On recursion relations in topological string theory

TL;DR

This paper explores the connection between algebraic recursion relations and the holomorphic anomaly equation in topological string theory, introducing an operator that unifies these concepts within a generalized algebraic framework.

Contribution

It defines an operator within a generalized algebra that links topological recursion relations to the holomorphic anomaly equation in string theory.

Findings

The operator ${\

The algebra generalizes tt* equations

Reproduces recursion relations and anomaly equations

Abstract

We discuss a link between the topological recursion relations derived algebraically by Witten and the holomorphic anomaly equation of Bershadsky, Cecotti, Ooguri and Vafa. This is obtained through the definition of an operator ${\cal{W}}_s$ that reproduces the recursion relations for topological string theory coupled to worldsheet gravity a la BCOV. This operator is contained inside an algebra that generalizes the tt* equations and whose direct consequence is the holomorphic anomaly equation itself.