We show that the Wilson Dirac operator in lattice gauge theory can be identified as a mathematical object in <italic>K</italic>-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&#8211;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 <italic>K</italic> and <inline-formula><tex-math id='TM0004' notation='LaTeX'>$KO$</tex-math></inline-formula> group formulas reproduce the known index theorems in continuum theory. We examine the Atiyah&#8211;Singer index on a flat 2D torus and, for the first time, demonstrate that the Atiyah&#8211;Patodi&#8211;Singer index with nontrivial curved boundaries, as well as the mod-two versions, can be computed on a lattice.

Japan

Shoto Aoki

Hidenori Fukaya

Mikio Furuta

Shinichiroh Matsuo

Tetsuya Onogi

Satoshi Yamaguchi

<italic>K</italic>-Theoretic Computation of the Atiyah(&#8211;Patodi)&#8211;Singer Index of Lattice Dirac Operators

We study the <inline-formula><tex-math id='TM0001' notation='LaTeX'>$N \times N$</tex-math></inline-formula> Hermitian one-matrix model modified by the double-trace interaction. It is known that the coupling for the double-trace interaction can control the weight for the microscopic wormholes if interpreting the matrix model as the lattice model of a random surface; tuning the coupling to its critical value, the effect of wormholes becomes substantial to change the critical behavior of the pure 2D quantum gravity, which is characterized by a certain positive value of the string susceptibility. In the large- <italic>N</italic> limit, we calculate the continuum limit of the disk amplitude in which the wormhole effects are important. The resulting continuum disk amplitude is the same as that of the pure 2D quantum gravity. We also introduce the renormalized coupling for the double-trace interaction, and show that the newly introduced renormalized coupling can alter the renormalized bulk cosmological constant effectively.

Tsunehide Kuroki

Yuki Sato

Properties of the Wormhole-Dominant Phase in 2D Quantum Gravity

We propose a novel fermionic model on the graph. The Dirac operator of the model consists of deformed incidence matrices on the graph and the partition function is given by the inverse of the graph zeta function. We find that the coefficients of the inverse of the graph zeta function, which is a polynomial of finite degree in the coupling constant, count the number of fermionic cycles on the graph. We also construct the model on grid graphs by using the concept of the covering graph and the Artin–Ihara <inline-formula><tex-math id='TM0001' notation='LaTeX'>$L$</tex-math></inline-formula>-function. In connection with this, we show that the fermion doubling is absent, and the overlap fermions can be constructed on a general graph. Furthermore, we relate our model to statistical models by introducing the winding number around cycles, where the distribution of the poles of the graph zeta function (the zeros of the partition function) plays a crucial role. Finally, we formulate gauge theory including fermions on the graph from the viewpoint of the covering graph derived from the gauge group in a unified way.

So Matsuura

Kazutoshi Ohta

Fermions and Zeta Function on the Graph

Entanglement asymmetry is a measure of symmetry breaking in quantum subsystems, inspired by quantum information theory, particularly suited to study out-of-equilibrium states. We study the entanglement asymmetry of a class of excited “coherent states” in conformal quantum field theories with a <italic>U</italic>(1) symmetry, employing Euclidean path-integral methods with topological symmetry defects and the replica formalism. We compute, at leading order in perturbation theory, the asymmetry for a variety of subsystems, including finite spherical subregions in flat space, in finite volume, and at positive temperature. We also study its Lorentzian time evolution, showcasing the dynamical restoration of the symmetry due to thermalization, as well as the presence of a quantum Mpemba effect. Our results are universal, and apply in any number of dimensions. We also show that the perturbative entanglement asymmetry is related to the Fisher information metric, which has a known holographic dual called the Hollands&#8211;Wald canonical energy, and that it is captured by the anti-de-Sitter bulk charge contained in the entanglement wedge.

Italy

Francesco Benini

Victor Godet

Amartya Harsh Singh

Entanglement Asymmetry in Conformal Field Theory and Holography

We derive the probability for chiral oscillation of Majorana neutrinos based on quantum field theory. Since the Hamiltonian under the Majorana mass term does not conserve the lepton number, the eigenstates of the lepton number change continuously over time. Therefore, the transition amplitude is described by the inner product of the eigenstates of the lepton number at the time of the neutrino production and the detection. With the Bogoliubov transformation, we successfully relate the lepton number eigenstates at different times. This method enables us to understand the time variation of the lepton number induced by chiral oscillations in terms of transition probabilities. We also present the physical picture that emerges through the Bogoliubov transformation.

