Integrability and conformal invariance

Modern Physics Letters A, 1994

We construct infinite sets of local conserved charges for the conformal affine Toda model. The technique involves the abelianization of the two-dimensional gauge potentials satisfying the zero-curvature form of the equations of motion. We find two infinite sets of chiral charges and apart from two lowest spin charges, all the remaining ones do not possess chiral densities. Charges of different chiralities Poisson commute among themselves. We discuss the algebraic properties of these charges and use the fundamental Poisson bracket relation to show that the charges conserved in time are in involution. Connections to other Toda models are established by taking particular limits.

2019

Based on the diffeomorphism group vector fields on the complexified torus and the related Lie-algebraic structures, we study multi-dimensional dispersionless integrable systems, describing conformal structure generating equations of mathematical physics. An interesting modification of the devised Liealgebraic approach subject to the spatial dimensional invariance and meromorphicity of the related differential-geometric structures is described and applied to proving complete integrability of some conformal structure generating equations. As examples, we analyzed the Einstein–Weyl metric equation, the modified Einstein–Weyl metric equation, the Dunajski heavenly equation system, the first and second conformal structure generating equations, the inverse first Shabat reduction heavenly equation, the first and modified Plebański heavenly equations and its multi-dimensional generalizations, the Husain heavenly equation and its multi-dimensional generalizations, the general Monge equation ...

Journal of High Energy Physics, 2002

We review concepts of integrability in higher dimensions and apply them to construct Lorentz invariant field theories with an infinite number of local conserved currents.

2019

Based on the diffeomorphism group vector fields on the complexified torus and the related Lie-algebraic structures, we study multi-dimensional dispersionless integrable systems, describing conformal structure generating equations of mathematical physics. An interesting modification of the devised Lie-algebraic approach subject to the spatial dimensional invariance and meromorphicity of the related differential-geometric structures is described and applied to proving complete integrability of some conformal structure generating equations. As examples, we analyzed the Einstein--Weyl metric equation, the modified Einstein--Weyl metric equation, the Dunajski heavenly equation system, the first and second conformal structure generating equations, the inverse first Shabat reduction heavenly equation, the first and modified Pleba\'nski heavenly equations and its multi-dimensional generalizations, the Husain heavenly equation and its multi-dimensional generalizations, the general Monge ...

Based on the vector elds on the complexi ed torus and the related Lie-algebraic structures, we devise an approach to constructing multidimensional dispersionless integrable systems, describing conformal structure generating equations of mathematical physics. As examples, we have analyzed Einstein–Weyl metric equation, the modi ed Einstein– Weyl metric equation, the Dunajski heavenly equations, rst and second conformal structure generating equations, inverse rst Shabat reduction heavenly equation, rst Plebański heavenly equation, modi ed Plebański equation and Husain heavenly equation. 1. Vector fields on the complexified torus TC and the related Lie-algebraic properties It is well known [13] that the loop Lie algebra ~ G := ] diff(T); consisting of the set of smooth mappings fC S ! G = diff(Tg; extended, respectively, holomorphically from the circle S C on the disc D+ of the internal points 2 D and on the disc D of the external points 2 CnD; can be centrally extended as b G := (...

Physics Letters B, 1992

We present a superconformally invariant and integrable model based on the twisted affine Kac-Moody superalgebraôsp(2|2) (2) which is the supersymmetrization of the purely bosonic conformal affine Liouville theory recently proposed by Babelon and Bonora. Our model reduces to the super-Liouville or to the super sinh-Gordon theories under certain limit conditions and can be obtained, via hamiltonian reduction, from a superspace WZNW model with values in the corresponding affine KM supergroup. The reconstruction formulae for classical solutions are given. The classical r-matrices in the homogeneous grading and the exchange algebras are worked out.

Symmetry, Integrability and Geometry: Methods and Applications

Using diffeomorphism group vector fields on C-multiplied tori and the related Lie-algebraic structures, we study multi-dimensional dispersionless integrable systems that describe conformal structure generating equations of mathematical physics. An interesting modification of the devised Lie-algebraic approach subject to spatial-dimensional invariance and meromorphicity of the related differential-geometric structures is described and applied in proving complete integrability of some conformal structure generating equations. As examples, we analyze the Einstein-Weyl metric equation, the modified Einstein-Weyl metric equation, the Dunajski heavenly equation system, the first and second conformal structure generating equations and the inverse first Shabat reduction heavenly equation. We also analyze the modified Plebański heavenly equations, the Husain heavenly equation and the general Monge equation along with their multi-dimensional generalizations. In addition, we construct superconformal analogs of the Whitham heavenly equation.

Journal of Physics A: Mathematical and Theoretical, 2018

We explain how to incorporate the action of local integrals of motion into the fermionic basis for the sine-Gordon model and its UV CFT. The examples up to the level 4 are presented. Numerical computation support the results. Possible applications are discussed.

Physics Letters B, 1993

We present a new supersymmetric integrable model: the N = 2 superconformal affine Liouville theory. It interpolates between the N = 2 super Liouville and N = 2 super sine-Gordon theories and possesses a Lax representation on the complex affine Kac-Moody superalgebraŝl(2|2) (1) . We show that the higher spin W 1+∞ -type symmetry algebra of ordinary conformal affine Liouville theory extends to a N = 2 W 1/2+∞ -type superalgebra.

Theoretical and Mathematical Physics

The symmetry algebra of the real elliptic Liouville equation is an infinite-dimensional loop algebra with the simple Lie algebra o(3, 1) as its maximal finite-dimensional subalgebra. The entire algebra generates the conformal group of the Euclidean plane E2. This infinite-dimensional algebra distinguishes the elliptic Liouville equation from the hyperbolic one with its symmetry algebra that is the direct sum of two Virasoro algebras. Following a discretisation procedure developed earlier, we present a difference scheme that is invariant under the group O(3, 1) and has the elliptic Liouville equation in polar coordinates as its continuous limit. The lattice is a solution of an equation invariant under O(3, 1) and is itself invariant under a subgroup of O(3, 1), namely the O(2) rotations of the Euclidean plane.