Alexey Kokotov

A new hierarchy of integrable systems associated to Hurwitz spaces.

In this paper, we introduce a new class of integrable systems, naturally associated to Hurwitz spaces (spaces of meromorphic functions over Riemann surfaces). The critical values of the meromorphic functions play the role of ‘times’. Our systems give a natural generalization of the Ernst equation; in genus zero, they realize the scheme of deformation of integrable systems proposed by Burtsev, Mikhailov and Zakharov. We show that any solution of these systems in rank 1 defines a flat diagonal metric (Darboux–Egoroff metric) together with a class of corresponding systems of hydrodynamic type and their solutions.

On G-function of Frobenius manifolds related to Hurwitz spaces

The semisimple Frobenius manifolds related to the Hurwitz spaces H g, N (k 1 ,…,k l ) are considered. We show that the corresponding isomonodromic tau-function τ I coincides with (−1/2)-power of the Bergmann tau-function which was introduced in a recent work by the authors. This enables us to calculate explicitly the G-function of Frobenius manifolds related to the Hurwitz spaces H 0, N (k 1 , …, k l and H 1, N (k 1 , …, k l . As simple consequences we get formulas for the G-functions of the Frobenius manifolds 3/8N/W~k(AN−1) and 3/8×3/8N−1×{ℑz>0}/J(AN−1), where W~k(AN−1) is an extended affine Weyl group and J(A N−1 ) is a Jacobi group, in particular, proving the conjecture of Strachan (2003). In case of Frobenius manifolds related to Hurwitz spaces H g, N (k 1 , …, k l ) with g ≥ 2, we obtain formulas for ∣τ I ∣ 2 which allow to compute the real part of the G-function.

On the isomonodromic tau-function for the Hurwitz spaces of branched coverings of genus zero and one

The isomonodromic tau-function for the Hurwitz spaces of branched coverings of genus zero and one are constructed explicitly. Such spaces may be equipped with the structure of a Frobenius manifold and this introduces a flat coordinate system on the manifold. The isomonodromic tau-function, and in particular the associated

On the asymptotics of determinant of Laplacian at the principal boundary of the principal stratum of the moduli space of Abelian differentials

G

On the Spectral Theory of the Laplacian on Compact Polyhedral Surfaces of Arbitrary Genus

-function, are rewritten in these coordinates and an interpretation in terms of the caustics (where the multiplication is not semisimple) is given.

Metrics of Constant Positive Curvature with Conical Singularities, Hurwitz Spaces, and Determinants of Laplacians

Let X be a translation surface of genus g > 1 with 2g − 2 conical points of angle 4π and let γ, γ′ be two homologous saddle connections of length s joining two conical points of X and bounding two surfaces S+ and S− with boundaries ∂S+ = γ − γ′ and ∂S− = γ′ − γ. Gluing the opposite sides of the boundary of each surface S+, S− one gets two (closed) translation surfaces X+, X− of genera g+, g−; g+ + g− = g. Let ∆, ∆+ and ∆− be the Friedrichs extensions of the Laplacians corresponding to the (flat conical) metrics on X , X+ and X− respectively. We study the asymptotical behavior of the (modified, i. e. with zero modes excluded) zeta-regularized determinant det∗ ∆ as γ and γ′ shrink. We find the asymptotics

Determinants of pseudo-Laplacians

There are several well-known ways to introduce a compact Riemann surface which are also discussed in the present volume, e.g., via algebraic equations or by means of some uniformization theorem, where the surface is introduced as the quotient of the upper half-plane over the action of a Fuchsian group. In this chapter we consider a less popular approach which is at the same time, perhaps, the most elementary: one can simply consider the boundary of a connecter (but, generally, not simply connected) polyhedron in three dimensional Euclidean space.

Isomonodromic tau function on the space of admissible covers

Let

Krein Formula and S-Matrix for Euclidean Surfaces with Conical Singularities

f: X\to {\Bbb C}P^1

Polyhedral surfaces and determinant of Laplacian

be a meromorphic function of degree