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£/9Jt a map are given satisfying the following conditions for all a, b, 7 G T: (ri)
tpw)
=
) *} =
are not 0 for all k e Z>oSimilar argument using both bosonic and fermionic fields gives the "fundamental representations" of affine superalgebras, which is studied in [125]. 6.2
Boson-Fermion Correspondence
It is sometimes convenient to consider charged free fermionic fields written in the form ip(z) = V ] i/ijZ _J "2
and
tp*(z) =
V") ^ • z _ J ' _ 5 ,
with anti-commutation relations
for all j,ke\+Z. Let F denote the Fock space with the vacuum vector |0), and with the irreducible action of tpj's and V'j"si where ipj, ipj (j > 0) are annihilation, and ij)j, x/)j (j < 0) are creation operators. Namely F is a vector space with a basis
<-"^^V^-l°>.
( 6 - 138 )
6.2. Boson-Fermion
Correspondence
231
where m,n > 0 and 0 < i\ < • • • < in and 0 < j \ < • • • < j m . The charge and the energy of this element (6.138) is defined as in Example 6.1.6. Notice that formulas for operator products are quite the same with those in Example 6.1.6, since the operator products of tp and %p* are the same. Letting a = \ in (6.134), we consider the Virasoro field L(z)
=
^
(
#
W
:
+ : ^ ( # ) : ) .
Then, by (6.135a), (6.136a) and (6.136b), one has the following: 2\)L+±\3,
[LXL] = (d + [Lx1>] = (d + £\ rf>
and
[Lxp]
=
(d+£\p,
and so, by the skew-symmetry [1>\L] = [PxL]
-[L-a-x1>] = -(d+±(-d-\)\^
=
\{-d
+
= i ( - f l + A)^,
\)r-
We now consider the field H(z)
:= :^(z)r(z):
#»*~B_1 •
= £
The commutation relations of this field H with fermionic fields ip and ip* are as follows: [Hxj>] =
[: VV>* -x V>] = : i> [ $ M • = V>,
(6.139a)
l
[Hxp]
=
[: W* :x V] = - • [W]i>*
•= ~P ,
l
so, by the skew-symmetry, WxH] = ->,
WxH\
= rP*.
Then one has [HxH]
=
[Hx:W:] [H^r-
+ ^[Hxri-+J -V>*
[[Hxj>]prl>']dn
(6.139b)
232
6. Operator Calculus /•A
namely [H(z)H(w)}
(z — w)2 '
This formula, in view of (6.94) in Example 6.1.4, implies that H(z) is a free bosonic field, and that operators Hn satisfy the commutation relations [Hm, Hn]
—
mSm+nfi.
Let L'(z)
:=
\:H{z)H{z):
be the Virasoro field associated to this free bosonic field H{z). Then, by (6.99b) and (6.104a), one has [L'XL'} = {d + 2\)L' +
~Xi
and [L'XH] = (d + X)H. The commutation relations of L(z) with H{z) and L'(z) are calculated as follows: [LXH]
-
[Lx-.i/nf*-]
=
: \Lxfl r
•• + •• ip [LlT]
(a+^)v =
d : n*
-+J
(a+£)V>* •• +A : VV'* : + / JO
s
[[LxW (a+^W-
[ W W ] dp + $ f v
'
-M[IM*]
(d + \): ipip* :- f ndn + 2
(d +
\)H,
and [LXL'}
=
±[LX:HH:}
pPW
l
JO
s
[^V*] dfi v 1
'
6.2. Boson-Fermion
233
Correspondence
{d+X)H
(d+\)H
(d+X)H (-/x+A)[i/„i/]
=
l(d + 2X):HH:+l *
=
(0 + 2A)L' + ^ A 3 .
z
f (X- p) [H^H\ dp v Jo —v—'
Then by the skew-symmetry of A-bracket, one has [L'XL] = =
-[L-8-xL'\
2(-d-\)}L'+{~duX)3
= -{d +
(a + 2 A ) L ' - ^ A 3 .
From these relations, one sees that the "coset Virasoro field"
L =
L-L'
satisfies [LXH] = 0
(6.140)
[LXL] = (d + 2X)L.
(6.141)
and
The formula (6.140) implies [Ln, Hk] = 0
foralln,fceZ,
and (6.141) implies that the central charge of L(z) is 0. Since the coset action L(z) of Virasoro algebra on the Fock space F is unitarizable, this means that the action of Ln is trivial, i.e., Ln
= 0
on F
and so Ln
= L'n
on F
for all n G Z, since the Virasoro action Ln is unitarizable on the space F and its central charge is equal to 0. By the definition of Hn and Ln, one notices the following:
234
6. Operator Calculus
L e m m a 6.2.1.
1) 0) (ii)
Ln = \ E ( 2 i - n ) : ^ n _ ^ ; : > L0 = E j : iP-jifi* : = J^ j(^-i^J? + ^-i^i) » jei+z
2)
(i)
Hn =
E
i>o ^n-i^i
/or a/Z n e Z - {0},
(ii) ff0 = E •*-&•= E ^ - ^ - E ^ - ^ Proof. 1) Differentiating
v>(z) = E ^i z _ j ' - 5
an
d
v^O2) = E $jz~j~*
jei+z
jei+z
by 2:, one has
0^(z) = - E (i + *W~ i _ § and
dV*(*) = - E (j' + fW;*-''-*So one has „
z
J
j-k-2
{k-j):Wi--z-J-k-2,
5 E 2 j,feei+z which implies L
"
=
fc n_ 9 *)) : ^»-*$t : 2 E , ( ~(
= \2 E (2fc-n):t/>„_feVJt:, , proving (i). (ii) follows immediately from (i). 2) is clear from the definition of H(z).
•
6.2. Boson-Fermion
Correspondence
235
Prom this lemma, it is easy to see the following Corollary 6.2.2.
Let u = ^*_ix • • • ip*_im^-h
1)
HQU
= (n — m)u,
2)
L0u
= (ii +
him+jiH
• • • V>-j„ |0> € F.
Then
jn)u-
So Ho is the charge operator and LQ is the energy operator, and the "character" of the Virasoro module F is given by
3=1
since the energy of ip-k and tp*_k is equal to k. Let fcez be the charge decomposition, where Fk is the space spanned by all elements (6.138) satisfying n — m = k. Introducing the "charge variable" £, we define the character of F, which contains more information than (6.142), as follows: ch(F)
:=
TrF(ZH°qL°)
=
$ ^ f c 1 r * {qL°) fcGZ
oo
= Hil + W-lW + C1?-*)-
(6-143)
3=0
Then, using the Jacobi triple product identity OO
i
n( +^"0( j=i
i+rv
1
2
"") = ^ z ^ ^^' fcez
(6-144)
and comparing the terms of £k in (6.143), one has
^(gLo)
=
WY
(6 145)
'
So the character of Fk is just equal to the character of the space C [ s i , x 2 > •••] = C f o ; j e N ] of polynomial functions in x^-'s with deg(xj) :— j . To describe more precisely, we define elements | ± k) G F±k, called the vacuum state in JF±k, as follows: |fc> :=
V'-(fc-i)-iV'-(fc-2)-iV'-(fc-3)-i---V'-i|0)
6. Operator Calculus
236 and \-k)
:=
^ ( f c _ 1 ) _iV'!( f e _ 2 ) _iV'l ( f c _3)-i---V'li|0)
for k G N . Lemma 6.2.3. -For each fceZ, Wiere exists a unique linear isomorphism Ok • Fk
—>
C[xl,x2,x3,---]
satisfying the conditions 1) 2)
ak(\k))
= 1,
<7fc o J/ n = a n o a t defined by (6.91).
/or ai/ n G Z — {0}, where an 's are operators
Proof. This lemma holds because (a)
H{z) is a free bosonic field,
(b)
Hn\k) = 0
(c)
foralln>0,
the coincidence (6.145) of characters.
• Thus one can identify each space Fk with the space C[XJ ; j e N]. This identification is called the boson-fermion correspondence. Note that the operator Ho on each space Fk (k G Z) is the scalar operator k • IdFk, and so, puting ao :— \ for each fixed k, one has the commutative diagram Fk - ^ ^ C[xn ; n G N]
*4 *fc ^ ^
1°" CK;nGN],
for all n G Z and A; G Z. With the operator ao defined in this way, the map Ok is an isomorphism as Virasoro modules. For each j G \ + Z, Vj (resp. ipj) is an operator from Fk to -Ffc+i (resp. Fk-i). The action of ^(^) and tp*(z) on the vacuum |fc) is given as follows: Lemma 6.2.4. 1)
i/>(z)\k) = \k+l)-zk
2)
ip*(z)\k)
= \k-\)-z~k
+ terms of zi (j > k) + terms of z> (j > k)
, .
6.2. Boson-Fermion
Correspondence
237
Proof. 1) First we notice that tf(*)|0> = $ > _ ; _ ! |0)-2' = V - i l O H ^ V - i - j l O ) - ^ . X
j>0
v
'
j>l
Lettng fc be a positive integer, we compute tp(z)\ ± k) as follows: V»(«)l*>
=
V'(z)V'-(fc-1)-iV'-(fc-2)-i---V'-i|o> j>k
=
V'-fe-iV-(fc-i)-i^-(*-2>-j • • • V»-i|o> -zk s
/
v
lfc+l>
+ 5 Z V'-i-iV'-(fe-i)-i^-(fe-2)-i---V'-i|0)-2; i ) j>fc+i
and
=
^ _ i C ( ) , _ 1 ) 4 ^ ( t _ 2 ) _ i • • • ^110) -^-fc
*:(fc_2)_i-^iio>H-fe+i> +
Yl
V' j -i^-( fe -i)-iV'-(fc-2)-i-"^-ii 0 )' 2;
j
j
proving 1). The proof of 2) is quite similar.
•
From (6.139a) and (6.139b) and Proposition 6.1.1.1), one has [Hm, V»H] = wmi)(w)
and
[Hm, ip*(w)} = -wmip*(w)
(6.146)
for all m e Z - {0}. We now look at formulas of vertex operators which will be discussed in §9.1. Then we see that the formula (6.146) coincides with the formula (9.4), by letting A = 1, under the identification ak = Hk- Then by using Lemma 14.5 in Kac's book [100], we obtain commutative diagrams Fk
Fk+1
-
^
C[xn ; n € N]
- ^ + C[xn ; n 6 N]
238
6. Operator Calculus
and Fk
—^-»
C[i„;«eN]
y.*(z)
"k
Fk-i
X
' C[x n ; n e N],
where V±i(,z) are the vertex operators defined by (9.2): V\(z)
9 zT> dxj j 7= 1
— exp( Ay~]xjz J )exp( — A ^ V
1= 1
^
'
for A = ± 1 . Namely the action of the fields ip{z) and ip*(z) on the space F is written in terms of vertex operators as follows: ip(z) = V>*(z) =
zk-aklx
oVi{z)ook,
(6.147a)
z- -a^\oV^{z)oak.
(6.147b)
k
Note that the factors z±k in the right sides of (6.147a) and (6.147b) arise from the coefficients of \k ± 1) in the right sides of the formulas in Lemma 6.2.4.
Chapter 7
Branching Functions Given a pair of finite-dimensional simple Lie algebras p C g, their affmization naturally satisfy p C g. In this chapter, we consider the structure of a 0-module V as a p-module. For this sake, an operator commuting with the action of p will play a role. But operators commuting only with the action of the derived algebra p' := [p, p] give us enough information since p = p' © Cd. Actually one can easily see that the coset Virasoro field L(z)
= L*{z)-L»{z)
^Lnz-n-2
= n€Z B
commutes with the derived algebra p', where L (z) (resp. Lp(z)) is the Virasoro field of g (resp. p) defined by (6.79). Namely the action of the Virasoro algebra spanned by {L„}„ e z commutes with the action of p'. Then the space of p'singular vectors in V is a Virasoro module, and one can study the p-module structure of V by using the representation theory of the Virasoro algebra. In this chapter, we give a quick review on the representation theory of Virasoro algebra in §7.1, and then explain its application to study the branching functions of representations of affine Lie algebras. 7.1
Virasoro Modules We consider vector fields dn
:= tn+1jt
(neZ)
on R>o- They satisfy the commutation relations [dm, dn] = (n-m)dm+n 239
(7.1)
7. Branching Functions
240 so Witt
:=
@Cdn
is a Lie algebra, called Witt algebra. Witt algebra is identified with the Lie algebra on the circle S1 spanned by
It is known by [72] that the non-trivial central extension of Witt algebra exists unique up to isomorphisms, which is called Virasoro algebra. A standard exposition of Virasoro algebra is given by rewriting the above commutation relations (7.1) as [-dm,-dn] = (m-n)(-dm+n) and writing £n in place of — dn as follows: Vir := ^nez
f^C£n)®Cc '
with brackets [tm,c]
=
[£m, ?~n\ =
0, (m-n)em+n-\
——<5m+„,oc,
(7-2)
for all m,n e Z. With this normalization, c is called the primitive central element of Vir. Putting Vir+ := 0 C I „ ,
Vir.
n>0
:= @ C £ n ,
Vir0 := C£0 + Cc,
n<0
one has the triangular decomposition of Virasoro algebra: Vir =
Vir- 0 Vir0 ® Vir+ ,
which gives the decomposition of the universal envelopping algebra U(Vir)
= U(Vir-)-U{Viro)-U(Vir+).
(7.3)
From this, the notion of highest weight modules is defined quite similarly with the usual cases as follows: Definition 7.1.1. A Vir-module V is called a highest weight module if there exists a non-zero vector vo in V and complex numbers h and c' satisfying the conditions
7.1. Virasoro Modules
241
(HI)
£0v0 = hvo
and
(H2)
env0
(H3)
U(Vir_)«b =
= 0
cv0 = C'VQ,
for all n > 0 V.
By conditions (HI) and (H3), the primitive central element c is a scalar operator on a highest weight module, namely c =
c' • Idy .
So, given a highest weight module, we may identify the primitive central element c with a complex number c', and write simply c
—
c•Idy
•
A successive use of toi-n
= £-ne0 + rdn
gives us 4(^-m---^-n f c wo)
=
(ft + niH
\-nk)£-ni
• • • t-nkvo,
(7.4)
for ni i • • • i nk € N . Namely ( " l , - - - ,njfe > 0 )
are eigenvectors of £Q. This means that all eigenspaces of £Q are finite-dimensional, and that V is a direct sum of ^0-eigenspaces. So one can define chV := Tr v (a;'°)
=
^TdimWa;*,
(7.5)
A
where Vx
:= {veV
; £0v = Xv}
is the eigenspace of £Q with eigenvalue A. chV is called the character of a Virasoro module V. Prom (7.4), one sees that all eigenvalues of (.Q on V are of the form h + n (n e Z>o), and so the above formula (7.5) is rewritten as follows: chF =
J ^ dimVfc+n • xh+n n>0
= xh ^
dimVh+n • xn .
n>0
A Hermitian form ( , ) on a Virasoro module V, not necessarily assumed to be positive definite, satisfying the conditions (£jU, v) = (u, i—jv)
and
(cu, v) = {u, cv),
(7.6)
7. Branching Functions
242
for all u, v G V and j G Z, is called invariant. A Virasoro module V is called unitarizahle if there exists a positive definite invariant Hermitian form on V. As to the unitarity of a highest weight module, we notice the following: V(h, c) : unitarizable =>• h > 0, c > 0.
(7.7)
Actually this is shown as follows. Let vo be the highest weight vector. Then, for each j G N, one has
(2jh+1^Lc)vo
2Jh+J-^-c\(v0,vQ), >o namely '3
2jh + 3—ji^-c
> 0
for all j € N .
This implies h, c > 0, proving (7.7). The converse "<$=" in (7.7) does not hold. As to the converse, one easily sees that c = 1 and h > 0 =$• V(h, 1) : unitarizable, since the Fock representations of Virasoro algebra in Example 6.1.4 are unitary for all /x G R. Moreover, by making use of a deformation (6.110) of the Virasoro operators with £ G R, one sees that c > 1 and h >
c— 1
=>• V(h, c) : unitarizable.
But a stronger result holds. It is known (see e.g., Kac's book [115]) by a more detail analysis using the Kac determinant formula that h > 0 and c > 1 = >
V(h, c) : unitarizable.
(7.8)
And actually it is known that there exist unitarizable representations in the domain {(h, c) ; h > 0, 0 < c < 1} and that the numbers h and c, for which V(h, c) is unitarizable, are described with descrete parameters. They are as follows. We put c(m)
._
i
2
(m + 2)(m + 3)
(7 Q-) K
'*>
243
7.1. Virasoro Modules and him) . _ ((m + 3 ) r - ( m + 2 ) s ) 2 - l ' 4(m + 2)(m + 3) '
hr s
U
">
for m € Z>o and r, s G N satisfying 1 < r < m + 1 and 1 < s < m + 2. Since fc(m) _ L ( " » ) — "r,« "m+2-r,m+3-s '
we introduce the equivalence relation (r,s)
~ (m + 2 — r, m + 3 — s)
in the set ^(m) . = { ( r ) S ) e N 2 ;
l < r < m + l, l < s < m + 2 } .
Then the quotient set S)/ ~ is identified with £(m)
.
=
^
S
)
G N
2
.
l <
s
<
r
<
T O +
l } ,
or with {(r,s) G N 2 ; 1 < r < m + 2, 1 < s < m + 3, r = s mod2}. T h e o r e m 7.1.1. =
{(h,c)
; 0 < c < 1, V(h, c) is unitarizable}
{(him\c^)
; m€Z>0,
(r,«)e^ra>}.
These unitarizable irreducible Virasoro modules V(h^', c^™)) are called representations of discrete series. Verma modules and their homomorphisms of the Virasoro algebra are studied in detail in [51]. From their results, we can deduce the following formula for the character of an irreducible Virasoro module. T h e o r e m 7.1.2. Consider the irreducible Virasoro module V(h,c) c = l - f c ^ pq
and
with
a
-fr-g>a,
(7-19)
mew
where W is the Weyl group of A\ ' and X and p, are dominant integral forms of A\ defined by X := A p _ 2 ; s _! = p := A , ^ ^ =
(p-s-^Ao + ^-^Ai (g-r-^Ao + ^ - ^ A !
P?'2, P^2.
£ e
Proof. We compute the right side of (7.19) and show that it is just equal to the right side of (7.16). Notice that X+p
=
p+ p
=
(p- s)A 0 + sAi = pA0 + -OJI , r (q- r)A 0 + rA x = qA0 + -ax,
and that the Weyl group W is given by W
=
{tjai,
tjairi
; j
eZ}.
Since A0 + jot! -
j2\a-i\2
tjaiA0
=
tjaiai
= ax - (ja^a^S
b* =
2
=
A0 + j a i -
j26,
a1-2j6,
one has
W * + P) =
p(ko+jai-j28)
+
^(a1-2j8)
pA0 + (pj + - J a i - (pj 2 + sj)«, *j a i ri(A + p)
=
p(A0 + jjax a i - j26) - - ( a i - 2j«5) pA0 + [pj - - J a i - (pj 2 - sj)<5,
and so p
g
[pV
2/
2qr J
" .pi^-^-^r1
modc
*-
7. Branching Functions
248 So one has tjai (A + p)
/i + p
{q(2pj + s)-pr}
2(M) 2
,
and tjairi(X
+ p)
1
n+p
2(M)
j{q{2pj-s)-pr}
.
Thus one has 2
2_] e{w)x
i p l-{,(2pj-S)-pr}2
jGZ
jGZ
proving the proposition.
•
As we shall see later in Theorem 7.1.5, the modular transformation of Xrfi (T) is described by matrices «
:
=
M
( - D ^ H ^ ^ a i n ^ t p - ^ . r i n ^ ^ - , ) , (7.20)
which is well defined for (r, s), (r', s') G $)(p'q)/ ~ , since
x sin
7rr(g — r') {p-q)
_(_!)r(p-9)
. • sin
ns(p-s')
(p - 9)
P sin
2rmi(p_q) _(_l)»(p-<,)
sin
I^i(p_q) S
=
r+.)(r'+.') JTfaHI^f (_l)(r+.)(r'+.') J^_ g i n : ^ ( p _ g j*. s m• ^™^ (' p _ q ) Vpq q ' " q
=
A(r,s),(r',s')
Theorem 7.1.5. Let p,q e N>2 6e mutually prime positive integers, and r and s be positive integers such that 1 < r < q — 1 and 1 < s < p — 1. 7%en 1)
there exist unique positive integers r0 and SQ satisfying 1 < ro < q — 1,
2)
i) minffij
1 < So < p — 1,
; (r',s') e #*•'>} = /><™>
and
r0p — s0q = 1. l-(p-q)2
4pg
7.1. Virasoro Modules
249
ii) m i n { / # $ - ^z<™> ; (r',s') e S ( p ' 9 ) } -
I,(M) _ _1 Z (M) ft
" Q\
Y(M)
(_±\ ^
4)
r0,.o
-
xipiq)(r)
T
JL f JL _ 1^
-
24 \pq
A(P'9)
V
'
-
24
J'
Y^M
(r',s')ei5(P^)
i° A*p'9>
-e*d('"9,, d(M)
:=
where
i
.
(7.21)
pq Proof. 1) In this proof, we may assume that p > q. Then, for each integer n such that 1 < n < q — 1, there exist positive integers j n and kn satisfying np = jnq + kn
and
1 < fc„ < q - 1.
Here we notice that 1 < J„
(7.22)
To prove (7.22), let us assume the converse, namely fcn = &m
f° r some l < m < n < g — 1.
Then, from np = jnq + kn
and
mp = jmq + km,
one has (n - m)p = (jn - jm)q, which implies
q\(n-m) since p and g are coprime. But this is impossible since \<m
= {1,2,-
— 1.
-,9-l}.
So there exists n such that fcn = 1. Thus we have proved the existence of a pair of integers (r, s) satisfying 1
7. Branching Functions
250
The uniqueness follows from the coprimeness of p and q. 2) follows immediately from 1) and the following calculation: 2
_
/,(M)
Z
=
(M)
(p-q) i-W-W
_ _1 [(.1
4pq
6(p-q)
24 y
pg
= 4pq J - - 124 = 24 ±(°--l \pg It is possible to show 3) by using the transformation formulas (Proposition 2.4.16) of the modular transformation of theta functions / j > m denned by (2.74b). But we give here a proof by making use of the modular transfomation of admissible characters of an affine Lie algebra A{ '. Actually this is proved by Theorem 7.4.1 and (7.110) and by using the list (3.87) of a(A, /x) for A± as follows. By (7.18), one may assume that r—s G 2Z. Then, noticing that u(m+2) = q and u(m + 3) = p in (7.107), one has x (p,«)
(_1\
8Am;o,r-l ( /,Ao®A + 1;0,«-1 I
=
r
E E
j=0,10
_ _ \ TJ
E ^o^-iS
0
-l);(fc,n)
* V5
We now look at the list of a(A, /i) for A\ ' given in (3.87): «/n^.
iwt„\
=
(0,r-l);(fc,n) (m+1)
e
Sin-
/~2~«rka
•
1/
y _
Uq
q US
(n'
+ 1)W
oi/n IWL n = \ — e sin(0,a-l);(fc,n') y UJ9 p Since r — s € 2Z by our assumption, one has e27rife('—*) = l ; rewritten as follows: Y(p,g)
=
\-^ >
j=0,10
x6A3®Am;fc,„(r)
-
^ V(P.9)
X
(7.23) is
f_A
1 /~2~ v—> —\ — > ' ^
SO;
' (_\ lT
r.+ l,n' + l -'
v^> > 0
. ur(n + l)n . us(n' + l)7r sin sin *
^
7.1. Virasoro Modules 1 [2 ~\
— '
251
yr-y
-c-y
. ur{n + l)n s m. us(n' + l)n
/ / sin 0
—-
*
(
„Q\
..
— • X„li „>J-1 (T).
^
/~2~ ^ . ur{n+l)n . tts(n'+l)7r (pq) \ — > sm — — sin — — • \ , V / , , (r).
Putting r' := n + 1 and s' := n' + 1, this is rewritten as X^9) I —
= \/ —
>]
s
"i
sin
xj™/ r .
7.24
l
Since the set
^(P>«)
is the 2-fold covering of ij( p ' 9 \ this is rewritten as
*'(-?) =2\/I £ - T
, I ,
T ^
(7 25)
'
Notice that (—l)( r + s K r +3 ) = 1 since we are asuming r — s € 2Z and that p — q = u. Then the transformation formula (7.25) just gives the formula (7.20), proving 3). To prove 4), we look at the transformation formula in 3): •V<-P>I)(T)
-
V
X.r,s
—
2—t
\T)
(r',a')eii<','<<)/~
A{p'q)
>1
v(p,,)
(T-,s),(r',s');<-r-',s' I
f-~\ T
I •
Since
the minimal of leading powers in the right side, when T j 0, is achieved by the term (r', s') = (r0, s0), and the leading term of Xro'.so (— 7) is equal to
Thus we obtain 4).
•
The asymptotic behavior of Virasoro characters of the minimal series representations are given in Theorem 7.1.5.4). It is possible to compute the asymptotic bahavior of Virasoro characters other than minimal series by using the
252
7. Branching Functions
character formulas given in Theorem 7.1.2. The idea is as follows. First we look at the function l-xn
1
27TtT-
24
= e
•
r)(r)
Since TJ(T)
(-«-/ ,
TIO
i 2
7Tt
12T
(7.26)
-27rinr,
(7.27)
{-ir\
e
and 1-e2™7
~ T|0
one has ^—^ ¥>(z)
( i = e2**1").
~ W-*T)8e» no
(7.28)
When the asymptotic behavior of the character of V is given in the form chv(:E)
~
(x = e2viT),
A-{-ir)ie^
(7.29)
TJ.0
with some real numbers A, B and C, we call the triple of numbers (A, B, C) the asymptotic dimension of V and, in particular, the number C the growth of V. We notice that, writing T = iT with T > 0, the condition (7.29) is rewritten as ch v (a;)
~
{x^e~27TT,
A-T^e^
T > 0).
(7.30)
The formula (7.28) says that the growth of chy is equal to 1, in the case when xh(l — xn) the character is of the form chy(x) = — for some (n > 0). tp(x) We now compute the asymptotic behavior of the function
^v
where
^
' j : finite
means a finite sum. Then, rewriting this as
j : finite
fix)
—
> j:
Xa> finite
— ^K
'
7.1. Virasoro Modules
253
and using (7.28), one has /(*) ,-To
2n
(
E ^j:
(x = e 2 ™),
(6i-fli))(-ir)8e# finite
'
so the growth of /(a;) is equal to 1. We now consider a function
3<m or j > n
where ra, n G Z such thatTO< n, and a^ and 6j are quadratic forms in j . Then one can rewrite g{x) as follows:
where G{x) := y*(a: 0 i - a;6')
and
/(a:) := - J ^ r
(xa* - xb>).
V m<j
Since G(x) is a modular function, the growth of — ^ - is less than 1, so
= the growth of f(x)
= 1.
Thus, in view of Theorems 7.1.2 and 7.1.3, we obtain the following: Corollary 7.1.6. Let V is an irreducible Virasoro module of real central charge with growth less than 1. Then V is a representation of minimal series. Namely there exist mutually prime positive integers p,q G N>2 and (r, s) G fi(p'q) such thatV=V{h%?\z<*«'>). Example 7.1.1. We consider the case when c = —2, namely (p,q) = (2,1). Then (xh(l
- xn+1) , ,—xh(l (p[x) - a; 2 <" +1 >) V
chV(h, - 2 )
=«
xh
ip(x)
.t L (2n + 1 ) 2 - 1 / q „ . 3 ifh=± ^ neZ>0 .f u _ (2n) 2 -8 1 /3 ifh= ^—*( d n G Z> 0 ) otherwise. (7.31)
254
7. Branching
Functions
From this, one has
4k+l)2-l
+
2
E(-Dj+k
=
k>j
k>0
for j € Z>orepresentations 7.2
2
X
These functions
take place as string functions
of
fundamental
of affine superalgebras si{n, 1).
Virasoro Modules of Central Charge - ^
One among most interesting Virasoro modules is a module of central charge with (p,q) = (5,2), namely c ( p , 9 ) = — T2-. T h e "smallest" non-trivial unitarizable representations a r e representations with central charge | , namely {.Pi q) = (4,3). While t h e representations with (p, q) = (5,2) is much smaller, since their growth is equal t o d(s,2)
i _ A = ? < ^(4,3) 10 5 <
=
1 2'
=
a n d actually this is t h e smallest among t h e growth of non-trivial Virasoro modules. Notice t h a t SjW {(1,1), ( 1 , 2 ) } , = and t h a t
M?i 2) = o
h)1 A ' ~ — "i 5 T h e characters of these representations are given by (5,2)/
N
xTfir)
_
rC
(5.2)/ -,
Xl,2
(T)
x
and
eofix0) (p(x) 1 — 11 5 60
/
JJ(l-x 5n - 1 )(l-a: 5,, - 4 ) ! E:\
OO
na
=
(7.32a).
n=l „5n-2
)(l-^-
3
)
;
(7.32b)
a n d their modular transformation is obtained from Theorem 7.1.5 as follows:
r 5(5,2)
x£ 2) (r) N (7.33a)
Xff{r) where .
Q(5,2)
=
_£_
2TT
7T
\
- sin — sin .
27T
sin-
.
(7.33b)
7T
s m -
/
7.2. Virasoro Modules
of Central Charge
-22/5
255
T h e asymptotics of characters are obtained from Theorem 7.1.5.5) as follows: x i r ( T ) ~ ^ s i n — - e w s ,
[
x
5
f ( r ) ~ - s i n - . e ^ ' =,
(7.34)
since d ( 5 ' 2 ) = § and ( r 0 , s 0 ) = (1,2) when (p,q) = (5,2). To compute the modular transformation a n d asymptotics of characters, it is useful t o notice t h e following: .
7T
2 T T \ / 5 - 1
sin — = cos — = 10 5 / .
TT\2
sm
l To)
K)-
=
4
.3TT
, '
3-v^
/
-8~'
>2
.
sm
3TT\2
3 +
{ wJ
5 -y/5
f
8
T T \ / 5
+
sin —- = cos — = 10 5
, 4 '
V5
=^T~'
2TT\2
.
1
5 +
V5
sin V 5
'
and so 7T . 3n 1 sin —- sin —- = - , 10 10 4'
. 7T . 2?T V5 s i n - sin — = ——, 5 5 4 '
and
Sin
( S) ( sin D
2
+
/
.
ZTTV
'
Sin
ioJ
/
.
2
2
3
= 4'
2TT\2
5
Notice also t h a t
sin
( D
2
5-\/5 8
2TT\2
sin
yJ
5+\/5
=
= -r-
2
\/5-l 4
>/5 . 7T 2 sin 1 0 '
v^5
\/5 + l
\/5
N/5
.
3?r
sm
T ' — r - = T io-
So, from (7.34), one has (5,2)/ \2
Xi,i CO (5,2)/
Xy
W
^ /
~
\2
.
4 / .
~
2TT\
g fsin—1 5l
s m
JLL.4
2
.
e«- 5 =
7T\
2
5J
e l 2 r 5
s m
_»±.4
2 =
^
.
S m
37T
_ . 7T
i0'
_EL.4 e
i
(7.35a)
2 T 5
J>LL.4 e l 2 T
5
,-„,.,,
'
(
^
We now consider t h e tensor product of these representations. T h e growth of t h e character of tensor product V"A \' ® V^.,'J,y where (r, s),(r', s') = (1,1) or (1,2), is 2d<5'2) = 2 . | = | 5 5
<
1,
7. Branching Functions
256
so, by Corollary 7.1.6, this is a sum of minimal series representations: T/(5,2)
(r,s)
2_\
(r',a')
(irreducible Virasoro modules).
finite sum
Proposition 7.2.1. The decomposition of tensor product representations of Virasoro modules with central charge — T2- is given as follows: iras T/(10,3) "1,9 (skew-sym. part)
(10,3)
^i ( , B i 2 ) ®V§ 2 )
1)
V,1.1
(sj/m. part) /(5,2) "l,2
2)
"l,7 (skew-sym.
l,3 (sym. part)
1/5,2)^(5,2) v l,l ^ "l,2
3).
T/ (10,3)
T/ (10,3) V
(5,2)
®K
_ —
part)
^(10,3) "l,5
Namely
D'
22
V 0;
y
0;
22
=
44
V[0;
n*-T
"5"
sym. part
„w
2)
¥
,/
F
1
;
22 \
h -y J
T
0F
/
1
22 \
;
/
2
;
skew-sym.
part
44 \
/3
T
e F
("5 - y j = ^ ("5 - y j sym. part
;
44 \
U - yJ
skew-sym.
part
•>' ^-?)-H-?H(4-!)Proof. First let us find (p, g) satisfying d(P,<*) =
2d(5.2))
and
2
Z (P,«) =
2z&
\
namely 4 5'
pq
and
1-
6(p-) 2 P9
44
y-
There are two solutions for these equations : (p, q) — (10,3) and (3,10), so we may take (p, q) = (10,3). Next we compute h\ s' for 1 < s < 9. Then we have: s ,(10,3)
1 0
2 n "40
3
4
5
6
7
8
"I
3
I "5
1
3 5
49
"9
40
9 2
7.2. Virasoro Modules of Central Charge - 2 2 / 5
257
Prom this one sees that the tensor product decomposes in the following form:
v(,-fW,-f)
= v(*-$)+.v(*-%
where a, 6 6 Z>oTo determine non-negative integers a and 6, we compute the asymptotics of characters. For (p, q) = (10,3), one has j(10,3)
_
£
~ 5' and the number (ro,so) in Theorem 7.1.5.1) is (ro,So) = (1)3) when (p, q) = (10,3). So the asymptotics of the charcters become as follows : (10,3)/ N
A(10,3)
jEi_.i
2
.
7w .
27S7T
.JLL.4
x ( i, s 'V) ~ 4;),(i,3)- e l 2 T 5 = v ^ s m T s m - T f r ' e l 2 T 5 ' and so x (10,3)( r )-
~ ^sinio'e^'5
^" = 1 ' 9 ) '
X
~ ^sinT5-e*"
0 = 3 7)
™
^' 3 ) ( T )
(10,3)/ N Xl,5 (T)
1
. 37T
' '
1 •£*-•# ~ ^ e •
~
So one has (10,3)/ s ,
(10,3)/ N
l + O .
37T *_L.4
(10,3)/ s , ,
(10,3)/ x
1+ 6 .
7T _ M _ . J
xi.i (T) + axi,9 (T) ~ "7/H" s i n io' e Xi,3
( T ) + bx\,7
'(r)
~
'
—^-sin — • e"- s .
Then, by (7.35a) and (7.35b), one has a = b — 1, proving the proposition. Corollary 7.2.2. TTie linear span of
{X^'^W + X ^ M , X& 3) W + xi:70'3)(r), X i ^ M } is SL(2,
Z)-invariant.
•
258
7. Branching Functions
One can, of course, compute the explicit modular transformation of these functions by using ,(10,3) (r,a),(r',«')
(-1)
(r+S)(r'+s').
\ / l 5 sin
7irrr' . 7irss' sin- 10
It is also possible to compute it by using the modular transformation of Xr,s as follows. For simplicity, we put f(T) 9(T)
=
X?f(T)>,
X(I:3°'3)(T)+X^3)(T)
V2xt2)(r)-xt2Hr).
h(r) Then, from (7.33a) and (7.33b), one obtains '/(-l/r)N fl(-l/r)
(7.36a)
where
(Sinl)^ A
=
/-r
.
IT
.
2TT
-V2sin — sin ~5~ 5 5
K)'
2TT
sin
/- . n . 2n - v 2 sin — sin — 5 5 < 5 - V 5 5 + >/5 5 + V5 5-y/E -2v^0 2V^0
2TT
i- . 7r . 2n v 2 sin — sin — 5 5
\/2 sin — sin — 5 5 2n\ ( . ny
in
yJ -l s m 5J
-2A/10\
(7.36b)
2v/l0 J . 2>/5 /
In a similar way, one can calculate the decomposition of VPjS' ® ly/^, for (p, q) = (4,3) and (7,2). The result is as follows: P r o p o s i t i o n 7.2.3. 1)
(i) (ii)
(4,3^ c h (, (V5 ,^2 )®^ ,V ^) K5,2) ^
chy(12,5)
—
U11v
,(12,5) )
Vfi3)) = r(5,2) ,
K4,3)x
(i)
ch(V$" ® V ™ )
(ii)
ch(vf/>®V#3>)
_
=
_L „ W ( 1 2 . 5 )
chV^r^+chV^V
,(4,3)
16* ^ 2 1
(iii) 2)
f
_
(12,5)
cnV^+chV^ (12,5)
^(12,5)
chV 3 ^' o ; + chV^" B)
dnff- + d , ^ )
7.2. Virasoro Modules of Central Charge - 2 2 / 5 (iii)
ch(vg a >®vfi»>)
=
259 chV^+cW^K
Proposition 7.2.4. 1)
(i) ch(yft 2 ) ® V # 2 ) ) = chvff*
+ chVl$7)
+ c h V ® 7 * + chv£° 9 ' 7)
(ii) c h ( ^ 2 > ® V S « ) = c h v ' f 7> + c h V ^ 7 ) + c h K ^ + c h ^ (iii) ch(v£ 2 > ® l # a > ) = c h ^ 0 ' 7 ) + AVff* 2)
+ *V
+
7 )
cnV^
(i) c h ( v g 2 ) ® < 7 ' 2 ) ) = chvf 7 °' 7) + chvffi 7 * + c h v f ^ + chlf 2 0 /> (ii) c h ( v f / > ® y g a > ) = chV#°'7> + c h ^ 3 0 / * + ch^ 30 7 - 7) + c h l ^ 3 0 / ) (iii) ch(vf2'2> ® vf 3 ' 2 ) ) = c h V ™ + chVffi 7 ) + c h V ™ + chvf 2 °/> •
But here it is unclear whether the decompositions in Propositions 7.2.3 and 7.2.4 are direct sum as Virasoro modules since there is no unitarity, so the decomposition of the representation space may possibly be a "sub-quotient" decomposition. N o t e 7.2.1. It is interesting that the exponents of the simple Lie algebra E$ (resp. E6) appear in the right sides of Proposition 7.2.4 (resp. Proposition 7.2.3). It is also possible to calculate the decomposition of tensor products of unitarizable Virasoro modules with central charge \ by using their characters. The result is as follows: Proposition 7.2.5.
