Holomorphic Anomalies in Topological Field Theories

TL;DR

This paper investigates the genus one partition function in N=2 superconformal field theories, revealing how to compute it via anomalies, and applies this to count holomorphic elliptic curves on Calabi-Yau manifolds, advancing mirror symmetry understanding.

Contribution

It introduces a method to compute the stringy genus one partition function using anomalies and applies it to enumerate elliptic curves, linking topological field theories and mirror symmetry at the quantum level.

Findings

Computed genus one partition function using BRST anomaly

Counted holomorphic elliptic curves on Calabi-Yau manifolds

Linked topological theories with mirror symmetry at quantum level

Abstract

We study the stringy genus one partition function of $N=2$ SCFT's. It is shown how to compute this using an anomaly in decoupling of BRST trivial states from the partition function. A particular limit of this partition function yields the partition function of topological theory coupled to topological gravity. As an application we compute the number of holomorphic elliptic curves over certain Calabi-Yau manifolds including the quintic threefold. This may be viewed as the first application of mirror symmetry at the string quantum level.