Nonstandard KP evolution and the quantumτ-function

Kharchev, S.; Миронов, А.; Morozov, A.

doi:10.1007/bf02066659

Cited by 10 publications

(8 citation statements)

References 4 publications

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“…If at all resolvable this is the problem of a clever choice of the weight functions s R m,m (t,t) in (5.1). Following [56] we shall now demonstrate that the problem is resolvable in principle, though at the moment it is a quantum deformation of somewhat non-conventional description of KP/Toda system (with evolution, introduced differently from that in s.5.2, and it is not just a change of time-variables: the transformation is representation R-dependent).…”

Section: Comments On the Quantum Deformation Of Kp/toda τ -Functionsmentioning

confidence: 89%

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Matrix Models as Integrable Systems

Морозов¹

1999

Particles and Fields

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Section: Comments On the Quantum Deformation Of Kp/toda τ -Functionsmentioning

confidence: 89%

“…Now it remains to recall the basic relation between the infinitesimal norm and the measure: 53) provided c N in (2.1) and (2.46) is chosen to be 54) where the volume of unitary group in Haar measure is equal to 56) where…”

Section: -Matrix Model In Eigenvalue Representationmentioning

confidence: 99%

Matrix Models as Integrable Systems

Морозов¹

1999

Particles and Fields

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“…does not satisfy criteria (4.2) and (4.4) as a function of any time or time pairs, see [7,8] for a detailed consideration (it is not even clear if it fits into the wide class of the non-Abelian τ -functions of [73][74][75][76]). Notable exceptions are the cases when k = 1, 2 and when β ∆ are adjusted to provide any linear combination of the standard Casimir operators (3.4), which are nicknamed as complete cycles in [77,78].…”

Section: Representation Via Casimir Operatorsmentioning

confidence: 99%

On KP-integrable Hurwitz functions

Alexandrov

Миронов

Морозов

et al. 2014

J. High Energ. Phys.

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101

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There is now a renewed interest [1]-[4] to a Hurwitz τ -function, counting the isomorphism classes of Belyi pairs, arising in the study of equilateral triangulations and Grothiendicks's dessins d'enfant. It is distinguished by belonging to a particular family of Hurwitz τ -functions, possessing conventional Toda/KP integrability properties. We explain how the variety of recent observations about this function fits into the general theory of matrix model τ -functions. All such quantities possess a number of different descriptions, related in a standard way: these include Toda/KP integrability, several kinds of W -representations (we describe four), two kinds of integral (multi-matrix model) descriptions (of Hermitian and Kontsevich types), Virasoro constraints, character expansion, embedding into generic set of Hurwitz τ -functions and relation to knot theory. When approached in this way, the family of models in the literature has a natural extension, and additional integrability with respect to associated new time-variables. Another member of this extended family is the Itsykson-Zuber integral.

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“…Uncompactified D = 11 Mtheory was found to have an equivalence with the N = ∞ limit of supersymmetric matrix quantum mechanics describing D 0 branes. Matrix models of 2D gravity and Toda theory have been discussed by Gerasimov et al [105] and by Kharchev et al [103].…”

Section: Discussionmentioning

confidence: 99%

STRINGS AND MEMBRANES FROM EINSTEIN GRAVITY, MATRIX MODELS AND W_∞ GAUGE THEORIES AS PATHS TO QUANTUM GRAVITY

Castro

2008

Int. J. Mod. Phys. A

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It is shown how w∞, w1+∞ Gauge Field Theory actions in 2D emerge directly from 4D Gravity. Strings and Membranes actions in 2D and 3D originate as well from 4D Einstein Gravity after recurring to the nonlinear connection formalism of Lagrange-Finsler and Hamilton-Cartan spaces. Quantum Gravity in 3D can be described by a W∞ Matrix Model in D = 1 that can be solved exactly via the Collective Field Theory method. We describe why a quantization of 4D Gravity could be attained via a 2D Quantum W∞ gauge theory coupled to an infinite-component scalar-multiplet. A proof that non-critical W∞ (super) strings are devoid of BRST anomalies in dimensions D = 27 (D = 11), respectively, follows and which coincide with the critical (super) membrane dimensions D = 27 (D = 11). We establish the correspondence between the states associated with the quasi finite highest weights irreducible representations of W∞,W∞ algebras and the quantum states of the continuous Toda molecule. Schroedinger-like QM wave functional equations are derived and solutions are found in the zeroth order approximation. Since higher-conformal spin W∞ symmetries are very relevant in the study of 2D W∞ Gravity, the Quantum Hall effect, large N QCD, strings, membranes, ...... it is warranted to explore further the interplay among all these theories. In this introductory section we will review the work of [2] and afterwards we will discuss the recent work related the Hidden Symmetries of M theory. Keywords1.1 Gravity in D = m + n as an m-dim Gauge Theory of diffeomorphisms of an internal n-dim space and HolographySome time ago Park [1] showed that 4D Self Dual Gravity is equivalent to a WZNW model based on the group SU (∞). Namely, 4D Self Dual Gravity is the non-linear sigma model based in 2D whose target space is the "group manifold" of area-preserving diffs of another 2D-dim manifold. Roughly speaking, this means that the effective D = 4 manifold, where Self Dual Gravity is defined, is "spliced" into two 2D-submanifolds: one submanifold is the original 2D base manifold where the non-linear sigma model is defined. The other 2D submanifold is the target group manifold of area-preserving diffs of a two-dim sphere S 2 . The authors [2] went further and generalized this particular Self Dual Gravity case to the full fledged gravity in D = 2 + 2 = 4 dimensions, and in general, to any combinations of m + n-dimensions. Their main result is that m + n-dim Einstein gravity can be identified with an m-dimensional generally invariant gauge theory of Dif f s N , where N is an n-dim manifold. Locally the m + ndim space can be written as Σ = M × N and the metric G AB decomposes as: [62]. The decomposition (1.1) must not be confused with the Kaluza-Klein reduction where one imposes an isometry restriction on the γ AB that turns A a µ into a gauge connection associated with the gauge group G generated by isometry. Dropping the isometry restrictions allows all the fields to depend on all the coordinates x, y. Nevertheless A a µ (x, y) can still be identified as a conne...

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Nonstandard KP evolution and the quantumτ-function

Cited by 10 publications

References 4 publications

Matrix Models as Integrable Systems

Matrix Models as Integrable Systems

On KP-integrable Hurwitz functions

STRINGS AND MEMBRANES FROM EINSTEIN GRAVITY, MATRIX MODELS AND W_∞ GAUGE THEORIES AS PATHS TO QUANTUM GRAVITY

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Nonstandard KP evolution and the quantumτ-function

Cited by 10 publications

References 4 publications

Matrix Models as Integrable Systems

Matrix Models as Integrable Systems

On KP-integrable Hurwitz functions

STRINGS AND MEMBRANES FROM EINSTEIN GRAVITY, MATRIX MODELS AND W∞ GAUGE THEORIES AS PATHS TO QUANTUM GRAVITY

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Product

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About

STRINGS AND MEMBRANES FROM EINSTEIN GRAVITY, MATRIX MODELS AND W_∞ GAUGE THEORIES AS PATHS TO QUANTUM GRAVITY