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Tetsuya Onogi

Publications and source records attributed to Tetsuya Onogi.

At least 19 recordsLinked to original sources
arXiv2026

Symmetric Mass Generation for Domain-Wall Fermions

We report the first numerical examination of symmetric mass generation (SMG) for domain-wall fermions. Our simulation on a two-dimensional Euclidean lattice computes the path-integral of the Fidkowski-Kitaev Majorana chain with two boundaries, corresponding to the two (physical and mirror) walls. We demonstrate by evaluating the lattice Dirac operator eigenvalue spectrum that the local four-Fermi interaction selectively gaps the edge fermions on the mirror wall while leaving the physical wall intact. We further identify a characteristic signature of SMG in the path-integral: the gap of the Dirac operator opens along the imaginary axis of the complex spectrum, in contrast to the standard mass shift in the real direction. The conventional hybrid Monte Carlo algorithm remains practical with reweighting the complex phase of the Pfaffian, which is found to be remarkably small throughout the simulation. These results provide a practical nonperturbative framework for interaction-induced mirror-fermion decoupling of lattice domain-wall fermions.

hep-lat↗
arXiv2026

Taste-splitting mass and edge modes in $3+1$ D staggered fermions

We investigate the symmetry structure of the $3+1$ D staggered fermion Hamiltonian and its implications for anomalies. Since the spin and flavor degrees of freedom of Dirac fermions are distributed over the lattice, in addition to the standard on-site mass term, the staggered fermion system also admits one-, two-, and three-link bilinear terms within a unit cube as local, charge conserving mass terms with different spin and flavor dependence. We identify the spin flavor structures of all those bilinear mass terms and determine the symmetries preserved by each of them. Among them, one of the one-link mass terms preserves a larger residual symmetry associated with conserved charges that generate the Onsager algebra. Motivated by this structure, we consider a kink profile of the one-link mass and analyze the resulting domain-wall system. In the low-energy limit, the $3+1$ D bulk becomes gapped, while two-flavor massless Dirac fermions appear as localized modes on the $2+1$ D domain wall. We show that the bulk conserved charges act on the wall as generators of a flavor $\mathrm{SU}(2)$ symmetry, and that no symmetric mass gap is allowed for the boundary theory when this $\mathrm{SU}(2)$ symmetry and space reflection symmetry are both imposed. This realizes the parity anomaly of the boundary theory and shows that the boundary flavor symmetry and anomaly descend from the ultraviolet staggered-fermion Hamiltonian rather than emerging only in the infrared.

hep-lat↗
arXiv2026

Conserved Non-Singlet Charges for Staggered Fermion Hamiltonian in 3+1 Dimensions

We study conserved charges of the staggered fermion Hamiltonian in 3+1 dimensions. By decomposing staggered fermions into Majorana components and exploiting lattice translation symmetries, we construct a set of conserved non-singlet charges. We analyze their algebra andshow that, although the charges exhibit nontrivial non-commutativity on the lattice, they generate axial SU(2)_L \tines SU(2)_R transformations for low-energy degrees of freedom in the continuum limit. Possible implications for anomalies are discussed.

hep-lat↗
arXiv2026

How to formulate the $\mathbb{Z}_8$ topological invariant of Majorana fermion on the lattice

Topological invariants and their associated anomalies have played a crucial role in understanding low-energy phenomena in quantum field theories. In lattice gauge theory, the standard $\mathbb{Z}$-valued Atiyah-Singer index is formulated via the overlap Dirac operator through the Ginsparg-Wilson relation, but extensions to more general topological invariants have remained limited. In this work, we propose a lattice formulation of the Arf-Brown-Kervaire (ABK) invariant, which takes values in $\mathbb{Z}_8$. The ABK invariant arises in Majorana fermion partition functions with reflection symmetry on two-dimensional non-oriented manifolds, and its definition involves an infinite sum over Dirac eigenvalues that must be properly regularized. By carefully treating the boundary conditions, with and without a domain-wall mass term, we demonstrate that the ABK invariant can be extracted from Pfaffians of the Wilson Dirac operator. We further provide numerical verification on two-dimensional lattices, showing that the $\mathbb{Z}_8$-valued results on the torus, Klein bottle, real projective plane, and M\"obius strip agree with those in the continuum theory.

