An index for flux vacua

TL;DR

This paper introduces a topological index based on the winding number of the gradient of a scalar potential, linking critical points inside the field space to boundary behavior, demonstrated in M-theory flux compactifications.

Contribution

It presents a novel topological method to analyze flux vacua by relating interior critical points to boundary conditions in scalar field spaces.

Findings

The winding number index effectively characterizes flux vacua.

Application to M-theory on Calabi-Yau four-folds demonstrates practical utility.

The Fermat sextic example illustrates the method's simplicity and relevance.

Abstract

We propose to use the winding number of the gradient of a scalar potential as a simple topological index that relates critical points in the interior of the scalar field space to the behavior of the potential at the (asymptotic) boundary of the field space. We demonstrate this technique for supersymmetric flux compactifications of M-theory on Calabi-Yau four-folds, and use the Fermat sextic as a simple, one-parameter example.