Takuya Morozumi

Tomoharu Tahara

The Probability for Chiral Oscillation of Majorana Neutrino in Quantum Field Theory

The ’t Hooft partition function <inline-formula><tex-math id='TM0001' notation='LaTeX'>$\mathcal {Z}_{\text{tH}}[E;B]$</tex-math></inline-formula> of an <inline-formula><tex-math id='TM0002' notation='LaTeX'>$SU(N)$</tex-math></inline-formula> gauge theory with the <inline-formula><tex-math id='TM0003' notation='LaTeX'>$\mathbb {Z}_N$</tex-math></inline-formula> 1-form symmetry is defined as the Fourier transform of the partition function&#160; <inline-formula><tex-math id='TM0004' notation='LaTeX'>$\mathcal {Z}[B]$</tex-math></inline-formula> with respect to the spatial-temporal components of the &#8217;t&#160;Hooft flux&#160; <inline-formula><tex-math id='TM0005' notation='LaTeX'>$B$</tex-math></inline-formula>. Its large volume behavior detects the quantum phase of the system. When the integrand of the functional integral is real-positive, the latter partition function&#160; <inline-formula><tex-math id='TM0006' notation='LaTeX'>$\mathcal {Z}[B]$</tex-math></inline-formula> can be numerically computed by a Monte Carlo simulation of the <inline-formula><tex-math id='TM0007' notation='LaTeX'>$SU(N)/\mathbb {Z}_N$</tex-math></inline-formula> gauge theory, just by counting the number of configurations of a specific &#8217;t&#160;Hooft flux&#160; <inline-formula><tex-math id='TM0008' notation='LaTeX'>$B$</tex-math></inline-formula>. We carry out this program for the <inline-formula><tex-math id='TM0009' notation='LaTeX'>$SU(2)$</tex-math></inline-formula> pure Yang&#8211;Mills theory with the vanishing <inline-formula><tex-math id='TM0010' notation='LaTeX'>$\theta$</tex-math></inline-formula>-angle by employing a newly developed hybrid Monte Carlo (HMC) algorithm (the halfway HMC) for the <inline-formula><tex-math id='TM0011' notation='LaTeX'>$SU(N)/\mathbb {Z}_N$</tex-math></inline-formula> gauge theory. The numerical result clearly shows that all nonelectric fluxes are &#8220;light&#8221; as expected in the ordinary confining phase with the monopole condensate. Invoking the Witten effect on&#160; <inline-formula><tex-math id='TM0012' notation='LaTeX'>$\mathcal {Z}_{\text{tH}}[E;B]$</tex-math></inline-formula>, this also indicates the oblique confinement at&#160; <inline-formula><tex-math id='TM0013' notation='LaTeX'>$\theta =2\pi$</tex-math></inline-formula> with the dyon condensate.

Okuto Morikawa

Hiroshi Suzuki

Direct Monte Carlo Computation of the ’t Hooft Partition Function

We carry out a hybrid Monte Carlo (HMC) simulation of the <inline-formula><tex-math id='TM0002' notation='LaTeX'>$SU(2)/\mathbb {Z}_2$</tex-math></inline-formula> Yang&#8211;Mills theory in which the <inline-formula><tex-math id='TM0003' notation='LaTeX'>$\mathbb {Z}_N$</tex-math></inline-formula> 2-form flat gauge field (the &#8217;t&#160;Hooft flux) is explicitly treated as one of the dynamical variables. We observe that our HMC algorithm in the <inline-formula><tex-math id='TM0004' notation='LaTeX'>$SU(2)/\mathbb {Z}_2$</tex-math></inline-formula> theory drastically reduces autocorrelation lengths of the topological charge and of a physical quantity which couples to slow modes in the conventional HMC simulation of the <inline-formula><tex-math id='TM0005' notation='LaTeX'>$SU(2)$</tex-math></inline-formula> theory. Provided that sufficiently large lattice volumes are available, therefore, the HMC algorithm of the <inline-formula><tex-math id='TM0006' notation='LaTeX'>$SU(N)/\mathbb {Z}_N$</tex-math></inline-formula> theory could be employed as an alternative for the simulation of the <inline-formula><tex-math id='TM0007' notation='LaTeX'>$SU(N)$</tex-math></inline-formula> Yang&#8211;Mills theory, because local observables are expected to be insensitive to the difference between <inline-formula><tex-math id='TM0008' notation='LaTeX'>$SU(N)$</tex-math></inline-formula> and&#160; <inline-formula><tex-math id='TM0009' notation='LaTeX'>$SU(N)/\mathbb {Z}_N$</tex-math></inline-formula> in the large volume limit. A possible method to incorporate quarks [fermions in the fundamental representation of&#160; <inline-formula><tex-math id='TM0010' notation='LaTeX'>$SU(N)$</tex-math></inline-formula> with the baryon number&#160; <inline-formula><tex-math id='TM0011' notation='LaTeX'>$1/N$</tex-math></inline-formula>] in this framework is also considered.

