论文标题
Casimir Energy and Modulinity在更高维度的保形场理论中
Casimir Energy and Modularity in Higher-dimensional Conformal Field Theories
论文作者
论文摘要
量子场理论（QFT）中的一个重要问题是了解在非平凡拓扑的时空流形上可观察到的结构。当在有限的温度和/或有限体积研究物理系统时，自然会出现这种可观察到的物质，并编码基础微观理论的微妙特性，这些特性在平坦的时空上通常被遮盖。 QFT的局部性意味着这些可观察物可以通过沿空间切片切割和胶片来构建这些可观察的构造，其中至关重要的成分是空间歧管上的希尔伯特空间。在共形场理论（CFT）中，由于操作员状态的对应关系，我们对空间球上的希尔伯特空间有非扰动的理解。但是，考虑更一般的空间流形仍然是一个挑战。在这里，我们在空间歧管$ t^2 \ times \ mathbb {r}^{d-3} $上研究cfts $ d> 2 $，这是球形拓扑以外的最简单歧管之一。我们专注于这个希尔伯特空间中的基态，并分析基态能量的通用性能（也称为Casimir Energy），这是圆环的复杂结构模量$τ$的非平凡函数。 Casimir Energy受到模块化不变性的限制，我们使用$ PSL（2，\ Mathbb {Z}）$光谱理论拼写出来。此外，我们使用kaluza-klein降低CFT的有效野外理论（EFT）在薄的圆环限制中得出了一个简单的通用公式，并从全球intsantons缩小了指数的小校正。我们用众所周知的CFT的明确示例说明了我们的公式，包括$ d = 3 $中的关键$ o（n）$模型和$ d \ geq 3 $中的全息cfts。
An important problem in Quantum Field Theory (QFT) is to understand the structures of observables on spacetime manifolds of nontrivial topology. Such observables arise naturally when studying physical systems at finite temperature and/or finite volume and encode subtle properties of the underlying microscopic theory that are often obscure on the flat spacetime. Locality of the QFT implies that these observables can be constructed from more basic building blocks by cutting-and-gluing along a spatial slice, where a crucial ingredient is the Hilbert space on the spatial manifold. In Conformal Field Theory (CFT), thanks to the operator-state correspondence, we have a non-perturbative understanding of the Hilbert space on a spatial sphere. However it remains a challenge to consider more general spatial manifolds. Here we study CFTs in spacetime dimensions $d>2$ on the spatial manifold $T^2\times \mathbb{R}^{d-3}$ which is one of the simplest manifolds beyond the spherical topology. We focus on the ground state in this Hilbert space and analyze universal properties of the ground state energy, also commonly known as the Casimir energy, which is a nontrivial function of the complex structure moduli $τ$ of the torus. The Casimir energy is subject to constraints from modular invariance on the torus which we spell out using $PSL(2,\mathbb{Z})$ spectral theory. Moreover we derive a simple universal formula for the Casimir energy in the thin torus limit using the effective field theory (EFT) from Kaluza-Klein reduction of the CFT, with exponentially small corrections from worldline instantons. We illustrate our formula with explicit examples from well-known CFTs including the critical $O(N)$ model in $d=3$ and holographic CFTs in $d\geq 3$.
