On the sub-KP hierarchy and its constraints, revisited

2014 ◽  
Vol 26 (10) ◽  
pp. 1450019 ◽  
Author(s):  
Dafeng Zuo ◽  
Ling Zhang ◽  
Qing Chen
Keyword(s):  
Witt Algebra ◽  
Kp Hierarchy ◽  
Additional Symmetries

In this paper, we are interested in a series of sub-hierarchies of the KP hierarchy introduced by Date et al. in [4], which we call the ℬ𝒞r-KP hierarchy for r ∈ ℤ≥0. In a unified way, we construct additional symmetries of the ℬ𝒞r-KP hierarchy and show that all of them form a [Formula: see text]-algebra. Then, we introduce the constrained ℬ𝒞r-KP hierarchy, and construct its additional symmetries and show that all of them form a Witt algebra.

scholarly journals Additional symmetries and solutions of the dispersionless KP hierarchy

2003 ◽  
Vol 44 (8) ◽  
pp. 3294-3308 ◽  
Author(s):  
Luis Martı́nez Alonso ◽  
Manuel Mañas
Keyword(s):  
Kp Hierarchy ◽  
Additional Symmetries

scholarly journals Constrained KP Hierarchies: Additional Symmetries, Darboux–Bäcklund Solutions and Relations to Multi-Matrix Models

1997 ◽  
Vol 12 (07) ◽  
pp. 1265-1340 ◽  
Author(s):  
H. Aratyn ◽  
E. Nissimov ◽  
S. Pacheva
Keyword(s):  
Matrix Models ◽  
Lattice Structure ◽  
Discrete Symmetry ◽  
Partition Functions ◽  
Specific Class ◽  
Kp Hierarchy ◽  
Tau Functions ◽  
Additional Symmetries ◽  
Kdv Hierarchy ◽  
Close Relationship

This paper provides a systematic description of the interplay between a specific class of reductions denoted as cKP r,m(r,m ≥ 1) of the primary continuum integrable system — the Kadomtsev–Petviashvili (KP) hierarchy and discrete multi-matrix models. The relevant integrable cKP r,m structure is a generalization of the familiar r-reduction of the full KP hierarchy to the SL (r) generalized KdV hierarchy cKP r,0. The important feature of cKP r,m hierarchies is the presence of a discrete symmetry structure generated by successive Darboux–Bäcklund (DB) transformations. This symmetry allows for expressing the relevant tau-functions as Wronskians within a formalism which realizes the tau-functions as DB orbits of simple initial solutions. In particular, it is shown that any DB orbit of a cKP r,1 defines a generalized two-dimensional Toda lattice structure. Furthermore, we consider the class of truncated KP hierarchies (i.e. those defined via Wilson–Sato dressing operator with a finite truncated pseudo-differential series) and establish explicitly their close relationship with DB orbits of cKP r,m hierarchies. This construction is relevant for finding partition functions of the discrete multi-matrix models. The next important step involves the reformulation of the familiar nonisospectral additional symmetries of the full KP hierarchy so that their action on cKP r,m hierarchies becomes consistent with the constraints of the reduction. Moreover, we show that the correct modified additional symmetries are compatible with the discrete DB symmetry on the cKP r,m DB orbits.

scholarly journals A COMMENT ON THE ODD FLOWS FOR THE SUPERSYMMETRIC KdV EQUATION

1994 ◽  
Vol 09 (35) ◽  
pp. 3235-3243 ◽  
Author(s):  
EDUARDO RAMOS
Keyword(s):  
Conservation Laws ◽  
Kdv Equation ◽  
Two Dimensional ◽  
Kp Hierarchy ◽  
Additional Symmetries ◽  
Type Reduction ◽  
Lax Operator ◽  
Nonlocal Conservation Laws ◽  
Fourth Root ◽  
Supersymmetric Kdv Equation

In a recent paper Dargis and Mathieu introduced integrodifferential odd flows for the supersymmetric KdV equation. These flows are obtained from the nonlocal conservation laws associated with the fourth root of its Lax operator. In this note I show that only half of these flows are of the standard Lax form, while the remaining half provide us with Hamiltonians for an SKdV-type reduction of a new supersymmetric hierarchy. This new hierarchy is shown to be closely related to the Jacobian supersymmetric KP-hierarchy of Mulase and Rabin. A detailed study of the algebra of additional symmetries of this new hierarchy reveals that it is isomorphic to the super-W1+∞ algebra, thus making it a candidate for a possible interrelationship between superintegrability and two-dimensional supergravity.

scholarly journals Genus expansion of matrix models and ћ expansion of KP hierarchy

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
A. Andreev ◽  
A. Popolitov ◽  
A. Sleptsov ◽  
A. Zhabin
Keyword(s):  
Matrix Models ◽  
Matrix Model ◽  
Special Interest ◽  
Hermitian Matrix ◽  
Geometric Meaning ◽  
Kp Hierarchy ◽  
Hurwitz Numbers ◽  
Kp Equations ◽  
Genus Expansion ◽  
Hermitian Matrix Model

Abstract We study ћ expansion of the KP hierarchy following Takasaki-Takebe [1] considering several examples of matrix model τ-functions with natural genus expansion. Among the examples there are solutions of KP equations of special interest, such as generating function for simple Hurwitz numbers, Hermitian matrix model, Kontsevich model and Brezin-Gross-Witten model. We show that all these models with parameter ћ are τ-functions of the ћ-KP hierarchy and the expansion in ћ for the ћ-KP coincides with the genus expansion for these models. Furthermore, we show a connection of recent papers considering the ћ-formulation of the KP hierarchy [2, 3] with original Takasaki-Takebe approach. We find that in this approach the recovery of enumerative geometric meaning of τ-functions is straightforward and algorithmic.

scholarly journals Intertwining operator and integrable hierarchies from topological strings

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Jean-Emile Bourgine
Keyword(s):  
Topological Strings ◽  
Vertex Operator ◽  
Integrable Hierarchies ◽  
Kp Hierarchy ◽  
Tau Function ◽  
Time Deformation ◽  
Toroidal Algebra ◽  
Crystal Model ◽  
Melting Crystal ◽  
Definition Of

Abstract In [1], Nakatsu and Takasaki have shown that the melting crystal model behind the topological strings vertex provides a tau-function of the KP hierarchy after an appropriate time deformation. We revisit their derivation with a focus on the underlying quantum W1+∞ symmetry. Specifically, we point out the role played by automorphisms and the connection with the intertwiner — or vertex operator — of the algebra. This algebraic perspective allows us to extend part of their derivation to the refined melting crystal model, lifting the algebra to the quantum toroidal algebra of $$ \mathfrak{gl} $$ gl (1) (also called Ding-Iohara-Miki algebra). In this way, we take a first step toward the definition of deformed hierarchies associated to A-model refined topological strings.

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