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تسجيل مستخدم جديدThe Atiyah-Patodi-Singer index and domain-wall fermion Dirac operators
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We introduce a mathematician-friendly formulation of the physicist-friendly derivation of the Atiyah-Patodi-Singer index of our previous paper. Our viewpoint sheds some new light on the interplay among the Atiyah-Patodi-Singer boundary condition, domain-wall fermions, and edge modes.
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The Atiyah-Patodi-Singer(APS) index theorem attracts attention for understanding physics on the surface of materials in topological phases. The mathematical set-up for this theorem is, however, not directly related to the physical fermion system, as
it imposes on the fermion fields a non-local boundary condition known as the APS boundary condition by hand, which is unlikely to be realized in the materials. In this work, we attempt to reformulate the APS index in a physicist-friendly way for a simple set-up with $U(1)$ or $SU(N)$ gauge group on a flat four-dimensional Euclidean space. We find that the same index as APS is obtained from the domain-wall fermion Dirac operator with a local boundary condition, which is naturally given by the kink structure in the mass term. As the boundary condition does not depend on the gauge fields, our new definition of the index is easy to compute with the standard Fujikawa method.
We compute the index of a Callias-type operator with APS boundary condition on a manifold with compact boundary in terms of combination of indexes of induced operators on a compact hypersurface. Our result generalizes the classical Callias-type index theorem to manifolds with compact boundary.
We propose a non-perturbative formulation of the Atiyah-Patodi-Singer(APS) index in lattice gauge theory, 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 fi
nite 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.
We study the index of the APS boundary value problem for a strongly Callias-type operator D on a complete Riemannian manifold $M$. We show that this index is equal to an index on a simpler manifold whose boundary is a disjoint union of two complete m
anifolds $N_0$ and $N_1$. If the dimension of $M$ is odd we show that the latter index depends only on the restrictions $A_0$ and $A_1$ of $D$ to $N_0$ and $N_1$ and thus is an invariant of the boundary. We use this invariant to define the relative eta-invariant $eta(A_1,A_0)$. We show that even though in our situation the eta-invariants of $A_1$ and $A_0$ are not defined, the relative eta-invariant behaves as if it was the difference $eta(A_1)-eta(A_0)$.
We study the Cauchy data spaces of the strongly Callias-type operators using maximal domain on manifolds with non-compact boundary, with the aim of understanding the Atiyah-Patodi-Singer index and elliptic boundary value problems.
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