1)
(i)
S>v(0;\)
= (
V>2;1)W
0
n2n2;l)),
0 even
(ii)
&v(\;\)
V(n2;l)W
= ( 0
0
V(2n>; 1)) , even
2)
A2l/(0;i) ^
= A2v(y~)
'
^
3
> m ^{TA) V
/
= '
0 nGNodd
= © "(T!1 nGNodd
V
n==tl mod 8
«» >»(&)x
'
© v(£, neN„dd n = ± 3 mod 8
V
n2n2;l),
260
7. Branching Functions
o a. " H ) . ^ ) - ©/OH' n€Nodd
(«) M ^27 V~ ' CV 1^6 ' 2 , =
©
n€N0dd
v(=i')
n = ± l mod 8
n = ± 3 mod 8
Proof. To prove this proposition, we first notice the following: chs2v(a;)
=
-ichv{x)2
+ chv(z2)}
ch A 2 V (x)
=
Uchv(x)2-chv(x2)Y
(7.37a) ^7'37b^
Actually these formulas are shown as follows. Let chy (x) = xh^2 dimV} • xj. jez Then, since dim(S2Vj)
=
^dimVj • (dimV,- + 1),
dim(A2V9)
=
^dimV^ • (dimV,- - 1),
one has ch^z)
=
x2h • ^-i
V
(dimV, • ^(dimVk
Y. (d\mVjxi)(d\mVkxk)i.fcez
+ ^(dimV' j )(dimV j + jez =
=
l)xj\ J
\{chv{x))2+\Y,^™Vi)x2* jez ^(cM^Of+^chytx2),
proving (7.37a). The proof of (7.37b) is quite similar.
• xk)
£ (dimV i ) 2 z : u jez
7.2. Virasoro Modules of Central Charge -22/5
261
To compute characters of tensor products of Virasoro modules of central charge | , we note that l_ 1 16'2
V 3) V
V™ = V (0; 0 , Wr = ^ ( H ) .
£ =
and that their characters are given by <3>(z)±ch^>(x)
=
g d i x - l )
=
n=l
W
^
+ *") = 4 ? w -- ftn (!(!+-") =
(7.38a)
< ^ ,
^
c4^W
^T>
(7-38b)
so
chi^W
=
1 f °°
°°
*• n = l
n=l
n=l
We first compute the characters of S2V1^
2
J
J JJti + s n - i j - J I f i - s - i U . ^ n=l
3)
1
2
(7.39b)
J
. Prom (7.39a), one has
2
i a;2n 1
,
1
4 _^
oo
n=l OO oo
1
r
Z
^ n=l
n=l oo
oo
n=l
J
N
n=l
J
So ch(S 2 V# 3) )(x) /
oo
2
2
oo
i ;c2
oo
i
N
= \\ n(i+x-i) +n(i-^^) +2n( - "" )[ 1
^ n=l
^
+
*
n=l
OO
n=l
OO
-v
l|jj(l
+ x2n-l)+^(1_x2„-1)l
^ n=l
n=l
^n=l ^ oo
+JiK n=l
i+a;2
i
n=l oo
1
J J
i a;2n_i
"" )+^n( n=l
)
v
'
262
7. Branching
Functions
and ch(A2l#3>) -j
•
OO
OO
OO
1+xn i 2
= i{U(
~ ) + U( - ~ )
^ n=l
-*
1 xn i 2+2
1 x2n l
Il( - ~ )\
n=\
n=l
1
1
i af
J a
'
- i l l l d + ^ ^ + IK -^"" )} ^n=l oo
-I •
= si ll(
i+a!
J a
n=l oo
J v
oo
1
"" ) +n( - "" ) }- \li(i+x2n~1y
"- n = l
n=l
•*
n=l
To continue, we notice t h e following:
^V
n=l
7
00
?(z) n=l
^
neZ
2 '
2 1
= E(-Dn-2n2
^na+z - ) = S r n=l
" ^
••OO
OO
n
2
>
*• n = l
neZ
-v
n
2
x)l l[(i +x ~i) +l[(i-x ~i) \
¥>(
n€Z
n=l
= ^ x ^ + E(-i)"^ J
n£Z
n£Z
*E
2n
nez T h e n one has
ch(S»V<*3))W ^v
' ^
n€Z >
n£Z v
n6Z
2 £) x 2 " 2 =2 53 x 8 " 2 n£2Z
^V
'
v
n£Z
1+2 53 x 8 ^ «—1
'
n£Z
n£Z
- 1 + 53 (-i)"x" 2 n—0
J
263
7.3. Branching Functions
^K
'
k
n=l
D n£2Z>0 OO
o„2
Y^
x
n—1
v
v-^ '
J
n=0
s
>
v ' / (2r.)2 (2n+2)2 \ (x 4 -X 4 1 V / (2r»)2 (2n+2) 2
a;
4
—x
nfZf-.m
4
'
chV(8„2;l)
y
chV(fi^;l)
OO
=
^chV(8n2;l) + n=l
chy(n2;l).
]T n62Z>0
Thus we have proved l)(i). Similar calculation proves the rests. 7.3
•
Branching Functions We recall Example 6.1.3 which proves the following thoerem:
Theorem 7.3.1. Let g be a non-twisted affine Lie algebra over a finite-dimensional simple Lie algebra g, and (n, V) be a highest weight g-module of level m such that m 7^ — hv. Define the operators Ln (n £ Z) by dim§
L(z) = J2L"z~n~2 = £:u*(*H(*):, nGZ
i=l
1
where {u } and {ut} are bases ofg satisfying (ui\v?) = 6ij. Then V is a Virasoro module by letting 7r(£n) := Ln (n 6 Z)
[Tr(£n),ir(ti®X)}
2
*<*> =
>
= -jw(ti+n
w(c) :=
TTl
——-dimg. m + nv And this Virasoro action has the following properties: 1)
and
® X)
{X
eg)
2^)-n{dh
Let
fl' := [0,0] =
(($V®g\®CK •jez
be the derived algebra of 0. It is sometimes convenient and important to look at a highest weight g-module V as g'-module. We remark here that, in this case, V is irreducible as 0-module <==> V is irreducible as 0'-module,
(7.40)
264
7. Branching Functions
since B = g'®Cd, and d is a scalar operator on each weight space. In this section, we let g be a finite-dimensional simple Lie algebra and p =
po©Pi©---©ps
its reductive subalgebra, where po is the center of p and pi (i = 1, • • • , s) are simple Lie algebras. We assume that
i)
Poffi(y]iJi) E5«) C jj. i=l
ii)
Ylpi+
C 0
'
+'
i=i
where hj (resp. h) is a Cartan subalgebra of pj (resp. g), and p j + (resp. g+) is the sum of positive root spaces of p» (resp. g). For simplicity, we put
6:=Po©(5^JJt) ^ i=l
and
p+:=^pi+i=\
'
We do not assume that dimrj — dimrj, so we deal with dimf) < dimrj generally. Here are some of examples of such pairs (g, p). Example 7.3.1. 1) Let p := rj is a Cartan subalgebra ofg. (p, g) satisfies the condition.
Then the pair
2) p := so(k) c sl(k) —: g satisfies the condition. Example 7.3.2. A finite-dimensional simple Lie algebra p is a subalgebra of End(p) by the adjoint representation, which gives the inclusion p C so (p) since the adjoint action preserves the Killing form. satisfies the condition.
Then the pair (p,so(p))
We consider the affinization
0 = p =
tDt3®B]®CK®Cd,
I v
iez
(0t»'®pJeCA'©Cd.
7.3. Branching Functions
265
Note that the affinization p is the sum of the affinization of p i ; i.e., a
p= £>. i=0
with the common center, where the affinization pi's are constructed by using the standard inner product on pi. Since p C fl, one has p C fl, but here we have to be careful about the relation of the primitive central elements K and K. E x a m p l e 7.3.3. We consider the affinization of Example 7.3.1, namely sb(n) C sl(n). This inclusion is constructed in terms of the Dynkin diagram as follows. sl(n) is a simple Lie algebra of type A^^. Let ei,fi,aY be its canonical Chevalley generators. Namely these elements satisfy l e i) Jji
°i,j°-i
[ai i e j ]
)
where
^tj&jj
[Q^ , JjJ
( 2 -1 0 -1 2 -1 0 - 1 2
0 0
O-ijJjj
... ... ...
_i\ o o
2 -1
-1 2/
(7.41)
\Q'ij)i1j=0,--- ,n—l
0 \-l is the Cartan matrix of A^^. relation „„. _ (adei) 1i - o"*'e
0 0
••• 0
Here we do not need to think about the Serre {adfi)i-aijf.
=
0
{ i ¥ : j ) t
since the the fundamental relations (7.41) together with the existence of nondegenerate invariant bilinear form determine uniquely a Lie algebra up to isomorphisms. To see the subalgebra sb(n) in An_x, we devide the cases according as n is even or odd. Case(I) : n = 21. In this case, we put e0 := [e 0 ,ei] + [e 0 ,e 2 £-i], ei:=ei + e2e-i, ee := [ee, e*_i] + [et, et+l],
/o := [/o,/i] + [fo, f2t-i], fi := fi + fu-i, {l
and A.V
a,
2
" 0 + al + a2t-l '
(7.42a)
266
7. Branching Functions V
i
V
(l
a," + a 2l-i
(7.42b)
Then it is easy to see that these elements satisfy [eii Jj\
=
"i,jai
[ai i ej\ — aijeji
>
[ai ) / j j
=
(7.43)
~aijJ3i
where /2 0 0 2 -1-1 («tj)i,j=0,-,/
-1 -1 2
0 0 -1
••• ••• ...
0 0 0
0\ 0 0
-1 2 -1 -1 0 - 1 2 0
= 0
0
•••
0 \ 0
0 0
••• 0
•••
-1
0
2)
zs #ie Cartan matrix of D\i ( i ) . So the subalgebra g of g generated by these elements ii,fi,hi and d is the affine Lie algebra of tyle Dt , since the induced inner product on g is non-degenerated. In the Dynkin diagram, this inclusion looks as follows: Ctt-3
<*l-2
Oti-i
•• — o — o {1)
A
Ct2t-1
Ctlt-1
«i (i)
-o—o
OL-it-Z
a2
ote+3 ae+2 <*e+i
0:3
Ctt-3
o—o—o-
D\
a
i-2
Ott-l
-o—o—o
6«o
O«/
By (7.42b), the central element c of g is given as follows: 2l-\
e-2
c=
a0v
v
V
+ dr + 2 ^ d t + a , - i + ^ i=2
V
= 2 j > . Y = 2c.
(7.44)
i=0
So a highest weight g-module of level m is a module of level 2m when it is looked as g-module. Case(II)
: n=
2£+l.
7.3. Branching
Functions
267
In this case, we put e0 := [e 0 , e x ] + [e 0 ,e 2 <], ki := e» + e 2 £+i_t, e f + ef+1, e*
/ o := [/o, / i ] + [fo, fi •= fi +fa+i-i,
fa], (1 < t < ^ - 1)
(7.45a)
// := 2(// + / £ + 1 )
and
2a v + a v + an ,
=
v • — "i +
a)
va 2t+l-i
(l
v
a;
(7.45b)
= 2(a, +ay +1 ).
JTien it is easy to see that these elements
satisfy
[«, v , ej] = d ^ e , ,
[e<, /j] = ^ j ^ ,
[cVv, /,•] = - a ^ / j ,
(7.46)
where
-1
0 2 -1
0 0
0 0
Vo
o
/2 0
-1
0
0 0 0
0 \ 0 0
(
is the Cartan
matrix
of B\
.
0
-1 0
2 -1
...
o
0 -1
2/
So the subalgebra g of g generated
by these
elements &i,fi, hi and d is the affine Lie algebra of tyle Bt , since the induced inner product on g is non-degenerated. In the Dynkin diagram, this inclusion looks as follows: cte-2
cte-i
cte
-o—o- o id) *2£
a2e ct\
a
2t-\ 0.2
j(D
ct2e-2 «3
• o
-o—o—6 oce+3 at+2 cti-2
oti-i
a>e+i at
-o—a=»
O"o 5 y (7.45b), the central element c of g is given as follows: £—1
21
c = d£ + 22>y+dy = 2 f > y = 2c. i=l
t=0
(7.47)
268
7. Branching Functions
So a highest weight g-module of level m is a module of level 2m when it is looked as g-module. Let V be a highest weight g-module. We are going to study how is the structure of V as a p-module. For this purpose, we make use of the Virasoro fields in Example 6.1.3 of §6.1, namely ditng
2 n€Z
v=l
and L
1
]z n 2
LM(z) = J2 % ~ ~
uniip
= ±£:«*(*X(*):,
where {M*} and {ui} (resp. {u"} and {u^} ) are bases of g (resp. p) satisfying n (U 1 |UJ) = 6ij (resp. (u \u'j)' = Sij), where ( | ) (resp. ( | )') is the standard bilinear form on g (resp. p). Then, by (6.80a), these operators satisfy
[L-WL-wi
=
^M
z7. — —w in
+
^M
2+ (z iy— — inw)i"
±^L
4
— w)1* I(z z —in
(7.48a)
and
m^M}
= ^ M + 2^M + ± ^ L , z — io
(z — io) 2
(z — wp
(7.48b)
where K K ~ „ , L »dimi and c'"' := ^ ^dimp, K + hy K + hy and hw is the dual Coxeter number of p. From (6.88b), one also has the following: c[B]
(7.49)
[L^,tj®X] = -jtj+n®X
(Xeg, n j e Z )
(7.50a)
[L]»\tj®X]
( l e p , n,jeZ).
(7.50b)
and = -jti+n®X
So the field L^(z)
:= L l » l ( z ) - I " l ( z ) .
satisfies [Lto>\#<8>XI
= 0
(Xep, n,jGZ).
(7.51)
7.3. Branching Functions
269
By a similar calculation as we did in Example 6.1.6, one has [Ltoirt(2)Lto*](„)] =
W + ^ _ J g z—w (z — w)z
+
^ - ^ , (z — to) 4
(7.52)
where ctoiPl
:==
C[B]_C[P)
g =
dimfl- .
K
.
dimp.
(7.53)
So L\a'v^{z) is a Virasoro field, called the coset Virasoro field, with the central charge c'fl;|,l. We now assume that V = L(A) (A € P+) is an integrable irresducible g-module of level m. Let \) (resp. p + ) denote the Cartan subalgebra (resp. positive part) of p. For v e \f, let V(B,»
.=
L
e L ( A )
.
*(X)v = 0(XeP+),
1
be the space of p-singular vectors. Then, by (7.51), Vu is stable under the 8,p) action of coset Virasoro operators Lj? ( n £ Z), so Vj is a coset Virasoro module, and L(A) decomposes as Vir © p-module in the following form: L A
()
=
©
(Vj-^OLM).
(7.54)
For i/ G ()*, the function c£( 9 ) := Trv
(7.55)
is called the branching function. By this definition, the decomposition (7.54) gives the following decomposition of characters: ch L(A ) =
^
c£(g)-ch i ( l / ) ,
(7.56)
where q = e~6. Defining the normalized branching function
btir)
~
g**-*" •<£(«)
the formula (7.56) is rewritten as follows:
(q = e2nil,
(7-57)
270
7. Branching Functions
N o t e 7.3.1. If A € f)* is not o dominant integral form, then L(A) is not unitarizable. In this case, L(A) may not necessarily be completely reducible as a p-module and the decomposition formula (7.54) of the representation space may fail to hold, but the decomposition formula (7.55) of characters still holds by replacing the definition ofVv with the space of primitive vectors of weight When A G P™, L(A) is integrable as g-module and so integrable as pmodule. So ch^A> and ch'-. 's in (7.58) are modular functions. From this, one sees that b${r) are modular functions, and can calculate their modular transformation. Theorem 7.3.2. (cf. Theorem 13.10 in [100]) : Let g be a non-twisted affine Lie algebra over a finite-dimensional simple Lie algebra g. Let p be a reductive subalgebra ofg andp its affinization with the canonical central element K. For A e P p , we put m := (A\K). Then
1)
# ( ~ ) =
2)
b*(T + 1) =
E
E
«(A,A')^X)e'(r),
e 2^i(«A-i,) 6 A ( r )
Example 7.3.4. We consider the restriction of the fundamental Ar2^_x-modules L(AJ; Aix_i) (0 < j < 2£ — 1) to its affine subalgbra D\ '. Since c = 2c by (7.44), L(A_,; A^_j) is a sum of D\ -modules of level 2. Notice that the Coxeter number h and the dual Coxeter number hv are h
(4u-i)
=
fcV
(4w-i)
=
2i
and
KD?)
= ^{D^)
= 2(*-l),
and that dimA M _i dimD*
= =
(2t-l)(h+l) t(h + l)
= =
(2*-l)(2£+l) £{21-1).
So the central charges of Virasoro fields are given by
z(S]
}
1
/i\
= ^dimD™ = |
1
• l(2t - 1) =
2t-l,
(7.59b)
and so the central charge of the coset Virasoro fields is equal to 0. This means (a ti\
that Vu is finite-dimensional for each v, since the only unitarizable Virasoro representation of central 0 is the trivial representation, namely L(Aj) is a sum of finite numbers of irreducible D\ -modules of level 2.
7.3. Branching Functions
271
To obtain their decomposition more exactly, we compute the asymptotic behavior of characters. First we consider L(A0; A"2I-\)Let v\0 be the highest weight vector of L(A 0 ; ^2^-1 )• ^ * s vector, of course, is a singular vector as D\ -module, and so produces a submodule L(2A0, D\ '), since (Ao, 6%) = 28ifi. But there exists another singular vector, namely faV\0, because for
ei(/o^A0) = °
al1
*•
Since the weight O//O«A 0 is A0 — ot0, and (A0 - a0, d^) = 26itl
for all i
by (7.42b), this vector produces a submodule L(2Ai; D\ ). Thus one has L(A0, 4 f - i ) =» L(2A 0 , £»f ) )©L(2A 1 ,£>| 1 ) ).
(7.60)
We now compute the asymptotic behavior of characters given in Theorem 2.4-6.6). By (2.59) in Example 2.4-1 and (7.59a), the asymptotic behavior of level 1 characters of A^£1 is given by
<=%,„.»,,M.o) ~ ^ . . * < - • > . From the list of a(A) in [170], one has d(2A0; D™) = a(2Ai;D™)
= -^=,
(7.61)
and so
<*WiW T ' 0 ' 0 > ~ 272l- e ^' (2f " 1)
0 = 0 1)
' '
by (7.59b). Thus one sees that the asymptotics of the characters of both sides of (7.60) are just equal, and so there can appear no more representations in the right side of (7.60). Hence " D " is " =", and one obtains L(A0;4i-i)
= L(2A0;D^1))®L(2A1;D^)).
(7.62)
Applying a similar argument, one obtains the following: L(A i ; A^ld
= LiAo +
M-D^),
L(Aj; 4J>_i) - ^ ( A , ; ^ ) i(AM;41-i) = LiAt^+At-D^),
(2<j
7. Blanching
272 L(A,; 4 i - i )
Functions
L{2kl-i\D^))®L(2\i;D{p).
=
So the representations L(A j ; A$_x) (1 < j < I - 1 and £ + 1 < j < 2t - 1) remain irreducible when restricted to D\ and the representations L(AQ\ - A ^ - I ) and L(Af, ^ f - i ) are ^e sum °ftwo irreducible D\ -modules. Similar argument applied to Bt C A^ gives us the following: L(A 0 ; A™) L(A1;A^) L(Aj;A$) L(Af,A™)
= = = =
L(2A0;Bie1))®L(2A1;B{e1)), LiAo + A^B^), LfaB™) (2<j
N o t e 7.3.2. In particular in the case when n = 3 in Example 7.3.3, the inclusion sb(3) C sl(3) is nothing else but sl(2) c sl(3), namely A\' c A j , and one can choose the Chevalley generators {ei,/i,d^}i = o,i of A\ as follows : eo /o
= = =
[eo,ei] + [eo,e2], [/o,/i] + [/o,/ 2 ], 2(2a 0 v +c*v + a v ) ;
&\ A dv
ei+e2, A + /2, 2(ai'+«2/)-
.FVom this one sees that c = 4c and obtains the decomposition of fundamental J4 2 ' -modules, in a similar way as above described, as follows : L(A0;A^) LiAi-,4^)
= =
L(4A0;41))©i(4Ai;A(11)), L(2A0 + 2 A 1 ; 4 1 ) ) (. = 1,2).
Branching functions are studied in [119] with their modular properties and conformal properties. In the next section, we consider the branching functions of g C g © g, namely branching functions of the decomposition of tensor products. 7.4
Tensor Product Decomposition
We now consider an affine Lie algebra g and its irreducible highest weight modules L(X) and L{y) where A,/x G h*. The whole story works for all affine Lie algebras but, for the sake of simplicity of explanation, we assume that g is a non-twisted affine Lie algebra over a finite-dimensional simple Lie algebra g: g =
(C[M _ 1 ] ® fl) ® CAT 0 CeL
To give a precise description, we denote the action of g on L(X) and L(/i) by TT\ and ftp respectively. We put m :— (A|6) and m' := (n\8), and assume that m + hy =£ 0,
m' + /i v ^ 0
and
m + m! + hy ^ 0.
7.4. Tensor Product
273
Decomposition
Then, as is shown in Example 6.1.3, the Virasoro algebra Vir acts on L(\) and L(/x), namely, by (6.80a), we have Virasoro fields dimfl
L Z
^)
=
— ^ JT^AM^TTAKC*)):, m + nv f—• dimg
1
i=l
satisfing the following commutation relations: (z — w)z
z—w
(z — wp
{z — w)z
z —w
(7.63)
{z — w) 4
We consider the tensor product L{\) ® L(fi). Notice that there are two actions of the Virasoro algebra on this space L(\) <S> L(fi); the one is the tensor product as Virasoro modules Lx>"(z)
:= Lx{z) ® IdLM
+ IdLW
® W{z)
and the other is the Virasoro fields defined from the g-action on L(X)
1
LXm
W
:
= -^
L(/J):
:
,.hV E
(** ® *„)(««(*))(*> ® TT^XuV)) : .
t= l
Then, from (7.63) and (6.80a), one has {L^(z)L^H]
= M ^ 2 — 10
+
2 ^
+
H ^
( z — W)^
+
(2: — t o )
^)
d i n
4
(7.64a) {L^(z)L^(W)}
=
aL
">M
+^
H
z —w (z — w)z By (6.88b), these fields have the property [L%", (TA ® wM)(in ® X)] [ C
n
(7TA®7rM)(i ®X)]
= =
+ *• f ^ f " *
.(7.64b)
(z — io) 4
-n(7r A ® 7r /x )(t n+m ® X ) -n(7TA®7r M )(t"
+m
®A:) >
(7.65a) (7.65b)
and so [ L ^ - L £ ® M * A ® *,.)(*" ® * ) 1 = 0,
(7.66)
*
|
7. Branching Functions
274
for all X e g and m, n e Z. Namely the coset Virasoro field L(z) := L A '"(z) - LA®"(z) =
] T Lnz""-2
commutes with the action of the derived Lie algebra (C[t,t-1]<»g)®CK,
fl' = [0,0] =
and so we consider the decomposition of L(X) ® L(/i) as g'-module. A fl-module V is called \)-diagonolizable if each weight space is finite-dimensional and V is the direct sum of weight spaces. Notice that a h-diagonalizable flmodule V is irreducible if and only if it is irreducible as fl'-module, since the element d is a scalar operator on each weight space. For v e h* such that (y\8) — m + m', let
f V^
fa®
*„)(*> = 0
:= \v e L(X) ® L(ji) ;
(wx®nlt)(H)v
(XeQ+) }
= v(H)v
\
(7.67)
be the space of all singular vectors of weight v. Then, by (7.66), these spaces V*'* are Virasoro modules and L(X) ® L(fi) decomposes as follows
L(X) ® L{ji) = Y, *?'" ® L W .
( 7 - 68 )
as V*r © fl'-modules. N o t e 7.4.1. Jra tfie above explanation, we are assuming that L(X) ® ^(M) *S completely reducible. In the case when this assumption fails to hold, the definition ofV^'^ and the decomposition formula (7.68) should be replaced as follows: 3
U
v.y :=iveL(X)®m
: invariant subspace ofL(X) ® L(fi) such that
• w ^U^){X)V&u (iii) fa ® ^)(H)v
Cxefl+);
- v{H)v e (7
v
( /fGE-=0Catv) (7.69) and chL(A) • chi(/i) = ^ T c h V ^ • chL(i/),
(7.70)
namely the decomposition of characters. In (7.69), V*'1* is the space of primitive vectors of weight v.
7.4. Tensor Product
Decomposition
275
IfX,v^P+, then both L(X) and L(fi) are unitarizable (cf. Theorem 11.7 in [100]), so L(X) ® L{n) is completely reducible. It is proved in Corollary 4-1 in [122] that L(\)®L(n) is completely reducible if A € P+ and \i is an admissible weight. Function <**"(q) := Trv^(q-d)
(7.71)
are called the branching functions of L(A) ® L(fi). By this definition, the decomposition (7.68) gives the following decomposition of characters: = J2CZ@fl(
chLW(T,Z,t)-chL{ll)(T,Z,t)
(7.72)
v
where q — e2™T. Defining the normalized branching function by 6*®"(r)
: =
(q = e2"iT),
q'^-o-c^iq)
(7.73)
the formula (7.72) is rewritten as follows: ch'L{X)(T,z,t)
• ch'L(li)(T,z,t)
= Y,bt°»(T)ch'L(v)(T,z,t).
(7.74)
In the case when A and fi are integrable or admissible weights, the left side of (7.74) is a modular function and so is the right side. If one of A and fi is integrable and the other is principal admissible, then v is also principal admissible, so all of ch^,A, and ch^, •. and ch^- ^ are modular functions, where we make use of the terminology "principal admissible" including integrable. Since the modular transformation of these functions is known, one can easily compute the modular transformation of the branching functions 6^®/1(r). Detail explanation and calculation of the transformation matrices, when A , / J £ P+, are given in [101] and [119] and [170], and a similar argument works when one of A and n is a principal admissible weight. So we give here only the resulting formula as follows: Theorem 7.4.1. Let g be a non-twisted affine Lie algebra, and X be a dominant integrable form of level m and fj, be a principal admissible weight of level m'. Then 1) TO +TO'is a principal admissible number,
2)
b^(~l)
= E
E
E
«(A,A')«(M,M')^X)^>'(r)
where a(X, A'), • • • are transformation matrices given by (2.54) and (3.78).
276
7. Branching Functions
We now consider the structure of V*'1* as a Virasoro module. From (7.64a), (7.64b) and (7.65a), the coset Virasoro fields L(z) satisfies [L{z)L(w)\
=
+ z—w
r^ + —: r^- , (z — wy (z —to)4
(7.75)
where , ,. c(m, m):=[ v '
/ m m! — — v H \m + h m' + hv
m + ml \ ,. _ — dimn . m + m' + h^J
._ „„.
v(7.76) ;
In particular when A , / J G P+, L(X) and L(fj.) are unitarizable and so are all V^'^'s. If, in addition, c(m,m') is less than 1, then each V^ should be the finite sum of Virasoro discrete series. Under this situation, with the help of the modular properties in Theorem 7.4.1 if necessary, one can know the branching functions explicitly. For details, readers are expected to refer to [119] and Chapter 6 of [170]. In the rests of this section, we give an explicit formula of the branching functions of the tensor product representation L(A) ® L(/J,) in terms of string functions of L(A), where A is an integrable weight and /x is an integrable or a principal admissible weight. Since an integrable weight is regarded as an admissible weight of integral level, we consider the case when fi is a principal admissible weight. Let m = - be a principal admissible number, and consider PZ
= {2/-(A° - (« - l)(m + ftv)A0) ; A0 e p«<™+" v >-" v }>
where y = tpy (@ 6 M, y G W) satisfies the condition (3.24). Let W(«,,,) := yW{u)y~x
= {ra,ae
y(S ( u ) ))
denote the Weyl group of y(S(u)) as in (3.26). In this section, we write an element w G W(UtV) always w
= yt^&y
1
— ytua.wy
1
,
without making use of the notation to since, in this decomposition, w is not the W-component of an element to with respect to the decomposition W = t\i * W but is the ^-component of y~1wy G W(uy Lemma 7.4.2. Let w G W(UiV) and 7 G h*; and write w as follows: w = yt^ivy'1 Then
= t/i u a tut/ _ 1 wyt1y~1w~1
(a G M and w G W). =
ty^.
7.4. Tensor Product
277
Decomposition
Proof. Writing y as y — tpy (f3 e M, y € W), one has = ytuaw = tpytuaw = tp • ytuay~l — tptugayw = tp+Uyayw.
wy
• yiv
So one has wyt^y^w'1
wytyiwy)'1
= =
t/S+uyaywt-yiywyH-p-uya
—
tp-\-Uyaty.tf)jt
— p^Uya
=
tyW*fl
proving the lemma.
•
Theorem 7.4.3. Let g be an arbitrary affine algebra, m € Z>o and ml = ^ be a principal admissible number. Then for an integrable weight A € P™ and a principal admissible weight + hv)A0)
PL = y.(Li°-(u-l)(m'
e P£y,
(7.77)
the following formula holds: ch^T.Z.O.ch^T.M)
-
J2
b^{T)-cb!„{T,Z,t),
epm+m'
s.t. i/=A+/i modQ
w/iere &A®"(T)
.
^
=
£(„,),
X
(m' + h V )(m+m'+h V ) I t»(.>°+p) ^ |m+m'+fcv
|.°+p f ^ + ^ |
S(^("0+p)-(^0+A')-(«-l)TOAo)('r)-
(7-78)
Proof. The normalized character ch^ is given by Theorem 3.3.1.2) as follows: ch'
A
=
"+"
^
A
'
Zip
where Afi+p
=
e
-2:(m>)*
^
e(u>)e ,o( ' 1+ ' ,) .
(7.79)
The normalized character chA of an integrable representation -L(A) is given in terms of string functions as follows: chA =
£ £SA+Q+C« mod(mM+C«)
Qs4=
£
Ee-^'eMO^.
f6A+Q+C6 7 6 M mod(mM+C5)
(7.80)
7. Branching Functions
278
To prove the theorem, it suffices to show that ch^ • A^p of the functions b^(r) denned by (7.78) as follows:
ch^-A^ =
J2
is written in terms
b^(r)Au+p.
(7.81)
s.t. v=K-\-ii modQ
We calculate the left side of (7.81) as follows. By (7.79) and (7.80), one has
mod(mM+C6)
(7.82) Writing w G W(u^ = yW^y-1 as w = yta wy-1 (a G M and w G W) and replacing 7 by 2/107, the last term in the above equality (7.82) is rewritten as follows: £ e - ^ V ^ > c £ = £ e-££*eW«c£ . Using Lemma 7.4.2, this becomes
7€M
Putting £' := y _ 1 u; _ 1 £, this is equal to
7GM
76M
since string functions are V7-invariant. So (7.82) is rewritten as follows: ^ A'
M+Z* — ^
\»°+p\l s 2(m' + hV)«
» 6 f ( « , , ) f'e(«ij/)_1A+Q+C«7€M mod(mM+C«) fi-2(m'+hV)*
:
5;
^
E^--^^1)^.
mod(roM+C«)
(7.83) We note that
MO = «, + m7-((f|7) + ! ^ ) * ,
7.4. Tensor Product
Decomposition
279
since (£'\8) = m. So writing £ in place of £', the above formula (7.83) is rewritten as follows: „0j.„|2
cK-\+P = e~£Mn< E
E
E
mod(roM+C«)
x e(w)ew^+p+^+m™e-^-',e —
2 ( m ' + hV)0
g
X
V X
^ / c;j / « X
mod(mM+C«)
x
e(«,)e»<"
+
'^'«
+m
T))e-J^?lificA€.
(7.84)
Here one notices that p + p + y(£ + rwy) is an integral form with respect to the "simple" coroot system y(S(u)) and that the summation as to £ and 7 can be taken over (£, 7) such that p + p + y(£ + rwy) is regular with respect to W^y). Then there exist a unique a G W(UiV) — 2/W(„)j/_1 and a unique v&PZ{m+m'+hV)-hV(y(Siu)))
(7.85)
which is a dominant integral form with respect to the simple coroot system y(S{u)), satisfying p + p + y(£ + rwy) = a{y + p{u'y)), where p("'») := y(p^)
is the Weyl vector of y(S^),
(p(u'y\
y(ij))
=
1
U=
(7.86)
namely o,-•-,£)•
So the condition (7.85) is equivalent to (P + P^Kvili))
e
N
O' = 0, • • - , , € ) ,
t
E«*>K(7i)>
= u{m + m' +
hy)-hv.
3=0
Prom this, one has (y-^
+ pM,^)
e N
Q'= 0, • " , , * ) •
So v' := y-xv e and one can write 1/ as follows:
P?m+m'+hV)-h\s{u)).
v' = E ™ ^ ' t=0
(7-87)
280
7. Branching Functions
where rrii € Z>o satisfying t
J2atmi
= u(m + m' + hv)-hv.
(7.88)
i=0
Define v° and v by i
v° := Y,miAi
e
Pl{m+m'+hV)-hV
(7.89a)
i=0
^+ p
:=
u + pl">v> = y(i/ + PM),
(7.89b)
and rewrite the formula (7.87), using (3.16) and (7.88) and (7.89a), as follows:
i=0
-
U
\
/
(u(m + m' + /i v ) - hv) A0
y ^ WiAi H i=0
=
z/) + ( l - u ) ( m + m' + / i v ) A 0 -
— V A
0
.
So one has v' + p(u)
= v° + p-(u-\){m
+ m' +
ftv)A0,
(7.90)
since
p(«) =
P + ^ V A
u by (3.17). Prom (7.89b) and (7.90), one has v + p = y(v° + p-(u-l)(m
0
+ m' + hv)A0),
(7.91)
hence v is a principal admissible weight of level m + m', namely v e P^1^"1 • Then by (7.86), (7.89b) and (7.91), one has V + P + yit + mj)
a(v + P(u'y)),
= (•.• (7.86))
= ^ ^
+ P)-
(V (7.89b))
Rewriting (7.92) as y£
= =
a{v
+ p)-(/j, + p)- ym-y t-y-y (a(v + p) — (p, + p))
one has • v£
~
C
^("+P)-(/'+P)'
mod C6,
( 7 - 92 )
7.4. Tensor Product
Decomposition
281
so (7.84) is rewritten as follows:
ch^JW = e~&S
E
E
w W
CT
£ (u,y)
6W(u,v)
E ^gpm+m'
v=A+/J
£ "^*
modQ
J ] e(a) W ^ c ^ - M - ^ ) . (7.93)
u,y
v=A+n modQ
where 2(m + /i v )
2m
2(m + m' + / i v ) '
^ '
;
To compute a, we decomposeCTe W^u.y) = J/W(u)J/-1 as follows: o- = yt^Zy'1
ytuaZy'1
=
where a £ M and
=
ay (w° + p - (u - l)(m + m' +
(V (7.91)) N - ^ ytuafi
V
= ytua(a(v°
+ p)-(u-l)(m
hv)A0) '
+ m' + hy)Ao).
(7.95)
To compute the right side, we note the following: tua (o{y° + p) - (u - l)(m + m' + /i v )A 0 ) ff(iy° + p) - (u - l)(m + m! + /i v )A 0 + (m + m' + hv)ua ,21^,12
-l^Y^(m
+ m' + hv)+u(a(iy°
+
p)\a)\s
a{v° + p) + (m + m' + hw)ua - (u - l)(m + m' + /i v )A 0 - ubS (7.96) since =
(fr(i/° + p) - (u - l)(m + m! + h")A0 \S) u(m + m'+ hs/)-(u-l)(m + m'+ hy) = m + m' + hv,
where we put \a\2 b := >-p-u(m + m' + hy) + (a{u° + p)\a).
7. Branching Functions
282 Notice also that p + p = y(p° + p-(u-l)(m'
+ hs/)A0)
(7.97)
by (7.77). Then by (7.95), (7.96) and (7.97), one has o(v + p)-(n + p) = 2/(#("° + p) + u(m + m' + /i v )a - {u - l)(m + m' + hv)A0) - ub6 -y(p° + p-(u-l)(m' + hv)A0) 1/0 — v(^( + P)~ (M° + P)+ u(m + m' + hy)a - (u - l)mA 0 ) - ubS. So the square length of this element is given by \a{v + p)-(p + p)\2 = \a(v° + p) - (p° + p) + u(m + m' + hv)a -{ul)mA 0 - ubS\2 = \a(j/° + p) - (p° + p)+ u(m + m' + h^)a - (u - l)mA 0 | 2 - 2ubm = \a(i/° + p)-(u° + p)+u(m + m'+ hy)af-2umb, (7.98) since (a(i/° + p)- (p° + p)+ u(m + m' + hv)a + (1 - u)mA 0 \6) = u(m + m' + /i v ) - u(m' + hv) + 0 + (1 - u)m = m. Next we compute tQCT(^° + p) : taa{y° + p)
=
a(f° + p) + u(m + m' + hv)a - | i^L • u(m + m' + hv) + {6(y° + p)\a) \ 6
=
a(i/° +p) + u(m + m' + hv)a - b6.
So we have \taa(v0 + p)-(p° + p)\2 = \a{v° + p) + u(m + m' + hv)a - (p.0 + p) - b6\2 = |tf (i/° + p) + u(m + m' + hv)a - {pa + p) | 2 - 2umb, since (a(u° + p) + u(m + m' + hv)a - (p° + p)\6) = u(m + m' + /i v ) + 0 — u(m' + /i v ) =
um.
Then, from (7.98) and (7.99), one has \a{v + p)-{p
+ p)\2
=
\ta6{v° + p) - (M° + p)\2
(7.99)
7.4. Tensor Product
Decomposition
283
Plugging this into (7.94), one has \taa(i,° + p) - (p° + p)\2 2m
°
(m' + hy)(m + m' + hy) 2m
\p° + p\2 2(m' + hv) tao{i/> + p) m + m' + /i v
| M ° + /9|2 2(m + m' + hv) pP + p (7.100) m' + h>
Now the formula (7.93), combined with (7.94) and (7.100), gives us cn
A
•
A
H+P 2
m+m' I/ cp"»+"»'
+ h^
~ TTL' + h v
^giyaeM
f=A+/i modQ X
S(*C*(" 0 +P)-(/XO+P)-(«-1)TOAO) J'
proving the theorem. • Corollary 7.4.4. Lei fl(A) 6e a symmetric or twisted affine algebra of type Xjy , and m' — ^ be a principal admissible number. We define positive integers p and p' by p' := u(m' + hv)
p := p' + u = u(m' + 1 + hw),
and
namely m'
+ hy = -^—
and l + m' + hv = —^—.
p-tf
(7.101)
v-tf
Let p = y(/x0 + p - ( U - l ) ( m ' + / i v ) A o ) - p
€
P$y
and V = y(„° + p-(u-l)(l + m' + hS/)A0)-p €P™^ + 1 be principal admissible weights, and A be a dominant integral form of level 1 uniquely determined by the condition A-y(v°-p0-{u-l)Ao) e Q. (7.102) Then
where GA{T) is a modular function, in view of (2.10), defined by
SAT) := {"%,
,«=,
1 V(T) V\TT) r~1
''''I' if
r>2.
