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.

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

References in corpus (7)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Witten Index and Wall Crossing
• Wall-Crossing from Boltzmann Black Hole Halos
• A fixed point formula for the index of multi-centered N=2 black holes
• BPS Wall Crossing and Topological Strings
• General instanton counting and 5d SCFT
• Geometric engineering of (framed) BPS states

Cited by in corpus (11)

• Witten Index and Wall Crossing
• Defects and Quantum Seiberg-Witten Geometry
• Duality walls and defects in 5d N=1 theories
• General instanton counting and 5d SCFT
• New AdS${}_2$ backgrounds and $\mathcal{N}=4$ Conformal Quantum Mechanics
• 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
• Surface defects and instanton partition functions
• Mutation, Witten Index, and Quiver Invariant