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Mikio Furuta

Publications and source records attributed to Mikio Furuta.

18 recordsLinked to original sources
arXiv2026

Generalization of lattice Dirac operator index

We provide a comprehensive lattice formulation of various types of the Dirac operator indices, employing $K$-theory to classify the Wilson Dirac operator via its spectral flow. In contrast to the index of the overlap Dirac operator defined through the Ginsparg-Wilson relation, which is restricted to flat tori in even dimensions, our formulation offers several key advantages: 1) It can be applied straightforwardly to the Atiyah-Patodi-Singer index for manifolds with boundary. 2) The boundary can be curved, allowing for the inclusion of gravitational background effects. 3) The mod-2 index in both even and odd dimensions can be defined as a natural extension of the same formulation. In this talk, we present the mathematical proof and provide numerical evidence supporting the formulation.

hep-lat↗
arXiv2026

Capturing the Atiyah-Patodi-Singer index from the lattice

We construct a formulation of the Atiyah-Patodi-Singer index of Dirac operators in lattice gauge theory for domains with compact boundaries in a flat torus. The key idea is to exploit its equality to the spectral flow of the domain-wall fermion Dirac operators, which we generalize in this work to cases without product structure near the boundary. We prove that, for sufficiently small lattice spacings, this formulation correctly captures the continuum Atiyah-Patodi-Singer index.

math.DG↗
arXiv2025

$K$-theoretic computation of the Atiyah(-Patodi)-Singer index of lattice Dirac operators

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, our formulation does not require the Ginsparg-Wilson relation and has broader applicability to systems with boundaries and to the mod-two version of the indices in general dimensions. We numerically verify that the $K$ and $KO$ group formulas reproduce the known index theorems in continuum theory. We examine the Atiyah-Singer index on a flat two-dimensional torus and, for the first time, demonstrate that the Atiyah-Patodi-Singer index with nontrivial curved boundaries, as well as the mod-two versions, can be computed on a lattice.

hep-th↗
arXiv2025

The index of lattice Dirac operators and $K$-theory

We mathematically show an equality between the index of a Dirac operator on a flat continuum torus and the $η$ invariant of a lattice Dirac operator known as the Wilson Dirac operator with a negative mass when the lattice spacing is sufficiently small. Unlike the standard approach, our formulation using $K$-theory does not require modified chiral symmetry on the lattice. We prove that a one-parameter family of continuum massive Dirac operators and the corresponding Wilson Dirac operators belong to the same equivalence class of the $K^1$ group at a finite lattice spacing. Their indices, which are evaluated by the spectral flow or equivalently by the $η$ invariant at a finite mass, are proved to be equal.

math.KT↗
arXiv2025

$η$ invariant of massive Wilson Dirac operator and the index

We revisit the lattice index theorem in the perspective of $K$-theory. The standard definition given by the overlap Dirac operator equals to the $η$ invariant of the Wilson Dirac operator with a negative mass. This equality is not coincidental but reflects a mathematically profound significance known as the suspension isomorphism of $K$-groups. Specifically, we identify the Wilson Dirac operator as an element of the $K^1$ group, which is characterized by the $η$-invariant. Furthermore, we prove that, at sufficiently small but finite lattice spacings, this $η$-invariant equals to the index of the continuum Dirac operator. Our results indicate that the Ginsparg-Wilson relation and the associated exact chiral symmetry are not essential for understanding gauge field topology in lattice gauge theory.

hep-lat↗
arXiv2021

A physicist-friendly reformulation of the mod-two Atiyah-Patodi-Singer index

Gauge anomaly in 4-dimensions can be viewed as a current inflow into an extra-dimension, where the total phase of the fermion partition function is given in a gauge invariant way by the Atiyah- Patodi-Singer(APS) eta-invariant of a 5-dimensional Dirac operator. However, this formalism requires a non-local boundary condition, with which the physical roles of edge/bulk modes are unclear and how the causality of the theory is maintained is not obvious. In this work, we consider a special case where the Dirac operator is in a real representation and its eta invariant becomes the mod-two type APS index. We propose a physicist-friendly reformulation of the mod-two index using domain-wall fermion formalism, which naturally describes how the global anomaly is canceled between edge and bulk.

