arXiv subjects

Takuya Tsukioka

Publications and source records attributed to Takuya Tsukioka.

At least 19 recordsLinked to original sources

arXiv2026

Non-perturbative on-shell multiplet structure of SU(N) Yang-Mills fields

Color multiplets of the gauge fields and fermions on mass shell in SU(N) Yang-Mills theory are classified according to representations of the Weyl group W(SU(N)). The multiplet structure of quark and gluon multiplets has been studied in the framework of a non-perturbative approach by considering complete exact equations of motion of the SU(N) Yang-Mills theory with matter fields. An important case of singlet non-Abelian gluon solutions corresponding to one-dimensional singlet representations of the Weyl group is revised on a rigorous mathematical basis. We demonstrate that Weyl group as a finite color subgroup of SU(N) reveals an inherent color symmetry of quarks and gluons on mass shell which determines a universal color miltiplet structure of quark-gluon solutions in a pure SU(N) Yang-Mills theory and in Abelian projected Yang-Mills theories with quarks. The obtained results allow to introduce strict concepts of fundamental particles, quarks and gluons, which differ drastically from the particle definitions in the conventional perturbative Yang-Mills theory. Possible applications of our results in non-perturbative quantum chromodynamics and hadron physics are discussed.

hep-th↗

arXiv2025

Generalized mKdV Equation and Genus Two Jacobi Type Hyperelliptic Differential Equation

We generalized the mKdV equation in order that the static equations include ${\rm sn}$ differential equation. As a result, a good correspondence was obtained between the KdV equation and the mKdV equation.For general genus two hyperelliptic curves, we obtained differential equations for Weierstrass type and Jacobi type hyperelliptic functions. Considering the special case of $λ_6=0, λ_0=0$, Weierstrass type and Jacobi type hyperelliptic functions are different solutions to the same hyperelliptic differential equations. Then these solutions are connected by the special ${\rm Sp(4, {\bf R})}$ Lie group transformation.

math.CA↗

arXiv2025

Weyl multiplet structure of QCD

Weyl group symmetric structure of SU(3) quantum chromodynamics (QCD) with one flavor quark is considered. It has been demonstrated that Weyl group as a finite color subgroup of SU(3) provides an intrinsic color symmetry of quark and gluon fields on mass shell. We show that standard QCD equations of motion lead to systems of equations for quarks and gluons forming non-trivial irreducible multiplets in one- and two-dimensional representations of the Weyl group. Some implications of Weyl multiplet structure in QCD and hadron physics are considered.

hep-th↗

arXiv2024

Gauss Metric on the Kummer Surface

On the Kummer surface, we have obtained two different Gauss metrices by parametrizing it in two ways. We have found that these Gauss metrices are not Ricci flat. The double sphere, which is the special case of the Kummer surface, has the Kähler metric and the first Chern class of it does not vanish. Its metric is the Einstein metric which is not Ricci flat.

math.CA↗

arXiv2024

A Quadratic Curve Analogue of the Taniyama-Shimura Conjecture

For quadratic curves over $F_p$, the number of solutions, which is governed by an analogue of the Mordell-Weil group, is expressed with the Legendre symbol of a coefficient of quadratic curves. Focusing on the number of solutions, a quadratic curve analogue of the modular form in the Taniyama-Shimura conjecture is proposed. This modular form yields the Gaussian sum and also possesses some modular transformation structure.

math.NT↗

arXiv2024

The Lie Group Structure of Elliptic/Hyperelliptic $\wp$ Functions

We consider the generalized dual transformation for elliptic/hyperelliptic $\wp$ functions up to genus three. For the genus one case, from the algebraic addition formula, we deduce that the Weierstrass $\wp$ function has the SO(2,1) $\cong$ Sp(2,$\R$)/$\Z_2$ Lie group structure. For the genus two case, by constructing a quadratic invariant form, we find that hyperelliptic $\wp$ functions have the SO(3,2) $\cong$ Sp(4,$\R$)/$\Z_2$ Lie group structure. Making use of quadratic invariant forms reveals that hyperelliptic $\wp$ functions with genus three have the SO(9,6) Lie group and/or it's subgroup structure.

