A note on the quantization of M2-branes on local Calabi-Yau manifolds

Abstract

In this note we present a detailed computation of the BPS states from M2-branes wrapping 2-cycles in local Calabi-Yau manifolds, at the level of brane quantization. For the cases of local Calabi-Yau threefolds, the supersymmetric quantum mechanics is the 1d $\mathcal{N}=4$ non-linear sigma model. We achieve systematic, physical interpretations of refined Gopakumar-Vafa (GV) invariants, first for the curve classes with fixed geometric genus, and then for curves whose geometric genus jumps. The BPS states of 5d $\mathcal{N}=1$ superconformal field theories are interpreted from the local geometries, which are presented in a number of case studies. In particular, we compute the refined GV invariants for generalized conifolds, local CY3 giving 5d $\mathcal{N}=1$ gauge theories and a singularity whose crepant resolution contains a singular divisor. For the cases of local Calabi-Yau fourfolds, we point out the supersymmetric quantum mechanics is of type $\mathcal{N}=2B$. We set up the framework for studying M2-branes wrapping 2-cycles, and discuss a few cases of M2-brane wrapping genus-0 curves with this new approach.