translated to the language of modules this property reads as follows: for the module £ over A there exists another module £' such that £ © £' = FN = A®N. We have denoted by FN = A ®c CN the so-called free module over A of rank N. Unless otherwise stated the symbol below will be used for tensor products over C. The modules with this property are called projective. The reason for them to be called in such a way is that £ is an image of the free module F/v under the projection which is identity on £ and zero on £'. In other words, for each projective module £ there exists N and an operator P £ Hom(i AV The algebras of smooth functions are mapped in the opposite

144

way: g* : C°°{X2) ->• C°°(Xi), g*{f){xx) algebras is the algebra homomorphism:

= /(gfa)).

The induced map of the

<7*(/i/2) = S*(/i)s*(/ 2 ), 9*(h + / 2 ) = g_1ag The infinitesimal diffeomorphisms of the ordinary manifolds are generated by the vector fields Vldi, which differentiate functions, f^f

+ eV'dif

In the noncommutative setup the vector field is replaced by the derivation of the algebra V £ Der(A): a H-> a + eV(a),

V(a) e A

and the condition that V(a) generates an infinitesimal homomorphism reads as: V{ab) = V{a)b + aV(b) which is just the definition of the derivation. Among various derivations there are internal ones, generated by the elements of the algebra itself: Vc(a) = [a,c] := ac— ca,

c£ A

These infinitesimal diffeomorphisms are absent in the commutative setup, but they have close relatives in the case of Poisson manifold X. 2.2

Flat noncommutative

space

The basic example of the noncommutative algebra which will be studied here is the enveloping algebra of the Heisenberg algebra. Consider the Euclidean space R d with coordinates x',i = 1 , . . . , d. Suppose a constant antisymmetric matrix 8lj is fixed. It defines a Poisson bi-vector field ^6ljdt A dj and therefore the noncommutative associative product on Rd. The coordinate functions x% on

145

the deformed noncommutative manifold will obey the following commutation relations: [x\xi] = iBi> , (1) We shall call the algebra Ag (over C) generated by the x% satisfying (1), together with convergence conditions on the allowed expressions of the xx - the noncommutative space-time. The algebra Ag has an involution a 4 a ' which acts as a complex conjugation on the central elements ( A - l ) - A - l , A G C and preserves x1: (x 2 ) t = xl. The elements of Ag can be identified with ordinary complex-valued functions on R d , with the product of two functions / and g given by the Moyal formula (or star product): f*g(x)

= exp

2

f(xi)g(x2)\Xl-_

dx\ dx32

(2)

Fock space formalism By an orthogonal change of coordinates we can map the Poisson tensor 8ij onto its canonical form: xl H-» za,za,

a=l,...,r;

yb,

b = 1 , . . . ,d - 2r,

so that:

[ya,yb] = [yb,Za] = [yb,za] = o,

[za,zb} = -29a5ab,

6a>0

(3)

ds2 - dxj + dyl = dzadza + dy\. Since z{z) satisfy (up to a constant) the commutation relations of creation (annihilation) operators we can identify functions f(x,y) with the functions of the ya valued in the space of operators acting in the Fock space %r of r creation and annihilation operators:

~Hr = (J) C | n i , . . . , n r ) 4 _

VMa -

a

ca\n) = ,fna~\n - la),

zZ

V20a

ai

\pat C(J

(4)

Sab

c\\n) = \/na + l|n + 10

Let na = c\ca be the o'th number operator.

146

The Hilbert space TLr is the example of left projective module over the algebra Ae- Indeed, consider the element Po = |0)(0| ~ exp — J2a ^f53-. It obeys PQ = PQ, i.e. it is a projector. Consider the rank one free module Fi = Ae and let us consider its left sub-module, spanned by the elements of the form: f * Po. As a module it is clearly isomorphic to 7ir, isomorphism being: \n) H-» |n)(0|. It is projective module, the complementary module being

Ae(l-Po)

cAe.

The procedure that maps ordinary commutative functions onto operators in the Fock space acted on by za, za is called Weyl ordering and is defined by: f(x) H- f(za,za)

= j f{x) ^ 0 ^

e *<*.*.+».*.-p-).

(5)

•Show that if / H* / , g H-» g then / * g >-» fg. Integration over the noncommutative space Weyl ordering also allows to express the integrals of the ordinary functions over the commutative space in terms of the traces of the operators, corresponding to them: Jd2rxf(x) = (2ne)TTTHJ, (6) as follows immediately from (5). Sometimes the integral reduces to the boundary term. What is the boundary term in the noncommutative, Fock space setup? To be specific, let us consider the case r = 1. The general case follows trivially. Consider the integral Jd2x

(ft A+82

f2) = (27ri)Tvn {[f\,x2] + [x\f2])

=

TTV26TT

([c,f] - [c+,ft])

f = / i + if2- In computing the trace Tr w [c,f] = 5 > | [ c , f ] | n ) ,

(8)

n

we get naively zero, for the trace of a commutator usually vanishes. But we should be careful, since the matrices are infinite and the trace is an infinite sum. If we regulate it by restricting the sum ton < N, then the matrix element (iV|c|iV + l)(N + l|f|iV) is not cancelled, so that the regularized trace is Trn„[c,f\ = VNTl{N

+ l\f\N)

(9)

147 and similarly for c*\ Thus: /

/idx 2 - }2Axl =

2TT^/26(N

+ l)Re{N + 1| / i + if2 \N)N^oo

(10)

J oo

• Consider f = -j-c . Compute the integral (7) directly and via (10). Let us conclude this section with the remark that (6) defines the trace on a part of the algebra Ag consisting of the trace class operators. They correspond to the space L 1 (R 2 r ) of integrable functions. Even though the trace is taken in the specific representation of Ag it is defined intrinsically, thanks to (6). In what follows we omit the subscript T-ir in the notation for the trace on A$. Symmetries of the flat noncommutative space The algebra (1) has an obvious symmetry: xl H-> xl + e% with e1 £ R. For invertible Poisson structure 0, such that 6ij6^k — 8f this symmetry is an example of the internal automorphism of the algebra: a^eUi'£ixiae-'l9i'eixJ

(11)

In addition, there are rotational symmetries which we shall not need. 2.3

Gauge theory on noncommutative space

In an ordinary gauge theory with gauge group G the gauge fields are connections in some principal G-bundle. The matter fields are the sections of the vector bundles with the structure group G. Sections of the noncommutative vector bundles are elements of the projective modules over the algebra Ag. Gauge fields, matter fields In the ordinary gauge theory the gauge field arises through the operation of covariant differentiation of the sections of a vector bundle. In the noncommutative setup the situation is similar. Suppose M is a projective module over Ag. The connection V is the operator V : Kd x M -»• M,

V e (m) € M,

e e Rd, m £ M ,

where R d denotes the commutative vector space, the Lie algebra of the automorphism group generated by (11). The connection is required to obey the Leibnitz rule: V e (ami) = el{dia)m\ + aV E mi (12)

148

Ve{mTa) = m r £'(9 ; a) + (Vemr)a

.

(13)

Here, (12) is the condition for left modules, and (13) is the condition for the right modules. As usual, one defines the curvature Fij = [Vj,Vj] - the operator A 2 R d x M -» M which commutes with the multiplication by a € Ag. In other words, F^ £ End^(M). In ordinary gauge theories the gauge fields come with gauge transformations. In the noncommutative case the gauge transformations, just like the gauge fields, depend on the module they act in. For the module M the group of gauge transformations QM consists of the invertible endomorphisms of M which commute with the action of ^ on M: GM=GLAe(M) Its Lie algebra End^(M) will also be important for us. All the discussion above can be specified to the case where the module has a Hermitian inner product, with values in Agin addition to gauge fields gauge theories often have matter fields. One should distinguish two types of matter fields. First of all, the elements tp of the module M where V acts, can be used as the matter fields. Then V;

Tir, i = 1 , . . . , 2r, such that: [V;,a] = did

149 for any a £ A. Using the fact t h a t dia = i8ij[xj,a] %r we conclude that: Vi = i6ijxj

+ m,

and the irreducibihty of

Ki e C

(14)

If we insist on unitarity of V , then k ; £ R . Thus, the space of all gauge fields suitable for acting in the Fock module is rather thin, and is isomorphic t o the vector space R d (which is canonically dual to the Lie algebra of the automorphisms of Ag). T h e gauge group for the Fock module, again due t o its irreducibihty is simply the group U(l), which multiplies all the vectors in Qr by a single phase. In particular, it preserves K^'S, SO they are gauge invariant. It remains to find out what is the curvature of the gauge field given by (14). T h e straightforward computation of the commutators gives: Fij = iezj

(15)

i.e. all connections in t h e Fock module have t h e constant curvature. Free m o d u l e s a n d c o n n e c t i o n s t h e r e If the right (left) module M is free, i.e. it is a sum of several copies of the algebra Ag itself, then the connection Vi can be written as Vi = di + Ai where Ai is the operator of the left (right) multiplication by the matrix with Ae-valued entries: Vimi = dim\ + m\Ai,

V i m r = dimT + Aimr

(16)

In the same operator sense the curvature obeys the standard identity: Fij ~ OiJ\j

UjJ\i T J\-iJ±j

/\.j/±i

.

Given a module M over some algebra A one can multiply it by a free module A®N to make it a module over an algebra Mat NXN(A) of matrices with elements from A. In the non-abelian gauge theory over A we are interested in projective modules over Matjvxjv(- / l)- If the algebra A (or perhaps its subalgebra) has a trace, Tr, then the algebra Matjv X iv(-4) has a trace given by t h e composition of a usual matrix trace and Tr. It is a peculiar property of the noncommutative algebras t h a t the algebras A and Mat;vxiv(-4) have much in common. These algebras are called Morita equivalent and under some additional conditions the gauge theories over A and over M a t y v x i v ^ ) are also equivalent. This phenomenon is responsible for t h e similarity between the "abelian noncommutative" and "non-abelian commutative" theories.

150

Observables from gauge fields Just like in the commutative case the difference of two connections in the module M is the operator from Endy\g(M). Thus we can write: Vi = idijxj + Di,

Di G End^ 9 (M)

(17)

The curvature of the connection Vi is, therefore: Fij = [V,, Vj] = iOtj + [Du Dj] e End^ e (Af)

(18)

For free modules the operators Di appear from the formulae (16) if we represent di as i9ij[xj, •]: Vimi = i6ijXjmr + mTDi,

V;m r = -mridijX:' + D\mT

(19)

The relation of the operators Di and the conventional gauge fields Ai is Di = -i6zjxj

+ Ai

(20)

Going back to the generic case, we shall now describe a (overcomplete) set of gauge invariant observables in the gauge theory on the module M. The gauge transformations act on the operators Di "locally", i.e. for g G GL^ e (M): Di H+ g-lDi9

(21)

Hence the spectrum of the operators Di, or of any analytic function of them is gauge invariant. In particular, the following observables are the noncommutative analogues of the Wilson loops in the ordinary gauge theory. Choose a contour on the ordinary space R 2 r : 7*(t), 0 < t < 1. Define an operator WT = P e x p / dtDif(t) ./o Jo which also transforms as in (21). The following observable W 7 = Tr M U-y

(22)

(23)

is gauge-invariant. It is typically a distribution on the space of contours 7. For example, for the Fock module W 7 = e

*,(7'(l)-7'(0))

x T r

^

1

while for the free module in the vacuum (Di = —\6ijxi): W7=<52r(^(7i(l)-7i(0)))exPi

0ii7W / ,7

The operators (23) are closely related to the "noncommutative momentum carrying Wilson loops" considered in 5 0 .