nlin.SI↗

arXiv2022

Inherent color symmetry of quantum Yang-Mills theory

We present the basic non-perturbative structure of the space of classical dynamical solutions and corresponding one particle quantum states in SU(3) Yang-Mills theory. It has been demonstrated that the Weyl group of su(3) algebra plays an important role in constructing non-perturbative solutions and leads to profound changes in the structure of the classical and quantum Yang-Mills theory. We show that the Weyl group as a non-trivial color subgroup of SU(3) admits singlet irreducible representations on a space of classical dynamical solutions which lead to strict concepts of one particle quantum states for gluons and quarks. The Yang-Mills theory is a non-linear theory and, in general, it is not possible to construct a Hilbert space of classical solutions and quantum states as a linear vector space, so, usually, a perturbative approach is applied. We propose a non-perturbative approach based on Weyl symmetric solutions to full non-linear equations of motion and construct a full space of dynamical solutions representing an infinite but countable solution space classified by a finite set of integer numbers. It has been proved that the Weyl singlet structure of classical solutions provides the existence of a stable non-degenerate vacuum which serves as a main precondition of the color confinement phenomenon. Some physical implications in quantum chromodynamics are considered.

hep-th↗

arXiv2022

The Half-period Addition Formulae for Genus Two Hyperelliptic $\wp$ Functions and the Sp(4,$\mathbb{R}$) Lie Group Structure

In the previous study, by using the two-flows Kowalevski top, we have demonstrated that the genus two hyperelliptic functions provide the Sp(4,$\mathbb{R}$)/$Z_2$ $\cong$ SO(3,2) Lie algebra structure. In this study, by directly using the differential equations of the genus two hyperelliptic $\wp$ functions instead of using integrable models, we demonstrate that the half-period addition formula for the genus two hyperelliptic functions provides the order two Sp(4,$\mathbb{R}$) Lie group structure.

nlin.SI↗

arXiv2022

Two Flows Kowalevski Top as the Full Genus Two Jacobi's Inversion Problem and Sp(4,$\mathbb{R}$) Lie Group Structure

By using the first and the second flows of the Kowalevski top, we can make the Kowalevski top into the two flows Kowalevski top, which has two time variales. Then we show that equations of the two flows Kowalevski top become those of the full genus two Jacobi inversion problem. In addition to the Lax pair for the first flow, we costruct Lax pair for the second flow. Using the first and the second flows, we show that the Lie group structure of these two Lax pairs is Sp(4,$\mathbb{R}$) $\cong$ SO(3,2). Through the two flows Kowalevski top, we can conclude that the Lie group structure of the genus two hyperelliptic function is Sp(4,$\mathbb{R}$) $\cong$ SO(3,2).

nlin.SI↗

arXiv2021

Differential Equations of Genus Four Hyperelliptic $\wp$ Functions

In order to find higher dimensional integrable models, we study differential equations of hyperelliptic $\wp$ functions up to genus four. For genus two, differential equations of hyperelliptic $\wp$ functions can be written in the Hirota form. If the genus is more than one, we have KdV equation. If the genus is more than two, we have KdV and another KdV equations. If the genus becomes more than three, there appear differential equations which cannot be written in the Hirota form, which means that the Hirota form is not enough to characterize the integrable differential equations. We have shown that some differential equations are satisfied for general genus. We can obtain differential equations for general genus step by step.

nlin.SI↗

arXiv2020

Color structure of quantum SU(N) Yang-Mills theory

Color confinement is the most puzzling phenomenon in the theory of strong interaction based on a quantum SU(3) Yang-Mills theory. The origin of color confinement supposed to be intimately related to non-perturbative features of the non-Abelian gauge theory, and touches very foundations of the theory. We revise basic concepts underlying QCD concentrating mainly on concepts of gluons and quarks and color structure of quantum states. Our main idea is that a Weyl symmetry is the only color symmetry which determines all color attributes of quantum states and physical observables. We construct an ansatz for classical Weyl symmetric dynamical solutions in SU(3) Yang-Mills theory which describe one particle color singlet quantum states for gluons and quarks. Abelian Weyl symmetric solutions provide microscopic structure of a color invariant vacuum and vacuum gluon condensates. This resolves a problem of existence of a gauge invariant and stable vacuum in QCD. Generalization of our consideration to SU(N) (N=4,5) Yang-Mills theory implies that the color confinement phase is possible only in SU(3) Yang-Mills theory.

hep-th↗

arXiv2020

Color confinement and color singlet structure of quantum states in Yang-Mills theory

