Unitarity of the Knizhnik-Zamolodchikov-Bernard connection and the Bethe Ansatz for the ell(12)

We work out finite-dimensional integral formulae for the scalar product of genus one states of the group G Chern-Simons theory with insertions of Wilson lines. Assuming convergence of the integrals, we show that unitarity of the elliptic KnizhnikZamolodchi

where h j n h n j,= ( 1;1;:::; 1;K; 2;1;::::::; N;KN ) P quence of K= n Kn simple roots satisfying1

( 1;:::; K ) is a se(20)

X

K

s=11

s

=

X

N

n=1

n;

y= (y1;1;:::; y1;K; y2;1;::::::; yN;KN ) (y1;:::; yK ) is a sequence of K points in T and the kernels Fp; are composed of the Green functions of twisted@ FK; (; u; z; y)=Y

n

Phu;

n;1+:::+ n;Kn i (zn

? yn;1 ) Phu;

Phu;

n;2+:::+ n;Kn i (yn;1 n;Kn i (yn;Kn?1

? yn;Kn ): (21)

? yn;2)

" . . ." contains the terms that will drop under the renormalization of the -integral.3.2 Functional integration

The above use of eq.(15) to express the 0 dependence of the insertions reduces the 0 integral to the formZ Y

K

s=1

+2 _ X (@z 0 s )(ys )(@z 0 s )(vs ) exp?i(k2 h ) s e?h u; i@ 0@ 0]

Y

= ( k+2h_ )K

X

K Y

2SK s=1

>0 e h u (ys ); si s; (s)

>0; zY

d2(e?h

u; i=2

0)

(ys? v (s) )

>0

det(@ y@ )?1 (22)

with running over the permutations of K points and with@ determinants are well knownY2

e?h u; i=2@ e h u; i=2 . The

>0

Y det(@ y@ )?1= const: e? r=6 j (u)j?2 j1? q l j2r

1

_ _ exp ih sh@;@ i? 2h ju? uj2]: 42

l=1

After the 0 -integration and an easy combinatorial manipulation, see 10], trading the sum over root sequences into sums over permutations (two 's satisfying eq. (20) di er necessarily only by a permutation), the scalar product formula (11) becomes

k k2= const: e? r

Z 1 K N=6 Y j1? q l j2r Y e?h u (zn ); n i Y e h u (ys ); si s=1 n=1 l=1 r Y i exp? 4 sh@;@ i+ 2 2 ju? uj2] ( (0)) D j j (u)j2 j=1 K 2 Y X X FK; (; u; z; y) h j n (e( )n;1 e( )n;Kn )(n) (u) i d2r u d2ys s=1 K 2SK2

(23)

where

k+ h_ and is a xed sequence of K simple roots satisfying (2

0).8

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