Integrable hierarchies of Frobenius manifolds and Integrable hierarchies of Toda type joint work with B. Dubrovin Piergiulio Tempesta SISSA - Trieste Gallipoli, June 28, 2006

Integrable hierarchies of PDEs (’60) Topological field theories (WDVV equations) 1990 Witten, Kontsevich (1990-92) Integrable hierarchies of PDEs (’60) Frobenius manifolds (Dubrovin, 1992) Manin, Kontsevich (1994) Singularity theory (K. Saito, 1983) Gromov-Witten invariants (1990)

Topological field theories in 2D Simplest example: the Einstein-Hilbert gravity in 2D. Euler characteristic of Consider a TFT in 2D on a manifold, with N primary fields: The two-point correlator: determines a scalar product on the manifold. The triple correlator defines the structure of the operator algebra A associated with the model:

Problem: how to formulate a coherent theory of quantum gravity in two dimensions? 1) Matrix models of gravity (Parisi, Izikson, Zuber,…) Discretization: polyhedron : the partition is an integral in the space of N x N Hermitian matrices function of a solution of the KdV hierarchy. 2) Cohomological field theory (Witten, Kontsevich, Manin): : moduli space of Riemann surfaces of genus g with s “marked points” (stability) : Deligne-Mumford compactification : line bundles over Fiber over

Gromov-Witten theory Gromov-Witten invariants of genus g X : smooth projective variety : moduli space of stable curves on X of genus g and degree with m marked points basis Gromov-Witten invariants of genus g total Gromov-Witten potential Witten’s conjecture: the models 1) and 2) of quantum gravity are equivalent. = log of the -function of a solution of the KdV hierarchy

GWI and integrable hierarchies (Witten): The generating functions of GWI can be written as a hierarchy of systems of n evolutionary PDEs for the dependent variables and the hamiltonian densities of the flows given by WDVV equations (1990) Crucial observation:

Frobenius manifold Definition 1. A Frobenius algebra is a couple where A is an associative, commutative algebra with unity over A field k (k = R, C) and is a bilinear symmetric form non degenerate over k, invariant: Def. 2. A Frobenius manifold is a differential manifold M with the specification of the structure of a Frobenius algebra over the tangent spaces , with smooth dependence on the point . The following axioms are also satisfied: FM1. The metric over M is flat. FM2. Let . Then the 4-tensor must be symmetric in x,y,z,w. FM3. vector field F(t) FM WDVV

Bihamiltonian Structure (Casimir for ) : primary Hamiltonian; : descendent Hamiltonians Tau function: (1983) Dispersionless hierarchies and Frobenius manifolds Frobenius manifold solution of WDVV eqs. an integrable hierarchy of quasilinear PDEs of the form

Dispersionless hierarchies Frobenius manifold Tau structure, Virasoro symmetries Whitham averaging Full hierarchies Topological field theories Witten, Kontsevich Problem of the reconstruction of the full hierarchy starting from the Frobenius structure Result (Dubrovin, Zhang) For the class of Gelfand-Dikii hierarchies there exists a Lie group of transformations mapping the Principal Hierarchy into the full hierarchy if it admits: a tau structure; Simmetry algebra of linear Virasoro operators, acting linearly on the tau structure 3) The underlying Frobenius structure is semisimple.

Frobenius manifolds and integrable hierarchies of Toda type B. Dubrovin, P. T. (2006) Problem: study the Witten-Kontsevich correspondence in the case of hierarchies of differential-difference equations. Toda equation (1967) Bigraded Extended Toda Hierarchy G. Carlet, B. Dubrovin 2004 Two parametric family of integrable hierarchies of differential- difference equations It is a Marsden-Weinstein reduction of the 2D Toda hierarchy. Def. 7. is a shift operator: Def. 8. The positive part of the operator is defined by: Def. 9. The residue is

