Unitarity of the Knizhnik-Zamolodchikov-Bernard connection and the Bethe Ansatz for the ell(16)

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

blocks of the WZW theory satisfy the (Knizhnik-Zamolodchikov) equations r= 0. In fact, the WZW conformal blocks are horizontal sections of a (generally proper) subbundle 0 W 0 XN V G. The bers Wz0 of W 0 may be identi ed by the assignment ! (0) with the genus zero CS state spaces. The subbundle W 0 may be described by giving explicit algebraic conditions, depending holomorphically on z, on the invariant tensors in V G 0 29, 16]. The KZ connection r preserves the subbundles XN V G and W 0 of the trivial 0 bundle XN V . The extension of the Knizhnik-Zamolodchikov connection to the genus one case was rst obtained in ref. 3] and elaborated further in 8, 6, 14, 12]. We shall use the description of the genus one CS states by the theta functions (u)= (u) (u). The spaces W;z of states form a holomorphic bundle over the space of pairs (; z) with no coincidences in zn 's viewed as points in T . The holomorphic sections of W correspond to holomorphic families (; z) ! (; z; ). The KZB connection is given by the formulae

r=@; rzn=@zn; r=@+ 1 H0(; z); rzn=@zn+ 1 Hn(; z); (36)compare to eqs. (1). Explicitly 6, 14, 12],

H0 (; z)=

i 4

u

+

Hn (; z)=?

X

r

i X@ P x hu; i (zn? zn0 ) (e )(n) (e? )(n0 ) 4 n;n0=1 r X+ 8i ( (zn? zn0 )2+ 0 (zn? zn0 )) hjn) hjn0 ) ( ( j=1X X X

N

; (37)

j=1

hjn@uj?( )

n0 6=n

Phu; i(zn? zn0 ) (e ) n (e? ) n0( ) (

)

+P

X

r

j=1

(zn? zn0 )) hjn hjn0( ) (

)

(38)

2 where u#01=#1. The expressions for Hn; n 1; reduce to the j (@uj ) and ones for the Gaudin Hamiltonians (35) in the limit ! i1. The operators Hn; n 0; acting, say, on meromorphic functions of u 2 hC with values in V 0 commute, see 14]. Their commutation forms part of the conditions assuring the atness of the KZB connection (36). Although the coe cient functions in Hn have poles on the hyperplanes hu; i 2 Z+ Z the, connection maps holomorphic families of genus one CS states into families with the same property, due to the increased regularity (6) of the CS states on the singular hyperplanes, see 8, 14]. 0 The commuting operators Hn for genus zero or Hn at genus one are quantizations of classical Poisson-commuting Hamiltonians of the, respectively, genus zero and genus one Hitchin integrable systems 22, 4, 24, 5]. Let us brie y recall this relation. Given a Riemann surface and a group G, let A denote the corresponding space of 0; 1-gauge elds and G the group of complex gauge transformations, as in the beginning of Sect. 2. Both are (in ni

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