Bosonic Field Propagators on Algebraic Curves

TL;DR

This paper develops a new method to compute scalar field propagators on algebraic curves using polynomial parameters, avoiding theta functions, and analyzes their properties and applications in field theory.

Contribution

It introduces an alternative approach to propagator calculation on algebraic curves by expressing it through polynomial parameters, with detailed analysis and a special case involving symmetry and branch points.

Findings

Derived the third kind differential with purely imaginary periods.

Expressed scalar field correlators in terms of the differential.

Provided a simplified propagator expression for symmetric algebraic curves.

Abstract

In this paper we investigate massless scalar field theory on non-degenerate algebraic curves. The propagator is written in terms of the parameters appearing in the polynomial defining the curve. This provides an alternative to the language of theta functions. The main result is a derivation of the third kind differential normalized in such a way that its periods around the homology cycles are purely imaginary. All the physical correlation functions of the scalar fields can be expressed in terms of this object. This paper contains a detailed analysis of the techniques necessary to study field theories on algebraic curves. A simple expression of the scalar field propagator is found in a particular case in which the algebraic curves have $Z_n$ internal symmetry and one of the fields is located at a branch point.