hep-lat↗
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

The Arf-Brown-Kervaire invariant on a lattice

We propose a lattice formulation of the Arf-Brown-Kervaire (ABK) invariant which takes values in $\mathbb{Z}_8$. Compared to the standard $\mathbb{Z}$-valued index, the ABK invariant is more involved in that it arises in Majorana fermion partition functions with reflection symmetry on two-dimensional non-orientable manifolds, and its definition contains an infinite sum over Dirac eigenvalues that requires proper regularization. We employ the massive Wilson Dirac operator, with and without domain-walls, on standard two-dimensional square lattices, and use its Pfaffian for the definition. Twisted boundary conditions and cross-caps, which reverse the orientation, are introduced to realize nontrivial topologies equipped with nontrivial $\mathrm{Pin}^{-}$ structures of Majorana fermions. We verify numerically (and partly analytically) that our formulation on a torus, Klein bottle, real projective plane (as well as its triple connected sum), and two types of M\"obius strip reproduces the known values in continuum theory.

hep-lat↗
arXiv2025

Non-singlet conserved charges and anomalies in 3+1 D staggered fermions

In this paper, we show that the 3+1 D staggered fermion Hamiltonian possesses, in addition to the conserved charge $Q_0$ that generates the vector $\mathrm{U}(1)_V$ transformation, conserved charges $Q_F$ that generate the $\mathrm{SU}(2)_A$ transformations in the continuum limit, acting simultaneously on left- and right-handed Weyl fermions in opposite directions. Each conserved charge $Q_F$ can be regarded as the generator of a $\mathrm{U}(1)_F$ subgroup of $\mathrm{SU}(2)_L \times \mathrm{SU}(2)_R \times \mathrm{U}(1)_A$. One of these lattice charges satisfies the Onsager algebra. On the lattice, the charges $Q_F$ do not commute with $Q_0$, and no symmetric mass term exists that commutes with both $Q_0$ and $Q_F$. This signals the presence of a mixed anomaly. Remarkably, however, in the continuum limit, a symmetric mass term commuting with both $Q_0$ and $Q_F$ can be constructed. % This implies that the non-trivial lattice anomaly becomes a trivial anomaly in the continuum QFT. This means that the mixed anomaly that is nontrivial on the lattice becomes trivial in the IR QFT obtained in the continuum limit, which is consistent with the analysis of the Ward--Takahashi (WT) identity on the lattice. Indeed, by evaluating this identity associated with the $\mathrm{U}(1)_F$ transformation on the lattice, we confirm that $\mathrm{U}(1)_F$ symmetry is exactly conserved.

hep-lat↗
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

Discrete symmetry and 't Hooft anomalies for 3450 model

We report our study of the discrete symmetry for lattice 3450 model proposed by Wang and Wen. Lattice 3450 model is expected to describe the anomaly free chiral U(1) gauge theory in 1+1 dimension using 2+1 dimensional domain-wall fermion with gapping interactions for the mirror sector. We find that the lattice model has exact discrete symmetry in addition to U(1) x U(1) symmetry. Assuming the Zumino-Stora procedure works also for discrete symmetry, we compute the full 't Hooft anomaly for the target continuum U(1) chiral gauge theory with the same discrete symmetry. We show that the mixed and self anomalies involving the discrete symmetry are absent, which is consistent with the expectation that the lattice 3450 produces chiral U(1) gauge theory in the continuum limit.

hep-lat↗
arXiv2025

SU(6) model revisited

We discuss the vacuum structure of the SU(6) model, a chiral gauge theory, from the perspective of anomaly matching. To this end, we first identify all possible 't Hooft anomalies in the UV theory using the Stora-Zumino procedure. Subsequently, we construct an effective theory by applying the idea of the Wess-Zumino-Witten action to derive the topological terms that encode the 't Hooft anomalies. As a result, we demonstrate that a low-energy effective theory reproducing one of the anomalies, namely the mixed anomaly, is described by a Z3-valued scalar field. On the other hand, the effective theory that accounts for the discrete chiral self-anomaly is significantly more intricate, and elucidating its structure remains an ongoing challenge.