151

2.4

Lagrangian, and couplings

In order to write down the Lagrangian for the gauge theory in the commutative setup one needs to specify a few details about the space-time and the gauge theory: the space-time metric GM„, gauge coupling
£{A) = - *

J2 VGGii'G»'TrFijFi,j,

.

(24)

If additional adjoint Higgs matter fields $ are present then the (24) becomes:

£(A, #) = £{A) + Y^ \/GG" TrVi$Vi-* + ...

(25)

These formulae make sense for the constant Euclidean metric Gij only. String theory allows more general backgrounds, where the closed string metric g and the B field both are allowed to be non-constant. They are presumably described by more abstract techniques of noncommutative geometry14 which we shall not use in these lectures. We now proceed with the exposition of how the associative algebras, their deformations, and gauge theories over them arise in the string theory. 3 3.1

Noncommutative geometry and strings in background B-fields Conventional strings and D-branes

Let us look at the theory of open strings in the following closed string background: Flat space X, metric g^, constant Neveu-Schwarz B-field B^; Dpbranes are present, so that B^ jt 0 for some i, j along the branes. The presence of the Dp-branes means that the B; 3 cannot be gauged away - if we try to get rid of B^ by means of the gauge transformation B -> B 4- dk then we create a gauge field Ai on the brane, whose field strength F^ is exactly equal to B^. The bosonic part of the worldsheet action of our string is given by: S = j—j

f (gijdaxidaxj

- 2Tria'Bijeabdaxidbxj)

(26)

152

Here we denote by dt the tangential derivative along the boundary dT, of the worldsheet £ (which we shall momentarily take to be the upper half-plane). We shall also need the normal derivative dn. The boundary conditions which follow from varying the action (26) are6: gijdnx* + 2Tvia'Bijdtxj\d-£

= 0

(27)

Now, on the upper half-plane with the coordinate z = t + iy, y > 0 we can compute the propagator: {xi{z)xj{w)} = (28) -a' U l o g (Z-^-\ + G'Hoglz - w\2 + - ^ ' l o g ( ^ - ) + D" \z — w J 1-KO! \z — w) where D^ is independent of z, w, Gij = (

eij = i-Kci

^

^]

(29)

1 g + 2ira'B

where S and A denote the symmetric and antisymmetric parts respectively. The open string vertex operators are given by the expressions like

: / (x(t), dtx(t), dfx(t), ...):

(30)

where everything is evaluated at y = 0. The properties of the open strings are encoded in the operator product expansion of the open string vertex operators. By specifying (28) at y = 0 we get b

It was observed by S. Shatashvili in 1995 right after the appearance of 6 that the Bfield (or equivalently the constant electromagnetic field) as well as the tachyon interpolates between the Dirichlet and von Neumann boundary conditions, and as a consequence between different solutions of the target space lagrangian of background independent open string field theory 9 4 , 8 5 , and that by taking some of the components of Btj —> oo one creates lower dimensional D-branes. Similar properties of the tachyon backgrounds are now extensively investigated 34 - 56 ' 35 - 7 . 83 - 89

153

the expression for the propagator of the boundary values of the coordinates (i*(*)a^'(s)> = -a'Glj\og{t

- sf + \oije{t

- s)

(31)

where e(t) = - 1 , 0 , + 1 for t < 0, t = 0, t > 0 respectively. Prom this expression we deduce : [x\xj]

:= T(x\t

- 0)xj(t) - x\t + 0)xj(t))

= i0ij

(32)

It means that the end-points of the open strings live on the noncommutative space where: [x\xj]=i6ij with 0li being a constant antisymmetric matrix. Similarly, the OPE of the vertex operators: Vp(t) =: e 1 "- : (t) is given by : VP(t)Vq(s) = (t- s)2a'G'ip^e-iei'p^Vp+q(s)

(33)

Seiberg and Witten suggested32 to consider the limit a' —> 0 with G, 0 being kept fixed. In this limit the OPE (33) goes over to the formula for the modified multiplication law on the ordinary functions on a space with the coordinates x: VpVq=e-ie'lp'VVp+q (34) (the appearance of the noncommutative algebra (34) in the case of compactification on a shrinking torus was observed earlier by M. Douglas and C. Hull in 2 4 ). The algebra defined in (34) is isomorphic to Moyal algebra (2). The product (34) is associative but clearly noncommutative. Witten 96 remarked that the a' —> 0 limit with fixed gij,0lj makes the algebra of open string vertex operators (which is associative as the vertex algebra) to factorize into the product of the associative algebra of the string zero modes and the algebra of string oscillators (excited modes). This allows to see some stringy effects already at the level of the noncommutative field theories. 3.2

Effective action

Vertex operators (30) give rise to the space-time fields $& propagating along the worldvolume of the Dp-brane. Their effective Lagrangian is obtained by

154

evaluating the disc amplitudes with (30) inserted at the boundary of the disc:

/ '

dp+1xVdetGTrdniidn22

• • -dnk§k

~

(35)

fc

( J ] : P m (dx(tm), d2x(tm),...) 771 =

<>*».•*('".) : )

1

If we compare the effective actions of the theory at 6 = 0 and 0 / 0 the difference is very simple to evaluate: one should only take into account the extra phase factors from (33): k

(H

Pm (&c(t m ), 3 2 i ( t m ) , . . . ) e'P™ -<*•»> :) G , 9 =

(36)

771=1

e-*E„>m*r^r*»-«-)([[ m=l

:Pm(dx(tm),^x{tm),...)^-'^

:)G,o

(the exponent in (36) is depends only on the cyclic order of the operators, due to the antisymmetry of 9 and the momentum conservation:

m

Using (34),(2) we can easily conclude that the terms like (35) go over to the terms like: ' dp+1xVdetGTrdni$! *dn2$2 * • • • *dnk$k (37) /• This result holds even without taking the Seiberg-Witten limit. In general, the relation between the off-shell string field theory action and the boundary operators correlation functions in the wolrdsheet conformal theory depends on the type of string theory we are talking about. In the case of bosonic string in the framework of Witten-Shatashvili background independent open string field theory this relation reads as 94 ' 85 : S^Z-F-^Z where Z(t) is the generating function of the boundary correlators: Z{t) = (exp
eOi)

(38)

155 with d running through some basis in the space of the open string vertex operators, tl being the corresponding couplings, and (5l the /^-function of the coupling tl:

? = Aif The relation(38) derived i n 8 5 is very restrictive, for ft1 vanish exactly where dS does, and allows to calculate the exact expression for S u p to two space-time derivatives 3 4 ' 5 6 . For the superstring it seems 57 t h a t S = Z. This approach was successfully applied to the study of D-branes with the B-field turned o n 1 8 , 7 5 , 7 . 3.3

Gauge theory from string

theory

Open strings carry gauge fields. This follows from the presence in the spectrum of the allowed vertex operators of the operator of the following simple form: dip)

: dtzV*"* : ^ ^

A = Ai(x)dxi

(39)

It has (classical) dimension 1 and deforms the worldsheet conformal field theory as follows: S = S0-i
AiWdtx'dt

(40)

This deformation has the following naive symmetry: SAj = d3e

(41)

e(x) is the tachyonic vertex operator. Let us see, whether (41) is indeed a symmetry. To this end we estimate the (5-variation of the disc correlation function: 6 [ Dx e~s = {ij> dtedt)s

= (i I dtedt)So

dtdsAjixitydtx^^dseis^So

(42)

+ ••-

Let us regularize (42) by point splitting, i.e. by understanding the s,t integration with s / t (of course, there are other regularizations, their equivalence leads t o important predictions concerning the noncommutative gauge theory, s e e 8 2 ) . Then the total s-derivative needs not to decouple. Instead, it gives: -{
f d8Aj(x(t))dtxi(t)dte{x(a)))s0 Js^t

dtAjixWdtx^t)

=

(e(x(t + 0)) - e(x(t - 0 ) ) ) ) S o =

156

=

((be-kA~A-ks)i

This calculation shows that the naive transformation (41) must be supplemented by the correction term: 6Aj =djE + Aj-ke-e*

Aj

(43)

The formula (43) is exactly the gauge transformation of the gauge field in the gauge theory on the noncommutative space. Similarly, the effective action for the gauge fields in the presence of 6 ^ 0 becomes that of the Yang-Mills theory on the noncommutative space plus the corrections which vanish in the a ' -> 0 limit. In what follows we shall need the relation between various string theory moduli in the case of D3-branes in the background of the constant B-field: Open and closed string moduli. We want to consider the decoupling limit of a D3-brane in the Type IIB string theory in a background with a constant Neveu-Schwarz B-field. Let us recall the relation of the parameters of the actions (24), (25) and the string theory parameters, before taking the Seiberg-Witten limit 8 2 . We start with the D3-brane whose worldvolume is occupying the 0123 directions, and turn on a B-field: ^Bdx1

A dx2

(44)

The indices i,j below will run from 1 to 3. We assume that the closed string metric gij is flat, and the closed string coupling gs is small. According to 82 the gauge theory on the D3-brane is described by a Lagrangian, which, when restricted to time-independent fields, equals (25) with the parameters Gij,0lJ,9YM

i

which are related to 9ij 5 *->i j •> 9 s

via (29) as follows:

G , = 9lJ - (W)» (Bg^B),., & = -(W)» ( ^ ^ J — J L ^ ) ' ' (45) g$M=4n2gs(det{l

+ 2Tra'g-1B))i

.

157

Suppose the open string metric is Euclidean: dj

= 6~ij, then (44),(45) imply:

ff = £fa +

» (wpT^^ + ^ '

B=

(2*Jr

+ *'

(46)

and 9YU

2TT v /(2 7 ra') 2 + #2

(47)

The Seiberg-Witten limit is achieved by taking a' ->• 0 with G,#,5y M kept fixed. In this limit the effective action of the D3-brane theory will become that of the (super)Yang-Mills theory on a noncommutative space Ae • The relevant part of the energy functional is: £ = -A~

[ TiiBi + Bi + ViQ + ViQ) ,

(48)

where i

Bi = -EijkFjk •

(49)

The fluctuations of the D3-brane in some distinguished transverse direction are described by the dynamics of the Higgs field. Of course the true D3-brane theory will have six adjoint Higgs fields, one per transverse direction. We are looking at one of them. Also notice that the expression for the energy (48) goes over to the case of N D3-branes, one simply replaces Ae by Matyv(-4e) 3.4

Noncommutative geometry from topological string theory

String theory can be well-defined even in the presence of non-constant B-field. The absolute minima of the B-field energy are achieved on the flat B-fields, i.e. if dB = 0. In this case, if in addition the tensor Bij is invertible one can define the inverse tensor 9lj — (J3-1)1-7' which will obey Jacobi identity 038 = 0. It is not widely appreciated in the string theory literature that a string can actually propagate in the background of the non necessarily invertible Poisson bi-vector #1J (we leave the question of whether the Poisson property can also be relaxed to future investigations). We shall now describe a version of a string theory, which is perfectly sensible even for the non-constant Poisson bi-vector field 0 t J , not necessarily invertible. To do this we shall start with the bosonic string action with the target space X and then take the a' —¥ 0 limit a la Seiberg and Witten. For simplicity we make an analytic continuation 9^ —• i# 1J (cf. 54 ' 13 ).

