# M-theoretic Lichnerowicz formula and supersymmetry

## TL;DR

This paper derives an M-theoretic Lichnerowicz formula linking supersymmetric operators to eleven-dimensional supergravity, including higher-derivative corrections and implications for compactifications.

## Contribution

It presents the first M-theoretic Lichnerowicz formula for 11D supergravity with higher-derivative terms and supersymmetric invariants, extending previous geometric frameworks.

## Key findings

- Derived the M-theoretic Lichnerowicz formula for 11D supergravity.
- Identified two supersymmetric invariants with free coefficients.
- Discussed implications for G2 holonomy compactifications.

## Abstract

A suitable generalisation of the Lichnerowicz formula can relate the squares of supersymmetric operators to the effective action, the Bianchi identities for fluxes, and some equations of motion. Recently, such formulae have also been shown to underlie the (generalised) geometry of supersymmetric theories. In this paper, we derive an M-theoretic Lichnerowicz formula that describes eleven-dimensional supergravity together with its higher-derivative couplings. The first corrections to the action appear at eight-derivative level, and the construction yields two different supersymmetric invariants, each with a free coefficient. We discuss the restriction of our construction to seven-dimensional internal spaces, and implications for compactifications on manifolds of $G_2$ holonomy. Inclusion of fluxes and computation of contributions with higher than eight derivatives are also discussed.