2019/09/10 by Hu, Shengda, Moraru, Ruxandra, Svoboda, David
#FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.1909.04646
In this paper, we study the geometries given by commuting pairs of generalized endomorphisms \cal 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.