Supersymmetric Particles in Four Dimensions

Abstract

In this dissertation we study supersymmetric particles in four spacetime dimensions and their relations to other physical observables. For a large class of four-dimensional N=2 systems, the supersymmetric particles are described by the ground states of certain quiver quantum mechanics in the low energy limit. We derive a localization formula for the index of quiver quantum mechanics with four supercharges. Our answer takes the form of a residue integral on the complexified Cartan subalgebra of the gauge group. The wall-crossing phenomenon appears as discontinuities in the value of the residue integral as the integration contour is varied. We then move on to study the ground states in the Kronecker model of quiver quantum mechanics. This is the simplest quiver with two gauge groups and bifundamental matter fields, and appears universally in four-dimensional N=2 systems. The ground state degeneracy may be written as a multi-dimensional contour integral, and the enumeration of poles can be simply phrased as counting bipartite trees. We solve this combinatorics problem, thereby obtaining exact formulas for the degeneracies of an infinite class of models. For large ranks, the ground state degeneracy is exponential with the slope being a modular function that we are able to compute at integral values of its argument. We also observe that the exponential of the slope is an algebraic number and determine its associated algebraic equation explicitly in several examples. The speed of growth of the degeneracies, together with various physical features of the bound states, suggests a dual string interpretation. In the last part of the dissertation, we conjecture a precise relationship between a limit of the superconformal index of four-dimensional N=2 field theories, which counts local operators, and the spectrum of BPS particles on the Coulomb branch. We verify this conjecture for the case of free field theories, N=2 QED, and SU(2) gauge theories coupled to matter. Assuming the validity of our proposal, we compute the superconformal index of all Argyres-Douglas theories. Our answers match expectations from the connection of Schur operators with two-dimensional chiral algebras.