Generalized Euler Index, Holonomy Saddles, and Wall-Crossing

We formulate Witten index problems for theories with two supercharges in a Majorana doublet, such as in $d=3$ $\mathcal N=1$ or in $d=1$ $\mathcal N=2$. 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.