Generalized Hitchin systems and Knizhnik-Zamolodchikov-Bernard equation on elliptic curves

TL;DR

This paper derives the KZB equation on elliptic curves via classical Hamiltonian reduction and quantization, introducing generalized Hitchin systems with noncommutative integrals and exploring their quantization leading to the KZB equation.

Contribution

It extends Hitchin systems to generalized Hitchin systems on elliptic curves, incorporating moduli of curves and noncommutative integrals, and explicitly constructs their quantization and associated KZB equation.

Findings

Derived the KZB equation from classical Hamiltonian reduction.

Constructed higher quantum Hitchin integrals explicitly.

Connected generalized Hitchin systems to the Belinson-Drinfeld algebra.

Abstract

Knizhnik-Zamolodchikov-Bernard (KZB) equation on an elliptic curve with a marked point is derived by the classical Hamiltonian reduction and further quantization. We consider classical Hamiltonian systems on cotangent bundle to the loop group $L(GL(N,{\bf C}))$ extended by the shift operators, to be related to the elliptic module. After the reduction we obtain the Hamiltonian system on cotangent bundle to the moduli of holomorphic principle bundles and the elliptic module. It is a particular example of generalized Hitchin systems (GHS), which are defined as hamiltonian systems on cotangent bundles to the moduli of holomorphic bundles and to the moduli of curves. They are extensions of the Hitchin systems by the inclusion the moduli of curves. In contrast with the Hitchin systems the algebra of integrals are noncommutative on GHS. We discuss the quantization procedure in our example. The quantization of the quadratic integral leads to the KZB equation. We present the explicite form of higher quantum Hitchin integrals, which upon on reducing from GHS phase space to the Hitchin phase space gives a particular example of the Belinson-Drinfeld commutative algebra of differential operators on the moduli of holomorphic bundles.