Motokazu Abe

Monte Carlo Simulation of the <italic>SU</italic>(2)/&#8484; <sub>2</sub> Yang&#8211;Mills Theory

We analyze the correlation function in Jackiw–Teitelboim gravity using three approaches: by summing over all geodesics connecting boundary operators, integrating over the region of moduli space determined by the “no-shortcut condition,” and using the formula for the universal spectral density correlation in the <inline-formula><tex-math id='TM0001' notation='LaTeX'>$\tau$</tex-math></inline-formula>-scaling limit. We find that the behaviors of the three results coincide at late times: they all exhibit a &#8220;ramp&#8221; instead of permanent decay. Using the third approach we also confirm that the &#8220;plateau&#8221; appears after <inline-formula><tex-math id='TM0002' notation='LaTeX'>$T_H=2\pi e^{S_0}\widehat{\rho }_0(E)$</tex-math></inline-formula>. Overall, our results are consistent with the spectral form factor analysis. We also calculate the Einstein&#8211;Rosen bridge length <inline-formula><tex-math id='TM0003' notation='LaTeX'>$\langle \ell (T) \rangle$</tex-math></inline-formula> using the three approaches and find that the results are in good agreement with each other. We also find that the <inline-formula><tex-math id='TM0004' notation='LaTeX'>$\langle \ell (T) \rangle$</tex-math></inline-formula> grows as a cubic function in <inline-formula><tex-math id='TM0005' notation='LaTeX'>$T$</tex-math></inline-formula> due to the contribution from geometry including one observable baby universe, and converges to a constant after <inline-formula><tex-math id='TM0006' notation='LaTeX'>$T=T_H$</tex-math></inline-formula>. For the geometry with one baby universe, we compute the size <inline-formula><tex-math id='TM0007' notation='LaTeX'>$\langle b(T) \rangle$</tex-math></inline-formula> of the baby universe and find that it is of the same order as <inline-formula><tex-math id='TM0008' notation='LaTeX'>$\langle \ell (T) \rangle$</tex-math></inline-formula>. This result is consistent with the baby universe emission mechanism previously claimed.

Masayoshi Sato

What Happens Due to the Baby Universe Effect in Jackiw–Teitelboim Gravity? Analysis of Correlation Functions and Einstein–Rosen Bridge Length at Late Time Using Three Approaches

If a scalar field couples to the Ricci scalar with a large nonminimal coupling, the Standard Model coupling parameters can differ above and below an intermediate field range of the scalar due to the nonrenormalizability. In this paper, we study, for the first time, the threshold effects on a gauge coupling in both the metric and Palatini formulations of gravity. We find that the gauge coupling naturally jumps around this intermediate scale since counterterms for the renormalization behave so. If the gauge coupling becomes strong with a large scalar field value due to this effect, there can be an enhanced small instanton contribution, the dilute gas approximation of which is justified because the gauge sector decouples when the scalar wave mode is very short. Using these findings, we discuss the QCD axion quality problem, the heavy QCD axion, the axion abundance, and the suppression of isocurvature perturbations. We show that axion physics may differ substantially from naïve expectations when we introduce a large nonminimal coupling for any scalar field.

Juntaro Wada

Wen Yin

Gauge Coupling Jump and Small Instantons from a Large Nonminimal Coupling

We present an extension of the 3-3-1 gauge model supplemented by an <inline-formula><tex-math id='TM0001' notation='LaTeX'>$A_4$</tex-math></inline-formula> flavor symmetry and cyclic discrete symmetries, including <inline-formula><tex-math id='TM0002' notation='LaTeX'>$Z_2$</tex-math></inline-formula>, <inline-formula><tex-math id='TM0003' notation='LaTeX'>$Z_2^{\prime }$</tex-math></inline-formula>, <inline-formula><tex-math id='TM0004' notation='LaTeX'>$Z_3$</tex-math></inline-formula>, <inline-formula><tex-math id='TM0005' notation='LaTeX'>$Z_4$</tex-math></inline-formula>, <inline-formula><tex-math id='TM0006' notation='LaTeX'>$Z_7$</tex-math></inline-formula>, and <inline-formula><tex-math id='TM0007' notation='LaTeX'>$Z_{10}$</tex-math></inline-formula>. The model successfully reproduces the observed Standard Model fermion mass hierarchies and mixing patterns in the quark and lepton sectors. The smallness of the active neutrino masses is explained through an inverse seesaw mechanism, enabled by the introduction of right-handed and sterile Majorana neutrinos. In the quark sector, flavor-changing neutral currents arise at tree level exclusively for up-type quarks via the new heavy neutral gauge boson exchange, leading to strong constraints from <inline-formula><tex-math id='TM0008' notation='LaTeX'>$D^0$</tex-math></inline-formula>&#8211; <inline-formula><tex-math id='TM0009' notation='LaTeX'>$\bar{D}^0$</tex-math></inline-formula> mixing. The charged lepton sector exhibits sizeable flavor-violating effects, especially in the <inline-formula><tex-math id='TM0010' notation='LaTeX'>$\mu \rightarrow e\gamma$</tex-math></inline-formula> decay, mediated by loops involving heavy neutrinos and new charged gauge bosons as well as charged scalars. We perform a detailed numerical fit of fermion masses and mixing parameters and identify viable regions of parameter space consistent with experimental data on the Cabibbo&#8211;Kobayashi&#8211;Maskawa and Pontecorvo&#8211;Maki&#8211;Nakagawa&#8211;Sakata mixing matrices. The model predicts branching ratios for charged lepton flavor-violating decays and <inline-formula><tex-math id='TM0011' notation='LaTeX'>$\mu$</tex-math></inline-formula>&#8211; <inline-formula><tex-math id='TM0012' notation='LaTeX'>$e$</tex-math></inline-formula> conversion rates within the sensitivity of future experiments.