Abstract
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.
Highlights
The topology of gauge fields, which plays a special role in particle physics, is not well defined in lattice gauge theory, where the space-time is discretized and the notion of the “manifold” is lost
A breakthrough was made by Hasenfratz et al [2] where they formulated the Atiyah–Singer (AS) index theorem [3] on a finite lattice, using a lattice Dirac operator obeying the Ginsparg–Wilson (GW) relation [4]
We have shown that the η invariant of the domain-wall fermion Dirac operator converges to the Atiyah–Patodi–Singer index in the continuum limit
Summary
The topology of gauge fields, which plays a special role in particle physics, is not well defined in lattice gauge theory, where the space-time is discretized and the notion of the “manifold” is lost. The APS index theorem has been drawing attention from physics, as it was pointed out that the theorem describes the bulk–edge correspondence of topological insulators [16], through the cancellation of the time-reversal symmetry anomaly Motivated by this fact, three of the authors proposed a new formulation [17] of the APS index in four-dimensional continuum theory. Eq (4) gives a good definition of the η invariant of HW, which is a regularized number of positive eigenmodes minus the number of negative eigenmodes This fact is suggestive, in that the index may be defined by the massive Wilson–Dirac operator, without chiral symmetry or the GW relation at all. We treat the momenta in other directions as continuous
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