P.103)
284
7. Branching Functions
Proof. Notice that pP and v° are dominant integral forms of level u(m' + hv) - hv = p' - hv
and
u(m' + 1 + /i v ) - h" = p-
hy
respectively. Then, by Theorem 7.4.3, one has
bm„{T)
^2£{w)q
= X
C
(m' + h v ) ( l + m ' + h v ) 2
tK(^Q+p) l+m' + hv
M°+P m'+hv
s/(W(^+p)-(/xO+p)-(«-l)Ao)(r) pp' I ix(v0+p)
2 ^ e(w)<2> wew
I
p
/x° + p I
"I
cy{w{l/0+p)_{li0+p)_(u_1)Ao)(T). (7.104)
We now compute the string function. Since = =
w(v° + p)- (p° + p) - (u - 1)A0 (i/° + p) - (p° + p)-(u1)A0 mod Q i/° - /i° - (u - 1)A0
for any w G W, this together with (7.102) gives us
A = y(w(v0 + p)-(p0
+
p)-(u-l)A0),
and so Cy(w(v°+p)-(v,°+p)-(u-l)K0)(T)
=
C
A(T) =
~ , .
for any w G M^. Now (7.104) and (7.105) prove the corollary.
(7.105)
•
Example 7.4.1. We compute the formula (7.78) in the case of A\ ' = sl(2, C). We let A :— A0 and y := t_k and p := (t_kai).((m-n)A0+nA1
=
Ara.fc,n
as is defined by (3.37). Then v in (7.78) is written as v := ( i _ i Q l ) . ( ( m + l - n ' ) A 0 + n'Ai =
ATO+i;fci„-,
where n' G Z SMC/I tta£ 0 < n' < u(m + 3) — 2 and n' — n G 2Z fcj/ the condition A + p — v £ Q. In this case, p° and i/° are given by M° == K(m+2)-2;n
•= («(«* + 2) - fl - 2)A0 + 12Ai ,
7.4. Tensor Product v°
Decomposition
:= A„ (m+3) _2; n <
285
:= (u(m + 3) - n ' - 2)A0 + n'A!.
Since the string function is -rpr, the formula (7.78) gives ,
1 —r^- ^ e(iw)x rj(r) ^ -' w€W U
"(A"(™+3);T>' + I )
u(m+2)n(Trt+3) - y I - / -
A
tt(m+2);n+l
"( A ti(m+3);T»' + l J ti(m+3)
A
n(m+2);T>+1 u(m+2)
' w€W
(7.106) where x = e2mT. Putting p := u(m + 3) and q := u(m + 2) and comparing this formula with Proposition 7.1.4, one obtains C J "
= xiTiUiW
fom = n'mod2,
(7.107)
which shows that the branching functions are irreducible characters of minimal series representations of the Virasoro algebra. In this example, we have proved the following: Proposition 7.4.5. Let A — A\' and m = - be a principal admissible number. Then, forfc,n e Z such that 0
L(A0)®L(Am;fe,n) =
Yl
i(Am+i:fc,n') ® K!+T2<+I .
0
(7.108a) L(A1) ® L(A m;fc , n )
=
^
i ( A m + l i f c , n ' ) ® ^ . t 2 ' + l,
0
(7.108b) where p := u(m + 3) and p' := u(m + 2). Since integrable weights are principal admissible weights with u = 1, this proposition gives the following: Corollary 7.4.6. Let A = A\' and m e Z> 0 . Then, for j e Z such that 0 < j < m, the decomposition of L(Ai) ® L(A m ; j ) (i — 0,1) is given as follows: L(A 0 )®L(A m ; j )
=
£
£(A„, +1;fc ) ® ^
f c + 1
,
(7.109a)
0
LiAJVLiA^j)
=
Yl 0
i(Am+i ifc ) ® ^+i > f c + i • (7.109b)
7. Branching Functions
286
In concluding this chapter, we remark that the formula (7.19) shows that the Virasoro characters of minimal series representations are just the functions ¥>A,/*("?") for the affine Lie algebra A\ ' = sl(2, C) introduced by (4.26b) in §4.2. Then it is of course possible to compute their modular transformations by Theorem 4.2.5. Namely one can deduce Theorem 7.1.5.3) from Theorem 4.2.5. Also fusion coefficients of Virasoro algebra are obtained from those of integrable A\ -modules and Theorem 4.3.13. Since \J\ = 2 for A\ ', the condition on p,p' in Theorem 4.3.13 is gcd(p, p') = 1
and
p' = odd,
which does not put any further restriction than in Proposition 7.1.4 by exchanging p and p1 if necessary. The fusion coefficients for integrable sl(2, C)-modules are easily calculated by using Lemma 5.4 in Chapter 5 of [170] and the Clebsch-Gordan's rule for the tensor product decomposition of finite-dimensional sl(2, C)-modules. The result is well-known as follows: 1 N.A m ; i , A m ; j , A m ; f c 0
if i + j + k e 2Z, \j - k\ < i < j + k and i + j + k < 2m, otherwise.
(7.110)
Then, from (7.110) and Theorem 4.3.13, one obtains the following: Theorem 7.4.7. Let p,q G N>2 be mutually prime positive integers and (j + 1,fc+1), ( j ' + l,A;' + l ) , (j" + l,k" + l) be representatives of £,£',£" € &(*>«)/ ~ respectively. Then the fusion coefficient N(,,(,<,(," of the Virasoro minimal series is given as follows: 1
N(£',t"
— «
0
if one can choose representatives of £,£',£" satisfying j + f+j", k + k' + k" G2Z, j+ j'+j"<2p, k + k' + k" < 2q, \j'-j"\<j
Actually this theorem follows from Theorem 4.3.13 when p e N>2 and q G N0dd,>2 axe mutually prime positive integers and then, exchanging p and q if necessary, holds for all co-prime integers p and q greater than 1.
Chapter 8
W-algebra In this chapter, we describe an approach to the theory of W-algebra associated to an affine Lie algebra via quantized Drinfeld-Sokolov reduction. This theory still contains lots of problems to be investigated, and the materials in this chapter are now in progress, including the vanishing and non-vanishing of the cohomology spaces. So the exposition in this chapter is just a beginning and an introduction to the theory of W-algebras. 8.1
Free Fermionic Fields ij)(z) and rp*(z) Let us consider charged fermionic fields
n€Z
neZ
of conformal weights 1 and 0 respectively, with anti-commutation relations
These fields are studied in Example 6.1.6 of §6.1, and we follow notations from Example 6.1.6. The space F is naturally identified with the exterior algebra A^
:= / \ (>_!,V-2, ••• , ^ o - V ' - i , •••)
over {ipj}j
287
8. W-algebra
288
with V'-i! A • • • A C
i m
A V-i! A • • • A V-i„
in A ¥ , which we shall simply write as
r-il---r-imi>-n--^-in,
(8.i)
where m,n > 0 and 0 < ii < ••• < i m and 1 < j \ < ••• < j n . When m — n = 0, the element (8.1) is understood to be the element 1 in A ^ . Then the action of the Clifford algebra Cliff {tp, ip*) on this wedge space is given by tp-j, ip*_k i—• V'fe, il>j '—>
^ - j A , ^-fcA a§r-k, a$—
: •
left multiplication contraction
for ji > 0 and A; > 0. The space A~2~ is called semi-infinite wedge space in literatures. Recall that the creation and annihilation operators are given in Example 6.1.6 as follows: creation operators annihilation operators
: ipn (n < 0) : ipn (n > 0)
and and
ip^ (n < 0) Vn (n > 0)
,
and that the charge and the energy of the element (8.1) are denned to be charge energy
:= :=
n — m, «H h im + j i -\
jn.
The commutation relations of ip(z) and ip*{w) are given by (6.128a) as follows: Mz)p(w)]
= [p(z)rl>{w)\ = j
^
.
(8.2)
For a complex number a, we consider a field Ta(z) := a : dip(z)ip*(z) : +(1 - a) : dip*(z)tp{z) : . Then the operator product of this field is given by (6.135b) as follows:
z—w
(z — w)z
(z — iy) 4
Note also that the following formulas are obtained from (6.136a) and (6.136b): [TaxiP] [Taxip*}
= =
dip+(l-a)\ij>, dip*+a\ip*.
(8.4a) (8.4b)
8.1. Free Fermionic Fields tp(z) and tp*(z)
289
We now compute each component of this field:
T°(z)=£T°z-"- 2 . Since
and iez
are fields of conformal weights 2 and 1 respectively, the normally ordered products : dtp{z)ip*(z) : and : dtp*{z)tp{z) : are obtained from (6.126) as follows:
:dil,{z)P(z): = - W £ t f
+ 1
> : ^«-; : } z " n " 2 -1
nGZ ' ' j e Z
E {Ec? +!): ^ - ^ : V- n - 2 J
nez *• j e z and
:¥WW: = -£{£'' : W.-i : K B " : nGZ '•j'GZ
"* 2
"-2.
J
nez *-iez So one has T
n
=
=
J2(aU+1) + (I-"•)(!-"))•• Vn-jVj-iez E^' + a + ( a - 1 ) n ) : ' , / ' » - i ^ : ' iez
and in particular (label=eqn:(10.1.4)) T
s
= E(j'+a):^-i^: iez i>o
i>o
Prom this, the action of Tft on the space F is computed as follows: To°V-„|0) =
(n-a)t/._„C-V-n|0) i-V-„V£
=
(n-a)^_„|0)
(8.6a)
8. W-algebra
290 and TSP-JO)
=
(m + a)r-m^m-r-m\0)
= (m + aW-JO),
(8.6b)
where n > 0 and m > 0. So one has
To^_ni---V-„rCmi---CroJ0> =
{(ni — a) H
(rar — a) + (mx + a) H
(m s + a)}
x K - ^ ^ - ^ j 0 >
(8-7)
We read this formula as follows. Let h* be a 3-dimensional complex vector space equipped with a non-degenerate bilinear form ( | ) and a basis {Ao, a, 6} satisfying (A0|<5) = 1 and (a\a) = 2 and all other inner product being equal to 0. Given a complex number a, we aa put x :— — G rj so that we have (a\x) = a
and
(A 0 |x) = (S\x) = 0.
Then the numbers n — a and m + a in the above formulas (8.6a) and (8.6b) are written as follows: n — a — (n6 — a\Ao + x),
m + a= (m8 + a\Ao + x).
So the formulas (8.6a) and (8.6b) give qTSV>_n|0) = q(nS-a\Ko+x)il)_n\0)
= e -(»*-«l-2^(Ao+*))^_ n | 0 )
(8
8a)
and
for n > 0 and m > 0, where q = e2niT. So the alternating sum of the trace of the operator qT° with respect to the charge decomposition
jez is given by the following formula:
jez _
T T (l _ n=l
v
' '
e-(n«-a|-2wiT(Ao+x))'\
oo TT / j _ m=0
e-(m«+a|-2iriT(A0+x))\
(g g a )
8.2. Free Fermionic Fields <j>(z) and <)>*{z)
291
One may write this formula as
£(-irir(>o| jGZ
^
(fj(l-e—+-)(l-e-<-1>'-)}(T>-rx>0)
\ = A
/
3
L
n =
i
>
(8.9b) in terms of the coordinate (r,z,t)
~2m(-TA0
+ z + tS)
(r G C+, z G f)*, i G C).
We note that the formula (8.5) is written also in the following form: T
o = " 5 > ' + «) = ^ j C , : = £ ( J " «) : *-& jez
8.2
••
(8-10)
jez
Free Fermionic Fields (j)(z) and ^*(z) Another useful fermionic fields for our purpose are
<M*) = ]•>„*-" and ^(z) = 5 3 ^ - » - 1 , nez
nez
of conformal weights 0 and 1 respectively, with anti-commutation relations (t>j
=
which are quite similar with those of ipj and ip%. In this case, we define the creation and annihilation operator as follows: creation operators annihilation operators
: <j)n (n < 0) :
and and
<j>^ (n < 0) 0„ (n > 0)
.
Let Cliff {(f), <j)*) be the Clifford algebra over {
over {*j}j<0. Elements in A'^r are linear combinations of 4>-n A • • • A
A
• • • A 0-*..
which we shall simply write as tfV^-i^-ii
"•*-*..
(8-n)
8. W-algebra
292
where m,n > 0 and I < i\ < ••• < im and 0 < j \ < • • • < j n . When m = n = 0, the element (8.11) is understood to be the element 1 in A'~2~. The charge and the energy of the element (8.11) are defined quite similarly with the case of tp and ip*: charge energy
:= :=
n — m, h +•••+im+ji-\
jn,
and the commutation relations of (j){z) and
[*(w)] = [F(z)4>(w)] - j ^ -
(8-12)
One may look at these fields under the correspondence : (j>{z) <—• i}>*(z)
and
<j>*(z) <—> ip(z)
except that the charge is converted. So just corresponding to Ta(z), we consider a field T'b(z) := b : d<jf(z)4(z) : +(1 - b) : d<j>{z)cj>* (z) : for a complex number b. Then the operator product of this field is given by the same formula with (6.135b) and (8.3): r>(2)T>)1 =
ar>, z—w
+
2TM _ e ^ a (z — w)1
+i
(z — wp
Notice that (6.136b) and (6.136a) in Example 6.1.6 of §6.1 give the following formulas in this case: [T'bx
d
(8.14a)
d(j>* + (1 - b)\cj>* .
(8.14b)
Translating formulas in §8.1 into <j> and <j>*, one has T
n = 5 > - + &+(&-l)n):<^_^*:,
and iez
Y^U + bW-jfi + Y,(J - W-jhj>0
j>0
(8-15)
8.2. Free Fermionic Fields <j>(z) and 4>*(z)
293
And the action of Tg6 on the space F' is ^V*-„|0> =
(n-6)^_n|0)
(8.16a)
(m + b)4>-m\0),
(8.16b)
and T^-m|0)
=
where n > 0 and m > 0. And (label=eqn:(10.2.6)) =
{(mi + b) + ---(ms + b) + (nl-b) + --- (n r - 6)} x 0 _ m i • • • <j>-mA-ni • • •
(8-17)
We rewrite these formulas in terms of a 3-dimensional complex vector space h* with a non-degenerate bilinear form ( | ) and a basis {Ao, a, 6} as in §8.1. _ ba , Let j/ := — 6 rj so that we have (a|y) = 6
and
(A0|y) = (6\y) = 0.
Then the numbers m + b and n — b in the above formulas (8.16a) and (8.16b) are written as follows: m + b = (m8 + a\A0 + y),
n — b= (n6 — a\A0 + y).
So formulas (8.16a) and (8.16b) give T 9
«V-m|0> = g< m * +0 l Ao+ *ty_ TO |0) = e -< m 5 + a l- a , r i 7 " ( A o + » ) ) ^_ m |0)
(8.18a)
and g ^ V - j O ) = g( n *- a l A o + «V-„|0) = e - ( " * - « l - 2 ^ ( A o + « ) ) ^ n | o ) ;
( 8 .i8b)
for m > 0 and n > 0. So the alternating sum of the trace of the operator qT° with respect to the charge decomposition iez is given by the following formula: x
jez _
oo TT A n=l / oo
=
_
7
e-(n«-a|-27rtT(Ao+»))
I ] i1 -
^ n=l
'
,
\
oo TT
(]_ _
e-(m6+a\-2*iT(h.0+v))\
m=0 N
e_
"* +a ) {X - e-(n~1)s-a)
\(T, -TV, 0). J
(8.19)
8. W-algebra
294 8.3
Ghost Field Associated to a Simple Lie Algebra
Let g be an affine Lie algebra of non-twisted type over a finite dimensional simple Lie algebra g of rank I, with the standard bilinear form ( | ). Let f) be the Cartan subalgebra of g, and A + the positive root system. Let 8 denote the primitive imaginary root, and A + the positive root system of g as usual. Then the set A ^ of all positive real roots of g is given by A!j_e — {n6 + ct; a G A+, n G Z> 0 } U {n6 - a ; a G A + , n G N } . For each a G A + , we consider fermionic fields
and
4T{z)
ra(z)
-i-i
TP(Z)
= ^ p -
1
,
j Z
,
iez
iez
J6Z
satisfying anti-commutation relations ISP
*&
t*Pjfi'
-
and all others being equal to 0, and with creation and annihilation operators defined as follows: creation n<0 n<0
annihilation n>0 n>0
creation n<0 n<0
annihilation n>0 n>0
The conformal weights of fields ipa(z) and
w and all other products are 0, for a, (3 G A + . Let A ^ + ) (resp. A ^ ( - ) ) be the semi-infinite wedge space generated by all elements ip" and ipja (resp. >" and ^ a ) over the vacuum |0) = 1. Fix an element x e f), and put (a|x)
(8.20)
8.3. Ghost Field
295
for each a £ A + . The energy-momentum tensors
Tx+(z)
:= =
J ] K = d^a{zWa{z) ^T^+2-"-2 nez
: +(1 - aa) : dr°{zW{z)
on A^(+)
:} (8.21)
and
T*-(z) := Y, K : ^"WW
:
+(1 - O : 3*a W ( z ) :}
a€A+
=
J^T*"*-""2
on A ^ ( - )
(8.22)
n€Z
are called the ghost fields in some of physical literatures. By (8.3) and (8.13), one has the following: (6a 2 - 6aa + 1)
£
lT*±(z)T*±H] = ^ ^ M
+
z—w
^ M
*GA _ ff*i_
{8.23)
+
(z — w)2
(z — w)4
and a € A + J€Z
= E {Ew +a -)^->^ + E(j-°-^vr} (8-24) and
T
j>0
The alternating sums of the trace of qT» composition A ¥<+> =
0Af(+)
and
J
with respect to the charge de-
A*<-> = ® A? (_)
jez
jez
are given by (8.9a) and (8.19) as follows:
£(-iyTr(>0-| j€Z
^
A A
'
'
=
JJ ^_ e -H- 2 ^(A 0 + ,))\ aGA"
296
8. W-algebra
=
Y[ (l - 9HA°+*)) .
(8.26a)
Or one may write this formula in the following form :
^(-1)^(^1
\= Jl (l- e -«)(r,-rz,0)
in terms of the cordinate (r, z, t) := 27ri(—TAQ + z + td). To compute the central charge of the ghost field Tx±(z), number
(8.26b)
we calculate the
J2 (6a« - 6a« + 1). aGA+
First, by the Lemma 1.3.5, one has
Y^ o-l = 5Z (a\x)(<*\x) = hv(x\x). aeA+
a£A+
Also one has a£A+
a6A+
and so J2
(Gat - 6aa + 1)
=
6hv(x\x)
- 12(p\x) + |A+|
6hv(x\x)
- 12{p\x) + i(dimfl - €).
o£A+
Using the strange formula i m
2/i v
- 24 idi"*
this is rewritten as follows: £
(6a* - 6aa + 1)
=
6/i v (z|z) - 12(/>|i) + ^ ( p | p ) - ^
aeA+
= ±\Vx-tf-L.
(8.27)
We notice the following formulas which easily follow from (8.4a), (8.4b), (8.14a) and (8.14b): [Tx+Xtpa] [Tx+Xip*a]
= =
di>a + (1 - aa)\ipa dxp*a + \aai>*a
,
(8.28a) (8.28b)
8.4. BRST
Complex
297
and [Tx-X(/>a} [Tx~X(j>*a} 8.4
= =
d
(8.29a) (8.29b)
B R S T Complex
Let g be an affine Lie algebra of non-twisted type as in the previous section, and V be a highest weight {(-module. Using the charge decomposition of the semi-infinite wedge space
introduced in the previous section, we consider the following spaces for each jeZ: C ( ± ) (fl)j := il(fl) ® hf(±),
C ( ± ) ( V ) ; := V ® A f ( ± )
and put C ( ± ) (fl)
:=
H(fl)®A¥(±)
=
0 ^ ' ^
,
JGZ
^'(v)
:=
V®A*(±)
= 0c(±)(y)j . iez
For each a € A + , we choose and fix a non-zero element ea in the root space ga, and let c^ „, for a, /?,7 € A + be the structure constants defined by
We define the odd field d s t ± (z) acting on the spaces C^{g) as follows:
and
C^(V)
n£Z
2 aGA+
a,^,7GA +
(8.30)
and
n£Z
298
8. W-algebra
(8.31) Notice that dat+{z) and so, writing dst+(z)
(resp. d3t~{z)) is a field of conformal weight 1 (resp. 2)
= J2 d^+z-"-1
d3t~(z)
and
= £ d^~ vz~- n - 2
nGZ
nEZ
one has jat+
_
"n
— «(„)
J»*+
anr
l
a n Q
jet-
—
«n
-
jst-
"(n+1)-
For each n G Z, operators d^4± are written as follows:
d +
- = E E^-^r - \ E_ E <*: aeA + J'6Z
rt-j-^r^
••
a ,/3, 7 eA + i.feez
and
#- = E B o - ^ r -1 E a£A+ i ^ z
a,/3,yeA+
E 'I? •• K-j-^rtf
J.fcGZ
•• •
Since, for each n e Z , the operator d^ decreases the charge of each particle by 1, they are linear maps from C^(g)j to C^HfOj-i, and from C^\V)j to
c<±>(vviFor their actual calculation, it is convenient to put datW+(z)
:=
J2 ea(z)TJ)*a{z)
(8.32a)
o£A+ dst(2)+{z) :=
1
clp:r(z)ra(z)rP(z):
£
(8.32b)
a,/3,7GA +
and d^W-fc)
:=
d°W-(z)
:=
Y
ea(z)(j>*a(z)
(8.33a)
£ _ c^:^(z)^(z)^(z):, a,/3,7eA +
so that we have and
d3t+(z) = dst^+(z) - d*n+{z) dst-(z)
= d3t^-{z)
-
d3t(2)~(z).
(8.33b)
8.4. BEST
Complex
299
N o t e 8.4.1. In the second term of the above formulas (8.30) and (8.31), the normal products are actually the "usual" product since the structure constant C I p vanishes if ^ is equal to a or j3. So, for example, one may write (8.30) as follows:
dat+(z) = £ ea(z)ra(z) - ^ £ _ ci0r(z)ra(zwp(z) a£A+
(8.34)
a,,8,7£A +
without taking care of the normal ordering. By an easy but somehow tiresome calculation, one sees the following: [dat(l)+A^(l)+]
=
^
CI^W? ,
[dst(l)+xdst(2)+]
=
[dt(2)
[dst(2)+xdst(2)+]
=
0>
[d rt(2)- Ad rt(2)-]
=
Q)
a,/3, 7 6A + + Arfrt(l)+]
=
1
^
cljfrVpl',
a,/3,7eA +
and
which give the following : Lemma 8.4.1.
[d a ' + (z)d s t + (w)] =
[d s <-(z)d s '-(u;)] =
0.
This Lemma implies, with Proposition 6.1.1.2), that
for all j , fceZ. Since d a
's axe odd operators, this means that (df±)2
for all j € Z. So, in particular, (d^)2 C (±) (fl) = 0 C ( ± ) ( f l ) i
= 0
(8.35)
= 0, and and
C^(V)
= ®C(
1 are cochain complexes with the coboundary operators dj•«t±
±
^)
j
8. W-algebra
300
We deform this "standard" coboundary operator dat±(z) by adding fields corresponding to simple roots. Namely we consider the field p^(z) defined by
p+{z) := Z"1 £>*«*(*) = E E C r i _ 1 i=i
i=ijez
and
p~(z) := z - ^ ^ - W = EEC^"'" 2 i = l j'eZ
i=l
Obviously these field p^(z)
satisfy
[dat{1'>±(z)p±(w)} = 0
and
[dat^±(z)p±(w)\
= 0.
Actually the first relation is clear and the second relation follows from the fact that, in the definition (8.32b) and (8.33b) of d*t^±(z), the structure constant c a a *s e Q u a l to 0 if 7 is a simple root. So one has [dat+{z)p+{w)\ = [dat-{z)p-{w)\
= 0,
and the fields d+{z)
:= dat+(z)+p+(z)
"£d+z ra-l
=
nn €€ Z Z
and
d-(z) := dat-(z)+p-(z)
= £d,
^-n-2
nGZ
satisfy [d+(z)d+(w)] = [dT(z)d-(w)]
= 0.
So the coboundary operators d^ define the cohomology spaces tfWfo)
= ©tfj^Cg) fcez fcez
and
/f(±>(V) =
© ^ ( V ) .
These cohomology spaces are called the BRST-complexes. One sees that the 0-th cohomology spaces HQ '(g) are naturally Lie algebras, called the Walgebra of g, and H^. (V)'s are its modules. The detail analysis on the structure of these cohomology spaces, including the vanishing or non-vanishing and irreducibility, remains important problems. In the next section, we see that HQ (g) always contain the Virasoro algebra, and calculate the EulerPoincare characteristics of the Virasoro operators Lf on the cohomology spaces
H^{V).
8.5. Euler-Poincare 8.5
Characteristics
301
Euler-Poincare Characteristics
Let WQ denote the longest element in the finite Weyl group W, and consider the following transformation of A: A 3a
i—> a := -w0a
G A.
(8.36)
Namely a is the so called "transposed root" of a, and a is positive (resp. negative) if a is. Notice that hta =
hta
for all a 6 A, since
hta
=
(pv\a) = -(F\woa)
=
(/5v|a) = hta.
= -(^V)|a)
We introduce an important linear transformation w of the dual space
r>* = ( C A 0 © cs) © i)* of the Caxtan subalgebra \) denned by
_
;=
{
W0tpV
. ^-WP0tpV
= t-pvw0
on CA 0 © C<S,
._„ „„
_ -^
,
"' -1 —t-pvw0 on h . This transformation w preserves the inner product in f), since CA 0 © C6 is orthogonal to h*. The action of w is written explicitly as follows : Lemma 8.5.1. 2) 3) 4)
1)
wh = tip" wa
wA0
= A0 — p v —
-w0h + {pv\h)8
for
6, heif^i)*,
= pv + \py\26. — a + (hta)£
for
a € A.
Proof. 1) follows from |«V|2
iwA0 = i_pVio0A0 = t_pvA 0 =
A0-ps/-}—~6.
2) follows from wh = —wotpvh = —w0(h — (pv\h)8) 3) and 4) are immediate consequences from 2).
— —w0h + (pA/\h)6. D
302
8. W-algebra,
The formula 1) in the above Lemma 8.5.1 implies that w(d)
= d-p^-^-K,
(8.38)
under the canonical identification of ()* with h. Since to0 is an element in the finite Weyl group W, it extends to an automorphism of g, which we denote by u>o using the same notation. We choose a family of non-zero elements ea e ga satisfying (i) (ii) (iii)
6_aJ
[6Q;,
w0(ea) r<*+p
-' _
= a
namely
(e Q |e_ Q ) — 1,
&WQOL
C
-a,-P
for all a, ft 6 A (as to the condition (iii), see e.g., Theorem 5.5 in Chap.Ill of [81]). Then S+P
_
^a,P
-w0(a+P)
_
-w0a,—w0P
~
u
_
w0(a+P)
_
°-w0a,iDoP
~
_ C
a+p
/ „ ,Q%
a,/3 '
\O.OV)
and so we define the action of w on the affine algebra g as follows : w(ea(z)) w{h{z))
:= :=
-zhta • e^oa(z) -(w0h)(z) +
= -zhta-es(z), (p \h{0))Kz-1
(8.40a)
s/
"
v
'
(Pvl>>)
= -(w0h)(z) + (ps/\h)Kz-1 for all a € A and h G i). By this definition, in particular one has i5(pv(z))
= ps/(z) + \ps/\2Kz~1
(8.41)
since woPv = — py • L e m m a 8.5.2. 1)
w(ea
2)
u>(/i ® tj) = -w0h
3)
to zs an automorphism of g.
foraeA
and j e Z, /or ft e \),
Proof. 1) follows immediately from (8.40a):
€}(j2(ea®tj)z-j-1} S"ez
= ^(egOf')* 1 * 1 -''- 1 '
jez
(8.40b)
8.5. Euler-Poincare
Characteristics
303
=
5^(e5®t,'+hta)*~i~1.
2) follows from (8.40b), and 3) is shown by easy calculation. For example, relations [w(ea ® tj), w{e_a ® tk)\ [u5(d))u?(ea
=
w([ea
=
J
w([d, e Q ®t '])
(8.42a) (8.42b)
are shown from 1) and (8.38) as follows: \w(ea ® t>), w(e-a ® *fe)] = [e_^0« ® * j + h t Q , e ^ ® i fc ~ hta ] = [e_iDoo.. etDoa] ®tJ+k + (J + hta) (e-^oa\e^oa) K • 6j+k,o >
=
v
'
"
s,
'
-w0a ® *>'+* + (j + hta)K • 6j+kfi,
and w([ea®tj,e-a®tk])
u>([eQ, e_ a ] ®i j+fc ) + j (e a |e_ Q )/f • <5 j + M
=
a
1
w(a ® ti+k) v •* '
+jK • 8i+kfi
-w0a+(pv\a)K6j+kt0
— -w0a + ((p v \a) +j)K • Sj+k,o. hta
Thus we have (8.42a). And also we have [w(d), w(ea®tj)\ =
-[d,e_floa®t»'
=
d py
-
-^Y-K>-e-«o°®*i+hta]
+hte
(j+hto)e_ f i ) 0 „«8,ti+^
_ (w0a\pW)
e^oa
J^—WQtx §9 ^
and
w([d, ea ® *»]) = jw(ea ® tj) = - j e - ^ a ® t J + h t o
proving (8.42b).
•
L e m m a 8.5.3. Let a G A and /*, ft' 6 f). Then 1)
w(: e a (z)e_ Q (z) :) = : e s (z)e_ s (z) : + z - 1 h t a • a(z) + z-2 • h t a ( h t Q ~
1}
K,
8. W-algebra
304 2)
w(: h{z)h'(z) :) =: (w0h)(z)(w0h')(z)
:
- z-xK{py\h){w0ti){z)
-z^Ki^lh'XrvohXz)
+ z~2 K2 (py \h){py \h').
Proof. This lemma is seen easily from the {y-action (8.40a) and (8.40b) and Proposition 6.1.9 and the following operator products : r
, s
/
a(w) z—w
M
K (z — w)2'
Actually 2) is shown, using (8.40b), as follows : w{: h(z)h'(z)
:)
= =
: w(h(z))w(h'(z)) : : (w0h)(z)(w0h')(z) : -z-\py\h')K • (w0h)(z) x y -z- {p \h)K • (w0h')(z) + z-2(py\h)(pV\h')K2.
• We notice here the following formulas
X>ta = Y, ^» = 2^» a£A+
£
( 8 - 43a )
a€A+
(hta)
2
=
a6A+
(py\a)2 = hy\py\2
£
(8.43b)
a6A+
£>ta)a
=
a£A+
£ > » a
= /*V,
(8.43c)
a£A+
which follow from Lemma 1.3.5 and will be often used for calculation of Virasoro fields in the sequel of this section. Proposition 8.5.4. Let
Le(z) = 2(K + hv){ ^ '' e^e~a^ a€A
2)
w(Ls(z)) ti(L»)
= LB(z) + z~1py(z)
= L»+py
+
±K\py\2.
J2
:u z u z
i( ) i( )
:
|
*—1
be the Virasoro field of g, where {ui,--Then 1)
:+
+
,ue} is an orthonormal basis oft). ^z~2K\py\2,
8.5. Euler-Poincare
Characteristics
305
Proof. By Lemma 8.5.3, one has w(L«{z)) = L»(z) + {T> where
-2KZ-1
Y^(PW\ui)(w0Ui) (z) + i f 2 z - 2 $ > i=l
2(^
|^)
2
t=l
=(iDopv)W=-pv(^)
1
V
i-^^z"
1
|pv|
hta-a(z)+z-2X £
£ aGA+
(hta) 2
a6A+
+2^-1pv(z) + K2Z-2|pv|2|, Using (8.43b) and (8.43c), this is written as follows :
+2Kz-1pw(z)
+
K2z-2\pw\2\
= z"V(z) + ^ - 2 ^ V | 2 , proving 1). 2) follows from 1) by comparing the coefficients of z~2 in both sides.
•
S
We note that W(£Q) * computed also by using the exphcit formula (6.84)
Since the Casimir element il is fixed by w, one has
™ ( '' 8,
=
2(JC + k") ~ **° 2{K + W)
which gives just the formula in Proposition 8.5.4.2). In view of Lemma 8.5.1.4), we define the action of w on fermionic elements
C and VC by ^ n
•= C+ht«
and
wrna
:= f-hta-
(8-44)
8. W-algebra
306 Prom this definition, one notices that V>™ :
creation
<$=> w«/>" :
creation
(resp. annihilation)
(resp. annihilation)
nor •*/£" :
creation (resp. annihilation)
w>i/£a :
•<=>•
creation (resp. annihilation)
does not hold except the case when a is a simple root. So the transformation w on fermionic fields does not preserve the normal product and so, in particular, w{: ipa(z)i{)*a(z) :) and : wffl* (z))w(il;*a (z)) : are not equal unless a is a simple root. The action (8.44) of w on fermionic fields ipa{z) and ip*a{z) is written, in terms of fields, as follows:
VneZ
=
E
^2
/ U o M
""
1
neZ
= z
htQ
~V5(z),
(8.45a)
m£Z
\n€Z
=
/
J ^ ^ mGZ
n€Z
hta m
z~ 2
=
z-hta+V5(,z)-
(8.45b)
—hta+l z —m-l
The following proposition is immediate from (8.40a), (8.45a), (8.45b) and (8.39) and (8.44) : Proposition 8.5.5. 1)
w(dst+(z))
2)
w{d+(z))
= =
-z-dst-{z), -z-d-(z).
And also, from Propositions 6.1.9 and 6.1.10, one sees easily the following : Lemma 8.5.6. 1)
w(: i>a{z)xl>*a{z) :) = : (j?{z)(j>*«(z) : +(hto - l)z~x ,
2)
w(: dipa(z)ip*a(z)
:) = : d(j>s(z)(j>*5(z) :
+ ( h t a - l ) * " 1 : rM*{z)
: +0*°
~ Q0*« ~ *)z-2 ,
8.5. Euler-Poincare
307
Characteristics
w(: i>a(z)dip*a{z) :) = : (j>a(z)d
3)
J5I„\J*ZI„\ 5{z) :. +(1 - h t a ) * " 1i :. (f{z)(j>*
h t
« ( h t o - 1) „-2
Letting x := p v for Tx+(z) and x := 0 for T x - ( z ) in the definition (8.21) and (8.22) of Tx±(z), we define "ghost fields" L&h±(z) = £ „ e Z £ £ h ± z ~ " ~ 2 as follows : L g h + (z)
:=
T(x=ijV)+(z) a£A+
=
J ] aad : ^(z)^*«*(z) : " E : ^W^'°W : • ( 8 - 46 ) aGA+
h
L& -(z)
aSA+
x
:= T< =°)-(z) =
a
J ] :^ (z)^a(z):,
(8.47)
a£A+
where aQ
:= h t a = (a|(5 v ).
(8.48) e
Using these ghost fields and the Virasoro field L (z) of g, we define the "energy momentum fields" L^{z) — X)nez L^z~n~2 as follows: L+(z)
:= L 8 ( z ) + L s h + ( z ) + dp v (,2),
L-(z)
:= L8(,) + JL^-(,) + z- 2 {(p v |p)-^±^|^| 2 } +z-1dfzpv(z) ^
(8.49)
+ z £ ) (a 0 - 1) : <j>a(z)(f>*a{z) : Y (8.50) aSA+
The definition of L~ (z) looks less beautiful because of the existence of 3rd and 4th terms, but these terms axe required as we shall see later in the proof of Proposition 8.5.8. The 0-th components of these fields axe given as follows : Lemma 8.5.7.
1)
i) Lf+ = £ £ ( n - (a|pv)) : C X = <*eA+ « 6 Z
£ {E
308
8. W-algebra
ii)
Lf~= £
^n:f_nC:
a€A + 2
)
0
^
=
«
ft
U
n
n>0
^°
^ - A O - P
V
2(if+/iv) v
U
L?+,
+
Proof. Formulas in 1) are shown, from (8.24) and (8.25), as follows :
L +
t
= E E( n -( a ^ V )) : ^-n^ a : a £ A + «€Z
= E {E( n - Hpv))^-nCa - E( n - («i/5v))ca^„| a€A +
n
>°
™<0
v
a
^
+
v
= E {E("-Hp ))^„c + E(" wp )¥-^i „g/i , *• n>0
L h
n>0
n:
o - = E E *-»#
a:
E {E^-n^-E^nV-n} E {E^-n^ + E^-nCJagA+
n>0
n>0
To prove 2) we notice that T*
L
°
Q
-
"
2{K +
A
hy)~A°
by (6.84), and -71-2
apv(z) = afv E(^ V ® in)-2~n_1) = - E ( n + X ) ^ V ® *n)z~ n£Z
'
n€Z
Then, taking the coefficients of z~ 2 , we have L+ = Lg - p v + Lf+
=
n 2{K +hV)
~ Ao - Pv + L?\
(8.51)
8.5. Euler-Poincare
Characteristics
309
proving i). To prove ii), one has only to note that + z J2 (a- - 1) :
z^dizp'iz)
does not contain the ,z _2 -component, since the complete derivative does not contain the term of z~x. • w{L+(z))
Proposition 8.5.8.
Proof. First we compute w(Lsh+(z))
=
L~(z).
as follows:
w(L^h+(z))
= 5 3 a<*d€i(:www••)-
E «K:^a(z)ora(z):)
a€A+
-
glr^^a^^r+a-ht^z-
1
:^^)^
5
^):-^^""
1
^-
}. J
«eA+
Since aa = hta = hta, this is rewritten as follows: w(L^b+(z))
=
J2
ao.d:<j>a(z)
aeA+
-
aa{aa -
l)z~2
aGA+
: F(z)d4>*a(z) :
j ; a eA
' +d:<^(z)<£*<«(z):-:a<£<*(z)4>*<»(z):
+ E <«« " I)*"1 : *"<*)*"(*) = + £ a€A+
=
2
aGA+ a
^
: +z~* : <j>°'(z)
(a a - 1 ) | fl :
Q G A
htQ(ht a 1) 2 2 ~ ^
+
«-10(*:0°(Z)^-"(Z):)
+ ^
:a^(z)^(z):-i ^
aGA+
aQ(aa-l)z-2
a£A+ L«l>-(2)
aGA+ h
+L« "(z)-i 53 aGA+
aQ(aa-l)2-2.
310
8. W-algebra
Also notice, by (8.41), that w(dpw{z)) = d(wpw(z)) = d (p(z) + IpVfAz- 1 ) = dp(z) -
\pv\2Kz~2
Then proposition follows from these and Proposition 8.5.4.
•
Proposition 8.5.9. z—w 2)
| i ± W i ± ( r o ) |
=
^
2
+
(z — wy L
^
£W2
+
(Z — W)"2
2 — 10
(z — wp
(2 — U>)4
uViere cW := e-^^+24(p\pV)-12(K
+ hV)\pV\*. (8.52)
Proof. 1) follows from (8.23) and (8.27), since aa = (/0 v |a). 2) The operator product of L+(z) = LB(z) + Lsh+(z)
+ dp(z)
is easily calculated by using (6.85a), (6.85b), (6.86) and (8.27), namely [L'W]
= (0 + 2
^+ ^ . - ^ — d i m f l , 2
[LKd^)} l(dpV)xL»] \(dp^x(d^)}
= (d + \) p\ = -\*p\ = -\p^2KX3:
and [jL gh +AL gh +] = {d + 2 A ) L g h + + A3 (J_
_l
f e
V - ^ .
