Speaker
Description
We propose a construction of fermions on arbitrary graphs, regarded as discrete space-time. In our framework, the Dirac operator for fermions on the graph is formulated in terms of an incidence matrix deformed by specific parameters. Consequently, the partition function of the model is given by the inverse of the graph zeta function (Ihara zeta function). We explore the physical significance of the graph zeta function, showing that it provides a generating function for the possible fermion cycles (states) on the graph. Furthermore, utilizing concepts such as covering graphs and Artin-Ihara $L$-functions, we analyze fermions on lattices (grid graphs) with periodic boundary conditions and demonstrate that the fermion-doubling problem is avoided in our construction. Additionally, we reveal a significant connection between the partition function of this model and that of the Ising model by introducing winding numbers associated with graph cycles. Finally, we examine the index theorem applicable to graphs within our framework. (This talk is based on https://arxiv.org/abs/2501.08803 .)