Unitarity of the Knizhnik-Zamolodchikov-Bernard connection and the Bethe Ansatz for the ell(29)

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

in the notations of eq. (43) and with yn;0 zn; which follows easily form the de nitions of the forms and !q . Eq. (52) may be rewritten asX

n

j dzn n

?

X

s

j dys s

^=( )

i 2

d^@uj?

X

n

dzn^ hjn;( )

(53)

with the contragradient action of hjn on the V -valued form . The last relation, upon substitution to eq. (51), yields

@ k k2= const: 2?r=2

Z

UY;z+ 4i dZ

e

2 2

jw?wj2 e? 1 S 22

@ 0 h; i? 1@S^ h; i d^X

^ uh; i?X

i

+1?= const: 2 r=2UY;z

e

n2 2

jw?wj e? 1 S 22

dzn^@uj ( h hjn; i )^ d2r u( )

j

@uj (h@uj; i)

@ 0? 1@S^+1X

? 4i d^4

u

+(?1)K

;@0+and

i

d^

u

?

1

X

n

n dzn^ hjn)@uj (

dzn^ hjn@uj;( )

^ d2r u:(54)

The crucial result are the following equalities:

@0= 0

(@S )^+ d^ H0+

X

n

dzn^ Hn= 0

with the contragradient action of Hn 's. The rst of these equalities is a straightforward consequence of the closedness of the forms !q from which is built. The second, more technical one, has been announced in 12] (as Prop. 9). Using these relations, we nally obtain

@k

k2=

? const: 2 r=2

Z

UY;z

e

2 2

jw?w j2 e? 1 S 2 (?1)K h

;@ 0+ 1 d^ H0

+1

X

n

dzn^ Hn i^ d2ru= (; r )

which proves the unitarity of the KZ connection w.r.t. the scalar product (26) modulo the control of convergence of the integrals.

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