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

TL;DR

This paper analyzes the scalar product in elliptic Chern-Simons theory with group SU(2), connecting it to elliptic Wess-Zumino-Witten models and providing a framework for understanding the Hilbert space structure and conformal blocks.

Contribution

It extends the analysis of scalar products in Chern-Simons theory to the elliptic case, linking it with elliptic Wess-Zumino-Witten models and conformal field theory.

Findings

Scalar product expressed as a finite-dimensional integral

Convergence checked for states with a single Wilson line

Scalar product relates to Bethe-Ansatz solutions of the Lame equation

Abstract

This letter continues the program aimed at analysis of the scalar product of states in the Chern-Simons theory. It treats the elliptic case with group SU(2). 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 Lame 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.