Prom these, one has [L+xL+]
= (8 + 2A)(LB + L&h+) + {(d + A)2 - A2 }py d(d+2X)
Then K + h,v^3 ia|p|2 " ^
Ir
'
ftv " i2/tv|pv|2_24(p|pv)+i2i^
8.5. Euler-Poincare
Characteristics
311
K \ 12|p|2 s v,2 , - w , , v^ , / I - 12{K + ftv)|pY + 24(p|p ) + - ^ — - 1 proving the formula for £+(2). Then the formula for L~(z) follows since the transformation w preserves operator products. • We notice that the formula (8.52) is rewritten as follows: i 24
dimg 24
K + hv._Vl2V 2 -|p | 2
+
since
, ,_v. (p|p V ) =
1 /.,. K ^ ( ^ ) - ^ - d i m , ) ,
dimfl 24
Jpp 2/i v '
=
(8.53)
This proposition states that L s (z) and i> ± (2) are Virasoro fields acting on the spaces C^(g) and C^(V). To calculate the commutation relations of these Virasoro fields with the coboundaxy operators dat± and d^, we start from noticing the following properties: L e m m a 8.5.10. 1)
[Lgh+AV>a] = fy" + (1 - hta)AV>Q ,
2)
[Lgh+AT/>*a] = di/>*a + Ahta • ip*a ,
3)
[L Bh+ A (^* a ^* /S )] -
d{i>*ail>*0) + A(hta + ht/9W*«V" ,
Proof. We need only to check 3) since others are immediate from (8.28a) ~ (8.28b), and this is shown by using 2) as follows: [Lsh+x(iP*a^)\
[lP+xifi*a]
=
>
v
i>*p + V>*"
[L^+x^]
'
v
+ f [ y[ I 8 h + A f 1 Jo * '
'
^]d/i
dip*" +Ahta-0* 0
=
d(Y>*"V*/3) + A(hta + ht/3)ip*atp*P.
• Using this lemma, one can prove the following : L e m m a 8.5.11. 1)
[L g h + A d s t ( 1 ) + ] =
Yl o£A+
eadpa
+ \ J2 aGA+
hta-eaxP*a,
312
8. W-algebra
2)
{Lsh+xdat{2)+}
= (d + A)d s t ( 2 ) +.
Proof. 1) is quite easy to see as follows: [ L gh + A d at(i) + ]
£
=
[L^+X(eara)} = £
aeA+
ea \L^+Xra]
a€A +
To prove 2), we first compute [Lsh+\(: ip"ltp*a%j}*^ :)] by using Lemma 8.5.10 as follows: [L*h+A(: W°if,*f>
:)]
= : [L^+x^] V'*aV*/3 : + : V'7 >
v
'
[i gh+ \{^* a ^* P )] v
3
+ / [ [L^+xr\
v ' dil>~i+ (\-hti)Xil>i
= {d + A(l + bta + ht/3 - ht7)} : ^ ^ 7
o
M*arp)w
N
a
^ :
a /3
+ /" [W )^(V'* ^ )]^+A(i-ht7) / Jo
v
:
'
a(V>*°'V'*' )+A(hta+ht/9)V'* "0*'3
3V>T+(l-ht7)Ai/>''
JO
v
v
-M[V-VV>*"'/>*'3)]
'
&\(rarp)w.
Jo
The last two terms are equal to 0 unless 7 = a or 7 = /?. So one has
[Lgh+A(: V7V'*QV'*/5 01 = {# + A (l + h t o + h t 0 - h t 7)} : ^WV3 :, (8.54) if 7 ^ a and 7 ^ /?. Then, by (8.32b), one has
2[^h+Adst(2)+] =
^
c 7 ;/3 [^ h + A (: rrar0
oi
a,/3,7eA+
a,/3, 7 €A +
+A Since
£
<£ij8(l + hto + ht/3 - ht 7 ) : ^^*a^*13 : .
a,0,7eA +
c„»/0
=>• 7 = a + ,9
=>
M7 = hta + ht/3,
8.5. Euler-Poincaxe Characteristics
313
the second term in the above is equal to A
c
l,H : ^ i > *
Y,
a
^ • = 2Ad st < 2 >+ ,
a,P,-y€A+
hence 2[L g h + Atf' ( 2 ) + ]
\)datW+,
= 2(d +
proving 2).
D
Lemma 8.5.12. [Lsxdat(1)+]
1)
] T (dea)i>*a +
=
\dstW+,
a£A+
2)
[(dps/)xdst^+}
= - A j h t a • eara
•
a£A +
Proof. 1) holds since [L*xdst(i)+]
J2lL»x(eara)}=
=
J2[L*xea]ra-
"eAV
a€A
~+ (9+A)e„
2) is seen from the following calculation: [(0p v )Ad rf(1)+ ] = E
[(9pv)A(eQV*a)] = E
a€A +
A
= - E
0
[^A^IV-*
<xe~A+
[(^PV)Ae«]V*a
a£A+
v
e
Q
= -A E [^> <^* = - A E wp,e^*a a€A+
aeA+
•
h t Q
D Using these lemmas, one easily sees the following: [L+xd^+j = (d + \)d3t+ .
Lemma 8.5.13.
Proof. Prom Lemmas 8.5.11 and 8.5.12, one has [L+Xdst+] =
= [(L* + dp + Lsh+)x(dstW+
[L«xdst^+]
- d3tW+)}
+ l(dp)xdst{1)+} + [L 6 h + A d s t ( 1 ) + ] - [Lgh+A<2st(2)+] s (1)+ htQ e a
= [ E (teaW + Ad ' 1 - A E a€A+
+{ E e«w*a+A E htQ • e^*a] ' aeA+
• ^*
a€A+
06A4-
-(d+wt{2)+
8. W-algebra
314
=
eaifi*Q ~(d + Wt{2)+ = (d + X)dst+,
(.d + ^ Y .
e*€A+ * v <*»'(!) +
'
proving the lemma.
•
One notices also the following: [L&+XP+] = (d + X)p+,
(8.55)
which follows immediately from the definition of p+(z) and Lemma 8.5.10. Then one has P r o p o s i t i o n 8.5.14. 1)
i) ii)
[L+Xd+] = (d + \)d+, [L-X(zd-)} = (d+\)(zd-),
2)
i) ii)
[d+xL+] = Xd+, [(zd-)xL-) = XzdT.
Proof. 1) i) follows from Lemmas 8.5.11 and 8.5.12 and (8.55). Then, applying w and using Propositions 8.5.5 and 8.5.8, one obtains ii). 2) follows from 1) by skew-symmetry. • Prom this proposition and Proposition 6.1.4.1) in §6.1, one obtains the following: Corollary 8.5.15. 1)
[di,L±{z)\
= 0,
2)
[df, L±] = 0
forallneZ.
Thus the Virasoro fields L^(z) commute with the boundary operators d0 . This means that L^'s, for all n € Z, belong to the 0-th cohomology spaces HQ '(g) respectively, and that L+ (resp. L~) act on the spaces HJ.+'(V) (resp. H^ (V)). In the sequel of this section, we compute the Euler-Poincare characteristics of the operator qLo on H^(V). For each element x G rj, we consider fields xsh±(z) = $Z xf~ z~i~x and j€Z
x^(z)
" '
= 5Z ifr-\Z~*~x defined as follows : iez
xgh+(z) :==
Y,(x\a)-.r(z)ra(z)--
(8.56a)
aGA+
xsh-(z)
= Yl (*ia): ^w-c*) : -
:--
QGA+
(8.56b)
8.5. Euler-Poincare
Characteristics
315
and x+{z)
:= x ( z ) + x g h + ( z )
x~(z) Prom this definition,
:= x ( z ) - x
gh
(8.57a)
~(z)
(8.57b)
written explicitly as follows:
= E (*l«){ £V£„lC " £ ^ C
(8.58a)
*8)~ = £ £(*I«)^-„
= E (*ia){ E «« a - E » « } •n>0
a€A+
(8.58b)
n>0
Remark 8.5.1. VFe note that the fields x ± ( z ) , or even their 0-th components i i , , do not commute with the coboundary operators d^. So the operators xr^, do not act on cohomology spaces H^. (V). For a highest weight g-module V, we define the Euler-Poincare characteristics of complexes C^\V) as follows: chH^\V)
:= limV(-l)'Tr H «, v .
(8.59)
JGZ
To compute these alternating sums, we first notice the following : Lemma 8.5.16. Let x be an element in i) and £ be a complex number. Then OO
x
\
1)
5>iyTrAf(+)gLch++<)+
2)
E("1)JTrAf<-)9Lf""£X"0>~ - ( I I ( l - e - a ) ) ( r , e r x , 0 ) .
=
( Y[ (l-e- Q ))(r,r(ex-^),0), /
OO
x
Froo/. By Lemm 8.5.7 and (8.58a) and (8.58b), Lg h ± ± e x 8 ^ are written as follows: Lf+
+ £xg+
=
£
£
a£A+ »6Z
(„ + (ex - p v |a)) : C V C =
316
8. W-algebra
Then the lemma follows from (8.26b) in view of (8.10) and (8.15). Lemma 8.5.17. Let h = (r,TZ,Tt) e h* and q — e2nlT.
•
Then
(n a-.-))
<w)eW"
= e-^Se"
w£W
J J (1 " e - Q ) m u l t a , a£A+
one has = q^e2™^+hV»
Ap(T,TZ,Tt) =
^«(*W+fcV*(«-*!?(T))Y
[J
J J (1 - e " Q ) V r , T 2 , r t ) (l-e-Q)Vr,r0,rt),
proving the lemma.
•
We are now in a position to compute the the Euler-Poincare characteristics of H^ (V) for any highest weight g-module V. Lemma 8.5.18. Let V be any highest weight g-module of level m. Then 1)
ch//(+)(F) = l i m 9 ^ & v > f c h v -
2)
chHl-\V)
=
JJ
(l-e-Q)Vr,T(ea;-/5v),0),
q-^W+W)
x limqr2(".+"v) ( c h v -
J J (1 - e~a)\(T,ETX,0)
Proo/. 1) By (8.59), (8.57a) and Lemma 8.5.7, one has OO
chH^(V)
=
l i m ^ ( - l ) J ' T r c ( + ) ( v ) g L o +I-+ e x ( 0 ) j=0
.
8.5. Euler-Poincare Characteristics
317
oo
=
limE(- 1 ) i ^c(+)(v)« 5 S : : ? S V T ~ ( A 0 + , ! V ) + 6 V S , , + + e ^. 3=0
Since C^iV) — V ® A ^ - 1 ^ and q = e2"^11", this is rewritten as follows : =
g2(^vy
. lim J chv ( e 2-(-xA„+x (e x-pV))N ^ ( . ^ T r E_>0 l-N » 'j^o chv(T,T(ex-ps/),0)
¥ ( + ) Aj
>
^h++<,+ I > ,
V
(na^+°{l-e-a))(T,T(£x-Pv),0)
e_>
aeAf6
^
'
proving 1). 2) By (8.59), (8.57b) and by Lemma 8.5.7, one has
l h n ^ - l ^ T r ^ - , ^ „L/ n" + e £ ( 0 )
chff(-)(V) = €
^
3=0
oo
' •
_Ao_-±^|pV|2+W^V)+£a;
L
g h - _ eh-
(8.60) Notice that CJ - ) (V) = V
r r v q - A o + £ X = c h V 9 " A o + £ x = ch v e 2 , r i T ( - A o + £ : i : ) =
chv(T,Tex,0).
Then the above equation (8.60) becomes as follows : Chtf<-)(V)
=
g^lsSsn-^^l^l'+W^)
•
OO
Ao+ex
x lim I TrvqO g 2(m+hV)
m-fh
v
^ - V ^ A ^ I
proving 2).
^ ' * ^ ^
\
|=V|2 . /=|=V\
V - l ? I +(H*» )
x limchv(T,T£x,0)(
=
-v 1
JJ
(1 -
e"Q)\(T,ETX,0)
qf5?=Ss^-1B:^l'»vla+Wv)iim^chv
J J (l-e-")Vr,£Ta;,0), D
In particular when V is an admissible fl-module L(A), the Euler-Poincare characteristics of H^(L(A)) are obtained as follows.
318
8. W-edgebra
Proposition 8.5.19. Let m = - be a principal admissible number and A € F™y be a principal admissible weight. Then + t3),T-{t+
AAo+p (uT,Ty-\z 1)
ch^ (A) (r, r z , rt) =
2)
q^fr)
Ap(r,Ap(r,rz,rt) Tz,rt) Yl (1 - e~a)\ ( r , r z , r t )
fchL{A) ^
!|2
(z\0) + ^ ) )
r.cA«
^ - ( P I ^ + Z R ^ I / ^ I
2
-
=
/
-hvt
AAo+p (UT,ry~\z
+ 0), ~{t+ (z\(3) +
V
"J
^-))
2 y
T]{T)1
Proof. 1) is already known by Theorem 3.3.4 (i.e., the character formula for admissible representations). 2) Since ftA = |A + p\2 - \p\2 and |A+p| 2
. |pl 2
ch A (r,T2:,Ti) = q 2<™+hV> " " " ^ c h ^ r , TZ, rt), the left side of 2) is written, by using 1) and Lemma 8.5.17, as follows: LHS of 2) lA+pi'-iri"
IA+PI»
| IPI2 ^ A ° + P (tiT.TjrH* + P),TA*
+ (*lff) + ffi))
r)(T)e
x 0q A - ^ - w » ) - f c v * ^ I i I £ i I ? ) V(T)'
J.
•,•»
— a2* 2(m+hV)
(
^
h*t
lP|z; "• i
^AO+P («r, T ^ ( * + P), I (t + W) + ¥ ) ) V
/_
9
»/(r)' (8.61)
Since 24
2(m + /i v )
24
lPIP j +
2
lP
' '
by Proposition 8.5.9, the power of in the right side is rewritten as follows: power of q in the right side of (8.61) t \P\2 (p|z)-/*vi v
24
c
2(m + fc )
= ^-{p\pv) +
n
^\py\2-m-vt.
Then, substituting this into (8.61) proves 2).
•
8.5. Euler-Poincare
Characteristics
319
Prom this, one has the following: Theorem 8.5.20. Let m — J be a principal admissible number and A e Pf£ be a principal admissible weight. Then ch.H^(L(A)) is written, in terms of the function
chH^'(L(A)) = £(c)g~ sr (pA 0 ,^; where p, is an element in and a eW satisfying p + pv — a(uAo — y~x{P — Pv))-
2)
ch.H^~\L{A)) — q 24, <^AoiM, fined by p + pv := uAo — y - 1 /?.
p^u~h
where p is the element in p^u~h
de-
Proof. 1) Using Lemma 8.5.18.1) and applying Proposition 8.5.19.2) to z = —pv and t = 0, one has chH(+HL(A)) = q * ^
(chL{A)
(l-e-Q)V,-rpv,0)
• []
c a ) +2 ^ lh y |2 A^P (^-nr^-P
+ 0.5 ( - WW + ^ ) ) T?(T)'
qi 24 T/(T)
A A o + ^ t i r . r y " 1 ^ - />v), ^ [ p v | 2 + ^ ( - (p v |/?) + ^ )
),
X|g-Pvl2 2u
since 2(m + /i v ) = ~24~ + _ 2 ~ ~ ' ' ' and the level of A0 + p is equal t o « ( m + / j v ) . So 1) follows from the definition (4.26b) of ipx^. To prove 2), we make use of the formula in Lemma 8.5.18.2): chif(")(L(A))
=
q-^W+W?) x lim
fl^+^T
ch L ( A )
Yl
(1 - e~a)
(r, 6TX, 0) .
Applying Proposition 8.5.19.2) by letting z = ex and t = 0, this is rewritten as follows: chir<->(L(A))
=
q
-^ipvi2+(^%^-(^v)+=^vi
1
proving 2).
(
-
2
r
l/?l2\
D
320
8. W-algebra
Prom this theorem and Theorem 4.2.2.2) in §4.2, one obtains the following: Corollary 8.5.21. Let g be a non-twisted affine Lie algebra and A be a principal admissible weight of level m. Then q-^chH^iV)
= VA(T),
where V'AC7") * S the residue of normalized character of L(A) defined by (4.31). If only one among H^~'(V)'s is not {0} with all others vanishing and it is an irreducible HQ_)(g)-module, then the above corollary states that tp^(T)'s are normalized characters of irreducible HQ (fl)-modules. Assuming this, the associated fusion coefficients are calculated in §4.3.
Chapter 9
Vertex Representations for Affine Lie Algebras In this chapter, we explain two kinds of natural construction of the basic representation L(A0) for a symmetric affine Lie algebra, in terms of vertex operators. 9.1
Simple Examples of Vertex Operators
Given a real or complex number A, we consider the operator e ^ acting on a real or complex valued function f(x) in one variable x:
Then one easily sees from the Taylor's expansion formula that the right side is nothing else but f(x + A), and so
ex&f(x) = f(x + X).
(9.1)
Prom this one has the following formula, which is very useful in the calculation of vertex operators: e A £ (g(x)f(x)) so, in particular,
= g(x + X)f(x + X) = g(x +
e^-t (e»xf(x))
= 321
e^e^e^fix)
\)ex&f(x),
322
9. Vertex
Representations
and eA£(x/(x))
\)ex&f(x).
= (x +
Namely Lemma 9.1.1. Let X andu- be complex numbers, andg(x) be a function. 1)
ex&og(x)
2)
ex&oe»x
3)
eA=oi
\)ex£,
= g(x + = =
Then
e^-e^e^, xeAa= + AeA3s.
It is also easy to prove the following: Lemma 9.1.2. For A,A',/x, y! e C, £Ae following formulas hold: 1)
(e^e**) o (e"'BeV*)
2)
[ e ^ e A * , e"'*e A '^] =
3)
[^,
4)
[x, e"xex&]
Proof. 1)
e"*eA^] =
=
c V c (M+M')- e (A+V)i j
( C V _ cAV)c(i«+*»')*e(A+V)ij
fie"*^,
-Ae"xeA£.
=
(e»xex£)(e>*'xex'£)
= e»x ex£e"'x
ex'&
2) follows from 1). \4~, e ^ c A * l = lax J
3)
4-°e"x da; Me
4)
oeA^-e"xeA^— = ax
/le^e**.
MX+eMX^
[x, e ^ e * ^ ] = x e ^ V ^ - e ^
eA^ox A
d
X
=
-\eTex^.
d
xe 3x -|-Ae 3x
PJ.l
ITie Space C[Xj
; j G N]
We consider the space # := C[xi, X2,£3, • ••] and operators I + Z
A (- )
-A
a
— ne_r3sjz
= exp
D
9.1. Simple Examples of Vertex Operators
323
and Vx~(z)
oo
/ oo
:= Y[e^*'
= exp M T A ^ \j=l
3=1
where z is a formal variable and A € C. The operator OO
Vx(z)
:= Vt(z)V+(z)
=
/
n(e*X,Ve' 3=1
=
A d e x p l ^ A x ^ l e x p f - g ^ ^ j
(9.2)
is called a vertex operator. Putting ak := -x— and a_fe := kxk (9.3) dxk for fc e N , the commutation relation of ak and V\(z) is given by 3) and 4) of Lemma 9.1.2 as follows: [ak,Vx(z)}
= \zkVx{z),
(9.4)
for k € Z - {0}. To compute the product of vertex operators, we introduce the following operator V^(z,w)
:= e x p ( f > * J + / z t ^ x ^ e x p ^ - f ) ^ " 3 + ^
^
.
(9.5)
L e m m a 9.1.3. Let A and /x 6e complex numbers such that Xfj, € Z. T/ien
i)
vs+(z)v;-H = ^((-^j
2)
VxWV^H = ^ f ( ^ )
]v-(w)v+(z), "W/.M-
Proof. 1) We compute the left side of 1) as follows:
+
Vx (z)v-(w) = n ( e ^ ^ z rr 00
3=1
i=i
/ _x a -j\
V
^
°°
j=l
/JXjJUJ
324
9. Vertex Representations
n
| e
*£,-:> ,„J
i
(•.• L e m m a 9.1.1.2)) •? J=l
e oo
'-1
Vytli(z,w).
^
The infinite series 2 . —z~*vP converges to —log (1 3=1
J in the domain \w\ <
J
\z\. So, using the notation introduced in section 4, this is written as
-VE.KV
//
i
\"A"\
//
z
\~x^
proving 1). 2) follows from 1).
•
The Taylor expansion of the operator V\;il(z, w) with respect to the complex variable z around the point w is calculated as follows: Lemma 9.1.4. Let X and fi be complex numbers. Then 1)
VX;il(z,w)
=
Vx+Ii(w)
+ (z — w)AJ + 2)
( f ; jxjv?-1^ Vx+I1{w) + ^ +M M f V ' " 1 ^ - }
0((z-wf),
VX;-x(z,w)
= I + X(z-w)
Yl
ajW^-1
+ 0((z - w)2).
j6Z-{0}
Proof. 1) Let / be a function in xi, X2, • • • . Then V\;li(z,w)f(x1,x2,---) oo
=
e
3=1
/[•••,xj
—
»•••)•
The Taylor expansion of each facter in the right side is given as follows:
e
J-1
=
e
3=1
+ (2; - w)X
+ 0((z-«,)*),
^jw1
Xj I e ' = 1
(9-6)
9.1. Simple Examples of Vertex Operators
325
so e'- 1
e3=1
=
+(z — w)\\2^jw3
XJ I e' = 1
V=i
/
+ 0((z - w)2).
(9.7)
And ,/
Xz
j
/ { • • • , X J
+ fiw
j
-•
\ , • • • )
+ (,-.)£A.--g(...,, i -^±^ ) ...) + 0 ( (z-.)
2
)
= ^ H / + (* - H A ^ H E ^ ' " 1 ! : + o«z - wf). t=i
*
(9.8) Then, substituting (9.7) and (9.8) into (9.6) proves the lemma. 2) follows from 1).
•
Notice that
dVx(z)
=
^X^jx^~^x{z)
=
X:(
Yl
+
Vx{z)[xY^~z-i-^
ajZ-i-Avx(z)
:.
(9.9)
J
Sez-{o}
Proposition 9.1.5. Let X be a real number such that X2 6 N , and zr, wr (r — 1, • • • , N) be complex numbers satisfying \zr\ < \zs\,
\zr\ < \ws\,
\wr\ < \za\,
\wr\ < \w3\
for 1 < r < s < N. Then the composition of operators V\--\(zr, wT) 's is given by V\.-x(zN,wN)
l
• • • VA ; _A(ZI, WI)
(Zsa-Wr)(w r / v s-Zr) s
v
rJ
"^
x
^ j=lr=l ^ ^
>< -(->f;fM£)
~
^
'
j
9. Vertex Representations
326
V\;-\(ZN, WN) • • • Vx--\(Zl,
n
Wl)/(l)
(za - zT)(wa - wr)
/ ^
\
/or/(i)€ff=C[xi,a;2>---]. Proof. First we consider the case N = 2. Then we have V(z 2 ,w 2 )V(2; 1 ,toi)/(i) =
V ( z 2 , t « 2 ) e x p ( A ^ ( ^ - w ^ Z j J / f ••• . i j - A ^
-
exp f A ^ ( z ^ - wjjxj J exp
7^— , ••• J
A j ^ ( z j - w{) (Xj - A ^
^
-JJ
/ ( . . . ^ . - A ^ - ^ - A ^ ^ , . . . )
' 2 exp(Af;X:^-^)x J )/(..., a ; i -A^^^,---), \
j=lr=l
'
^
r=l
•*
where A := V
3
.1=1
'
Since exp I — 2_. — ) = 1 — x
for |x| < 1,
one has
=
1
=
- zxw^)-x{l
(1 - ziz^)(l 1
(1-Zl^2" )( -
W
w
-
1
l 2" )
(1 - ZilWj J ) ( l - W i ^ 1 )
=
Wlz^)-\1
-
wxw^)
{*2 - Zl)(w2 - Wl) (t"2 - Zi)(z 2 - Wl) '
'
9.1. Simple Examples of Vertex Operators
327
proving the formula when N — 2. Then the proposition follows by induction on N. • In particular letting N = 2 and taking the limit (z2,W2) —• {z\,W\), one obtains the following: Corollary 9.1.6. Let X be a real number such that X2 € N , and z and w be complex numbers satisfying \z\ < \w\. Then Vx.-X(z,w)2 9.1.2
= 0.
The Space C[XJ ; j e N o d d ]
For another construction of the basic representation of sl(2, C), we consider the space 3odd := C[xx, x3, •••] = C[XJ ; j e NDdd] and operators
vrd)+w - n ***->'" = «p f- E -A and r(odd)-/ x
TT e^AJ.-ZJ
vi°dd>-(.) := n
^ = exP £
j G Nodd
^
Vie N0dd
and V^A\z)
:=
y A ( o d d ) -(^)F A ( o d d ) + (z)
n (eAx^'e^^*"i) jeNodd ^ '
=
expf ^ Az^jexpf- £ ~ * - ' " ) , J \jeNodd / \ jeNoddJ )
=
(9.10)
for A e C. To compute the product of vertex operators, it is useful to introduce the operator
:=
exp I
2_]
y,jeNodd
(^
Xz~i + /iiy--7 d
+ ^W^)XJ
j exp I —
/
\
2_\
j'eNodd
3 1
for A,/J € C. In this case, the product of these vertex operators are given by the following formula:
328
9. Vertex
Representations
Lemma 9.1.7. Let X and /J, be complex numbers such that Xfj, € 2Z. Then
i) v^d)+(z)v^-(W) =
,Lz wU^yx"/2yiodd)-WV(od6)+{z)
2) ^ w ^ ) ( » ) = ^((J^)" v/ Vi ; ; dd) (^-)Proof. 1) vlodd)+(z)V^-(w)
=
J]
(e^A''^
jeNodd ^
'
V M < o d d )-(u;)V r A ( o d d ) + (0).
;eNodd
e
(e<-^)
'
jeNodd ^
=
JJ
(9.11)
Since 2j.„2j j£Nodd
J
j=l
J
j=l
J
Z + U)
to
1
+
- «( "*) 5
IOg 1
( "^)
=
S
108
z— w
in the domain \z\ > \w\, one has
e «-
= ^ ((-±_)
Then 1) follows from (9.11) and (9.12). 2) is an immediate consequence from 1).
j .
(9.12, •
In particular, by letting A = fi in Lemma 9.1.7.2), one immediately deduces the following important property of vertex operators, which will be used in Example 10.3.2 to compute solutions of a soliton equation. Proposition 9.1.8. Let X be a real number such that X2 € 2N, and e be a complex number. Then
1)
V{°dd)(z)2
= 0,
9.1. Simple Examples of Vertex Operators 2)
exp( £ V A (odd) (z))
329
eV{odd\z).
= l+
The commutation relation of these vertex operators is obtained immediately from Lemma 9.1.7:
Lemma 9.1.9. Let X and /x be complex numbers such that Xfi € 4Z. Then
We notice that V^°x
(—w,w) is the identity operator on 3cxid and that /
^iGN odd
S'€N odd
j /
So the Taylor expansion of V^°x '(—w,w) around z = —w is given as follows: Lemma 9.1.10. Let X be a complex number. Then VA(°Add) (z,w)
= 1 + (z + w)Aa(°dd> (w) + 0((z + ^ ) 2 ) ,
iw/iene a<°dd>(iu) := ^
afctw - * -1 .
feez0dd
and ak :— -£— dxk
an
d
a
-k •= kxk
for k € N o d d .
Corollary 9.1.11. Let X = ±2. T/ien
1)
[Vfdd\z),
V^(-w)] f4Aa(° dd )(w)
2)
[z-'V^Hz),
4
1
(-w)-lV?dd)(-w)]
= -4{Xa(°dd)(w)6(z-w) + dw6(z-w)}.
330
9. Vertex
Representations
Proof. 1) By Lemma 9.1.9, one has [v Hodd) (z); V(^){_w)]
=
_
{latu
( g ± ^ ) ! ) V^d\z,
lwz)
-w).
(9.13)
Since (z + w)2 (z — w)2
(z — w)2 + Azw = (z — w)2
4zw (z — w)2'
1+
one has I
\l>z w
-
^f(Z + W)2\ \o \(z-w)2J
t"W.z) I /
_
}
—
/
\^z.w
_
\(
4ZW
t"U).z) I /
\ \o I • \{z-w)2J
(QIA\ \^'^)
Noticing, by Lemma 9.1.10, that V^dd) (z, -w)
= =
l + (z- io)Aa(°dd> (-to) + 0((z - w)2) l + (z-w)\a.(°dd)(w)+0((z-w)2),
(9.15)
one obtains 1) from (9.13), (9.14) and (9.15). 2) follows immediately from 1).
•
The following formulas are quite similar with (9.4):
[xk,vfdd\z)]
=
-£*-*V A ( o d d ) (z)
for k e N0dd and A e C, and so [a^V^iz)}
= \zkviodA\z)
(fc€Zodd),
(9.16)
Prom this, one has [a (odd) (z)>
vi°dd\w))
=
J2 fceZodd
[ak,vtdd)(w)]z-k-1 AtD*VA(odd)(iu)
J2 ^~k~1wk-V^odd)(w).
=
(9.17)
fcez0dd
Let us recall the definition (6.9) of the normally ordered product _ \akV^odd){z) (z) : - j ^ o d d ) ^
T/ (odd), A
: akVx
if A; < 0 =
0fcVMd> ( z )
= akvfdd)(z)-\zkvfddHz)Y+(k),
_ ^ ( o d d ) ^
i f fc > Q
(9.18)
9.1. Simple Examples of Vertex Operators
331
where Y+ is the Heaviside function defined by (6.17):
!%(*):= j 1 +V ;
iffc 0
"'
[0
iffc< 0.
Notice that the relation (9.18) is rewritten as akV
= : akv{odd)(z)
: +XzkVJiodd)(z)Y+(k),
(9.19)
and that VA(odd)(2)afc
=
akV
(9.20)
since Y+(k) + Y-(k) = 1
forA;eZodd
by (6.17) and (6.105). Note also that Y+(k)
= Y_(-k)
for fceZodd-
Using these, one has the following: [:ajak:,vtdd\w)] = [ajak, V A ( o d d ) M] = aj [ak, V!°dd\w)} + [aj, vl°dd)(w)} ak >
= = =
v Atu fc V,< odd) (w)
'
*
v \wivlodd)(w)
'
XwkajVJiodd)(w) + AiwJVA(odd)(«;)afe Xwk{ : a,-VA(odd)(t») : +AwJVA(odd)(w)Y+(J)} +A«^'{ : akV^odd)(w) : - A w V f d ) (ui)y.(it)} Awfc : ajVJ:odd\w) : +Xwj : akVJ;odd\w) : +X2wj+kV^odd)(W){Y+(j) - Yl(fc)} .
(9.21)
So the operators r(odd)
^2„
:
= 2« 5 3 :«2n-itafc:, fcez„dd
defined for n G Z, satisfy the following commutation relation: (odd) ^ (odd)^ ] 2[L(odd) T/ ( o d d )iA M]
_
=
y » \[.: a„_2 n _.f„e a. f.e :T/(°dd), ^ ,yr d ) H] fcez„
(9.22)
332
9. Vertex =
Representations
{^••a2n_kV^dd\w):+w2n-k:akV^dd)(w):}
A J2 fc<EZ0dd
+ ^w2nV^odd)(w) =
=
4
^
{Y+(2n-k)-Y_(k)}
feeZ dd
°
w2n k
2A J2
:
~
' v_(k-2n) ( dd)
«fe^ °
'
H=
fcGZodd
+4w2nV<;odd)(w)
J^
{Y_(fc-2n)-YL(fc)}
fcGZodd v
=
2n k
2A J2
v odd)
w - :akV<;
'
2n
odd
(w):+4nw v( \w).
(9.23)
fcGZodd
Since the differential of the vertex operator is given by dv£odd\z)
= A: (
J2
« i ^ " J ' " 1 ) ^ A 0 d d ) W •= x-
^jGZodd
a,(°dd)(z)vlodd)(z):,
'
(9.24) (9.23) is rewritten as [4°„ dd) . ^A° dd) (^)]
= =
*™2n+1 • ^odd\w)VJiodd)(w) : +2nw2nvlodd\w) 2n+1 odd) 2n odd w dv{ (w) + 2nw VJ: \w). (9.25)
Prom this, the commutation relation of the vertex operator with the Virasoro field L(°dd)(2).= ^L(odd)0-2n-2
is obtained as follows: Lemma 9.1.12. Let A = ± 2 . Then [L (odd >(*), V<°dd\w)]
1) = 2)
{dV
- w-'V^iw^o^z
-w)
+ V
- w),
[L^dd\z),w-1V^odd)(w)] = 8 (w-xviodd\w))
• 6odd(z -w)+
w^V^iw)
• d6odd(z - w),
where 6odd(z — w) is the "odd" 6-function defined by Sodd(z-w)
:=
]T jGZ0dd
z-l-iwi.
(9.26)
333
9.1. Simple Examples of Vertex Operators Proof. 1) By (9.25), one has [H°dd\z),
Vfdd)H]
E^2n"2[L2°ndd)^A(°dd)H]
=
nGZ 2n 2
2n+1
Y^ z~ ~ w \ dV$°dd) (w) + ( Y, „2n, z~2n~2w2n\ Vl°dd) (w) v nez ' nez(2n+1)_1 (odd) dd) ay A (to) • 6odd(z - w) + Vl° (w) £ (2n + l ) ^ 2 " " V " v
=
nGZ
_9_ £
-vf dd) M ^ z-22T»-2„..2n "-2^ nGZ ">
v
z-2«-2w2„+l
r»ez
'
nEZ
=
c>vA(odd)H • 1wdd)* -w)+ vf d d ) H w- vt (w)Sodd(z-w),
• a w * - to)
proving 1). 2) Multiplying w~x to both sides of 1), one has [L(° dd )( Z ), t o ^ v f = { w-'dV^iw)
dd)
(to)] - w-2vlodd\w)
}6odd(z - w)
a(»-'v<° dd >( w )) +w-1V^odd)(w)d6odd(z-w), proving 2).
D
We now compute the commutators [L2°m > a n] and [L2m i ^2n ] ^ f°l~ lows. Quite similarly with (6.102) and (6.103) in Example 6.1.4 in §6.1, one sees the following formulas [: ajdk :, a„] =
ka,j6k+n,o + jakt>j+n,o ,
(9.27)
and [4°^ d ) >«n] =
-na2m+n,
(9.28)
and lL2m
>
:a a
3 k :1 =
~J : a2m+jak • ~k : aja2m+k • +jk{Y_(k) - K.(2m + k) }62m+j+k,o •
(9-29)
334
9. Vertex
Representations
So one h a s
2[4°rodd). 4 ^ =
[4°mdd),:«2n-fca*:]
£ /cGZodd
=
-
£ (2n - A;) : a2(m+n)-kak feGZ0dd fcez0dd
•-
/L
* : a2n-kO-2m+k •
£ (i-2m):"2(m+n)-i°3: J £ Z odd
+
£
(2n-fc)A;{y_(A;)-Y'-(2m + A;)}^m +n,0
fcGZ0dd
:a
2(m - n) £
2(m+n)-feafe
:
feGZodd „ j (odd) • a Y y 2(m+n)
-
^
(2m + A;)fc{y_(A;)-y_(2m +
fc)}(5m+n,o.
(9.30)
fc^Zodd
Just like (6.108), the following holds: 1 -1 y _ ( f c ) - r _ ( 2 m + A;) = « 0
if m > 0 and - 2m < k < 0, if m < 0 and 0 < k < - 2 m , otherwise.
First consider the case when m > 0. Then putting —A; — 2j — 1, one has -2m < k <0
<=^
0<-k<2m
<==>
l<j<m,
and so £
(2m + fc)A;{r_(A;)-Y_(2m + fc)}
fcGZodd m
=
m
£ ( 2 j - l)(2j - 2m - 1) = £ i=i
= _
{4J2 - 4(m + l)j + (2m + 1)}
i=i
2m(m+l)(2m+l) „ . „., ^ - - 2m(m + l ) 2 + m(2m + 1) -m(2m2 + 1 ) 3 '
The same formula holds also when m < 0. So one obtains r r (odd)
[L2m
r (oddh
>£2„
0
/
\r(odd)
J = 2 ( m - n)L\(mln)
, 2m3 + m
+
-
6m+nfi .
(9.31)
9.1. Simple Examples of Vertex Operators
335
One can rewrite this formula as follows: 1 (m_n\. r(oM) (
rlr(odd) i r M d h _ l2^2m > 2 2n J ~
2m3+ m ro n 24 + >° 3 , fm -m < m +
2 m
' 2 ( +") l r (odd)
= (—•0-5*S=U+ = V =
7 «->ro+n,0(9.32)
Then, putting 4odd)
== ^4°n d d ) + ^ n , o ,
(9.33)
one obtains [Z (odd) ; L (od d)]
=
(m_n)L(odd) +
m__Inw^;
( 9 3 4 )
namely £(odd) (2) .
=
J-£(odd)z-n-2
is a Virasoro field of central charge 1. The field Z,(odd) (z) satisfies the following commutation relations with a( odd ) (z) and V{°dd)(z): L e m m a 9.1.13.
a ( o d d > M • dw6odd(z
= 2)
[L ( o d d ) (z), a ( o d d ) (w)]
1)
[a<°dd>(z), a(°ddHw)]
- tii) + aa(° dd )(w) • W * - w),
= dw6odd(z
- w).
Proof. 1) is shown from (9.28) as follows:
[L<°dd>(,), «<«")(«,)] = E E l^ d d ) > - i ] * - * - 8 ^ - 1 m € Z j'GZodd
E
$Z
(2m-A;) afcz" • 2 m - 2
2m—k-1
meZfeeZ dd
° (2mTlM^l) E (2m + l ) z - 2 r o - 2 i y 2 m J^ mez fc6Z0dd
v
v
z-2m-2w2m+1)
\m£Z
/
(fc + l)a fc ur*- 2 £y
£ fcGZodd v
m£Z v
'
-8»[ f c zE a*™-*-1) \ 6 odd
kV>-k-1
'
9«»[ E -
a
/
2m-2
2m+l
336
9. Vertex =
dw6odd(z
Representations
- w) • a(° d d >M + dwa(°ddHw) • 6odd(z - w).
