The Atiyah–Patodi–Singer index on a lattice

Abstract

We propose a nonperturbative formulation of the Atiyah–Patodi–Singer (APS) index in lattice gauge theory in four dimensions, in which the index is given by the $\eta$ invariant of the domain-wall Dirac operator. Our definition of the index is always an integer with a finite lattice spacing. To verify this proposal, using the eigenmode set of the free domain-wall fermion we perturbatively show in the continuum limit that the curvature term in the APS theorem appears as the contribution from the massive bulk extended modes, while the boundary $\eta$ invariant comes entirely from the massless edge-localized modes.

A lattice formulation of the Atiyah-Patodi-Singer index

“…The fact that the index of the overlap Dirac operator is the same as the η-invariant of the massive Wilson-Dirac operator strongly supports a hypothesis that the naive discretization of the η-invariant of the domain-wall Dirac operator agrees with the APS index in the continuum limit. As shown in [15], the original APS boundary condition is difficult to realize on a lattice with the overlap Dirac operator. Also, any boundary condition would break the GW relation, which makes it impossible to define the APS index on a lattice by the chiral zero modes.…”

“…Since the index is defined on a compact manifold, we should consider in a compact space but here we proceed as if we were on an infinite lattice to make the presentation simpler. See [15] for more precise treatment. The key of this work is to find a good complete set to evaluate the η-invariant as…”

A physicist-friendly reformulation of the Atiyah-Patodi-Singer index and its mathematical justification

“…Nevertheless, as we will show below, there is no problem in giving a fermionic integer, which coincides with the original index. This is true even on a lattice, and we proposed a non-perturbative formulation of the APS index in lattice gauge theory in [10]. In fact, the η invariant of the massive Dirac operator gives a unified view of the index theorems including their lattice version.…”

“…In fact, the η invariant of the massive Dirac operator gives a unified view of the index theorems including their lattice version. See also Kawai's contribution [11] to these proceedings.…”

The Anomaly Inflow of the domain-wall fermion in odd dimension

“…To realize this situation, the domain-wall fermion [2,3] was used and it produced the edge-localized mode and bulk Chern-Simons action. The anomaly inflow now covers a wide area of physics such as chiral gauge theory on a lattice [4,5], topological matters [6,7] and index theorem [8,9,10].…”