Connecting the Kontsevich-Witten and Hodge Tau-functions by the {\widehat{GL(\infty)}} Operators

In this paper, we present an explicit formula that connects the Kontsevich-Witten tau-function and the Hodge tau-function by differential operators belonging to the \({\widehat{GL(\infty)}}\) group. Indeed, we show that the two tau-functions can be connected using Virasoro operators. This proves a conjecture posted by Alexandrov in (From Hurwitz numbers to Kontsevich-Witten tau-function: a connection by Virasoro operators, Letters in Mathematical physics, doi:10.1007/s11005-013-0655-0, 2014).     

Non-linear PDEs for gap probabilities in random matrices and KP theory

Airy and Pearcey-like kernels and generalizations arising in random matrix theory are expressed as double integrals of ratios of exponentials, possibly multiplied with a rational function. In this work it is shown that such kernels are intimately related to wave functions for polynomial (Gelfand–Dickey) reductions or rational reductions of the KP-hierarchy; their Fredholm determinant also satisfies linear PDEs (Virasoro constraints), yielding, in a systematic way, non-linear PDEs for the Fredholm determinant of such kernels. Examples include Fredholm determinants giving the gap probability of some infinite-dimensional diffusions, like the Airy process, with or without outliers, and the Pearcey process, with or without inliers.     

Deformation of a Virasoro algebra and the integrals of motion of the quantum sine-Gordon equation

A special deformation of a Virasoro algebra such that the screening operator is not deformed (the space where it operates is deformed) is studied. This deformation leads to a 3-index algebra. The residue of the generating function of the generators of this algebra is a generating function of the integrals of motion for the quantum sine-Gordon model. The algebra of generating functions is calculated. Explicit formulas are presented for the first few integrals of motion. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 5, 375–380 (10 March 1996)     

Conformal blocks and generalized Selberg integrals

Operator product expansion (OPE) of two operators in two-dimensional conformal field theory includes a sum over Virasoro descendants of other operator with universal coefficients, dictated exclusively by properties of the Virasoro algebra and independent of choice of the particular conformal model. In the free field model, these coefficients arise only with a special “conservation” relation imposed on the three dimensions of the operators involved in OPE. We demonstrate that the coefficients for the three unconstrained dimensions arise in the free field formalism when additional Dotsenko–Fateev integrals are inserted between the positions of the two original operators in the product. If such coefficients are combined to form an n-point conformal block on Riemann sphere, one reproduces the earlier conjectured β-ensemble representation of conformal blocks. The statement can also be regarded as a relation between the 3j  -symbols of the Virasoro algebra and the slightly generalized Selberg integrals IY$I_Y$, associated with arbitrary Young diagrams. The conformal blocks are multilinear combinations of such integrals and the AGT conjecture relates them to the Nekrasov functions which have exactly the same structure.     

Geometry of Virasoro constraints in nonperturbative 2-d quantum gravity

The string equation and the Virasoro constraints for arbitrary hermitian multimatrix models are derived using the Lie-Bäcklund symmetries of the generalised KdV equations. From this point of view the origin of the string equation and the meaning of the Virasoro constraints are explained. Some speculation about the appearance of extra constraints, which we suspect to be theW-constraints, is also given.     

The Toda Equations and the Gromov–Witten Theory of the Riemann Sphere

Consequences of the Toda equations arising from the conjectural matrix model for the Riemann sphere are investigated. The Toda equations determine the Gromov–Witten descendent potential (including all genera) of the Riemann sphere from the degree 0 part. Degree 0 series computations via Hodge integrals then lead to higher-degree predictions by the Toda equations. First, closed series forms for all 1-point invariants of all genera and degrees are given. Second, degree 1 invariants are investigated with new applications to Hodge integrals. Third, a differential equation for the generating function of the classical simple Hurwitz numbers (in all genera and degrees) is found – the first such equation. All these results depend upon the conjectural Toda equations. Finally, proofs of the Toda equations in genus 0 and 1 are given.     

Localization and Gluing of Topological Amplitudes

We develop a gluing algorithm for Gromov-Witten invariants of toric Calabi-Yau threefolds based on localization and gluing graphs. The main building block of this algorithm is a generating function of cubic Hodge integrals of special form. We conjecture a precise relation between this generating function and the topological vertex at fractional framing.     

Renormalised Multiple Integrals of Symbols with Linear Constraints

Given a holomorphic regularisation procedure (e.g. Riesz or dimensional regularisation) on classical symbols, we define renormalised multiple integrals of radial classical symbols with linear constraints. To do so, we first prove the existence of meromorphic extensions of multiple integrals of holomorphic perturbations of radial symbols with linear constraints and then implement either generalised evaluators or a Birkhoff factorisation. Renormalised multiple integrals are covariant and factorise over independent sets of constraints.     

Canonical quantization of interacting WZW theories

Using canonical quantization we find the Virasoro centre for a class of conformally invariant interacting Wess-Zumino-Witten theories. The theories have a group structure similar to that of Toda theories (both abelian and non-abelian) but the usual Toda constraints on the coupling constants are relaxed and the theories are not necessarily integrable. The general formula for the Virasoro centre is compared to that derived by BRST methods in the Toda case, and helps to explain the structure of the latter.     

