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Yosuke Kubota

Publications and source records attributed to Yosuke Kubota.

At least 19 recordsLinked to original sources
arXiv2025

Stable homotopy theory of invertible gapped quantum spin systems I: Kitaev's $Ω$-spectrum

We provide a mathematical realization of a conjecture by Kitaev, on the basis of the operator-algebraic formulation of infinite quantum spin systems. Our main results are threefold. First, we construct an $Ω$-spectrum $\mathit{IP}_*$ whose homotopy groups are isomorphic to the smooth homotopy group of invertible gapped quantum systems on Euclidean spaces. Second, we develop a model for the homology theory associated with the $Ω$-spectrum $\mathit{IP}_*$, describing it in terms of the space of quantum systems placed on an arbitrary subspace of a Euclidean space. This involves introducing the concept of localization flow, a semi-infinite path of quantum systems with decaying interaction range, inspired by Yu's localization C*-algebra in coarse index theory. Third, we incorporate spatial symmetries given by a crystallographic group $Γ$ and define the $Ω$-spectrum $\mathit{IP}_*^Γ$ of $Γ$-invariant invertible phases. We propose a strategy for computing the homotopy group $π_n(\mathit{IP}_d^Γ)$ that uses the Davis--Lück assembly map and its description by invertible gapped localization flow. In particular, we show that the assembly map is split injective, and hence $π_n(\mathit{IP}_d^Γ)$ contains a computable direct summand.

math-ph↗
arXiv2025

Tensor category equivariant KK-theory

In this paper, we introduce Kasparov's bivariant K-theory that is equivariant under symmetries of a C*-tensor category. It is motivated by some dualities in quantum group equivariant KK-theory, and the classification theory of inclusions of C*-algebras. The fundamental properties of the KK-theory, i.e., the existence of the Kasparov product, Cuntz's picture, universality, and triangulated category structure, hold true in this generalization as well. Moreover, we further prove a new property specific to this theory; the invariance of KK-theory under weak Morita equivalence of the tensor categories. As an example, we study the Baum-Connes type property for $3$-cocycle twists of discrete groups.

math.OA↗
arXiv2024

Groupoid homology and K-theory for algebraic actions from number theory

We compute the groupoid homology for the ample groupoids associated with algebraic actions from rings of algebraic integers and integral dynamics. We derive results for the homology of the topological full groups associated with rings of algebraic integers, and we use our groupoid homology calculation to compute the K-theory for ring C*-algebras of rings of algebraic integers, recovering the results of Cuntz and Li and of Li and L\"uck without using Cuntz-Li duality. Moreover, we compute the K-theory for C*-algebras attached to integral dynamics, resolving the conjecture by Barlak, Omland, and Stammeier in full generality.

math.OA↗
arXiv2022

Delocalized spectra of Landau operators on helical surfaces

On a flat surface, the Landau operator, or quantum Hall Hamiltonian, has spectrum a discrete set of infinitely degenerate Landau levels. We consider surfaces with asymptotically constant curvature away from a possibly non-compact submanifold, the helicoid being our main example. The Landau levels remain isolated, provided the spectrum is considered in an appropriate Hilbert module over the Roe algebra of the surface delocalized away from the submanifold. Delocalized coarse indices may then be assigned to them. As an application, we prove that Landau operators on helical surfaces have no spectral gaps above the lowest Landau level.

math-ph↗
arXiv2021

Band width and the Rosenberg index

A Riemannian manifold is said to have infinite $\mathcal{KO}$-width if it admits an isometric immersion of an arbitrarily wide Riemannian band whose inward boundary has non-trivial higher index. In this paper we prove that if a closed spin manifold has inifinite $\mathcal{KO}$-width, then its Rosenberg index does not vanish. This gives a positive answer to a conjecture by R. Zeidler. We also prove its `multi-dimensional' generalization; if a closed spin manifold admit an isometric immersion of an arbitrarily wide cube-like domain whose lowest dimensional corner has non-trivial higher index, then the Rosenberg index of $M$ does not vanish.

math.KT↗
arXiv2021

Codimension 2 transfer of higher index invariants

This paper is devoted to the study of the higher index theory of codimension $2$ submanifolds originated by Gromov-Lawson and Hanke-Pape-Schick. The first main result is to construct the `codimension $2$ transfer' map from the Higson-Roe analytic surgery exact sequence of a manifold $M$ to that of its codimension $2$ submanifold $N$ under some assumptions on homotopy groups. This map sends the primary and secondary higher index invariants of $M$ to those of $N$. 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 $ρ$-number of $M$ with those of $N$.

