Generalization of lattice Dirac operator index

Abstract

We provide a comprehensive lattice formulation of various types of the Dirac operator indices, employing $K$-theory to classify the Wilson Dirac operator via its spectral flow. In contrast to the index of the overlap Dirac operator defined through the Ginsparg-Wilson relation, which is restricted to flat tori in even dimensions, our formulation offers several key advantages: 1) It can be applied straightforwardly to the Atiyah-Patodi-Singer index for manifolds with boundary. 2) The boundary can be curved, allowing for the inclusion of gravitational background effects. 3) The mod-2 index in both even and odd dimensions can be defined as a natural extension of the same formulation. In this talk, we present the mathematical proof and provide numerical evidence supporting the formulation.

Figures (3)

• Figure 1: Left panel : continuum eigenvalue spectrum of the massive Dirac operator where the mass term is varied by $-sm$ with $s\in [-1,1]$. Right panel : an example of deformed spectrum by chiral symmetry breaking.
• Figure 2: The eigenvalue spectrum the domain-wall Dirac operator with a $U(1)$ flux $Q'=2$ (left panel) and $Q'=-1.75$ (right) inside a circular domain-wall at $r=10$. We assign $\epsilon=+1$ for $r<10$ (and $\epsilon=-1$ for $r\ge 10$). In the both cases the APS index theorem on a disk is well reproduced.
• Figure 3: The eigenvalue spectrum of the free domain-wall Dirac operator with periodic boundary conditions. For the left panel, the domain-wall mass term is assigned as $\epsilon=+1$ for $r<10$ (and $\epsilon=-1$ for $r\ge 10$)(Disk). For the right-panel, we set $\epsilon=-1$ for $r<10$ (and $\epsilon=+1$ for $r\ge 10$) ($T^2$ with $S^1$ boundary).