Knizhnik-Zamolodchikov-Bernard equations on Riemann surfaces

TL;DR

This paper derives explicit forms of Knizhnik-Zamolodchikov-Bernard equations for twisted conformal blocks on Riemann surfaces, demonstrating their role as flat connections with spectral parameters in the moduli space.

Contribution

It provides a general explicit formulation of these equations in terms of twisted b-c system correlation functions on Riemann surfaces with marked points.

Findings

Equations are explicitly written in a general projective structure.

Confirmed that equations define a flat connection on the moduli space.

Demonstrated the spectral parameter role in the equations.

Abstract

Knizhnik-Zamolodchikov-Bernard equations for twisted conformal blocks on compact Riemann surfaces with marked points are written explicitly in a general projective structure in terms of correlation functions in the theory of twisted b-c systems. It is checked that on the moduli space the equations provide a flat connection with the spectral parameter.