Towards quantum elliptic cohomology from GLSMs

Abstract: This paper briefly outlines a proposal for a GLSM Coulomb-branch interpretation of the quantum elliptic cohomology rings currently being developed in mathematics by Bouaziz-Huq-Kuruvilla-Lee. The proposal is modeled on the established GLSM computations of quantum cohomology and quantum K theory of Fano toric varieties, where ring relations arise as critical loci of twisted effective superpotentials. In the case of quantum elliptic cohomology, the natural generalization would be to encode ring relations in Coulomb branch computations in two-dimensional theories obtained by KK reductions from four-dimensional theories, describing OPEs of parallel surface defects, which we review. However, in some basic examples, this interpretation is obstructed by four-dimensional gauge anomalies. In such cases, we provide a classical superpotential whose critical loci generate the expected ring relations, but which is not obtained by KK reduction from four dimensions, though it shares many characteristics with such KK reductions. We describe some simple examples, and make some predictions for the case of gerbes.