2) is easily seen as follows: [a ( o d d ) (2), a(°dV(w)}
=
[aj,ak]z-j-1w-k-1
Yl j,fcez0dd
£
2 3 1
Yl J " ' ™1'1 =
=
dw6odd(z-w).
jez0dd D 9.2
Basic R e p r e s e n t a t i o n s of sl(2,C)
There are several ways to construct the basic representation. It is known in [113] and [135] that there exists a construction of a basic representation L(A0) corresponding to each conjugate class in the finite Weyl group W. Most familiar and simplest ones among them are homogeneous realization corresponding to the unit element and the principal realization corresponding to the Coxeter element. Others are less easy to describe (cf. [113], [135]). Before going to affine algebras in general, we look at them in the case of the simplest affine Lie algebra A\ — sl(2, C), where necessary calculations can be easily worked out. We take the following basis of sl(2, C) :
"=-G - U M o s) -d M ° o ) - <»•»> Then these elements satisfy the commutation relations [H, E] = 2E, 9.2.1
[H, F] = -2F
and
[E, F] = H.
Homogeneous Picture
The affine Lie algebra A\ ' — sl(2, C) is the Lie algebra 0 =
(©
sl 2 C
( ' ) ®tJ ) e
CK
®Cd'
with commutation relations lu, v}(j+k) + 3 • ti(uv)6j+kfi
[d, u(j)] [K,g]
U
J U)>
{0}
• K,
9.2. Basic Representations of sl(2, C)
337
for u,v G g and j,k € Z, where UQ) := u<2>f as usual. Then the currents
H(z) := Y,!!^-1, E(z) := J ^ z " ^ 1 jez
md F z :
( ) = E F 0) z " i _ 1
jez
jez
satisfy the following operator products:
[BW%)] = - # - + 7 - ^ 2 , tu
2
\Z
[ S ( z ) £ H ] = [F(z)F(w)] = 0,
IDj
namely [HXH] = 2K\, [HXE\ = 2E, [HXF] = -2F, [EXF] = H + K\, [EXE] = [FXF] = 0, ^•ao) in the terminology of A-bracket. We consider a Z-lattice Za of rank 1 with the bilinear form defined by (a\a) — 2, and a linear space
U := 53e"°0C[xi,a:2, • • • ] , nez and, for A € C, operators := z ± a e ± ( V A ( z )
Tla{z)
(9.37)
acting on the space U, where Vx{z) is the vertex operator defined by (9.2). The action of these operators is described more precisely as follows: r±a{z)(ena
® /)
=
z ±(«*|na) c (n±l)«*
±2n
=
n± a
z e^ ^
®
y ^ f
®Vx{z)f.
(9.38)
From (9.4), one has [ak, Tia(z)] (e™ ® / )
=
z ± 2 » e ("± 1 )« ® [flfe, VA(z)] /
= =
Az f c ± 2 n e ( n ± 1 > Q ig>^(z)/ AzfcrA:Q(^(e"a®/),
namely [ak,Tia(z)]
= Az f c I^ a (z)
(9.39)
for Jfc G Z - {0}. We compute the commutation relation of vertex operators. First we notice the following:
9. Vertex
338
Representations
Lemma 9.2.1. Let A = ±\/2- Then
[r*(z), r*M] = [r*a(z), ri a («o] = o. Proof. We note the following:
r*(z)r*(w)(ena ® /) = ™2nr*(z)(e("+1>Q ® vx{w)f) =
-
2 2(n+l) w 2n e (n+2)a 0
VA(Z)VA(W)/ v '
x
(™) 2 "e<" + 2 ) a ® i,,„((z -
w)2)Vx,x{z,w)f,
and, exchanging z and w, r » r ^ ) ( e » a ® / ) = (ztB) a "e(" +2 > 0 '®t w ,,((z-r«) 2 )VA i A(z,ti;)/. So one has
[r*(z),r*(w)](e™
i)
r*(z)rz*(w)(e™ ® /)
2)
r:^Hr^(z)(e" Q ®/) [TXa(z),TZXa(w)](ena®f)
3) =
A(i,,„ - w ) ( j Z - ) f^"" 1 ^Z W ' ^
+
S aito-'-1j(ena®/) jez-{o} J
(z — w)2 Proof. 1) By (9.38), one has T*(z)TZxa(w)(ena
®/)
=
u T ^ T ^ z ) ( e ^ " 1 * " ® V_ A (u;)/)
=
u r 2 n z 2 ( n - 1 ) e n a ® Vj(«)y.i(ti/)/.
•
9.2. Basic Representations ofsl(2, C)
339
Using Lemma 9.1.3.2) and A2 = 2, this becomes 2
w-2nz2n~2ena
=
® t2,w ( ( — ? - )I z —w
)VX;-x(z,w)f
^T^i^r*^-*'™"' proving 1). 2) is obtained in a similar way: rz*(w)T*(z)(ena
® f)
22nw-2(n+l)e„a
z2nrz*(w)(e(n+1>®Vx(z)f)
= Q
V-\(w)VX(z)f 2
z2nw-2n-2ena®Lw,z\[-^-\
=
)Vx.,-X(z,w)f
3) By 1) and 2), one has
[rxa(z),rzxa(w)](ena®f) =
( ^ ) 2 n • (t,,„ - w ) ( ( z _ 1 t t > ) 2 ) e " a ® ^ ; - A ( ^ , « ; ) / .
(9.40)
We compute the Taylor expansion of the right side as follows. First notice that (^)2n = (i+^^]n \wJ \ w J
= i+^(2_w) + 0 ( ( 2 - ™ ) 2 ) w
(9.41)
Then, by using Lemma 9.1.4.2), one has / y \ 2TI
(-) \wJ
VX;-x(z,w)
=
(l+—(z-w)\S.l
=
l+ (2-iw)|2nw_1 + A
Y, ajw-f-1}
+ \(z-w) ^
ajW-J-H
So the right side of (9.40) is rewritten as follows:
[r^),r:;>)](e" a ®/) =
(*,,. - *.„) ( ( ^ 2 ) (e na ® / )
+ 0((z - w)2)
+ 0((z - w)2).
340
9. Vertex
+(i,,„ - w )
ynw~l
(JZT^) ^Z
W
'
+ A J2
*•
Representations
a w j 1 ena
3 ~ ~ \(
j€Z-{0}
® /) •
-*
Thus we obtain 3), since A2 = 2.
D
We now fix A as A = y/2 and, hereafter, we write simply
ra(z) := Tf(z)
and
r_ a (z) :=
Fzf(z),
and define the operator a^ by ao(ena®/)
:= \nena®f
:= V2nena ® f.
(9.42)
Then Lemma 9.2.2.3) gives us the following: [ r a ( z ) , r_ a («;)]
V^(4,,„-t,,,,»)(^—)53oJ-«;--''-1
=
^
Z
+(tz,™ - i™,z) ( 7
W
\(z-w)2J
'
j€Z
^
) •
(9-43)
Thus we have proved the following: Lemma 9.2.3.
[ra(z)r_a(u;)]
L
namely
[(ra),\r-a]
l ^™1 + z—w (z — w)z
=
J
=
v 2 a + A in the terminology of X-bracket.
Notice that a0r±a(z)(ena
® /)
ao(z±2ne(n±1)a®y±^(z)/) V2(n ± l)z ± a n e< n ± 1 > a ® V±^(z)f V2(n±l)T±a(z)(ena®f)
= = =
and r±a(z)a0(ena®/)
= = =
V2nT±a(z)(ena ® /) ±2n (n±1)a \/2nz e ® 14^(2)/ v/2nr±Q(z)(e"Q®/).
So one has [ao,r±a(z)]
=
±V2T±a(z).
Then (9.39) and (9.44) give the following:
(9.44)
9.2. Basic Representations ofsl(2, C) Lemma 9.2.4. Z, 2)
1)
[ak, T±a(w)]
[a{z)T±a(w)\ namely
=
[a\T±a\
341 = ±V2wkT±a(w)
for all k e
j - ^ ,
= ±V2T±a
in the terminology of X-bracket.
Proof. We need only to prove 2) and it is shown, using 1), as follows: [a(z),T±a(w)]
= =
J^z-'-^afc.TiaH] fcez ±\/2^2z-k-1wk-r±a(w)
=
±y/26(z -
fcez
w)T±a(w).
a We note also the following: Lemma 9.2.5.
dT±a(z)
= ± \ / 2 : a(z)T±a(z)
: .
Proof. Differentiating both sides of
one has
=
±2n z ±2n-l e (n±l)a 3 V ^ f c ) / -1r±„(z)(e»»«8/) (V (9.38)) flV^z)/ +z ±2r, e (n±l)a 0
±^|E»i«-'-1)v±v5W: (•• 0-9))
=
± v ^ z - 1 r ± a ( z ) ( > / 2 n e n a ® / ) ± >/2 : ( j ^ z " ' - ^ ^ ^ ) oo(e»«®/)
J
^U
=
± v / 2 ^ - 1 r ± Q ( z ) a 0 ( e n a ® / ) ± y/2 : (^ajz^-Ar^z)
=
±>/2 : j £ ajz'i-1
=
±v^:a(z)r±a(2):(enQ®/),
proving lemma.
: (ena ® / ) : (ena ® / )
) I±a(z) : (ena ® / )
•
342
9. Vertex
Representations
We now consider the Virasoro field L(z)
:= - : a(z)a(z) : .
Then 1
°°
Q
and so, by the definition (9.42) of a 0 , one has L0{ena®f)
=
{n 2 + d e g ( / ) } - e " " ® / ,
(9.46)
where the degree of an element in C[xj,x 2 , •••] is counted by deg(xj) := j . Lemma 9.2.6.
2)
1)
[Lxa] = {d + X)a.
[Lxr±a] = (d +
\)r±a.
Proof. 1) is clear from (6.104a) in §6.1. To show 2), we apply Proposition 6.1.6 to b(z) = T±a(z) and s = ±y/2. Then we have [: aa :x T± J
= 2 (±V2) : aT±a : + 2 A r ± a = 2(d + A ) r ± a , ar±„
proving 2).
•
Summing up the above Lemmas 9.2.3, 9.2.4 and 9.2.6, one obtains the following Theorem 9.2.7. The map H(z)^V2a(z),
E(z)^Ta(z),
F(z)^r_a(z)
and do i—> — Lo,
K i—> identity operator
is a representation sentation of A\ ' — sl(2, C) on the space U, where
r«(*)/ :=
®exp V~2± ££ ^z*»e^ e^1)- ® exp ((V2± V 5XkJzA > z * ) exp ( - V 2 £ ^ * J-z~*) -fc)/» Xk JJ nez .ez V fe=i / V k=\ (9.47a) (9,
9.2. Basic Representations of sl(2, C)
343
r-a(*)/
J] z-*.e<-«- ® exp ( - V§5>**) «P ( ^ E *L^ k )/. £
z - * - V » ® exp ( - V ^ f > f e ) exp {V~2± J ^ z ^ v
n£Z
fe=l
v
'
fn,
fc=l
(9.47b) for f = ^Znez e " Q ® fn £ U. The sl(2, C)-module U is an irreducible highest weight module with a highest weight vector 1 of weight Ao, and so is isomorphic to the basic representation L(AQ). The field d{z) = Yl dnz~n~2 acts on this space U by n€Z
d{z)
i—> -L(z)
= - - : a{z)a{z) : .
We note that vertex operators satisfy r±„(z)2 = 0
(9.48)
by Corollary 9.1.6. It is sometimes useful to rewrite the above representation into the following form to avoid the appearence of the number \/2 in vertex operators. Putting V2xk=:x'k
(* = 1, 2, 3, • • • ) ,
their partial derivatives are transformed as
A.
dxk
^*i.JL
=
dxk
and so vertex operators T±a(z) follows:
dx'k
T/Q
=
d
dx'k'
and operators y/2ak (k G Z) are written as
r ± a ( z ) ( e ~ ® / ) - z ± 2 n e< B ± 1 > a ®exp f ± f > ' f c / ) exp ^
fe=l
'
f ^ E ^ z " * ) / , ^
h=\
k
and =
V2a-k s/2ak V2a0(ena
® /)
=
V2kxk dxk 2nena ® f,
-
kxk
(k e N), (k e N),
2
d^k
by (9.42). Writing x'k simply as xk, one obtains the following:
'
344
9. Vertex
Corollary 9.2.8. Let t±a(z) Ta(z)(ena®f)
Representations
be vertex operators defined by k z2ne(n+Va®ew(^xkzk\^p(-2JT^^z-k)f \
:=
(9.49a) (9.49 f_a(z)(ena
® f)
(9.49b) for n 6 Z and f 6 C[xi,X2, • ••], namely Ta{z)F
:=
E^e^)«®exp(5:xfc,'=)exp(-2$:-^-2-'=)Fn
=
£
z * " V » ® exp ( E * ^ ) ex P ( -
neZ
^fc=l
2
V
'
f;
^ - z - ^ ^ x ,
fc=l
7
*
(9.50a) f-a(z)F
E z - a » e ( " - 1 ) a ® e x p ( - £ z f c z * J exp [ 2 £ - r ^ - z - f c j F n
:=
n€Z nSZ
^ ^
fe=l fc=l
'
^ V
'
Xk
fc=l
fc
fc=l
' '
(9.50b) /or F = J2nez E(z) H®t~k
e
" a ® f n e ( / ' T / i e n the maV .—> r „ ( z ) , F(z) .—> f_„(z) fc i—> fexfc, #®i i—> 21r— dxk
(forked)
and (H®t°){ena®f)
:= 2nena ® / , /•
d(ena®f)
f
-r)
z-2ne(n-^a®e^(-^xkzk\exp(2Yj-^-z-k\
:=
OO
:= ~ena ® U2+ Ykxk—
PI
if
is the basic representation L(Ao) of A\' = sl(2, C) on the space U. One sees from (9.48) that these vertex operators T± a (z) also satisfy f± Q (z) 2
= 0.
(9.51)
9.2. Basic Representations of sl(2, C) 9.2.2
345
Principal Picture
Let a denote the involution of sl(2, C) defined by a(x)
:= —*x
for all x € sl(2, C),
and consider its eigenspaces So 01
t {xesl(2,C); x = -x}, {x G «I(2, C ) ; fx = x}.
{x e sl(2, C ) ; a(x) = x} {x G *[(2, C ) ; a{x) = -x}
Then cx
00 =
and
o
CHQCXi,
0i
where Xn
0 -1
E-F
1 0
E+ F
and
-
)
•
and i?, E and -F are elements defined by (9.35). It is easy to check the commutation relations of these elements XQ,X\ and H: [H,X-0] [H, Xi]
= 2Xi, — 2XQ, = [E-F,
[XQ, X J ]
E + F] =
2H.
Consider the Lie algebra
0 := ( 0
0o ® A e ( 0
0i ® ^ 0 0 ^ + Cd0
j'6Z 0 dd
3 c ^even
with commutation relations = = =
[dp, uU)] [K,~B]
[u, v]{j+k) + I • Tr(uv)<5J+fe,o • K, J«0)» {0}
for j,fc G Z and u G 0j m o d2 a n ( l w € 0jtmod2> where uyj := u®P. check that the inner product ( | ) defined by («(j)N*))
(*H)
(/f|A-) (K|tiW))
=
tfi+k>0Tr(m>) 2 (do Mo) {d0\uU))
(= := :=
It is easy to
M 0 0
is a non-degenerate invariant bilinear form on g, and that the elements din
'•— t 2n+l
d dt
(neZ)
9. Vertex
346
Representations
act on the derived Lie algebra
fl' : = [§.«] = ( ©
flo®*J)®(
^jEZeven
'
©
91®A®CK
^jeZodd
'
by [d2n. «(j)]
:
= JU0'+2n) •
We now look at the Chevalley generators of g. For this sake, we put b! := iX,
=
(_?.
*)
(9.52)
and e' := ^(H-iX-J, f := ^ ( H + i X i ) . The brackets of these elements look as follows: [h',e']
=
^[iX^H-iXi]
= i { t [ X 0 , i r ] + [X0>JfI]} -2Xj
=
[h',f]
H-iXi
=
2e',
= ^[iX^H + iXi] = i{t[X 0> /r]-[Jf D ,JTi]} -2Xj
=
[e',f]
2tf
- ( t f + iXi) =
= -[H-iX-^H 4L "
2H
-2/',
+' iX^1J = 2, IfrXt]
= iX0 = /i'.
2 Xo
Since T r ( # 2 ) = Tr(X?) = 2
and
Tr(X?) = - 2 ,
and H, XQ, XI are mutually orthogonal with respect to the inner product defined by the trace, one has Tr(e7') =
llt^H-iXiHH
+ iXiJ)
=
\TT(H2
+ Xf)
= 1. (9.53)
So the elements eo:=f'®t,
/0:=e'
ej := e'
/ i := / '
and and
satisfy the commutation relations [ei, fj]
= tfijOti'
c# := -h' ® t° + — , a £ := /i'
(9.54)
9.2. Basic Representations of sl(2, C)
347
and k y > eA = aijej
and
\ai> /j] =
~aijfj,
for i, j = 0,1, where (aij)i,j=o,i
-
(_2
2
)
is the Caxtan matrix of the affine Lie algebra A\ , which implies that g is the afRne Lie algebra of type A\ ', since g is generated by do and e,, / , (i = 0,1) and the invariant bilinear form ( | ) on g is non-degenerate. This Lie algebra g is called the principal realization of A^ . Notice that d0 = 2d+\aX-\K. z o
(9.55)
Actually the element do given by the right side of (9.55) satisfies (d0\d0) = 0,
(d0\K) = 2,
(dolctf) - (do\ai) = 1,
since
1
2
(doK) = 2(dK2+iKK2 = i, 1
(d 0 K)
-2
= 2(dK) + i ( a y | a v )
The brackets of H(j)'s and ^Q(j)'s (i € Z/2Z, j e Z) are easily computed and obtained as follows: [HQ), X 6(fc) ] = 2Xi(j+k), [#o(j)> ^i(fc)] = 2H(j+k),
[H(j), ^i(fc)] =
2X 5 (j +fc) , (9.56a)
and [iJ ( j ) , H(k)\ = j6j+k,oK, [X1(j), Xm] = jSj+kjaK. For simplicity we put
[Xo(j), xo(k)] — -Jtij+kflK, (9.56b)
348
9. Vertex
Representations
for all j ' e Z . Then [H{j), X(k)]
=
2X{j+k).
and U-l)j-2Hu+k) \(-l)i-i . j6j+kt0K
rx ^ i I 0). (*)J
if j + Zeis odd, i f j + fciseven.
We now consider the fields X(z)
:= J2XU)^J~^ jez
H(z)
:=
£ jez o d d
HU)z~'-\
iez Then commutation relations of these fields are given as follows: Lemma 9.2.9. 1)
2)
(i)
[H(z),H(w)]
=
Kdw6odd(z-w),
(ii)
[H(z),X(w)]
=
26odd(z-w)X(w),
(iii)
[X(z),X(-w)]
(i)
+
K-dw6(z-w).
[d(°ddHz), H(w)] =
(ii)
= 2H(w)6(z-w)
[dl°MHz), =
-dH(w)
• 6odd(z -w)-
H(w) • dw6odd(z
- w),
• Sodd(z -~w)-
X(w) • dw6odd(z
- w),
X(w)] -dX(w)
where 6odd(z — w) is the odd S-function defined by (9.26). Proof. Formulas in 1) are shown as follows: (i)
[H(z),H(w)]
=
lHUhHw]z~i~lw~k~1
Yl j,fc€Z odd '
7 ^
_,
J°3 + Jfc,0"
=
(ii)
K ^ jz-'^w3'1 jez o d d
[H(z),X(w)] = Yl
=
J2iHUhX(^z~J~lw"k'1
jez o d d fcezv —>
j € Z o d d nSZ
dw6odd(z-w).
'
9.2. Basic Representations of sl(2,C) 2 J ] z - ' ' - V ^ X ^ t i ; - " - 1 = 26 odd (z - w)X(w). jez o d d «ez
=
(iii)
349
[*(*), X(-w)] = J ] [X(j), X(fc)]z-^1(-«;)-fe-1 j.fcez
=
2(-l)^0+fc)^-1(-«;)-fc-1
^ j,fcez j+fc—odd
-K 52 (-ly^+^-'-^-ti,)-*- 1
=
2
j.fcez
H(n)z->-w-»-i-A'52(-i)j>-i-i(-«»)3'-1
E E jeZnez o d d z_j
= 252 '~
jez
ltui
n i
- E ^(n)«'~ ~ +ii'52^~,'~lu,,'~1
jez nez odd jez = 26(z - w)H(io) + K • dw6(z - w). Next we show the formulas in 2). (i) [d
2w-k-i
v
52 52
(n-2j)
j€ZneZodd
„..„ ( n + l ) - ( 2 j + l)
kH,
(.k+2j)
^ ^ - v -
1
52 (n + l)H(n)w-n-2Y,*-2i-2"li+1
nez odd
jez
"ezodd
-52(2j+i)z-«-2«,« 52 ^n)^-"-1 j'eZ
nGZ o d d 2
9«, £ z - « - t o 2 j + i iez
-dH(w)
(ii)
• Sodd(z -w)-
dw6odd(z
dd
- w) • H(w).
}z-2i- 2 w - f c - l
[d(° >(z), X H ] = 52 [d2j,X{k) J.fceZ
=
E
(»- ^2 j)^
i l l
„ K X
-2,„ (n)Z "2 i "">' 2 2 n r
2j
-n-l
52(n+i)x(B)u,-»- 52z- ''-v + i nez
jez n£2
9. Vertex
350
Representations
- £(2j + l)z-*-*vF J2 X(^-n~1 jeZ
n€Z
>
v
'
jez
=
-dX(w)
• 6odd(z -w)-
dw6odd(z
- w) • X(w).
a Comparing Lemma 9.2.9 with (9.17) and Corollary 9.1.11 and Lemma 9.1.13, one obtains the following: Theorem 9.2.10. The map n : H{z)
__> a (odd) W )
x{z)
l
v^\z)z-1
__>
(9.57)
and do i—> — L0°
— - • Id,
K i—> Id = identity operator
o
is a representation of A\' — sl(2, C) on £fce space # = C[XJ ; j G N 0 jd]. 77ie sl(2, C)-module $ is an irreducible highest weight module with a highest weight vector 1 of weight A0, and so is isomorphic to the basic representation L(Ao). We note that the above (9.57) is written as follows: H{z)
_>
The term —\-Idm
fl(odd)(z)>
Y,X{j)z-i jez
—> \ v ^ \ z ) .
(9.58)
7r(do) is due to (9.55).
N o t e 9.2.1. The appearence of a purely imaginary number | in the coefficient of the vertex operator in the above theorem may look curious at a glance, but it
is quite natural in view of (9.52) and (9.54). To see it, we compute the action of the element X(0) on the highest weight vector 1 € 3". Noticing that
vra\z).i = i+o(z), r(odd)
one has
$>-M*o))i = 5 - i + o(z) l
jez by (9.58), and so 7r(X (0) )l =
\ \ .
9.3. Construction of Basic
Representation
351
Then, from (9.52) and (9.54), one has 7r(/i'®t°)l = «rpr ( ( ,))l and so 7r«)l TT«)1
= =
I.l+I 1 •2.1+1.1
-Tr(h'®t°)l+y(K)l n(h'®1?)l + lir(K)l
2
x
1,
o,
x
^ 2
which show that the weight of the highest weight vector 1 e 3 is AoWe note also that the factor | in Theorem 9.2.10 is, of course, in good coincidence with Theorem 9.3.17 in §9.3.2. To see it, we calculate the number ca in (9.108) for A\ ' as follows. Since 9
=
1 2ai
=
1 v 2ai (
V®t° +
(9.54))
2
1
(
-XQ
(9.52))
®t° +
2
4 K
7'
one has (Pl*(0)) =
l(Xo\X-0)
= ^lr(X 0 2 )
=
-i.
(9.59)
-2
So the number ca in (9.108) becomes
__ Ca
~
(pi^(o)) _ _ ^ h
2
_
i 2'
(9.60)
which coincides with the coefficient of the vertex operator in (9.58). 9.3
C o n s t r u c t i o n of Basic R e p r e s e n t a t i o n
It is known in [113] and [135] that, for an affine Lie algebra g over a finite dimensional simple Lie algebra g, the basic representation has various kinds of explicit construction corresponding to the conjugacy classes of the finite Weyl group. In this picture, the homogeneous realization is the construction corresponding to the unit element in the Weyl group, and the principal realization is the one corresponding to the Coxeter element. Also the explicit construction of the basic representation for all elements of the Weyl group is achieved for affine algebras sl(n,C) in [108] and [132]. In this section, we give a sketch on the homogeneous and the principal constructions of the basic representation of a non-twisted affine Lie algebra
352
9. Vertex
Representations
g over a finete-dimensional simple Lie algebra g of rank t with a symmetric Cartan matrix. Let A — (aij);,j=o,--,f be the Cartan matrix of g. Then our assumption is that A is symmetric and A := (a,ij)ij—it... j is the Cartan matrix of g. We fix a Cartan subalgebra f) of g, and let A (resp. A + ) denote the set of all roots of g with respect to f), and II = {ai, • • • , on} be the set of simple roots. Let ( | ) be the non-degenerate invariant bilinear form on g such that (a\a) = 2 and Ei,Fi,Hi
for all a G A ,
(9.61)
(i — 1, • • • , £) be Chevalley generators of g, namely
[Ei, Fj] = SijHi,
[Hi, Ej] = ciijEj,
[Hi, Fj[ = —aijFj.
Let 0 G A + be the highest root, and choose root vectors E±g G g±e such that (Ee\E^0) = 1, namely [Eg, E-9] = 9. 9.3.1
(9.62)
Homogeneous Picture
In this section, we describe the homogeneous construction of basic representation. For this sake, we consider the homogeneous realization of an affine Lie algebra g:
g := f ^2 « ® *" ) ® CK ® VnGZ
Cd
'
/
with the bracket [u(j), U(fc)] [d, uU)] [K,g]
:= [u, u](j+/b) + 3 • (u\v)Sj+kt0 := ju(j), := {0},
• K,
for all u, v G g and j , k G Z, where UQ) :—U® P as usual. It is the homogeneous Heisenberg algebra \) :=
(fj®C[i,f~ 1 ])©CA'
with its center CK and commutation relations [u(j),V(k)]
•=
j6j+k,o-K
that plays an important role. We consider two representations it\ and -KQ of this Heisenberg algebra. -K\ is the so called Schrodinger representation on a
9.3. Construction of Basic
Representation
353
certain symmetric algebra and n0 is a representation on a "twisted lattice" with a non-abelian product. The basic representation is constructed on the tensor product of this symmetric algebra and the twisted lattice. First we construct the representation m. Let C[t _ 1 ] be the ring of polynomials in t _ 1 , and C[£ _ 1 ]_ := i _ 1 C [ t - 1 ] be its subring consisting of polynomials without constant terms. We consider an abelian subalgebra oo
of h, and its symmetric algebra S'(f)_). This symmetric algebra S'(h_) has the natural gradation
S(i)-) = 0^ (fc) (^-), fc=0
where S^k\\j-) ments in h - :
is the linear span of symmetric products of fc-numbers of ele-
{U1®t-n)---(uk®t-jk)
=
Ul{_hy-UH_jk)
for « i , - - - ,Ufc € f) and ji,---,jk € N . In the calculation involving vertex operators, it sometimes is necessary to consider the "completion" of S'(h_), consisting of all elements of formal infinite sum oo
5> fc
(IkeSwi)),
fe=0
or, in other words, it is the projective limit:
Sfi-) := Um05^(F-). fc=o For M € I and 0 / j e Z, we define the operator wi(u^)) follows: (i)
When j > 0, we consider a map
A,0) : ^ q r ^ ^ c defined by A.0)(w®*"*)
:= «j,*(«|«).
on S(f)_) as
9. Vertex
354
Representations
This map DUu) is uniquely extended to a derivation of 5(h_), namely A.0)((«i®t~fcl)"-(t*.®t_fc")) «
=
A
£ W«l«i)("i ® *~fel) • • • («* ® *"fci) • • • K ® *"fc"), t=i
where the hat "A" on the top indicates to remove this element. We put (ii)
When j < 0, wi(u^)) is the multiplication operator by —juy).
(iii)
When j = 0, 7Ti(u(j)) := 0.
In other words, the operator Du... (j > 0) may be viewed as the "contraction" operator by the element u^ or the differential operator dua . Then one has M « ( j ) ) . Mv(k))]
= jSj+k,o • {u\v)
(9.63)
for w, v € ^ and j , k G Z, and so, by letting 7ri(K") be the identity operator, K\ is a representation of the Heisenberg algebra h. We write simply u^ in place of 7Ti(u(j-)) if there may be no confusion. For u et) and j € N , the operator
n=0
is denned on both spaces 5(h) and its completion 5(h), and
e«<-» := £ (*!(«(_,-)))" n=0
n:
is an operator from 5(h) to 5(h). It is easy to see that these operators satisfy the following commutation relations: H j = k, c«o>6«(-*> = M"l">e-(-«e««> | e «(_ f c ) e « U ) if J V A; and e"u>*n(t>(_fc))
=
7ri(« ( _ fc) )e'*o)+i(«|t;)e , 'o).« i , fc
for u, v e h and j , A; e N . Then the formulas of Lemma 9.1.2 are written in this formulation as follows:
9.3. Construction of Basic
Representation
355
Lemma 9.3.1. Let u, v, u', v' eb and j , k G N . Then 1)
(e^-^e"")) o (e"'(-i)e"o)) =
2)
[e"<-'>e"«), e ^ - ^ e " ^ ) ]
e i ( " l "' ) e u (-^e u o) )
— ^gi(«l«') _ e i M « ' A e(«+"')(-.j)e(t'+,'')
[ffi(«'0))» eu<-k>e"
4)
K K - j ) ) , eu<-*)e"(fc']
fc
j(u'\u)eu(-»ev^6jtk, -j(u'\v)eu^-^ev^6jtk.
=
Lemma 9.3.2. Let a, 0 € fj and j G N. 7%en e ^(
a
1)
e /"ri(^(-i)) e ^l(<*(i))
2)
e** 1 ^")^!^-,-)) = Tr1(/3{_j))ex"1(a^
=
Wg^l(a(j)) e ("fiW(-i))
)
\(a\f3)ex^a^K
+
For the description of the representation 7r0, we consider the root lattice
Q
= Y,Za = tE= i Z a a£A
and a map e : QxQ
- -
{±i>
satisfying the following conditions (i) and (ii): (i)
(bi-multiplicativity) e(a + a',0) e(a, 0 + 0')
= =
e(a, 0)e(a', 0) e(a, 0)e(a, 0')
for all a, a',0,0'
'(_1)(««K) (ii)
(asymmetry)
e(cti,aj)
_ e Q,
if i < j if i = j if i > j .
= <-1 1
From this condition, one easily sees the following: e(a,a) e(a,-a) e(a,0) e(a,0)
= = = =
(-l)iH-) e(-o,a) e(0,a) e(-a,0)
= = =
(-l)iH«) l e(a,-/3)
=
e(-a,-/3)
356
9. Vertex
Representations
and e(a,f3)E(J3,a) for a,/3eQ.
=
(-l)Wfl
It follows that e(±a,±a)
= -1
(9.64)
for any choice of ± , if a is a root. The importance of this function e is that this gives the structure constants of a simply-laced finite dimensional Lie algebra g. Namely it is known (see e.g. [100] §7.8) that one can choose a family of root vectors {Xa}ae^ satisfying [Xa, Xp] [Xa, X-a\
= =
if a + (3e
£(a,(3)Xa+p —a.
A,
Let aeQ
denote the (non-commutative) associative algebra over• C with the twisted multiplication eaep
:= e(c*,/?)e a+/3 .
(9.65;
The Heisenberg algebra rj acts on the space Ce{Q} by 7r
o(«(j))(eQ) := 6jt0(u\a)ea
and
ir0(K) := 0,
so the space &
••=
C£{Q}®S(t)-)
is a f)-module by n := -KQ ® ~K\. Notice that uyj (j ^ 0) act on the second component S(t)-), whereas W(0) a£^s o n the first component Ce{Q}. So one may simply write
"Kfl) = \wii;u)\
*i#°
(9 66)
-
The action 7r(it(j)) on each element in §e is explicitly given as follows:
={? 7 ® t y> / f
for u e i), j e Z, 7 e Q and / e S(i)-).
*j*°n
(9-67)
Then, by (9.63), it is easy to see that
[*•(««))» T(«(fc))] = J*j+fc,o(«|w)
(9.68)
9.3. Construction of Basic Representation
357
for it, v G t) and j , k G Z. For u G J), we consider a field := J ] 7 r ( u 0 ) ) z - ' » - 1 .
uW(z)
(9.69)
Then (9.68) gives the following commutation relation of fields: [u(ir){z), v^\w)}
Lemma 9.3.3. nameij/
= (u|v) • dw6{z - w),
[ ( u ^ ) ^ ] = M«)A, /or u, v G J). This lemma shows that «'''(z)'s are free fields and that their normal products are commutative . nW(z)vW(z)
:
=
: B W(«) B W( Z )
:
(9.70)
for all u,w G f) by Corollary 6.1.13. For a G Q, we consider the following operators on the space $£ :
exp
V
i<0
(
OO
J
J
V
j=l
^
x
-Y,°fz~j)'
(9 71a)
(g^M-gT*")'
-
and ra(z)
:= zaeaVa(z),
(9.71b)
namely rQ(2)(eT®/)
:=
a+
Z^^e{a^)e
^
®Va{z)f.
(9.72)
These operators Ya (z) are called vertex operators. Differentiating both sides of (9.72) by z, one has ~Ta(z)(e^®f) 02
=
(a|7)2-1z(aWe(a,7)ea+-1'®yQ(z)/ >
v
r„a(0)(e-»®/)
'
358
9. Vertex
Representations
z< a M £ (a, 7 ) e Q + 7 ® ~^v^)f v 2 /
+
N
J"€Z
Thus we have proved the following property of Ta(z). Lemma 9.3.4.
= : a ( , r ) (z)r Q (z) :
dTa(z)
foraeQ.
Vertex operators satisfy the following commutation relations with fields Lemma 9.3.5. Let u 6 i), j; e Z and a e Q . TTien
i)
[^K)), ra(z)] = («|a)z* • ra(*),
2)
[u ( , ) (2), r a ( w ) ]
= (u\a)Ta(w)6(z
[(«W) A r a ] = ( « | a ) r a
and
- w),
rcameZy
[(r a ) A «W] = - ( « | a ) r Q
(9.73)
in the terminology of X-bracket. 3)
: u<*>(z)r a (z) : - : Ta(z)u^(z)
: =
4)
: r a (z)a<*>(z) : = {l - ( a | a ) } 0 r Q ( z ) .
{u\a)dTa(z).
Proof. 1) First we consider the case when j ^ 0. Then 7r( U(i) )
=z(o|7)e(a,7)eo-HY®7r1(ti0))Va(z)/>
ra(z)(e*®/)
z<"lT)e(a,7)e°+-'®V < i t (z)/
and rQ(z)7r(u(j))(e7®/)
= =
^(^(e^TnC^))/) z
And so, by using Lemma 9.3.1.3), one has [*(«(,-)), r a ( z ) ] (e 7 ® / )
=
z<«l7)e(a> 7 ) e a + 7 ® M W ( j ) ) , Va(z)] f v
«, zi(u\a)Va(z)
= =
zj(u\a) • z( Q W£(a,7)e a + 7 ® Va(z)f zJ'Ha)-ra(z)(e7®/).
'
9.3. Construction of Basic
Representation
359
Next in the case j ; = 0, one has 7r(«(0))
= H a + 7)-2(aHe(a,7)e^®Va(z)/
ra(z)(eT®/)
z(°\~/'>e(a,'f)ea+-i®Va(z)f
= (u|a + 7 ) - r Q ( z ) ( e 7 ® / ) and ra{z)*(u(0))(e'®f) "
=
v («|7)eT®/
(u\>y)-ra{z)(ei®f)
'
and so [7r( W(0) ),r Q (*)](eT
2) is easily shown from 1) as follows:
[««(*), ra(«o] = £ Huu))> r«M>- j_1 =
(«|a) y j i w ^ ' z - ^ - 1
= (u\a)6(z — w).
3) and 4) follow from 2) and Proposition 6.1.12 and Lemma 9.3.4. Let {«t}i=i,—,< and {u*}»=i,—,* D e consider the field L{z)
Dases
•
of fj satisfying («i|uJ') = £i,j, and
:= ^ ^ - W W t t f ' W : .
(9.74)
This field L(z) does not depent on the choice of bases, so one may put L{z) := \Y,:u
(9.75)
by making use of an orthonormal basis {tii}i=i,... / oil). Then 1
e
e
°°
Ft
L0 = - E ^ W 2 + £!>*-»> &rrr
( 9 - 76 )
and, by (9.67), one has Lo(ef®/)
= (^+deg(/))e^®/,
(9.77)
where the degree of an element in 5(f)-) is defined by deg(u(_ n) ) :— n
for u e 5
an
d n G N.
(9.78)
360
9. Vertex Representations
Lemma 9.3.6. Let u e | and a 6 Q. Then [ L A u w ] = (d + \)uM,
1)
[(u w ) A L] = Au (7r) .
and
2) (i) [Lxra] = j d + ^ A J r Q , (ii)
KTa)AL] = / ^ - l W
^ . A T
a +
0
[LAL] = (d + 2 A ) L + - A 3 .
3)
Proof. 1) We notice that [ ( «
=
W
) A : ^ : ]
: [(u^hu^} >•
v
uf
: + / * [[(«<')) A «J r) ] „«<*>] dM
: + : uf {(u^uf] N
'
\{U\UJ)
v
'
Jo
\(U\UJ)
^
s/
'
X(U\UJ)
2\{u\ui)u^).
= Then
£
£
[(u^)xL] = ^ £><*>)* :«<*>«<-> :] = A]T>| U >f> = A«W. i=i
i=i
Then, by the skew-symmetry of A-bracket, one has
[ W > ] = -[(u(,r))_a_AL] = -(-a-AJnW = (3 + A)«W, proving 1). To prove 2), we apply Proposition 6.1.6.1) to a(z) := Uj (z), b{z) :— Ta(z)
and s :— (uj\a),
in view of (9.73). Then we have [: ufuf
:A r a ]
= 2 ( u » : ufva
and so e
[LxTa] = £ [ : u^uf :A Ya\
: +X(Uj\a)2Ta
,
9.3. Construction of Basic Representation
361
t
i
= £ ( « > ) : «Wra:+^£K-|a)2r0 :a(l)ra: 9r„
+^(a|a)2ra,
(V Lemma 9.3.4)
proving (i). (ii) is shown from (i) and the skew-symmetry:
[(Ta)xL] = -[L-a-xTa] = - Id -
{
^-(d +\)\ra
To prove 3), we notice that
[Lx:ufu^:] = : [Lxuf] u^ : + : «<*> [ L ^ ] : + 1 * [[L^™] ^ l d " (d+\)u^
(9+A)^.< , r ,
(0+A) u <*>
0Ov„("), (-M+A)[(«f )„«V
(d + 2A) : t i ^ u W :+f(X-li)
[(uf)^}
JO
>
d»
v
'
/ 0 0 i 0O\
= (d + 2X):u(f)u
2[LAL] = £[L A : t^u™ :] = {8 + 2A) £ : ufuf i=i
: +^A3 ,
j=i 2L
proving 3).
