Unitarity of the Knizhnik-Zamolodchikov-Bernard connection and the Bethe Ansatz for the elliptic Hitchin systems

TL;DR

This paper establishes a connection between the unitarity of the elliptic Knizhnik-Zamolodchikov-Bernard connection and the Bethe Ansatz for elliptic Hitchin systems, using integral formulas for Chern-Simons states.

Contribution

It provides finite-dimensional integral formulas for scalar products of genus one Chern-Simons states and links unitarity to the Bethe Ansatz in elliptic Hitchin systems.

Findings

Integral formulas for scalar products of genus one states.

Unitarity of the KZB connection is related to the Bethe Ansatz.

Quantization of elliptic Hitchin Hamiltonians via Bethe Ansatz.

Abstract

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 Knizhnik-Zamolodchikov-Bernard connection with respect to the scalar product of CS states is closely related to the Bethe Ansatz for the commuting Hamiltonians building up the connection and quantizing the quadratic Hamiltonians of the elliptic Hitchin system.