Properties of the series solution for Painlevé I

We present some observations on the asymptotic behaviour of the coefficients in the Laurent series expansion of solutions of the first Painlevé equation. For the general solution, explicit recursive formulae for the Taylor expansion of the tau-function around a zero are given, which are natural extensions of analogous formulae for the elliptic sigma function, as given by Weierstrass. Numerical and exact results on the symmetric solution which is singular at the origin are also presented.     

The Hurwitz enumeration problem of branched covers and Hodge integrals

We use algebraic methods to compute the simple Hurwitz numbers for arbitrary source and target Riemann surfaces. For an elliptic curve target, we reproduce the results previously obtained by string theorists. Motivated by the Gromov–Witten potentials, we find a general generating function for the simple Hurwitz numbers in terms of the representation theory of the symmetric group S_n. We also find a generating function for Hodge integrals on the moduli space of Riemann surfaces with two marked points, similar to that found by Faber and Pandharipande for the case of one marked point.     

Representations of the Witt Algebra and Gl(n)-Opers

We exhibit a new link between certain representations of the Witt algebra and some Gl(n)-opers on the punctured disc. As applications, we discuss the connection with the KdV hierarchy and Virasoro constraints and how the Virasoro constraints of the so-called topological recursion fit in our approach.     

Virasoro Symmetries of the Extended Toda Hierarchy

We prove that the extended Toda hierarchy of [1] admits a nonabelian Lie algebra of infinitesimal symmetries isomorphic to half of the Virasoro algebra. The generators L_m, m–1 of the Lie algebra act by linear differential operators onto the tau function of the hierarchy. We also prove that the tau function of a generic solution to the extended Toda hierarchy is annihilated by a combination of the Virasoro operators and the flows of the hierarchy. As an application we show that the validity of the Virasoro constraints for the CP¹ Gromov-Witten invariants and their descendents implies that their generating function is the logarithm of a particular tau function of the extended Toda hierarchy.Acknowledgements.The research of B.D. was partially supported by Italian Ministry of Education research grant Cofin2001 Geometry of Integrable Systems. The research of Y.Z. was partially supported by the Chinese National Science Fund for Distinguished Young Scholars grant No.10025101 and the Special Funds of Chinese Major Basic Research Project Nonlinear Sciences. Y.Z. thanks Abdus Salam International Centre for Theoretical Physics and SISSA where part of the work was done for their hospitality. The authors are grateful to the referee for the suggested improvements of the paper.     

Modular Constraints on Calabi-Yau Compactifications

We derive global constraints on the non-BPS sector of supersymmetric 2d sigma-models whose target space is a Calabi-Yau manifold. When the total Hodge number of the Calabi-Yau threefold is sufficiently large, we show that there must be non-BPS primary states whose total conformal weights are less than 0.656. Moreover, the number of such primary states grows at least linearly in the total Hodge number. We discuss implications of these results for Calabi-Yau geometry.     

Analytical study of a potential model of tsunami with a simple source of piston type. 1. Exact solution. Creation of tsunami

A linear hydrodynamic problem concerning the generation of gravitational waves on the free surface of a liquid by a source (defined as an initial instant vertical displacement of the bottom of the basin) is studied, where the displacement is defined by a rather simple axially symmetric function of the horizontal coordinates. A solution to the problem is obtained in the form of single integrals and is regarded as a distribution (a ??generalized function??) with respect to time. These integrals are evaluated numerically and asymptotically. In this part of the paper, using the results of numerical evaluation carried out for each source (having a given characteristic radius in a wide range of values), we find the initial instantaneous displacement of the fluid, determine the parameters of the leading crest of the created surface wave, and estimate the minimal radius which a source must have to be referred to tsunami generators.     

On Darboux–Bäcklund Transformations for the Q-Deformed Korteweg–de Vries Hierarchy

We study Darboux–Bäcklund transformations (DBTs) for the q-deformed Korteweg–de Vries hierarchy by using the q-deformed pseudodifferential operators. The elementary DBTs are triggered by the gauge operators constructed from the (adjoint) wave functions of the associated linear systems. Iterating these elementary DBTs, we obtain not only q-deformed Wronskian-type but also binary-type representations of the tau-function of the hierarchy.     

Analytical expressions for momentum autocorrelation function of a classic diatomic chain

We use recurrence relations method to study a classical harmonic diatomic chain. The momentum autocorrelation function results from contributions of acoustic and optical branches. By use of convolution theorem, analytical expressions for the acoustic and optical branches are derived as even-order Bessel function expansions. The expansion coefficients are given in terms of integrals of real and complex elliptic functions for the acoustic and optical branches, respectively. Double convolution results respectively in integrals of trigonometric and hyperbolic functions for expansion coefficients of acoustic and optical branches.     

A generalization of the Virasoro algebra to arbitrary dimensions

Colored tensor models generalize matrix models in higher dimensions. They admit a 1/N expansion dominated by spherical topologies and exhibit a critical behavior strongly reminiscent of matrix models. In this paper we generalize the colored tensor models to colored models with generic interaction, derive the Schwinger Dyson equations in the large N limit and analyze the associated algebra of constraints satisfied at leading order by the partition function. We show that the constraints form a Lie algebra (indexed by trees) yielding a generalization of the Virasoro algebra in arbitrary dimensions.