Laplacians in Various Dimensions and the Swampland

TL;DR

The work identifies a unifying principle: protected higher-derivative gravitational couplings in supersymmetric theories are governed by second-order elliptic operators on moduli space, often taking the Laplace-Beltrami form. In many maximal and half-maximal cases, the R^4 or R^2 coefficients satisfy a Laplace equation with eigenvalues tied to operator dimension and moduli count, with dualities guiding the exact operator (sometimes requiring dilaton-augmented forms). By analyzing Type IIB, IIA, heterotic, F-theory, and M-theory compactifications across dimensions 8–10 and models with 8 supercharges, the authors show how solving these Laplace equations constrains the species hull and informs the asymptotic behavior of the quantum gravity cutoff, \\Lambda_{QG}. The results illuminate a deep link between automorphic forms, dualities, and the Swampland, and provide a bottom-up framework to infer asymptotic towers of states from moduli dynamics, with potential implications for SWGC and entropy-like interpretations of species scales.

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.

Figures (5)

• Figure 1: Example of convex hull generated by tower (blue) and species vectors (red). This example is realized in the string theory Landscape via M-theory compactification on an attractive K3 surface Castellano:2023jjt. The dashed line separates different duality frames, wherein a different tower provides the dominant decay rate. Notice that the two polytopes are dual to each other with respect to a ball of radius $1/\sqrt{5}$, i.e., the black (half-)circumference centered at the origin.
• Figure 2: The $d=10$ Laplace equation translated as a geometric constraint imposed on the saxionic tangent space parametrized by $\lambda$. The red points represent the solutions to $2\lambda\left(\lambda-\sqrt2\right)=3$. The duality symmetry acts on this space as a discrete $\mathbb{Z} _2$ remnant sending $\lambda \to -\lambda$, which leads to the dual Laplace solutions captured by the blue points. All solutions of both Laplace equations are characterized by leading to a decay rate of the species scale $|\lambda_{\text{sp}}| \le 1/\sqrt{d-2}$, which is indicated by the dashed interval. The outer dots reconstruct the Type IIB species polytope.
• Figure 3: Laplace circles and species polytope for the case of maximal supergravity in $d=9$. The outer dashed circle is the upper bound \ref{['eq:upperbound9d']} derived in the main text. The two circles are centered respectively in ${\vec{\xi}}_\pm=\left( -\frac{1}{\sqrt{14}},\pm\frac{1}{\sqrt{2}}\right)$, with radius $4/\sqrt7$.
• Figure 4: 9d Laplace equation (orange), 10d IIB (green) and 10d IIA (blue) equations plotted together. The $x$ axis is included in the 10d IIB solutions. In the upper half plane the intersection between all constraints is given by $M$ and $A$, which together the dual point $B$ reconstructs the species polytope. In the lower half-plane the point $O$ also solves all equations and recovers the missing constant term contribution from the $E_{3/2}^{sl_2}$ expansion in \ref{['eq:9dwilsoncoeff']}.
• Figure 5: Laplace spheres in the 8d theory. The one relative to the original ansatz \ref{['eq:8dlapleq']} is depicted in purple. The pink one is generated by acting with the $S_2$ premutation element of the duality remnant $S_3 \times S_2$. The other two are generated by acting with the order two element $b \in S_3$. To avoid cluttering, we did not plot the other 8 spheres generated by the order three element $a \in S_3$, which action rotates by $120^\circ$ in the plane $y-z=0$ fixed by the $S_2$ action. In the lower image we depict the species polytope, whose vertices always touch the Laplace spheres.