Wall-crossing made smooth

TL;DR

This paper verifies a recent formula for the BPS index in 4D N=2 theories, showing it accurately accounts for multi-particle states and remains smooth across walls of marginal stability, with implications for black hole physics.

Contribution

It provides a physical validation of a new formula for the BPS index that includes multi-particle contributions and is smooth across walls of marginal stability.

Findings

Two-particle contributions match spectral asymmetry in supersymmetric quantum mechanics.

Supports the use of error function profiles in indefinite theta series and black hole partition functions.

Confirms the formula's consistency with physical scattering states.

Abstract

In $D=4,N=2$ theories on $R^{3,1}$, the index receives contributions not only from single-particle BPS states, counted by the BPS indices, but also from multi-particle states made of BPS constituents. In a recent work [arXiv:1406.2360], a general formula expressing the index in terms of the BPS indices was proposed, which is smooth across walls of marginal stability and reproduces the expected single-particle contributions. In this note, I analyze the two-particle contributions predicted by this formula, and show agreement with the spectral asymmetry of the continuum of scattering states in the supersymmetric quantum mechanics of two non-relativistic, mutually non-local dyons. This provides a physical justification for the error function profile used in the mathematics literature on indefinite theta series, and in the physics literature on black hole partition functions.