Mathematical Application Research Team Meeting #18

Mathematical Application Research Team is delighted to welcome Prof. Shinichiroh Matsuo from Nagoya University for an upcoming team meeting. We warmly invite everyone to join us to hear his talk.

Please note that, exceptionally, this meeting will be held at Okochi Hall. Immediately following Prof. Matsuo’s talk, Prof. Hiraku Nakajima will give a seminar entitled “Coulomb branches in supersymmetric gauge theories.” We encourage participants to join us for both talks.

Title: The Index of Wilson--Dirac Operators Revisited: Beyond Regular Lattices on Flat Tori

Abstract:
This is joint work with Mikio Furuta.

Our ultimate goal is to discretise Seiberg--Witten theory. In view of PL = DIFF in dimension four, we would like to construct something like a PL Seiberg--Witten theory.

As a first step towards this goal, we have studied the discretisation of the analytic index of Dirac operators. In earlier work, using K-theoretic methods, we showed that the index of the Wilson--Dirac operator associated with a sufficiently fine regular lattice on a flat torus agrees with that of the corresponding continuum Dirac operator. However, this lattice-index construction was confined to regular lattices on flat tori.

In this talk, I will introduce “mesoscopic” Wilson--Dirac operators on arbitrary closed Riemannian manifolds. They form a family of bounded integral-difference operators on \(L^2\), parametrised by a mesoscopic length \(a>0\), and they converge, in a suitable sense, to the continuum Dirac operator as \(a \to 0\).

The mesoscopic operators are self-adjoint and Fredholm, but not odd, so the usual graded Fredholm index is unavailable. We prove that the “index” of a mesoscopic Wilson--Dirac operator agrees with the index of the corresponding continuum Dirac operator when \(a\) is sufficiently small.

Replacing the integrals in our mesoscopic operators by Riemann sums produces finite-dimensional lattice approximations on general closed Riemannian manifolds. The mesoscopic Wilson--Dirac operators thus provide a bridge between the microscopic continuum Dirac operator and the macroscopic lattice Dirac operators.