On Solutions to the String Equation

Schwarz, Albert

doi:10.1142/s0217732391003171

Cited by 48 publications

(62 citation statements)

References 10 publications

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“…It was a remarkable observation of Schwarz [24] that actually Burchnall-Chaundy locus can be described by means of matrices with properties as in proposition 6.5 modulo the action of ∆. Here we adapt the results of [24] to our situation. Namely we explain how, given C(z), one can recover W(z) and the corresponding spectral curve.…”

Section: Proposition 62 ([23])mentioning

confidence: 91%

Block Toeplitz Determinants, Constrained KP and Gelfand-Dickey Hierarchies

Cafasso

2008

Math Phys Anal Geom

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We propose a method for computing any Gelfand-Dickey τ function living in SegalWilson Grassmannian as the asymptotics of block Toeplitz determinant associated to a certain class of symbols W(t; z). Also truncated block Toeplitz determinants associated to the same symbols are shown to be τ function for rational reductions of KP. Connection with Riemann-Hilbert problems is investigated both from the point of view of integrable systems and block Toeplitz operator theory. Examples of applications to algebro-geometric solutions are given. IntroductionThis paper deals with the applications of block Toeplitz determinants and their asymptotics to the study of integrable hierarchies. Asymptotics of block Toeplitz determinants and their applications to physics is a developing field of research; in recent years it has been shown how to compute some physically relevant quantities (e.g. correlation functions) studying asymptotics of some block Toeplitz determinants (see [25], [26], [27]). In particular in [25] and [26] authors, for the first time, showed effective computations for the case of block Toeplitz determinants with symbols that do not have half truncated Fourier series. This is of particular interest for us as, with our approach, we will be able to do the same for certain block Toeplitz determinants associated to algebro-geometric solutions of Gelfand Dickey hierarchies. Let us mention some theoretical results about (block) Toeplitz determinants we will use in this paper. Given a function γ(z) on the circle we denote T N (γ) the Toeplitz matrix with symbol γ given byWe use the term block Toeplitz for the case of matrix-valued symbol γ(z). In that case the entries γ (j−i) of the above matrix are n × n matrices themselves. We denoteand we use the notation T (γ) for the N × N matrix obtained letting N go to infinity. where G(γ) is a normalizing constant and the operator T (γ)T (γ −1 ) is such that its determinant is well defined as a Fredholm determinant (see section 2 for the precise statement). Once the asymptotics had been computed the next quite natural question was to find an expressions relating directly D N (γ), and not just its asymptotics, to certain Fredholm determinants. The problem was solved many years later by Borodin and Okounkov in [9] for the scalar case and generalized, in the same year, for matrix case by E. Basor and H.Widom in [10]. For matrix valued case Borodin-Okounkov formula reads(here we assume G(γ) = 1). The operator (I − K γ,N ) can be written explicitely in coordinates knowing certain Riemann-Hilbert factorizations of γ. Its Fredholm determinant is well defined (see section 2 for details). Now many proofs of Borodin-Okounkov formula are known (for instance [11] contains another proof of the same formula, see also the earlier paper [12]).In this paper we apply block-Toeplitz determinants to the computation of τ function of an (almost) arbitrary solution of Gelfand-Dickey hierarchy(L differential operator of order n, j = nk). More precisely to a given pointin the big cell of Segal-...

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Section: Proposition 62 ([23])mentioning

confidence: 91%

Block Toeplitz Determinants, Constrained KP and Gelfand-Dickey Hierarchies

Cafasso

2008

Math Phys Anal Geom

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“…We shall demonstrate below that just in this case the algebra (2.8) is associative and the WDVV equations are fulfilled. The pair of differentials in this case is essentially the same as arising in many related contexts: (p, q) in minimal conformal models, a pair of polynomials in associated matrix models [24,25,22], a pair of operators in the Kac-Schwarz problem [26,22] and in a bispectral problem in the theory of KP/Toda hierarchies [27] etc. The simplest example of this 4 In the conventional Landau-Ginzburg topological models the spectral curve is Riemann sphere.…”

Section: The Full Set Of the Wdvv Equationsmentioning

confidence: 99%

MORE EVIDENCE FOR THE WDVV EQUATIONS IN ${\mathcal N} = 2$ SUSY YANG–MILLS THEORIES

Marshakov

Миронов

Морозов³

2000

Int. J. Mod. Phys. A

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We consider 4d and 5d N = 2 supersymmetric theories and demonstrate that in general their Seiberg-Witten prepotentials satisfy the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. General proof for the Yang-Mills models (with matter in the first fundamental representation) makes use of the hyperelliptic curves and underlying integrable systems. A wide class of examples is discussed, it contains few understandable exceptions. In particular, in perturbative regime of 5d theories in addition to naive field theory expectations some extra terms appear, like it happens in heterotic string models. We consider also the example of the Yang-Mills theory with matter hypermultiplet in the adjoint representation (related to the elliptic Calogero-Moser system) when the standard WDVV equations do not hold.

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“…The Virasoro constraints thus constitute a fundamental notion of the theory and are the heart of matrix models' applications to both gravity-coupled theories and the intersection theory 1 . A major task is to find their general solution (see, for instance, [14]). …”

Section: Introduction and Discussionmentioning

confidence: 99%

Solving Virasoro constraints on integrable hierarchies via the Kontsevich-Miwa transform

Semikhatov

1992

Nuclear Physics B

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We solve Virasoro constraints on the KP hierarchy in terms of minimal conformal models. The constraints we start with are implemented by the Virasoro generators depending on a background charge Q. Then the solutions to the constraints are given by the theory which has the same field content as the David-Distler-Kawai theory: it consists of a minimal matter scalar with background charge Q, dressed with an extra 'Liouville' scalar. In particular, the Virasoro-constrained tau function is related to the correlator of a product of (dressed) '21' operators. The construction is based on a generalization of the Kontsevich parametrization of the KP times achieved by introducing into it Miwa parameters which depend on the value of Q. Under the thus defined Kontsevich-Miwa transformation, the Virasoro constraints are proven to be equivalent to a master equation depending on the parameter Q. The master equation is further identified with a nullvector decoupling equation. We conjecture that W (n) constraints on the KP hierarchy are similarly related to a level-n decoupling equation. We also consider the master equation for the N -reduced KP hierarchies. Several comments are made on a possible relation of the generalized master equation to scaled Kontsevich-type matrix integrals and on the form the equation takes in higher genera.

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On Solutions to the String Equation

Cited by 48 publications

References 10 publications

Block Toeplitz Determinants, Constrained KP and Gelfand-Dickey Hierarchies

Block Toeplitz Determinants, Constrained KP and Gelfand-Dickey Hierarchies

MORE EVIDENCE FOR THE WDVV EQUATIONS IN ${\mathcal N} = 2$ SUSY YANG–MILLS THEORIES

Solving Virasoro constraints on integrable hierarchies via the Kontsevich-Miwa transform

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