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

Chronopotentiometry at platinum electrode in KF-NaF-AlF3-Al2O3 melt

Some features of the mechanism of the anode process on platinum in KF–NaF–AlF3–Al2O3 melt at 750–780 °C depending on the of anodic current density (0.5 mA/cm2 to 2.0 A/cm2) and anodic pulse duration have been studied using chronopotentiometry method. In curves of change in the platinum anode potential a small peak at current densities of 10–30 mA/cm2 and a clear peak at current densities of 0.5–2.0 A/cm2 are recorded when the current is cut on. Analysis of dependencies of the transition time on the current density indicates that the first peak in curve is associated with the formation of an oxide compound on the platinum surface, and the second one is related to hindering the diffusion for delivery of electroactive particles to its surface.The study was supported by the Russian Foundation for Basic Research (grant № 13–03–00829 A)

Electronic structure of FeSe monolayer superconductors

We review a variety of theoretical and experimental results concerning electronic band structure of superconducting materials based on FeSe monolayers. Three type of systems are analyzed: intercalated FeSe systems A_xFe_2Se_{2-x}S_x and [Li_{1-x}Fe_xOH]FeSe as well as the single FeSe layer films on SrTiO_3 substrate. We present the results of detailed first principle electronic band structure calculations for these systems together with comparison with some experimental ARPES data. The electronic structure of these systems is rather different from that of typical FeAs superconductors, which is quite significant for possible microscopic mechanism of superconductivity. This is reflected in the absence of hole pockets of the Fermi surface at \Gamma-point in Brillouin zone, so that there are no "nesting" properties of different Fermi surface pockets. LDA+DMFT calculations show that correlation effects on Fe-3d states in the single FeSe layer are not that strong as in most of FeAs systems. As a result, at present there is no theoretical understanding of the formation of rather "shallow" electronic bands at M points. LDA calculations show that the main difference in electronic structure of FeSe monolayer on SrTiO_3 substrate from isolated FeSe layer is the presence of the band of O-2p surface states of TiO_2 layer on the Fermi level together with Fe-3d states, which may be important for understanding the enhanced T_c values in this system. We briefly discuss the implications of our results for microscopic models of superconductivity.Comment: 21 pages, 13 figures, minor typos correcte

Trigonometric Sutherland systems and their Ruijsenaars duals from symplectic reduction

Besides its usual interpretation as a system of $n$ indistinguishable particles moving on the circle, the trigonometric Sutherland system can be viewed alternatively as a system of distinguishable particles on the circle or on the line, and these 3 physically distinct systems are in duality with corresponding variants of the rational Ruijsenaars-Schneider system. We explain that the 3 duality relations, first obtained by Ruijsenaars in 1995, arise naturally from the Kazhdan-Kostant-Sternberg symplectic reductions of the cotangent bundles of the group U(n) and its covering groups $U(1) \times SU(n)$ and ${\mathbb R}\times SU(n)$, respectively. This geometric interpretation enhances our understanding of the duality relations and simplifies Ruijsenaars' original direct arguments that led to their discovery.Comment: 34 pages, minor additions and corrections of typos in v

Multiloop Superstring Amplitudes from Non-Minimal Pure Spinor Formalism

Using the non-minimal version of the pure spinor formalism, manifestly super-Poincare covariant superstring scattering amplitudes can be computed as in topological string theory without the need of picture-changing operators. The only subtlety comes from regularizing the functional integral over the pure spinor ghosts. In this paper, it is shown how to regularize this functional integral in a BRST-invariant manner, allowing the computation of arbitrary multiloop amplitudes. The regularization method simplifies for scattering amplitudes which contribute to ten-dimensional F-terms, i.e. terms in the ten-dimensional superspace action which do not involve integration over the maximum number of $\theta$'s.Comment: 23 pages harvmac, added acknowledgemen

On Microscopic Origin of Integrability in Seiberg-Witten Theory

We discuss microscopic origin of integrability in Seiberg-Witten theory, following mostly the results of hep-th/0612019, as well as present their certain extension and consider several explicit examples. In particular, we discuss in more detail the theory with the only switched on higher perturbation in the ultraviolet, where extra explicit formulas are obtained using bosonization and elliptic uniformization of the spectral curve.Comment: 24 pages, 1 figure, LaTeX, based on the talks at 'Geometry and Integrability in Mathematical Physics', Moscow, May 2006; 'Quarks-2006', Repino, May 2006; Twente conference on Lie groups, December 2006 and 'Classical and Quantum Integrable Models', Dubna, January 200