An Index Formula for Supersymmetric Quantum Mechanics

Clay Córdova and Shu-Heng Shao

Journal of Singularities
volume 15 (2016), 14-35

Received: 29 September 2014. Received in revised form: 3 July 2015.

DOI: 10.5427/jsing.2016.15b

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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.

Author(s) information:

Clay Córdova Shu-Heng Shao
Society of Fellows Jefferson Physical Laboratory
Harvard University Harvard University
Cambridge, MA, USA Cambridge, MA, USA
email: email: