arXiv 2008.018292020-08hep-thmath-phmath.MP

四维Chern-Simons理论的同伦分析与可积场论

Homotopical analysis of 4d Chern-Simons theory and integrable field theories

Marco Benini, Alexander Schenkel, Benoit Vicedo

备注：: 27 pages; v2: Final version accepted for publication in Communications in Mathematical Physics

摘要：

本文对$\mathbb{C}P^1$上的任意亚纯$1$ -形式$ω$给出了$\mathbb{R}^2 \times \mathbb{C}P^1$上的$4$ -维Chern-Simons理论的详细研究.使用同伦理论的技术，有限规范变换下的适当正则化版本的科斯特洛和山崎提出的行动的行为进行了研究。它的规范不变性与位于$ω$极点的表面缺陷上的边界条件有关，这些边界条件由某一缺陷李代数的各向同性李子代数确定。证明了满足这种边界条件的域的广群等价于通过同伦拉回实现边界条件的广群，从而导致边缘模式的出现。后一种观点是用来澄清如何从$4$维陈-西蒙斯理论产生的可积场论。

英文摘要：

This paper provides a detailed study of $4$-dimensional Chern-Simons theory on $\mathbb{R}^2 \times \mathbb{C}P^1$ for an arbitrary meromorphic $1$-form $ω$ on $\mathbb{C}P^1$. Using techniques from homotopy theory, the behaviour under finite gauge transformations of a suitably regularised version of the action proposed by Costello and Yamazaki is investigated. Its gauge invariance is related to boundary conditions on the surface defects located at the poles of $ω$ that are determined by isotropic Lie subalgebras of a certain defect Lie algebra. The groupoid of fields satisfying such a boundary condition is proved to be equivalent to a groupoid that implements the boundary condition through a homotopy pullback, leading to the appearance of edge modes. The latter perspective is used to clarify how integrable field theories arise from $4$-dimensional Chern-Simons theory.

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