arXiv subjects

Kiyonori Gomi

Publications and source records attributed to Kiyonori Gomi.

At least 19 recordsLinked to original sources

arXiv2026

Magnitude and matrix inequalities

We present several applications of matrix-theoretic inequalities to the magnitude of metric spaces. We first resolve an open problem by showing that the magnitude of any finite metric space of negative type is less than or equal to its cardinality. This is a direct consequence of Styan's matrix inequality involving the Hadamard product of matrices. By related methods we also show a subadditivity property for the magnitude function of negative type compact metric spaces, and prove a convexity property for the magnitude for metrics interpolating in a natural way between two given, comparable metrics on a given set.

math.MG↗

arXiv2026

On the monotonicity of magnitude functions of negative type finite semimetric spaces

Motivated by the conjectured monotonicity of the magnitude functions of metric spaces of negative type, we investigate the monotonicity problem for finite semimetric spaces, namely, ``metric spaces without the triangle inequality''. We completely resolve this problem by proving that the monotonicity holds for negative type semimetric spaces with at most three points, while constructing negative type semimetric spaces with at least four points whose magnitude functions are not monotonically increasing.

math.MG↗

arXiv2026

Hilbert Grassmannians as classifying spaces

In this short work we prove that the Hilbert Grassmannians endowed with the weak topology are models for the classifying spaces of the unitary groups. As application of this result one can use Hilbert Grassmannians for the presentation of the $K$-theory of topological spaces by computing equivalences classes of homotopy equivalent maps.

math-ph↗

arXiv2025

Geometric interpretation of magnitude

For an $n\times n$ positive definite symmetric matrix $Z$ with $Z_{ii} = 1$ for all $i$, we show that there exists a set of vectors $V_Z\subset \mathbb{R}^n$ such that the radius $R$ of the circumsphere of $V_Z$ satisfies ${\rm Mag}\ Z = (1-R^2)^{-1}$. This leads us to interpret geometrically several known and new facts on magnitude. In particular, we show that ${\rm Mag}\ Z_{X}< n$ for an $n$-point metric space $X$ of negative type with $n>1$. This result gives a negative answer to a problem given by Gomi--Meckes \cite{GM}. Furthermore, we also have a similar geometric description of magnitude for general real symmetric matrix $Z$ with $Z_{ii} = 1$ for all $i$. In this case, the radius corresponds to that of a circum-quasi-sphere, namely the set of points having a prescribed norm in a vector space endowed with an indefinite inner product.

math.MG↗

arXiv2025

Magnitude homology of geodesic space

This paper studies the magnitude homology groups of geodesic metric spaces. We start with a description of the second magnitude homology of a general metric space in terms of the zeroth homology groups of certain simplicial complexes. Then, on a geodesic metric space, we interpret the description by means of geodesics. The third magnitude homology of a geodesic metric space also admits a description in terms of a simplicial complex. Under an assumption on a metric space, the simplicial description allows us to introduce an invariant of third magnitude homology classes as an intersection number. Finally, we provide a complete description of all the magnitude homology groups of a geodesic metric space which fulfils a certain non-branching assumption.

math.AT↗

arXiv2023

A direct proof for the positive definiteness of four point metric spaces

We provide a direct and elementary proof for the fact that every four point metric space is positive definite, which was first proved by Meckes based on some embedding theorems of metric spaces. As an outcome of the direct proof, we also provide a condition for the magnitude of a finite metric space to obey the inclusion-exclusion principle with respect to a specific choice of subspaces.

math.MG↗

arXiv2023

S-folds and 4d N=3 superconformal field theories

S-folds are generalizations of orientifolds in type IIB string theory, such that the geometric identifications are accompanied by non-trivial S-duality transformations. They were recently used by Garcia-Etxebarria and Regalado to provide the first construction of four dimensional N=3 superconformal theories. In this note, we classify the different variants of these N=3 preserving S-folds, distinguished by an analog of discrete torsion, using both a direct analysis of the different torsion classes and the compactification of the S-folds to three dimensional M-theory backgrounds. Upon adding D3-branes, these variants lead to different classes of N=3 superconformal field theories. We also analyze the holographic duals of these theories, and in particular clarify the role of discrete gauge and global symmetries in holography. In the main part of the paper, certain properties of cohomology groups associated to the S-folds were conjectured and used. This arXiv version includes an appendix written by Kiyonori Gomi in 2023 providing the proofs of the required properties using the technique of Borel equivariant cohomology, whose brief review is also provided.

hep-th↗

arXiv2023

Generalized homology and Atiyah-Hirzebruch spectral sequence in crystalline symmetry protected topological phenomena

We propose that symmetry protected topological (SPT) phases with crystalline symmetry are formulated by equivariant generalized homologies $h^G_n(X)$ over a real space manifold $X$ with $G$ a crystalline symmetry group. The Atiyah-Hirzebruch spectral sequence unifies various notions in crystalline SPT phases such as the layer construction, higher-order SPT phases and Lieb-Schultz-Mattis type theorems. Our formulation is applicable to interacting systems with onsite and crystalline symmetries as well as free fermions.

