Description
Abstract: Recasting of geometric index theorems into path integral of nonlinear sigma models has a long history, dating back to early 1980's. The idea has been revived in the last decade for gauge theories with complex supersymmetries, under the banner of localization, and resulted in several systematic formulae for Hirzebruch genus, Elliptic genus, and other higher-dimensional twisted partition functions.
Here, we turn to gauge theories with two real supercharges, as in d=3 N=1 and d=2 N=(1,1). The holomorphy is no longer
available and chemical potentials drop out universally. Instead, winding numbers of the real superpotential W enter as key quantities, incompleteness of which implies wall-crossing and is prevalent in all spacetime dimensions, unlike the D-term wall-crossing with complex supersymmetries. We introduce the notion of holonomy saddles which allows systematic constructions of d=1,2,3 twisted partition functions from d=0 Gaussian building blocks. d=3 Chern-Simons terms require extra care, but contribute additive pieces computable again from winding numbers. We close with comparisons against Intriligator-Seiberg and checks of several d=3 N=1 dualities.
