Supervertices and Non-renormalization Conditions in Maximal Supergravity Theories

TL;DR

This work analyzes higher derivative deformations of maximal supergravity using on-shell superamplitudes, classifying D-term and F-term supervertices across dimensions D=9..3 and focusing on 8-, 12-, and 14-derivative couplings. By exploiting soft scalar limits, factorization, and dimensional lifting, it derives non-renormalization conditions in the moduli space, including Laplacian-type equations for the $R^4$, $D^4R^4$, and $D^6R^4$ coefficients and additional higher-representation Hessian constraints. The results align with toroidal Type II string theory and M-theory expectations, with explicit differential equations fixed by perturbative data and U-duality considerations, and a lifting framework that explains which F-term vertices can extend to 11D. Overall, the paper provides a structured, symmetry-based approach to determining moduli-dependent higher derivative couplings in maximal supergravity and their string/M-theory implications, up to 14-derivative order. The methods enable potential full determination of M-theory effective actions via controlled supersymmetry constraints and cross-checks with perturbative string theory.

Abstract

We construct higher derivative supervertices in an effective theory of maximal supergravity in various dimensions, in the super spinor helicity formalism, and derive non-renormalization conditions on up to 14-derivative order couplings from supersymmetry. These non-renormalization conditions include Laplace type equations on the coefficients of $R^4$, $D^4R^4$, and $D^6R^4$ couplings. We also find additional constraining equations, which are consistent with previously known results in the effective action of toroidally compactified type II string theory, and elucidate many features thereof.

Figures (6)

• Figure 1: Single soft limit of a superamplitude and its relation to the lower point supervertex.
• Figure 2: Lorentz rotations of an $n$-point amplitude ${\mathcal{A}}$ with fixed momenta in a $d$-dimensional sub-spacetime.
• Figure 3: Factorizations of the two independent 6-point 8-derivative superamplitudes through $R^4$ supervertex and ${\delta}\phi^I R^4$ supervertex respectively.
• Figure 4: Factorization of the five dimensional 7-point 12-derivative superamplitude through one $D^4({\delta}\phi^{\cal I}{\delta}\phi^{\cal J})_{[0200]}R^4$ supervertex.
• Figure 5: Factorizations of the 6-point 14-derivative superamplitude through one $D^6R^4$ supervertex or two $R^4$ supervertices.