Related papers: A noncommutative Atiyah-Patodi-Singer index theore…
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…
The Atiyah-Patodi-Singer index theorem describes the bulk-edge correspondence of symmetry protected topological insulators. The mathematical setup for this theorem is, however, not directly related to the physical fermion system, as it…
The Atiyah-Singer index theorem on a closed manifold is well understood and appreciated in physics. On the other hand, the Atiyah-Patodi-Singer index, which is an extension to a manifold with boundary, is physicist-unfriendly, in that it is…
We define an analytical index map and a topological index map for conical pseudomanifolds. These constructions generalize the analogous constructions used by Atiyah and Singer in the proof of their topological index theorem for a smooth,…
Recently, the Atiyah-Patodi-Singer(APS) index theorem attracts attention for understanding physics on the surface of materials in topological phases. Although it is widely applied to physics, the mathematical set-up in the original APS…
We give an Atiyah-Patodi-Singer index theory construction of the bundle of fermionic Fock spaces parametrized by vector potentials in odd space dimensions and prove that this leads in a simple manner to the known Schwinger terms…
The Atiyah-Patodi-Singer (APS) index theorem relates the index of a Dirac operator to an integral of the Pontryagin density in the bulk (which is equal to global chiral anomaly) and an $\eta$ invariant on the boundary (which defines the…
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…
We give a very simple derivation of the Atiyah-Patodi-Singer (APS) index theorem and its small generalization by using the path integral of massless Dirac fermions. It is based on the Fujikawa's argument for the relation between the axial…
K-theory allows us to define an analytical condition for the existence of `false' gauge field copies through the use of the Atiyah-Singer index theorem. After establishing that result we discuss a possible extension of the same result…
In [Wu], the noncommutative Atiyah-Patodi-Singer index theorem was proved. In this paper, we extend this theorem to the equivariant case.
We show that the Wilson Dirac operator in lattice gauge theory can be identified as a mathematical object in $K$-theory and that its associated spectral flow is equal to the index. In comparison to the standard lattice Dirac operator index,…
We give a cohomological formula for the index of a fully elliptic pseudodifferential operator on a manifold with boundary. As in the classic case of Atiyah-Singer, we use an embedding into an euclidean space to express the index as the…
We use techniques from functorial quantum field theory to provide a geometric description of the parity anomaly in fermionic systems coupled to background gauge and gravitational fields on odd-dimensional spacetimes. We give an explicit…
We formulate and prove a lattice version of the Atiyah-Singer index theorem. The main theorem gives a $K$-theoretic formula for an index-type invariant of operators on lattice approximations of closed integral affine manifolds. We apply the…
Recently, examples of an index theory for KMS states of circle actions were discovered, \cite{CPR2,CRT}. We show that these examples are not isolated. Rather there is a general framework in which we use KMS states for circle actions on a…
We discuss the behaviour of the signature index class of closed foliated bundles under the operation of cutting and pasting. Along the way we establish several index theoretic results: we define Atiyah-Patodi-Singer (APS) index classes for…
Atiyah-Singer index theorem on a lattice without boundary is well understood owing to the seminal work by Hasenfratz et al. But its extension to the system with boundary (the so-called Atiyah- Patodi-Singer index theorem), which plays a…
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…
This paper reviews recent work on a new geometric object called a bundle gerbe and discusses some new examples arising in quantum field theory. One application is to an Atiyah-Patodi-Singer index theory construction of the bundle of…