Two-Sphere Partition Functions and Gromov-Witten Invariants

TL;DR

This paper proposes that the two-sphere partition function of certain gauge theories exactly computes the Kähler potential and Gromov-Witten invariants of Calabi-Yau threefolds, enabling calculations without mirror symmetry.

Contribution

It introduces a novel method to compute Gromov-Witten invariants directly from gauge theory partition functions for Calabi-Yau threefolds, including cases without known mirrors.

Findings

Computed Gromov-Witten invariants for the quintic and Rødland's Calabi-Yau.

Predicted invariants for a determinantal Calabi-Yau in P^7 without a known mirror.

Confirmed consistency with existing geometric results.

Abstract

Many N=(2,2) two-dimensional nonlinear sigma models with Calabi-Yau target spaces admit ultraviolet descriptions as N=(2,2) gauge theories (gauged linear sigma models). We conjecture that the two-sphere partition function of such ultraviolet gauge theories -- recently computed via localization by Benini et al. and Doroud et al. -- yields the exact K\"ahler potential on the quantum K\"ahler moduli space for Calabi-Yau threefold target spaces. In particular, this allows one to compute the genus zero Gromov-Witten invariants for any such Calabi-Yau threefold without the use of mirror symmetry. More generally, when the infrared superconformal fixed point is used to compactify string theory, this provides a direct method to compute the spacetime K\"ahler potential of certain moduli (e.g., vector multiplet moduli in type IIA), exactly in {\alpha}'. We compute these quantities for the quintic and for R{\o}dland's Pfaffian Calabi-Yau threefold and find agreement with existing results in the literature. We then apply our methods to a codimension four determinantal Calabi-Yau threefold in P^7, recently given a nonabelian gauge theory description by the present authors, for which no mirror Calabi-Yau is currently known. We derive predictions for its Gromov-Witten invariants and verify that our predictions satisfy nontrivial geometric checks.