Gauss-Manin connection in disguise: Calabi-Yau threefolds

TL;DR

This paper explores the algebraic structure of Calabi-Yau threefold moduli spaces using the Gauss-Manin connection, relating it to topological string theory and providing algebraic proofs of known results.

Contribution

It introduces an algebraic framework for the Gauss-Manin connection on Calabi-Yau threefolds, connecting it to topological string partition functions and the holomorphic anomaly equation.

Findings

Recovered Yamaguchi-Yau and Alim-Länge results algebraically

Described algebraic topological string partition functions

Linked special geometry coordinates to moduli space

Abstract

We describe a Lie Algebra on the moduli space of Calabi-Yau threefolds enhanced with differential forms and its relation to the Bershadsky-Cecotti-Ooguri-Vafa holomorphic anomaly equation. In particular, we describe algebraic topological string partition functions $F_g^{alg}, g\geq 1$, which encode the polynomial structure of holomorphic and non-holomorphic topological string partition functions. Our approach is based on Grothendieck's algebraic de Rham cohomology and on the algebraic Gauss-Manin connection. In this way, we recover a result of Yamaguchi-Yau and Alim-L\"ange in an algebraic context. Our proofs use the fact that the special polynomial generators defined using the special geometry of deformation spaces of Calabi-Yau threefolds correspond to coordinates on such a moduli space. We discuss the mirror quintic as an example.