Self-dual Yang-Mills Research Articles

Nonperturbative aspects of spontaneous symmetry breaking

Spontaneous symmetry breaking is studied in the ultralocal limit of a scalar quantum field theory, that is when [Formula: see text] (or infrared limit). In this infrared approximation the theory [Formula: see text] is formally two-dimensional and its Euclidean solutions are instantons. For BPST-like solutions with [Formula: see text], the map between [Formula: see text] in two dimensions and self-dual Yang–Mills theory is carefully discussed.

On the symbolic computation of exact solutions and conservation laws of a generalized (2+1)-dimensional Calogaro-Bogoyavlenskii-Schiff equation

New exact solutions of the (2+1)-dimensional Calogaro-Bogoyavlenskii-Schiff equation are presented. This equation was first erected by Bogoyavlenskii and Schiff in unique ways, namely Bogoyavlenskii used the modified Lax formalism, whereas Schiff employed the same equation by reducing the self-dual Yang Mills equation. We also derive conserved vectors of the above mentioned equation by invoking the multiplier method.

Isospectral deformation of the reduced quasi-classical self-dual Yang–Mills equation

We derive a new four-dimensional partial differential equation with the isospectral Lax representation by shrinking the symmetry algebra of the reduced quasi-classical self-dual Yang–Mills equation and applying the technique of twisted extensions to the obtained Lie algebra. Then we find a recursion operator for symmetries of the new equation and construct a Bäcklund transformation between this equation and the four-dimensional Martínez Alonso–Shabat equation. Finally, we construct extensions of the integrable hierarchies associated to the hyper-CR equation for Einstein–Weyl structures, the reduced quasi-classical self-dual Yang–Mills equation, the four-dimensional universal hierarchy equation, and the four-dimensional Martínez Alonso–Shabat equation.

Optimal Łojasiewicz–Simon inequalities and Morse–Bott Yang–Mills energy functions

Abstract For any compact Lie group 𝐺 and closed, smooth Riemannian manifold ( X , g ) (X,g) of dimension d ≥ 2 d\geq 2 , we extend a result due to Uhlenbeck (1985) that gives existence of a flat connection on a principal 𝐺-bundle over 𝑋 supporting a connection with L p L^{p} -small curvature, when p > d / 2 p>d/2 , to the case of a connection with L d / 2 L^{d/2} -small curvature. We prove an optimal Łojasiewicz–Simon gradient inequality for abstract Morse–Bott functions on Banach manifolds, generalizing an earlier result due to the author and Maridakis (2019), principally by removing the hypothesis that the Hessian operator be Fredholm with index zero. We apply this result to prove the optimal Łojasiewicz–Simon gradient inequality for the self-dual Yang–Mills energy function near regular anti-self-dual connections over closed Riemannian four-manifolds and for the full Yang–Mills energy function over closed Riemannian manifolds of dimension d ≥ 2 d\geq 2 , when known to be Morse–Bott at a given Yang–Mills connection. We also prove the optimal Łojasiewicz–Simon gradient inequality by direct analysis near a given flat connection that is a regular point of the curvature map.

Solitons of Some Nonlinear Sigma-Like Models

We present a set of differential identities for some class of matrices. These identities are used to derive the N-soliton solutions for the Pohlmeyer nonlinear sigma-model, two-dimensional self-dual Yang-Mills equations and some modification of the vector Calapso equation.

Manifestly-invariant action for self-dual supersymmetric Yang-Mills theories coupled to auxiliary supergravity

We propose a new action-formulation for the N=(1,1) multiplet (AμI,λI) of a self-dual (SD) supersymmetric Yang-Mills (YM) theory in D=2+2 dimensions using an auxiliary supergravity multiplet. We first consider non-supersymmetric case with constraint lagrangians that restrict the metric gμν to be (ημν)=diag.(+,+,−,−). We add the kinetic-term of a purely-bosonic YM-field, and show that the YM-field in the D=2+2 space-time allows only SD or anti-SD (ASD) configurations. In other words, the condition of vanishing energy-momentum (EM) tensor restricts the YM-field to be either SD or ASD, but not an admixture of them. Finally, we generalize this result to N=(1,1) supersymmetric YM multiplet (AμI,λI;DI) in D=2+2 with the off-shell supergravity multiplet (eμm,ψμ;S,P,am) as an ‘auxiliary’ multiplet without its kinetic terms. Special constraint lagrangians restrict the supergravity multiplet to be auxiliary, i.e., both the Riemann-tensor and the gravitino field strength to vanish. We show that the eμm and ψμ-field equations produce the vanishings of both the EM-tensor and spinor-current, which in turn lead only to either SD or ASD supersymmetric YM multiplet, as the only Lorentz and gauge-covariant supersymmetric solutions. A broad class of similar action-formulations with auxiliary supergravity multiplets yielding SD or ASD systems in D=2+2 or higher-dimensions may be constructed in a similar fashion.

