arXiv subjects

So Matsuura

Publications and source records attributed to So Matsuura.

At least 19 recordsLinked to original sources

arXiv2026

Generalized Kazakov-Migdal Models on Graphs via Artin-Ihara $L$-function and Random Partitions

We introduce Kazakov-Migdal (KM)-type gauge theories on graphs via the Artin-Ihara $L$-function, providing a unified description of the models proposed in prior works. Using harmonic analysis on the group manifold, we reformulate the KM-type model on the cycle graph as a random partition model governed by the Schur measure. We exactly solve the KM-type model on the cycle graph in the fundamental representation as the random partition model in the large $N_c$ limit and demonstrate that the Gross-Witten-Wadia phase transition occurs precisely when the limiting shape of the Young diagram touches the boundary of the allowed representation space. We further clarify that this phase transition is intimately related to Bose-Einstein condensation, and the strong/weak coupling duality possesses a natural combinatorial interpretation as the exchange between the Young diagram and its complement, reflecting the functional equation of the Artin-Ihara $L$-function. We also establish a definitive relationship between the eigenvalue density of the unitary matrix and the Maya diagram density of the random partitions by deriving a droplet picture from the spectral curve.

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arXiv2025

Fermions and Zeta Function on the Graph

We propose a novel fermionic model on the graphs. 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 $L$-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.

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arXiv2024

Functional Equations and Pole Structure of the Bartholdi Zeta Function

In this paper, we investigate the Bartholdi zeta function on a connected simple digraph with $n_V$ vertices and $n_E$ edges. We derive a functional equation for the Bartholdi zeta function $ζ_G(q,u)$ on a regular graph $G$ with respect to the bump parameter $u$. We also find an equivalence between the Bartholdi zeta function with a specific value of $u$ and the Ihara zeta function at $u=0$. We determine bounds of the critical strip of $ζ_G(q,u)$ for a general graph. If $G$ is a $(t+1)$-regular graph, the bounds are saturated and $q=(1-u)^{-1}$ and $q=(t+u)^{-1}$ are the poles at the boundaries of the critical strip for $u\ne 1, -t$. When $G$ is the regular graph and the spectrum of the adjacency matrix satisfies a certain condition, $ζ_G(q,u)$ satisfies the so-called Riemann hypothesis. For $u \ne 1$, $q=\pm(1-u)^{-1}$ are poles of $ζ_G(q,u)$ unless $G$ is tree. Although the order of the pole at $q=(1-u)^{-1}$ is $n_E-n_V+1$ if $u\ne u_* \equiv 1-\frac{n_E}{n_V}$, it is enhanced at $u=u_*$. In particular, if the Moore-Penrose inverse of the incidence matrix $L^+$ and the degree vector $\vec{d}$ satisfy the condition $|L^+ \vec{d}|^2\ne n_E$, the order of the pole at $q=(1-u)^{-1}$ increases only by one at $u=u_*$. The order of the pole at $q=-(1-u)^{-1}$ coincides with that at $q=(1-u)^{-1}$ if $G$ is bipartite and is $n_E-n_V$ otherwise.

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arXiv2024

Gross-Witten-Wadia Phase Transition in Induced QCD on the Graph

In this paper, we examine a modification of the Kazakov-Migdal (KM) model with gauge group $U(N_c)$, where the adjoint scalar fields in the conventional KM model are replaced by $N_f$ fundamental scalar fields (FKM model). After tuning the coupling constants and eliminating the fundamental scalar fields, the partition function of this model is expressed as an integral of a graph zeta function weighted by unitary matrices. The FKM model on cycle graphs at large $N_c$ exhibits the Gross-Witten-Wadia (GWW) phase transition only when $N_f > N_c$. In the large $N_c$ limit, we evaluate the free energy of the model on a general graph in two distinct parameter regimes and demonstrate that the FKM model generally consists of multiple phases. The effective action of the FKM model reduces to the standard Wilson action by taking an appropriate scaling limit when the graph consists of plaquettes (fundamental cycles) of the same size, as in the square lattice case. We show that, for the FKM model on such a graph, the third-order GWW phase transition universally occurs in this scaling limit.

