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

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

3. 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:

4. 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

5. 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

6. 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:

7. 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 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

8. 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

9. 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.

10. 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 is a shift operator:
Def The positive part of the operator
is defined by:
Def. 9. The residue is

11. 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

12. 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

13. 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

14. 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)

15. 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)

16. 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:

17. Realization of the Virasoro algebra

18. 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:

19. 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

20. 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.

21. 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

22. 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.