Sphere Partition Function of Calabi–Yau GLSMs

Abstract

The sphere partition function of Calabi–Yau gauged linear sigma models (GLSMs) has been shown to compute the exact Kähler potential of the Kähler moduli space of a Calabi–Yau. We propose a universal expression for the sphere partition function evaluated in hybrid phases of Calabi–Yau GLSMs that are fibrations of Landau–Ginzburg orbifolds over some base manifold. Special cases include Calabi–Yau complete intersections in toric ambient spaces and Landau–Ginzburg orbifolds. The key ingredients that enter the expression are Givental’s I/J-functions, the Gamma class and further data associated to the hybrid model. We test the proposal for one- and two-parameter abelian GLSMs, making connections, where possible, to known results from mirror symmetry and FJRW theory.

On genus-0 invariants of Calabi-Yau hybrid models

“…In [29], it was first observed that the sphere partition function of a Calabi-Yau GLSM computes the Kähler potential on the associated Kähler moduli space. In [9] it was shown that for a GLSM that has a phase that is a "good" hybrid with orbifold group G, the sphere partition function reduces to…”

“…Finally, I δ (u(t), H) is the component of the I-function associated to the δ-twisted sector that can be expanded further in powers of H ∈ H 2 (B). This structure has been used in [9] to identify the I-functions of Calabi-Yau hybrid models.…”

“…In this section, we extract enumerative invariants of hybrid phases of a class of one-parameter models. The corresponding GLSMs and hybrid phases have been described at length in [9] and the I-functions were extracted from the sphere partition function and shown to match with results in the mathematics literature where available. We use the same labelling for the models as in [9].…”

“…To our knowledge, the invariants for these specific examples have been defined but have not been written down explicitly. Making use of the mathematical structure that is already in place and results from supersymmetric localisation in GLSMs [9], we use the following approach to extract the invariants. The invariants have a generating function given by the J-function [10].…”

“…Then we specialise to hybrid theories and state some further properties of the correlators that are specific for hybrid models [4]. Furthermore we briefly remind the reader of the connection between the GLSM sphere partition function and the I-function for the case of hybrids [9]. In Section 3 we follow [1] to give a review of N = (2, 2) hybrid CFTs and their state spaces.…”