158

Take the action (26) not assuming that gij, £?;., are constant and rewrite it in the first order form: S = - ^

f {gijdaxidaxj

- 2TTia'Bijeabdaxidbxj)

o

(50)

Pi A dxr - 7ra' Gnpi A *pj - -0lJpi A pj where 2ira'G + 6 =

2-KOL

(g +

l

2-KOL B)~

Now suppose we take a' —> 0 limit keeping 6 and G fixed. The remaining part Jp A da; + %9p A p of the action (50) immediately exhibits an enhancement of the (gauge) symmetry 13 : Pi

H-> pi - dXi - diejkpj

Afc,

xi M- x i + 0ij'Aj

(51)

This symmetry must be gauge fixed, at the cost of introduction of a sequence of ghosts, anti-ghosts, Lagrange multipliers and gauge conditions13. It is not the goal of these lectures to do so in full generality with regards to various types of target space diffeomorphisms invariance one might want to keep track of8. Let us just mention that the result of this gauge fixing procedure is the topological string theory, which has some similarities both with the type A and type B sigma models. The field content of this sigma model is the following. It is convenient to think of p, and xl as of the twisted super-fields, which simply mean that both pi and x% are differential forms on E having components of all degrees, 0,1,2. The original Lagrangian had only 0-th component of x1 and 1st component of pi- In addition, there are auxiliary fields x\-^% which are zero forms of opposite statistics - x °f the fermionic while H of the bosonic one. The gauge fixing conditions restrict the components of the superfields p and x to be:

4)=*d*\

pf]=Q

(52)

We are discussing an a' —> 0 limit of an open string theory. It means that the worldsheet S has a boundary, 9E. The fields x,p,x,H obey certain boundary conditions: x obeys Neumann boundary conditions, H vanishes at 9E, i.e. it obeys Dirichlet boundary conditions. In this sense the theory we are studying is that of a D-brane wrapping the zero section (H = 0) X of the tangent bundle TX to the target space X. The rest of boundary conditions is summarized in the formula for the propagator below. The field \ l is constant at the boundary (fermionic Dirichlet condition).

159

The further treatment of the resulting theory is done in the perturbation series expansion54 in 8 around the classical solution xl(z) = x1 = const. We shall work on E being the upper half-plane Imz > 0. The propagator is most conveniently written in the superfield language:
^

z

^

= kl0g{l-l]{Zz-^)

(53)

where d is the deRham differential on E x E. Bulk observables The theory can be deformed by adding observables to the action. Any polyvector field Q ^ - ^ A A . . . -?A- € A T X on X defines an observable in the theory: OX

1

OX

9

Oa=ail-i»pil...pi,

(54)

which is again an inhomogeneous form on E. Its degree 2 component can be added to the action. In particular, the 0-term in the original action corresponds to the bi-vector \9l^di A dj. If we write the action (before the gauge fixing) in the form: Pi A da;1 + Oa / where a is a generic polyvector field on X then the only condition is that [a, a] = 0 where [, ] is the Schoutens bracket on the polyvector fields:

(the sign is determined by the parity of the degrees of a, (3, in such a way that [,] defines a super-Lie algebra). Boundary observables The correlation functions of interest are obtained by inserting yet another observables at the boundary of E. These correspond to the differential forms onX: uh...iqdx11

A . . . A dx1" H-> u>il..Aqxil

•••Xiq

Actually, one is interested in computing string amplitudes. It means that the positions of the vertex operators at the boundary must be integrated over, up to the SL2(R) invariance (the gauge fixed action still has (super)conformal invariance).

160

In particular, the three-point function on the disc (or two-point function on the upper-half-plane, as a function of the boundary condition at infinity, x(oo) = x), produces the deformed product on the functions: (f(x(0))g(x(l))

[h(x)X ...x] (oo)) e , E = d i . c = [ f*gh Jx

(56)

for f,g£ Fun(X), h G fldirnX{X). The ^-product defined by (56) turns out to be associative: / * (g * /) = (f*g)*l. This is a consequence of a more general set of Ward identities obeyed by the string amplitudes in the theory. For their description s e e 5 4 , 1 3 . The perturbative expression for the *-product following from (56) turns out to be non-covariant with respect to the changes of local coordinates in X. This is a consequence of a certain anomaly in the theory. It will be discussed in detail in 8 . From now on we go back to the case where X is flat and 6 is constant.

161

4

Instantons in noncommutative gauge theories

We would like to study the non-perturbative objects in noncommutative gauge theory. In this lecture specifically we shall be interested in four dimensional instantons. They either appear as instantons themselves in the Euclidean version of the four dimensional theory (theory on Euclidean D3-brane), as solitonic particles in the theory on D4-brane, i.e. in 4+1 dimensions, or as instanton strings in the theory on D5-brane. They also show up as "freckles" in the gauge theory/sigma model correspondence61. We treat only the bosonic fields, but these could be a part of a supersymmetric multiplet. A D3-brane can be surrounded by other branes as well. For example, in the Euclidean setup, a D-instanton could approach the D3-brane. In fact, unless the D-instanton is dissolved inside the brane, the combined system breaks supersymmetry 8 2 . The D3-D(-l) system can be rather simply described in terms of a noncommutative U(l) gauge theory - the latter has instanton-like solutions74. More generally, one can have a stack of k Euclidean D3-branes with ./VD(-l)branes inside. This configuration will be described by charge N instantons in U(k) gauge theory. Let us work in four Euclidean space-time dimensions, i, j = 1,2,3,4. As we said above, we shall look at the purely bosonic Yang-Mills theory on the spacetime Ae with the coordinates functions xl obeying the Heisenberg commutation relations: [xi,xj]=i9ij (57) We assume that the metric on the space-time is Euclidean: Gij — Oij

(58)

The action describing our gauge theory is given by: S =

1— TTF A *F + — TrF A F

(59)

where gyM is the Yang-Mills coupling constant, i? is that gauge theory theta angle, and F = ^Fijdx1 A dxj,

Fa = [Vi, Vj]

Here the covariant derivatives act Vi in the free module Fk-

(60)

162

4-1

Instantons

The equations of motion following from (59) are:

Y,[Vi,Fij]=Q

(61)

i

In general these equations are as hard to solve as the equations of motion of the ordinary non-abelian Yang-Mills theory. However, just like in the commutative case9, there are special solutions, which are simpler to analyze and which play a crucial role in the analysis of the quantum theory. These are the so-called (anti-)instantons. The (anti-)instantons solve the first order equation: Fij = ± - 2_^£ijkiFki

(62)

k,i

which imply (61) by virtue of the Bianci identity: [Vi,Fjk]+

cyclic permutations = 0

These equations are easier to solve. The solutions are classified by the instanton charge: N = —^TTFAF

(63)

and the action (59) on such a solution is equal to: SN = 27rir7V,

T

=

47ri

+

9YM

$

(64)

2

*

Introduce the complex coordinates: z\ = x\ + 1x2 = x+, z
FZlXl+FZaS3=0

(65)

The first equation in (65) can be solved (locally) as follows:

A-Za = C'dzJ,

AZa = -&zM~l

•

(66)

with £ a Hermitian matrix. Then the second equation in (65) becomes Yang's equation: 2 2

£ < M ^ r ) = o.

(e?)

0=1

In the noncommutative case this ansatz almost works globally (see below).

163

4.2

ADHM

construction

In the commutative case all solutions to (62) with the finite action (59) are obtained via the so-called Atiyah-Drinfeld-Hitchin-Manin (ADHM) construction5. If we are concerned with the instantons in the U(k) gauge group, then the ADHM data consists of 1. the pair of the two complex vector spaces V and W of dimensions N and k respectively; 2. the operators: BUB2

<E Hom(V, V), and J £ HormW, V), J £

Eom(V,W);

3. the dual gauge group Gyv = U(N), which acts on the data above as follows: Ba .-*• g-'Bag, I.-» g^I, J .-> Jg (68) 4. Hyperkahler quotient47 with respect to the group (68). It means that one takes the set Xk,N = AV'HO) I"1 / i 7 1 (0) °f the common zeroes of the three moment maps: fir = [Bi,Bl] + [B2,B\]+IP Hc = [B1,B2]+IJ

-J^J (69)

fic = [Bl,Bl] + Jll1 and quotients it by the action of GNThe claim of ADHM is that the points in the space Mk,N = X% NI^N parameterize the solutions to (62) (for 6%i = 0 ) up to the gauge transformations. Here X% N C Xk,N is the open dense subset of Xk,N which consists of the solutions to fl = 0 such that their stabilizer in GN is trivial. The explicit formula for the gauge field Ai is also known. Define the Dirac-like operator: * = ( ' £ - £ Here zi,z2

Bl'-l

j . ^ V a C e H ' - V , *

(70)

denote the complex coordinates on the space-time:

z\ = x\ + ix2,

z2 = xz + ixi,

z\ — X\ — ix2,

z2 = x$ — ix±

The kernel of the operator (70) is the ^-dependent vector space £x C V C2(BW. For generic x, £x is isomorphic to W. Let us denote by $ = $(x) this isomorphism. In plain words, $ is the fundamental solution to the equation: £>+# = 0,

$ :W ^>V ®C2®W

(71)

164

If the rank of "3/ is x-independent (this property holds for generic points in M), then one can normalize: * f * = Id*

(72)

which fixes * uniquely up to an x-dependent U(k) transformation ^(x) H-> ^(x)g(x), g(x) e U(k). Given * the anti-self-dual gauge field is constructed simply as follows: Vi = di + Al,

Az = ^(x)-^-${x)

(73)

The space of (£?o, B\, I, J) for which ty(x) has maximal rank for all x is an open dense subset MN,U — X^^/GN in M. The rest of the points in X^^/GN describes the so-called point-like instantons. Namely, \t(ir) has maximal rank for all x but some finite set {x\, . . . , £ ; } , I < k. The (72) holds for x ^ xt, i — 1,... ,1, where the left hand side of (72) simply vanishes. The noncommutative deformation of the gauge theory leads to the noncommutative deformation of the ADHM construction. It turns out to be very simple yet surprising. The same data V, W, B,I,J,... is used. The deformed ADHM equations are simply Mr = C r , fJ-c = (c

(74)

where we have introduced the following notations. The Poisson tensor 9^ entering the commutation relation [a;1, a;-7'] = i9lj can be decomposed into the self-dual and anti-self-dual parts ^ ± . If we look at the commutation relations of the complex coordinates z\, z2, z\, z2 then the self-dual part of 9 enters the following commutators: [z\,z2] = -Cc

[zi,z1] + [z2,z2] = ~(r

(75)

It turns out that as long as \(\ = Cr + CcCc > 0 one needs not to distinguish between -Xjv,* and X^ k, in other words the stabilizer of any point in Xw,jb = /i~ 1 ( — (r) O ACH - CC) is trivial. Then the resolved moduli space is M.n,k = By making an orthogonal rotation on the coordinates a;M we can map the algebra Ag onto the sum of two copies of the Heisenberg algebra. These two algebras can have different values of "Planck constants". Their sum is the norm of the self-dual part of 9, i.e. \(\, and their difference is the norm of anti-self-dual part of 9: [zi,^i] = - 6 ,

[z2,z2] = - C 2

(76)

165

where Ci + C2 = |# + |, (1 - C2 = \0~\- By the additional reflection of the coordinates, if necessary, one can make both (1 and (2 positive (however, one should be careful, since if the odd number of reflections was made, then the orientation of the space was changed and the notions of the instantons and anti-instantons are exchanged as well). The next step in the ADHM construction was the definition of the isomorphism * between the fixed vector space W and the fiber £x of the gauge bundle, defined as the kernel of the operator T>+. In the noncommutative setup one can also define the operator T>+ by the same formula (70). It is a map between two free modules over Ae: D+: {V®C2®W)(g)A8-*

