Integration of the continuous Heisenberg spin chain through the inverse scattering method

The inverse scattering method is applied to the Heisenberg chain. We give the general scheme of the solution of the equations of motion. We describe the process of solitons scattering and show the existence of an infinite series of constants of motion.     

Hyperbolic -spheres with conical singularities, accessory parameters and Kähler metrics on

We show that the real-valued function on the moduli space of pointed rational curves, defined as the critical value of the Liouville action functional on a hyperbolic -sphere with conical singularities of arbitrary orders , generates accessory parameters of the associated Fuchsian differential equation as their common antiderivative. We introduce a family of Kähler metrics on parameterized by the set of orders , explicitly relate accessory parameters to these metrics, and prove that the functions are their Kähler potentials.

     

The Selberg zeta function and a new Kähler metric on the moduli space of punctured Riemann surfaces

A local index theorem for families of -operators on Riemann surfaces with functures is proved. A new Kähler metric on the moduli space of punctured surfaces is described in terms of the Eisenstein-Maass series.     

Asymptotic behavior of the solutions of the Kadomtsev-Pyatviashvili equation (two-dimensional Korteweg-De Vries equation)

The asymptotic form of the solutions of the Kadomtsev-Pyatviashvili equation for t → ± ∞ is presented. The reverse problem of reconstructing the solution from its asymptotic form is also solved.     

The first Chern form on moduli of parabolic bundles

For moduli space of stable parabolic bundles on a compact Riemann surface, we derive an explicit formula for the curvature of its canonical line bundle with respect to Quillen’s metric and interprete it as a local index theorem for the family of \(\overline\partial\)-operators in the associated parabolic endomorphism bundles. The formula consists of two terms: one standard (proportional to the canonical Kähler form on the moduli space), and one nonstandard, called a cuspidal defect, that is defined by means of special values of the Eisenstein–Maass series. The cuspidal defect is explicitly expressed through the curvature forms of certain natural line bundles on the moduli space related to the parabolic structure. We also compare our result with Witten’s volume computation.     

Liouville Action and Weil-Petersson Metric on Deformation Spaces, Global Kleinian Reciprocity and Holography

We rigorously define the Liouville action functional for the finitely generated, purely loxodromic quasi-Fuchsian group using homology and cohomology double complexes naturally associated with the group action. We prove that classical action – the critical value of the Liouville action functional, considered as a function on the quasi-Fuchsian deformation space, is an antiderivative of a 1-form given by the difference of Fuchsian and quasi-Fuchsian projective connections. This result can be considered as global quasi-Fuchsian reciprocity which implies McMullen's quasi-Fuchsian reciprocity. We prove that the classical action is a Kähler potential of the Weil-Petersson metric. We also prove that the Liouville action functional satisfies holography principle, i.e., it is a regularized limit of the hyperbolic volume of a 3-manifold associated with a quasi-Fuchsian group. We generalize these results to a large class of Kleinian groups including finitely generated, purely loxodromic Schottky and quasi-Fuchsian groups, and their free combinations.     

Generating Functional in CFT and Effective Action for Two-Dimensional Quantum Gravity on Higher Genus Riemann Surfaces

We formulate and solve the analog of the universal Conformal Ward Identity for the stress-energy tensor on a compact Riemann surface of genus g > 1, and present a rigorous invariant formulation of the chiral sector in the induced two-dimensional gravity on higher genus Riemann surfaces. Our construction of the action functional uses various double complexes naturally associated with a Riemann surface, with computations that are quite similar to descent calculations in BRST cohomology theory. We also provide an interpretation of the action functional in terms of the geometry of different fiber spaces over the Teichmüller space of compact Riemann surfaces of genus g > 1. Received: 12 September 1996 / Accepted: 6 January 1997     

A local index theorem for families of\bar \partial -operators on punctured Riemann surfaces and a new Kähler metric on their moduli spaceson punctured Riemann surfaces and a new Kähler metric on their moduli spaces

We prove a local index theorem for families of \(\bar \partial \) -operators on Riemann surfaces of type (g, n), i.e. of genusg withn>0 punctures. We calculate the first Chern form of the determinant line bundle on the Teichmüller spaceT _g,n endowed with Quillen's metric (where the role of the determinant of the Laplace operators is played by the values of the Selberg zeta function at integer points). The result differs from the case of compact Riemann surfaces by an additional term, which turns out to be the Kähler form of a new Kähler metric on the moduli space of punctured Riemann surfaces. As a corollary of this result we derive, for instance, an analog of Mumford's isomorphism in the case of the universal curve.