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 ∞ -algebra underlying the tensor hierarchy.The AKSZ construction identifies that same hamiltonian with the lagrangian of a 7dimensional 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.

Higher curvature Bianchi identities, generalized geometry, and L∞ algebras

“…Courant algebroids were shown to have an L 3 -algebra in [49]. In the case of [52,53], where the correct L ∞ algebra was demonstrated to follow from interpreting the M theory geometries as dg-symplectic manifolds. More broadly speaking, the higher structures that feature in string and M theory are known to be classified by super homotopy theory, see the review [54] and references therein.…”

Pre‐NQ Manifolds and Correspondence Spaces: the Nilmanifold Example

“…whose interpretation in terms of symplectic graded geometry is not known so far. First steps were achieved in [57], but the general case is an open problem. Studying derived brackets, algebraic conditions as (56) might give a more systematic insight into dualities of such theories (U-duality).…”

Generalising Courant algebroids to M-theory

“…3 We will proceed to describe all three algebroids of the table at once using the language of NPQ manifolds (used also in e.g. the original AKSZ paper [8]), which is how the M2 algebroid (which appeared in 2010 in [9]) and M5 algebroid (in [1]) were originally written down anyway. The relation of the M2 algebroid to SL(5)-exceptional generalised geometry was first mentioned in [10], while the general picture of algebroids and their associated branes sketched in the above paragraph in generality is one first main point of [1] which this contribution is summarising.…”

“…the original AKSZ paper [8]), which is how the M2 algebroid (which appeared in 2010 in [9]) and M5 algebroid (in [1]) were originally written down anyway. The relation of the M2 algebroid to SL(5)-exceptional generalised geometry was first mentioned in [10], while the general picture of algebroids and their associated branes sketched in the above paragraph in generality is one first main point of [1] which this contribution is summarising. (The other points raised therein are the novel appearance of the worldvolume gauge field for the M5 which has no analogue in the string or M2 cases, and the mathematical interpretation of the "tensor hierarchy" in exceptional generalised geometry as an L ∞ -algebra obtained from the NPQ manifold structure.…”