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arXiv2024

Phases and Duality in Fundamental Kazakov-Migdal Model on the Graph

We examine the fundamental Kazakov-Migdal (FKM) model on a generic graph, whose partition function is represented by the Ihara zeta function weighted by unitary matrices. The FKM model becomes unstable in the critical strip of the Ihara zeta function. We discover a duality between small and large couplings, associated with the functional equation of the Ihara zeta function for regular graphs. Although the duality is not precise for irregular graphs, we show that the effective action in the large coupling region can be represented by a summation of all possible Wilson loops on the graph similar to that in the small coupling region. We estimate the phase structure of the FKM model both in the small and large coupling regions by comparing it with the Gross-Witten-Wadia (GWW) model. We further validate the theoretical analysis through detailed numerical simulations.

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arXiv2022

Graph Zeta Functions and Wilson Loops in Kazakov-Migdal Model

In this paper, we consider an extended Kazakov-Migdal model defined on an arbitrary graph. The partition function of the model, which is expressed as the summation of all Wilson loops on the graph, turns out to be represented by the Bartholdi zeta function weighted by unitary matrices on the edges of the graph. The partition function on the cycle graph at finite $N$ is expressed by the generating function of the generalized Catalan numbers. The partition function on an arbitrary graph can be exactly evaluated at large $N$ which is expressed as an infinite product of a kind of deformed Ihara zeta function. The non-zero area Wilson loops do not contribute to the leading part of the $1/N$-expansion of the free energy but to the next leading. The semi-circle distribution of the eigenvalues of the scalar fields is still an exact solution of the model at large $N$ on an arbitrary regular graph, but it reflects only zero-area Wilson loops.

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arXiv2022

Kazakov-Migdal model on the Graph and Ihara Zeta Function

We propose the Kazakov-Migdal model on graphs and show that, when the parameters of this model are appropriately tuned, the partition function is represented by the unitary matrix integral of an extended Ihara zeta function, which has a series expansion by all non-collapsing Wilson loops with their lengths as weights. The partition function of the model is expressed in two different ways according to the order of integration. A specific unitary matrix integral can be performed at any finite $N$ thanks to this duality. We exactly evaluate the partition function of the parameter-tuned Kazakov-Migdal model on an arbitrary graph in the large $N$ limit and show that it is expressed by the infinite product of the Ihara zeta functions of the graph.

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arXiv2022

Supersymmetric Gauge Theory on the Graph

We consider two-dimensional N=(2,2) supersymmetric gauge theory on discretized Riemann surfaces. We find that the discretized theory can be efficiently described by using graph theory, where the bosonic and fermionic fields are regarded as vectors on a graph and its dual. We first analyze the Abelian theory and identify its spectrum in terms of graph theory. In particular, we show that the fermions have zero modes corresponding to the topology of the graph, which can be understood as kernels of the incidence matrices of the graph and the dual graph. In the continuous theory, a scalar curvature appears as an anomaly in the Ward-Takahashi (WT) identity associated with a U(1) symmetry. We find that the same anomaly arises as the deficit angle at each vertex on the graph. By using the localization method, we show that the path integral on the graph reduces to an integral over a set of the zero modes. The partition function is then ill-defined unless suitable operators are inserted. We extend the same argument to the non-Abelian theory and show that the path integral reduces to multiple integrals of Abelian theories at the localization fixed points.

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arXiv2018

Commutative Geometry for Non-commutative D-branes by Tachyon Condensation

There is a difficulty in defining the positions of the D-branes when the scalar fields on them are non-abelian. We show that we can use tachyon condensation to determine the position or the shape of D0-branes uniquely as a commutative region in spacetime together with non-trivial gauge flux on it, even if the scalar fields are non-abelian. We use the idea of the so-called coherent state method developed in the field of matrix models in the context of the tachyon condensation. We investigate configurations of noncommutative D2-brane made out of D0-branes as examples. In particular, we examine a Moyal plane and a fuzzy sphere in detail, and show that whose shapes are commutative $\mathbb{R}^2$ and $S^2$, respectively, equipped with uniform magnetic flux on them. We study the physical meaning of this commutative geometry made out of matrices, and propose an interpretation in terms of K-homology.

