Wall-Crossing in Supersymmetric Gauge Theories

Abstract: We study $\mathcal{N}=2$ supersymmetric Yang--Mills theory in four dimensions and then compactify it on $\mathbb{R}^{3}\times S^{1}$. The gauge symmetry of the theory is broken by a vacuum expectation value of the scalar field, which parametrises the moduli space. The spectrum of BPS states, carrying electric and magnetic charges, is piece-wise constant, changing only when the vacuum expectation value crosses the so-called walls of marginal stability. Kontsevich and Soibelman proposed an algebraic construction relating BPS spectra on both sides of a wall of marginal stability. These formulae are known to correctly relate the strong- and weak-coupling spectra in theories with gauge group SU(2) with and without fundamental flavours; we generalise this result to gauge group SU(n) without flavours in the weak-coupling regime. In addition, we find the walls of marginal stability in the SU(n) theory at the root of the Higgs branch and, employing the wall-crossing formula, determine the BPS spectrum in all regions of the moduli space. Gaiotto, Moore, and Neitzke (GMN) proposed an ansatz expressing the moduli space metric of $\mathcal{N}=2$ theory on $\mathbb{R}^{3}\times S^{1}$ in terms of a set of integral equations. Using the GMN ansatz, we find perturbative and instanton corrections in $\mathbb{R}^{3}\times S^{1}$ for gauge group SU(2) with or without flavours and for gauge group SU(n) without flavours. For gauge group SU(n), we also demonstrate that the predicted two-instanton metric is continuous across the walls. Then, we calculate instanton corrections from first principles. We find perfect agreement between the GMN prediction and the first-principles result. We also take the limit of small radius of the compactified dimension finding one- and two-instanton corrections in three dimensions, recovering some of the previously known semiclassical results.