An Index Formula for Supersymmetric Quantum Mechanics

Abstract

We derive a localization formula for the refined index of gauged quantum mechanics with four supercharges. Our answer takes the form of a residue integral on the complexified Cartan subalgebra of the gauge group. The formula captures the dependence of the index on Fayet-Iliopoulos parameters and the presence of a generic superpotential. The residue formula provides an efficient method for computing cohomology of quiver moduli spaces. Our result has broad applications to the counting of BPS states in four-dimensional N=2 systems. In that context, the wall-crossing phenomenon appears as discontinuities in the value of the residue integral as the integration contour is varied. We present several examples illustrating the various aspects of the index formula.

arXiv comment30 pages, 12 figures, v2 references added, minor clarifications, and changes to sign conventions for FI parameters

References in corpus (22)

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• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Four-dimensional wall-crossing via three-dimensional field theory
• Wall-crossing, Hitchin Systems, and the WKB Approximation
• Elliptic genera of 2d N=2 gauge theories
• Elliptic genera of two-dimensional N=2 gauge theories with rank-one gauge groups
• The spectrum of BPS branes on a noncompact Calabi-Yau
• Witten Index and Wall Crossing
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• N=2 Quantum Field Theories and Their BPS Quivers
• Dying Dyons Don't Count
• Quantum Wall Crossing in N=2 Gauge Theories
• 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
• From Black Holes to Quivers
• BPS Wall Crossing and Topological Strings
• Counting Yang-Mills Dyons with Index Theorems
• Ab Initio Wall-Crossing
• Localization for nonabelian group actions
• A theory of generalized Donaldson-Thomas invariants
• General instanton counting and 5d SCFT
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Cited by in corpus (30)

• Schur Indices, BPS Particles, and Argyres-Douglas Theories
• Witten Index and Wall Crossing
• Defects and Quantum Seiberg-Witten Geometry
• Classification of Argyres-Douglas theories from M5 branes
• Duality walls and defects in 5d N=1 theories
• A slow review of the AGT correspondence
• Attractor flow trees, BPS indices and quivers
• Fundamental Vortices, Wall-Crossing, and Particle-Vortex Duality
• The Coulomb Branch Formula for Quiver Moduli Spaces
• Integrability from 2d N=(2,2) Dualities
• 6d strings and exceptional instantons
• New AdS${}_2$ supergravity duals of 4d SCFTs with defects
• General instanton counting and 5d SCFT
• New AdS${}_2$ backgrounds and $\mathcal{N}=4$ Conformal Quantum Mechanics
• BPS Hall Algebra of Scattering Hall States
• Black hole microstates and supersymmetric localization
• Coulomb Branch Localization in Quiver Quantum Mechanics
• ABCD of 't Hooft operators
• Regge Trajectories in $\mathcal{N}=2$ Supersymmetric Yang-Mills Theory
• Counting Trees in Supersymmetric Quantum Mechanics
• Discrete Integrable Systems, Supersymmetric Quantum Mechanics, and Framed BPS States - I
• Asymptotics of Ground State Degeneracies in Quiver Quantum Mechanics
• A New Integrable Ising-type Model from 2d $\mathcal{N}$=(2,2) Dualities
• SUSY in Silico: numerical D-brane bound state spectroscopy
• Surface defects and instanton partition functions
• Connecting Localization and Wall-Crossing via D-Branes
• Mirror Symmetry And Loop Operators
• Hamiltonian Truncation Study of Supersymmetric Quantum Mechanics: S-Matrix and Metastable States
• Non-Perturbative Schwinger-Dyson Equations for 3d $\mathcal{N}=4$ Gauge Theories
• Mutation, Witten Index, and Quiver Invariant