Commuting Pairs, Generalized para-Kähler Geometry and Born Geometry

Abstract

In this paper, we study the geometries given by commuting pairs of generalized endomorphisms A ∈ End(T ⊕ T * ) with the property that their product defines a generalized metric. There are four types of such commuting pairs: generalized Kähler (GK), generalized para-Kähler (GpK), generalized chiral and generalized anti-Kähler geometries. We show that GpK geometry is equivalent to a pair of para-Hermitian structures and we derive the integrability conditions in terms of these. From the physics point of view, this is the geometry of 2D (2, 2) twisted supersymmetric sigma models. The generalized chiral structures are equivalent to a pair of tangent bundle product structures that also appear in physics applications of 2D sigma models. We show that the case when the two product structures anti-commute corresponds to Born geometry. Lastly, the generalized anti-Kähler structures are equivalent to a pair of anti-Hermitian structures (sometimes called Hermitian with Norden metric). The generalized chiral and anti-Kähler geometries do not have isotropic eigenbundles and therefore do not admit the usual description of integrability in terms of the Dorfman bracket. We therefore use an alternative definition of integrability in terms of the generalized Bismut connection of the corresponding metric, which for GK and GpK commuting pairs recovers the usual integrability conditions and can also be used to define the integrability of generalized chiral and anti-Kähler structures. In addition, it allows for a weakening of the integrability condition, which has various applications in physics.

Integrable deformation of $\mathbb{CP}^n$ and generalised Kaehler geometry

“…As it turns out, complex projective spaces CP n are the natural setting where to carry out our investigation. They are exemplars of a broader class of algebraic varieties, known as generalised flag manifolds, which have recently benefited, alongside para-complex manifolds, from a surge of interest in the physics literature as they materialise in rich examples [26][27][28][29][30][31].…”

D-Branes in Para-Hermitian Geometries

“…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.…”

“…Let (E, η, ρ, • , • ) be an exact pre-Courant algebroid. In Appendix A.5 we describe the natural para-Hermitian structures (K σ , η) on E. More generally, a para-complex structure K ∈ Aut1(E) which is compatible with the metric η, in the sense of Definition A1, is the analogue in generalised geometry of an almost para-Hermitian structure on a manifold and is called an almost generalised para-complex structure [18].…”

“…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.…”

“…At the purely bosonic level, all that is needed is the boundary conformality condition, and indeed our Born D-branes are conformally invariant. It would be interesting to develop supersymmetric extensions of the Born sigma-model, and to discern what new types of target space para-complex structures appear; (generalised) para-complex structures appear on target spaces for topologically twisted and supersymmetric sigma-models [12,18,43], as well as in target spaces for N = 2 hypermultiplets in Euclidean signature [44]. Another subtle issue we do not address in this paper is the extension to (multiple) non-abelian D-branes.…”