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 Kähler potential on the conformal manifold. We show how the Kähler potential can be assembled from classical, perturbative, and nonperturbative 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

Sphere Partition Function of Calabi–Yau GLSMs

“…In the following we will discuss the Landau-Ginzburg, the geometric, and the hybrid phase. In the context of supersymmetric localisation this model has also been discussed in [8,12,64].…”

“…For a discussion of the sphere partition function of this phase, see also [64]. After defining z i = iσ i − q i , the poles of the sphere partition functions are determined by the following divisors…”

“…We divide M by 8π 3 in order to get a canonically normalised ζ(3) term in the geometric phase. See also[5,64] where similar normalisations have been applied.…”