Hyperbolic Monopoles, (Semi-)Holomorphic Chern-Simons Theories, and Generalized Chiral Potts Models (2502.17545v1)

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.

A Systematic Exploration of the Hyperbolic Monopole/gCPM Correspondence

The paper investigates the intriguing connection between hyperbolic monopoles and generalized chiral Potts models (gCPM) through the prism of field theory, particularly extending Atiyah and Murray's original observations to a broader class of configurations and systems. Such connections are pivotal in understanding integrable systems' latent symmetries and influence advancing research in both mathematical physics and integrable systems.

Structure of the Paper

The paper is divided into key explorative sections: establishing the correspondence between hyperbolic monopoles and gCPM, integrating gCPM with four-dimensional (4d) Chern-Simons (CS) theory, and probing a theoretical origin of the correspondence rooted in higher-dimensional field theories. Each section aims to bridge traditional methods with modern theoretical frameworks.

Correspondence Between Hyperbolic Monopoles and gCPM

The study begins by formalizing the notion that, while Atiyah and Murray's initial work focused on $SU(2)$ monopoles, similar correspondences hold for $SU(n)$ groups. The authors map out the hyperbolic monopole's spectral data using the spectral parameter of the gCPM. A significant contribution here is demonstrating that the curves associated with hyperbolic $SU(n)$ monopoles can be associated with gCPM configurations, leading to the proposal of magnetic charges $m_1 = \cdots = m_{n-1} = N$, directly linking the physical features of monopoles to the combinatorial structure of gCPMs.

Implementation in 4d CS Theory

Subsequent sections discuss embedding the gCPM framework within the 4d CS theory. The research outlines how such structures on gCPM can be seen as a perturbation of the theory — notably explaining the distinctive non-rapidity difference property of gCPMs, a deviation from conventional model behaviors. Explicit assertions about gauge Lie algebras reveal insights into symmetry properties crucial for solving integrable spin models.

Theoretical Underpinnings from Higher Dimensions

Most innovatively, the paper attempts to elucidate these correspondences' theoretical origins by considering the ramifications of six-dimensional (6d) and ten-dimensional (10d) field theories in a multi-dimensional twist. This consideration of holomorphic Chern-Simons theories, manifested from 6d and 10d dimensions, offers a master framework linking both sides of the correspondence— hyperbolic monopoles and gCPMs— as two realizations of a 10d physics reality.

Key Implications and Future Directions

The paper engages with broad terrains—lie algebras, gauge theory, and integrable models—showing their interwoven nature at the interface of physical and mathematical theory. It highlights the potential of 4d CS theory to encompass integrable models with higher-genus spectral parameters, suggesting a profound way forward in quantum integrability research. There is also speculation about new models stemming from hyperbolic monopoles's symmetry-breaking patterns, hinting at fresh vistas for theoretical and computational exploration.

Conclusions

The exploration successfully broadens the understanding of Atiyah and Murray's monopole correspondence, making a notable advance in integrable systems. By seamlessly weaving together field theory, geometric insights, and algebraic classifications, the paper sets the stage for novel approaches to historically complex problems in mathematical physics. Future work could dive deeper into the ramifications of these frameworks, exploring unknown integrable models' landscape, thus enriching both theoretical insights and applications in modern physics.