Counting Strings, Wound and Bound

TL;DR

This paper investigates zero mode counting for Dirac operators in string theory backgrounds, analyzing bound states in various geometries, and explores implications for dualities and special functions.

Contribution

It introduces new methods for counting bound states via Dirac zero modes in string-inspired geometries, including deformed backgrounds and duality transformations.

Findings

Counted bound states in asymptotically linear dilaton backgrounds.

Identified multiple pole behavior in U(1) charge fugacities.

Connected results to Appell-Lerch sums and domain wall states.

Abstract

We analyze zero mode counting problems for Dirac operators that find their origin in string theory backgrounds. A first class of quantum mechanical models for which we compute the number of ground states arises from a string winding an isometric direction in a geometry, taking into account its energy due to tension. Alternatively, the models arise from deforming marginal bound states of a string winding a circle, and moving in an orthogonal geometry. After deformation, the number of bound states is again counted by the zero modes of a Dirac operator. We count these bound states in even dimensional asymptotically linear dilaton backgrounds as well as in Euclidean Taub-NUT. We show multiple pole behavior in the fugacities keeping track of a U(1) charge. We also discuss a second class of counting problems that arises when these backgrounds are deformed via the application of a heterotic duality transformation. We discuss applications of our results to Appell-Lerch sums and the counting of domain wall bound states.