We consider two fundamental long-standing problems in quantum chromodynamics (QCD): the origin of color confinement and structure of a true vacuum and color singlet quantum states. There is a common belief that resolution to these problems needs a knowledge of a strict non-perturbative quantum Yang-Mills theory and new ideas. Our principal idea in resolving these problems is that structure of color confinement and color singlet quantum states must be determined by a Weyl symmetry which is an intrinsic symmetry of the Yang-Mills gauge theory, and by properties of a selected class of solutions satisfying special requirements. Following this idea we construct for the first time a space of color singlet one particle quantum states for primary gluons and quarks and reveal the structure of color confinement in quantum Yang-Mills theory. As an application we demonstrate formation of physical observables in a pure QCD, pure glueballs.

hep-th↗

arXiv2020

Elliptic Solutions for Higher Order KdV Equations

We study higher order KdV equations from the GL(2,$\mathbb{R}$) $\cong$ SO(2,1) Lie group point of view. We find elliptic solutions of higher order KdV equations up to the ninth order. We argue that the main structure of the trigonometric/hyperbolic/elliptic $N$-soliton solutions for higher order KdV equations is the same as that of the original KdV equation. Pointing out that the difference is only the time dependence, we find $N$-soliton solutions of higher order KdV equations can be constructed from those of the original KdV equation by properly replacing the time-dependence. We discuss that there always exist elliptic solutions for all higher order KdV equations.

nlin.SI↗

arXiv2020

Common Hirota Form Bäcklund Transformation for the Unified Soliton System

We study to unify soliton systems, KdV/mKdV/sinh-Gordon, through SO(2,1) $\cong$ GL(2,$\mathbb R$) $\cong$ Möbius group point of view, which might be a keystone to exactly solve some special non-linear differential equations. If we construct the $N$-soliton solutions through the KdV type Bäcklund transformation, we can transform different KdV/mKdV/sinh-Gordon equations and the Bäcklund transformations of the standard form into the same common Hirota form and the same common Bäcklund transformation except the equation which has the time-derivative term. The difference is only the time-dependence and the main structure of the $N$-soliton solutions has same common form for KdV/mKdV/sinh-Gordon systems. Then the $N$-soliton solutions for the sinh-Gordon equation is obtained just by the replacement from KdV/mKdV $N$-soliton solutions. We also give general addition formulae coming from the KdV type Bäcklund transformation which plays not only an important role to construct the trigonometric/hyperbolic $N$-soliton solutions but also an essential role to construct the elliptic $N$-soliton solutions. In contrast to the KdV type Bäcklund transformation, the well-known mKdV/sinh-Gordon type Bäcklund transformation gives the non-cyclic symmetric $N$-soliton solutions. We give an explicit non-cyclic symmetric 3-soliton solution for KdV/mKdV/sinh-Gordon equations.

nlin.SI↗

arXiv2019

The Unified Soliton System as the ${\rm AdS_2}$ System

We study the Riemann geometric approach to be aimed at unifying soliton systems. The general two-dimensional Einstein equation with constant scalar curvature becomes an integrable differential equation. We show that such Einstein equation includes KdV/mKdV/sine-Gordon equations.

nlin.SI↗

arXiv2019

The Static Elliptic $N$-soliton Solutions of the KdV Equation

Regarding $N$-soliton solutions, the trigonometric type, the hyperbolic type, and the exponential type solutions are well studied. While for the elliptic type solution, we know only the one-soliton solution so far. Using the commutative Bäcklund transformation, we have succeeded in constructing the KdV static elliptic $N$-soliton solution, which means that we have constructed infinitely many solutions for the $\wp$-function type differential equation.

math-ph↗

arXiv2019

The Construction of the mKdV Cyclic Symmetric $N$-soliton Solution by the Bäcklund Transformation

We study group theoretical structures of the mKdV equation. The Schwarzian type mKdV equation has the global Möbius group symmetry. The Miura transformation makes a connection between the mKdV equation and the KdV equation. We find the special local Möbius transformation on the mKdV one-soliton solution which can be regarded as the commutative KdV Bäcklund transformation can generate the mKdV cyclic symmetric $N$-soliton solution. In this algebraic construction to obtain multi-soliton solutions, we could observe the addition formula.

nlin.SI↗

arXiv2018

Stable spherically symmetric monopole field background in a pure QCD

We consider a stationary spherically symmetric monopole like solution with a finite energy density in a pure quantum chromodynamics (QCD). The solution can be treated as a static Wu-Yang monopole dressed in time dependent field corresponding to off-diagonal gluons. We have proved that such a stationary monopole field represents a background vacuum field of the QCD effective action which is stable against quantum gluon fluctuations. This resolves a long-standing problem of existence of a stable vacuum field in QCD and opens a new avenue towards microscopic theory of the vacuum.

hep-th↗