Topological entanglement and number theory

Abstract

The recent developments in the study of topological multi-boundary entanglement in the context of 3d Chern-Simons theory (with gauge group $G$ and level $k$) suggest a strong interplay between entanglement measures and number theory. The purpose of this note is twofold. First, we introduce a $q$-deformed version of the Witten zeta function using the Chern-Simons theory at level $k$. We analyze the large $k$ limit of this function and show that it converges to an integer multiple of the classical Witten zeta function of $G$, where the integer multiple is precisely the order of the center of the group. This analysis provides an alternative way to compute the classical zeta functions, and we present some examples. Next, we study the quantum state associated with the $S^3$ complement of torus links of type $T_{p,p}$ and show that we can write the Rényi entropies at finite $k$ in terms of $q$-deformed Witten zeta functions. Using our first result, we obtain the $k\to \infty$ limit of the Rényi entropies and find that the entropies converge to finite values, which can be written in terms of the classical Witten zeta functions evaluated at positive integers. Since Witten zeta functions naturally appear in the symplectic volumes of moduli spaces of flat connections on Riemann surfaces, we give a geometric interpretation of the $k\to \infty$ limit of the Rényi and entanglement entropies in terms of these volumes. The results of this paper reveal an intriguing connection between topological entanglement, number-theoretic structures arising from Witten zeta functions, and the geometry of moduli spaces.

Version-3: Matches with published version, Fig-4 and Fig-5 corrected

References in corpus (18)

• Quantum entanglement
• Topological entanglement entropy
• Detecting topological order in a ground state wave function
• Topological Entanglement Entropy in Chern-Simons Theories and Quantum Hall Fluids
• Torus knots and mirror symmetry
• Multi-Boundary Entanglement in Chern-Simons Theory and Link Invariants
• Chern-Simons Invariants of Torus Links
• Entanglement Entropy and the Colored Jones Polynomial
• Partition functions on 3d circle bundles and their gravity duals
• Entanglement on linked boundaries in Chern-Simons theory with generic gauge groups
• Double parton distributions in the pion in the Nambu and Jona-Lasinio model
• Linking Entanglement and Discrete Anomaly
• Circuit Complexity of Knot States in Chern-Simons theory
• Entanglement on multiple $S^2$ boundaries in Chern-Simons theory
• $3d$ One-form Mixed Anomaly and Entanglement Entropy
• Semiclassical limit of topological Rényi entropy in $3d$ Chern-Simons theory
• Musings on SVD and pseudo entanglement entropies
• Multi-boundary entanglement in Chern-Simons theory with finite gauge groups