Sokolov Hierarchies Research Articles

BCFG Drinfeld–Sokolov hierarchies and FJRW-theory

According to the ADE Witten conjecture, which is proved by Fan, Jarvis and Ruan, the total descendant potential of the FJRW invariants of an ADE singularity is a tau function of the corresponding mirror ADE Drinfeld-Sokolov hierarchy. In the present paper, we show that there is a finite group $\Gamma$ acting on a certain ADE singularity which induces an action on the corresponding FJRW-theory, and the $\Gamma$-invariant sector also satisfies the axioms of a cohomological field theory except the gluing loop axiom. On the other hand, we show that there is also a $\Gamma$-action on the mirror Drinfeld-Sokolov hierarchy, and the $\Gamma$-invariant flows yield the BCFG Drinfeld-Sokolov hierarchy. We prove that the total descendant potential of the $\Gamma$-invariant sector of a FJRW-theory is a tau function of the corresponding BCFG Drinfeld-Sokolov hierarchy.

Erratum to: Classical $${\mathcal {W}}$$ W -Algebras and Generalized Drinfeld–Sokolov Hierarchies for Minimal and Short Nilpotents

Erratum to: Classical $${\mathcal {W}}$$ W -Algebras and Generalized Drinfeld–Sokolov Hierarchies for Minimal and Short Nilpotents

Dirac Reduction for Poisson Vertex Algebras

We construct an analogue of Dirac’s reduction for an arbitrary local or non-local Poisson bracket in the general setup of non-local Poisson vertex algebras. This leads to Dirac’s reduction of an arbitrary non-local Poisson structure. We apply this construction to an example of a generalized Drinfeld–Sokolov hierarchy.

Classical $${\mathcal{W}}$$ W -Algebras and Generalized Drinfeld–Sokolov Hierarchies for Minimal and Short Nilpotents

We derive explicit formulas for lambda-brackets of the affine classical W-algebras attached to the minimal and short nilpotent elements of any simple Lie algebra g. This is used to compute explicitly the first non-trivial PDE of the corresponding intgerable generalized Drinfeld-Sokolov hierarchies. It turns out that a reduction of the equation corresponding to a short nilpotent is Svinolupov's equation attached to a simple Jordan algebra, while a reduction of the equation corresponding to a minimal nilpotent is an integrable Hamiltonian equation on 2h-3 functions, where h is the dual Coxeter number of g. In the case when g is sl_2 both these equations coincide with the KdV equation. In the case when g is not of type C_n, we associate to the minimal nilpotent element of g yet another generalized Drinfeld-Sokolov hierarchy.

Block (or Hamiltonian) Lie Symmetry of Dispersionless D-Type Drinfeld–Sokolov Hierarchy

In this paper, the dispersionless D-type Drinfeld–Sokolov hierarchy, i.e. a reduction of the dispersionless two-component BKP hierarchy, is studied. The additional symmetry flows of this hierarchy are presented. These flows form an infinite-dimensional Lie algebra of Block type as well as a Lie algebra of Hamiltonian type.

Tau function of the CKP hierarchy and nonlinearizable virasoro symmetries

We introduce a single tau function that represents the C-type Kadomtsev–Petviashvili (CKP) hierarchy in a generalized Hirota ‘bilinear’ equation. The actions on the tau function by additional symmetries for the hierarchy are also calculated, which involve strictly more than a central extension of the -algebra. As an application, for Drinfeld–Sokolov hierarchies associated to affine Kac–Moody algebras of type C, we obtain a formula to compute the obstacles in linearizing their Virasoro symmetries and hence prove the Virasoro symmetries to be nonlinearizable when acting on the tau function.

From additional symmetries to linearization of Virasoro symmetries

We construct the additional symmetries and derive the Adler–Shiota–van Moerbeke formula for the two-component BKP hierarchy. Considered as certain reductions of the two-component BKP hierarchy, the Drinfeld–Sokolov hierarchies of type D are proved to possess symmetries written as the linear action of a series of Virasoro operators on the tau function. It results in that the Drinfeld–Sokolov hierarchies of type D coincide with Dubrovin and Zhang’s hierarchies associated to the Frobenius manifolds for Coxeter groups of type D, and that every solution of such a hierarchy together with the string equation is annihilated by certain combinations of the Virasoro operators and the time derivations of the hierarchy.

Drinfeld–Sokolov hierarchies on truncated current Lie algebras

Drinfeld–Sokolov hierarchies on truncated current Lie algebras

On the Drinfeld-Sokolov Hierarchies of D Type

We extend the notion of pseudo-differential operators that are used to represent the Gelfand–Dickey hierarchies and obtain a similar representation for the full Drinfeld–Sokolov hierarchies of D n type. By using such pseudo-differential operators, we introduce the tau functions of these bihamiltonian hierarchies and prove that these hierarchies are equivalent to the integrable hierarchies defined by Date–Jimbo–Kashiware–Miwa and Kac–Wakimoto from the basic representation of the Kac–Moody algebra D n (1) .

The Sixth Painlevé Equation as Similarity Reduction of $${\widehat{\mathfrak{gl}}_3}$$ Generalized Drinfel’d–Sokolov Hierarchy

Scaling symmetry of \({\widehat{\mathfrak{gl}}_n}\) -type Drinfel’d–Sokolov hierarchy is investigated. Applying similarity reduction to the hierarchy, one can obtain the Schlesinger equation with (n + 1) regular singularities. Especially in the case of n = 3, the hierarchy contains the three-wave resonant system and the similarity reduction gives the generic case of the Painleve VI equation. We also discuss Weyl group symmetry of the hierarchy.

