Generating functions for $\mathcal{N}=2$ BPS structures

TL;DR

This paper introduces generating functions that encode BPS state degeneracies and wall-crossing phenomena in $ =2$ theories, linking representation theory, Lie algebras, and known BPS spectra, with applications to various models.

Contribution

It proposes a new class of generating functions for $ =2$ BPS structures based on Lie algebra representations, extending known formulas and testing them on multiple theories.

Findings

Successfully reproduces BPS spectra of Seiberg-Witten SU(2) theory

Captures the D6-D2-D0 BPS structure of the resolved conifold

Reproduces wall-crossing structures of the Argyres-Douglas A2 theory

Abstract

We propose generating functions which encode the degeneracies and wall-crossing phenomena of $\mathcal{N}=2$ BPS structures. The generating functions have a representation-theoretic origin and are the analogs of the 1/4-BPS dyon counting formula in $\mathcal{N}=4$ theories involving the Weyl denominator formula of a Borcherds-Kac-Moody Lie algebra. A general form of the generating function is suggested based on the Lie algebra associated to the adjacency matrix of the BPS quiver whenever the BPS spectrum of the $\mathcal{N}=2$ theory admits such a description. This proposal is tested for the BPS spectrum of Seiberg-Witten SU(2) theory as well as for the $D6$-$D2$-$D0$ BPS structure of the resolved conifold which are both captured by an affine $A_1$ Lie algebra and are obtained from limits of the $\mathcal{N}=4$ generating function. The general proposal also reproduces the correct BPS spectra and wall-crossing structures for the Argyres-Douglas $A_2$ theory. We further discuss connections to scattering diagrams studied in the context of stability structures.