Laplacians in Various Dimensions and the Swampland

TL;DR

This paper explores how certain eigenfunctions of elliptic operators over moduli space, particularly Laplacians, impose restrictions on quantum gravitational theories with various amounts of supersymmetry, linking higher-derivative corrections to the Swampland.

Contribution

It demonstrates that Wilson coefficients of higher-derivative operators are eigenfunctions of Laplacians on moduli space, providing new constraints on the Swampland in theories with different supersymmetries.

Findings

Eigenfunctions of elliptic operators constrain moduli space in supergravity.

Laplace equations restrict species hull vectors in various dimensions.

Proposes symmetry-based criteria for when Laplacian operators are relevant.

Abstract

The species cutoff is a moduli-dependent quantity signaling the onset of quantum gravitational phenomena, whose form can be oftentimes determined from higher-derivative and higher-curvature corrections within low-energy gravitational EFTs. In this work, we point out that these Wilson coefficients are eigenfunctions of an appropriate second-order elliptic operator defined over moduli space in theories with more than four supercharges. This was already known to be the case for the leading $\mathcal{R}^4$-correction to the two-derivative (bosonic) action of maximal supergravity in $d\leq 10$. Here, we reconsider this fact from the Swampland point of view and show how, in $d=10,9,8$, solving a Laplace equation imposes non-trivial restrictions on the species hull vectors. We further argue that this property is also satisfied in settings with less supersymmetry. In particular, we focus on the $\mathcal{R}^4$-operator in minimal supergravity theories in $d=10,9$, and on the leading $\mathcal{R}^2$-term in setups with 8 supercharges in $d=6,5,4$. Finally, we provide a symmetry-based criterion for determining when the relevant elliptic operator should be the Laplacian. A bottom-up rationale for this constraint remains to be fully understood, and we conclude by outlining some compelling possibilities.