2023/09/21 by Davide Gaiotto, Gaiotto, Davide, Ahsan Z. Khan +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #FOS: Physical sciences #High Energy Physics - Theory (hep-th)
paper · pdf · doi:10.48550/arxiv.2309.12103
openalex publication_date 2023/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Kontsevich-Soibelman wall-crossing formula is known to control the jumping behavior of BPS state counting indices in four-dimensional theories with N=2 supersymmetry. The formula can take two equivalent forms: a ``fermionic'' form with nice positivity properties and a ``bosonic'' form with a clear physical interpretation. In an important class of examples, the fermionic form of the formula has a mathematical categorification involving PBW bases for a Cohomological Hall Algebra. The bosonic form lacks an analogous categorification. We construct an equivalence of chain complexes which categorifies the simplest example of the bosonic wall-crossing formula: the bosonic pentagon identity for the quantum dilogarithm. The chain complexes can be promoted to differential graded algebras which we relate to the PBW bases of the relevant CoHA by a certain quadratic duality. The equivalence of complexes then follows from the relation between quadratic duality and Koszul duality. We argue that this is a special case of a general phenomenon: the bosonic wall-crossing formulae are categorified to equivalences of A_∞ algebras which are quadratic dual to PBW presentations of algebras which underlie the fermionic wall-crossing formulae. We give a partial interpretation of our differential graded algebras in terms of a holomorphic-topological version of BPS webs.