Phases and Duality in Fundamental Kazakov-Migdal Model on the Graph






松浦 壮 教授(慶応義塾大学)






In this talk, we will explore the fundamental Kazakov-Migdal (FKM) model on a generic graph, where the partition function is represented by the Ihara zeta function weighted by unitary matrices. The effective action of the FKM model is described by a summation of all Wilson loops on the graph, which can be regarded as an extension of the usual Wilson action in lattice gauge theory. We show that the FKM model on regular graphs exhibits an exact strong/weak coupling duality, reflecting the functional equation of the Ihara zeta function. Although this duality is not precise for irregular graphs, we show that the effective action in the large coupling region can also be represented by a summation of Wilson loops, similar to that in the small coupling region. We also discuss the stability of the model and show that the FKM model becomes unstable in the critical strip of the Ihara zeta function. Interestingly, the FKM model universally exhibits the so-called Gross-Witten-Wadia (GWW) phase transitions. By comparing the FKM model with the GWW model, we estimate the phase structure of the FKM model in both small and large coupling regions, which is validated through numerical simulations. For the benefit of graduate students, we will begin the seminar with an introductory explanation of lattice gauge theory and graph theory, providing a foundation for understanding the subsequent discussion.