阿贝尔规范理论的对偶协变几何与DSZ量化

IF 1 4区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL Advances in Theoretical and Mathematical Physics Pub Date : 2021-01-18 DOI:10.4310/atmp.2022.v26.n7.a5
C. Lazaroiu, C. Shahbazi
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引用次数: 1

摘要

在任意拓扑的有向洛伦兹四流形$(M,g)$上发展了具有一般对偶结构的经典阿贝规范理论的Dirac-Schwinger-Zwanziger (DSZ)量子化,从而通过主束上的连接得到了这类理论的对偶协变几何表达式。我们通过将理论的场强限定为那些定义了在积分辛空间的群上取值的局部系统的二次上同调中起源的类的场强来实现DSZ条件。我们证明了这样的场强是定义在主束P上的连接$\mathcal{A}$的曲率,其结构群$G$是整仿射辛环面的自同构的不连通群。$G$的单位元的连通分量是一个环面群,而它的连通分量群是一个修正的Siegel模群。该公式包含了相等基础上的电磁和磁电规范电位,并通过曲率$\mathcal{a}$的一阶极化自对偶条件描述了运动方程。该条件涉及$(M,g)$的Hodge算子与由$P$确定的对偶结构的驯服的组合,其选择编码了理论的自耦合。这种描述让人想起四维欧几里得瞬子理论,尽管我们考虑的是洛伦兹特征中的二阶导数理论。我们用这个公式刻画了阿贝尔规范论中对偶群的层次,给出了电磁对偶群作为P规范群的离散残馀的规范理论描述。我们还对极化自对偶条件进行了类时化简,得到了一类新的Bogomolny方程,并在特定情况下得到了显式解。
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The duality covariant geometry and DSZ quantization of abelian gauge theory
We develop the Dirac-Schwinger-Zwanziger (DSZ) quantization of classical abelian gauge theories with general duality structure on oriented Lorentzian four-manifolds $(M,g)$ of arbitrary topology, obtaining, as a result, the duality-covariant geometric formulation of such theories through connections on principal bundles. We implement the DSZ condition by restricting the field strengths of the theory to those which define classes originating in the degree-two cohomology of a local system valued in the groupoid of integral symplectic spaces. We prove that such field strengths are curvatures of connections $\mathcal{A}$ defined on principal bundles $P$ whose structure group $G$ is the disconnected group of automorphisms of an integral affine symplectic torus. The connected component of the identity of $G$ is a torus group, while its group of connected components is a modified Siegel modular group. This formulation includes electromagnetic and magnetoelectric gauge potentials on an equal footing and describes the equations of motion through a first-order polarized self-duality condition for the curvature of $\mathcal{A}$. The condition involves a combination of the Hodge operator of $(M,g)$ with a taming of the duality structure determined by $P$, whose choice encodes the self-couplings of the theory. This description is reminiscent of the theory of four-dimensional euclidean instantons, even though we consider a two-derivative theory in Lorentzian signature. We use this formulation to characterize the hierarchy of duality groups of abelian gauge theory, providing a gauge-theoretic description of the electromagnetic duality group as the discrete remnant of the gauge group of $P$. We also perform the time-like reduction of the polarized self-duality condition to a Riemannian three-manifold, obtaining a new type of Bogomolny equation which we solve explicitly in a particular case.
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来源期刊
Advances in Theoretical and Mathematical Physics
Advances in Theoretical and Mathematical Physics 物理-物理:粒子与场物理
CiteScore
2.20
自引率
6.70%
发文量
0
审稿时长
>12 weeks
期刊介绍: Advances in Theoretical and Mathematical Physics is a bimonthly publication of the International Press, publishing papers on all areas in which theoretical physics and mathematics interact with each other.
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