Fractional derivative of inverse matrix and its applications to soliton theory

In this paper, a formula of the local fractional partial derivative of inverse matrix is presented and proved. With the help of the derived formula, two new non-linear PDE are derived including the local fractional non-isospectral self-dual Yang-Mills equation and the local fractional principal chiral field equation. It is shown that the formula of the local fractional partial derivative of inverse matrix can be used to derive some other local fractional non-linear PDE in soliton theory.

Theory of Multidimensional Delsarte–Lions Transmutation Operators. II

The differential-geometric and topological structures related to the Delsarte transmutation operators and the Gelfand–Levitan–Marchenko equations that describe these operators are studied by using suitable differential de Rham–Hodge–Skrypnik complexes. The correspondence between the spectral theory and special Berezansky-type congruence properties of the Delsarte transmutation operators is established. Some applications to multidimensional differential operators are presented, including the three-dimensional Laplace operator, the two-dimensional classical Dirac operator, and its multidimensional affine extension associated with self-dual Yang–Mills equations. The soliton solutions are discussed for a certain class of dynamical systems.

Minitwistors and 3d Yang-Mills-Higgs theory

We construct a minitwistor action for Yang–Mills–Higgs (YMH) theory in three dimensions. The Feynman diagrams of this action will construct perturbation theory around solutions of the Bogomolny equations in much the same way that MHV (maximally helicity violating) diagrams describe perturbation theory around the self-dual Yang Mills equations in four dimensions. We also provide a new formula for all tree amplitudes in YMH theory (and its maximally supersymmetric extension) in terms of degree d maps to minitwistor space. We demonstrate its relationship to the Roiban-Spradlin-Volovich-Witten (RSVW) formula in four dimensions and show that it generates the correct MHV amplitudes at d = 1 and factorizes correctly in all channels for all degrees.

5a-A-10 2次元Derivative Nonlinear Schrodinger方程式と自己双対Yang-Mills方程式

5a-A-10 2次元Derivative Nonlinear Schrodinger方程式と自己双対Yang-Mills方程式

Ultraviolet properties of the self-dual Yang-Mills theory

We compute the ultraviolet divergences in the self-dual Yang-Mills theory, both in the purely perturbative (zero instanton charge) and topologically non-trivial sectors. It is shown in particular that the instanton measure is precisely the same as the one-loop result in the standard Yang-Mills theory.

Yang–Mills instantons in Kähler spaces with one holomorphic isometry

We consider self-dual Yang–Mills instantons in 4-dimensional Kähler spaces with one holomorphic isometry and show that they satisfy a generalization of the Bogomol'nyi equation for magnetic monopoles on certain 3-dimensional metrics. We then search for solutions of this equation in 3-dimensional metrics foliated by 2-dimensional spheres, hyperboloids or planes in the case in which the gauge group coincides with the isometry group of the metric (SO(3), SO(1,2) and ISO(2), respectively). Using a generalized hedgehog ansatz the Bogomol'nyi equations reduce to a simple differential equation in the radial variable which admits a universal solution and, in some cases, a particular one, from which one finally recovers instanton solutions in the original Kähler space. We work out completely a few explicit examples for some Kähler spaces of interest.

On a reduction of the generalized Darboux–Halphen system

The equations for the general Darboux–Halphen system obtained as a reduction of the self-dual Yang–Mills can be transformed to a third-order system which resembles the classical Darboux–Halphen system with a common additive terms. It is shown that the transformed system can be further reduced to a constrained non-autonomous, non-homogeneous dynamical system. This dynamical system becomes homogeneous for the classical Darboux–Halphen case, and was studied in the context of self-dual Einstein's equations for Bianchi IX metrics. A Lax pair and Hamiltonian for this reduced system is derived and the solutions for the system are prescribed in terms of hypergeometric functions.

Chiral higher spin theories and self-duality

We study recently proposed chiral higher spin theories — cubic theories of interacting massless higher spin fields in four-dimensional flat space. We show that they are naturally associated with gauge algebras, which manifest themselves in several related ways. Firstly, the chiral higher spin equations of motion can be reformulated as the self-dual Yang-Mills equations with the associated gauge algebras instead of the usual colour gauge algebra. We also demonstrate that the chiral higher spin field equations, similarly to the self-dual Yang-Mills equations, feature an infinite algebra of hidden symmetries, which ensures their integrability. Secondly, we show that off-shell amplitudes in chiral higher spin theories satisfy the generalised BCJ relations with the usual colour structure constants replaced by the structure constants of higher spin gauge algebras. We also propose generalised double copy procedures featuring higher spin theory amplitudes. Finally, using the light-cone deformation procedure we prove that the structure of the Lagrangian that leads to all these properties is universal and follows from Lorentz invariance.

Finite-action solutions of Yang-Mills equations on de Sitter dS4 and anti-de Sitter AdS4 spaces

We consider pure SU(2) Yang-Mills theory on four-dimensional de Sitter dS4 and anti-de Sitter AdS4 spaces and construct various solutions to the Yang-Mills equations. On de Sitter space we reduce the Yang-Mills equations via an SU(2)-equivariant ansatz to Newtonian mechanics of a particle moving in {mathbb{R}}^3 under the influence of a quartic potential. Then we describe magnetic and electric-magnetic solutions, both Abelian and non-Abelian, all having finite energy and finite action. A similar reduction on anti-de Sitter space also yields Yang-Mills solutions with finite energy and action. We propose a lower bound for the action on both backgrounds. Employing another metric on AdS4, the SU(2) Yang-Mills equations are reduced to an analytic continuation of the above particle mechanics from {mathbb{R}}^3 to {mathbb{R}}^{2,1} . We discuss analytical solutions to these equations, which produce infinite-action configurations. After a Euclidean continuation of dS4 and AdS4 we also present self-dual (instanton-type) Yang-Mills solutions on these backgrounds.

