Operator Formalism on the $Z_n$ Symmetric Algebraic Curves

TL;DR

This paper proves conjectures relating $b-c$ systems on Riemann surfaces with abelian symmetry groups to multivalued field theories on the complex plane, using operator formalism and solutions to the Riemann-Hilbert problem.

Contribution

It establishes a connection between $b-c$ systems on algebraic curves and holonomic quantum field theories via operator formalism and Riemann-Hilbert solutions.

Findings

$b-c$ systems are equivalent to multivalued field theories on the complex plane.

Amplitudes can be derived from operator product expansions.

Normal ordering rules recover amplitudes on Riemann surfaces.

Abstract

In this work, the following conjectures are proven in the case of a Riemann surface with abelian group of symmetry: a) The $b-c$ systems on a Riemann surface $M$ are equivalent to a multivalued field theory on the complex plane if $M$ is represented as an algebraic curve; b) the amplitudes of the $b-c$ systems on a Riemann surface $M$ with discrete group of symmetry can be derived from the operator product expansions on the complex plane of an holonomic quantum field theory a la Sato, Jimbo and Miwa. To this purpose, the solutions of the Riemann-Hilbert problem on an algebraic curve with abelian monodromy group obtained by Zamolodchikov, Knizhnik and Bershadskii-Radul are used in order to expand the $b-c$ fields in a Fourier-like basis. The amplitudes of the $b-c$ systems on the Riemann surface are then recovered exploiting simple normal ordering rules on the complex plane.