Unitarity of the Knizhnik-Zamolodchikov-Bernard connection and the Bethe Ansatz for the ell(8)

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 takes values in the Cartan algebra h and U in the compact group G. Upon the change of variables, eld U decouples from the functional integral (7) due to the gauge invariance leaving us with a WZW-type functional integral over the GC=G-valued elds b 21]

k k2= h (Au);Q

Z

?1 n (bb )(n) (zn ) (Au )i ik R ekS (bb;Au+Au )? 2 tr(Au Au )

j (u; b) ( (0)) Db d2u

(10)

where Db= z db(z ) is the formal product of GC -invariant measures on GC=G, the delta function xes the remaining freedom in the parametrization (8) and the Jacobian of the change of variablesy j (u; b)= const: 2?2r e 2h_ S(bb; Au+Au ) det(@Au@Au ):

Above, r stands for the rank of G, h_ for its dual Coxeter number and@Au The last determinant was computed in 21]:y 2 det(@Au@Au )= const: 2 r e h_ ju?u j= j (u)j42 2

@+ Au;].

where

is the Kac-Weyl denominator:Y Y Y (u)= e id=12 (e ihu; i? e? ihu; i ) (1? q l )r (1? q l e 2 ihu; i )

>0

l>0

with d denoting the dimension of the group and q parametrization (9) takes the form

e2 i . The WZW action in the

S (bb; Au+ Au )=? 4iwhere2

Z

h@;@ i? 2i2

Z

he u (n?1@nu ) e? u; n?1@nu i u u

nu= e? (z?z )u= n e (z?z )u=;Note that

u

=? (z? z )(u? u)= 2: expX

nu= exp

X

>0

e? hu; i(z?z )= v e]2

>0

v0 e]:

In terms of

the Iwasawa variables, the invariant measure on GC=G is

db=

Y

r

j=1

d

j

Y

>0

d2(e?h; i=2v )

where j= hhj; i are the coordinates of w.r.t. an orthonormal basis (hj ) of h. Using the holomorphic functions (u) to represent and the parametrization (9), we obtain

h (Au); (bb )?1(zn ) (Au) i= e nY

n

( )

2

k Re(juj2 )

h (u); n((nu)?1e? u n?1 ) n (zn) (u) i: u( )

5

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