D
Thus L(z) is a Virasoro field. In particular when a € A, the square length (a\a) is equal to 2 by (9.61), and then 2) of the above lemma gives the following operator product of the Virasoro field L(z) with the vertex operator Ta. Corollary 9.3.7. Let a eQ. Then [LxTa]
= (d + \)Ta
and
[(Ta)xL]
= XTa.
362
9. Vertex
Representations
To compute the commutation relations of vertex operators, we introduce the following operator: •3
Va-p{z,w)
:= exp ^ - ^ V=l
r — ^ — I exp - ^ 3 J \ 3=1
• 3
}
for a, ft eQ. Lemma 9.3.8. Let a,fieQ. 1)
Va{z)Vp{w)
2)
Va;(,(z,w)
=
Then lZjW((-±—\
jVa;0(z,w), (z-w)'^2'ir(ai_j))w:'-1Va+l3(w)
= Va+/3(w) +
3= 1 OO
+(z - w)Va+fi(w)
Y^A^i))™'3'1
+ 0((z - w)2).
i=i
Using this lemma, the product of vertex operators are given as follows: = = =
=
ra(z)T0(w)(e^ 0 /) w^h(/3, 7) • Ta{z){e^ ® Vfi(w)f) w^We(/3,7) • zlaW-*h(a,0 + j)ea+0+^ <2> Va(z)Vp(w)f (a z l"+T V M e ( a , p)e(a, -y)e(p, -y)ea+0+^ (( z \ ~ ^ \ ^ ( a | 7 ) wWMefa, (l)e{a + f3,7)ea+"+T ® t,,„ ((z - w )< a l">) Va;/3(z, w)f. (9.79a)
Replacing a <—> /3 and z <—• w, one has
r>HrQ(z)(e^®/) =
^ h f ) ^ ) ^
a
) e ( a + ft 7 ) e «+/3+7 0
iwz
(^w _ 2 )<«lfl) v a ; / ,( z >
w)f.
(9.79b) Prom these formulas, we obtain the following Lemma 9.3.9. Let a, (3 eQ. ?a(z), i y ™ ) ] ^ ® / )
Then =
2(
a
W f f l < ^ £ ( a ^ ) £ ( a + i3, 7 )
x ( t , , „ - w ) ((« - ™) W / 3 ) ) • e a + / 3 + 7 ®
VaiP(z,w)f.
9.3. Construction of Basic
Representation
363
Proof. By (9.79a) and (9.79b), one has
=
ra(z),rp{w)\(e>®f) Q h V / 3 M e ( a + £,7)e a + / 3 + T
2(
®{*»,«(e(a,/3)(*-«0(a|/S)) - *»,.( ^ ( M (tfl-2)(a|"))Wai/J(z>«>)/ (-l)<«l0) e (a,/3)
=
z(«WratfWe(ai
/3) e (a + / 3 , 7 ) e
»{*.,« ( ( z - H
W/3)
Q+
^
+7
) - W ((-l)(a|^-z)H^)}v^(z,™)/
» ( * , , « - t»,«) ((« - ™ ) H / J ) ) V a # ( * , t i ; ) / ,
proving the lemma.
D
By Lemmas 9.3.8 and 9.3.9, we obtain the following: Proposition 9.3.10. Forct,f3 £ A, the commutation relations of vertex operators Ta(z) and Tp(w) are given as follows: 1)
If {a\j3) = - 1 , then [La(Z),
2)
i-0(W)\
— {Lz,w-lw,z)\
—
I.
/ / (a|/3) = - 2 i.e., 0 =-a , then
[ra(z),i>H] = - ( ^ - - w ) ( ^ + ( ^ p ) , 3)
[ra(z), T^H] = 0
*/ (a|/3)>0.
Proof. 1) If (a|/3) = —1 then a + /? is a root and, by Lemma 9.3.9 and Lemma 9.1.4, one has
[ra(z),rp{w)](e'®f) z
W7)
(/,hr) w
e ( a , /?)e(a + 0,7)e Q + / 3 + 7
« ; ( < » I T ) + 0 ( Z — U))
® (<•*,«, - V * ) ( \z-wj
)
Va-p(z,w) ^ ^ ,
f
Va+f3(w)+0(z-w)
r„ +3 (iu)(eT(8/)
364
9. Vertex Representations
proving 1). 2) By Lemma 9.3.9, one has
[raw,r_a(«))](e''®/) =
zHT)«;-H7) £ ( a _ a ) e7 g, (4
_
)f
*
) Va;-a(z, W)f
=
/Z\H7) / 1 \ - (- ) e 7
-1
(9.80)
We rewrite this right side as follows. By Lemma 9.3.8, one has Va;-a(z,w) oo
=
I + (z - w)^27rl(a(-j))wJ1 3=1
7+(z-«;)
w
)/;wl(a(j))w~jl
J=l
+ 0 « Z - W)2)
=
oo
+ (z ~
^("(j))™^"1+£>((•* "™) 2 )-
JZ J€Z-{0}
Also notice that (-)
=
1+
+ 0((z-w)2).
= l + a 7
Substituting these into (9.80), one has
[rQ(z),r_aM](e7®/) = -u -t )( \ 2 + W ^ _ 1 (z — w)
jez-{o} \ ( ! \(z — w)z
i
z—w J
7r(a(u>))\ z—w J
since 7r(a( 0 ))(e 7 ® / ) = (a|7)e 7 ® / . Thus 2) is proved.
•
Theorem 9.3.11. Letg be a symmeric affine Lie algebra over a finite-dimensional simple Lie algebra g. Then the map n defined by u(z)
.—> «<*>(*),
Xa(z)
h—»
Ta(z)
for u £ h and a € A, and d i—> — LQ,
K
I—> Id = the identity operator
9.3. Construction of Basic
Representation
365
is a representation of Q on $E = Ce{Q} ® S(f)_). The g-module ffe is an irreducible highest weight module with a highest weight vector 1 := e° ® 1 € $e of weight A0, and so isomorphic to the basic representation L(AQ). n 2 The field d(z) = 5Z acts on this space $e by n ez ^n^ -mez' d{z)
_>
-I(,),-1^: 2
W(^W
U
W
:
i=i
where {UJ}J=I,...
,e is an orthonormal basis of I).
Example 9.3.1. In the case g = A\' — sl(2, C), t/ie twisted lattice CE{Q} = ^2 Cenai is a commutative associative algebra with multiplication nez e
mai n o i
_
/_j\mne(m+n)ai
since e(mai, n a a )
mn
= e(au
ai)
(-l)r
=
Putting
e""1 := ( - ^ ^ e " 0 "
/orneZ,
this multiplication is written as V
'
"" 2
— 1)
i
v
e(m+n)ai
' _
g{m+n)ai
Namely the twisted multiplication turns out to be a usual "non-twisted" multiplication with respect to a basis { e n a i } n g z We now look at the definition (9.72) of the vertex operator. In the case of g = sl(2, C), this is written as follows: rQl(z)(e"ai ® /)
=
rai(z)((-l)
2
nll 11
f
e(ai, na{) e ^ s
= Multiplying (—1)
z^na^
v (-1)"
1
^ ® Vai(z)f
'
( - l ) n z 2 n e ( n + 1 ) a i ® Vai (z)f .
to 6o#i sides, one has
e"^®/) = z2"(-l)^(-l)"/"+1>"i®yai(z)/, >
v
'
366
9. Vertex
Representations
namely rai(z)(e"Q1®/)
z2ne
=
which is the same formula with the vertex operator action (9.38) given in §9.2.1. Thus one sees that Theorem 9.2.7 for g — sl(2,C) is deduced from Theorem 9.3.11 by changing a basis ofCe{Q} into e" Ql = ( - l ) n i ¥ i l e " Q l . 9.3.2
Principal Picture
For a root a = E i < i < £ m * a * or"fl>w e P u * ht(aO := J2i
g =
8U),
Y.
( 9 ' 81 )
j€Z/fcZ
by putting gU)
06A
ht(a)=i mod h
if 3 = 0, for each j e Z/hZ. The element
E := J2Ei
+ E
-o>
t=i
called the cyclic element in [129], is a regular semisimple element, and so its centralizer 5 in g is a Cartan subalgebra, called the principal Cartan subalgebra. Let A denote the set of all roots of g with respect to this principal Cartan subalgebra s. For each root a e A 1 ^, we denote its root space by g^, i.e., fl£" := { l e g ; [H, X] = a(H)X
for all H e s}.
The linear transformation w := e x p ( - p a d ( p v ) J
(9.82)
of g satisfies w(X)
= e^X
for all X e jjU).
(9. 83 )
9.3. Construction of Basic Representation
367
Notice that w(E) = e~rrE since E G g^K So w stabilizes the principal Cartan subalgebra s and defines an element in the Weyl group of (fl, s), called the Coxeter element. Since E e fl^1) is a homogeneous element with respect to this gradation, the principal Cartan subalgebra I is the direct sum
J2 *ng ( j ) .
s =
jez/hz Let 1 = mi < mi < ••• < mt-i
<me = h — 1
be the set of exponents of g. It is known in [129] that (i)
s n g^ ^ {0}
(ii)
dim(infl(j))
(iii)
mi 4- me+i-i
<=>•
j is an exponent of g ,
= fl{» ; = h
rrn=j}, for all i.
Since the invariant inner product ( | ) satisfies (fl(i) lfl(j)) = {0}
if i + j & 0 mod h,
(9.84)
one can choose a basis S^ (i = 1, - • • , i) of I such that S^esng{mi) for i,j = l , - -
(S [ i ] |S [ i l ) = M i + j , f ,
and
(9.85)
,£.
Since Xa G g ^ is a root vector, one has [S®,Xa]
=
a(S&)Xa,
namely
J2 [S®,X&] = a(S®) Yl jez/hz
X{ }
J-
( 9 - 86 )
jez/hz
So, comparing the components in g(m*+-?), one has [SM,XW]
= a(5(il)xim-+j).
(9.87)
Note 9.3.1. The exponents for simply-laced Lie algebras are listed as follows:
368
9. Vertex Representations
Ae
:
1,2,3,-- , £ + 1 ,
Dt
:
1,3,5,-•• , 2 ^ - 3
E6
:
1,4,5,7,8,11,
E7
:
1,5,7,9,11,13,17,
E8
:
1,7,11,13,17,19,23,29.
So the exponents
and i - \ ,
{mi}i
1,3,5, • • • , 2n - 3, In - 1,2n - 1, In + 1, • • • , An - 3. Namely mn = m„+i = 2n — 1, and so, in this case, both S' n ' and S^™"1"1' belong to g ( 2 n _ 1 \ 77ws is the reason why we cannot use a better notation S^"1^, but have to make use of the notation 5'*' in this book. Lemma 9.3.12. Let a G A** and
j'GZ/hZ
6e zis roo£ vector. Then 1) 2)
(SMpd^) ^
= 0
e^I«>
/ o r * = !,••• ,1. e 0
jez/hz
Proof. 1) Since each root space g ^ *s orthogonal to the Cartan subalgebra, one has 0 = (SW\Xa)
=
V .J=ii,
(5w|Xij))
(SW\Xi-mi>),
= b
(••• y (
984
»
proving 1). 2) follows from
^*£~^
jez/hz
• It is known in [129] that A ^ is the union of ^-numbers of (lo)-orbits, and that wk(a) ^a
if l
for all roots a. Prom this, one sees the following:
(9.88)
9.3. Construction of Basic Representation
369
L e m m a 9.3.13. Let a e A1", and X±a — Z}jez//iZ^±a tors. Then 1)
Xa
(j € Z/hZ)
2)
(XW\X™)
$±a be root vec-
are linearly independent.
=
\{Xa\X_a)6j+kfimodh.
Proof. 1) We consider wk(Xa)
e g^ka for A; = 0,1, • • • , h — 1. Then we have
k w
e
e
{xa) = 53 " W ) = £ iez//iZ
^^')
(9-89)
jGZ/hz
by (9.83), since Xaj) eoU)k Notice that w a (0 < k < h — 1) axe distinct roots, and so wk(Xa) (0 < k < h — 1) axe linearly independent which implies, by (9.89), that Xa3' (j G Z//iZ) are linearly independent, proving 1). 2) Since wr(Xa) e Q^ra and w s (X_ Q ) e fl?^^ and wra ^ w3a unless r = s modh, one has (wr(Xa)\wa(X_a))
= 6r,smodh(Xa\X„a).
(9.90)
Putting C := e"^1, these elements are decomposed, by (9.89), as follows: wr(Xa)
=
Y.
C J > ^i J )
and
w°(X_a)
jSZ/hZ
=
^
C*«A1*2,
JfeGZ/hZ
and so
= 53 cj>+fcs(^j)i^)
K(xa)i^(x_a))
j',fc6Z/hZ (v(9 84))
-
53 C^-H^l^ia^)-
(9-91)
jeZ/,z
Prom (9.90) and (9.91), one has
(Xa\X„a)6r,s = 53 C ^ - ' W l * ^ ) jez/hz In particular letting s = 0, this gives
(Xa\X-a)6r,0 =
53 Cjr(*£)l*iai)) iez/hz
(9-92)
9. Vertex
370
Representations
for r = 0,1, • • • , h — 1. In matrix form, this is written as follows:
/
f(Xa\X-a)\ 0 =
B-
(XW\X«1) \
(xP\x
(9.93)
where /I 1 -B := (CJ )j,fe=o,i, - ,fc-i
1
c2 c
V Ch - 1
1
1
c 42 c
£2(h-l)
£2(fc-l)
£(h-l){h-l) J
Notice that £?*.B is a scalar matrix, namely B*B = h • I, where B* := lB is the Hermitian conjugate of B. So multiplying B* to both sides of (9.93), one obtains 1
i i ^1
i 2
1 1
r
r r2
\ £-2(/»-l)
C-4
^-(fc-1)
£-2(h-l)
(•-(h-ixh-i)J
/(Xa\X.a)\ 0 0 y
( (x^\x^l) \ (xPix") 2) 2 (xi |xi 2)
yixf-^x^)) proving 2)
D
It follows from this lemma that X& / 0
for all j e
(9.94)
Z/hZ.
Let {71, • • • , je} be a set of all representatives of (lo)-orbits in A . Then, by Lemma 9.3.12.2) and Lemma 9.3.13, one sees that {S [i] , X«> ;
i, k = 1, • • • , I, and j e
Z/hZ}
(9.95)
9.3. Construction of Basic Representation
371
is a basis of g. We notice that it is not always possible to choose representatives {7i i """ J 7*} from the set of simple roots as one sees from Examples 9.3.3, 9.3.4 and 9.3.5 at the end of this section. A Coxeter element w is not uniquely determined, but it is characterized to be a product of all simple reflections:
where a is an arbitrary permutation of 1, • • • , L It is known that these elements are W^-conjugate. The so called "principal affinization" of g is the Lie algebra
with bracket [u(i)> v(k)] [d0, uU)} [K,B]
1
=
[«» v] (j + k) + - • (u\v)6j+kt0
• K,
U
= 3 U)> = 0, where u^ :— u ® P. To show that this Lie algebra g p r is an affine Lie algebra of type X\ , we ve put i := Ei{1) = Ei®t,
fi := F i ( _i) = ^ i ® * " 1
and aVi •= Hm + ^-K = Hi®t° + ^-K
(9.96)
for i= 1,- •• ,£, and e0 /o
:= -B-e(i) := £7«(_i)
<*o == -«(o) + ^--K"
= E-g®t = Eg^t'1 =
-0®*° + ^ " ^ ,
where Ei,Fi,Hi (i = 1, • • • , ^) are Chevalley generators of g, and # is the highest root in A + , and E±g are the root vectors satisfying the condition (9.62). Then it is easy to see that these elements satisfy the commutation relations [ei, fj] = 6itjay,
[0%, ej] = Oijej,
[a-,ft]= -(Hjfj
(9.97)
372
9. Vertex
Representations
for i, j — 0, • • • , I, where (fflij)»,i=o,— ,i is the affine Cartan matrix of type X(t '. Notice that the symmetric bilinear form on g1"- defined by :•-= (u(i)\do) :--= (d0\K) :-. (do Mo) :==
(u{i)\vU))
Si+jfi(u\v), (.u(i)\K) h, (K\K)
= 0, = 0
is g^-invariant and non-degenerate. This, together with (9.97), shows that g1"" is an affine Lie algebra of type X\ , and is called the principal affinization of g. Since g is simply-laced, the Cartan matrix of gw is symmetric, so the dual Coxeter number hv is equal to the Coxeter number h, and the square length of all real roots of g*"" is 2, and simple coroots a^ are naturally identified with simple roots a,. Notice that the element d0, called the energy operator, satisfies &i{do) = 1
f° r i = 0,1, • • • ,1
since all e^ belong to g^ ® t1 which implies [do, e<] =
e<
ati(do)ei.
So (9.98)
do = p mod C6.
This relation is more exactly written as follows with respect to the usual choice of p: p = hAo + p i.e., (p\p) = (p\p). Lemma 9.3.14. 1)
do =
2)
(Ao|do)
p-—(p\p)K,
2ft (P\P) • Proof. 1) It suffices to show that the square length of the right side is 0, which is shown as follows: -i
-,
2
P-Yh(P\P)K
(pM-^miPiK)
= o.
2) holds since (Ao|do) =
{A0[p)--±(p\p)(A0\K)
=
-^(P|P). D
9.3. Construction of Basic
Representation
L e m m a 9.3.15. Let a e A
373
and
J2 x « j )
xa=
e
«sr
jez/hz 6e its root vector. Then (dolxM)
= 0.
Proof. Since X a e g ^ , one has (d 0 |[S W ,X Q ])
= a(5W)(do|X a )
=
a(SM)(do|X<°>).
(do|xi0))
Using the fj-invariance of the inner product ( | ), the left side of this equation becomes as follows: (do\[S®,Xa])
= ([d 0 , S®]\xa)
=mi{S®\Xa)
= 0.
rmSW
So one has a(SW)(do\xM)
= 0
for all SM, which implies (do\XJ?}) = 0, proving the lemma.
D
L e m m a 9.3.16. Identifying an element H in t) with an element H ® t° € i) ® t° c g1"", the following holds: i
H+\{p\H)K
e
£ ) C a >v i=l
Proof Since if 6 f), one can put if = N ^ i # i i.xi £ C). Then, by (9.96), one has
i=i
^
n
/
i = 1
«i=1
(PIH)
=
Y,xi<$-\iP\H)K, i=l
proving the lemma.
D
9. Vertex
374
Representations
We put :
m+nit+i)
= rrii + nh
(9.99)
and Si+n{i+i)
S®®tmi+nh
~
G 5
(9.100a)
and S[i+n(i+i)]
._
s\i]
e
5
(9.100b)
for i = 1, • • • ,£ and n G Z. Then mi's and ,%'s and S^'s are defined for all t e Z - ( £ + l ) Z . We put /
:=
Z-(*-r-l)Z,
/+ := / n N = N - ( £ + l ) N . Then rrii =
— m-i
for i G / .
(9.101)
and, by (9.85), these elements satisfy the commutation relation [Si, Sj] =
mi6i+ji0K,
(9.102a)
and
[SitxW]
=
aiS^X^+V
(9.102b)
for each a € A . For the principal realization, it is convenient to consider the "shifted" current X^{z) defined as follows:
••= Y,x°)z~k
xzw
( 9 - 103 )
feez for each a G A
, multiplying z to the usual current. Then
[SitX?(z)] =
aiSW^XJr^*-*
=
a(SW)J2xLk)z~k+mi
=
kez a{S®)zmiX%-{z).
(9.104)
9.3. Construction of Basic
Representation
375
We now consider the linear space V*
:= C[Xj
;
j€l+],
with a Z-gradation deg(a;j)
:=
nij.
Then the q-dimension of this space is given by oo
1
9-dim(^)
f
= n r—^ = n n H
jel+
H
n=Oj=l
j—^-
For each element in g*"", we correspond a linear operator on $pr as follows. First we let •K : Sj i—• —— and
S-j i—• rnjXj
(9.105a)
for j € 7 + , and 7r : K
*
:
i—> Jd := the identity operator,
d
(9.105b)
-^h-(p\P)Id-L%'>
° ^
( 9 - 105c )
where
^r == E m^ohr
For a G A
•=
A
\
(9106)
n o -i™
, we consider a vertex operator
:= exp(V £ aOSbW-'W -£ / V je/+
a
jeI+
^ [ ~ j il ) ~ ^i m
ox
(9.107) Theorem 9.3.17. The space y - is an irreducible highest weight flpr-module with the QW-action n defined by (9.105a) and (9.105c) and
for each a G A1"". The constant function 1 is a highest weight vector satisfying TTKV)1
=
6i,0l
(i = 0,
so this representation is the basic representation
•••,£), L(AQ).
9. Vertex
376
Representations
Proof. Prom the principally specialized characters, one sees that L(A0) is sirreducible and that #*"" is isomorphic to L(Ao) as s-modules. Namely one can identify L(A0) with y - as an s-module. But i(Ao) is a {('""-module, so we need to know the action of other elements in fl*"-, namely elements Xa (a e A , j e Z). The action of these elements on y is known from the commutation relation (9.104) and the Schur's lemma. The commutation relation (9.104) implies that the action X1^r(z) on y coincides with the vertex operator Vgr(z) up to a scalar multiple, namely
Tr(xnz)) = cavr(z). To know these coefficients ca, we compute the action of the constant term of the vertex operator V^r(z) on the highest weight vector Vo = 1 in 3*"": V2""(z)l = exp ( J2
aiS^xjz"
so, by comparing the constant terms in both sides,
v
(A0, aV)v 0 =
0,
Lemma 9.3.16 gives
X^vo
=
-l(P\X^)v0.
So CB =
_(P!^)
(9108)
n and the theorem is proved.
•
We remark here that, in order to write down the action of all elements in fl*"", it is not necessary to know the action of Xg'(z) for all roots a e A , but it is enough to know those for a representative ±ji of (uj)-orbits. Example 9.3.2. In the case g = A\' elements in s are given by
I = Zodd, /+ = Nodd
and
= sl(2, C), the sets of exponents and
S^ =H®tj=(l
_°J ® tj ,
377
9.3. Construction of Basic Representation
in the notation of §9.2.2. So one has on(S^) = 2 for j G I+ — N 0 dd, and the vertex operator defined by (9.107) is given as follows:
l^(*)=exp(2 V
£
«^)exp(-2
j6Nodd
V
'
£
^ £ . ) .
j6Nodd
7
J /
Thus one sees that Theorem 9.3.17 covers Theorem 9.2.10 for sl(2, C). Closing this section, we give some examples of the (w)-orbits for several choices of Coxeter elements w in the case of exceptional Lie algebras Ei (£ = 6,7,8). In the following examples, we make use of the notation i
(mi,---
,me)
:=
y^mjaj.
Example 9.3.3. We consider E& with the Dynkin diagram ot\
ot2
c*3
<*4
<*5
o—o—o—o—o (ja6 1) First we take w := rxr^r^r^r^r^. as follows: i)
Then (w)-orbits through simple roots are
The orbit through a±: wax v?ai w3ai w*ai vfiai w6ai
= = = = = =
(0,1,0,0,0,0), (0,0,1,0,0,0), (1,1,1,1,0,1), (0,1,1,1,1,0), (0,0,1,0,0,1), (0,0,0,1,0,0),
iy 7 «i wsai w9ai wwai w11^ w12ax
= = = = = =
ii)
The orbit through a?.:
wka2 =
wk+1ct\.
iii)
The orbit through a$:
wka3 —
wk+2a\.
iv)
The orbit through a 4 :
wkon =
wk+6ai.
v)
The orbit through 0J5:
wka§ —
wk+7a\.
(0,0,0,0,1,0), -(1,1,1,1,1,0), -(0,1,1,0,0,1), -(0,0,1,1,0,0), -(1,1,1,1,1,1), (1,0,0,0,0,0).
9. Vertex Representations vi)
The orbit through a^:
wa6 w2a6 w3a6 w*a6 w5a6 w6a6
= = = = = =
-(1,1,1,0,0,1), -(0,1,1,1,0,0), -(1,1,2,1,1,1), -(0,1,1,1,0,1), -(0,0,1,1,1,0), -(0,0,0,0,0,1),
So the set of simple roots cannot be a . for this w. 2) Let us take w := rir3r5r2r4r6. Then i)
(1,1,1,0,0,1), (0,1,1,1,0,0), (1,1,2,1,1,1), (0,1,1,1,0,1), (0,0,1,1,1,0), (0,0,0,0,0,1).
\em of representatives of (w) -orbits
= = = = = =
(0,1,1,0,0,0), (0,0,1,1,1,1), (1,1,1,1,0,0), (0,1,1,0,0,1), (0,0,0,1,1,0), -(0,0,0,0,1,0),
io 7 ai w8ai
=-== -= == == -=
-(0,0,1,1,0,0), -(1,1,1,0,0,1), -(0,1,1,1,1,0), -(0,0,1,1,0,1), -(1,1,0,0,0,0), (1,0,0,0,0,0).
w7a2 = w*a2 = wQa2 = w10a2 = wlla2 = w12a2 =
(0,0,1,1,1,0), (1,1,2,1,0,1), (1,2,2,1,1,1), (0,1,2,2,1,1), (1,1,1,1,0,1), (0,1,0,0,0,0).
W9Oti
wwai wnai w12ai
The orbit through a2:
wa2 w2a2 w3a2 w4a2 w5a2 wea2 iii)
= = = = = =
The orbit through ct\ :
wUl w2ax w3ai w*ai w5ai wsai ii)
w7a6 w*a6 w»a6 w10a6 wlxaG w12a6
= = = = = =
-(1,1,1,0,0,0), -(0,1,2,1,1,1), -(1,1,2,2,1,1), -(1,2,2,1,0,1), -(0,1,1,1,1,1), -(0,0,0,1,0,0),
The orbit through a3:
wa3 w2a3 w3a3 w4a3 w5a3 w6a3
= = = = = =
(1,1,2,1,1,1), (1,2,3,2,1,1), (1,2,3,2,1,2), (1,2,2,2,1,1), (0,1,1,1,0,1), -(0,0,1,0,0,0),
io 7 a3 w8a3 w9a3 w10a3 wua3
== == == == ==
19
w a3
=
-(1,1,2,1,1,1), -(1,2,3,2,1,1), -(1,2,3,2,1,2), -(1,2,2,2,1,1), -(0,1,1,1,0,1), (0,0,1,0,0,0).
9.3. Construction of Basic iv)
v)
379
The orbit through a±: watt w2a± w3a4 w4a^ w5an
= = = = =
W6Ct4
—
== io 8 a 4 == ty 9 a 4 == w 10 a4 == wnct4, -= 19 = ur a 4 = W70t4
-(0,0,1,1,1,0), -(1,1,2,1,0,1), -(1,2,2,1,1,1), -(0,1,2,2,1,1), -(1,1,1,1,0,1), -(0,1,0,0,0,0),
(1,1,1,0,0,0), (0,1,2,1,1,1), (1,1,2,2,1,1), (1,2,2,1,0,1), (0,1,1,1,1,1), (0,0,0,1,0,0).
The orbit through a^: war, = W2Ol5
=
w3a^
=
4
=
W «5
w 5 a5 — w6a$ = vi)
Representation
(0,0,1,1,0,0), (1,1,1,0,0,1), (0,1,1,1,1,0), (0,0,1,1,0,1), (1,1,0,0,0,0), -(1,0,0,0,0,0),
=
-(0,1,1,0,0,0), -(0,0,1,1,1,1), -(1,1,1,1,0,0), -(0,1,1,0,0,1), -(0,0,0,1,1,0), (0,0,0,0,1,0).
== -= == == == =
(0,0,1,0,0,1), (1,1,1,1,1,0), (0,1,2,1,0,1), (1,1,1,1,1,1), (0,1,1,1,0,0), (0,0,0,0,0,1).
11/05
=
wsa5
=
t09O!5
—
w10at5 wua5
= =
12 WLZOt5
The orbit through ae: was w2ae w3ae w4ae w5a& w6a6
=-== == -= == -=
w7ae wsae w9a6 w10a6 w 11 a6
-(0,0,1,0,0,1), -(1,1,1,1,1,0), -(0,1,2,1,0,1), -(1,1,1,1,1,1), -(0,1,1,1,0,0), -(0,0,0,0,0,1),
So the set of simple roots {ai}i=i,.,6 (w)-orbits for this w.
19 1W can
Q!6 =
be a system of representatives of
Example 9.3.4. We consider E-j with the Dynkin diagram Ct\
a2
o—o
"3
«4
OC5
a6
o—o—o—o
0«7 1) First we take w := rxrir^r^r^r^TT. Wk+9Cli
Then -wkcti
for all k G Z and i = 1, • • • , 7, and wkai (1 < k < 8) are as follows:
380 i)
9. Vertex wka1
(l
wax w2ai w3ai w4ai
8):
= = = =
(0,1,0,0,0,0,0), (0,0,1,0,0,0,0), (1,1,1,1,0,0,1), (0,1,1,1,1,0,0),
ii)
wk(*2 = wk+1a\
for all k e Z.
iii)
wkct3 = wk+2ai
for all k G Z.
iv)
wka4
(\
wa4 w2a4 w3a4 w4a4
= = = =
(0,0,0,0,1,0,0), (0,0,0,0,0,1,0), -(1,1,1,1,1,1,0), -(0,1,1,0,0,0,1),
wka§ = wk+1a4
for all k e Z.
vi)
wkae = wk+2a4
for all k G Z.
vii)
wka7 wa7 a7 w3a7 w^a7
= = = =
W5OCi
==
w6ai
==
W7Ot\
(1,1,2,1,1,1,1), (0,1,1,1,0,0,1), -= (0,0,1,1,1,0,0),
w8ai
-=
(1,1,1,1,1,1,1).
w5a4 w6a4 w7a4 w8a4
= = = =
-(0,0,1,1,0,0,0), -(1,1,1,1,1,0,1), -(0,1,1,1,1,1,0), -(0,0,1,0,0,0,1).
8):
v)
(l
Representations
8): -(1,1,1,0,0,0,1), -(0,1,1,1,0,0,0), -(1,1,2,1,1,0,1), -(1,2,2,2,1,1,1),
w5a7 w6a7 w7a7 w8a7
=•• -(0,1,2,1,1,0,1), —: -(1,1,2,2,1,1,1), =: -(0,1,1,1,1,0,1), == -(0,0,1,1,1,1,0).
So the set of simple roots cannot be a system of representatives of (w) -orbits for this w. 1) Let us take w := rir3r5r2r4rer7. Then wk+9oti =
-wkai
for all k e Z and i = 1, • • • ,7, and wkai (1 < k < 8) are as follows: i)
wkai
(l
wa\ w2ai w3a.\ w4a-L
-= == -= ==
8): (0,1,1,0,0,0,0), (0,0,1,1,1,0,1), (1,1,1,1,1,1,0), (0,1,2,1,0,0,1),
w5ati w6a\ w7ai w8ai
(1,1,1,1,1,0,1), (0,1,1,1,1,1,0), (0,0,1,1,0,0,1), (1,1,0,0,0,0,0).
9.3. Construction of Basic ii)
wka2
(l
wa2 w2a2 w3a2 w*a2 iii)
wka3
= = = =
iv)
wka4
(l
wa4 w2a4 w3a4 w4a4 v)
wka5
= = =
= = = =
wa5 w2a5 w3a5 w4a5 vi)
wka6 wa6 w2a6 w3a& w*a6
vii)
wka7 wa7 w2a7 w3a7 w4a7
= = = =
(l
w5a2 w6a2 w7a2 wsa2
= = = =
-(1,2,2,2,1,0,2), -(1,2,2,2,2,1,1), -(0,1,2,2,1,1,1), -(1,1,1,1,0,0,1).
w5a3 w6a3 w7a3 w8a3
= = = =
(1,3,4,3,2,1,2), (1,2,3,3,2,1,2), (1,2,2,2,1,1,1), (0,1,1,1,0,0,1).
w5a4 w6a4 w7a4 w*a4
= = = =
-(1,2,3,3,2,1,1), -(1,2,3,2,1,1,2), -(1,2,2,2,1,0,1), -(0,1,1,1,1,1,1).
w5a5 w6a5 w7a5 w8a5
= = = =
(1,1,2,2,1,1,1), (1,2,2,1,0,0,1), (0,1,1,1,1,0,1), (0,0,0,1,1,1,0).
w5a6 w6a6 w7a6 w8a6
= = = =
-(0,0,1,1,1,1,1), -(1,1,1,1,0,0,0), -(0,1,1,0,0,0,1), -(0,0,0,1,1,0,0).
w5a7 «,%, w7a7 w»a7
= = = =
-(1,2,2,1,1,1,1), -(0,1,2,2,1,0,1), -(1,1,1,1,1,1,1), -(0,1,1,1,0,0,0).
8): (1,1,2,1,1,0,1), (1,2,3,2,2,1,1), (1,2,4,3,2,1,2), (2,3,4,3,2,1,2), 8): -(0,0,1,1,1,0,0), -(1,1,2,1,1,1,1), -(1,2,3,2,1,0,1), -(1,2,3,2,2,1,2),
(l
381
8): -(1,1,1,0,0,0,0), -(0,1,2,1,1,0,1), -(1,1,2,2,2,1,1), -(1,2,3,2,1,1,1),
(l
wa3 w2a3 w3a3 ™4a3
Representation
8): (0,0,1,1,1,1,0), (1,1,2,1,0,0,1), (1,2,2,1,1,0,1), (0,1,2,2,2,1,1), 8): -(0,0,0,0,1,1,0), -(0,0,1,1,0,0,0), -(1,1,1,0,0,0,1), -(0,1,1,1,1,0,0), 8): -(0,0,1,0,0,0,1), -(1,1,1,1,1,0,0), -(0,1,2,1,1,1,1), -(1,1,2,2,1,0,1),
382
9. Vertex
Representations
So the set of simple roots {ai}i=i,..,7 can be a system of representatives of (w)-orbits for this w. Example 9.3.5. We consider Eg with the Dynkin diagram ai
«*2
<*3
<*4
«5
ote
a7
o—o—o—o—o—o—o 6»8 and let w := rir3r5rTr2r4rers-
Then h+15
w
=
ai
-W
Cti
k
for all k e Z and i = 1, • • • ,8, and w a>i (1 < k < 14) are as follows: i)
wkon (l
ii)
iii)
k wu a2 wa2 w2a2 w3a2 w4a2 w5a2 w6a2 w7a2
= = = = = = =
(0,1,1,0,0,0,0,0), (0,0,0,1,1,0,0,0), (0,0,0,0,1,1,1,1), (0,0,1,1,1,1,0,0), (1,1,1,1,1,0,0,1), (0,1,1,1,1,1,1,0), (0,0,0,1,2,1,0,1),
(l
2
w a3 w3a3 w4a3 W5a3 6
w a3 w7a3
wsai w9at\ wwot\ wxla\ 19
w a.\ w13ai w14ot\
(0,0,1,1,1,1,1,1), (1,1,1,1,1,1,0,0), (0,1,1,1,1,0,0,1), (0,0,0,1,1,1,1,0), (0,0,0,0,1,1,0,1), (0,0,1,1,0,0,0,0), (1,1,0,0,0,0,0,0).
U):
-(1,1,1,0,0,0,0,0), -(0,1,1,1,1,0,0,0), -(0,0,0,1,2,1,1,1), -(0,0,1,1,2,2,1,1), -(1,1,2,2,2,1,0,1), -(1,2,2,2,2,1,1,1), -(0,1,1,2,3,2,1,1),
wka3 (l
U):
w8a2 w9a2 w10a2 wna2 19
wLZa2 w13a2 w14a2
-(0,0,1,2,3,2,1,2), -(1,1,2,2,2,2,1,1), -(1,2,2,2,2,1,0,1), -(0,1,1,2,2,1,1,1), -(0,0,0,1,2,2,1,1), -(0,0,1,1,1,1,0,1), -(1,1,1,1,0,0,0,0).
14):
(1,1,1,1,1,0,0,0), (0,1,1,1,2,1,1,1), (0,0,1,2,3,2,1,1), (1,1,2,2,3,2,1,2), (1,2,3,3,3,2,1,1), (1,2,2,3,4,2,1,2), (0,1,2,3,4,3,2,2),
w8a3 w9a3 w10a3 wna3 19
w a3 w 1 ^ a3 14
w
a3
(1,1,2,3,4,3,1,2), (1,2,3,3,3,2,1,2), (1,2,2,3,3,2,1,1), (0,1,1,2,3,2,1,2), (0,0,1,2,2,2,1,1), (1,1,1,1,1,1,0,1), (0,1,1,1,0,0,0,0).
9.3. Construction of Basic iv)
wka4 wa4 w2a4 w3a4 w4a4 w5a4 w6a4 w7a4
v)
wka5 wa5 w2a5 w3a5 w4a5 w5a5 w6a5 w7a5
vi)
wka6
wa6 w2a6 w3a6 w*a6 w5a6 w6a6 w7a6 vii)
wka7
wa7 w2a7 w3a7 w4a7 w5a7 w6a7 w7a7
(l
= = = = = = =
w8a4 w9a4 wwa4 w11a4 w12a4 w13a4 w14a4
= — — = — = =
-(1,2,3,4,5,4,2,2), -(1,2,3,4,5,3,1,3), -(1,2,3,4,4,3,2,2), -(1,2,2,3,4,3,1,2), -(0,1,2,3,3,2,1,2), -(1,1,1,2,2,2,1,1), -(0,1,1,1,1,1,0,1).
U): (0,0,1,1,2,1,1,1), (1,1,2,2,3,2,1,1), (1,2,3,3,4,2,1,2), (1,2,3,4,5,3,2,2), (1,2,3,4,6,4,2,3), (1,2,4,5,6,4,2,3), (2,3,4,5,6,4,2,3),
(l
383
14):
-(0,0,1,1,1,0,0,0), -(1,1,1,1,2,1,1,1), -(0,1,2,2,3,2,1,1), -(1,1,2,3,4,2,1,2), -(1,2,3,3,4,3,2,2), -(1,2,3,4,5,3,1,2), -(1,2,3,4,5,3,2,3),
(l
= = = = = = =
Representation
w8a5 w9a5 w10a5 w11^ w12a5 w13a5 w14a5
= = = = = = =
w8a6 w9a6 wwa6 wna6 w12a6 w13a6 w14a6
= = = = = = =
(1,3,4,5,6,4,2,3), (1,2,3,5,6,4,2,3), (1,2,3,4,5,4,2,3), (1,2,3,4,4,3,1,2), (1,2,2,3,3,2,1,2), (0,1,1,2,2,2,1,1), (0,0,0,1,1,1,0,1).
