Constructive Wall-Crossing and Seiberg-Witten

TL;DR

This paper provides a first-principles solution to the wall-crossing problem in 4D N=2 Seiberg-Witten theories, deriving low energy dynamics of BPS states and connecting index computations with geometric and algebraic structures.

Contribution

It introduces a comprehensive method to analyze wall-crossing in Seiberg-Witten theories, linking index calculations to geometric moduli spaces and algebraic invariants.

Findings

Low energy dynamics derived from Seiberg-Witten theory near stability walls.

Index can be expressed in terms of the classical moduli space, despite dynamics not reducing to it.

Rational invariants naturally emerge from orbifolding due to quantum statistics.

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.