Constructive Wall-Crossing and Seiberg-Witten

Abstract

We outline a comprehensive and first-principle solution to the wall-crossing problem in D=4 N=2 Seiberg-Witten theories. We start with a brief review of the multi-centered nature of the typical BPS states and recall how the wall-crossing problem thus becomes really a bound state formation/dissociation problem. Low energy dynamics for arbitrary collections of dyons is derived, from Seiberg-Witten theory, with the proximity to the so-called marginal stability wall playing the role of the small expansion parameter. We find that, surprisingly, the ${\mathbb{R}}^{3n}$ low energy dynamics of n+1 BPS dyons cannot be consistently reduced to the classical moduli space, $\CM$, yet the index can be phrased in terms of $\CM$. We also explain how an equivariant version of this index computes the protected spin character of the underlying field theory, where $SO(3)_\CJ$ isometry of $\CM$ turns out to be the diagonal subgroup of $SU(2{)}_L$ spatial rotation and $SU(2{)}_R$ R-symmetry. The so-called rational invariants, previously seen in the Kontsevich-Soibelman formalism of wall-crossing, are shown to emerge naturally from the orbifolding projection due to Bose/Fermi statistics.

25 pages, conference proceeding contribution for "Progress of Quantum Field Theory and String Theory," Osaka, April 2012

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• Wall-Crossing from Boltzmann Black Hole Halos
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