An index for flux vacua

TL;DR

The paper proposes a topological index for flux vacua based on the winding of the gradient of the scalar potential, connecting interior critical points to asymptotic boundary data via a boundary integral $I$. It specializes to M-theory flux vacua with the GVW superpotential $W(z)=\int_{X_4} G_4\wedge\Omega$ and argues that the boundary index computed from $DW$ yields a sufficient condition for the existence of SUSY minima, particularly in no-scale regimes. The Fermat sextic Calabi–Yau four-fold serves as a tractable test case, where the complex-structure moduli reduce to a one-parameter space and the period vector satisfies a Picard–Fuchs system; the authors construct a contour around the boundary of moduli space, incorporating large complex structure, conifold, and Landau–Ginzburg regions with their monodromies to evaluate the index. They demonstrate several explicit flux choices yielding LG, LCS, or LG-point vacua, and show consistency with direct numerical solutions of $D_i W=0$, highlighting both the power and limitations of inferring interior vacua from asymptotic data. The work suggests possible generalizations to higher-dimensional moduli spaces and connections to asymptotic Hodge theory and the tadpole bound, with potential implications for the swampland program and moduli stabilization strategies.

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.

Figures (3)

• Figure 1: The complex structure moduli space $\mathcal{M}_\psi$ of the mirror sextic (the covering space with respect to the LG-monodromy). The Landau-Ginzburg (LG) point sits at the origin, there are six conifold (C) points, and the large complex structure (LCS) point is located at complex infinity. The conifold points are connected with the large complex structure point by six branch cuts (dashed radial lines). In red the integration contour encircling these points as well as the branch cuts. It is to be completed around the LCS point at infinity, indicated by the dashed red line.
• Figure 2: Plots of $\psi^{-1} D_\psi W$ on the $\psi$-plane for different choices of the flux vector $g^I$. Colors encode the complex phase.
• Figure 3: Left: The ratio of the imaginary parts of the periods $D_\psi \Pi^2$ and $D_\psi \Pi^4$. Right: The real part of the period $- \left(D_\psi \Pi^1 + D_\psi \Pi^5\right)$. Both are evaluated on the first branch cut at ${\rm Im\,} \psi = 0$ and ${\rm Re\,} \psi \geq 1$.