On exceptional QP-manifolds

TL;DR

This paper establishes a precise connection between tensor hierarchies and QP-manifolds, revealing new insights into p-branes, exceptional spaces, and Leibniz algebras within a unified geometric framework.

Contribution

It introduces a duality-covariant formulation of p-brane QP-manifolds and explores their relation to tensor hierarchies and Leibniz algebras, proposing new geometric realizations.

Findings

Solutions correspond to 1/2-BPS p-branes.

Reduction reproduces known p-brane QP-manifolds.

Speculations on exceptional extended spaces and Leibniz algebra manifolds.

Abstract

The connection between two recent descriptions of tensor hierarchies - namely, infinity-enhanced Leibniz algebroids, given by Bonezzi & Hohm and Lavau & Palmkvist, the p-brane QP-manifolds constructed by Arvanitakis - is made precise. This is done by presenting a duality-covariant version of latter. The construction is based on the QP-manifold $T^\star[n]T[1]M \times \mathcal{H}[n]$, where $M$ corresponds to the internal manifold of a supergravity compactification and $\mathcal{H}[n]$ to a degree-shifted version of the infinity-enhanced Leibniz algebroid. Imposing that the canonical Q-structure on $T^\star[n] T[1] M$ is the derivative operator on $\mathcal{H}$ leads to a set of constraints. Solutions to these constraints correspond to $\frac{1}{2}$-BPS p-branes, suggesting that this is a new incarnation of a brane scan. Reduction w.r.t. to these constraints reproduces the known p-brane QP-manifolds. This is shown explicitly for the SL(3)$\times$SL(2)- and SL(5)-theories. Furthermore, this setting is used to speculate about exceptional 'extended spaces' and QP-manifolds associated to Leibniz algebras. A proposal is made to realise differential graded manifolds associated to Leibniz algebras as non-Poisson subspaces (i.e. not Poisson reductions) of QP-manifolds similar to the above. Two examples for this proposal are discussed: generalised fluxes (including the dilaton flux) of O(d,d) and the 3-bracket flux for the SL(5)-theory.