192 research outputs found
Controlled topological phases and bulk-edge correspondence
In this paper, we introduce a variation of the notion of topological phase
reflecting metric structure of the position space. This framework contains not
only periodic and non-periodic systems with symmetries in Kitaev's periodic
table but also topological crystalline insulators. We also define the bulk and
edge indices as invariants taking values in the twisted equivariant
-groups of Roe algebras as generalizations of existing invariants
such as the Hall conductance or the Kane--Mele -invariant. As a
consequence, we obtain a new mathematical proof of the bulk-edge correspondence
by using the coarse Mayer-Vietoris exact sequence.Comment: 30 pages, Section 2 is completely rewritte
The Joint Spectral Flow and Localization of the Indices of Elliptic Operators
We introduce the notion of the joint spectral flow, which is a generalization
of the spectral flow, by using Segal's model of the connective -theory
spectrum. We apply it for some localization results of indices motivated by
Witten's deformation of Dirac operators and rephrase some analytic techniques
in terms of topology.Comment: 35 page
Notes on twisted equivariant -theory for -algebras
In this paper, we study a generalization of twisted (groupoid) equivariant
-theory in the sense of Freed-Moore for -graded
-algebras. It is defined by using Fredholm operators on Hilbert
modules with twisted representations. We compare it with another description
using odd symmetries, which is a generalization of van Daele's
-theory for -graded Banach algebras. In particular,
we obtain a simple presentation of the twisted equivariant -group
when the -algebra is trivially graded. It is applied for the
bulk-edge correspondence of topological insulators with CT-type symmetries.Comment: 26 pages, minor correction
Compact Lie group actions with continuous Rokhlin property
In this paper, we study continuous Rokhlin property of
-dynamical systems using techniques of equivariant
-theory and quantum group theory. In particular, we determine the
-equivalence class and give a classification of Kirchberg
-algebras when the is a compact Lie group with Hodgkin condition.Comment: 18 pages. This paper is separated from Section 7 of arXiv:1508.0681
A categorical perspective on the Atiyah-Segal completion theorem in -theory
We investigate the homological ideal , the kernel of the
restriction functors in compact Lie group equivariant Kasparov categories.
Applying the relative homological algebra developed by Meyer and Nest, we
relate the Atiyah-Segal completion theorem with the comparison of
with the augmentation ideal of the representation ring.
In relation to it, we study on the Atiyah-Segal completion theorem for
groupoid equivariant -theory, McClure's restriction map theorem,
permanence property of the Baum-Connes conjecture under extensions of groups
and a class of -injective objects coming from
-dynamical systems, continuous Rokhlin property.Comment: 37 pages. Section 7 is separated to arXiv:1512.06333, minor
corrections in Section
The relative Mishchenko--Fomenko higher index and almost flat bundles I: The relative Mishchenko--Fomenko index
In this paper, the first of two, we introduce an alternative definition of
the Chang--Weinberger--Yu relative higher index, which is thought of as a
relative analogue of the Mishchenko--Fomenko index pairing. A main result of
this paper is that our map coincides with the existing relative higher index
maps. We make use of this fact for understanding the relative higher index.
First, we relate the relative higher index with the higher index of amalgamated
free product groups. Second, we define the dual relative higher index map and
show its rational surjectivity under certain assumptions.Comment: 30 pages. Section 5, 6, 7, 8 of the previous version are separated to
other article
Almost flat relative vector bundles and the almost monodromy correspondence
In this paper we introduce the notion of almost flatness for (stably)
relative bundles on a pair of topological spaces and investigate basic
properties of it. First, we show that almost flatness of topological and smooth
sense are equivalent. This provides a construction of an almost flat stably
relative bundle by using the enlargeability of manifolds. Second, we show the
almost monodromy correspondence, that is, a correspondence between almost flat
(stably) relative bundles and (stably) relative quasi-representations of the
fundamental group.Comment: 25 pages, separated from Section 5, 6.2 of arXiv:1807.03181,
corrected minor error
Codimension 2 transfer of higher index invariants
This paper is devoted to the study of the higher index theory of codimension
submanifolds originated by Gromov-Lawson and Hanke-Pape-Schick. The first
main result is to construct the `codimension transfer' map from the
Higson-Roe analytic surgery exact sequence of a manifold to that of its
codimension submanifold under some assumptions on homotopy groups. This
map sends the primary and secondary higher index invariants of to those of
. The second is to establish that the codimension 2 transfer map is adjoint
to the co-transfer map in cyclic cohomology, defined by the cup product with a
group cocycle. This relates the Connes-Moscovici higher index pairing and
Lott's higher -number of with those of .Comment: 39 pages; some minor revisions, added Subsection 4.3 and a more
detail in Subsection 5.
The index theorem of lattice Wilson--Dirac operators via higher index theory
We give a proof of the index theorem of lattice Wilson--Dirac operators,
which states that the index of a twisted Dirac operator on the standard torus
is described in terms of the corresponding lattice Wilson--Dirac operator. Our
proof is based on the higher index theory of almost flat vector bundles.Comment: 20 page
The bulk-dislocation correspondence for weak topological insulators on screw-dislocated lattices
A weak topological insulator in dimension is known to have a
topologically protected gapless mode along the screw dislocation. In this paper
we formulate and prove this fact with the language of C*-algebra K-theory. The
proof is based on the coarse index theory of the helical surface.Comment: 17 page
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