Four ways across the wall

Abstract

An important question in the study of N=2 supersymmetric string or field theories is to compute the jump of the BPS spectrum across walls of marginal stability in the space of parameters or vacua. I survey four apparently different answers for this problem, two of which are based on the mathematics of generalized Donaldson-Thomas invariants (the Kontsevich-Soibelman and the Joyce-Song formulae), while the other two are based on the physics of multi-centered black hole solutions (the Coulomb branch and the Higgs branch formulae, discovered in joint work with Jan Manschot and Ashoke Sen). Explicit computations indicate that these formulae are equivalent, though a combinatorial proof is currently lacking.

17 pages, 2 figures, proceedings of the workshop "Algebra, Geometry and Mathematical Physics", Tjärnö, Sweden, 25-30 October 2010

References in corpus (12)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• R-Twisting and 4d/2d Correspondences
• Wall-Crossing from Boltzmann Black Hole Halos
• Mergers and Typical Black Hole Microstates
• Plumbing the Abyss: Black Ring Microstates
• D-instantons and twistors
• A fixed point formula for the index of multi-centered N=2 black holes
• Wall crossing in local Calabi Yau manifolds
• Wall-crossing from supersymmetric galaxies
• Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants
• Search for Light Resonances Decaying into Pairs of Muons as a Signal of New Physics
• Framed BPS States, Moduli Dynamics, and Wall-Crossing

Cited by in corpus (24)

• Wall-Crossing from Boltzmann Black Hole Halos
• Scaling BPS Solutions and pure-Higgs States
• From Black Holes to Quivers
• On the Coulomb and Higgs branch formulae for multi-centered black holes and quiver invariants
• Quiver Invariants from Intrinsic Higgs States
• Spectral Networks with Spin
• Ab Initio Wall-Crossing
• TASI lectures on complex structures
• Attractor flow trees, BPS indices and quivers
• Exploring 5d BPS Spectra with Exponential Networks
• $R^3$ index for four-dimensional $N=2$ field theories
• The Coulomb Branch Formula for Quiver Moduli Spaces
• S-Duality and Modular Transformation as a non-perturbative deformation of the ordinary pq-duality
• Spectral Networks and Snakes
• BPS Hall Algebra of Scattering Hall States
• On the BPS Spectrum of the Rank-1 Minahan-Nemeschansky Theories
• BPS Spectrum, Indices and Wall Crossing in N=4 Supersymmetric Yang-Mills Theories
• Multi-centered black holes, scaling solutions and pure-Higgs indices from localization
• Wall Crossing from Dirac Zeromodes
• Equivalence of Three Wall Crossing Formulae
• Four-dimensional N=2 Field Theory and Physical Mathematics
• Joyce-Song wall-crossing as an asymptotic expansion
• Categorical Wall-Crossing in Landau-Ginzburg Models
• Quiver Structure of Heterotic Moduli