{V®C2)§§Ae

(77)

which commutes with the right action of Ae on the free modules. Clearly, £ = Ker£>+ is a right module over Ae, for if V+s = 0, then V+{s • a) = 0, for any a G Ae£ is also a projective module, for the following reason. Consider the operator V+V. It is a map from the free module V ® C 2 (g) Ae to itself. Thanks to (74) this map actually equals to A ® Idps where A is the following map from the free module V Cg> Ae to itself: A = (Bt - zx){B\ - -Zl) + (B2 - z2)(B\ - z2) + J / t

(78)

We claim that A has no kernel, i.e. no solutions to the equation A n = 0, v G V Ae. Recall the Fock space representation % of the algebra Ae • The coordinates za,za, obeying (76), with Ci,C2 > 0, are represented as follows: Z\ = y/Cl c\, Zi = y/ClCi,

Z2 = \/C2c2> Z2 = \/C2C2

(79)

where Cit2 are the annihilation operators and c\ 2 are the creation operators for the two-oscillators Fock space

^ = 0

C|ni,n2)

Let us assume the opposite, namely that there exists a vector v G V (8) Ae such that Av = 0. Let us act by this vector on an arbitrary state |ni,7i2) in H. The result is the vector v^ G V H which must be annihilated by the operator A, acting in V <8> U via (79). By taking the Hermitian inner product

166

of the equation Ai/n = 0 with the conjugate vector v\ we immediately derive the following three equations: (B\ - Z2)Vri

=

0

(Bl-ziK

=

0

/^n

=

(80)

0

Using (74) we can also represent Ax in the form: A = (B\ -

Zl){Bx

-

Zl)

+ (B\ - z2)(B2 - z2) + Jtj

.

(81)

From this representation another triple of equations follows: {B2 - z2)v*.

=

0

(Bi-zi)i/fl

=

0

JVf,

0

=

(82)

Let us denote by e;, i = 1 , . . . , N some orthonormal basis in V. We can expand vn in this basis as follows: JV

i=\

The equations (80),(82) imply: (Bay3vl

= zavi,

{Bi))vi=zav^

a = 1,2

(83)

in other words the matrices Ba, B\ form a finite-dimensional representation of the Heisenberg algebra which is impossible if either Ci or (2 ^ 0. Hence i/fi = 0, for any n = {n\,n2) which implies that v = 0. Notice that this argument generalizes to the case where only one of ("a ^ 0. Thus the Hermitian operator A is invertible. It allows to prove the following theorem: each vector rp in the free module (V <8> C 2 © W) ® Ae can be decomposed as a sum of two orthogonal vectors: iP = $^@Vx^,

D+*^=0,

2

X^£{V®C

)®Ag

(84)

167

where the orthogonality is understood in the sense of the following ^ - v a l u e d Hermitian product: (V'I , 1P2) = T r y 0 C 2 f f i W

(
The component *,/, is annihilated by V+, that is ^ e £. The image of V is another right module over Ae (being the image of the free module (V <8> C 2 ) ® As): £' = V{V C 2 ® A ) and their sum is a free module:

£®£'

=T:={V®C2@W)®Ae

hence £ is projective. It remains to give the expressions for $ ^ , x v :

x* = P ^ + t f . ** = nv, n = ( i - i ? ^ 2 ? + )

(85)

The noncommutative instanton is a connection in the module £ which is obtained simply by projecting the trivial connection on the free module T down to £. To get the covariant derivative of a section s G £ we view this section as a section of J7, differentiate it using the ordinary derivatives on Ae and project the result down to £ again: Vs = IIds (86) The curvature is defined through V 2 , as usual: VVs = F • s = d n A dll • s

(87)

where we used the following relations:

n2 = n, Us = s

(88)

Let us now show explicitly that the curvature (87) is anti-self-dual, i.e. [V i ,V j ] + ie ijfci [V fc ,V ; ] = 0

(89)

First we prove the following lemma: for any s G £: d n A dlls = nd£> — — <W+ s Indeed,

dn A dn = d [ v—^~v+ ) A d (v-^--v+ v+v 7 V v+v

(90)

168

V v+v

J

v+v

v+v

x>+n = o hence + d \f^^i^^ ) A dV(v-^—vA = v+v V+V J J + + = ndz>—^—d2> n + v-^— v+v v+v dv n dv—^~v+ v+v

and the second term vanishes when acting on s £ £, while the first term gives exactly what the equation (90) states. Now we can compute the curvature more or less explicitly: Fs

= 2Tl

|/_

-i/3

V 0

0

o\.s

(91)

0/

where /3, /+, /_ are the basic anti-self-dual two-forms on R 4 : h = i (da;i A dzi - dz 2 A dz 2 ), /+ = d^i A dz 2 , / - = d^i A dz 2 (92) Thus we have constructed the nonsingular anti-self-dual gauge fields over AeThe interesting feature of the construction is that it produces the non-trivial modules over the algebra A$, which are projective for any point in the moduli space. This feature is lacking in the £ —>• 0, where it is spoiled by the point-like instantons. This feature is also lacking if the deformed ADHM equations are used for construction of gauge bundles directly over a commutative space. In this case it turns out that one can construct a torsion free sheaf over C 2 , which sometimes can be identified with a holomorphic bundle. However, generically this sheaf will not be locally free. It can be made locally free by blowing up sufficiently many points on C 2 , thereby effectively changing the topology of the space n . The topology change is rather mysterious if we recall that it is purely gauge theory we are dealing with. However, in the supersymmetric case this gauge theory is a n a ' ^ 0 limit of the theory on a stack of Euclidean D3-branes. One could think that the topology changes reflect the changes of topology of D3-branes embedded into flat ambient space. This is indeed the case for monopole solutions, e.g. 12>45>63>62. It is not completely unimaginable possibility, but so far it has not been justified (besides from the fact that the DBI solutions 82 ' 91 are ill-defined without a blowup of the space). What makes this unlikely is the fact that the instanton backgrounds have no worldvolume scalars turned on.

169

At any rate, the noncommutative instantons constructed above are welldefined and nonsingular without any topology change. Also note, that the formulae above define non-singular gauge fields for any 6 ^ 0. For 9+ ^ 0 these are instantons, for 9~ ^ 0 they define anti-instantons (one needs to perform an orthogonal change of coordinates which reverses the orientation of R 4 ). The identificator * In the noncommutative case one can also try to construct the identifying map $ . It is to be thought as of the homomorphism of the modules over A:

* :W

®Ae^£

The normalization (72), if obeyed, would imply the unitary isomorphism between the free module W ® Ae and E. We can write: II = *k^ and the elements s of the module £ can be cast in the form: s = *-c7,

a eW ®Ae

(93)

Then the covariant derivative can be written as: Vs = ILi(* • a) = * * f d (*
(94)

where A = ¥dV

(95)

just like in the commutative case. For Pf(0) ^ 0, after having introduced the "background independent" operators 81 Di = idijX^ + Ai} we write: Di = i&0ijx>*

5

(96)

Abelian instantons

Let us describe the case of U(l) instantons in detail, i.e. the case k — 1 in our notations above. Let us assume that Q+ ^ 0. It is known, from68, that for Qr > 0, ( c = 0 the solutions to the deformed ADHM equations have J = 0. Let us denote by V the complex Hermitian vector space of dimensionality N, where Ba, a = 1,2 act. Then i" is identified with a vector in V. We can choose our units and coordinates in such a way that (r = 2, £c = 0.

170

5.1

Torsion free sheaves on C 2

Let us recall at this point the algebraic-geometric interpretation 68 of the space V and the triple (Bi,B2,I). The space XN,I parameterizes the rank one torsion free sheaves on C 2 . In the case of C 2 these are identified with the ideals 1 in the algebra O ss C[zi,z2] of holomorphic functions on C 2 , such that V = O/X has dimension N. An ideal of the algebra O is a subspace I c O , which is invariant under the multiplication by the elements of 0 , i.e. if g G 1 then fg 6 1 for any O. An example of such an ideal is given by the space of functions of the form: f(zi,z2)

= z"g(z1,z2)

+

z2h(z1,z2)

The operators Ba are simply the operations of multiplication of a function, representing an element of V by the coordinate function za, and the vector / is the image in V of the constant function / = 1. In the example above, following11 we identify V with C[z\]/z^, the operator B2 = 0, and in the basis e; = y/(i — l ) ! ^ ^ - 1 the operator B\ is represented by a Jordan-type block: Biei = A/2(I - l)ei_i, and / = V2NeN. Conversely, given a triple (B\,B2,I), such that the ADHM equations are obeyed the ideal I is reconstructed as follows. The polynomial / G C [2:1,22] belongs to the ideal, / G I if and only if f(Bi,B2)I = 0. Then, from the ADHM equations it follows that by acting on the vector / with polynomials in B\, B2 one generates the whole of V. Hence C[zi, z2]/X « V and has dimension N, as required. 5.2

Identificator ty and projector P

Let us now solve the equations for the identificator: "D^$ = 0, * * * decompose:

=

1. We

(97)

* (t) where I/J± £ V ® A$, £ £ Ae- The normalization (72) is now:

^++VU- +^

= 1

(98)

It is convenient to work with rescaled matrices B: Ba = \fC^pa, OL = 1,2. The equation V^ty = 0 is solved by the substitution:

+ = - \ / G ( 4 - C 2 K

V- = \fcM

- ci)*>

(99)

171

provided Av + I£ = 0,

A = Y, UP* - <£)(/£ - ca)

(100)

a

Fredholm's alternative states that the solution £ of (100) must have the property, that for any v e %, x £ ^ \ s u c n t n a t A ( ^ ® x ) = 0,

(101)

the equation (v*®X*)lt = 0 (102) holds. It is easy to describe the space of all v ® \ obeying (101): it is spanned by the vectors: c i = l,...,N (103) eE^ °|0,0)®ei, where e^ is any basis in V. Let us introduce a Hermitian operator G in V: G = (0,0|eD' 3 - c '"J/t e E' 3 i c i|0 ) 0>

(104)

It is positive definite, which follows from the representation: G = (0,0|eX> c <* £ & ( / £ ~ ca){Pa

~4)eE^4|0,0)

and the fact that (3a — cj, has no kernel in H ® V. Then define an element of the algebra Ag P = jteI>i4|o,0)G-1(0,0|e5>°c"J (105) which obeys P2 — P, i.e. it is a projector. Moreover, it is a projection onto an ./V-dimensional subspace in TL, isomorphic to V. 5.3

Dual gauge invariance

The normalization condition (72) is invariant under the action of the dual gauge group GN ~ U(N) on Ba,I. However, the projector P is invariant under the action of larger group - the complexification Cft « GLN(C): ( £ « , / ) - • O r 1 ^ , a " 1 J),

(Bi./t^tetB^t.-i^t.-i)

(106)

This makes the computations of P possible even when the solution to the fir — (r part of the ADHM equations is not known. The moduli space Mw,k can be described both in terms of the hyperkahler reduction as above, or in terms of the quotient of the space of stable points Yft k C M(T1(0) by the action of G% (see 2 0 , 6 8 for related discussions). The stable points (Bi,B2,I) are the ones where B\ and B2 commute, and generate all of V by acting on I: C[B\,B2\I = V, i.e. precisely those triples which correspond to the codimension N ideals in C[zi,Z2].

