交换对、广义para-Kähler几何和Born几何
Commuting Pairs, Generalized para-Kähler Geometry and Born Geometry
摘要：
本文研究了广义自同态交换对${\cal A} \in \text{End}(T\oplus T^*)$所给出的几何，并证明了它们的乘积定义了一个广义度量。有四种类型的交换对：广义Kähler几何（GK）、广义para-Kähler几何（GpK）、广义手征几何和广义反Kähler几何。我们证明了GpK几何等价于一对仿埃尔米特结构，并由此导出了可积性条件。从物理学的角度来看，这是$2D$ $(2,2)$扭曲超对称西格玛模型的几何结构。广义手征结构等价于一对切丛积结构，这对切丛积结构也出现在$2D$西格玛模型的物理应用中.我们表明，当两个产品结构的情况下，反交换对应于玻恩几何。最后，广义反Kähler结构等价于一对反厄米结构（有时称为厄米-诺登度量）。广义手征几何和反凯勒几何没有各向同性的本征丛，因此不允许用多尔夫曼括号来描述可积性。因此，我们使用另一种可积性定义，即相应度量的广义Bismut联络，对于GK和GpK交换对，它恢复了通常的可积性条件，也可以用来定义广义手征和反Kähler结构的可积性。此外，它允许弱化可积性条件，这在物理学中有各种应用。
英文摘要：
In this paper, we study the geometries given by commuting pairs of generalized endomorphisms ${\cal A} \in \text{End}(T\oplus T^*)$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
