Hyperbolic Monopoles, (Semi-)Holomorphic Chern-Simons Theories, and Generalized Chiral Potts Models

Abstract

We study the relation between spectral data of magnetic monopoles in hyperbolic space and the curve of the spectral parameter of generalized chiral Potts models (gCPM) through the lens of (semi-)holomorphic field theories. We realize the identification of the data on the two sides, which we call the hyperbolic monopole/gCPM correspondence. For the group $\text{SU}(2)$, this correspondence had been observed by Atiyah and Murray in the 80s. Here, we revisit and generalize this correspondence and establish its origin. By invoking the work of Murray and Singer on hyperbolic monopoles, we first generalize the observation of Atiyah and Murray to the group $\text{SU}(n)$. We then propose a technology to engineer gCPM within the 4d Chern-Simons (CS) theory, which explains various features of the model, including the lack of rapidity-difference property of its R-matrix and its peculiarity of having a genus$\,\ge 2$ curve of the spectral parameter. Finally, we investigate the origin of the correspondence. We first clarify how the two sides of the correspondence can be realized from the 6d holomorphic CS theory on $\mathbb{P}S(M)$, the projective spinor bundle of the Minkowski space $M=\mathbb{R}^{1,3}$, for hyperbolic $\text{SU}(n)$-monopoles, and the Euclidean space $M=\mathbb{R}^4$, for the gCPM. We then establish that $\mathbb{P}S(M)$ can be holomorphically embedded into $\mathbb{P}S(\mathbb{C}^{1,3})$, the projective spinor bundle of $\mathbb{C}^{1,3}$, of complex dimension five with a fixed complex structure. We finally explain how the 6d CS theory on $\mathbb{P}S(M)$ can be realized as the dimensional reduction of the 10d holomorphic CS theory on $\mathbb{P}S(\mathbb{C}^{1,3})$. As the latter theory is only sensitive to the complex structure of $\mathbb{P}S(\mathbb{C}^{1,3})$, which has been fixed, we realize the correspondence as two incarnations of the same physics in ten dimensions.