A Klein TQFT: the local Real Gromov-Witten theory of curves

TL;DR

This paper develops a 2D Klein TQFT framework for local Real Gromov-Witten invariants of curves, providing explicit formulas and connecting to real Gopakumar-Vafa and Chern-Simons theories.

Contribution

It introduces a Klein TQFT approach to local Real Gromov-Witten theory and derives closed-form formulas extending previous results.

Findings

Explicit formula for local RGW invariants in terms of representation data

Connection between real Gromov-Witten invariants and SO/Sp Chern-Simons theory

Derivation of the local real Gopakumar-Vafa formula

Abstract

In this paper we study the Real Gromov-Witten theory of local 3-folds over Real curves. We show that this gives rise to a 2-dimensional Klein TQFT defined on an extension of the category of unorientable surfaces. We use this structure to completely solve the theory by providing a closed formula for the local RGW invariants in terms of representation theoretic data, extending earlier results of Bryan and Pandharipande. As a consequence we obtain the local version of the real Gopakumar-Vafa formula that expresses the connected real Gromov-Witten invariants in terms of integer invariants. In the case of the resolved conifold the partition function of the RGW invariants agrees with that of the SO/Sp Chern-Simons theory.