Index and localization for type B superconformal mechanics on singular spaces

TL;DR

This paper explores the computation of superconformal indices in type B superconformal quantum mechanics on singular spaces, proposing a resolution approach and analyzing subtleties related to selfadjoint extensions and special geometries.

Contribution

It introduces a method to compute superconformal indices on singular target spaces by resolution, and analyzes the index's behavior in models with different geometric structures.

Findings

Regularized index matches the actual index in physical models.

Index ambiguity arises in non-selfadjoint supercharge cases.

Type B index relates to the Hilbert series for Calabi-Yau cones.

Abstract

Type B superconformal quantum mechanical sigma models are of physical interest as they arise in the description of D-brane bound states forming an AdS$_2$ throat. In this work we discuss the applicability of localization methods to compute the superconformal index in these theories, despite the fact that their target spaces are generically singular. Similar in spirit to recent works on type A models, we propose to work on a suitably resolved target space to compute a regularized index. While this regularized index correctly captures the actual index unambiguously in models of physical interest, we do uncover a subtlety in more pathological examples. This occurs in situations where the supercharge is not essentially-selfadjoint, in which case the index becomes ambiguous and depends on the chosen selfadjoint extension. We also discuss the special class of models with K\"ahler target spaces, which can accommodate both type A and type B models, and show that the type B index is a particular limit of the type A index. For Calabi-Yau cones, the type B index coincides with the Hilbert series of the unresolved space.