Riemann Surfaces and Vertex Operator Algebras

TL;DR

This paper explores the relationship between Riemann surfaces and vertex operator algebras, demonstrating how cocycle realizations relate to Heisenberg VOAs at a fixed point.

Contribution

It establishes a correspondence between cocycle realizations on Riemann surfaces and specific Heisenberg vertex operator algebras.

Findings

Cocycle realizations correspond to Heisenberg VOAs at a fixed point

Distinct cocycles relate to different VOA structures

Provides a geometric interpretation of VOAs on Riemann surfaces

Abstract

We show that for any fixed point P on a Riemann surface S the distinct realizations of cocycles in H^1(S,O) correspond to the natural appearence of the standard Heisenberg vertex operator algebra II(P) and to the commutative Heisenberg vertex operator algebra II*(P) respectively