Four ways across the wall

TL;DR

This paper surveys four different methods for computing the change in BPS spectra across walls of marginal stability in N=2 theories, highlighting their apparent equivalence despite different mathematical and physical foundations.

Contribution

It compares four approaches—two based on generalized Donaldson-Thomas invariants and two on multi-centered black hole physics—showing their potential equivalence.

Findings

The four formulas are likely equivalent.

Explicit computations support their equivalence.

A combinatorial proof is still needed.

Abstract

An important question in the study of N=2 supersymmetric string or field theories is to compute the jump of the BPS spectrum across walls of marginal stability in the space of parameters or vacua. I survey four apparently different answers for this problem, two of which are based on the mathematics of generalized Donaldson-Thomas invariants (the Kontsevich-Soibelman and the Joyce-Song formulae), while the other two are based on the physics of multi-centered black hole solutions (the Coulomb branch and the Higgs branch formulae, discovered in joint work with Jan Manschot and Ashoke Sen). Explicit computations indicate that these formulae are equivalent, though a combinatorial proof is currently lacking.