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

We reinterpret the generalised Lie derivative of $\mathrm{M}$‑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 $\mathrm{M}$‑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 $\mathrm{M}5$‑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 $\mathrm{D}3$‑brane.

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Published 12 February 2020