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arXiv2018

O(a) Improvement of 2D N=(2,2) Lattice SYM Theory

We perform a tree-level O(a) improvement of two-dimensional N=(2,2) supersymmetric Yang-Mills theory on the lattice, motivated by the fast convergence in numerical simulations. The improvement respects an exact supersymmetry Q which is needed for obtaining the correct continuum limit without a parameter fine tuning. The improved lattice action is given within a milder locality condition in which the interactions are decaying as the exponential of the distance on the lattice. We also prove that the path-integral measure is invariant under the improved Q-transformation.

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arXiv2017

Spherical D-brane by Tachyon Condensation

We find a novel tachyon condensation which provides a D-brane system with spherical worldvolume in the flat spacetime. The tachyon profile is a deformation of a known D0-brane solution on non-BPS D3-branes in type IIA superstring theory, which realizes a bound state of a spherical D2-brane and a D0-brane. The D0-brane is resolved into the sphere as a $U(1)$ monopole flux of the unit magnetic charge. We show that the system has the correct tension and the RR-coupling. Although the low energy effective action of the system is the same as that of the dual description of the fuzzy sphere solution of multiple D0-branes, our system cannot be equivalent to the fuzzy sphere. We argue that the sphere can be stable in a certain RR-flux background. The use of projective modules in describing the tachyon condensation is emphasized.

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arXiv2016

Numerical Analysis of Discretized ${\cal N}=(2,2)$ SYM on Polyhedra

We perform a numerical simulation of the two-dimensional ${\cal N}=(2,2)$ supersymmetric Yang-Mills (SYM) theory on the discretized curved space. The $U(1)_{A}$ anomaly of the continuum theory is maintained also in the discretized theory as an unbalance of the number of the fermions. In the process, we propose a new phase-quenched approximation, which we call the "anomaly-phase-quenched (APQ) method", to make the partition function and observables well-defined by $U(1)_{A}$ phase cancellation. By adopting APQ method, we estimate the Ward-Takahashi identity for exact SUSY on lattice and clarify contribution of the pseudo zero-modes to the pfaffian phase.

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arXiv2016

Anomaly and Sign problem in $\mathcal{N}=(2,2)$ SYM on Polyhedra : Numerical Analysis

We investigate the two-dimensional $\mathcal{N}=(2,2)$ supersymmetric Yang-Mills (SYM) theory on the discretized curved space (polyhedra). We first revisit that the number of supersymmetries of the continuum $\mathcal{N}=(2,2)$ SYM theory on any curved manifold can be enhanced at least to two by introducing an appropriate $U(1)$ gauge background associated with the $U(1)_{V}$ symmetry. We then show that the generalized Sugino model on the discretized curved space, which was proposed in our previous work, can be identified to the discretization of this SUSY enhanced theory, where one of the supersymmetries remains and the other is broken but restored in the continuum limit. We find that the $U(1)_{A}$ anomaly exists also in the discretized theory as a result of an unbalance of the number of the fermions proportional to the Euler characteristics of the polyhedra. We then study this model by using the numerical Monte-Carlo simulation. We propose a novel phase-quench method called "anomaly-phase-quenched approximation" with respect to the $U(1)_A$ anomaly. We numerically show that the Ward-Takahashi (WT) identity associated with the remaining supersymmetry is realized by adopting this approximation. We figure out the relation between the sign (phase) problem and pseudo-zero-modes of the Dirac operator. We also show that the divergent behavior of the scalar one-point function gets milder as the genus of the background increases. These are the first numerical observations for the supersymmetric lattice model on the curved space with generic topologies.

