On the Coulomb and Higgs branch formulae for multi-centered black holes and quiver invariants

Abstract

In previous work we have shown that the equivariant index of multi-centered N=2 black holes localizes on collinear configurations along a fixed axis. Here we provide a general algorithm for enumerating such collinear configurations and computing their contribution to the index. We apply this machinery to the case of black holes described by quiver quantum mechanics, and give a systematic prescription -- the Coulomb branch formula -- for computing the cohomology of the moduli space of quiver representations. For quivers without oriented loops, the Coulomb branch formula is shown to agree with the Higgs branch formula based on Reineke's result for stack invariants, even when the dimension vector is not primitive. For quivers with oriented loops, the Coulomb branch formula parametrizes the Poincaré polynomial of the quiver moduli space in terms of single-centered (or pure-Higgs) BPS invariants, which are conjecturally independent of the stability condition (i.e. the choice of Fayet-Iliopoulos parameters) and angular-momentum free. To facilitate further investigation we provide a Mathematica package "CoulombHiggs.m" implementing the Coulomb and Higgs branch formulae.

45 pages, 1 figure; v2 (after publication in JHEP): New recursion scheme introduced in Note Added, 4.31-4.34; Appendix updated to document new features of the Mathematica package "CoulombHiggs.m" v2.0, available along with example files from the submission source, or from http://www.lpthe.jussieu.fr/~pioline/computing.html

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• Quantum Entropy Function from AdS(2)/CFT(1) Correspondence
• Wall-Crossing from Boltzmann Black Hole Halos
• Mergers and Typical Black Hole Microstates
• Arithmetic of Quantum Entropy Function
• A fixed point formula for the index of multi-centered N=2 black holes
• Scaling BPS Solutions and pure-Higgs States
• From Black Holes to Quivers
• Quiver Invariants from Intrinsic Higgs States
• Framed BPS States, Moduli Dynamics, and Wall-Crossing
• BPS States, Refined Indices, and Quiver Invariants
• Abelian quiver invariants and marginal wall-crossing

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• The Coulomb Branch Formula for Quiver Moduli Spaces
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• Black holes and higher depth mock modular forms
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• Generalized quiver mutations and single-centered indices
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• Quiver indices and Abelianization from Jeffrey-Kirwan residues
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• Discrete Integrable Systems, Supersymmetric Quantum Mechanics, and Framed BPS States - I
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• About the Absence of Exotics and the Coulomb Branch Formula
• SUSY in Silico: numerical D-brane bound state spectroscopy
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• On the existence of scaling multi-centered black holes
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