Algebroids, AKSZ Constructions and Doubled Geometry

Abstract

We give a self-contained survey of some approaches aimed at a global description of the geometry underlying double field theory. After reviewing the geometry of Courant algebroids and their incarnations in the AKSZ construction, we develop the theory of metric algebroids including their graded geometry. We use metric algebroids to give a global description of doubled geometry, incorporating the section constraint, as well as an AKSZ-type construction of topological doubled sigma-models. When these notions are combined with ingredients of para-Hermitian geometry, we demonstrate how they reproduce kinematical features of double field theory from a global perspective, including solutions of the section constraint for Riemannian foliated doubled manifolds, as well as a natural notion of generalized T-duality for polarized doubled manifolds. We describe the L ∞-algebras of symmetries of a doubled geometry, and briefly discuss other proposals for global doubled geometry in the literature.

D-Branes in Para-Hermitian Geometries

“…Combining (27) with (25) shows that the overall effect is to shift the 2-form B can to B can + F, which is also defined only on W. Repeating the arguments above shows that now:…”

“…We refer to these branes as 'generalised para-complex D-branes'. In particular, we show that the Born D-branes on an almost para-Hermitian manifold fit into this picture in a natural way: Any two-dimensional non-linear sigma-model naturally corresponds to an exact Courant algebroid (see, e.g., [38][39][40]); on an almost para-Hermitian manifold this is called the 'large Courant algebroid' [9,16,18,25]. Seen in this way, our D-branes provide natural para-complex versions of the A-branes and Bbranes of topological string theory.…”

“…Hence we proceed to develop a theory of branes on the slightly stronger structure of a pre-Courant algebroid. One advantage provided by this restricted class of metric algebroids is that the notion of D-integrability leads to Frobenius integrability: If L ⊂ E is a D-structure, then the bracket morphism property of the anchor (see Definition A5) implies that its image ρ(L) ⊂ TM is involutive, and hence induces a foliation of M. Pre-Courant algebroids constitute a physically meaningful intermediary step between the Courant algebroids of generalised geometry, wherein the section constraint is imposed and solved, and the more general metric algebroids of a fully unconstrained doubled geometry; see [9,16,23,25] for detailed descriptions of the chain of metric algebroids involved between type II supergravity and double field theory.…”

“…We work in the setting of para-Hermitian geometry and metric algebroids, which encompasses the standard flat space treatments (recovered as instances of para-Kähler geometry) and known examples of global doubled geometries (typically involving non-integrable almost para-complex structures). This approach was originally proposed by Vaisman [6,7], and has recently undergone a renewed flurry of activity, see, e.g., [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] for a host of developments in this direction.…”

“…With these connections in mind, we extend Gualtieri's approach [35] to viewing abelian generalised complex D-branes as 'generalised submanifolds' into the framework of metric algebroids and almost generalised para-complex structures. The setting provided by a generic metric algebroid turned out to be too coarse to provide any meaningful analysis, hence we restrict to the intermediary structure of a pre-Courant algebroid, which represents a structure along the way between the metric algebroids of double field theory and the Courant algebroids of supergravity arising from imposition of the section constraint [9,16,25]. We refer to these branes as 'generalised para-complex D-branes'.…”