Brane current algebras and generalised geometry from QP manifolds. Or, “when they go high, we go low”

Abstract

We construct a Poisson algebra of brane currents from a QP-manifold, and show their Poisson brackets take a universal geometric form. This generalises a result of Alekseev and Strobl on string currents and generalised geometry to include branes with worldvolume gauge fields, such as the D3 and M5. Our result yields a universal expression for the ’t Hooft anomaly that afflicts isometries in the presence of fluxes. We determine the current algebra in terms of (exceptional) generalised geometry, and show that the tensor hierarchy gives rise to a brane current hierarchy. Exceptional complex structures produce pairs of anomaly-free current subalgebras on the M5-brane worldvolume.

Brane wrapping, AKSZ sigma models, and QP manifolds

“…There are also similar structures that appear in higher algebroids. That is, one can define the notion of a Dirac structure for these higher algebroids and define the associated differential [2,[30][31][32][33][34][35]. These can be embedded into the Q-structure of the QP manifolds associated to these higher algebroids.…”

“…which induces a Poisson bracket [•, •] on M X . This Poisson bracket can be conveniently expressed in terms of 'test functions' as in [2]. Given arbitrary functions , η on X -which correspond to differential forms on X since X = T [1]X -they write…”

“…In a series of recent papers [1][2][3] we have been establishing a correspondence between (BPS) p-branes in String/M-theory on one hand and symplectic L p+1 -algebroids on the other hand. The latter can be thought of as appropriate generalisations of the (exact) Courant algebroid that encodes the generalised geometry of type II backgounds with Neveu-Schwarz 3-form H (but without Ramond-Ramond fluxes); a Courant algebroid is precisely a symplectic L n -algebroid of degree n = 2 [4].…”

“…When the n-brane has an (n−1)-brane boundary, an inflow-type argument with an appropriate boundary condition produces the WZ term that couples those same fluxes to the (n − 1)-brane [1,3] 1 . The algebroid also determines the corresponding (n − 1)-brane more directly via the brane phase space construction that yields the Poisson algebra of brane currents on phase space [2] (i.e. in the hamiltonian formulation of brane dynamics).…”

On exceptional QP-manifolds

“…E d(d) -covariant p-brane world-volume theories have been constructed via actions [28][29][30][31][32][33] or via Hamiltonian formulations [10,[34][35][36][37][38][39][40]. Topological terms and topological field theories to such p-branes, that are the dynamical objects in type II and eleven-dimensional supergravities, have been considered, partly including tensor hierarchies [41][42][43][44][45], but not in an E d(d) -covariant way. In these cases the underlying E d(d) -structure is rather hidden, and also the construction of the appearing manifolds is somewhat ad hoc.…”