Bihamiltonian Cohomologies and Integrable Hierarchies I: A Special Case

We present some general results on properties of the bihamiltonian cohomologies associated to bihamiltonian structures of hydrodynamic type, and compute the third cohomology for the bihamiltonian structure of the dispersionless KdV hierarchy. The result of the computation enables us to prove the existence of bihamiltonian deformations of the dispersionless KdV hierarchy starting from any of its infinitesimal deformations.Comment: 43 pages. V2: the accepted version, to appear in Comm. Math. Phy

Universality of a double scaling limit near singular edge points in random matrix models

We consider unitary random matrix ensembles Z_{n,s,t}^{-1}e^{-n tr V_{s,t}(M)}dM on the space of Hermitian n x n matrices M, where the confining potential V_{s,t} is such that the limiting mean density of eigenvalues (as n\to\infty and s,t\to 0) vanishes like a power 5/2 at a (singular) endpoint of its support. The main purpose of this paper is to prove universality of the eigenvalue correlation kernel in a double scaling limit. The limiting kernel is built out of functions associated with a special solution of the P_I^2 equation, which is a fourth order analogue of the Painleve I equation. In order to prove our result, we use the well-known connection between the eigenvalue correlation kernel and the Riemann-Hilbert (RH) problem for orthogonal polynomials, together with the Deift/Zhou steepest descent method to analyze the RH problem asymptotically. The key step in the asymptotic analysis will be the construction of a parametrix near the singular endpoint, for which we use the model RH problem for the special solution of the P_I^2 equation. In addition, the RH method allows us to determine the asymptotics (in a double scaling limit) of the recurrence coefficients of the orthogonal polynomials with respect to the varying weights e^{-nV_{s,t}} on \mathbb{R}. The special solution of the P_I^2 equation pops up in the n^{-2/7}-term of the asymptotics.Comment: 32 pages, 3 figure

State Sum Models and Simplicial Cohomology

We study a class of subdivision invariant lattice models based on the gauge group $Z_{p}$, with particular emphasis on the four dimensional example. This model is based upon the assignment of field variables to both the $1$- and $2$-dimensional simplices of the simplicial complex. The property of subdivision invariance is achieved when the coupling parameter is quantized and the field configurations are restricted to satisfy a type of mod-$p$ flatness condition. By explicit computation of the partition function for the manifold $RP^{3} \times S^{1}$, we establish that the theory has a quantum Hilbert space which differs from the classical one.Comment: 28 pages, Latex, ITFA-94-13, (Expanded version with two new sections

Geometry and Integrability of Topological-Antitopological Fusion

Integrability of equations of topological-antitopological fusion (being proposed by Cecotti and Vafa) describing ground state metric on given 2D topological field theory (TFT) model, is proved. For massive TFT models these equations are reduced to a universal form (being independent on the given TFT model) by gauge transformations. For massive perturbations of topological conformal field theory models the separatrix solutions of the equations bounded at infinity are found by the isomonodromy deformations method. Also it is shown that ground state metric together with some part of the underlined TFT structure can be parametrized by pluriharmonic maps of the coupling space to the symmetric space of real positive definite quadratic forms.Comment: 30 pages, plain TEX, INFN-8/92-DS

Radiative decays of light vector mesons

The new data on $\rho,\omega,\phi$ radiative decays into $\pi^0\gamma,\eta\gamma,\eta'\gamma$ from SND experiment at VEPP-2M $e^+e^-$ collider are presented.Comment: 5 pages, 2 figures, talk given at 8th International Conference on Hadron Spectroscopy (HADRON 99), Beijing, China, 24-28 Aug 199

Seiberg-Witten Theory and Extended Toda Hierarchy

The quasiclassical solution to the extended Toda chain hierarchy, corresponding to the deformation of the simplest Seiberg-Witten theory by all descendants of the dual topological string model, is constructed explicitly in terms of the complex curve and generating differential. The first derivatives of prepotential or quasiclassical tau-function over the extra times, extending the Toda chain, are expressed through the multiple integrals of the Seiberg-Witten one-form. We derive the corresponding quasiclassical Virasoro constraints, discuss the functional formulation of the problem and propose generalization of the extended Toda hierarchy to the nonabelian theory.Comment: 32 pages, LaTe

New experimental data for the decays ϕ→μ+μ−\phi\to\mu^+\mu^- and ϕ→π+π−\phi\to\pi^+\pi^- from SND detector

The processes $e^+e^-\to\mu^+\mu^-$ and $e^+e^-\to\pi^+\pi^-$ have been studied with SND detector at VEPP-2M $e^+e^-$ collider in the vicinity of $\phi(1020)$ resonance. The branching ratios $B(\phi\to\mu^+\mu^-)=(3.30\pm 0.45\pm 0.32)\times 10^{-4}$ and $B(\phi\to\pi^+\pi^-)=(0.71\pm 0.11\pm 0.09)\times 10^{-4}$ were obtained.Comment: 5 pages, 4 figures, talk given at 8th International Conference on Hadron Spectroscopy (HADRON 99), Beijing, China, 24-28 Aug 199

New Data from SND Detector in Novosibirsk

The current status of experiments with SND detector at VEPP-2M e^+e^- collider in the energy range 2E_0=400-1400 MeV and recent results of data analysis for $\phi$, $\omega$ and $\rho$ decays and e^+e^- annihilation into hadrons are presented.Comment: 7 pages, 8 figures, Latex2e, uses espcrc1.sty. Talk given at 8th International Conference on Hadron Spectroscopy (HADRON 99), Beijing, China, 24-28 Aug 199

On universality of critical behavior in the focusing nonlinear Schr\uf6dinger equation, elliptic umbilic catastrophe and the Tritronqu\ue9e solution to the Painlev\ue9-I equation

We argue that the critical behavior near the point of "gradient catastrophe" of the solution to the Cauchy problem for the focusing nonlinear Schrodinger equation i epsilon Psi(t) + epsilon(2)/2 Psi(xx) + vertical bar Psi vertical bar(2)Psi = 0, epsilon << 1, with analytic initial data of the form Psi( x, 0; epsilon) = A(x)e(i/epsilon) (S(x)) is approximately described by a particular solution to the Painleve-I equation

Measurable versions of the LS category on laminations

We give two new versions of the LS category for the set-up of measurable laminations defined by Berm\'udez. Both of these versions must be considered as "tangential categories". The first one, simply called (LS) category, is the direct analogue for measurable laminations of the tangential category of (topological) laminations introduced by Colman Vale and Mac\'ias Virg\'os. For the measurable lamination that underlies any lamination, our measurable tangential category is a lower bound of the tangential category. The second version, called the measured category, depends on the choice of a transverse invariant measure. We show that both of these "tangential categories" satisfy appropriate versions of some well known properties of the classical category: the homotopy invariance, a dimensional upper bound, a cohomological lower bound (cup length), and an upper bound given by the critical points of a smooth function.Comment: 22 page