Tensor hierarchies, Borcherds algebras and E11

TL;DR

This paper links tensor hierarchies in supergravity to Borcherds superalgebras and E11, providing a unified algebraic framework to understand the structure of p-form fields and their representations.

Contribution

It derives the tensor hierarchy formulas from Borcherds superalgebras and demonstrates the equivalence with E11 for determining representations in supergravity.

Findings

Tensor hierarchies correspond to Borcherds superalgebra structures.

E11 algebra reproduces the hierarchy representations up to p=D.

Large rank E(r) algebras extend the hierarchy analysis for higher p.

Abstract

Gauge deformations of maximal supergravity in D=11-n dimensions generically give rise to a tensor hierarchy of p-form fields that transform in specific representations of the global symmetry group E(n). We derive the formulas defining the hierarchy from a Borcherds superalgebra corresponding to E(n). This explains why the E(n) representations in the tensor hierarchies also appear in the level decomposition of the Borcherds superalgebra. We show that the indefinite Kac-Moody algebra E(11) can be used equivalently to determine these representations, up to p=D, and for arbitrarily large p if E(11) is replaced by E(r) with sufficiently large rank r.