Generating functions and attractor flow for N = 2 BPS structures

Abstract

This thesis studies wall crossing phenomena in BPS structures associated to 4d N = 2 QFTs including both Argyres-Douglas and Seiberg-Witten theories. The BPS spectrum is determined in each region of the moduli space, bounded by walls of marginal stability, both by using attractor flow methods and by deriving a generating function with Fourier coefficients that jump as a wall is crossed. The attractor flow methods are applied using existence conditions for the BPS states on the endpoints of the flow lines which split when a composite line flows into a wall. For N = 4 dyonic black holes the wall crossing of 1/4 BPS states is known to be determined by the Weyl denominator of a Borcherds-Kac-Moody algebra. This has a different Fourier expansion in the different chambers, which represents a jump in the degeneracies of black holes with specific charges. In this work, analogs of these counting functions are found for N = 2 BPS structures. These correspond to the Weyl denominator formulae in the case of ADE type Lie algebras, where the root system describes the charges of the BPS particles. The resulting formulae contain information about the spectra of BPS and framed BPS states in Weyl chambers within the moduli spaces of these theories. The regions in the moduli space with fixed spectra are found to be bounded by walls, including the Weyl chamber boundaries and an additional wall of marginal stability. In some examples of uncoupled BPS structures this can then be reproduced by the Stokes phenomena of the Borel summation of the topological string free energy.