hep-lat↗
arXiv2025

$\eta$ 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 $\eta$ 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 $\eta$-invariant. Furthermore, we prove that, at sufficiently small but finite lattice spacings, this $\eta$-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↗
arXiv2024

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 $\eta$ 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 $\eta$ invariant at a finite mass, are proved to be equal.

math.KT↗
arXiv2023

Energies and a gravitational charge for massive particles in general relativity

In this paper, we investigate relations or differences among various conserved quantities which involve the matter Energy Momentum Tensor (EMT) in general relativity. These charges include the energy with Einstein's pseudo EMT, the generalized Komar integral, or the ADM energy, all of which can be derived from Noether's second theorem, as well as an extra conserved charge recently proposed in general relativity. For detailed analyses, we apply definitions of these charges to a system of free massive particles. We employ the post-Newtonian (PN) expansion to make physical interpretations. We find that the generalized Komar integral is not conserved at the first non-trivial order in the PN expansion due to non-zero contributions at spatial boundaries, while the energy with Einstein's pseudo EMT at this order agrees with a total energy of massive particles with gravitational interactions through the Newtonian potential, and thus is conserved. In addition, this total energy is shown to be identical to the ADM energy not only at this order but also all orders in the PN expansion. We next calculate an extra conserved charge for the system of massive particles, at all orders in the PN expansion, which turns out to be a total number of particles. We call it a gravitational charge, since it is clearly different from the total energy. We finally discuss an implication from a fact that there exist two conserved quantities, energy and gravitational charge, in general relativity.

gr-qc↗
arXiv2023

Special flow equation and GKP-Witten relation

We develop a framework for the reconstruction of the bulk theory dual to conformal field theory (CFT) without any assumption by means of a flow equation. To this end we investigate a minimal extension of the free flow equation and find that at a special parametrization the conformal transformation for a normalized smeared operator exactly becomes the isometry of anti-de Sitter space (AdS). By employing this special flow equation to O$(N)$ vector models, we explicitly show that the AdS geometry as well as the scalar field satisfying the GKP-Witten relation concurrently emerge in this framework.

hep-th↗
arXiv2022

Conserved non-Noether charge in general relativity: Physical definition vs. Noether's 2nd theorem

In this paper, we make a close comparison of a covariant definition of an energy/entropy in general relativity, recently proposed by a collaboration including the present authors, with existing definitions of energies such as the one from the pseudo-tensor and the quasi-local energy. We show that existing definitions of energies in general relativity are conserved charges from the Noether's 2nd theorem for the general coordinate transformation, whose conservations are merely identities implied by the local symmetry and always hold without using equations of motion. Thus none of existing definitions in general relativity reflects the dynamical properties of the system, need for a physical definition of an energy. In contrast, our new definition of the energy/entropy in general relativity is generically a conserved non-Noether charge and gives physically sensible results for various cases such as the black hole mass, the gravitational collapse, and the expanding universe, while existing definitions sometimes lead to unphysical ones including zero and infinity. We conclude that our proposal is more physical than existing definitions of energies. Our proposal makes it possible to define almost uniquely the covariant and conserved energy/entropy in general relativity, which brings some implications to future investigations.

hep-th↗
arXiv2021

Charge Conservation, Entropy Current, and Gravitation

We propose a new class of vector fields to construct a conserved charge in a general field theory whose energy momentum tensor is covariantly conserved. We show that there always exists such a vector field in a given field theory even without global symmetry. We also argue that the conserved current constructed from the (asymptotically) time-like vector field can be identified with the entropy current of the system. As a piece of evidence we show that the conserved charge defined therefrom satisfies the first law of thermodynamics for an isotropic system with a suitable definition of temperature. We apply our formulation to several gravitational systems such as the expanding universe, Schwarzschild and BTZ black holes, and gravitational plane waves. We confirm the conservation of the proposed entropy density under any homogeneous and isotropic expansion of the universe, the precise reproduction of the Bekenstein-Hawking entropy incorporating the first law of thermodynamics, and the existence of gravitational plane wave carrying no charge, respectively. We also comment on the energy conservation during gravitational collapse in simple models.

gr-qc↗