Currents, charges and algebras in exceptional generalised geometry

TL;DR

This paper develops an $E_{d(d)}$-invariant Hamiltonian framework for p-branes in supergravity, revealing non-geometric currents and flux effects, with detailed analysis of membranes in SL(5) theory and reduction to string theory.

Contribution

It introduces a Hamiltonian formulation with a generalised metric and current algebra for p-branes, extending previous results to $d \,\leq\, 6$, and explores non-geometric currents and fluxes in exceptional geometry.

Findings

Current algebra is twisted by generalised fluxes in the SL(5) membrane.

Membrane reduction reproduces string theory results.

Currents for p>2 are generally non-geometric due to U-duality.

Abstract

A classical $E_{d(d)}$-invariant Hamiltonian formulation of world-volume theories of half-BPS p-branes in type IIb and eleven-dimensional supergravity is proposed, extending known results to $d \leq 6$. It consists of a Hamiltonian, characterised by a generalised metric, and a current algebra constructed s.t. it reproduces the $E_{d(d)}$ generalised Lie derivative. $E_{d(d)}$-covariance necessitates the introduction of so-called charges, specifying the type of p-brane and the choice of section. For p>2, currents of p-branes are generically non-geometric due to the imposition of U-duality, e.g. the M5-currents contain coordinates associated to the M2-momentum. A derivation of the $E_{d(d)}$-invariant current algebra from a canonical Poisson structure is in general not possible. At most, one can derive a current algebra associated to para-Hermitian exceptional geometry. The membrane in the SL(5)-theory is studied in detail. It is shown that in a generalised frame the current algebra is twisted by the generalised fluxes. As a consistency check, the double dimensional reduction from membranes in M-theory to strings in type IIa string theory is performed. Many features generalise to p-branes in SL(p+3) generalised geometries that form building blocks for the $E_{d(d)}$-invariant currents.