U):
-(0,0,0,0,1,1,1,0), -(0,0,1,1,2,1,0,1), -(1,1,2,2,2,1,1,1), -(1,2,2,2,3,2,1,1), -(0,1,2,3,4,2,1,2), -(1,1,2,3,4,3,2,2), -(1,2,3,3,4,3,1,2),
-(1,2,3,4,4,2,1,2), -(1,2,2,3,4,3,2,2), -(0,1,2,3,4,3,1,2), -(1,1,2,3,3,2,1,2), -(1,2,2,2,2,2,1,1), -(0,1,1,2,2,1,0,1), -(0,0,0,1,1,1,1,1).
(1 < k < 14j:
= = = = = = =
(0,0,0,0,1,1,0,0), (0,0,1,1,1,0,0,1), (1,1,1,1,1,1,1,0), (0,1,1,1,2,1,0,1), (0,0,1,2,2,1,1,1), (1,1,1,1,2,2,1,1), (0,1,2,2,2,1,0,1),
w8a7 w9a7 wwa7 wna7 19
li w w13aa77 w14a7
== -= -= == = =-= -=
(1,1,1,2,2,1,1,1), (0,1,1,1,2,2,1,1), (0,0,1,2,2,1,0,1), (1,1,1,1,1,1,1,1), (0,1,1,1,1,1,0,0), (0,0,0,1,1,0,0,1), (0,0,0,0,0,1,1,0).
9. Vertex
384 viii)
wka8 wa8 w2a8 w3a$ w4ag w5atg w6ag w7ag
Representations
(1 < k < \A): =-== -= == -= -= -=
-(0,0,0,0,1,0,0,1), -(0,0,1,1,1,1,1,0), -(1,1,1,1,2,1,0,1), -(0,1,2,2,2,1,1,1), -(1,1,1,2,3,2,1,1), -(0,1,2,2,3,2,1,2), -(1,1,2,3,3,2,1,1),
w8ag w9ag w10as wnag 12
w'-'atg w13as w14ag
--== == -=
= =-= -=
-(1,2,2,2,3,2,1,2), -(0,1,2,3,3,2,1,1), -(1,1,1,2,3,2,1,2), -(0,1,2,2,2,2,1,1), -(1,1,1,2,2,1,0,1), -(0,1,1,1,1,1,1,1), -(0,0,0,1,1,1,0,0).
So the set of simple roots {on}i=i,...,7 can be a system of representatives of (w)-orbits for this w.
Chapter 10
Soliton Equations
In this chapter, we explain the Hirota bilinear differential equations of KdV equations and then deduce Hirota bilinear differential equations from homogeneous and principal realization of basic representations of simply-laced affme Lie algebras. 10.1
Hirota Bilinear Differential Operators
In this section we review the theory of R. Hirota to obtain and write down exact solutions for the Korteweg-de Vries equation. First we consider two functions f(x) and g(x) in one variable x e R. The operator Dx defined by Dx(fog)
:= ^-f(x +
y)g(x-y)
oy
=
fxg-fgx
v=o
is called the first order Hirota bilinear differential operator. The "higher order" Hirota bilinear differential operators D* (k € Z> 0 ) are defined by k
D
x(f°9)
== h r - f(* + V)9(x-V) \dy)
i^{-iy(%kx-if.dig. W
J=0
385
(10.1a) v=o
(io.ib)
10. Soliton Equations
386
It is easy to see that these operators satisfy the following: dxDkx
Dkxdx,
=
k
D x(f°9) Dkx{fo\) Dkx{log)
k
= = =
(10.2a) k
(-l) D (gof), 8kf, {-\)kdkg.
(10.2b) (10.2c) (10.2d)
The following formula is also easy but very important: Dk(eaxoebx)
=
(a - b)keaxebx,
(10.3)
where a, b € C. Actually this formula is checked using (10.1b) as follows: Dk(eaxoebx)
Y^(-l)j(k}dk-jeax-diebx
=
J2(-l)j =
(k^)ak-jV 1 eax • ebx
(a - b)keaxebx.
(10.4)
Notice also that Dk(fof)
= 0
if k is odd,
(10.5)
by (10.2b). We now consider a multi-variable case, namely Hirota bilinear differential operators acting on functions with n-variables (x\, ••• , xn), where n is any positive integer and fixed. Let P € C[xi, • • • , x n ]. Then the Hirota bilinear differential operator P(Di, ••• , Dn) is denned as follows: P(D1,---,Dn)fog
:= p
{wi''''^){f{xi+yu'''jXn
+ yn)
*9(X1 ~ Vl, • • • , Xn ~ VnUl
•
(10.6)
/ li/ 1 =...=j / „=0
The following formula is proved by a quite similar calculation with (10.4): =
P(DU ••• ,Dn) (e a i x i+-+°« x " oe b l X l + - + b '> x ") P ( a i - 6i, • • • , an - bn)eaiXl+-+a"x" • e^i+-+Kxn
^ ^
Just like (10.5), one has P(Dlt
• • • , Dn)f of
= 0
if deg(P) is odd,
(10.8)
387
10.2. KdV Equation since P(D1,---,Dn)fog
= P{-Du...,-Dn)g0f,
(10.9)
by the definition (10.6) of bilinear differential operators, where the degree of polynomial function P(x\, • • • ,xn) is the degree with respect to the usual gradation deg(xi) = • • • = deg(x„) = 1. Let us consider two functions u and / related to each other as follows: u(x,y,z,
••• ,t) = log/(x, y, z, • • • ,t).
Then the following simple formulas play a quite important role to connect the bilinear differential equations for / with the usual partial differential equations for u: is.
+ 3uxuxx + ux, ^p = Uxyz + UxUyz + UyUzx + UzUxy + UxUyUz, f - = uxxxx + 4uxuxxx + Zulx + 6uluxx + u%,
(10.10)
Is s
^jr1
=
Uxyzt + UxyzUt + UxyUzt
+ UxytUz
+ UxztUy
+ UyztUx
+ UXZUyt + UyZUxt
+ UXUyUzt + UXUZUyt + UyUZUxt
+uxutuyz 10.2
+ uyutuxz
+ uzutuxy
+
uxuyuzut.
K d V Equation and Hirota Bilinear Differential Equations
The Korteweg-de Vries equation, usually called "KdV equation", is the following non-linear partial differential equation for the unknown function u(x,t): 3 1 ut — -uux - -uxxx
— 0.
(10.11)
Putting u(x,t) = ~v{x,t),
(10.12)
the above equation (10.11) is rewritten as follows for v(x,t):
i (* - i«2 - r*-) = °-
(1013)
10. Soliton Equations
388 This equation (10.13) means 3 1 vt — -vx — -vxxx
= independent of x,
namely 3 1 t ~ -^i - ^xxx
v
=
(10.14)
where (p(t) is an arbitrary function of t. We put w(x,t)
:= v(x,t) — J ip(t)dt. Jo
Then wt = vt — (f{t),
wx = vx
and
wxxx = vxxx,
so this equation (10.14) is rewritten as 3 1 t ~ Jwl ~ jwxxx — 0, (10.15) and a solution u(t, x) of the KdV equation is obtained from w(t, x) by w
u(x,t)
= —w(x,t).
(10.16)
To solve the equation (10.15), we introduce a function f(x,t) w(x,t)
=
2^1og/(:M)
=
2-^.
defined by (10.17)
Then, by using (10.1b), one has the following: ^x\J
° J)
JxxxxJ
=
^Jxxxjx
2(ffxxxx-4fxfxxx
~T~ VJxxJxx
^Jxjxxx
+ 3fL),
r JJxxxx
(10.18a)
and
DxDt(fof)
=
Dx(ftof-fofx)
=
{ftxf
— ftfx)
=
2(ftxf - ftfx).
Since fx — -xfw by (10.17), ftx, fxx, fxxx
— (fxft
and fxxxx
(10.18a) and (10.18b) are calculated as follows: ftx
=
-^{ftw +
fwt),
—
fftx)
(10.18b) in the right sides of
10.2. KdV Equation fxx
=
389 ^(fxW
+ fwx)
2 ( ~2~ + ^
=
) '
ifw
fx fw2 fxxx
=
-J
\
f
[^-+Wxj+-(WWX
+ Wxx)
ifw
f Uw3 =
n,
\
/ (w3
fxxxx
A
-7 < I ~Y + " . I + 2(wwx + wxx) S
1
=
7 < - y + 3 1 0 ^ + 2iu xa: ^ ,
=
-T- S ~Y + 3WWX + 210^ V
/ f3 o
2
1
+ - < -w*wx + Zwx + 3wwxx + 2wx:r:i; > =
=
| 1 f y + 3w2wx + 2u«Bra J + (2>w2wx + 6wx + 6wwxx + 4wxxx) > f f w4 1
- < — + 6w2wx + 8wwxx + 6tox + 4wxxx V .
Substituting these into the right sides of (10.18b) and (10.18a), and Dx(f o / ) are obtained as follows: DxDt(f o / ) Dt(fof)
= =
2(ftxf - ftfx) 2(ffxxxx-4fxfxxx
= (ftw + fwt)f -ft-fw + 3f2x) = f(3wl
DxDt(fof)
= fwt, + wxxx).
So one has (Di - 4DxDt)(f
o f)
= f2(3wl+wxxx-4wt).
(10.19)
Hence w(x, t) is a solution of the equation (10.15) if and only if a function f(x) with values in C* is a solution of the bilinear differential equation (D4x-4DxDt)(fof)
=0.
(10.20)
And then a solution u(x,t) of the KdV equation (10.11) is obtained by u(x,t)
= 2~logf(x,t)
(10.21)
is a solution of the KdV equation (10.11). A solution f(x,t) of the Hirota bilinear differential equation (10.20) is often called a r-function.
390
10. Soliton Equations
We now describe the Hirota's method to solve the equation (10.20). It is clear that the constant function / — 1 is a solution of (10.20). Starting from this trivial solution and applying the method of perturbation, we consider the following function: = l + ef1+£2f2
/
+ e3f3 + ---,
(10.22)
where e is a parameter with its value in complex numbers. Then fof
= (i + ef1+e2f2 + e3f3 + ---)o(l + ef1 + e2f2 + e3f3 + ---) = l o l + e ( l o / i + / l 0 l ) + e 2 (l o f2 + / i o / j + f2 o 1) +£3(1 o / , + / i o / j + + / 2 o / 1 + / , o l ) + . . .
so the equation (10.20) is equivalent to a system of bilinear differential equations
(D£ - ADxDt) 11 o fk + Y, fi o fk-i + fk o 1J = 0,
(10.23)
for all k € Z>o- Since (D*-4DxDt)(lol) (D* - 4DxDt)(l o fk)
= 0, = (Di - 4DxDt)(fk
o 1) = ( ^ -
4dxdt)fk,
the first few terms of these bilinear differential equations are written explicitly as follows:
{dZ-4axdt)h = o, 2{8$ - idxdt)f2 + (Di - 4DxDt) (h 0 / 0 = 0 , (dx-4dxdt)f3 + {Dx-4DxDt)(f1of2)=0,
^1U-^
First we want to find a solution such that f1 = e2ax+bt
and
/< = 0 if » > 2,
where a, b 6 C, namely /
= 1 + e/i = l +
2ax+bt Ee
.
In this case, all equations which should be satisfied are (8* - 4dx&t)h - 0
and
( 7 ^ - 4 D x A ) ( / i o / i ) = 0,
whose left sides are computed as follows: (fi£ - 4dxdt)fi
= {&£ - 4dxdt)e2ax+bt
= 8(2a 4 -
ab)e2ax+bt
391
10.2. KdV Equation and (i£-4£>xA)(/i°/i) =
D*{e2ax+bt
=
ebtD*(e2axoe2ax)-4DxDt(e2ax+btoe2ax+bt) N
o e2ax+bt)
- WxDt(e2ax+bt
'
v
V
o
) = 0 .
'
v
0
2ax+bt e
0 (V (10.7))
Thus 2a 4 — ab = 0 is the only condition for / to be a solution in this case. Namely / = 1 + ef\ is a solution if and only if a = 0 or 2a 3 = b. If a = 0, then / = 1 + eebt is a function only in t and w = 2dx(logf)
= 0.
So this is a trivial solution. If 2a 3 = b, then so, by putting e = e 2c , one obtains a so-called "one-soliton solution"
„(*,«) = 2-^jPl
= ^laH
+ c)
<""»>
of the KdV equation (10.11). Next we go further to find a solution such that fx =
2aix+blt
Cle
+
c2e2a2X+b2t
where a,i,bi,Ci (i = 1,2) are a certain suitable complex numbers, and fi = 0 for all i > 3 ; namely /
=
l + eh + e2f2
=
1 + e (Cle2aix+blt
+ c2e2a2X+b2t)
+ s2f2 .
In this case, bilinear differential equations which should be satisfied are as follows: (9 4 - 4dxdt)ft
= 0,
(8X - 4dxdt)f2 + (Di - 4DxDt){h
(£>4-4L»KA)(/io/2)-0,
o fx) = 0,
392
10. Soliton Equations
(D*-4DxDt)(f2of2)
= 0.
These equations are easily solved and one obtains r-functions of "2-soliton solutions" /
=
1 + ec1e2
ec2e2^x+a^ 3 f,2(a1+a2)x+2(a 1+a*)t e2(al+a2)X+i(a1+a2)t
(10.26)
\ai+a \a!+a2J where a.i, bi,Ci and e are arbitrary complex numbers. In a similar way, r-functions of 3-soliton solutions of the KdV equation are obtained as follows: T
=
ai,a2,a3;ci,C2,C3\%>t) 3 l + J2Cie2^x+a^+ i=i
+ClC2C3
Q
1
J2 i
e2^+a^x+2^+a^t
CiCj(^—^-) \ai
a
f ( l ~ 2)(Ql - a 3 ) ( a 2 - « 3 ) V
+
"jj
c2(a1+a2+tta)tt+2(a?+a;*+a^
\ (ai + a2)(ai + a3)(a2 + a3) J (10.27) For a positive integer n, r-functions of n-soliton solutions are obtained in a similar way and written explicitly as follows: Tax,-- >a„;ci,--
,c„(x,t)
)
fc=l l < i i < - < i f c < n
2
1
TTe2(airx+alt) r=l
where a, and Ci (i = 1, • • • , n) are arbitrary complex numbers. 10.3
(10.28)
Hirota Equations Associated t o the Basic Representation
Let Q be an affine Lie algebra and (TT, V) be an integrable highest weight g-module. Let {& ; i e Z} and {£*; i e Z} be bases of g satisfying
and consider the operator S on V ® V defined by
S := 5 > ( 6 ) ® * ( f ) iez
(10.29)
10.3. Hirota Equations Associated to Basic
Representation
393
This operator S commutes with the action of g and is written, in terms of the Casimir element ft of g, as follows: S = - {(TT ® IT)(ft) - vr(ft) ® Idv - Idv ® ?r(ft)} ,
(10.30)
where Idy id the identity operator on V. We simply write Xv in place of TT(A>.
Given an integrable highest weight g-module V, let Gy denote the subgroup of GL(V) generated by exp(ei) and exp(/i) (i — 0,••• ,£). Notice that, for X = ej and /i and ( E V , the element °° 1 exp(X)„ := ^ -X«„ Tl=0
makes sence since e± and /j are locally finite on integrable modules by its definition and so the summation in the right side is a finite sum. Let us consider the Gv-orbit through VQ in V. Let v be an element in Gy • Vo, namely v = g • VQ for some g e Gv- Then S(v®v)
= S(gv0®gv0)
= (g®
g)(S(v0®v0)),
since S commutes with the action of g p r . Let A be the highest weight of V. Then VQ ® VQ is a singular vector of weight 2A, and so ft(vo ® vo) = (2A|2A + 2p)v0 ® v0 = 4(A|A + p)v0 ® v0 . Then, by (10.30), one has S{v0 ® v0) = i {4(A|A + p) - 2(A|A + 2p)} = (A|A)«b ® " 0 .
(10.31)
So one has ueGv-wo
==>• ^(u ® u) = (A|A)v ® v
(10.32)
when V = L(A). The converse of this statement is proved in [114]; namely, an element 0 / t ) £ £(A) belongs to the Gy-orbit through the highest weight vector i?o if and only if it satisfies S(v®v)
= (A\A)v®v.
(10.33)
This equation (10.33) is called the Hirota equation associated to L(A). In the sequel, we calculate Hirota equations associated to the basic representation when g is a non-twisted affine Lie algebra over a finite-dimensional simple Lie algebra g with a symmetric Cartan matrix. We write simply G for Gv when V =
L(AQ).
394 10.3.1
10. Soliton Equations Homogeneous Case
We fix an (arbitrary) orthonormal basis ul (i = 1, • - • ,£) of f). Then the symmetric algebra S(i)-) is identified with the polynomial algebra
C [ i f ; 1 < i < £, ke NJ by and the representation w of the Heisenberg algebra r) on this space is given by 7r
Kfc)) = ^77T
m d
*(«<-*)) = kxk\
(10-34)
for A; € N . In this coordinates, the representation space of the basic fl-module L(Ao) in the homogeneous picture is given as follows: L(A 0 ) =
C e { Q } ® c [ 4 0 ; l
fceN],
so an element / in L(AQ) is a finite sum
/ = J2eP®f?>
( 10 - 35 )
/3GQ
where fp is a polynomial function in zjj. s. We count the degree of an element in this space by deg(e^)
:=
\B\2 ^ -
deg (xV)
:=
j .
— for p e Q,
For simplicity we write
/ = Ee%> in place of (10.35). For each a e A, we choose S a € g such that (£„|£_a)
=
I-
[Ea, E-a]
=
a,
This condition is equivalent to
10.3. Hirota Equations Associated to Basic Representation
395
under the identification h = f)* via the standard inner product ( | ). We consider the following two bases of g: {&}
=
u{®tk,
Ea®tk,
I
K,
d
til
{C} = ul®t~k, E-a®t~k, d, K where a and k run over all elements in A and Z respectively. Then the operator S, defined by (10.29), is written as follows: i
s
= J2J2 *(«<*)) ® *(«(-*>) + 5Z E *(£«<*)) ® *(£-«(-*>) i=i fcez a6 Afcez +7r(tf) ® vr(d) + 7r(d) ® TT(K)
(10.36)
We write the space of tensor product of two L(A 0 )'s as L(Ao) ® L(Ao)
=
(C £ {Q'} ® C [ x « ' ] ) ® ( c £ { Q " } ® C [ x « " ] )
=
( C £ { g ' } ® C £ {Q"}) ® (c[xW'] ® C [ x « " ] ) ,
and write an element
Z) (ef3'fp'®eP"9P")
f®9 = Yl P'ZQ'
P"&Q"
in this space also as
/®ff = Yl
(e/3'®e/3")®(/^/3")-
S
0'€Q'/3"eQ"
To compute the action of 5 on L(Ao) ® L(AQ) namely
S(f®9) = Y P'eQ'
E
S{e^fpi®eP"g0n),
P"eQ"
we calculate S(e& fp> ®e13 gp») as follows. By (10.36), this is the sum of three parts: s{tPfp®e?'gf,^ where
= (I) + (II) + (III),
(10.37)
396
10. Soliton Equations e oo
= E E (M*>)(^M) «=1
® [M-^'ap-))
fc=l I oo
+ E E (M-k))^'M) i=l
® (Mk))(^"9P"))
fe=l
+ E (^Ho))^'/^)) ® ( T K O ^ V ) ) £
oo
£ E ( ^ ' ® ^") ® ( ^aj 7 ® *««" ) (f„, ® «,,) i=lfc=l
V^fc
£
oo
i=l
fc=l
/
„
+ E E( e / 3 ' ®e/3") ® (kxk]' ®dx®" 7 ^ ) (//»' ® ^») +( E
(^>(0)) ) • e?fa ® e " V
(^>(O))
(10.38)
(/3'|/3»)
and := (7r(K) ® 7r(d))(e^'//3' ® e ^ ' V ) + (*r(d) ®jr{K)){efi'ff), = e^V/3' ® 7r(rf)(e^"9/3") + w(d)(e? fp) ® e ^ ' V '
(III)
"•
i=i
fc=i
"^fc
® " g0,,)
J
-f^ + E E ^ W ^ a . L
=
-
t=l
l/y|2
g
l/y,|2
fc=l
O'^fe
J
(^'(»e^)«8>(/^«8>g/?»)
-(^' ® e^') £ E f c {4iV-4)7 ® i+1 ® 4 " " - ^ } (//" ® ^") (10.39) and
(")
:=
E E (ff(£-(fc)) ® *(£-«<-*))) (c^/8' ® e^"^») aeA fe ez
=
the constant term of
E ( E ^ i ^ ^ E 7 ^ - ^ ) ) ^ ) (*"'//»'®e/,'V)
ceA S ' e z =
jez
2
'
z
the constant term of z Y^ {^a( ) ® r_ a (.z)) (e^' fp< ® e^"gpn )
10.3. Hirota Equations Associated to Basic =
Representation
the constant term of ~2 £(a,(3')e(-a,p") z J- ZW)Z(-°W") a€A
(ea+?
397
® e""^')
e(a,0") e(a,P'+P")
®(va(z)fo ®V-M)9p») =
£
Y,
( a ' & +13") ( e " + / 3 ' ® e " a + / 3 " ) ® (II)a ; /j^»
(10.40)
a€A
where r a ( z ) and Va(z) axe vertex operators defined by (9.71b) and (9.71a) respectively, and (£I)a;P',p" '•= the constant term of z 2 + H / " - / J " V a ( z ) / „ , ® V-a(z)gpH.
(10.41)
To compute these operators, we introduce new variables x^ and y^ defined
*V = *l?+V? -d
*«" = *«-„«,
namely i •fc
+4°" == 2 * «
and
i f - i f
= 2»«
Then one has
d
dxf
d
+
and
dxf
te«' te«" fly«'
and (I) + (III) is given by (10.38) and (10.39) as follows: (I) + (HI)
= (e^^'W-£f>(^^)(^ *-
i=l
fe=l
*~ 2y<*>
V
"^fc ^
'
B
B~J?
+w)
- \wv) - \{P"\nyfP'®9p» l
=
oo
{e? ®ee")®{-2J^kyki^-\\(3> ^
»=i/b=i
-n){fp>®9p»). J
oVk
(10.42) Next we compute the right side of (10.41). Since oo
Va{z)fr
=
e x p ^ ^ j e x p ^ ^ - j / , ,
10. Soliton Equations
398 e oo
=
\
ex P E E ^ l ^ i * V \i=i
fc=i
/ / oo ("I"*) 9 ^_ k exp - £ £ ^ ^ " f c P "
/
\
one can rewrite Va{z)fp> ®V~a{z)gp"
i=i
c x
fe=i
' fc
/
as follows:
Va{z)fp> ® V-a(z)gpn = \ i=i
eJ^f^ialu^x^'-x^zA ' r~ ' )
fc=i
2j,<*>
X
""l-S£—(s^ - ^)' J7''8*" «
oo
i=l
fc=l
\
/
I
oo
/
\
i=l
(a|u*)_a
fc=l
"
_fc
®Vk
(10.43) In order to expand this into the power series in z, we introduce the Schur function P„a'(- • • , xjj. , • • •) defined by oo
, oo
tt)
£
\
0
E^l (--- ,4 .-)*" == expfEEH^)^^) n=0
^Jfe=li=l
(10-44)
'
for each a € A and n e Z>o- We simply write P„ (x) or P„ (xjj/) for
^(•••,4 i ) ,.-.).
Then, from (10.43), one has Va(z)ff>,®V-a(z)gfi,,=
E
•PnQ)(2y)iJir)(-^-%)^n-m,
(10.45)
and so the right side of (10.41) is rewritten as follows: a) tt)
(nW',/»» =
E
„,m>0
^n (2y)Pi f~-%W® ff H (io.46) V
fc
%fe /
n-m+2+H/3'-0")=O
We notice that, in this new coordinates (xjj/) and (y£ ), a function / ^ (x')gp» (x") is rewritten as follows:
/*'(*V'(s") - /^(4 i) +2/i i) )^'(4 i) -2/i i) )
10.3. Hirota Equations Associated to Basic Representation
399
= ^(*if) + *?, + yf , )^(*i i, -('i. i, + rf)t=0)) oo
e
}+
l)
« P E E ^a*i * K ^)*- H -*i?>) >,i=l
fe=l
<=0
(10.47)
Using bilinear differential operators defined by D )
k f°9
:= -^f(x
+
t)g(x-t) t=o
d
(i)
, « _1_ +(*),
+W
t, at' n/(-.<'+' .-M-^ -*r.-) (i)
t=0
(10.48) the formula (10.47) is rewritten as follows:
U' ® 5/J" = //»' (4 f ) + Vk)) 90" ( 4 ° ~ Vk])
=
ex
P ( Z E ^ ^ l /P'(4i))^»(4i)) • (10-49) ^i=i fc=i
Prom this formula, one has
i} Ajifr*fp») = A « P ((v E E » M ) )//"(«* V(«i ) dVj Vj \i=ik=i ) d
/
t
oo
\
i) i) ^ W (EE^XM ^(4 )^(4 ), \%=\ fe=i (10.50) and more generally P ...
S a («)'
(fp®9f}")
^i
P(...,Df\...)eXp(j2f2yk)D]:))fp'(xki))g^(xki)),(10.51)
=
\t=i
fe=i
/
where P is a polynomial function. Proposition 10.3.1. Let g be an affine Lie algebra over a finite-dimensional simple Lie algebra g of rank £ with a symmetric Cartan matrix, and consider its basic representation constructed on the space L(A 0 )
=
Ce{Q}®S(h_)
400
10. Soliton Equations
= Ce{Q} ® C [ i f ; l
j e N] .
T/ien, /or elements
and g=Yet3®9p
/ = £ V ® /,»
e L(A0),
5 ( / ® #) is given by the following formula: S{f®g)
xexp I Y^ Eyk )D< k
J //9' °90"
\fe=li=l
/ oo (
+ J2_ E ^ ^ + 0 V
,+a
e/3
a
® "~ ® E n = 0
/3',/3"eQaeA
p a) 2
« ( y)
^
x^Ul^-^) (-• '4 D * } '-) }exp ( E E ^ ^ i 0 ) fr°9P (10.52a) f
oo i
1
^ y ®e/?" ® - 2 E E * » W - ^ - ^
=
/3',/3"eQ
*•
fe=1
i = 1
^pfEE^^V0^'' \fc=l1=1
+
^ Y/®^" ',0"€Q
XjP
/
£
® E W + / n { E (P«Q)^) n = 0
a€A
n - 2 + ( c-!3'-/3'')
(~jfcI,*)) ) e X P ( E E V f c l ) £ , * ) ) //»'-« °»/»"+« J (10.52b)
where Pna> are Schur functions defined by (10.44) and D^' 's are bilinear differential operators defined by (10.48). The second formula (10.52b) in the above proposition follows from the first one (10.52a) by putting /3' = i - a
and
/3" = 7 " + a.
10.3. Hirota Equations Associated to Basic
Representation
401
Then one has (a\(3' - /?")
=
(a|(Y - a) - (7" + a)) - ( a | 7 ' - 7") - 2 (o|a)
= (a|y-y)-4, since (a|a) = 2 for a G A, and so, writing @' and /3" in place of 7' and 7", the second formula (10.52b) follows. Prom this proposition, we obtain the Hirota equations associated to the homogeneous basic representation as follows: Theorem 10.3.2. Let g be an affine Lie algebra over a finite-dimensional simple Lie algebra g of rank £ with a symmetric Cartan matrix, and consider its basic representation constructed on the space L(A0)
=
Ce{Q}®S(t>-)
= Ce{Q}
JGN].
Then the Hirota equations are given by the following generating function:
{
oo
£
-.
I
Jfc=li=l
/
o
J
o
f
\
\fc=li=\
J
x exp ( JT J2 y^D® ) Tp.-a ° T0„+a} = 0, \ t — 1 i—t
I
for all (3', P" e Q, where r = £
(10.53)
) ef>
® TP e
L A
( o) •
PeQ
The above formula (10.53) is written in terms of Schur polynomials P„ defined by (10.34) corresponding to an orthonormal basis of I). In place of an orthonormal basis, one can make use of bases {u l } i = l i ... ^ and {ui}i = i j ... / of f) satisfying (U%\UJ) = Sij. Then the associated Schur functions are denned by 00
/
00
t
\
£ f f ) ( . . . ,*£>,... )* n := expf 5 3 5 ^ ( a | i i * ) x « z f c ) n=0
^fe=li=l
(10.54a)
'
and OO
n=0
,
OO
l
^ fe=l i = l
X
'
for each a G A and n G Z>o, and then, via similar arguments, one arrives at the following formula:
10. Soliton Equations
402
Corollary 10.3.3. Let g be an affine Lie algebra over a finite-dimensional simple Lie algebra g of rank t with a symmetric Cartan matrix, and consider its basic representation constructed on the space L(Ao) =
C£{Q} ® C [xf
• 1 < t < *, j € N] .
Then the Hirota equations for T — }T e^ ® TP
e
-k(Ao) are given by the fol-
P£Q
lowing generating function: oo
+
£
{- 2 £ 5 > M
" j / o o f
n
}
\
-\w-nWv E E ^
E^^L-W'W-^)
E^^'+^I fc=l i = l
J
\fe=l i = l
/
^or^
n = 0
aEA
x ex
p ( E E »* )D* } ) TF-° ° r'3"+«} = °' Vfe=ii=i
/ a)
/or all P',f3" € Q, wfcere Pi and (10.54b), and
(10-55)
J
and Q^
are Schur functions defined by (10.54a)
5 = (....J51V--) -
(-.^.-)-
Example 10.3.1. We consider the case A\ ' — sl(2, C) and write simply a for ai. Letting u1 := | a andu^ := a, equations (10.54a) and (10.54b) give /
OO
E^
± a )
(^
n
OO
n-0
^
fc=l
OO
/
OO
EQi
± a )
(^
n
\
exp(±E^fc),
= =
/ x
exp(±2E^
n=0
^
fe=l
f e
j'
So these functions are written as •f \
Pia)(x) P
= p n (x) = Pn(-x)
a
^
f Q£\x) \ Q(na\x)
= =
Pn(2x)
pn(-2x)
in terms of the usual "elementary Schur polynomials" pn(x) defined by
n=0
^
fc=l
'
10.3. Hirota Equations Associated to Basic
Representation
403
namely ™7 1 ~ . j 2 ~ J 3
pn(x)
= pn(xi,x2,---)
=
5Z
i 1 4
x
l •L2
x
3
]
a-7
Jl -hW--
ji+2j2+3j3+-=n
Also recall that one can let e(ma,na) — 1 for all m,n € Z as is noted in Example 9.3.1. Then, forr = X) n e Z e n a ®T n (a;) S L(A0), the equation (10.55) in Corollary 10.3.3 becomes as follows: < - 2 ^2 kykDk - (n - m) 2 ^ exp I ^ oo
2/fe£»jt J T„ O r m
/ oo
\
-2+2(ra-m)
(-27?)exp J=0 oo
^ Vfc=l
^
J /
/ oo
\
\fe=l
/
—2(n—m)
Tn+1 ° T m _ i
j=0
= 0, (10.56) for all n,m € Z. 77&e Ze/t side o/ (10.56) is a formal power series in yk 's, and coefficients of lower few terms are given as follows: constant term : m)2TnTm+
__
-
(n -
p2(n-m)-2(-2£>)rn_1 o T
+
P-2(n-m)-2(20)viorm_i,
m +
i
(10.57a)
coefficient of yk • {2k+(n-m)2}Dk(TnoTm) +
{2p2 ( n _ m ) + fc„ 2 (-25) + p 2 ( n _ m ) _ 2 ( - 2 5 ) £ » f c } T „ _ 1 o r m + i
+
{-2p_ 2 ( n _ T O ) + f c _ 2 (25) + p _ 2 ( n _ m ) - 2 ( 2 5 ) J C ) f c } r n + i or m _ij(10.57b)
coefficient of y2. : 2'
+
2|p 2 ( n _ T O ) + f e _ 2 (-25)£)fc +P 2 („-m)+2A;- 2 (-2.D)
+ ^P2(n-m)-2(-25)£>|JTn_i O r m+ i +
2 j - p_2(n_m)+fe_2(2Z>)£>fc + P- 2 („- m ) +2 fe- 2 (25) + \p-2(n-m)-2{2D)Dl}Tn+1
coefficient of yjyk
{j ^ fc) :
O T ra _i,
(10.57c)
404
10. Soliton Equations
-
(n-m)2}j?jDk(TnoTm)
{2(j + k) + n—m)+j+k—2
(-21?) + 2p 2 (n-m)+j-2(-2£>) £>fc —m)+k — +
+ p2( (-2£»)£»j£»fc} 2(-2D)Dj {4p-2(n-m)+j+fc-2(2£>) - 2 p _ 2 ( n _ m ) + .,_2(25).D fc —2(n—m)+fc~2 (2D)Dj + p_ 2 ( Tn+l ° T m - l -
(10.57d) These equations should be equal to 0 since the right side of (10.56) is, which produces infinite numbers of bilinear differential equations. One sees that letting k = 1 and m = n or n + 1 in (10.57a) and (10.57b) give the trivial equations and that letting m = n + 2 in (10.57a) gives 2D\Tn+1 o Tn+1 - 4 r „ r n + 2 = 0.
(10.58)
Also letting k = 1 and j = 2 and m — n orm = n-\-\, one obtains the following bilinear differential equations: k — 1, m = n fc=l, m - n + 1 j = 2, k = 1, m = n
in (10.57c) in (10.57c) in (10.57d)
£>?rn o r n - 2 r n _ 1 r n + 1 = 0 (10.59a) (D? + D2)Tn o T„+I = 0 , DiD2Tn
orn
(10.59b)
+ 2£>iT n _i o T „ + I = 0 .
(10.59c) We note that the equation (10.58) just coincides with (10.59a). 10.3.2
Principal Picture
Let A be a symmetric affine Cartan matrix of type X\ ' where X = A, D or E, and g p r be the principal realization of Q(A) constructed in §9.3.2. In this section we follow the notation from §9.3.2. Let L(Ao) = C[Xj ; j e / + ] be the space of the basic representation in the principal picture, and we write the space of tensor product of two L(Ao)'s as L(Ao) ® L(Ao) = C [x'j ; j e 1+] ® C [x'i ; j e / + ] . For simplicity, we make use of the notation
*' ^ 4
and
:=
^ 4'
10.3. Hirota Equations Associated to Basic
Representation
405
We fix a Coxeter element w in the finite Weyl group W, and let {7^; i = 1, • • • , 1} be a set of representatives of (ty)-orbits in A . For each i, we choose root vectors jez/hz such that ( X y J X _ 7 J = h, which implies (xV>\X§)y)
= 6j,k6w
(10.60)
by Lemma 9.3.13.2). We consider the following two bases {£ r }rez and {£ r }rez of g1"": {6}
=
±Si®tmi,
X%\
I
K,
I
&4>
I
I
where j e I = Z — /iZ, k G Z and i = 1, • • • ,£. Then the operator S1, defined by (10.29), is written as follows:
jei
i=ifcez
+ i (7r(do) ® 7r(if) + TT(X) ® 7r(do)). In terms of the "shifted" current
x£u(*) ••= E*K*~ f c fcez introduced in (9.103), this formula is rewritten as follows:
s
= ^£^K®3'+3®4} + \^ < the constant term of ir(Xyi(z))
® 7r(X_ 7i (^)) ^
(10.61)
406
10. Soliton Equations -,
*•
+the constant term of — ^P\X^)ip\X-\i)V^iz) h2
i=i
m x d
l\J2
® V^yAz)
i 'i 'i ®Id + Id®Y^
\ie/+
mjx'Jfff
jei+
lYi^M-x'm-d")-^-™ i
+the constant term of ^ ^
kV%(z)
(10.62)
where we put bt := (p\X^)(p\X^t).
(10-63)
We now change variables: x'j = Xj + yj
and
x" — Xj — yj ,
namely x^- + x" — 2XJ
and
x'j — x" = 2yj .
Then d dx'-
JL-A. dx'J dxj
A
A dx'j
d
d
-
dx'-
dyj '
and
exp [ £ exp [ - £
=
expf
7i(5W)^z™»
-*(sM)4z»* I exp I ^
£7i(S
\jei+
J exp ( - £
W
)(^-^V
T i ^ '
1
) ^ ° mj dx^
^ [ - J l ) ^ ^
10.3. Hirota Equations Associated to Basic Representation
=
exp ( 2 £
7i(SW)Wz"*]
exp ( - £
407
7i(5[-i])£-^
A (10.64)
We now define the Schur functions
p(nA\x)
piAHXl,--.,Xj,-..)
=
in variables Xj (j e 1+) associated to an affine Lie algebra g(A) by exp( $>,-*"*'•) ^jG/+
=: f^PiAHx)zn.
'
(10.65)
n=0
Then the right side of (10.64) is written as
^)(27i(5W)%)Pr)(-^|:^^-)^-mJ
*£•<*> ®n<*)= E m,n>0
^
^
°Vj/
and so the constant term of Vf?(z)
7i(s [ " jl ) a
n=l
Notice that r(0) J >"i = ^ I ^ XAP^ KI' ^ ) = W — E / A^K*"^ 1=1
( 10 - 66 )
1=1
since {-4? } . =li ... £ and { x i l j . } .=1>... ^ are bases of \) satisfying ( X ^ |xi°>.) Using these, the formula (10.62) is rewritten as follows:
2 ^ y>
da
^{EE^H 2 ^),)p-(-^4)}. (10.67)
10. Soliton Equations
408
So, applying this operator S to a function f(x') ® g(x") in L(AQ)
® i(Ao) = C[a£ ; j G /+] ® C[a#; j G 1+]
one has
= l - f E m i»
5(/®5)
"»j
^ i=l n=l
9
% ,
^W)
•
(10.68) Using bilinear differential operators D
jf°9
•= -^-f{x +
t)g{x-t) t=o
dt
'Jv '' ixi
' ''ji''')9\
'' •>&]
,
tji''')
(10.69)
t=o
one can rewrite / ® g as follows: f{x')®g{x") -
= / ( • • • ,Xj +Vj,-- -)®g{--- ,Xj - % , • • • )
/(••• ^ j + i , - + % , • • • ) ®5(--- .a;j-(*j+2/j).---)|t=0 exp
( X ] yjdf)^"
'xi
~*J'"")
t=0
(10.70)
yJDj)f°9-
expf 5 Z k
>xi+ti>"')®9('"
jeJ+
This gives P1
'"'^'"" Irt*')®^") '^•'
=
expf
^yjDjjfog
^ ( - - - , - D j , - - - ) e x p [ 5^3/i-Dj J / ° 5 .