172

5.4

Instanton gauge field

Clearly, P annihilates £, thanks to (102). Let S be an operator in % which obeys the following relations: SS] = 1,

5f5 = 1 - P

(107)

The existence of 5 is merely a reflection of the fact that as Hilbert spaces T-Li as H. So it just amounts to re-labeling the orthonormal bases in fix and % to construct S. The operators S, 5 + were introduced in the noncommutative gauge theory context in 4 0 and played a prominent role in the constructions of various solutions of noncommutative gauge theories, e.g. in4°.1.31.42,41,32 Now, A restricted at the subspace S^H ® / C U ® V, is invertible. We can now solve (100) as follows: Z = A-hS\v = -jIt

(108)

where A = 1 + 1*4/ (109) A A is not an element of Ag, but A - 1 and AS* are. Finally, the gauge fields can be written as: Da = J^SA~?caAis\

Da = -J^-SAlc^A-iSl

(110)

If (^£2 = 0 then the formula (110) must be modified in an interesting way. We leave this as an exercise. Notice that if S^S was equal to 1 then the expressions (110) had the Yang form (66). 5.5

Ideal meaning of P

We can explain the meaning of P in an invariant fashion, following^9. Consider the ideal I in C[21,-22], corresponding to the triple (Bi,B2,I) as explained above. Any polynomial fel defines a vector f(VCic\,y/(^cl)\0,0} and their totality span a subspace %x C % of codimension N. The operator P is simply an orthogonal projection onto the complement to Hz- The fact X is an ideal in C[2i,z 2 ] implies that cl(Hz) C Hz, hence:

45+77 = s y for any 77 £ Ag, and also A~iS^

= 5**7"forsome 77',77" e Ag.

173

Notice that the expressions (110) are well-defined. For example, the Da component contains a dangerous piece Aic^ . . . in it. However, in view of the previous remarks it is multiplied by S* from the right and therefore well-defined indeed. 5.6

Charge one instanton

In this case: / = y/2, one can take Ba = 0, A = J2 C,ana, and in addition we shall assume that (1C2 7^ 0.

M = S « Ca«a, J2a (a = 2 - Let us introduce the notation N = nx + n2- For the pair n = (ni, n 2 ) let pn = %N(N -l) + n1. The map n -O- pn is one-to-one. Let Si\pn) = \Pn + 1). Clearly, SS* = 1, S f S = 1 - |0,0)(0,0|. The formulae (110) are explicitly non-singular. Let us demonstrate the anti-self-duality of the gauge field (110) in this case.

Y,DaD& = -S-(na ^DsDa

+

l)M

+ 2

= S-n C« " M + 2 - C a

St

M + u

M

A simple calculation shows:

Y,[Da, D&] = --?„

=- ( 1 +1 ) ,

C1C2

\Ci

[DQ1 £>„] = 0,

(111)

C2/

hence

X^«a=0 as

:

£9--1

U

(112)

+ 1 42

This is a generalization of the charge one instanton constructed in 7 4 , written in the explicitly non-singular gauge. The explicit expressions for higher charge instantons are harder to write, since the solution of the deformed ADHM equations is not known in full generality. However, in the case of the so-called "elongated" instantons 11 the solution can be written down rather explicitly, with the help of Charlier polynomials. In the U(2) case it is worth trying to apply the results of35 for the studies of the instantons of charges < 3.

174

5.1

Remark on gauges

The gauge which was chosen in the examples considered in 74 and subsequently adopted in 2 9 , 3 8 had £ = £*. It was shown in 2 9 that this gauge does not actually lead to the canonically normalized identificator \I/: one had ^ " f = 1 — P. This gauge is in some sense an analogue of 't Hooft singular gauge for commutative instantons: it leads to singular formulae, if the gauge field is considered to be well-defined globally over A$ (similar observations were made previously in 30 ). However, as we showed above, there are gauges in which the gauge field is globally well-defined, non-singular, and anti-self-dual. They simply have

175

6 6.1

Monopoles in noncommutative gauge theories Realizations of monopoles via D-branes

Another interesting BPS configuration of D-branes is that of a D-string that ends on a D3-brane. The endpoint of the D-string is a magnetic charge for the gauge field on the D3-brane. In the commutative case, in the absence of the £?-field, the D-string is a straight line, orthogonal to the D3-brane. It projects onto the D3-brane in the form of a singular source, located at the point where the D-string touches the D3-brane. From the point of view of the D3-brane this is a Dirac monopole, with energy density that diverges at the origin. The situation changes drastically when the B-field is turned on. One can trade a constant background £?-field with spatial components for a constant background magnetic field. The latter pulls the magnetic monopoles with the constant force. As a consequence, the D-string bends 4 5 , in such a way that its tension compensates the magnetic force. It projects to the D3-brane as a halfline with finite tension. It is quite fascinating to see that the shadow of this string is seen by the noncommutative gauge theory. The U(l) noncommutative gauge theory with adjoint Higgs field has a monopole solution 39 , which is everywhere non-singular, and whose energy density is peaked along a half-line, making up a semi-infinite string. The non-singularity of the solution is nonperturbative in 8 and cannot be seen by the expansion in 6 around the Dirac monopole 51 . This analytic solution extends to the case of U(2) noncommutative gauge theory. In this case one finds strings of finite extent, according to the brane picture 45 , which has the D-string suspended between two D3-branes separated by a finite distance. The solitons in U(l) theory were localized in the noncommutative directions, but generically occupied all of the commutative space, corresponding to (semi)infinite D(p-2)-branes, immersed in a Dp-brane, or piercing it. In the case of several Dp-branes we shall describe solitons which, although they have finite extent in the commutative directions, are nevertheless localized and look like codimension three objects when viewed from far away. The simplest such object is the monopole in the noncommutative U(2) gauge theory, i.e. the theory on a stack of two separated D3-branes in the Seiberg-Witten limit 8 2 . The fact that all the fields involved are non-singular, and that the solution is in fact a solution to the noncommutative version of the Bogomolny equations everywhere, shows that the string in the monopole solution is an intrinsic object of the gauge theory. As such, one could expect that the noncommutative gauge theory describes strings as well. In fact, the noncommutative gauge theory has solutions, describing infinite magnetic flux strings, whose fluctuations match

176

with those of D-strings, located anywhere in the ten dimensional space around theD3-branes 4 1 . 6.2

Monopole equations

If we look for the solutions to (62), that are invariant under translations in the 4'th direction then we will find the monopoles of the gauge theory with an adjoint scalar Higgs field, where the role of the Higgs field is played by the component A4 of the gauge field. The equations (62) in this case are called the Bogomolny equations, and they can be analyzed in the commutative case via Nahm's ansatz 6 4 . For the ^-independent field configurations the action (24) produces the energy functional for the coupled gauge-adjoint Higgs system:

£=

4

™ g-

f dx3 VdetGTr (-G™ G" Ft] * Fvr + 2 G y V , $ * V,-*)

(113)

5YM

where for the sake of generality we have again introduced a constant metric Gij. As before, the factor 2IT9 comes from relating the integral over xl,x2 to the trace over the Fock space which replaces the integration over the noncommutative part of the three dimensional space. The trace also includes the summation over the color indices, if they are present (for several D3-branes). In terms of the three dimensional gauge fields and the adjoint Higgs the Bogomolny equations have the form: Vi$ = ±Bi,

i = l,2,3.

(114)

where (49) in the case of generic metric G has the form: Bi =

±eijkG»'Gkk'VGFj.k.

As in the ordinary, commutative case, one can rewrite (113) as a sum of a total square and a total derivative:

¥- J dx3VGGljTr (V,$ ± B{) * (V,$ ± Bj)

£=

9YM

Tdj [VGGljTr (Bi * $ + $ * B{)

(115)

The total derivative term depends only on the boundary conditions. So, to minimize the energy given boundary conditions we should solve the Bogomolny equations (114).

177

6.3

Nahm's

construction

Ordinary monopoles To begin with, we review the techniques which facilitate the solution of the ordinary Bogomolny equations: Vi^

+ Bl=Q,

1 = 1,2,3

(116)

They are supplemented with the boundary condition that at the spatial infinity the norm of the Higgs field approaches a constant, corresponding to the Higgs vacuum. In the case of SU(2) this means that locally on the two-sphere at infinity:

#c)~diagQ,-D .

(117)

The solutions are classified by the magnetic charge N. By virtue of the equation (116) the monopole charge can be expressed as the winding number which counts how many times the two-sphere S2^ at infinity is mapped to the coset space SU(2)/U(1) RS S 2 of the abelian subgroups of the gauge group. We shall present a general discussion of the charge N monopoles in the gauge group U(k), where k will be either 1 or 2. The approach to the solution of (116) is via the modification of the ADHM construction. After all, (116) are also instanton equations, with different asymptotic conditions on the gauge fields. The appropriate modification of the ADHM construction was found by Nahm. Nahm 6 4 constructs solutions to the monopole equations as follows. Consider the matrix differential operator on the interval / with the coordinate z: -iA = dz+Ti(Ti,

(118)

Ti = Ti(z)+xi.

(119)

where the coordinates in the physical space R 3 , and the N x N matrices Ti{z) = T}{z) obey Nahm's equations: dzTi = ±eijk[Tj,Tk\,

(120)

with certain boundary conditions. The range of the coordinate z depends on the specifics of the problem we are to address. For the SU{2) monopoles with the asymptotics(117) we take / = (—a/2,a/2) where a is given in (117). For the C/(l) Dirac monopoles

178

we would take / = (—oo,0). At z approaches the boundary of / , z —> z0 we require that : Ti

l

h reg., [U,tj] = \Sijktk , (121) z — ZQ i.e. the residues ti must form a iV-dimensional representation of SU(2) (irreducible if the solution is to be non-singular). Then one looks for the fundamental solution to the equation: -*At*(z) = a , * - 7 ; ( 7 i * = 0 ,

(122)

where

and \P± are N x k matrices, which must be finite at dl and normalized so that: ' d z * t * = lfcxt . / '

(123)

Then: Ai

=

fdz&diV,

(124)

$ = / dzz&V . / Notice that for k = 2 the interval / could be (ai,a 2 ) instead of (—a/2, + a / 2 ) . The only formula that is not invariant under shifts of z is the expression (124) for (f>. By shifting $ by a scalar (ai + a 2 )/2 we can make it traceless and map / back to the form we used above. Abelian ordinary monopoles In the case k (-oo,0). The monopole has this harmonic form:

= 1 Nahm's equations describe Dirac monopoles. Take / = equation (116) becomes simply the condition that the abelian a magnetic potential $, which must be harmonic. Let us find function. The matrices Ti can be taken to have the following Ti(z) = ~,

[tittj] = i£ijktk ,

(125)

where ti form an irreducible spin j representation of SU(2). Let V w CN, N = 2j + 1 , be the space of this representation. The matrices * ± are now V-valued.

179

By an SU{2) rotation we can bring the three-vector Xi to the form (0,0,r), i.e. X\ = X2 = 0,£3 > 0. Then in this basis: $_ = 0 ,

. ^2r

*+ =

(2rz)jerz\j)

,

(126)

where \j) 6 V is the highest spin state in V. From this we get the familiar formula for the singular Higgs field TV corresponding to TV Dirac monopoles sitting on top of each other. Nonabelian ordinary monopoles We now take k = 2, TV = 1. Let H = Ck be the Chan-Paton space, the fundamental representation of the gauge group. Let eo,ei be the orthonormal basis in H. Again, for TV = 1 the analysis of the equation (120) is simple: Ti = 0. We can take a± = ± | and

# =

( £ ; s W - <*=*• - 2 =i>--

The condition that $ is finite at both ends of the interval allows for two solutions of (70) in the form of (128): v—e

,

which after imposing the normalization condition,(186), leads to

v /2sinh(ra)

L

V V^+^

J

V

v^^I

where x± = x\±\x
r)^'

This is famous 't Hooft-Polyakov monopole of the higgsed SU(2) gauge theory.