math.KT↗
arXiv2021

Twisted crystallograpic T-duality via the Baum--Connes isomorphism

We establish the twisted crystallographic T-duality, which is an isomorphism between Freed-Moore twisted equivariant K-groups of the position and momentum tori associated to an extension of a crystallographic group. The proof is given by identifying the map with the Dirac homomorphism in twisted Chabert--Echterhoff KK-theory. We also illustrate how to exploit it in K-theory computations.

math.KT↗
arXiv2020

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.

math-ph↗
arXiv2020

The Gromov-Lawson codimension 2 obstruction to positive scalar curvature and the C*-index

Gromov and Lawson developed a codimension 2 index obstruction to positive scalar curvature for a closed spin manifold M, later refined by Hanke, Pape and Schick. Kubota has shown that also this obstruction can be obtained from the Rosenberg index of the ambient manifold M which takes values in the K-theory of the maximal C*-algebra of the fundamental group of M, using relative index constructions. In this note, we give a slightly simplified account of Kubota's work and remark that it also applies to the signature operator, thus recovering the homotopy invariance of higher signatures of codimension 2 submanifolds of Higson, Schick, Xie.

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arXiv2019

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.

math.KT↗
arXiv2019

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.

math.KT↗
arXiv2019

The relative Mishchenko--Fomenko higher index and almost flat bundles II: Almost flat index pairing

This is the second part of a series of papers which bridges the Chang--Weinberger--Yu relative higher index and geometry of almost flat hermitian vector bundles on manifolds with boundary. In this paper we apply the description of the relative higher index given in Part I to provide the relative version of the Hanke--Schick theorem, which relates the relative higher index with index pairing of a K-homology cycle with almost flat relative vector bundles. We also deal with the quantitative version and the dual problem of this theorem.

math.KT↗
arXiv2018

Reconstructing the Bost--Connes semigroup actions from K-theory

We complete the classification of Bost--Connes systems. We show that two Bost--Connes C*-algebras for number fields are isomorphic if and only if the original semigroups actions are conjugate. Together with recent reconstruction results in number theory by Cornelissen--de Smit--Li--Marcolli--Smit, we conclude that two Bost--Connes C*-algebras are isomorphic if and only if the original number fields are isomorphic.

math.OA↗
arXiv2016

Notes on twisted equivariant $\mathrm{K}$-theory for $\mathrm{C}^*$-algebras

In this paper, we study a generalization of twisted (groupoid) equivariant $\mathrm{K}$-theory in the sense of Freed-Moore for $\mathbb{Z}_2$-graded $\mathrm{C}^*$-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 $\mathrm{K}$-theory for $\mathbb{Z}_2$-graded Banach algebras. In particular, we obtain a simple presentation of the twisted equivariant $\mathrm{K}$-group when the $\mathrm{C}^*$-algebra is trivially graded. It is applied for the bulk-edge correspondence of topological insulators with CT-type symmetries.

math.KT↗
arXiv2016

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 $\mathrm{K}$-groups of Roe algebras as generalizations of existing invariants such as the Hall conductance or the Kane--Mele $\mathbb{Z}_2$-invariant. As a consequence, we obtain a new mathematical proof of the bulk-edge correspondence by using the coarse Mayer-Vietoris exact sequence.

math-ph↗
arXiv2015

A categorical perspective on the Atiyah-Segal completion theorem in $\mathrm{KK}$-theory

We investigate the homological ideal $\mathfrak{J}_G^H$, 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 $\mathfrak{J}_G^H$ with the augmentation ideal of the representation ring. In relation to it, we study on the Atiyah-Segal completion theorem for groupoid equivariant $\mathrm{KK}$-theory, McClure's restriction map theorem, permanence property of the Baum-Connes conjecture under extensions of groups and a class of $\mathfrak{J}_G$-injective objects coming from $\mathrm{C}^*$-dynamical systems, continuous Rokhlin property.

math.KT↗
arXiv2015

Compact Lie group actions with continuous Rokhlin property

In this paper, we study continuous Rokhlin property of $\mathrm{C}^*$-dynamical systems using techniques of equivariant $\mathrm{KK}$-theory and quantum group theory. In particular, we determine the $\mathrm{KK}$-equivalence class and give a classification of Kirchberg $G$-algebras when the $G$ is a compact Lie group with Hodgkin condition.

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