CONTINUUM REDUCTION OF VIRASORO-CONSTRAINED LATTICE HIERARCHIES

1991 ◽  
Vol 06 (28) ◽  
pp. 2601-2612 ◽  
Author(s):  
A. M. SEMIKHATOV
Keyword(s):  
Continuum Limit ◽  
Toda Lattice ◽  
Integrable Hierarchies ◽  
Scaling Limit ◽  
Explicit Construction ◽  
Kp Hierarchy ◽  
Highest Weight ◽  
Virasoro Constraints ◽  
Toda Lattice Hierarchy ◽  
The Continuum

Integrable hierarchies with Virasoro constraints have been observed to describe matrix models. I suggest to define general Virasoro-constrained integrable hierarchies by imposing Virasora-highest-weight conditions on the dressing operators. This simplifies the study of the Virasoro constraints and allows an explicit construction of a scaling which implements the continuum limit of discrete (lattice) hierarchies. Applied to the Toda lattice hierarchy subjected to the Virasoro constraints, this scaling leads to the Virasoro-constrained KP hierarchy. Therefore, in particular, the KP hierarchy is shown to arise as the scaling limit of a matrix model.

scholarly journals SUPERLOOP EQUATIONS AND TWO-DIMENSIONAL SUPERGRAVITY

1992 ◽  
Vol 07 (21) ◽  
pp. 5337-5367 ◽  
Author(s):  
L. ALVAREZ-GAUMÉ ◽  
H. ITOYAMA ◽  
J.L. MAÑES ◽  
A. ZADRA
Keyword(s):  
Partition Function ◽  
Discrete Model ◽  
Continuum Limit ◽  
Basic Assumption ◽  
Scaling Limit ◽  
Integrable Hierarchy ◽  
Two Dimensional ◽  
Virasoro Constraints ◽  
Double Scaling Limit ◽  
The Continuum

We propose a discrete model whose continuum limit reproduces the string susceptibility and the scaling dimensions of (2, 4m) minimal superconformal models coupled to 2D supergravity. The basic assumption in our presentation is a set of super-Virasoro constraints imposed on the partition function. We recover the Neveu-Schwarz and Ramond sectors of the theory, and we are also able to evaluate all planar loop correlation functions in the continuum limit. We find evidence to identify the integrable hierarchy of nonlinear equations describing the double scaling limit as a supersymmetric generalization of KP studied by Rabin.

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.

scholarly journals ON THE CONTINUUM LIMIT OF THE CONFORMAL MATRIX MODELS

1993 ◽  
Vol 08 (18) ◽  
pp. 3107-3137 ◽  
Author(s):  
A. MIRONOV ◽  
S. PAKULIAK
Keyword(s):  
Matrix Models ◽  
Continuum Limit ◽  
Scaling Limit ◽  
Universality Class ◽  
Partition Functions ◽  
Discrete Level ◽  
New Class ◽  
Double Scaling Limit ◽  
The Continuum

The double scaling limit of a new class of the multi-matrix models proposed in Ref. 1, which possess the W-symmetry at the discrete level, is investigated in detail. These models are demonstrated to fall into the same universality class as the standard multi-matrix models. In particular, the transformation of the W-algebra at the discrete level into the continuum one of the papers2 is proposed and the corresponding partition functions compared. All calculations are demonstrated in full in the first nontrivial case of W(3)-constraints.

N = ½ SUPERSTRING IN ZERO DIMENSION AND THE SPONTANEOUS BREAKDOWN OF THE SUPERSYMMETRY

1992 ◽  
Vol 07 (32) ◽  
pp. 2979-2989 ◽  
Author(s):  
SHIN’ICHI NOJIRI
Keyword(s):  
Matrix Models ◽  
Random Matrix ◽  
Continuum Limit ◽  
Scaling Limit ◽  
Partition Functions ◽  
Spontaneous Breakdown ◽  
Random Matrix Models ◽  
Double Scaling Limit ◽  
The Continuum ◽  
String Theories

We propose random matrix models which have N=1/2 supersymmetry in zero dimension. The supersymmetry breaks down spontaneously. It is shown that the double scaling limit can be defined in these models and the breakdown of the supersymmetry remains in the continuum limit. The exact non-trivial partition functions of the string theories corresponding to these matrix models are also obtained.

scholarly journals INTEGRABLE HIERARCHIES AND DISPERSIONLESS LIMIT

1995 ◽  
Vol 07 (05) ◽  
pp. 743-808 ◽  
Author(s):  
KANEHISA TAKASAKI ◽  
TAKASHI TAKEBE
Keyword(s):  
Canonical Transformation ◽  
Toda Lattice ◽  
Integrable Hierarchies ◽  
General Solutions ◽  
Similar Construction ◽  
Toda Hierarchy ◽  
Technical Details ◽  
Dispersionless Limit ◽  
Toda Lattice Hierarchy ◽  
Twistor Construction

Analogues of the KP and the Toda lattice hierarchy called dispersionless KP and Toda hierarchy are studied. Dressing operations in the dispersionless hierarchies are introduced as a canonical transformation, quantization of which is dressing operators of the ordinary KP and Toda hierarchy. An alternative construction of general solutions of the ordinary KP and Toda hierarchy is given as twistor construction which is quantization of the similar construction of solutions of dispersionless hierarchies. These results as well as those obtained in previous papers are presented with proofs and necessary technical details.

scholarly journals Continuum limit of the N=(1|1) supersymmetric Toda lattice hierarchy

2002 ◽  
Author(s):  
Alexander Sorin ◽  
V. G. Kadyshevsky
Keyword(s):  
Continuum Limit ◽  
Toda Lattice ◽  
Toda Lattice Hierarchy

scholarly journals DOUBLE SCALING LIMIT OF THE SUPER-VIRASORO CONSTRAINTS

1993 ◽  
Vol 08 (13) ◽  
pp. 2297-2331 ◽  
Author(s):  
L. ALVAREZ-GAUMÉ ◽  
K. BECKER ◽  
M. BECKER ◽  
R. EMPARAN ◽  
J. MAÑES
Keyword(s):  
Scaling Limit ◽  
Majorana Fermion ◽  
Two Dimensional ◽  
Genus Zero ◽  
Heuristic Argument ◽  
Kdv Hierarchy ◽  
Virasoro Constraints ◽  
Bosonic Part ◽  
Double Scaling Limit ◽  
The Continuum

We obtain the double scaling limit of a set of superloop equations recently proposed to describe the coupling of two-dimensional supergravity to minimal superconformal matter of type (2,4m). The continuum loop equations are described in terms of a [Formula: see text] theory with a Z2-twisted scalar field and a Weyl–Majorana fermion in the Ramond sector. We have computed correlation functions in genus zero, one and partially in genus two. An integrable supersymmetric hierarchy describing our model has not yet been found. We present a heuristic argument showing that the purely bosonic part of our model is described by the KdV hierarchy.

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