Elliptic Wess-Zumino-Witten Model from Elliptic Chern-Simons Theory

This letter continues the program [17][12][20][21] aimed at analysis of the scalar product of states in the Chern-Simons theory. It treats the elliptic case with group SU2. The formal scalar product is expressed as a multiple finite dimensional integral which, if convergent for every state, provides the space of states with a Hilbert space structure. The convergence is checked for states with a single Wilson line where the integral expressions encode the Bethe-Ansatz solutions of the Lamé equation. In relation to the Wess-Zumino-Witten conformal field theory, the scalar product renders unitary the Knizhnik-Zamolodchikov-Bernard connection and gives a pairing between conformal blocks used to obtain the genus one correlation functions.