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

TL;DR

The paper establishes a comprehensive link between hyperbolic SU(n) monopoles in $\mathbb{H}^3$ and generalized chiral Potts models (gCPM) via (semi-)holomorphic 4d CS/6d hCS theories, revealing a two-fold realization of the same physics in ten dimensions. It first generalizes the Atiyah–Murray observation from SU(2) to SU(n), showing a precise correspondence between the gCPM spectral curve $\Sigma_{N,n}$ and the monopole spectral data through $n-1$ curves $\Sigma_{N,n}^{(i)}$ on $\mathbb{P}^1\times\mathbb{P}^1$, with magnetic charges $m_i=N$ and vanishing asymptotic Higgs values. It then engineers the gCPM inside the 4d CS framework by encoding the spectral-parameter curve as a branched cover of $\mathbb{P}^1$ and constraining the 1-form $\omega_{\mathbb{P}^1}$ via $\mathbb{Z}_N^{n-1}$ symmetry, explaining the lack of rapidity-difference property and the higher-genus spectral curve. Finally, the work uncovers the origin of the correspondence through a chain of dimensional reductions: 6d holomorphic CS theory on projective spinor bundles yields hyperbolic monopoles and gCPM; this extends to a 10d holomorphic CS theory whose fixed complex structure makes the two pictures equivalent, providing a unifying ten-dimensional perspective.

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.

Figures (8)

• Figure 1: A graphical representation of the R-matrix of an integrable model.
• Figure 2: A graphical representation of the Yang--Baxter equation.
• Figure 3: The configuration of Wilson lines in the 4d CS theory for the computation of R-matrix. The wavy lines denote the gluon exchange and the gray circle shows the interaction in the bulk.
• Figure 4: The solid and dashed lines represent the lattice $\mathcal{L}$ and its median lattice $\mathcal{L}'$, respectively, for gCPM. The spins are located on the square lattice while rapidities are located on its medial lattice.
• Figure 5: The two possible Boltzmann weights of the gCPM.

Theorems & Definitions (9)

• Proposition
• Theorem 3.1: The Twistor Correspondence for Non-Integral Hyperbolic $\text{SU}(n)$-Monopoles
• Proposition 4.1
• Proposition 6.1
• proof
• proof
• proof
• Proposition 6.2
• proof