Frobenius manifolds Integrable hierarchies of Toda typePiergiulio TempestaSISSA - Trieste Gallipoli, June 28, 2006joint work with B. Dubrovinand

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

Topological field theories in 2DSimplest 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 correlatordefines 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: polyhedron2) Cohomological field theory (Witten, Kontsevich, Manin):: moduli space of Riemann surfaces of genus g with s marked points: the partition is an integral in the space of N x N Hermitian matricesfunction of a solution of the KdV hierarchy.(stability): Deligne-Mumford compactification: line bundles over Fiber over

Wittens conjecture: the models 1) and 2) of quantum gravity are equivalent. = log of the -function of a solution of the KdV hierarchy

Gromov-Witten invariants of genus gtotal Gromov-Witten potentialGromov-Witten theoryX : smooth projective variety: moduli space of stable curves on X of genus g and degree with m marked pointsbasis

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

Frobenius manifoldDefinition 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 manifoldM 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 FMWDVVF(t)

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

Frobenius manifoldDispersionless hierarchiesTopological fieldtheoriesFull hierarchies

Witten, KontsevichWhithamaveragingTau structure, Virasoro symmetries 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 typeB. 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 HierarchyDef. 7. is a shift operator: Two parametric family of integrable hierarchies of differential- difference equations

It is a Marsden-Weinstein reduction of the 2D Toda hierarchy.Def. 8. The positive part of the operator is defined by:Def. 9. The residue is G. Carlet, B. Dubrovin 2004

Def. 10. The Lax operator L of the hierarchy isDef 11. The flows of the extended hierarchy are given by:whereRemark. 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. q = 0, q = 0, q = 1,dove G.Carlet, B. Dubrovin, J. Zhang, Russ. Math. Surv. (2003) B Dubrovin, J. Zhang, CMP (2004)

Objective: To extend the theory of Frobenius manifolds to the caseof 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 Theorem 1. The Frobenius structure associated to the extended TodaHierarchy is isomorphic to the orbit space of the extend affine Weyl group .The bilinear symmetric form on the tangent planes isK. Saito, 1983 : flat structures in the space of parametersof the universal unfolding of singularities.

Bihamiltonian structure. Let us introduce the HamiltoniansTheorem 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)

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

Lie symmetries and Virasoro algebrasTheorem 4. There exists an algebraof linear differential operators of the second orderassociated with the Frobenius manifold . These operators satisfy the Virasoro commutation relationsThe generating function of such operators is:

Realization of the Virasoro algebra

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

1. Any solution of this hierarchy can be represented through a quasi-Miura transformation of the form The functions are universal: they arethe same for all solutions of the full hierarchy and depend only on the solution of the dispersionless hierarchy.

Main Theorem are infinitesimal symmetries of the hierarchy (k = 2, m = 1), in the sense that the functionssatisfy the equations of the hierarchy modulo terms of order 2. The transformations

3. For a generic solution of the extended Toda hierarchy, thecorrespondong 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 TodaHierarchy, 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 invariantsThe dispersionless classical Toda hierarchy (k = m = 1) is described by a 2-dimensional Frobenius manifoldsAlternatively, it can be identified with the quantum cohomology of the complex projective lineConjecture 2. The total Gromov-Witten potential for the weighted projectivespace is the logarithm of the tau function of a particular solution to the bigraded extended Toda hierarchy. GWI orbifold Integrable hierarchiesIn the bigraded case:

Conclusions 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.Future perspectivesGW invariants orbifold and integrable hierarchies.Toda hierarches associated to the orbit spaces of other extended affine Weyl groups.In particular, it represents a natural geometrical setting for the study of differential-difference systems of Toda type.FM and Drinfeld-Sokolov hierarchies.