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

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.

47 pages including appendices. v2: typos fixed, references added. Enlarged font size for legibility. v3: Reference added v4: Fixed formula A.24