[论文解读] Aspects of higher spin Hamiltonian dynamics: Conformal geometry, duality and charges

Amaury Leonard|arXiv (Cornell University)|Sep 3, 2017

Superconducting Materials and Applications被引用 3

一句话总结

本论文为平坦时空与反德西特（AdS）时空中的自由高自旋规范场构建了一个哈密顿框架，建立了共形几何、扭曲自对偶性与表面电荷之间的联系。论文引入了编码共形不变性与电-磁对偶性的预势（prepotentials），实现了对任意自旋的玻色子与费米子场的完整哈密顿形式化，并显式构造了AdS时空中的表面电荷，同时将形式化推广至混合对称张量。

ABSTRACT

We have studied free higher spin gauge fields through an investigation of their Hamiltonian dynamics. Over a flat space-time, their Hamiltonian constraints were identified and solved through the introduction of prepotentials, enjoying both linearized generalized diffeomorphism and linearized generalized Weyl rescaling gauge invariance, motivating our study of conformal invariants for higher spins. We built these with the Cotton tensor, whose properties (tracelessness, symmetry, divergencelessness; completeness, invariance) we established. With these geometric tools, a first order action was written down in terms of the prepotentials. It is manifestly invariant under electric-magnetic duality which, with the gauge freedom of the prepotentials, completely fixes the action. This action is associated to twisted self-duality conditions. With an interest in supersymmetric extensions, we began to extend this study to fermions, similarly analyzing the spin $5/2$ massless free field, whose prepotential also enjoys conformal gauge invariance. The spin $2$-spin $5/2$ supermultiplet was considered, and a rigid symmetry of its action (a chirality-duality rotation) was built to commute with supersymmetry. We also investigated the properties of a mixed symmetry field on a flat six-dimensional space-time, the so-called chiral $(2,2)$-form: Hamiltonian analysis, prepotentials, and a first order action associated to self-duality conditions. Finally, we studied both fermionic and bosonic higher spin surface charges over a constantly curved background space-time. The Hamitonian constraints are the generators of gauge transformations. Plugging into them appropriate values of the gauge parameters (imposing a physical variation of the fields), their finite and non-vanishing on-shell values were computed and recognized as conserved charges of the theory. Their algebra was checked to be abelian.

研究动机与目标

• 为平坦时空与AdS时空中的自由高自旋规范场发展系统性的哈密顿形式化，超越标准的Fronsdal作用量。
• 通过显现出共形规范不变性与电-磁对偶性的预势识别并求解约束条件。
• 将扭曲自对偶性条件推广至任意自旋场，以一阶形式重写运动方程。
• 计算AdS时空中高自旋场的表面电荷，将其与渐近对称性及共形基灵张量联系起来。
• 将形式化推广至费米子场与混合对称张量，包括六维中的(2,2)-形式，作为(4,0)超引力理论的关键实例。

提出的方法

• 通过狄拉克约束系统形式化，从Fronsdal作用量推导哈密顿量与约束，识别共轭动量及一、二级约束。
• 通过变量变换引入预势，以满足动量与哈密顿量约束，同时保持规范不变性。
• 利用预势构造一阶作用量，显式满足$SO(2)$电-磁对偶性与共形不变性。
• 将扭曲自对偶性条件应用于将运动方程重写为电场与磁场之间的对偶关系，消除拉格朗日乘子。
• 利用共形基灵张量与旋量-张量定义AdS时空中的边界条件与渐近对称性。
• 将哈密顿形式化应用于超重力（自旋-2与自旋-5/2）及六维中的(2,2)-形式，推导出具有手征与非手征形式的预势作用量。

实验结果

研究问题

• RQ1 如何通过保持共形不变性与对偶性的预势，在平坦时空中原则性地表述任意自旋玻色子场的哈密顿动力学？
• RQ2 在三维空间中，Cotton张量在高自旋场的共形不变量中扮演何种角色？
• RQ3 高自旋理论中的扭曲自对偶性条件如何与一阶哈密顿作用量及电-磁对偶性相关联？
• RQ4 预势形式化能否推广至费米子高自旋场（如超重力中的自旋-5/2场）？其如何编码超对称性？
• RQ5 AdS时空中的高自旋场表面电荷如何计算？其与共形基灵张量及渐近对称性的关系为何？

主要发现

• 证明Cotton张量是三维高自旋场的完整共形不变量集合：其为对称、无迹、无散，并完全表征共形平坦性。
• 对任意自旋$s$，通过变换于共形规范对称性的预势求解动量与哈密顿量约束，导出唯一在$SO(2)$电-磁对偶性下不变的哈密顿作用量。
• 高自旋理论中的扭曲自对偶性条件被重述为一阶系统，将电场与磁场（符号除外）等同，消除辅助场并获得一致的变分原理。
• AdS中高自旋场的表面电荷显式地以场的边界值与共形基灵张量表示，为研究Vasiliev理论中类黑洞解提供了框架。
• 预势形式化成功推广至费米子场，如超重力中的自旋-5/2场，其以统一方式编码超对称性与对偶性对称性。
• 六维中的(2,2)-形式被证明可导出基于扭曲自对偶性的预势作用量，具有手征与非手征形式，为(4,0)理论的量化提供了可能路径。