Aq-analogue of hierarchy andq-Painlevé VI

A q-analogue of the gl_3 Drinfel'd-Sokolov hierarchy is proposed as a reduction of the q-KP hierarchy. Applying a similarity reduction and a q-Laplace transformation to the hierarchy, one can obtain the q-Painleve VI equation proposed by Jimbo and Sakai.

The sixth Painlevé equation arising fromD(1)4hierarchy

The sixth Painleve equation arises from a Drinfel'd-Sokolov hierarchy associated with the affine Lie algebra of type D_4 by similarity reduction.

Affine lie group approach to a derivative nonlinear Schrödinger equation and its similarity reduction

A generalization of the Drinfel'd-Sokolov hierarchies is proposed using the framework of Sato-Wilson dressing method. In the case of the A1(1)-type affine Lie group, we obtain the hierarchy that includes the derivative nonlinear Schrodinger equation. We give two types of affine Weyl group symmetry of the hierarchy based on the Gauss decomposition. The fourth Painleve equation and its Weyl group symmetry are obtained as a similarity reduction. We also discuss the connection between the reduced system and monodromy-preserving deformations.

Constrained KP models as integrable matrix hierarchies

We formulate the constrained KP hierarchy (denoted by cKPK+1,M) as an affine sl̂(M+K+1) matrix integrable hierarchy generalizing the Drinfeld–Sokolov hierarchy. Using an algebraic approach, including the graded structure of the generalized Drinfeld–Sokolov hierarchy, we are able to find several new universal results valid for the cKP hierarchy. In particular, our method yields a closed expression for the second bracket obtained through Dirac reduction of any untwisted affine Kac–Moody current algebra. An explicit example is given for the case sl̂(M+K+1), for which a closed expression for the general recursion operator is also obtained. We show how isospectral flows are characterized and grouped according to the semisimple non-regular element E of sl(M+K+1) and the content of the center of the kernel of E.

The Hamiltonian Structure of Soliton Equations and Deformed [formula omitted]-Algebras

The Poisson bracket algebra corresponding to the second Hamiltonian structure of a large class of generalized KdV and mKdV integrable hierarchies is carefully analysed. These algebras are known to have conformal properties and their relation to W-algebras has been previously investigated in some particular cases. The class of equations that is considered includes practically all the generalizations of the Drinfel'd–Sokolov hierarchies constructed in the literature. In particular, it has been recently shown that it includes matrix generalizations of the Gelfand–Dickey and the constrained KP hierarchies. Therefore, our results provide a unified description of the relation between the Hamiltonian structure of soliton equations and W-algebras, and it comprises almost all the results formerly obtained by other authors. The main result of this paper is an explicit general equation showing that the second Poisson bracket algebra is a deformation of the Dirac bracket algebra corresponding to the W-algebras obtained through Hamiltonian reduction.

Generalized Drinfel'd-Sokolov hierarchies, quantum rings, and W-gravity

We investigate the algebraic structure of integrable hierarchies that, we propose, underlie models of W-gravity coupled to matter. More precisely, we concentrate on the dispersionless limit of the topological subclass of such theories, by making use of a correspondence between Drinfel'd-Sokolov systems, principal sl(2) embeddings and certain chiral rings. We find that the integrable hierarchies can be viewed as generalizations of the usual matrix Drinfel'd-Sokolov systems to higher fundamental representations of sl( n). Accordingly, there are additional commuting flows as compared to the usual generalized KdV hierarchy. These are associated with the enveloping algebra and account for degeneracies of physical operators. The underlying Heisenberg algebras are nothing but specifically perturbed chiral rings of certain Kazama-Suzuki models, and have an intimate connection with the quantum cohomology of grassmannians. Correspondingly, the Lax operators are directly given in terms of multi-field superpotentials of the associated topological LG theories. We view our construction as a prototype for a multi-variable system and suspect that it might be useful also for a class of related problems.

Groups of equivariant loops in SL(n+1,C) and soliton equations

This article studies a generalization of the Drinfel’d and Sokolov hierarchies for sln+1 and shows how the dressing transformation [as interpreted by Segal and Wilson, Publ. Math. IHES 61, 5 (1985)] produces a large class of almost globally analytic solutions. Many of these solutions are ‘‘of finite type,’’ i.e., related to the Jacobian of a complete algebraic curve.

Tau-functions and generalized intergrable hierarchies

The tau-function formalism for a class of generalized “zero-curvature” integrable hierarchies of partial differential equations is constructed. The class includes the Drinfel'd-Sokolov hierarchies. A direct relation between the variables of the zero-curvature formalism and the tau-functions is established. The formalism also clarifies the connection between the zero-curvature hierarchies and the Hirotatype hierarchies of Kac and Wakimoto.

Formula omitted]-algebra symmetries of generalised Drinfel'd-Sokolov hierarchies

Using the zero curvature formulation, it is shown that W -algebra transformations are symmetries of corresponding generalised Drinfel'd-Sokolov hierarchies. This result is illustrated with the examples of the KdV and Boussinesque hierarchies, and the hierarchy associated to the Polyakov-Bershadsky W -algebra.