# Branes and Polytopes

## TL;DR

This paper explores the relationship between half-supersymmetric branes in supergravity theories and the vertices of uniform polytopes, revealing algebraic rules, symmetries, and a triality among certain branes.

## Contribution

It establishes a universal algebraic framework linking branes to uniform polytopes and uncovers a triality relation among 0-, 1-, and 2-branes.

## Key findings

- Branes correspond to vertices of specific uniform polytope families.
- Universal algebraic rules describe the number of independent 1/2-BPS p-branes.
- A triality relation connects 0-, 1-, and 2-branes.

## Abstract

We investigate the hierarchies of half-supersymmetric branes in maximal supergravity theories. By studying the action of the Weyl group of the U-duality group of maximal supergravities we discover a set of universal algebraic rules describing the number of independent 1/2-BPS p-branes, rank by rank, in any dimension. We show that these relations describe the symmetries of certain families of uniform polytopes. This induces a correspondence between half-supersymmetric branes and vertices of opportune uniform polytopes. We show that half-supersymmetric 0-, 1- and 2-branes are in correspondence with the vertices of the $k_{21}$ , $2_{k1}$ and $1_{k2}$ families of uniform polytopes, respectively, while 3-branes correspond to the vertices of the rectified version of the $2_{k1}$ family. For 4-branes and higher rank solutions we find a general behavior. The interpretation of half-supersymmetric solutions as vertices of uniform polytopes reveals some intriguing aspects. One of the most relevant is a triality relation between 0-, 1- and 2-branes.

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Source: https://tomesphere.com/paper/1705.01294