Linearly degenerate hierarchies of quasiclassical SDYM type

We demonstrate that SDYM (self-dual Yang-Mills) equations for the Lie algebra of one-dimensional vector fields represent a natural reduction in the framework of a general linearly degenerate dispersionless hierarchy. We define the reduction in terms of wave functions and introduce a generating relation, Lax-Sato equations, and the dressing scheme for the reduced hierarchy. A multidimensional case is also discussed.

Wilson lines in the MHV action

The MHV action is the Yang-Mills action quantized on the light-front, where the two explicit physical gluonic degrees of freedom have been canonically transformed to a new set of fields. This transformation leads to the action with vertices being off-shell continuations of the MHV amplitudes. We show that the solution to the field transformation expressing one of the new fields in terms of the Yang-Mills field is a certain type of the Wilson line. More precisely, it is a straight infinite gauge link with a slope extending to the light-cone minus and the transverse direction. One of the consequences of that fact is that certain MHV vertices reduced partially on-shell are gauge invariant — a fact discovered before using conventional light-front perturbation theory. We also analyze the diagrammatic content of the field transformations leading to the MHV action. We found that the diagrams for the solution to the transformation (given by the Wilson line) and its inverse differ only by light-front energy denominators. Further, we investigate the coordinate space version of the inverse solution to the one given by the Wilson line. We find an explicit expression given by a power series in fields. We also give a geometric interpretation to it by means of a specially defined vector field. Finally, we discuss the fact that the Wilson line solution to the transformation is directly related to the all-like helicity gluon wave function, while the inverse functional is a generating functional for solutions of self-dual Yang-Mills equations.

Supersymmetric self-dual Yang–Mills theories from local nilpotent fermionic symmetry

We present a system of a self-dual vector-spinor and a self-dual Yang–Mills (YM) field with local nilpotent fermionic symmetry (but not supersymmetry) in D=2+2 dimensions that embeds self-dual supersymmetric YM theory as a special set of exact solutions. Our system has local nilpotent fermionic symmetry generator NαI satisfying the algebra {NαI,NβJ}=0 with the adjoint index I of an arbitrary gauge group. Our original field content in D=2+2 is (AμI,ψμI,χI), where AμI is the usual YM gauge field, ψμI is a Majorana–Weyl vector-spinor gauging NαI, while χI is a Majorana–Weyl spinor compensator field needed for consistency. This system embeds self-dual supersymmetric YM system with the field content (AμI,λ−I) in D=2+2. As other examples, we consider similar systems in D=7+0 and D=8+0 embedding respectively N=1/8+7/8 and N=(1/8,1) supersymmetric YM theories with generalized self-dualities, such as FμνI=(1/2)fμνρσFρσI with a generalized octonionic structure constant fμνρσ. This result strongly suggests that our local nilpotent fermionic symmetry is more fundamental than the supersymmetric self-dual Yang–Mills systems that are supposed to generate all supersymmetric integrable models in D<4.

SDYM equations on the self-dual background

We introduce the technique combining the features of integration schemes for self-dual Yang–Mills (SDYM) equations and multidimensional dispersionless integrable equations to get SDYM equations on the conformally self-dual background. Generating differential form is defined, and the dressing scheme is developed. Some special cases and reductions are considered.

Three New (2+1)-dimensional Integrable Systems and Some Related Darboux Transformations**Supported by the National Natural Science Foundation of China under Grant No. 11371361, the Shandong Provincial Natural Science Foundation of China under Grant Nos. ZR2012AQ011, ZR2013AL016, ZR2015EM042, National Social Science Foundation of China under Grant No. 13BJY026, the Development of Science and Technology Project under Grant No. 2015NS1048 and A Project

We introduce two operator commutators by using different-degree loop algebras of the Lie algebra A1, then under the framework of zero curvature equations we generate two (2+1)-dimensional integrable hierarchies, including the (2+1)-dimensional shallow water wave (SWW) hierarchy and the (2+1)-dimensional Kaup-Newell (KN) hierarchy. Through reduction of the (2+1)-dimensional hierarchies, we get a (2+1)-dimensional SWW equation and a (2+1)-dimensional KN equation. Furthermore, we obtain two Darboux transformations of the (2+1)-dimensional SWW equation. Similarly, the Darboux transformations of the (2+1)-dimensional KN equation could be deduced. Finally, with the help of the spatial spectral matrix of SWW hierarchy, we generate a (2+1) heat equation and a (2+1) nonlinear generalized SWW system containing inverse operators with respect to the variables x and y by using a reduction spectral problem from the self-dual Yang-Mills equations.