cond-mat.str-el↗

arXiv2023

Atiyah-Hirzebruch Spectral Sequence in Band Topology: General Formalism and Topological Invariants for 230 Space Groups

We study the Atiyah-Hirzebruch spectral sequence (AHSS) for equivariant K-theory in the context of band theory. Various notions in the band theory such as irreducible representations at high-symmetric points, the compatibility relation, topological gapless and singular points naturally fits into the AHSS. As an application of the AHSS, we get the complete list of topological invariants for 230 space groups without time-reversal or particle-hole invariance. We find that a lot of torsion topological invariants appear even for symmorphic space groups.

cond-mat.str-el↗

arXiv2022

A new light on the FKMM invariant and its consequences

"Quaternionic" vector bundles are the objects which describe the topological phases of quantum systems subjected to an odd time-reversal symmetry (class AII). In this work we prove that the FKMM invariant provides the correct fundamental characteristic class for the classification of "Quaternionic" vector bundles in dimension less than, or equal to three (low dimension). The new insight is provided by the interpretation of the FKMM invariant from the viewpoint of the Bredon equivariant cohomology. This fact, along with basic results in equivariant homotopy theory, allows us to achieve the expected result.

math-ph↗

arXiv2022

Differential $KO$-theory via gradations and mass terms

We construct models of the differential $KO$-theory and the twisted differential $KO$-theory, by refining Karoubi's $KO$-theory [Kar78] in terms of gradations on Clifford modules. In order for this, we set up the generalized Clifford superconnection formalism which generalizes the Quillen's superconnection formalism [Qui85]. One of our models can be regarded as classifying "fermionic mass terms" in physics.

math.KT↗

arXiv2021

'Real' gerbes and Dirac cones of topological insulators

A time-reversal invariant topological insulator occupying a Euclidean half-space determines a 'Quaternionic' self-adjoint Fredholm family. We show that the discrete spectrum data for such a family is geometrically encoded in a non-trivial 'Real' gerbe. The gerbe invariant, rather than a naïve counting of Dirac points, precisely captures how edge states completely fill up the bulk spectral gap in a topologically protected manner.

hep-th↗

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↗

arXiv2021

The cohomology invariant for class DIII topological insulators

This work concerns with the description of the topological phases of band insulators of class DIII by using the equivariant cohomology. The main result is the definition of a cohomology class for general systems of class DIII which generalizes the well known $\mathbb{Z}_2$-invariant given by the Teo-Kane formula in the one-dimension case. In the two-dimensional case this cohomology invariant allows a complete description of the strong and weak phases. The relation with the KR-theory, the Noether-Fredholm index and the classification of "Real" gerbes are also discussed.

math-ph↗

arXiv2021

Freed-Moore K-theory

The twisted equivariant K-theory given by Freed and Moore is a K-theory which unifies twisted equivariant complex K-theory, Atiyah's `Real' K-theory, and their variants. In a general setting, we formulate this K-theory by using Fredholm operators, and establish basic properties such as the Bott periodicity and the Thom isomorphism. We also provide formulations of the K-theory based on Karoubi's gradations in both infinite and finite dimensions, clarifying their relationship with the Fredholm formulation.

math.KT↗

arXiv2020

Homological bulk-edge correspondence for Weyl semimetal

For a certain translation invariant tight-binding model of three-dimensional Weyl semimetals, we establish a bulk-edge correspondence as an equality of two relative homology classes, based on an idea of Mathai and Thiang: From spectral information on the edge Hamiltonian, we construct a relative homology class on the surface momentum space. This class agrees with the image under the surface projection of a homology class on the bulk momentum space relative to the Weyl points, constructed from the bulk Hamiltonian. Furthermore, the relative homology class on the surface momentum space can be represented by homology cycles whose images constitute the Fermi arc, the locus where the edge Hamiltonian admits zero spectrum.

math-ph↗

arXiv2019

Eigenvalue crossings in Floquet topological systems

The topology of electrons on a lattice subject to a periodic driving is captured by the three-dimensional winding number of the propagator that describes time-evolution within a cycle. This index captures the homotopy class of such a unitary map. In this paper, we provide an interpretation of this winding number in terms of local data associated to the the eigenvalue crossings of such a map over a three dimensional manifold, based on an idea from Nathan and Rudner, New Journal of Physics, 17(12) 125014, 2015. We show that, up to homotopy, the crossings are a finite set of points and non degenerate. Each crossing carries a local Chern number, and the sum of these local indices coincides with the winding number. We then extend this result to fully degenerate crossings and extended submanifolds to connect with models from the physics literature. We finally classify up to homotopy the Floquet unitary maps, defined on manifolds with boundary, using the previous local indices. The results rely on a filtration of the special unitary group as well as the local data of the basic gerbe over it.

math-ph↗

arXiv2019

Smoothness filtration of the magnitude complex

We introduce an intrinsic filtration to the magnitude chain complex of a metric space, and study basic properties of the associated spectral sequence of the magnitude homology. As an application, the third magnitude homology of the circle is computed.

math.AT↗