180

Nahm's equations from the D-string point of view The meaning of the Nahm's equations becomes clearer in the D-brane realization of gauge theory and the D-string construction of monopoles. The endpoint of a fundamental string touching a D3-brane looks like an electric charge for the J7(l) gauge field on the brane. By S-duality, a D-string touching a D3brane creates a magnetic monopole. If one starts with two parallel D3-branes, seperated by distance a between them, one is studying the U(2) gauge theory, Higgsed down to U(l) x U(l), where the vev of the Higgs field is

\0

a2J

One can suspend a D-string between these two D3-branes, or a collection of k parallel D-strings. These would correspond to a charge k magnetic monopole in the Higgsed C/(2) theory. The BPS configurations of these D-strings are described the corresponding self-duality equations in the 1+1 dimensional U(k) gauge theory on the worldsheet of these strings 19 , z being the spatial coordinate along the D-string. The equations (120) are exactly these BPS equations. The presence of the D3-branes is reflected in the boundary conditions (121). The matrices T, correspond to the "matrix" transverse coordinates X1, i = 1,2,3 to the D-strings, which lie within D3-branes. One can also consider the BPS configurations of semi-infinite D-strings, in which case the parameter z lives on a half-line. For example a collection of N D-strings ending on the D3-brane forms the so-called Blon16, described by the solution(125). Old point of view The old-fashioned point of view at the equations (122),(71) is that they are the equations obeyed by the kernel of the family of the Dirac operators in the background of the gauge/Higgs fields obeying self-duality condition 64 ' 17 . This interpretation also holds in the noncommutative case 74,73 . Noncommutative monopole equations Now let us study the solutions to the Bogomolny equations for a gauge theory on a noncommutative three dimensional space. As before, we assume the Poisson structure (6) which deforms the multiplication of the functions to be constant. Then there is essentially a unique choice of coordinate functions £1,2:2,£3 s u c n that the commutation relations between them are as follows (cf. (3)): [2:1,0:2] = - i # , 0 > 0

[2:1,2:3] = [2:2,2:3] = 0 .

181

This algebra defines noncommutative R 3 , which we still denote by A$. Introduce the creation and annihilation operators c, c*: cf = - = = (xi + ix2) , y 20

c = - = (xi - ix2), \/2a

(129)

that obey [c, c t] = 1. We wish to solve Bogomolny equations(116), which can also be written as: i

r„

„

n

[Di,*] = -e6ijk[Dj,Dk]-6i37i, 2

„ 1

(130)

where $ and Di, % = 1,2,3 are the a^-dependent operators in H <8> H. Now the relation between Di and Aj is as follows: D3=d3

+ A3,

Da = i0-1eaPxP

+ Aa,

a,/3 = 1 , 2

(131)

Noncommutative N a h m equations We proceed a /a 74 by repeating the procedure that worked in the ADHM instanton case, namely we relax the condition that x^s commute but insist on the equation (120) with Ti replaced by the relevant matrices Ti = Ti + Xi. Then the equation (120) on Ti is modified: dzTi = ±eijk[Tj,Tk]

+ 6i3e .

(132)

It is obvious that, given a solution Ti(z) of the original Nahm equations(120), it is easy to produce a solution of the noncommutative ones: Ti(z)m=Ti(z)+0z6a.

(133)

From this it follows that, unlike the case of instanton moduli space, the monopole moduli space does not change under noncommutative deformation. Notice the similarity of (132) and (130), which becomes even more striking if we take into account that the spectral meaning of the coordinate z as the eigenvalue of the operator $. This similarity is explained in the framework of noncommutative reciprocity3, generalizing the commutative reciprocity17'71

182 The deformation(132) is exactly what one gets by looking at the D-strings suspended between the DS-branes5'63 (or a semi-infinite D-string with one end on a D3-brane) in the presence of a £?-field. One gets the deformation: [X\Xi)

-+ [X\X']

- i0« = [TuTj] - ±6elj3

(134)

The reason why 0,J', instead of Bij, appears on the right hand side of (134) is rather simple. By applying T-duality in the directions x\,X2,x3 we could map the D-string into the D4-brane. The matrices Xl,X2,Xz become the components Aj, A^, A^ of the gauge field on the D4-brane worldvolume, and the B-field would couple to these gauge fields via the standard coupling F-{-• — B-{~-, where B~-~- is the T-dualized 2?-field. It remains to observe that B--~- = 01-7', since l

J

'•J

the T-dualized indices i label the coordinates on the space, dual to that of Xj's. 6.4

Solving Nahm's equations for noncommutative

monopoles

To solve (132) we imitate the approach for the charge N = 1 ordinary monopole by taking Ti, 2 = 0, T3 = 6z. (135) To solve (70) for $> we introduce the operators b,tf: b=-^(dz+x3 \J29

+ 9z),

6f = -$= {-dz + x3 + Oz) , y29

(136)

which obey the oscillator commutation relations: [6,6+] = [c, C t] = 1 .

(137)

W = x3z + hz2

(138)

Introduce the superpotential

so that b = 7^e~wdzew, comes:

,

6+ = - J ^ e ™ ' d z e ~ w ' . 6t*++c*_ =0 c t $ + - &^r_ = o

Then equation (70) be(139)

It is convenient to solve first the equation tfe+ + ce_ = 0 , ch+ - 6e_ = 0 ,

(140)

183 with e±(z,x3) £ ft. The number of solutions to (140) depends on what the interval I is. If I = (a-,a+) is finite then (140) has the following solutions: ea = ( j ) ^8 = ( H U e - w | 0 ) )

, a = 0,l Co =

'

^-+ ^ e -

w

,ej = 0

(141)

where eo, ei will be the basis vectors in the two dimensional Chan-Paton space. The functions /3°,/?* solve (&tb + n ) / £ = 0,

(142)

and are required to obey the following boundary conditions: 6/3>+) = 0, /3M(o-) = - 1 ,

#(a_) - 0 «£(a+) = 1

(143)

which together with (142) imply that: r

dz{e^e1 = 6^Smn,

(144)

A solution to (139) is given by: *=

E

e£'
(145)

n>0, a = 0 , l

and by virtue of (144) it obeys (186). All other solutions to (139), which are normalizable on I = ( a _ , a + ) are gauge equivalent to (145). If I = (—oo,0) then the number of solutions to (140) is roughly halved.

eo = f _ ^ e ° W|0> ) ,

Co = / " „ ^ e - 2 H / ,

'&0„|n-l> Vn/3„|n> J'

n >

(146)

°'

The functions /3„ solve (6f 6 + n) /?„ = 0,

(147)

184

and are required to obey the following boundary conditions: PnbPrriO) = 1,

/?„(*) -»• 0, Z -> - 0 0

(148)

which together with (147) imply that: /•O

dz (e„) t e T n = Smn,

/

(149)

7 —oo

A solution to (139) is given by:

$ = Y, e " • ("I-

(150)

n>0,

and by virtue of (149) it obeys (186). All other solutions to (139) which are normalizable on I = (—oo,0) are gauge equivalent to (145). Generating solutions of the auxiliary problem: 17(1) case To get the solutions to (147) solve first the equation for n = 1 and then act on it by 6"" 1 to generate the solution for higher n's. The result is:

V(n(x3)(n~l{X3)

where (we set 29 = 1): P2

f

C„(z)= / Jo

n 2

p e "^^dp

(152)

The functions ( n obey39: C n + l ( z ) = 2^Cn(^) + nCn-l(z)

,

8z(n = 2C„+i , Cn(0) = 2 ^ ( S f i ) !

(153)

We have explicitly constructed /?„ and thus * ± , from which we can determine, using (124), the Higgs and gauge fields. To this end: •Show that:

f° /

/3„6/9„=C"-lCn + l - C ' = " C n - l - ( " - l ) C n C n - 2 J — oo

, n > 0

(154)

185

Introduce the functions £, £ and 77 - = 12 £

i(n) = J^-,

»?(«) = - £ - ,

V Cn+1

«n) = J ^ .

U+l

V

(155)

^n

We will need the asymptotics of these functions for large £3. Let r1n = x\+ n. For r n + £3 -> 00 we can estimate the integral in (152) by the saddle point method. The saddle point and the approximate values of (n and Tjn are: p Cn

=

x3+rn 71

n

J — •( x 3 + r n ") n+k+ 5 e 5 r

~

1

'i + i

x3 +

+

A(*3+Tn)(3x3-rn)

.. (156)

We shall also need: C

°(Z)

lid"1,

(15?)

z^-00

Generating solutions of the auxiliary problem: U(2) case It is easy to generate the solutions to (142): first of all, fn(z)

= bn~\ew)

= ew^hn.1(2x3

+ z),

hk(u) = e ~ ^ " | ^

is a solution . Then

fn~U(z)j

"

is the second solution. Notice that, for k even, hk{u) > 0 for all u and, for k odd, the only zero of /ik(u) is at u = 0, and hk{u)/u > 0 for all M. Therefore, hk(z) is well-defined for all z. Consequently,

Pn — "n

y

nfn

+ 1(z)

+ Jn\Zl

Jz

f„ + 1(u)2

J

'

186

where

"-' = 2

K =

+ (^a-)^(a-)^7rf^i-^)/:7^k' + +

(/n(«+)/n+i(a+)r_ 7^+i)r_ 7ffe-

(159)

(again, note that 0%(z) are regular at z = —20:3). 6.5

Explicit U(l) solution

Now we present explicit formulae for the C/(l) monopole solution and study its properties. Higgs/gauge fields The Higgs field is given by: 00

*=£#„(a;3)|n>(n|,

(160)

71=0

it has axial symmetry, that is commutes with the number operator c*c. Explicitly: $n

=

-z

-r- = d3\og£n

Cn—1

S>n

=

(n-l)T)n-2-nr]n-i,

=

_^. = _ 2 i 3 - i , Co

n>0

n = 0.

Co

(161)

To arrive at the third line we used the fact that 1

1

Vn

Vn+l

= nr? n _i - (n + 1)T?„ ,

which follows immediately from the recursion relation for the C's in (153). These fields are finite at xz = 0. Indeed as x3 —> 0,

187

At the origin: $o(z 3 = 0 )

=

-y \ •

(163)

In the gauge where $ is diagonal the component A3 vanishes. In the same gauge the components Ai, A2 (which we consider to be anti-hermitian) are given by: Ac = \ (Ax + iA2), Ac

=

Aci = i (ili - iA2) = ~A\

r1K,Ct] = c t ( l - | g ^ T )

(164)

Again we see that the matrix elements of Ac are all finite and non singular. From (164) we deduce: Fl2 = 2 (dcAci - dc,Ac + [Ac,Aei]) = z

U « ( " + i ) c ' c «(n+i)J

= 2 £ „ > o ( - 1 + (n + 1) ( ^ )

2

L

) -

- n ( ^ )

2

)

|n)(n| +

+2 ( - 1 + ( f g } ) 2 ) I0)(0| ,

(165)

from which it follows, that: B3(n)

=

Bc

=

2

l(B1+

( l - n ^ + ( n - l ) ^ ) iB2) = c t ^ g y (*(„) - *(n + 1)) .

(166)

with the understanding that at n = 0: B 3 (0) = 2 ( l - ^

.

