Four-dimensional wall-crossing via three-dimensional field theory

Abstract

We give a physical explanation of the Kontsevich-Soibelman wall-crossing formula for the BPS spectrum in Seiberg-Witten theories. In the process we give an exact description of the BPS instanton corrections to the hyperkahler metric of the moduli space of the theory on R^3 x S^1. The wall-crossing formula reduces to the statement that this metric is continuous. Our construction of the metric uses a four-dimensional analogue of the two-dimensional tt* equations.

62 pages; v2: typos fixed, references added, minor improvements to exposition and notation; v3: typos fixed and small additions, including remark on TBA

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• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Quantum integrability and supersymmetric vacua
• Holographic Action for the Self-Dual Field
• Properties of hyperkahler manifolds and their twistor spaces
• Stability conditions and Stokes factors
• Non-perturbative superpotentials across lines of marginal stability
• Linear perturbations of Hyperkahler metrics
• Superconformal R-charges and dyon multiplicities in N=2 gauge theories

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