Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Equivalence of Three Wall Crossing Formulae

Ashoke Sen

TL;DR

This work proves the equivalence of three formulations of wall-crossing for BPS indices: the KS implicit symplectomorphism relation and the two explicit expressions from MPS known as the Higgs-branch and Coulomb-branch formulæ. It develops a permutation-based Higgs description via Reineke’s quiver-moduli results and a KS description through a universal enveloping algebra of a quantum torus, then establishes their equality by a detailed combinatorial and deformation analysis, including an alternative proof via quantum torus algebra. A parallel equivalence between Higgs and Coulomb formulæ is shown using extremum conditions of a multi-center potential, with sign factors matched by determinant calculations. Together, these results solidify the consistency of wall-crossing across different physical pictures (Higgs, Coulomb, and KS) and provide a robust toolkit for computing BPS indices across walls.

Abstract

The wall crossing formula of Kontsevich and Soibelman gives an implicit relation between the BPS indices on two sides of the wall of marginal stability by equating two symplectomorphisms constructed from the indices on two sides of the wall. The wall crossing formulae of Manschot, Pioline and the author give two apparently different explicit expressions for the BPS index on one side of the wall in terms of the BPS indices on the other side. We prove the equivalence of all the three formulae.

Equivalence of Three Wall Crossing Formulae

TL;DR

This work proves the equivalence of three formulations of wall-crossing for BPS indices: the KS implicit symplectomorphism relation and the two explicit expressions from MPS known as the Higgs-branch and Coulomb-branch formulæ. It develops a permutation-based Higgs description via Reineke’s quiver-moduli results and a KS description through a universal enveloping algebra of a quantum torus, then establishes their equality by a detailed combinatorial and deformation analysis, including an alternative proof via quantum torus algebra. A parallel equivalence between Higgs and Coulomb formulæ is shown using extremum conditions of a multi-center potential, with sign factors matched by determinant calculations. Together, these results solidify the consistency of wall-crossing across different physical pictures (Higgs, Coulomb, and KS) and provide a robust toolkit for computing BPS indices across walls.

Abstract

The wall crossing formula of Kontsevich and Soibelman gives an implicit relation between the BPS indices on two sides of the wall of marginal stability by equating two symplectomorphisms constructed from the indices on two sides of the wall. The wall crossing formulae of Manschot, Pioline and the author give two apparently different explicit expressions for the BPS index on one side of the wall in terms of the BPS indices on the other side. We prove the equivalence of all the three formulae.

Paper Structure

This paper contains 14 sections, 105 equations, 1 figure.

Table of Contents

  1. Introduction
  2. 'Higgs branch' wall crossing formula
  3. KS wall crossing formula
  4. Recursion relations for the KS wall crossing formula
  5. Equivalence of KS and 'higgs branch' wall crossing formulæ
  6. $g_{\rm KS}$ as a sum over permutations
  7. Deforming the $\alpha_i$'s
  8. Constraining the permutations and partitions
  9. Comparison with the constraints on $\sigma$ for non-vanishing ${\rm N}^{(n+1)}_{\rm higgs}(\{\alpha_i\};\sigma)$
  10. Proof of ${\rm N}^{(n+1)}_{\rm KS}={\rm N}^{(n+1)}_{\rm higgs}$
  11. An alternative approach to proving the equivalence of the Higgs branch and KS wall crossing formulæ
  12. Equivalence of 'higgs' and 'coulomb' branch wall crossing formulæ
  13. Physical interpretation of $g_{\rm higgs}$
  14. Proof of the $\Theta$ identity

Figures (1)

  • Figure 1: Figure illustrating the definition of $\gamma_2>\gamma_1$ and $\Theta(\gamma_2,\gamma_1)$.