Seminars
We explore the relationship between lattice field theory and graph theory, placing special emphasis on the interplay between Dirac and scalar lattice operators and matrices within the realm of spectral graph theory. Beyond delving into fundamental concepts of spectral graph theory, such as adjacency and Laplacian matrices, we introduce a novel matrix named as "anti-symmetrized adjacency matrix", specifically tailored for cycle digraphs ($T^1$ lattice) and simple directed paths ($B^1$ lattice).
The nontrivial relation between graph theory matrices and lattice operators shows that the graph Laplacian matrix mirrors the lattice scalar operator and the Wilson term in lattice fermions, while the anti-symmetrized adjacency matrix, along with its extensions to higher dimensions, are equivalent to naive lattice Dirac operators.
Building upon these connections, we provide rigorous proofs for two key assertions:
(i) The count of zero-modes in a free lattice scalar operator coincides with the zeroth Betti number of the underlying graph (lattice).
(ii) The maximum count of Dirac zero-modes in a free lattice fermion operator is equivalent to the cumulative sum of all Betti numbers when the $D$-dimensional graph results from a cartesian product of cycle digraphs ($T^1$ lattice) and simple directed paths ($B^1$ lattice).
Semi-classical approximation is one of the most commonly used techniques in quantum theories. As it produces factorially divergent series, they have to be analyzed in the context of the resurgence theory. In this talk, I will start with a general discussion on divergent series. After explaining how the non-perturbative quantities are hidden in the divergent behaviour and how they can be obtained via Borel summation, as an example, I will discuss 1 loop effective action in Schwinger's proper time formalism and the non-perturbative pair production in the presence of uniform electric field. Then, in order to handle general non-uniform fields, I will introduce a recursive method to compute a (proper) time dependent perturbative expansion from which the non-perturbative effects are obtained by a Borel-like summation. In the second part of the talk, I will discuss how the recursive expansion is connected to the semi-classical expansion in the WKB formalism and how it offers a generalization to higher dimensional problems which can not be handled by the standard WKB integrals. I will finish the talk with an example on the effective action of the time dependent periodic electric fields in connection with the exact WKB formalism.