Kazakov-Migdal model on the Graph and Ihara Zeta Function

Abstract

We propose the Kazakov-Migdal model on graphs and show that, when the parameters of this model are appropriately tuned, the partition function is represented by the unitary matrix integral of an extended Ihara zeta function, which has a series expansion by all non-collapsing Wilson loops with their lengths as weights. The partition function of the model is expressed in two different ways according to the order of integration. A specific unitary matrix integral can be performed at any finite $N$ thanks to this duality. We exactly evaluate the partition function of the parameter-tuned Kazakov-Migdal model on an arbitrary graph in the large $N$ limit and show that it is expressed by the infinite product of the Ihara zeta functions of the graph.

33 pages, 6 figures, references added, a figure added, a comment on physical model added, typos corrected

References in corpus (14)

• Integration with respect to the Haar measure on unitary, orthogonal and symplectic group
• Free Fermions and Thermal AdS/CFT
• Exact Results in Quiver Quantum Mechanics and BPS Bound State Counting
• Topologically Twisted $N=(2,2)$ Supersymmetric Yang-Mills Theory on Arbitrary Discretized Riemann Surface
• Coulomb Branch Localization in Quiver Quantum Mechanics
• Divergences in QED on a Graph
• Anomaly and Sign problem in $\mathcal{N}=(2,2)$ SYM on Polyhedra : Numerical Analysis
• Vortices and Superfields on a Graph
• Supersymmetric Gauge Theory on the Graph
• Exact Results in Discretized Gauge Theories
• The Volume of the Quiver Vortex Moduli Space
• Graph Zeta Function and Gauge Theories
• Seiberg Duality, Quiver Gauge Theories, and Ihara Zeta Function
• Simple Models in Supersymmetric Quantum Mechanics on a Graph

Cited by in corpus (2)

• Equivalence of lattice operators and graph matrices
• Gross-Witten-Wadia Phase Transition in Induced QCD on the Graph