Equivalence of Three Wall Crossing Formulae

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.

LaTeX file, 45 pages; v2: added eq.(2.10) containing explicit wall crossing formula for numerical invariants and comments on its possible connection to Joyce-Song formula, added references and some additional minor clarifications; added a simpler proof suggested by the anonymous referee, added other clarifications, version to appear in CNTP, 51 pages

References in corpus (13)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• R-Twisting and 4d/2d Correspondences
• Wall-Crossing from Boltzmann Black Hole Halos
• BPS Quivers and Spectra of Complete N=2 Quantum Field Theories
• A fixed point formula for the index of multi-centered N=2 black holes
• Wall crossing in local Calabi Yau manifolds
• BPS Wall Crossing and Topological Strings
• Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants
• Crossing the Wall: Branes vs. Bundles
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• On a hyperholomorphic line bundle over the Coulomb branch
• Poisson automorphisms and quiver moduli
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Cited by in corpus (6)

• N=2 Quantum Field Theories and Their BPS Quivers
• From Black Holes to Quivers
• On the Coulomb and Higgs branch formulae for multi-centered black holes and quiver invariants
• The Coulomb Branch Formula for Quiver Moduli Spaces
• Abelianization of BPS Quivers and the Refined Higgs Index
• Constructive Wall-Crossing and Seiberg-Witten