Def. 10. The Lax operator L of the hierarchy is Def 11. The flows of the extended hierarchy are given by: where Remark. We have two different fractional powers of the Lax operator: which satisfy: Logaritm of L. Let us introduce the dressing operators such that The logarithm of L is defined by

Example. Consider the case k=m=1. G.Carlet, B. Dubrovin, J. Zhang, Russ. Math. Surv. (2003) B Dubrovin, J. Zhang, CMP (2004) q = 0, q = 0, q = 1, dove

Finite discrete groups and Frobenius structures Objective: To extend the theory of Frobenius manifolds to the case of differential-difference systems of eqs. Construct the Frobenius structure 2) Prove the existence of : A bihamiltonian structure A tau structure A Virasoro algebra of Lie symmetries. Finite discrete groups and Frobenius structures K. Saito, 1983 : flat structures in the space of parameters of the universal unfolding of singularities. Theorem 1. The Frobenius structure associated to the extended Toda Hierarchy is isomorphic to the orbit space of the extend affine Weyl group . The bilinear symmetric form on the tangent planes is

Bihamiltonian structure. Let us introduce the Hamiltonians Theorem 2. The flows of the hierarchy are hamiltonian with respect to two different Poisson structures. Theorem 3. The two Poisson structures are defined by: (R-matrix approach)

Tau structure Lemma 1. For any p, q, : Def. 12 (Omega function): Def. 13 For any solution of the bigraded extended Toda hierarchy there exists a function called the tau function of the hierarchy. It is defined by Lemma 2. The hamiltonian densities are related to the tau structure by Lemma 3. (symmetry property of the omega function)

Lie symmetries and Virasoro algebras Theorem 4. There exists an algebraof linear differential operators of the second order associated with the Frobenius manifold . These operators satisfy the Virasoro commutation relations The generating function of such operators is:

Realization of the Virasoro algebra

Theorem 5. The tau function admits the following genus expansion Consider the hierarchy (k = 2, m = 1) The first hamiltonian structure is given by whereas the other Poisson bracket vanish. The relation between the fields and the tau structure reads Theorem 5. The tau function admits the following genus expansion where represents the tau function for the solution of the corresponding dispersionless hierarchy:

Main Theorem 1. Any solution of this hierarchy can be represented through a quasi-Miura transformation of the form The functions are universal: they are the same for all solutions of the full hierarchy and depend only on the solution of the dispersionless hierarchy. 2. The transformations are infinitesimal symmetries of the hierarchy (k = 2, m = 1), in the sense that the functions satisfy the equations of the hierarchy modulo terms of order

3. For a generic solution of the extended Toda hierarchy, the correspondong tau function satisfes the Virasoro constraints Here is a collection of formal power series in . Conjecture 1. For any hierarchy of the family of bigraded extended Toda Hierarchy, i.e.for any value of (k, m): There exists a class of Lie symmetries generated by the action of theVirasoro operators. 2. The system of Virasoro constrants is satisfied.

Toda hierarchies and Gromov-Witten invariants The dispersionless classical Toda hierarchy (k = m = 1) is described by a 2-dimensional Frobenius manifolds Alternatively, it can be identified with the quantum cohomology of the complex projective line In the bigraded case: Conjecture 2. The total Gromov-Witten potential for the weighted projective space is the logarithm of the tau function of a particular solution to the bigraded extended Toda hierarchy. GWI orbifold Integrable hierarchies

Conclusions Future perspectives The theory of Frobenius manifolds allows to establish new connections between topological field theories integrable hierarchies of nonlinear evolution equations enumerative geometry (Gromov-Witten invariants) the topology of moduli spaces of stable algebraic varieties singularity theory, etc. In particular, it represents a natural geometrical setting for the study of differential-difference systems of Toda type. Future perspectives Toda hierarches associated to the orbit spaces of other extended affine Weyl groups. GW invariants orbifold and integrable hierarchies. FM and Drinfeld-Sokolov hierarchies.