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

中文翻译：

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广义欧拉指数、完整鞍点和穿墙

我们为在马约拉纳双峰中具有两个超荷的理论制定 Witten 指数问题，如 d = 3 N $$ \mathcal{N} $$ = 1 理论及其降维。不管时空维度如何，穿墙一般发生在实超势 W 的参数空间中。仅使用标量多重态，路径积分就 dW 减少为高斯积分，具有绕数解释，并允许对穿墙进行深入研究。在讨论了与著名数学方法（如莫尔斯理论）的联系之后，我们继续讨论阿贝尔规范理论。尽管后者的索引定理稍微复杂一些，但我们再次将其简化为 dW 的中性部分的绕数计数。