New non-linear equations and modular form expansion for double-elliptic Seiberg-Witten prepotential

Integrable N-particle systems have an important property that the associated Seiberg-Witten prepotentials satisfy the WDVV equations. However, this does not apply to the most interesting class of elliptic and double-elliptic systems. Studying the commutativity conjecture for theta-functions on the families of associated spectral curves, we derive some other non-linear equations for the perturbative Seiberg-Witten prepotential, which turn out to have exactly the double-elliptic system as their generic solution. In contrast with the WDVV equations, the new equations acquire non-perturbative corrections which are straightforwardly deducible from the commutativity conditions. We obtain such corrections in the first non-trivial case of N=3 and describe the structure of non-perturbative solutions as expansions in powers of the flat moduli with coefficients that are (quasi)modular forms of the elliptic parameter.Comment: 25 page

Fluctuation Induced Forces in Non-equilibrium (Diffusive) Dynamics

Thermal fluctuations in non-equilibrium steady states generically lead to power law decay of correlations for conserved quantities. Embedded bodies which constrain fluctuations in turn experience fluctuation induced forces. We compute these forces for the simple case of parallel slabs in a driven diffusive system. The force falls off with slab separation $d$ as $k_B T/d$ (at temperature $T$, and in all spatial dimensions), but can be attractive or repulsive. Unlike the equilibrium Casimir force, the force amplitude is non-universal and explicitly depends on dynamics. The techniques introduced can be generalized to study pressure and fluctuation induced forces in a broad class of non-equilibrium systems.Comment: 5 pages, 2 figure

Rational Top and its Classical R-matrix

We construct a rational integrable system (the rational top) on a coadjoint orbit of ${\rm SL}_N$ Lie group. It is described by the Lax operator with spectral parameter and classical non-dynamical skew-symmetric $r$-matrix. In the case of the orbit of minimal dimension the model is gauge equivalent to the rational Calogero-Moser (CM) system. To obtain the results we represent the Lax operator of the CM model in two different factorized forms -- without spectral parameter (related to spinless case) and another one with the spectral parameter. The latter gives rise to the rational top while the first one is related to generalized Cremmer-Gervais $r$-matrices. The gauge transformation relating the rational top and CM model provides a classical rational version of the IRF-Vertex correspondence. From a geometrical point of view it describes the modification of ${\rm SL}(N,\mathbb C)$-bundles over degenerated elliptic curve. In view of Symplectic Hecke Correspondence the rational top is related to the rational spin CM model. Possible applications and generalizations of the suggested construction are discussed. In particular, the obtained $r$-matrix defines a class of KZB equations.Comment: 19 page