Four-Dimensional Wall-Crossing via Three-Dimensional Field Theory

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 hyperkähler metric of the moduli space of the theory on $${\mathbb R^3 \times S^1}$$. 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.