Brane Wess-Zumino terms from AKSZ and exceptional generalised geometry as an $L_\infty$-algebroid

TL;DR

This paper reinterprets M-theory's exceptional geometry using AKSZ and $L_$-algebroids, revealing new geometric structures underlying brane Wess-Zumino terms and their relation to Chern-Simons theories.

Contribution

It introduces an $L_$-algebroid framework for M-theory's exceptional geometry and connects it to AKSZ constructions of brane Wess-Zumino terms.

Findings

Identifies the $L_$-algebra underlying the tensor hierarchy.

Shows the AKSZ construction yields a 7D Chern-Simons-like theory.

Relates the framework to D3-brane and M5-brane Wess-Zumino terms.

Abstract

We reinterpret the generalised Lie derivative of M-theory $E_6$ generalised geometry as hamiltonian flow on a graded symplectic supermanifold. The hamiltonian acts as the nilpotent derivative of the tensor hierarchy of exceptional field theory. This construction is an M-theory analogue of the Courant algebroid and reveals the $L_\infty$-algebra underlying the tensor hierarchy. The AKSZ construction identifies that same hamiltonian with the lagrangian of a 7-dimensional generalisation of Chern-Simons theory that reduces to the M5-brane Wess-Zumino term on 5-brane boundaries. The exercise repeats for the type IIB $E_5$ generalised geometry and we discuss the relation to the D3-brane.