Gromov-Witten invariants and localization

Abstract

We give a pedagogical review of the computation of Gromov-Witten invariants via localization in 2D gauged linear sigma models. We explain the relationship between the two-sphere partition function of the theory and the Kahler potential on the conformal manifold. We show how the Kahler potential can be assembled from classical, perturbative, and non-perturbative contributions, and explain how the non-perturbative contributions are related to the Gromov-Witten invariants of the corresponding Calabi-Yau manifold. We then explain how localization enables efficient calculation of the two-sphere partition function and, ultimately, the Gromov-Witten invariants themselves.

This is a contribution to the review volume `Localization techniques in quantum field theories' (eds. V. Pestun and M. Zabzine): arXiv:1608.02952 and at https://arxiv.org/src/1608.02952/anc/LocQFT.pdf or http://pestun.ihes.fr/pages/LocalizationReview/LocQFT.pdf This article reviews portions of hep-th/9412236, hep-th/9508107, arXiv:1208.6244, arXiv:1305.3278, arXiv:1308.2157, and arXiv:1509.08511

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• Localizing non-linear $\mathcal{N}=(2,2)$ sigma model on $S^2$
• Matrix models from localization of five-dimensional supersymmetric noncommutative U(1) gauge theory