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arXiv2016

A one-loop test for construction of 4D N=4 SYM from 2D SYM via fuzzy sphere geometry

As a perturbative check of the construction of four-dimensional (4D) ${\cal N}=4$ supersymmetric Yang-Mills theory (SYM) from mass deformed ${\cal N}=(8,8)$ SYM on the two-dimensional (2D) lattice, the one-loop effective action for scalar kinetic terms is computed in ${\cal N}=4$ $U(k)$ SYM on ${\mathbb R}^2 \times$ (fuzzy $S^2$), which is obtained by expanding 2D ${\cal N}=(8,8)$ $U(N)$ SYM with mass deformation around its fuzzy sphere classical solution. The radius of the fuzzy sphere is proportional to the inverse of the mass. We consider two successive limits: (1) decompactify the fuzzy sphere to a noncommutative (Moyal) plane and (2) turn off the noncommutativity of the Moyal plane. It is straightforward at the classical level to obtain the ordinary ${\cal N}=4$ SYM on ${\mathbb R}^4$ in the limits, while it is nontrivial at the quantum level. The one-loop effective action for $SU(k)$ sector of the gauge group $U(k)$ coincides with that of the ordinary 4D ${\cal N}=4$ SYM in the above limits. Although "noncommutative anomaly" appears in the overall $U(1)$ sector of the $U(k)$ gauge group, this can be expected to be a gauge artifact not affecting gauge invariant observables.

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arXiv2016

Ising Model on Twisted Lattice and Holographic RG flow

The partition function of the two-dimensional Ising model is exactly obtained on a lattice with a twisted boundary condition. The continuum limit of the model off the critical temperature is found to give the mass-deformed Ising conformal field theory (CFT) on the torus with the complex structure $τ$. We find that the renormalization group (RG) flow of the mass parameter can be holographically described in terms of the three-dimensional gravity including a scalar field with a simple nonlinear kinetic function and a quadratic potential.

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arXiv2015

Exact Results in Discretized Gauge Theories

We apply the localization technique to topologically twisted N=(2,2) supersymmetric gauge theory on a discretized Riemann surface (the generalized Sugino model). We exactly evaluate the partition function and the vacuum expectation value (vev) of a specific Q-closed operator. We show that both the partition function and the vev of the operator depend only on the Euler characteristic and the area of the discretized Riemann surface and are independent of the detail of the discretization. This localization technique may not only simplify numerical analysis of the supersymmetric lattice models but also connect the well-defined equivariant localization to the empirical supersymmetric localization.

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arXiv2014

Topologically Twisted $N=(2,2)$ Supersymmetric Yang-Mills Theory on Arbitrary Discretized Riemann Surface

We define supersymmetric Yang-Mills theory on an arbitrary two-dimensional lattice (polygon decomposition) with preserving one supercharge. When a smooth Riemann surface $Σ_g$ with genus $g$ emerges as an appropriate continuum limit of the generic lattice, the discretized theory becomes topologically twisted $\mathcal{N}=(2,2)$ supersymmetric Yang-Mills theory on $Σ_g$. If we adopt the usual square lattice as a special case of the discretization, our formulation is identical with Sugino's lattice model. Although the tuning of parameters is generally required while taking the continuum limit, the number of the necessary parameters is at most two because of the gauge symmetry and the supersymmetry. In particular, we do not need any fine-tuning if we arrange the theory so as to possess an extra global U(1) symmetry ($U(1)_{R}$ symmetry) which rotates the scalar fields.

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arXiv2014

Lattice Formulation for 2d N=(2,2), (4,4) Super Yang-Mills Theories without Admissibility Conditions

We present a lattice formulation for two-dimensional N=(2,2) and (4,4) supersymmetric Yang-Mills theories that resolves vacuum degeneracy for gauge fields without imposing admissibility conditions. Cases of U(N) and SU(N) gauge groups are considered, gauge fields are expressed by unitary link variables, and one or two supercharges are preserved on the two-dimensional square lattice. There does not appear fermion doubler, and no fine-tuning is required to obtain the desired continuum theories in a perturbative argument. This formulation is expected to serve as a more convenient basis for numerical simulations. The same approach will also be useful to other two-dimensional supersymmetric lattice gauge theories with unitary link variables constructed so far -- for example, N=(8,8) supersymmetric Yang-Mills theory and N=(2,2) supersymmetric QCD.

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