(167)

Instantons, monopoles, and Yang ansatz As in the ordinary gauge theory case the monopoles are the solutions of the instanton equations in four dimensions, that are invariant under translations in the fourth direction x4. We observe that the solution presented above (161),(164), can also be cast in the Yang form: Take £ = £(0:3,71) as in (164). Then 83^, commutes with £ and we can write c ^ f - 1 = c?3log£. The formulae (66) yield exactly (164) and (161) with $ = iAA.

188 All of the above and Toda lattice At this point it is worth mentioning the relation of the noncommutative Bogomolny equations with the Polyakov's non-abelian Toda system on the semiinfinite one-dimensional lattice. Let us try to solve the equations (116) using the Yang ansatz and imposing the axial symmetry: we assume that £(xi ,X2,x3) = £(n, x3), n = c^c. Then the equation (67)forthe a^-independent fields reduces to the system: dtidtgng'1) ~ 9n9n\x + dn-ign1 = 0

(168)

where

*•<«-£<•("• 5) • (notice that gn(t) are ordinary matrices). In the U(l) case we can write gn(t) = e a - ( t ) , and rewrite (168) in a more familiar form: dfan + e " - 1 - " " - e " " - " ^ 1 = 0

(169)

For n = 0 these equations also formally hold if we set g_i = 0 (this boundary condition follows both from the Bogomolny equations and the same condition is imposed on the Toda variables on the lattice with the end-points). Our Higgs field $ „ has a simple relation to the a's: ${x3,n)

= -2x3 + a'n(2x3) .

Our solution to (169) is:

It is amusing that Polyakov's motivation for studying the system (168) was the structure of loop equations for lattice gauge theory. Here we encountered these equations in the study of the continuous, but noncommutative, gauge theory, thus giving more evidence for their similarity. We should note in passing that in the integrable non-abelian Toda system one usually has two 'times' t,t, so that the equation (168) has actually the form 25 : dt{dtgng^) - gngn+i + gn-ig'1 = 0 . (171) It is obvious that these equations describe four-dimensional axial symmetric instantons on the noncommutative space with the coordinates t, i, c, c* of which only half is noncommuting.

189 The mass of the monopole In this section we restore our original units, so that 29 has dimensions of (length) 2 . From the formulae (156) we can derive the following estimates: *(n) ~

1 ~2^~

1 2y/xl + 26n

n^O,

(172)

x3 ->• +oo X3 -+ —oo .

(173)

, for n = 0 we have: *(0)~-^, $(0) ~ 2 | a1= | ' 3

The asymptotics of the magnetic field is clear from the Bogomolny equations and the behaviour of
=-?\,

n/0,

(174)

and similarly for the other components of B. This is easily translated into ordinary position space, since, for large n, Bi(n,x3) ~ Bi{x\ + x\ ~ 71,0:3). Therefore the magnetic field for large values of X3 and n, or equivalently large Xi is that of a pointlike magnetic charge at the origin. However the n = 0 component of B3 behaves differently for large positive x3:

B3(n = 0) = -d 3 $(0) = J .

(175)

Notice, that this is exactly the value of the B-field. Thus, in addition to the magnetic charge at the origin we have a flux tube, localized in a Gaussian packet in the (xi,x2) plane, of the size tx 9, along the positive x 3 axis. The monopole solution is indeed a smeared version of the Dirac monopole, wherein the Dirac string (the D-string!) is physical. To calculate the energy of the monopole we use the Bogomolny equations to reduce the total energy to a boundary term: £ = 2 p - / d3x (B-kB + V^-k V $ ) = igi

J
£ > l a 3 * 2 + 4dc (e (dc^2) r 2 ) In) ,

(176)

where in the last line we switched back to the Fock space. Here we meet the

190 noncommutative boundary term, discussed in the section 2. Let us choose as the infrared regulator box the "region" where \x3\ < L,0 < n < N, L ~ V26N > 1. With the help of (10) the total integral in (176) reduces to the sum of two terms (up to the factor -^-): SYM

+ E^o^l\xxlztLL

•

(177)

The first line in (177) is easy to evaluate and it vanishes in L —>• oo limit. The second line in (177) contains derivatives of the Higgs field evaluated at 13 = L > 0 and at £3 = — L -C 0. The former is estimated using the z > 0 asymptotics in(156), and produces:

f:*«(x, = i )~2S^« + 2 L 1=0

The diverging with L piece comes solely from the n = 0 term. Finally, the X3 — —L case is treated via z C O asymptotics in (157) yielding the estimate ~ 9N/L3 vanishing in the limit of large L, N. Hence the total energy is given by 2?r(9 x 2L _ 2TTL £

«

20.2

a2

9YMQ2

(178)

= -^r-« . 9YMQ

which is the mass of a string of length L whose tension is

T

(179)

=ih •

Magnetic charge It is instructive to see what is the magnetic charge of our solution. On the one hand, it is clearly zero:

Qoc J

B-dS=

J dAxV

B= 0

(180)

since the gauge field is everywhere non-singular. On the other hand, we were performing a ^-deformation of the Dirac monopole, which definitely had

191

magnetic charge. To see what has happened let us look at (180) more carefully. We again introduce the box and evaluate the boundary integral (180) as in (9):

9. =

J2 [ B 3(z3 =L,n)-

2TT

n=0

B3(x3 = -L,n)] + 47V / ~

dx3^—^-{^N

- $N+l)

VN

(181) It is easy to compute the sums

E:^N=0B3(x,n)=d3^ 4^/fL^33^i(^-*N+i)=4(JV + l ) / ^ d l o g ? ^ 7 = = 2(7V + l ) ( ^ ) 2 | - = + L =

3 =~+tiL 2NC^±1]ll -L '

(182)

and the total charge vanishes as: Q=

x3=+L

2NCN-l(N+l+d3((N+l

= 2(N+l)\X3=+L-2(N

eN

(N

x3=—L

+ l) \X3 =

-L

(183)

We can better understand the distribution of the magnetic field by looking separately at the fluxes through the "lids" x3 = ±L of our box and through the "walls" n = N. The walls contribute 27V

Gv-iGv+i

r

2

X3 = + L

VUTN

while the lids contribute ~ + 1 . Let us isolate the term B3(+L,n = 0) —> +2 (recall (173)). It contributes to the flux through the upper lid. The rest of the flux through the lids is therefore ~ — 1. Hence the flux through the rest of the "sphere at infinity" is —2 and it is roughly uniformly distributed (—1 contribute the walls and — 1 the lids). So we get a picture of a spherical magnetic field of a monopole together with a flux tube pointing in one direction. This spherical flux becomes observable in the naive 9 —> 0 limit, in which the string becomes localized at the point x3 = 0, n = 0 (since the slope of the linearly growing $ 0 ~ ^ becomes infinite). In the 9 = 0 limit we throw out this point and all of the string. Thus we found an explicit analytic expression for a soliton in the U(l) gauge theory on a noncommutative space. The solution describes a magnetic monopole attached to a finite tension string, that runs off to infinity tr an verse

192

to the noncommutative plane. This soliton has a clear reflection in type IIB string theory. If the gauge theory is realized as an a' —> 0 limit of the theory on a D3-brane in the IIB string theory in the presence of a background NS Bfield, then the monopole with the string attached is nothing but the Dl-string ending on the D3-brane. What is unusual about the solution that we found is that it describes this string as a non-singular field configuration. Moreover, one can show that the long wavelength fluctuations of this string are described by the gauge theory 4 0 ' 4 1 . 6.6

Noncommutative

U(2) monopole.

Now we shall describe the noncommutative version of the 't Hooft-Polyakov monopole49. We are interested in the U{2) gauge theory on the noncommutative three dimensional space. Recall that H ra C 2 is the Chan-Paton space, i.e. the fundamental representation for the commutative limit of the gauge group, and let eo,ei denote an orthonormal basis in H. The noncommutative version of the fundamental representation is infinite dimensional, isomorphic to H ® %. That is, the U(2) matter fields $ belong to the space H ® Fun(x3) ® {% ® H), where the first two factors make it a representation of the algebra Ag of noncommutative functions on R 3 , while the second two factors make it a representation of the U(2) noncommutative gauge group. Actually, the latter is isomorphic to the group of (^-dependent) unitary operators in the Hilbert space %®H. Now, the Hilbert space H<8>H is isomorphic to U itself: \n) ig> ea -B- |2n + a) .

(184)

Now the solutions can (and will) have a finite non-trivial magnetic BPS charge: Qm = J dx3Trndi

(TrH$Bi)

where B

i = y£ijk[Dj,Dk]

and the Higgs field $ approaches

as x\ + 2#c+c -> oo.

- Si30 ,

,

(185)

193 In (145) we already found a two-component spinor vector-function

*(*,*) = ( j + ) , which obeys the equation (70). The solution to (70) is defined up to right multiplication by an element of Mat2(^4e) ~ A$ <8> End(If): \t i-¥ $>u. Among these elements the unitary elements (i.e. the ones which solve the equation uv) = v)u = 1) are considered to be the gauge transformations. In the commutative setup one normalizes \P as follows: /

dz*t* = i 2 = (I

°A .

(186)

The normalization (186) only implies that the transformations u must obey: v)u = 1. The finite matrices u would then automatically obey uu* = 1. However, in the infinite-dimensional case this is not true. The operator uu^ is merely a projection, which may have a kernel. The discussion on whether such projections should be viewed as gauge transformations in noncommutative gauge theory can be found in 3 0 ' 3 1 . Higgs/gauge field Finally, given $ the solution for the gauge and Higgs fields is given explicitly by (124) where now one integrates over z from a_ to a+. We now are in position to calculate the components of the Higgs field and of the gauge field. We start with $ = / dzz&ty = = J dzze^el = -2x36a~' + J(bft){b + &t)(6#j[) + nft(b + tf)pn = -2x36°i + ((&/8«)(6flD - n/?«/#) \aa+_ . (187) The component A3 of the gauge field vanishes, just as in the case of the U(l) solution of39:

where

tf?

A3 = j &d3V

= I {{b(5Z)d3(bpl) + n / 3 ° % ^ ) - e a e ; ® | n - a > < n - 7 | = i<93 / V * = 0 .

(188)

The components A\, A2 can be read off the expression for the operator D: = E„>o,a, 7 =o,i Dny • eae\ ® |n + 1 - a){n - 7 | , where £ » r = - V n ( ) 3 S + 1 (6)82)) |S+ •

(189)

194

The solution (187),(189) has several interesting length scales involved (recall that our units above are such that 28 = 1): 6\a+ — a_|,

v6,

\a+ — a-\ .

By shifting x% we can always assume that a_ = 0, a+ = a > 0. Suspended D-string In this section we set 6 back to i. As discussed in41 the spectrum of the operators D^, A = 1,2,3,4 determines the "shape" of the collection of Dbranes the solution of the generalized IKKT modef 5 corresponds to. To "see" the spatial structure of our solution let us concentrate on the (0|$|0) piece of the Higgs field, for it describes the profile of the D-branes at the core of the soliton. From (187) we see that

where p+ = (fQ°,p_ = ipl1. Let us look specifically at the component p+ of the Higgs field: P+

=-i^log(/0-4,e-to»-^)

,

2 3

= -2*3 + <(p»2°+ * , =

(q + 2 * 3 ) 2 4

(190)

(2*3)2 4

- 2 x , - 2-e —~e — -Y(a+2x3)—Y(2x3)

where ({0))i

f Oe^ dp = J" , P p2 J^e~ -rdp

fi l{z) = / dpe-^ Jo

.

