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## Frobenius manifolds

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and

Integrablehierarchies of

Toda type

joint work with B. Dubrovin

Piergiulio Tempesta

SISSA - Trieste

Gallipoli, June 28, 2006

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 theoperator 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

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:

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.

vector field

FM3.

F(t)

FM

WDVV

(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 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

Objective: Toextend thetheory of Frobenius manifoldsto 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)

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 thetau 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:

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:

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.

Integrable hierarchies

GWI orbifold

- 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.

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