hep-th↗
arXiv2020

Mod-two APS index and domain-wall fermion

We reformulate the mod-two Atiyah-Patodi-Singer (APS) index in a physicist-friendly way using the domain-wall fermion. Our new formulation is given on a closed manifold, which is extended from the original manifold with boundary, where we instead give a fermion mass term changing its sign at the location of the original boundary. This new setup does not need the APS boundary condition, which is non-local. A mathematical proof of equivalence between the two different formulations is given by two different evaluations of the same index of a Dirac operator on a higher dimensional manifold. The domain-wall fermion allows us to separate the edge and bulk mode contributions in a natural but not in a gauge invariant way, which offers a straightforward description of the global anomaly inflow.

hep-th↗
arXiv2020

The Atiyah-Patodi-Singer index and domain-wall fermion Dirac operators

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.

math.DG↗
arXiv2020

A physicist-friendly reformulation of the Atiyah-Patodi-Singer index and its mathematical justification

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 imposes on the fermion fields a non-local and unnatural boundary condition known as the "APS boundary condition" by hand. In 2017, we showed that the same integer as the APS index can be obtained from the $η$ invariant of the domain-wall Dirac operator. Recently we gave a mathematical proof that the equivalence is not a coincidence but generally true. In this contribution to the proceedings of LATTICE 2019, we try to explain the whole story in a physicist-friendly way.

hep-lat↗
arXiv2016

The perturbation of the Seiberg-Witten equations revisited

We introduce a new class of perturbations of the Seiberg-Witten equations. Our perturbations offer flexibility in the way the Seiberg-Witten invariants are constructed and also shed a new light to LeBrun's curvature inequalities.

math.DG↗
arXiv2014

Torus fibrations and localization of index III

This paper is the third of the series concerning the localization of the index of Dirac-type operators. In our previous papers we gave a formulation of index of Dirac-type operators on open manifolds under some geometric setting, whose typical example was given by the structure of a torus fiber bundle on the ends of the open manifolds. We introduce two equivariant versions of the localization. As an application we give a proof of Guillemin-Sternberg's quantization conjecture in the case of torus action.

math.DG↗
arXiv2014

Torus fibrations and localization of index II

We give a framework of localization for the index of a Dirac-type operator on an open manifold. Suppose the open manifold has a compact subset whose complement is covered by a family of finitely many open subsets, each of which has a structure of the total space of a torus bundle. Under an acyclic condition we define the index of the Dirac-type operator by using the Witten-type deformation, and show that the index has several properties, such as excision property and a product formula. In particular, we show that the index is localized on the compact set.

math.DG↗
arXiv2010

Torus fibrations and localization of index I

We define a local Riemann-Roch number for an open symplectic manifold when a complete integrable system without Bohr-Sommerfeld fiber is provided on its end. In particular when a structure of a singular Lagrangian fibration is given on a closed symplectic manifold, its Riemann-Roch number is described as the sum of the number of nonsingular Bohr-Sommerfeld fibers and a contribution of the singular fibers. A key step of the proof is formally explained as a version of Witten's deformation applied to a Hilbert bundle.

math.SG↗
arXiv2003

Finite dimensional approximations in geometry

In low dimensional topology, we have some invariants defined by using solutions of some nonlinear elliptic operators. The invariants could be understood as Euler class or degree in the ordinary cohomology, in infinite dimensional setting. Instead of looking at the solutions, if we can regard some kind of homotopy class of the operator itself as an invariant, then the refined version of the invariant is understood as Euler class or degree in cohomotopy theory. This idea can be carried out for the Seiberg-Witten equation on 4-dimensional manifolds and we have some applications to 4-dimensional topology.

math.GT↗
arXiv2002

A stable cohomotopy refinement of Seiberg-Witten invariants: I

The monopole map defines an element in an equivariant stable cohomotopy group refining the Seiberg-Witten invariant. This first of two articles presents the details of the definition of the stable cohomotopy invariant and discusses its relation to the integer valued Seiberg-Witten invariant.

math.DG↗