论文标题
四维几何超重力和电磁二元性：数学家的简短指南
Four-dimensional geometric supergravity and electromagnetic duality: a brief guide for mathematicians
论文作者
论文摘要
我们轻轻介绍了在定向的四维超级重力的玻色粒部门的全球几何表述中，在定向的四个manifold $ m $的任意拓扑结构上，提供了其U-二元组的几何表征。 The geometric formulation of four-dimensional supergravity is based on a choice of a vertically Riemannian submersion $π$ over $M$ equipped with a flat Ehresmann connection, which determines the non-linear section sigma model of the theory, and a choice of flat symplectic vector bundle $\mathcal{S}$ equipped with a positive complex polarization over the total space of $π$, which encodes该理论的逆向耦合和theta角，并确定其量规部门。该理论的经典字段由$ m $上的洛伦兹指标，$π$的全球部分以及以$ \ mathcal {s} $价值的两种形式组成，这些{s} $满足了\ emph {emph {twisted}自我duality的概念的代数关系。我们使用这种几何公式来研究超级重力的电磁偶性转换组，也称为连续的经典U偶性组，我们使用一定的短矢量载体组的一定短序列来表征它们。此外，我们讨论了四维超级重力的杀戮旋转方程的一般结构，提供了几个明确的例子，并就一些开放的数学问题进行了评论。该演讲旨在用于从事差异几何的数学家。
We give a gentle introduction to the global geometric formulation of the bosonic sector of four-dimensional supergravity on an oriented four-manifold $M$ of arbitrary topology, providing a geometric characterization of its U-duality group. The geometric formulation of four-dimensional supergravity is based on a choice of a vertically Riemannian submersion $π$ over $M$ equipped with a flat Ehresmann connection, which determines the non-linear section sigma model of the theory, and a choice of flat symplectic vector bundle $\mathcal{S}$ equipped with a positive complex polarization over the total space of $π$, which encodes the inverse gauge couplings and theta angles of the theory and determines its gauge sector. The classical fields of the theory consist of Lorentzian metrics on $M$, global sections of $π$ and two-forms valued in $\mathcal{S}$ that satisfy an algebraic relation which defines the notion of \emph{twisted} self-duality in four Lorentzian dimensions. We use this geometric formulation to investigate the group of electromagnetic duality transformations of supergravity, also known as the continuous classical U-duality group, which we characterize using a certain short exact sequence of automorphism groups of vector bundles. Moreover, we discuss the general structure of the Killing spinor equations of four-dimensional supergravity, providing several explicit examples and remarking on a few open mathematical problems. This presentation is aimed at mathematicians working in differential geometry.
