GENERALIZED HITCHIN SYSTEMS AND THE KNIZHNIK-ZAMOLODCHIKOV-BERNARD EQUATION ON ELLIPTIC-CURVES

The Knizhnik-Zamolodchikov-Bernard (KZB) equation on an elliptic curve with a marked point is derived by classical Hamiltonian reduction and further quantization. We consider classical Hamiltonian systems on a cotangent bundle to the loop group L(GL(N,C)) extended by the shift op erators, to be related to the elliptic module. After reduction, we obt ain a Hamiltonian system on a cotangent bundle to the moduli of holomo rphic principle bundles and an elliptic module. It is a particular exa mple of generalized Hitchin systems (GHS) which are defined as Hamilto nian systems on cotangent bundles to the moduli of holomorphic bundles and to the moduli of curves. They are extensions of the Hitchin syste ms by the inclusion the moduli of curves. In contrast with the Hitchin systems, the algebra of integrals are noncommutative on GHS. We discu ss the quantization procedure in our example. The quantization of the quadratic integral leads to the KZB equation. We present an explicit f orm of higher quantum Hitchin integrals which, upon reducing from GHS phase space to the Hitchin phase space, gives a particular example of the Beilinson-Drinfeld commutative algebra of differential operators o n the moduli of holomorphic bundles.