scholarly journals Space Curves and Solitons of the KP Hierarchy. I. The l-th Generalized KdV Hierarchy

2021 ◽  
Author(s):  
Yuji Kodama ◽  
◽  
Yuancheng Xie ◽  
Keyword(s):  
Kp Hierarchy ◽  
Kdv Hierarchy ◽  
Space Curves

THE STRING DIFFERENCE EQUATION OF THE D=1 MATRIX MODEL AND W1+∞ SYMMETRY OF THE KP HIERARCHY

1992 ◽  
Vol 07 (20) ◽  
pp. 4791-4802 ◽  
Author(s):  
M.A. AWADA ◽  
S.J. SIN
Keyword(s):  
Free Energy ◽  
String Theory ◽  
Difference Equation ◽  
Matrix Model ◽  
Zero Mode ◽  
Legendre Transformation ◽  
Kp Hierarchy ◽  
Kdv Hierarchy ◽  
The Difference ◽  
Special Case

We give a connection between the D=1 matrix model and the generalized KP hierarchy. First, we find a difference equation satisfied by F, the Legendre transformation of the free energy of the D=1 matrix model on a circle of radius R. Then we show that it is a special case of the difference equation of the generalized KP hierarchy with its zero mode identified with the scaling variable of the D=1 string theory. We propose that the massive D=1 matrix model is described by the generalized KP hierarchy, which implies the manifest integrability of D=1 string theory. We also show that the (generalized) KP hierarchy has an underlying W1+∞ symmetry. By reduction, we prove that the generalized KdV hierarchy has a subalgebra of the above symmetry which again forms a W1+∞. We argue that there are no W constraints in D=1 string theory, which is in contrast to D<1 theories, where there are W1+∞ constraints.

CONTINUUM SCHWINGER-DYSON EQUATIONS AND UNIVERSAL STRUCTURES IN TWO-DIMENSIONAL QUANTUM GRAVITY

1991 ◽  
Vol 06 (08) ◽  
pp. 1385-1406 ◽  
Author(s):  
MASAFUMI FUKUMA ◽  
HIKARU KAWAI ◽  
RYUICHI NAKAYAMA
Keyword(s):  
Partition Function ◽  
Quantum Gravity ◽  
Matrix Models ◽  
Square Root ◽  
Two Dimensional ◽  
Kp Hierarchy ◽  
Kdv Hierarchy ◽  
The One ◽  
Matter Fields ◽  
The Continuum

We study the continuum Schwinger-Dyson equations for nonperturbative two-dimensional quantum gravity coupled to various matter fields. The continuum Schwinger-Dyson equations for the one-matrix model are explicitly derived and turn out to be a formal Virasoro condition on the square root of the partition function, which is conjectured to be the τ function of the KdV hierarchy. Furthermore, we argue that general multi-matrix models are related to the W algebras and suitable reductions of KP hierarchy and its generalizations.

scholarly journals THE LAX OPERATOR APPROACH FOR THE VIRASORO AND THE W-CONSTRAINTS IN THE GENERALIZED KdV HIERARCHY

1993 ◽  
Vol 08 (20) ◽  
pp. 3457-3478 ◽  
Author(s):  
SUDHAKAR PANDA ◽  
SHIBAJI ROY
Keyword(s):  
Symmetry Operator ◽  
String Equation ◽  
Kp Hierarchy ◽  
Operator Approach ◽  
Additional Symmetry ◽  
Kdv Hierarchy ◽  
Lax Operator

We show directly in the Lax operator approach how the Virasoro and W-constraints on the τ-function arise in the p-reduced KP hierarchy or generalized KdV hierarchy. In particular, we consider the KdV and the Boussinesq hierarchy to show that the Virasoro and the W-constraints follow from the string equation by expanding the "additional symmetry" operator in terms of the Lax operator. We also mention how this method could be generalized for higher KdV hierarchies.

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.

MATRIX MODELS WITHOUT MATRICES: FROM INTEGRABLE HIERARCHIES TO RECURSION RELATIONS

1992 ◽  
Vol 07 (01) ◽  
pp. 43-54 ◽  
Author(s):  
A. M. SEMIKHATOV
Keyword(s):  
Integrable Hierarchies ◽  
Scaling Limit ◽  
Symmetry Algebra ◽  
Kp Hierarchy ◽  
Matrix Formulation ◽  
R Matrix ◽  
Kdv Hierarchy ◽  
Virasoro Constraints ◽  
The Matrix ◽  
Dimensional Gravity

Virasoro constraints on integrable hierarchies and their consequences are studied using the formalism of dressing operators. The dressing-operator description allows one to perform entirely in intrinsically hierarchical terms a double-scaling limit which takes "discrete" (lattice) Virasoro-constrained hierarchies into continuum hierarchies subjected to their own Virasoro constraints. Certain equations derived as consequences of the constraints suggest an interpretation as recursion/loop equations, thus establishing a link with the field-theoretic description. Such a correspondence with two-dimensional gravity-coupled theories, which does not require going through the matrix formulation, is conjectured to hold for general integrable hierarchies of the r-matrix type (appropriately constrained). The example considered explicitly is that of the Virasoro-constrained Toda hierarchy which undergoes a scaling into the Virasoro-constrained KP hierarchy, which in turn can be reduced to N-KdV hierarchies subjected to a subset of the KP Virasoro constraints. The dressing-operator formulation also facilitates the analysis of symmetry algebras of constrained hierarchies. The Kac–Moody sl (N) algebra is identified as a symmetry of the N-KdV hierarchy, while for the Virasoro-constrained KP hierarchy its symmetry algebra is related to a member of the family of the W∞(J) algebras. In the supersymmetric case this method allows one to impose super-Virasoro constraints on the super-KP hierarchy consistently with all the SKP flows.

Particle and Astro-physics Challenge Kant’s Phenomenolism

1998 ◽  
pp. 323-333
Author(s):  
Lawrence H. Starkey
Keyword(s):  
General Relativity ◽  
Black Hole ◽  
Physical Basis ◽  
Primary Sources ◽  
One Dimension ◽  
Space Curves ◽  
Slowing Down ◽  
Parity Conservation ◽  
Charge Parity ◽  
Einstein’S General Relativity

For two centuries Kant's first Critique has nourished various turns against transcendent metaphysics and realism. Kant was scandalized by reason's impotence in confronting infinity (or finitude) as seen in the divisibility of particles and in spatial extension and time. Therefore, he had to regard the latter as subjective and reality as imponderable. In what follows, I review various efforts to rationalize Kant's antinomies-efforts that could only flounder before the rise of Einstein's general relativity and Hawking's blackhole cosmology. Both have undercut the entire Kantian tradition by spawning highly probable theories for suppressing infinities and actually resolving these perplexities on a purely physical basis by positing curvatures of space and even of time that make them reëntrant to themselves. Heavily documented from primary sources in physics, this paper displays time’s curvature as its slowing down near very massive bodies and even freezing in a black hole from which it can reëmerge on the far side, where a new universe can open up. I argue that space curves into a double Möbius strip until it loses one dimension in exchange for another in the twin universe. It shows how 10-dimensional GUTs and the triple Universe, time/charge/parity conservation, and strange and bottom particle families and antiparticle universes, all fit together.

scholarly journals On tau-functions for the KdV hierarchy

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Boris Dubrovin ◽  
Di Yang ◽  
Don Zagier
Keyword(s):  
Tau Functions ◽  
Kdv Hierarchy

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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