(10.71)
for any polynomial function P(- • • ,Xj, • • •). Then, from (10.68), (10.70) and (10.71), one obtains the following:
10.3. Hirota Equations Associated to Basic Representation
409
Proposition 10.3.4. Let g be a non-twisted affine Lie algebra over a finitedimensional simple Lie algebra g of rani £ with a symmetric Cartan matrix, and consider its basic representation constructed on the space L(Ao) = C[Xj ; j 6 /+]. Then, for elements f,g e L(AQ),
m
MDi + ^ E E biPiA)(2li(SW)yj)
S(f ® g) = \-iY, *•
x
S(f ® g) is given by the following formula:
jei+
t=i n=i
^(-^^)}exp(l?^)/09-
Prom this proposition, one obtains the following: Theorem 10.3.5. Let g(A) be a non-twisted symmetric affine Lie algebra over a finite-dimensional simple Lie algebra g, and w be a Coxeter element in the finite Weyl group W. Then the Hirota equations associated to the basic representation L(Ao) in the principal picture are given as follows:
{ " I £ m^Dj + 1 ±bi JTP^^SMP^ *•
J6/+
i=l
n=l
(-^D,) ^
xexpf £ ) % • # / ) / ° / = 0,
J
} ' >
(10.72)
where s = X } j = i ^ m ^ is the principal Cartan subalgebra ofg, and Sj € a(m*') are chosen such that (Sj\Sk) = h6j+k,e, and 74 (i = 1, • • • ,£) is a set of representatives of (w)-orbits in A1"", and bi (i = 1, • • • , £) are defined by bi := ( p | X W ) ( p | X ^ ) &2/ using roof vectors
X±„= Y,
X2leg^
j&Z/hZ
such that (X 7 i |X_ 7 i ) = h. We can view the Hirota bilinear differential operators in terms of the decomposition of the tensor product L(A) ® L{h) as follows. Since the representation
10. Soliton Equations
410
L(A), for A G P+, is unitarizable, its tensor product is completely decomposable, namely it decomposes into the direct sum of irreducible components. Let Lhigh denote its highest component, namely the irreducible component in L(A)
(10.73)
for v G L(A), which gives another charactelization of Hirota equations associated to L(A).. In the case of basic representation, the highest component of L(AQ) ® L(Ao) is L(2Ao). Denoting by U' the sum of all "lower components" in L(AQ)®L{AQ), one has the orthogonal decomposition, L(A 0 ) «> .L(Ao) =
L(2A0)®U'.
Or insteads, it may be better to consider the symmetric part S2(L(A0)) of L(AQ) ® L(Ao), since the highest component L(2Ao) is a g^-submodule contained in the symmetric part S2(L(AQ)), SO let S2(L(A0))
=
L(2Ao)(BU
be the orthogonal decomposition. Then, by (10.73), the Hirota equation for v G L(Ao) is equivalent to the requirement that v®v in S2(L(AQ)) is orthogonal tot/. We now consider the explicit realization of the basic representation on the space of polynomial functions: L(A0) =
C[xj;jeI+}.
Let ( , ) be the positive definite invariant Hermitian inner product such that (1, 1) = 1, where 1 in the left side stands for the constant function in L(A0). Then this inner product is explicitly described as
(f,g)
rrij oxj
'
(10.74)
where g( ) denotes the complex conjugate. This inner product naturally induces the positive definite g^-invariant Hermitian form on the space of tensor product:
10.3. Hirota Equations
i d ' rrij dx"
X02
/i(---
to Basic
Representation
411
/ 2 (x"=0
d_ ' 2mj \dxj
9i x
Associated
_d_ dyj
' 2771j \&Cj
3% (10.75)
,Xj+yj,---)f2(--x,y=0
Let g e U a n d / e L(A0).
0
=
Then
{f®f,9)
a ' 2m,- V 3a;,-
' 2mj \ 3 a ; j
dyj
3 9j/j (10.76)
•2/jv)
*/(••• . ^ + 2 / j . - ")/(•
x,i/=0
T h u s each element <; in C7 gives rise t o a Hirota bilinear operator. In other words, t h e space of Hirota bilinear differential operators characterizing t h e Gorbit through vo is the orthogonal complement U of L(2A) in S2(L(AQ)). A more detail explanation on Hirota equations from this viewpoint is given in §14.12 of t h e Kac's book [100], together with the formula on t h e dimension of the space of Hirota equations of each degree. We note t h a t , in this counting, one must b e careful about t h e "trivial" bilinear differential equations as is seen from (10.8). E x a m p l e 1 0 . 3 . 2 . Let us consider g = A^ = s[(2, C ) , where h — 2 and the set of exponents is / + = N 0 dd and Schur functions are given by
l} E « (^ n=0
exp |
E "jeNodd
XJZ1
) .
In this case, one also has >i by (10.66) and Example
and
(P\P) =
9.3.2. So the equation (10.72) becomes as follows:
{- E jy^ + xexpf
VjDj)f°f
E >k• -civr
liiSj)
,.
j'eN od d
/
ltp^yj^i-U)} = °>
10. Soliton Equations
412 namely
*• jeNodd
n=l
\
D
xexpf Yl Vj i)f°f
J
/
)
( 10 - 77 )
= 0-
Coefficients of y-1 • • -y,3 's in the Taylor series expansion of the left side of (10.77) give the Hirota equations associated to this representation. Schur functions Pn * (x) 's are given by
J1.J3)-"G^>0
Ji+3J3+5J5+---=n
namely P^\x)
= Xl,
P^\X)
= \xl
PiA[1)\x)
= ±x$ + x1x3, . . . .
P^)\x)
=
\x\+x3,
Using these, coefficients of y1^ y33 • • • are easily calculated and obtained as follows: vl
••
y\
: f(£»f-4D1£»3), :
o,
2/12/3 2/i
:
-f(Df-4L>1£»3), 2m5(13£>f + 100£»f£)3-288£>iL»5-320£>§),
y?y 3
: £(£>? + 1QD\D3 - 72£>!D 5 - 8L>|),
2/11/5
: - i f ( £ ' ? + 25£»?£> 3 -126D 1 £>5 + 10£'i),
y|
: -if(2L>f + 5£»?Z)3 + 1 8 £ > 1 £ » 5 - 7 0 D | ) ,
.FVom # m data, one sees that Hirota bilinear differential operators of degree 4 are scalar multiples of D\ — 4DiD3, and that the space of Hirota bilinear differential operator of degree 6 is 2-dimensional. Namely, putting A
:= D\ + 16DfD3 - 72DXD5 - 8£>|,
10.3. Hirota Equations Associated to Basic Representation
B
413
:= D\D3 - 6£>i£>5 + 2D\,
the coefficients ofyf, y\yz, 2/12/5 and 2/3 are scalar multiples oflZA— 108.B, A, A 4- 9B and 2A — 27B respectively. Notice that we do not need to consider the terms of odd degree by (10.8). Letting x\ = : x and x$ =: t, the bilinear differential operator Di-iDxIh
D4x-4DxDt
=
which takes place in the coefficients of yf and 2/12/3 is the Hirota bilinear differential operator of KdV equation as is discussed in §10.2. This means, by (10.32), that if a function f{x\,X3, • • •) G C[xi, X3, • • • ] belongs to the G-orbit G • 1 then the function f(x, t, X5, X7, • • •) satisfies the bilinear differential equation (10.20): {Dx-±DxDt)fof = 0. So all functions f(x\,Xa, • • •) in G • 1 are solutions of (10.20) by letting x :— x\, t:— £3 and all other Xj 's be arbitrary complex numbers, and then d2 := 2—"2 log/(a:, i,c 5 ,c 7 , •••)
u{x,t)
is a solution of KdV equation (10.11) for any c 5 , C7, • • • G C . One can produce some such solutions by applying vertex operators to the constant function 1. Namely, using notations in $9.2.2, a function exp(eV£f\z)).l is a solution with z and e being arbitrary complex numbers, which is written by Proposition 9.1.8 as follows: exp( e y A ( ° d 2 d) (z))-l = (l + eV^\z))-l
= l + e e
2
^ ^ .
Since x\ = x, x3 = t and other Xj 's and z and e are arbitrary complex numbers, this function is of the form 1
+
ce 2(ax+o
3
t)
(
a >
ceC),
which is just the r -function of a one-soliton solution. Applying again the exponential of a vertex operator to this function, one get a function in the G-orbit G • 1. Repeating this procedure, one has
exp (envfj«\zn))
• • • exp (e2V^\z2))
exp ( e i V ^ f a ) ) • 1
10. Soliton Equations
414 =
( l + enV^\zn))
•••(! + e2V^\z2))
(l + e i ^ f r ) )
•1 (10.78)
forneN, which converges in the domain \z\\ < ••• < \zn\. Functions (10.78) are r-functions of n-soliton solutions. 10.4
Non-Linear Schrodinger Equations
In the previous section we have seen that the principal construction of the basic sl(2, C)-module L(A0) gives rise to the KdV equation and its solutions. In this section, we calculate Hirota bilinear differential equations associated to the homogeneous construction of L(A 0 ; A\ ') and its connection with non-linear Schrodinger equations. We fix an integer n and consider the functions r n and T„±I satisfying the following bilinear differential equations from (10.59a) ~ (10.59c): D\rn o r n - 2 r n _ i r n + 1 {Dl + D2)TnoTn+1 (£»f + £> 2 )r„_i o r n + 2D1Tn-1oTn+1
D1D2TnoTn
= 0 = 0
(10.79a) (10.79b)
= =
(10.79c) (10.79d)
0 0.
Using (10.1b), bilinear differential operators appearing in these equations are written in terms of usual differentials as follows: D\rnoTn
= ^
£>?T„_i OT„ D\D2Tn OT„ D2Tn O T n + i D2Tn-i OT„ -DlTn-l°Tn+i
= = = — =
where T
.
-
2
^+^
=
2«'rn-<2),
T^_xTn - lT'n_xT'n + T n _ i < ' , t'nTn - fnT'n - T'nfn + Tnf'n — 2{f'nTn TnTn+i - T n f n + i , fn-XTn - T„_if n T^Tn+i -Tn-\T'n+l,
dr := ^— oxi
, and
fnT^),
. 8T r := ——. ax2
Then the bilinear differential equations (10.79a) ~ (10.79d) become as follows: •n'n
'n
T
n-lTn+l
— 0>
TnTn+i - 2T„T„+1 + TnT„+1 + fnTn+1 T^-iTn ~ ^T^T^ + Tn-XT^ + fn-XTn f'nTn - fnT'n + T'n_xTn+l - Tn-XT'nJrX
- TnTn+1 = 0, - T n _ i f•n„ = 0.
(10.80)
10.4. Non-Linear Schrodinger Equations
415
Dividing by T%, these equations become as follows: r'2
Tn-1
Tn+i
7~n
Tn
Tn+1
'n-1
-2-
n+1 'n-1
_» + Tn Tn_!
>n
= o,
T~n
'n+1
+
Tn T„
Tn-1 T, T n _|_i
^
Tn+l
Tn+i
Ti
7"n
^n ^"n—1 ^"n
+
+
7~n—1
7"n_i
'n+1
= o.
— (x,t,c3,c4,
•••),
= o,
(10.81)
o,
' n
We put q(x,t)
Tn+1
q*(x,t) u(x,t) where C3, C4?
rA
=
•••),
(10.82)
log(r n (x,t,c 3 ,c 4 , • • • ) ) ,
• are axe arbitrary arbit] complex numbers. Then by (10.10), one has ^ D
J"" < - i
(x,t,c3,c4,
=
uxx + uj.,
=
qxx + 2qxux + q(uxx + v%),
=
qxx + tq>x + q* (uxx + ul),
=
uxt + uxut,
=
qt + q*ut,
T
n-\
— qx + qux,
T
T T^
n+1 Tn Tn
Jn
=
q* + q*ux
n+\
=
ut,
Jn
=
qt + qut,
'n
fn-l
Tn+1
and, using these relations, differential equations in (10.81) are rewritten as follows:
Since uxx follows:
uxx - qq* = 0,
(10.83a)
-«? + £ x + 2«*u«x = 0, qt + qxx + 2quxx = 0,
(10.83b)
uxt + qxq* - qqx = 0.
(10.83d)
(10.83c)
qq* by (10.83a), equations (10.83c) and (10.83b) become as
qt + qxx + 2q2q* = 0,
-q*t + q*xx + 2qq*2 = 0.
(10.84)
416
10. Soliton Equations
We note here that equation (10.83d) follows from (10.84) and (10.83a). To see it, we differentiate (10.83d) by x. Then we have uxxt + qxxq* - qqxx
= °>
(10.85a)
= 0,
(10.85b)
namely (qq*)t + qxxq*-qq*xx
since uxx — qq* by (10.83a). And one easily sees that this equation (10.85b) follows from (10.84) since (qq*)t
= =
qtq* + qq*t = -fax* + V ? * )
More precisely the above calculation means the following. Assume that functions u, q and q* satisfy the equations (10.83a) ~ (10.83c). Then we have shown in (10.86) that -j^(uxt
+ qxq* -qq*x)
= 0,
namely uxt + qxq* — qqx
is independent of x.
So one can put uxt + qxq*-qq*x
=: ip(t).
(10.87)
Then the function v(x,t)
:= u(x,t)—x
I ip(t)dt Jo
satisfies vxx = uxx
and
vxt = uxt
-ip(t).
Substituting this into (10.87), one has vxt + qxq* - qq*x = o, and so the functions q, q* and v (in place of u) satisfy the equations (10.83a) ~ (10.83d). Thus the equation (10.83d) follows from (10.84) and (10.83a). One can construct solutions of (10.84) by making use of vertex operators in a similar way as was explained in Example 10.3.2 for KdV equation. Namely letting N be an arbitrary positive integer and ai and Zi (i = 1, • • • , N) be arbitrary complex numbers satisfying \z\\ < • •• < |ZJV|, we put J2
ema
® Tm{x) := exp (aNf±a(zN)^
- • -exp ( a i f ± a ( « i ) ) • 1,
(10.88)
10.4. Non-Linear Schrodinger Equations
417
where F±a(z) are vertex operators defined by (9.50a) and (9.50b). Then the functions q(x,i) and q*(x,t) denned by (10.82) using T„ and T„±I in (10.88) are solutions of (10.84). A solution thus obtained in terms of N-±-numbers of Ta(z) and JVVnumbers of T_ Q (z) is called a soliton solution of type (Ni, N2). To write down explicitly soliton solutions of type (N, N), we first notice the following formula for a, b,z,w £ C: exp (bf _„(!»)) exp (aTa(zj) =
l + afa(z)
=
+ bf-a(w)
(l + fcf _ 0 (t»)) ( l + a f a ( z ) )
+ abf-a(w)fa(z)
(10.89)
on T = Y^mez ema®Tm{x)
by (9.51). The action of r-a(w)Ta(z) from (9.50a) and (9.49b) as follows:
is calculated
f_ a (u;)r a (z)T =
Y, * 2 m " 2 r - « M ( ema® exp (f^xkzkym-1(---,xk-^—
=
J2 z2m-2w-2me^-^a XTm.1^..,xk
,•••")]
® exp ( - f > f c * / ) exp ( f ) (xfc +
mGZ
/
-k
^
2w- fe
2z" fe
+ —k
fc=l
'
'
\
—,...j OO
(
x
•
OO
fc=i
/
'
2w~fe
_fc
«
- j^Xfctfl* J exp ( X ) ( ^ + •..^
^ ~ ) A
^fc=l
^ - )
^t=i
2z~k
Since exp
£ ( - + T H ) = -p(E^)-(»E^(=)') y
fe=i
=
e x p ( f ; ^ ) . ( l - ^ ) -
v
2
y
fe=i
x
fc=i
= e x p ( f > * V ( - ^ )
2
y
(10.90)
in the domain |z| < |w|, the above formula is rewritten as follows: f_a(w)Ta(z)r
=
^ — Y ( - ) 2 m e (w — zY z—' \w/
m a
m(zZ
®
exp^xfc(^-t»
f e
)ymr--,xfc+2(w
Z
fc~
Thus from (10.89) and (10.91), one obtains the following:
\---VlO-91)
10. Soliton Equations
418
Lemma 10.4.1. Let a, b, z, w be complex numbers satisfying \z\ < \w\. Then for T — Ylm€Z ema ® Tm(x), the following formula holds: exp (6r_ a (u;)) exp (aT„(z)) T =
T+
a^z2m-2ema®exp(JTx>°zk)T™-i(--->x>°-^]r>---) ^fc=i ' ^ '
mez
+ bJ2
w-2m-2ema®expl
-^IfcMMrm+1
m€Z
v
^
fc=l
'
••• ,xfc + — — ,••• J ^
v
' mez ( X Tml---,Xk+
fc=i
k
k
2(w~ -z- )
'
'
\ ' " ' J"
Let N be a positive integer, and Oj, bi, Zi, u>i (i = 1, • • • , AT) be non-zero complex numbers satisfying kil < |«>i| < N l < \w2\ < ••• < |ZJV| < \WN\-
(10.92)
We put Ai := a,i exp f ^
xfe,zf j ,
£* := 6» exp I - ^
Xfciyf j
(10.93)
(2ip-Zi,)2Vii"--^v>
(W.94a)
for i = 1, • • • , N, and An , - ; » .
:=
(
II
^l
'
Bn,-,i, •= ( II K - ^ ) 2 ) % - ^ > l
K,-^h,-,is
•=
(10-94b)
'
1^—S
Ail,-,irBh.-J.
(10-94C)
n IK*,-™*) 2 p=l
9=1
for 1 < r , s < N and 1 < i\ < • • • < ir < N and 1 < ji < • •• < ja < N. In particular for r = 0 or s = 0, we put Ax,-,*, := 1 (if r = 0)
and
Bju...
tjs
Then these functions satisfy the following properties:
:= 1 (if s = 0).
10.4. Non-Linear Schrodiner Equations
419
Lemma 10.4.2.
1)
i
Ai\-,x
2z - f c N
k
k
(zN
Zi)2
-
J
'
H •>
72 Z
N
Ai\--,xk
-fc 2w AT
+
jfe
W%
i
Bj[--,Xk
k
Bj[~-,xk+—*-,
3)
i
il>*" i V I ' "
''
"
z2 R -w i)*B"
o_-fc
2)
A
(wN - z i)2
'
(zN
(WN - Wj)2
2wl
' *^fc
2z N k
1 ~~ z2r
'
2wn «!>•• i»r I ' ' ' ' • ' ' k "•"
A l , - ,ir,JV AAT ?w/ iN2 r
-
1.
D
" j i
w%
42
n (^JV - z«P) 4)
i
B,-
zjt
„2s Z AT
2z AT
n (ZN - wip)2
s,
p=l fe
, Zfc +
•"ii.— »ia
5)
i
A\\,—
2w7 "AT
1
A;
wN
2zTV
, i P a ' l , - ,ja I ^fe
AT
BN
1 -2(r-s)
2w , - f e
•^•l.— .vyii—,i» I •*-*; >
5'Ji,-,J.,JV
\
_
A
^i.-,v,JVai,iN
2(r-«)
WN
^i,-,v;ji,-,j.,N
BN
6) * , , , « „ „ - , « + 2 ^- 2 t2(r-s)
\ZN J Proo/. 1) Ai • • • , xfe
/_ (ZN
-
„„ \ 2 WN)
-^i,-,V,Af;ji,-,j.,Af — . -^AT-DAT
2zW
••J
=aiexp|^^
1
M-'gK^X -^'*
f c
-^-Jz?| ^AT — ^ i 1 ZJV
At,
10. Soliton Equations
420
proving i), and WN
exp < y^(x y^ k+(a M I ••• ,xk + - ^ - , ••• I = cii CLiexpl ^ fc=i ^
I ^fc\wJv7
J
V
-2
^JV/
Ufe
k /
\WN-ZiJ
proving ii). 2) is quite similar as follows: B j ( - , x
- ^ -
k
•)= 6,exp{-£(**-*£)«*}
proving i), and
proving ii). 3) is obtained from 1) as follows: A ( _^v! -^il," ,iT I ' ' ' ) Xk
^ ' J
(zip-zij2-(Ail..-Aj(---,3:fe-^,
n l
\
1 l_
1
72r
^
p=l
1 4 » r •-^ii.-.ir.AT)
proving i), and
I
2w k
N
k
' 2w7k
i
\
\ 2
10.4. Non-Linear Schrodinger Equations
421
= n (^-^) 2 -<-n (wjv ^, )2 -(^-^) l
p=l
V
%r
'
proving ii). 4) is quite similar and obtained from 2) as follows:
k
' 2z7k
l
=
\
J
n K-«i.)'^-n flw _v )a -(g*"gi.) l
J p
P=l
1
) 2 v*"^ •""•-J" = ^ •P=IiI ^r«„" ^ - «,.proving i), and 2^7*
= n ^u-wu)2-(Bh--Bim)(-,xk+^-,-.\ l
=
\
/
2
II
2
^ir-^iq) -^:-ll(^-wip) -(Bjl-Bim) N
l
P=l
J_ J_ R WN
proving ii). as follows:
BN
5) is obtained from 3) and 4) and the definition of Aiu... ,iT-jlt-. j ,
2z£_ \
/ ™i\,— i*riii.— ,js I ' " i
x
k
o~-fc ZZ
n n^p-^-,)
JV
2
p=l q=l 1 r
s
n nK-^-,)
2
_jL ^ii,--,ir,iV _2r ' A.. '
ZN
N
£iV
D
D
a
IKZN-VJ,)
2
ji,-,3s
422
10. Soliton Equations
J_
1 2 r S
°N( - ) ' A
il
'^^^'ir'N''jl'^'i°,
N
proving i), and
A -
•
•
•
(•••
xfc
n IK^-^J
+
2
^
(•^ii,—,ir-^ji,—J.)
I
"
> xk
+
,
p=l g=l W
J
/V
B
JL
A
J1,-,3„N
« #
2(r-a)
W
N
proving ii).
_j_
#JV
.
' D ' " > i r " ,*r-;ji,--- .i»>^V> iJjV
6) is obtained from 5) as follows:
2(r-a) AT
Z
A• « • • f--'1»l,-",tI.,JVjJi,-",J, I /•
rt. + ^ L ! -*"fc "I" £ 2i«
2(r+l-s)
2(r-s) Z N f/^N\2{rS) [ZN
J
)
^»i,-,v,iV;ji,-,j3,iV
„», , "™ (zN-WN)*AN (ZN ~ WN)2 '
A..T3.. ANBN
...^ I
BN
•^l.-.ir.Afjii,-,j„N-
D
Theorem 10.4.3. Lei N be a positive integer, and ai, fy, Z{, u>i (i = 1, • • • , JV) be non-zero complex numbers satisfying (10.92). Then the functions r m (x) (m e Z) defined by exp N>jvf _Q(iOiv)J e x P (ajvfa(zjv)J =:
^e mGZ
m a
®rm(x)
- e x p (&if_Q(iui)J
exp f a i r Q ( z i ) J •
423
10.4. Non-Linear Schrodinger Equations are given as follows: A
^2
iu-,ir;h,-J*
%f\m\
r,a€Z>o, i—3=rn l
Tm(x)
(10.95)
0
if \m\ > N.
Proof. To prove this theorem by induction on N, we put exp f6jv_if_ Q (wjv-i)J exp (aN-ifa(zN-i)J
x •••
• • • x exp (ftif _ a (u;i)) exp ( a i f a ( z i ) ) • 1 =: ^
e m a
and assume that
{
J2
^i.-,*riix,-j-
if|m|<JV-l
r,s€Z>0, r-s=m l
(10.96)
0 if |m| > N holds for allTOG Z. Since J2 ema ® 7V»(a:) = exp (&*?-„(«>*)) exp (ajyf a (zjv)) ( ^ e m a ® Gm(x) J, mGZ
^meZ
one has emQ
X!
m£Z
+
'
® *•».(*) = E e m Q ® ^ w mGZ
«JV E
4 m _ 2 e m a ® exp ( f > f c 4 ) G m - i ( • • -,xk-
mGZ
^fe=l
'
^ - , •
^
XkW Gm Xk + bNmYGlZ^N2m'^ma®^p(-f2 ^) +4--^ ^ fc=l ' ^
+
^k^^--)'
flAf&iV
x G
m
^ - - , x
f c
+ ^ L ^ , . . . j
(10.97)
by Lemma 10.4.1. Comparing the coefficients of ema in both sides of (10.97), one has Tm(x) =
Gm(x)
10. Soliton Equations
424 +aNz%n
2
exp(
+bNWn - exp apr_jr
{?0L\
(zN-wN)2
exp ( V xk(z% - w%) )
\wNJ
\^
x Gm[ ••• ,xk + Gm(x) + _i_R
2(u£* -
zj) k
Gm-il
...-2TO-2/-I
J
k
2
ANZ2^1
+BNwN
jf-'"')
••• ,xk
(
_
i
2wN
G m + i l ••• ,xk -\
—,•
2
ZJ..., +T(zAEM^(^) -w y \w j v Ik N
N
2w~k + —^-,
/ °° \ / f - ] T x k w % J G m + 1 l---,xk
2m 2
+
••• ' ^ ~ ^jf - '""' )
^XkzkjGm-J
N
+
^ ^ lk , . . . )J
( 1 o. 9 8 )
by (10.93). We note here that the following formulae hold from (10.96) and Lemma 10.4.2.5) and 6):
c ^m—
(
—
1I
j ^k
N j
i
/ ,, ^ t i , - - , v ; i i , - - ,5s I ' ' ' >xk r,s£Z>o, i—s=m—1 ^ l
-2(Si3I)-^ iV
E
.
)'
^i,-A.,";*,.-j.,
(10-99a)
r,sGZ>o, i—s=m—1 l
_ / 2wjf-fek \ Gm+i I ' • • , %k H 7—' " ' J
v
• •
A
7 . . . ** +
^
r,a€Z>o, i — s = m + l l
1
2(m+l)
BN
£ r,«€Z>o, i—a—m+1 l
G J...,x 1 +
2
<^i5
^n.-A-iii.-j.,".
(10.99b)
10.4. Non-Linear Schrodinger Equations
V —
(
Xk
U rJ x
(ZN-WN)2
, W-*^>
>Xfc^
2-/ - ^ l i — i*riJl.— .J»l r,s€Z>o, r —s=m ^ l < t i < —
fwN\2m
=
A-
425
£
^ '
/
y-^
A^JV
/mOQ^
^
^ , - , v , ^ 1 , - , ; „ i v . (10.99c)
r , s e Z > 0 , 7—s=m l < i i < —<»,<JV-1 l<jl<-<j.<JV-l
Then, substituting (10.96) and (10.99a) ~ (10.99c) into (10.98), one obtains the formula (10.95) for r m ( x ) , which completes the proof of this theorem. •
To write down (N, JV)-soliton solutions of partial differential equations (10.84) explicitly, we introduce the following functions: fi(x, t) := aieXZi+tz<,
gi(x,
t) := bie~(xWi+tw^,
(10.100a)
and
f fiu... ,ir(x,t)
(^-^)2-/ii"--/v
1= {l< P «7
{ and
II
II
ifr>0 (10.100b) if r = 0
(W3P - wu)2
• 9h •••9j„
l
/ * I , - , W I , - , J . 0 M ) := - T — i
n nfe P -^j
if s > 0 (10.100c) if s = 0
2
fii,-,ir9j1,-d.(x,t)
(lO.lOOd)
p=l g=l
Then one obtains Corollary 10.4.4. Let N be a positive integer, and a,i, bi (t = 1, • • • , N) be non-zero complex numbers, and zi, Wi (i = !,-•• ,N) be mutually different
10. Soliton Equations
426
complex numbers, and n be an arbitrary integer. Then the functions r,s€Z>o, r—s=n—1 1<»1< —
q(x,t)
1
:=
^
<
N
^
,
(10.101a)
r,sGZ>o, i—s=n l
g (x,t)
:=
^
-
(10.101b)
r,aGZ>o, i—s=n l
ane solutions of (10.84). We remark here that the condition (10.92) is not necessary in this Corollary, because functions defined by (10.100a) ~ (lO.lOOd) do not contain variables x 3, xii • • • and s o infinite sums as in (10.90) do no longer take place. We now proceed to consider the partial differential equation (10.84) with a boundary condition q(x,it)
= q*(x,it)
or
— q*(x,it)
for real variables (x,t). To describe soliton solutions, we let n = 0 henceforwards and consider the functions q(x, t) := — (x, t),
q*(x, t) := — (a;, t),
To
T0
where r0(x,t)
:=
Yl rez> 0
/n.-.*rJJi,-jV
(10.102a)
1<»1< —
r^(x,t) :=
J2
/*i.-.«r-Ji.-j-
(10.102b)
r,sGZ> 0 , i—s=—1 l
nfot) :=
J] r,a€Z>o, r—s—1 l
fii,-Mi,-J.
(10.102c)
427
10.4. Non-Linear Schrodinger Equations And we introduce functions i/)^(x,t)
defined by:
(x it) =• J ^ ( + ) ( x ' * ) | I/J(—^(a;, t)
if
g( x >^) = 1*(x,it), if q(x, it) = — q*(x,it),
where x,t G R. Then, from (10.84), one sees that these functions satisfy the partial differential equations
iV>!+) + v££ ) + 2V' (+) h/' (+) l 2 -;V>t(~) + V>ix ) -2V> ( - ) |> ( - ) | 2
= o, = o,
(10.103)
I2,
(10-104)
namely ^t(±)
= ^ ± 2 ^ 1 ^
called the non-linear Schrodinger equations. Prom (10.100a) ~ (lO.lOOd) and (10.102a) ~ (10.102c), one sees that frfait) Zj
= =
gj(x,it) -Wj
)1 J
(forall
= >
/ T0(x,it) = r0(x,it), \ T±1(x,it) = TT1(X,U) q(x,it) = q*(x,it)
and fj(x,it) Zj
= =
-gjix.it) ~9j(x,it) -Wj
(,, foral„ l
. JA J
_^ =>•
j r0(x,it) = T0(x,it) j T±1(x,it) = -T^i(x,it) q(x,it) = —q*(x,it),
where x , ( e R . So, given non-zero complex numbers aj and Zj (j = 1, - - • ,N), we define functions fj(x,t), fiu... iir(x,t) and fiu-,ir;ju-,j.(x>t) as follows: £•(*,*)
:=
aie"^fa">
/ii,-,tr(a;,*)
:=
< i
[ r
^ /»i.-" ,»r;ii,-" .is (x>*)
:=
1 s
if r = 0, 1
-
^
1 1 1 1 7T _i_ wr~\2 ' J*ii'".*r/ii.— .i»p = l q = l ^ 1 P "+" JV
428
10. Soliton Equations
Then functions
E r,s£Z>0,
:=
(±l)Vn, ••• , » r i J l , - "
-
-
E
>j»
r—3=—1
1
»-
.
(10.105)
(±i) s /u,.• , » » y ' i , - "
>j'»
(±i)'/u,... ,»T-iii>—
,js
sez> 0 Kil<-
£ *(±) (x,t)
r,s€Z>o, i—s=l Kii<-
E
(±i)"/n,- ,iatil,—
Js
s€Z>0 K t i < —
satisfy
#±>(M) -
±5*(±)0M)
and Thus we have proved Theorem 10.4.5. Given a positive integer N and non-zero complex numbers aj (j = 1, • - • , N) and mutually different complex numbers Zj (j = 1, • • • , N) satisfying Zi + z~j ^ 0 for all i and j , functions ip^ (x, t) := q^-' (x, t) defined by (10.105) are (N, N)-soliton solutions of the non-linear Schrodinger equations (10.104).
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Index admissible g-module, 77 admissible representation, 77 admissible weight, 77 admissible weight ^,y, 115 affine Cartan matrix, 3 affine Dynkin diagram, 3 affine Lie algebra, 3 affine type (Cartan matrix), 2 annihilation operator, 230, 288, 291 asymmetry, 355 asymptotic dimension (of Virasoro module), 252 bi-multiplicativity, 355 boson-fermion correspondence, 236 branching function, 269 branching function (of tensor product), 275 BRST-complex, 300 canonical central element, 9 Casimir element (of finite-dimensional Lie algebra), 27 Casimir element (on adjoint representation), 29 Casimir operator (of affine Lie algebra), 212 central function, 154 character formula (in terms of maximal weights), 36 character formula (of integrable highest weight ^-module), 33 character (of finite group), 154
character (of highest weight fj-module), 32 character (of Virasoro module), 241 charge, 226, 231, 288, 292 charge decomposition, 235, 290 charge operator, 235 charge variable, 235 charged fermionic field, 287 charged free fermion, 225 classical theta function, 59 Clifford algebra, 288, 291 co-Dynkin diagram, 13 co-integral form, 16 co-label, 5 co-tier number, 5 coboundary operator rig , 300 coboundary operator d^, 299 conformal weight, 181 constant field, 197 coset Virasoro field, 233, 269, 274 Coxeter element, 367 Coxeter group, 12 Coxeter number, 6, 366 creation operator, 230, 288, 291 current, 208 cyclic element, 366 degree of representation (of finite group), 155 ^-function, 175 denominator, 33 denominator identity, 33 derived Lie algebra, 274, 346
442 discrete series (of Virasoro algebra), 243 dominant co-integral form, 16 dominant integral form, 15 dual Coxeter number, 6, 366 dual tier number, 5 Dynkin diagram, 3 elementary Schur polynomial, 402 energy, 226, 231, 288, 292 energy momentum field, 307 energy operator, 213, 217, 235, 372 equivalent (of Cartan matrices), 1 equivalent representation (of finite group), 154 Euler-Poincare characteristics, 315 exponent, 367 extended affine Weyl group, 18 extended Weyl group, 18 field, 175 finite part, 10 finite part (of affine Cartan matrix), 10 finite part of H and A, 10 finite type (Cartan matrix), 2 Fock representation (of Virasoro algebra), 217 Fock space, 217, 230 free bosonic field, 215 free field, 357 Fresnel's integral, 167 Freudenthal de-Vries strange formula, 30 fundamental alcove, 14 fundamental chamber, 14 fundamental co-integral form, 16 fundamental co-weight, 16 fundamental imaginary root, 9 fundamental integral form, 15 fundamental weight, 15 fusion algebra, 163
Index fusion algebra (associated to fusion datum), 165 fusion coefficient, 128, 320 fusion datum, 162 GCM, 1 generalized Cartan matrix, 1 ghost field, 295, 307 growth (of Virasoro module), 252 ^-component, 10 f)-diagonalizable g-module, 274 f)*-component, 10 Heaviside function YL, 219, 226 Heaviside function Y+, 179, 331 height, 44, 366 Heisenberg group, 66 highest component, 410 highest root 6, 14, 20, 25, 29, 352, 366 highest weight, 31 highest weight g-module, 31 highest weight module (of Virasoro algebra), 240 highest weight vector, 31 Hirota bilinear differential operator, 385 Hirota equation (associated to -L(A)), 393 homogeneous construction (of basic representation), 352 homogeneous element, 178 homogeneous Heisenberg algebra, 352 homogeneous realization (of affine Lie algebra), 352 imaginary root, 12 indecomposable Cartan matrix, 2 indefinite type (Cartan matrix), 2 integrable highest weight module, 32 integral form, 15
Index intertwining operator, 155 invariant bilinear form, 3 invariant Hermitian form (on Virasoro module), 242 isotypic subspace, 161 Jacobi triple product identity, 33, 235 Kac-Moody Lie algebra, 2 Kac-Moody-Cartan matrix, 1 KdV equation, 387 Korteweg-de Vries equation, 387 label, 5 A-bracket, 185 level, 17 Lie superalgebra bracket, 198 Lie superalgebra End(V), 179 Lie superalgebra gl(m\n), 179 Lie superalgebra gi(V), 179 locality, 175 long root, 14 lower component, 410 maximal weight, 35 metaplectic group, 70 minimal series (of Virasoro algebra), 245 modular group, 67 multiplicity (of imaginary root), 12 multiplicity (of weight), 32 n-soliton solution (of KdV equation), 392 Neveu-Schwarz fermion, 221 (N, JV)-soliton solution, 425 (iV, iV)-soliton solution (of non-linear Schrodinger equation), 428 non-degenerate invariant bilinear form on principal A\ , 345 non-degenerate principal admissible weight, 113
443
non-linear Schrodinger equation, 414, 427 normal product, 177 normalized branching function, 269 normalized branching function (of tensor product), 275 normalized character (of integrable fl-module), 62 normalized character (of principal admissible representation), 95, 320 normalized character (of Virasoro module), 246 normalized character (of W-algebra), 320 normally ordered product, 177, 330 normally ordered product of JV-numbers of fields, 200 odd delta function 60dd(z — w)> 332 one-soliton solution, 413 operator product, 184 orthogonal decomposition, 410 orthogonality relation (of irreducible characters), 160 parity, 177, 178 partition number, 37, 215 positive real coroot, 13 primitive central element, 9 primitive imaginary root, 9 primitive vector, 274 principal admissible number, 83 principal admissible subset of coroots, 79 principal admissible weight, 79 principal admissible weight of level m and of type AY ,+ , 88 principal affinization, 371, 372 principal Cartan subalgebra, 366 principal realization of A\ , 347 principal specialization, 42
444 principally specialized character, 42 Ramond fermion, 225 real coroot, 13 real root, 12 representation matrix, 155 residue of principal admissible character, 113, 120, 320 scalar field, 197 Schrodinger representation, 352 semi-infinite wedge space, 288, 294 Serre relation, 265 shift transformation, 194 shifted current, 374, 405 short root, 14 simple coroot, 13 simply-laced Lie algebra, 3 soliton solutions of type (iV^A^) (of non-linear Schrodinger equation), 417 special index, 15 specialization, 42 specialized character, 42 standard bilinear form, 10, 25 strange formula, 30, 296 strange formula (for Lie superalgebra), 215 strictly positive integral form, 17 string function, 35 structure constant, 297, 356 super Jacobi identity, 179, 198 super skew-symmetry, 179, 198 super-bracket, 198 super-commutative field, 178, 187 symmetric Lie algebra, 3 symmetrizable Cartan matrix, 2 symplectic boson, 230 r-function, 389 theta function, 73
Index 3-soliton solution (of KdV equation), 392 tier number, 5 Tits cone, 14 translation operator ta, 11, 67 transposed Cartan matrix, 2 transposed Lie algebra, 13 transposed root, 301 twisted lattice, 353 twisted multiplication, 356 2-soliton solution (of KdV equation), _ 392 type Q (of affine Cartan matrix), 136 unitaxizable (of group representation), 155 unitaxizable Virasoro module, 242 unitary form, 155 vacuum state, 225 vacuum vector, 230 Verlinde's formula, 128 vertex operator, 237, 238, 323, 357, 362, 375 Virasoro algebra, 240 W-algebra, 300 weight (of highest weight fl-module), 32 weight space, 32 Weyl co-vector, 17 Weyl group, 11 Weyl vector, 17 Weyl-Kac character formula, 33 Weyl-Kac denominator identity, 33 Wick theorem, 185 Witt algebra, 240
Lectures on
Infinite-Dimensional Lie Algebra The representation theory of affine Lie algebras has been developed in close connection with various areas of mathematics and mathematical physics in the last two decades. There are three excellent books on it, written by Victor G Kac. This book begins with a survey and review of the material treated in Kac's books. In particular, modular invariance and conformal invariance are explained in more detail. The book then goes further, dealing with some of the recent topics involving the representation theory of affine Lie algebras. Since these topics are important not only in themselves but also in their application to some areas of mathematics and mathematical physics, the book expounds them with examples and detailed calculations.
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