洛伦兹流形上的格林双曲复形
Green hyperbolic complexes on Lorentzian manifolds
AI总结：
本研究提出格林双曲复形作为格林双曲算子的同调推广，通过格林同伦定义该复形并证明其唯一性，同时推导了其拟同构、泊松结构构造及相容性等关键同调性质，为规范理论导出临界轨迹研究提供新工具。
摘要：
我们对格林双曲算子进行了同调推广，提出格林双曲复形（Green hyperbolic complexes），它涵盖了洛伦兹号差下规范理论二次作用泛函的众多导出临界轨迹实例。我们通过推广推迟和超前格林算子（即推迟和超前格林同伦）来定义格林双曲复形，并证明这些同伦在可缩选择空间的意义下是唯一的。我们证明了格林双曲算子最相关特性的同调推广版本：（1）推迟减超前上链映射是拟同构；（2）格林双曲复形上的微分配对（推广了通常的逐纤维度量）可导出协变和固定时间泊松结构；（3）推迟减超前上链映射与这些泊松结构在同伦意义下相容。
英文摘要：
We develop a homological generalization of Green hyperbolic operators, called Green hyperbolic complexes, which cover many examples of derived critical loci for gauge-theoretic quadratic action functionals in Lorentzian signature. We define Green hyperbolic complexes through a generalization of retarded and advanced Green's operators, called retarded and advanced Green's homotopies, which are shown to be unique up to a contractible space of choices. We prove homological generalizations of the most relevant features of Green hyperbolic operators, namely that (1) the retarded-minus-advanced cochain map is a quasi-isomorphism, (2) a differential pairing (generalizing the usual fiber-wise metric) on a Green hyperbolic complex leads to covariant and fixed-time Poisson structures and (3) the retarded-minus-advanced cochain map is compatible with these Poisson structures up to homotopy.
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