arXiv subjects

A. Levin

Publications and source records attributed to A. Levin.

39 records · Page 3Linked to original sources

arXiv1999

Non-autonomous Hamiltonian systems related to highest Hitchin integrals

We describe non-autonomous Hamiltonian systems coming from the Hitchin integrable systems. The Hitchin integrals of motion depend on the W-structures of the basic curve. The parameters of the W-structures play the role of times. In particular, the quadratic integrals dependent on the complex structure (W_2-structure) of the basic curve and times are coordinate on the Teichmuller space. The corresponding flows are the monodromy preserving equations such as the Schlesinger equations, the Painleve VI equation and their generalizations. The equations corresponding to the highest integrals are monodromy preserving conditions with respect to changing of the W_k-structures (k>2). They are derived by the symplectic reduction from the gauge field theory on the basic curve interacting with W_k-gravity. As by product we obtain the classical Ward identities in this theory.

math-ph↗

arXiv1997

Painlevé - Calogero correpondence

It is proved that the Painlevé VI equation $(PVI_{\al,\be,\ga,\de})$ for the special values of constants $(\al=\frac{ν^2}{4},\be=-\frac{ν^2}{4}, \ga=\frac{ν^2}{4},\de=\f1{2}-\frac{ν^2}{4})$ is a reduced hamiltonian system. Its phase space is the set of flat $SL(2,C)$ connections over elliptic curves with a marked point and time of the system is given by the elliptic module. This equation can be derived via reduction procedure from the free infinite hamiltonian system. The phase space of later is the affine space of smooth connections and the "times are the Beltrami differentials. This approach allows to define the associate linear problem, whose isomonodromic deformations is provided by the Painlevé equation and the Lax pair. In addition, it leads to description of solutions by a linear procedure. This scheme can be generalized to $G$ bundles over Riemann curves with marked points, where $G$ is a simple complex Lie group. In some special limit such hamiltonian systems convert into the Hitchin systems. In particular, for $\SL$ bundles over elliptic curves with a marked point we obtain in this limit the elliptic Calogero N-body system. Relations to the classical limit of the Knizhnik- Zamolodchikov-Bernard equations is discussed.

alg-geom↗

arXiv1996

Double coset construction of moduli space of holomorphic bundles and Hitchin systems

We present a description of the moduli space of holomorphic vector bundles over Riemann curves as a double coset space which is differ from the standard loop group construction. Our approach is based on equivalent definitions of holomorphic bundles, based on the transition maps or on the first order differential operators. Using this approach we present two independent derivations of the Hitchin integrable systems. We define a "superfree" upstairs systems from which Hitchin systems are obtained by three step hamiltonian reductions. A special attention is being given on the Schottky parameterization of curves.

alg-geom↗