# Gravitation and quadratic forms

## TL;DR

This paper demonstrates that the Hamiltonians of pure gravity and N=8 supergravity in four dimensions can be expressed as quadratic forms, extending a feature previously known for Yang-Mills theories, and analyzes the impact of residual reparametrizations.

## Contribution

It shows that the quadratic form structure of Hamiltonians extends from Yang-Mills theories to gravity and supergravity, revealing a new mathematical property of these theories.

## Key findings

- Hamiltonians of pure gravity and N=8 supergravity are quadratic forms.
- Residual reparametrizations affect the Hamiltonian and its quadratic form.
- The quadratic form structure is consistent across different gravitational theories.

## Abstract

The light-cone Hamiltonians describing both pure Yang-Mills and N=4 super Yang-Mills may be expressed as quadratic forms. Here, we show that this feature extends to theories of gravity. We demonstrate how the Hamiltonians of both pure gravity and N=8 supergravity, in four dimensions, may be written as quadratic forms. We examine the effect of residual reparametrizations on the Hamiltonian and the resulting quadratic form.