Generalized Euler Index, Holonomy Saddles, and Wall-Crossing

We formulate Witten index problems for theories with two supercharges in a Majorana doublet, as in d = 3 $$ \mathcal{N} $$ = 1 theories and dimensional reduction thereof. Regardless of spacetime dimensions, the wall-crossing occurs generically, in the parameter space of the real superpotential W. With scalar multiplets only, the path integral reduces to a Gaussian one in terms of dW, with a winding number interpretation, and allows an in-depth study of the wall-crossing. After discussing the connection to well-known mathematical approaches such as the Morse theory, we move on to Abelian gauge theories. Even though the index theorem for the latter is a little more involved, we again reduce it to winding number countings of the neutral part of dW. The holonomy saddle plays key roles for both dimensions and also in relating indices across dimensions.