Operator Formalism on General Algebraic Curves

TL;DR

This paper extends the operator formalism used in string theory to general algebraic curves, providing explicit formulas and revealing a multi-branch Hilbert space structure for $b-c$ systems on Riemann surfaces.

Contribution

It generalizes the Laurent expansion and operator formalism to arbitrary algebraic curves, enabling new explicit formulas in string theory.

Findings

Hilbert space splits into n independent parts

Explicit formulas for string theory on algebraic curves

Operator formalism for $b-c$ systems on Riemann surfaces

Abstract

The usual Laurent expansion of the analytic tensors on the complex plane is generalized to any closed and orientable Riemann surface represented as an affine algebraic curve. As an application, the operator formalism for the $b-c$ systems is developed. The physical states are expressed by means of creation and annihilation operators as in the complex plane and the correlation functions are evaluated starting from simple normal ordering rules. The Hilbert space of the theory exhibits an interesting internal structure, being splitted into $n$ ($n$ is the number of branches of the curve) independent Hilbert spaces. Exploiting the operator formalism a large collection of explicit formulas of string theory is derived.