(191)

The ((•-•)) representation of the answer helps to analyze the qualitative behavior of the profile of p+. Clearly, the truncated Gaussian distribution which enters the expectation values ((...)) in (190) favors ptz0ifa<0<(3, p&a for a > 0 and p « /3 for /? < 0. Thus, p+ ~ 0, p+ 2a;3, p+ ~ a,

x3 > 0 0 > x3> -\a - i a > 2:3 •

(192)

195

This behavior agrees with the expectations about the tilted Dl-string suspended between two D3-branes separated by a distance \a\. The eigenvalue p+ corresponds roughly to the the transverse coordinate of the Dl string, that runs from a at large negative 0:3 to 0 at large positive x%. In between the linear behavior of the Higgs field corresponds to the Dl string tilted at the critical angle. Indeed, for large a 3> 1, in the region 0 > 2:3 > — Ja this solution looks very similar to that of a single fiuxon 40 . Another eigenvalue of (0|$|0), p_-, is given by: P where

M

_ 2x 3 (2x3+a)M+M 2 -(2a:3+a) 2 -e~ 2 x 3 °~'2'

M(2x3 + a-2x3M) e-2x3a-4

+

(193)

(2x3 + a) J0a e - 2 *3P-4 dp

At this point, however, we should warn the reader that only the eigenvalues of the full, 2oo x 2oo operator $ should be identified with the D-brane profile. The components p± do not actually coincide with any of them. The eigenvalues of $, as it follows from the representation (161), are located between 0 and a, which is also what we expect from the dual D-brane picture 4 5 . 6.7

Tension of the monopole string versus that of D-string

In this section we shall match the tension of the string we observed in the U(l) monopole solution to that of D-string ending on the D3-brane in the presence of the constant B-field. As already mentioned, a D-string ending on a D3-brane in the presence of the constant S-field bends. To analyze this bending one could use the exact solution of the Dirac-Born-Infeld theory 6 3 , the B-deformed spike solutions of12. However, for our qualitative analysis, it is sufficient to look at the linearized equations. If we replace the DBI Lagrangian by its Maxwell approximation, then the BPS equations in the presence of the .B-field will have the form: Bij + Fij + y/fetg£ijk

gkldi§ = 0 ,

(194)

where we should use the closed string metric (46). The solution of (194) is:

(195) The linearly growing piece in $ should be interpreted as a global rotation of the D3-brane, by an angle ip, tanip = ,2%a,*. This conclusion remains correct even after the full non-linear BPS equation is solved (see 6 3 . Notice however

196 that we fix Gij = Stj instead of gtj = Sij as in 6 3 ). The singular part of $, the spike, represents the D-string. If we rotate the brane, then the spike forms an angle \ - ip with the brane. If we project this spike on the brane, then the energy, carried by its shadow per unit length, is related to the tension of the D-string via: TDi_ = 1 y/(2*a')*+e* simp 2-Ka'gs 6

=

( 2 W ) 2 + 02 2ng2rM(a>)20 '

^

'

However, this is not the full story. The endpoint of the D-string is a magnetic charge, which experiences a constant force, induced by the background magnetic field. If we had introduced a box — L < X3 < x 3 of the extent 2L in the rr3-direction, then in order to bring a tilted D-string into our system from outside X3 > L of the box to 13 = 0 we would have had to spend an energy equal to ^QL, but we would have been helped by the magnetic force, which would decrease the work done by45 2TTL

9YM

_ 2TTL

„

22

1

(2Tra'gYM)2

9YM

In sum, the energy of the semi-infinite D-string in the box per unit length in the x3 direction will be given by 2?r

(197)

9YM&'

This expression coincides with our tension (179). On dimensional grounds, non-commutative gauge theory cannot produce any other dependence of the tension on 9 but that given in (179). 7

Conclusions a n d historic r e m a r k s

To conclude these lectures I would start with a very short survey of the topics not included into them. The abovementioned factorization of the OPE algebra of the open string vertex operators was argued to be useful in analyzing various instability issues, condensation of tachyons, responsible for the decays of the unstable D-branes 83 ' 7 . Moreover, noncommutative gauge theories allow to see both (many of) the non-BPS D-branes as classical solutions and the processes of their decays 1 ' 42 > 41 . My interest in noncommutative theories was prompted by the work with V. Fock, A. Rosly and K. Selivanov in 1991 on the geometric quantization

197

of higher-dimensional dimensional Chern-Simons theories28, where we encountered moduli spaces of codimension two four dimensional foliations with flat U(l) connections on the fibers as the classical phase space of the five dimensional theory (in modern terms these would be supersymmetric cycles, if they were holomorphic). Previously thee theories were encountered in the context of gauge anomalies in26. An example of such foliation would be an irrational foliation of a four-torus, the space suited for the analysis by noncommutative geometry. In 1994 with G.Moore and S. Shatashvili I tried to understand Sduality of the M = 4 gauge theory on ALE manifolds following92 and then together with A. Losev we came from that to the attempt of constructing the four dimensional analogue of the two dimensional RCFT 59 . In the course of this study we realized that the construction of37 of the instanton moduli spaces on ALE manifolds, whose Euler characteristics used in92 were generated by modular forms actually gave nontrivial answers already in the U(l) case, which is almost impossible to achieve by the ordinary gauge fields. With the help of I. Grojnowski we realized that Nakajima constructed not the instantons but the torsion free sheaves, who had no gauge theory interpretation (but were used by algebraic geometers to construct compactifications of instanton moduli spaces, e.g. Gieseker compactification). For some time these moduli spaces were a mystery. This mystery was getting deeper after discovery of D-branes by J. Polchinski76 and the realization by E. Witten and M. Douglas95 that the D(p-4)-branes within Dp-brane are instantons. Now, the D4-brane carries only a U(l) gauge field on it, so what is the DO-brane which is dissolved inside? This question is hard to ask if the gauge theory is not decoupled from the rest of the ten dimensional string theory, but it was soon realized that a constant .B-field may help (in the similar M-theory context turning on C-field helped to decouple fivebrane from the eleven dimensional sugra modes3). Things came together after we had a very fruitful lunch with A. Schwarz in ITP, Santa Barbara in the spring of 1998, where he explained to me his paper with A. Connes and M. Douglas, and I explained to him what we have learned about torsion free sheaves with other authors of9. After brief discussion we became confident that the gauge fields on the noncommutative R 4 must be i) constructed with the help of ADHM construction applied to the deformed ADHM equations 65 , which was shown by Nakajima to parameterize torsion free sheaves58 on C 2 ; ii) obey instanton equations on the noncommutative space. It was a matter of simple algebraic manipulations to check that this was the case74. But then the devil of doubts started to hunt me. Several people asked us whether the U(l) instantons were non-singular. The relation to large N gauge theory suggested they didn't exist, for the instanton effects usually die out in the large AT limit. Explicit computations 11 ' 29 seemed to imply that the C/(l) instantons had to

198

live over the space of complicated topology. Also, with D. Gross we were finding solutions to the BPS equations in three dimensions which seemed to follow from ADHM ansatz and yet did not solve Bogomolny equation everywhere in the space. Later on we realized that these solutions were obtained from the true solution by applying the "generalized" gauge transformation u, discussed below the equation(186), and which actually could produce a source. But in the spring of 2000 I was not yet aware of that and during the talk at CIT-USC center I was forced by E. Witten to announce that noncommutative instantons existed not on noncommutative R 4 but on some space, obtained from R 4 by a sequence of blowups, whose commutative limit was described in 1 1 . This point of view was immediately criticized by E. Witten himself, for it was not consistent with many results on the counting of states of D-branes. Then a week later D. Gross and I found true sourceless solution to the monopole equations 39 . And later on I realized that the explicit formulae for the f/(l) instanton gauge field presented in 74 were harmed by the same plague: they were written in the singular gauge. Almost at the same time this conclusion was reached by K. Furuuch? 1 - 72 . Independently of all this story, it is natural to look for the mechanism of the confinement in the noncommutative gauge theory, hoping to transport these results to the large N ordinary gauge theory. We find41 all sorts of magnetically charged objects in the gauge theory, whose mass/tension goes to zero in the limit 8 —> oo. Whether this could provide the sought for mechanism for the confinement via condensation of magnetic charges still remains to be seen. As far as other extensions of the work reported above are concerned let us mention a few. First of all, it is quite interesting to construct noncommutative instantons on other spaces, for example ALE manifolds74'58, tori or K3 manifolds33, or Del Pezzo surfaces or cotangent bundles to curves (see 69 ). In the latter three cases the very notion of the underlying noncommutative manifold is missing (for orbifold K3's one can make orbifolds of noncommutative tori, though 53 ). One could also try to construct supersymmetric gauge field configurations in higher dimensions (see 70 for some relevant algebraic geometric results). Also, SO(n) and Sp(n) theories 10 are quite curious to look at. The construction of the ordinary instantons 95 in these theories with D-brane techniques shows many interesting surprises22. It is also quite interesting to generalize the noncommutative twistor approach of52 to more general spaces. Another subject is the usage of the gauge theory/string duality in the form of supergravity duals of noncommutative gauge theories46. One can construct many sugra duals of the solitons in noncommutative theories and study their strong coupling behaviour79. Yet another interesting topic is the construction

199

of electrically charged solitons. They are important in the realizing of closed strings within open string field theories in the lines of87. Acknowledgements I thank all the people mentioned above and also H. Braden, S. Cherkis, P.-M. Ho, I. Klebanov, S. Majid, J. Maldacena, A. Polyakov, D. Polyakov, M. M. Sheikh-Jabbari, for various stimulating discussions and questions. I am especially thankful to my wife Tatiana Piatina for her understanding and support. The research was partially supported by NSF under the grant PHY9407194, by Robert H. Dicke fellowship from Princeton University, partly by RFFI under grant 00-02-16530, partly by the grant 00-15-96557 for scientific schools. I thank the organizers of the Trieste spring school on D-branes and strings for the kind invitation to present the results reported here. I thank the ITP at University of California, Santa Barbara, CIT-USC center for Theoretical Physics, String Theory Group at Duke University, Universities of Barcelona and Madrid, and Isaac Newton Institute for Mathematical Sciences (Cambridge, UK) for hospitality while these lectures were written up. References 1. M. Aganagic, R. Gopakurnar, S. Minwalla, A. Strominger, "Unstable Solitons in Noncommutative Gauge Theory", hep-th/0009142 2. O. Aharony, M. Berkooz, N. Seiberg, hep-th/9712117, Adv. Theor. Math. Phys.2 (1998), 119-153 3. O. Aharony, M. Berkooz, S. Kachru, N. Seiberg, E. Silverstein, hepth/9707079, Adv. Theor. Math. Phys.l (1998), 148-157 4. M. Alishahiha, Y. Oz, M.M. Sheikh-Jabbari, "Supergravity and Large N Noncommutative Field Theories", hep-th/9909215, JHEP 9911(1999), 007 5. M. Atiyah, V. Drinfeld, N. Hitchin, Y. Manin, "Construction of instantons", Phys. Lett. A65 (1978) 185-187 6. D. Bak, Phys. Lett. 471B (1999), 149-154, hep-th/9910135 7. I. Bars, H. Kajiura, Y. Matsuo, T. Takayanagi, "Tachyon Condensation on Noncommutative Torus", hep-th/0010101 M. Li, "Note on Noncommutative Tachyon in Matrix Models", hepth/0010058 Y. Hikida, M. Nozaki, T. Takayanagi, "Tachyon Condensation on Fuzzy Sphere and Noncommutative Solitons", hep-th/0008023

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