Unitarity of the Knizhnik-Zamolodchikov-Bernard connection and the Bethe Ansatz for the ell(22)

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

+ ( (zn? zn0 )2+ 0 (zn? zn0 )) h 0; 0 i: n n Hamiltonians hn, n 0 are the classical versions of the elliptic Gaudin Hamiltonians Hn of eqs. (37) and (38), see 24] 25]. The SU (n) elliptic Hitchin system corresponding to one insertion has unexpectedly appeared recently in the description of the low energy sector of supersymmetric YangMills theories 4].

5 Unitarity of the KZB connectionOne of the essential features of the structures discussed above should be the compatibility of the KZ and K

ZB connections with the scalar product of CS states. The integrals in eq. (31) have been conjectured in 17] (and proved in many cases) to converge precisely for invariant tensors 2 Wz0 and to equip bundle W 0 with the hermitian structure preserved by the KZ connection. The latter condition means that for all local holomorphic sections z 7! (z) of bundles W 0, corresponding to holomorphic families z 7! (z) of CS states, (41) In order to see why one should expect such a relation, it will be convenient to express the scalar product integral in the language of di erential forms, following refs. 27, 28]. Let !(y) 1 y?1 dy. Introduce the V -valued K forms0 K;

@zn k k2= (; rzn ):

(z; y)= ! (z1? y1;1 )^ ! (y1;1? y1;2)^:::^ ! (y1;K?1? y1;K )1 1 1

and

^:::::::::::::::::::::::::^!(zN? yN;1)^ !(yN;1? yN;2)^:::^ !(yN;KN?1? yN;KN ) h j n (e n;::: e n;Kn ) n0(z; y)

( )

=

X X

K 2SK

(?1)j j K; (z; y): e? S h1 0

We may rewrite the scalar product formula (31) as

k

k2=

const:

Z

Yz

0;

i2

2

where Yz stands for the space of y 's with ys 's not coinciding among themselves and with zn 's. We use the conventions that j 0j2 (?1)K(K?1)=2( i )K 0^ 0 and that the integral of the forms of degree lower than the dimension of the cycle is zero. Assuming a su ciently strong convergence of the integrals, we may enter with the holomorphic exterior derivative under the integral so that

@ k k2= const:@ e?Yz

Z

1

S0

h 0; i 215

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