拓扑量子引力中的 Perelman Ricci 流
Perelman's Ricci Flow in Topological Quantum Gravity
AI总结：
该研究在拓扑非相对论量子引力中找到使 Perelman Ricci 流成为路径积分局域化方程的区域，并建立熵泛函、τ 函数与场论结构的对应。
摘要：
我们找到了近期构建的拓扑非相对论量子引力中的一个区域，在该区域中，Riemann 流形上的 Perelman Ricci 流方程恰好作为路径积分中的局域化方程出现。在这种物理学与数学之间的映射中，Perelman 的伸缩子由我们的流逝函数扮演。当我们动能项中的 λ 参数趋近 λ→−∞ 时，Perelman 的局部固定体积条件会动力学地涌现。将度量流与伸缩子流解耦的 DeTurck 技巧，只不过是空间微分同胚规范对称性的一个规范固定条件。我们展示了 Perelman 的 F 和 W 熵泛函如何与我们的超势相关联。我们解释了 Perelman 的 τ 函数的起源，该函数出现在收缩孤子的 W 熵泛函中，它是与时间平移和空间重标度相关的 Goldstone 模式：事实上，在我们的量子引力中，Perelman 的 τ 最终扮演了各向异性尺度变换伸缩子的角色。Perelman 流与我们拓扑量子引力中局域化方程之间的映射，需要一次有趣的场重定义，其中包括度量的重新标架。随着 Perelman 方程被嵌入拓扑量子引力，关于 Ricci 流的大量数学结果现在可以被引入物理学，并用量子场论的语言重新表述。
英文摘要：
We find the regime of our recently constructed topological nonrelativistic quantum gravity, in which Perelman's Ricci flow equations on Riemannian manifolds appear precisely as the localization equations in the path integral. In this mapping between physics and mathematics, the role of Perelman's dilaton is played by our lapse function. Perelman's local fixed volume condition emerges dynamically as the $λ$ parameter in our kinetic term approaches $λ\to-\infty$. The DeTurck trick that decouples the metric flow from the dilaton flow is simply a gauge-fixing condition for the gauge symmetry of spatial diffeomorphisms. We show how Perelman's ${\cal F}$ and ${\cal W}$ entropy functionals are related to our superpotential. We explain the origin of Perelman's $τ$ function, which appears in the ${\cal W}$ entropy functional for shrinking solitons, as the Goldstone mode associated with time translations and spatial rescalings: In fact, in our quantum gravity, Perelman's $τ$ turns out to play the role of a dilaton for anisotropic scale transformations. The map between Perelman's flow and the localization equations in our topological quantum gravity requires an interesting redefinition of fields, which includes a reframing of the metric. With this embedding of Perelman's equations into topological quantum gravity, a wealth of mathematical results on the Ricci flow can now be imported into physics and reformulated in the language of quantum field theory.
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