Semisimple Frobenius Manifolds Research Articles

Riemann-Hilbert problem associated to Frobenius manifold structures on Hurwitz spaces: Irregular singularity

Solutions to the Riemann-Hilbert problems with irregular singularities naturally associated to semisimple Frobenius manifold structures on Hurwitz spaces (moduli spaces of meromorphic functions on Riemann surfaces) are constructed. The solutions are given in terms of meromorphic bidifferentials defined on the underlying Riemann surface. The relationship between different classes of Frobenius manifold structures on Hurwitz spaces (real doubles, deformations) is described at the level of the corresponding Riemann-Hilbert problems

Tau-functions on Hurwitz Spaces

We construct a flat holomorphic line bundle over a connected component of the Hurwitz space of branched coverings of the Riemann sphere P1. A flat holomorphic connection defining the bundle is described in terms of the invariant Wirtinger projective connection on the branched covering corresponding to a given meromorphic function on a Riemann surface of genus g. In genera 0 and 1 we construct a nowhere vanishing holomorphic horizontal section of this bundle (the ‘Wirtinger tau-function’). In higher genus we compute the modulus square of the Wirtinger tau-function. In particular one gets formulas for the isomonodromic tau-functions of semisimple Frobenius manifolds connected with the Hurwitz spaces Hg,N(1,⋯,1).

On G-function of Frobenius manifolds related to Hurwitz spaces

The semisimple Frobenius manifolds related to the Hurwitz spaces Hg, N(k1,…,kl) are considered. We show that the corresponding isomonodromic tau-function τI 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 H0, N(k1, …, kl and H1, N(k1, …, kl. As simple consequences we get formulas for the G-functions of the Frobenius manifolds ⅜N/W˜k(AN−1) and ⅜×⅜N−1×{ℑz>0}/J(AN−1)⁠, where W˜k(AN−1) is an extended affine Weyl group and J(AN−1) is a Jacobi group, in particular, proving the conjecture of Strachan (2003). In case of Frobenius manifolds related to Hurwitz spaces Hg, N(k1, …, kl) with g ≥ 2, we obtain formulas for ∣τI∣2 which allow to compute the real part of the G-function.

Frobenius manifolds: natural submanifolds and induced bi-Hamiltonian structures

Submanifolds of Frobenius manifolds are studied. In particular, so-called natural submanifolds are defined and, for semi-simple Frobenius manifolds, classified. These carry the structure of a Frobenius algebra on each tangent space, but will, in general, be curved. The induced curvature is studied, a main result being that these natural submanifolds carry a induced pencil of compatible metrics. It is then shown how one may constrain the bi-Hamiltonian hierarchies associated to a Frobenius manifold to live on these natural submanifolds whilst retaining their, now non-local, bi-Hamiltonian structure.

Symmetries and solutions of Getzler's equation forCoxeter and extended affine Weyl Frobenius manifolds

The G-function associated to the semi-simple Frobenius manifold C^n/W (where W is a Coxeter group or an extended affine Weyl group) is studied. The general form of the G function is given in terms of a logarithmic singularity over caustics in the manifold. The main result in this paper is a universal formula for the G-function corresponding to the Frobenius manifold C^n/W^(k)(A_{n-1}) where W^(k)(A_{n-1}) is a certain extended affine Weyl group (or, equivalently, corresponding to the Hurwitz space M_{0;k-1,n-k-1}), together with the general form of the G-function in terms of data on caustics. Symmetries of the G function are also studied.

Deformations of the Bihamiltonian Structures on the Loop Space of Frobenius Manifolds

We consider an important class of deformations of the genus zero bihamiltonian structure defined on the loop space of semisimple Frobenius manifolds, and present results on such deformations at the genus one and genus two approximations.

Semi-infinite variations of Hodge structures and integrable hierarchies of KdV type

We introduce integrable KdV type hierarchy associated naturally with arbitrary semi-simple Frobenius manifold. We present hierarchy in a Lax form and show that it admits bihamiltonian description.

Inverse Problem and Monodromy Data for Three-Dimensional Frobenius Manifolds

We study the inverse problem for semi-simple Frobenius manifolds of dimension 3 and we explicitly compute a parametric form of the solutions of the WDVV equations in terms of Painleve VI transcendents. We show that the solutions are labeled by a set of monodromy data. We use our parametric form to explicitly construct polynomial and algebraic solutions and to derive the generating function of Gromov–Witten invariants of the quantum cohomology of the two-dimensional projective space. The procedure is a relevant application of the theory of isomonodromic deformations.

Stokes matrices, Poisson Lie groups and Frobenius manifolds

We point out, and draw some consequences of, the fact that the Poisson Lie group G* dual to G=GL_n(C) (with its standard complex Poisson structure) may be identified with a certain moduli space of meromorphic connections on the unit disc having an irregular singularity at the origin. The Riemann-Hilbert map for such connections, taking the Stokes data, induces a holomorphic map from the dual of the Lie algebra of G to the Poisson Lie group G*. The main result is that this map is Poisson. First this leads to new, more direct, proofs of theorems of Duistermaat and Ginzburg-Weinstein (enabling one to reduce Kostant's non-linear convexity theorem, involving the Iwasawa projection, to the linear convexity theorem, involving the `diagonal part'). Secondly we obtain a new approach to the braid group invariant Poisson structure on Dubrovin's local moduli space of semisimple Frobenius manifolds: it is induced from the standard Poisson structure on G*.

THE TENSOR PRODUCT IN THE THEORY OF FROBENNIUS MANIFOLDS

We introduce the operation of forming the tensor product in the theory of analytic Frobenius manifolds. Building on the results for formal Frobenius manifolds which we extend to the additional structures of Euler fields and flat identities, we prove that the tensor product of pointed germs of Frobenius manifolds exists. Furthermore, we define the notion of a tensor product diagram of Frobenius manifolds with factorizable flat identity and prove the existence of such a diagram and hence a tensor product Frobenius manifold. These diagrams and manifolds are unique up to equivalence. Finally, we derive the special initial conditions for a tensor product of semi-simple Frobenius manifolds in terms of the special initial conditions of the factors.

Frobenius manifolds from Yang-Mills instantons

We present an elementary self-contained account of semisimple Frobenius manifolds in three dimensions, and exhibit a new family of explicit examples. These examples are constructed from Yang-Mills instantons with a certain symmetry.

Semisimple Frobenius (super)manifolds and quantum cohomology of $\mathbb P}^r$

We introduce and study a superversion of Dubrovin's notion of semisimple Frobenius manifolds. We establish a correspondence between semisimple Frobenius (super)manifolds and special solutions to the (supersymmetric) Schlesinger equations. Finally, we calculate the Schlesinger initial conditions for solutions describing quantum cohomology of projective spaces.