Hilbert Schemes, Separated Variables, and D-Branes

We explain Sklyanin's separation of variables in geometrical terms and construct it for Hitchin and Mukai integrable systems. We construct Hilbert schemes of points on $T^{*}\Sigma$ for \Sigma = {\IC}, {\IC}^{*} or elliptic curve, and on ${\bf C}^{2}/{\Gamma}$ and show that their complex deformations are integrable systems of Calogero-Sutherland-Moser type. We present the hyperk\"ahler quotient constructions for Hilbert schemes of points on cotangent bundles to the higher genus curves, utilizing the results of Hurtubise, Kronheimer and Nakajima. Finally we discuss the connections to physics of $D$-branes and string duality.Comment: harvmac, 27 pp. big mode; v2. typos and references correcte

Duality in Integrable Systems and Gauge Theories

We discuss various dualities, relating integrable systems and show that these dualities are explained in the framework of Hamiltonian and Poisson reductions. The dualities we study shed some light on the known integrable systems as well as allow to construct new ones, double elliptic among them. We also discuss applications to the (supersymmetric) gauge theories in various dimensions.Comment: harvmac 45 pp.; v4. minor corrections, to appear in JHE

Monte Carlo simulations of the classical two-dimensional discrete frustrated ϕ4\phi ^4 model

The classical two-dimensional discrete frustrated $\phi ^4$ model is studied by Monte Carlo simulations. The correlation function is obtained for two values of a parameter $d$ that determines the frustration in the model. The ground state is a ferro-phase for $d=-0.35$ and a commensurate phase with period N=6 for $d=-0.45$. Mean field predicts that at higher temperature the system enters a para-phase via an incommensurate state, in both cases. Monte Carlo data for $d=-0.45$ show two phase transitions with a floating-incommensurate phase between them. The phase transition at higher temperature is of the Kosterlitz-Thouless type. Analysis of the data for $d=-0.35$ shows only a single phase transition between the floating-fluid phase and the ferro-phase within the numerical error.Comment: 5 figures, submitted to the European Physical Journal

Many-body synchronization of interacting qubits by engineered ac-driving

In this work we introduce the many-body synchronization of an interacting qubit ensemble which allows one to switch dynamically from many-body-localized (MBL) to an ergodic state. We show that applying of $\pi$-pulses with altering phases, one can effectively suppress the MBL phase and, hence, eliminate qubits disorder. The findings are based on the analysis of the Loschmidt echo dynamics which shows a transition from a power-law decay to more rapid one indicating the dynamical MBL-to-ergodic transition. The technique does not require to know the microscopic details of the disorder.Comment: 5 pages, 4 figure

On the spatial structure of the Perseids meteor stream

The analysis of radar observations of the Perseid meteor stream conducted in an ionospherical laboratory in the period from 1964 to 1981 is presented. The Perseids meteor rates were determined by the fluctuation method. Analysis of their hourly distributions showed that the stream maximum position is different for different years, i.e., the stream nodal position is constantly changing. The results of the analysis are presented and discussed