Frobenius manifolds. and. Integrable hierarchies of Toda type. joint work with B. Dubrovin. Piergiulio Tempesta. SISSA - Trieste. Gallipoli, June 28, 2006. Topological field theories (WDVV equations) 1990. Witten, Kontsevich (1990-92). Integrable hierarchies of PDEs (’60).
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joint work with B. Dubrovin
SISSA - Trieste
Gallipoli, June 28, 2006
Manin, Kontsevich (1994)
(K. Saito, 1983)
Simplest example: the Einstein-Hilbert gravity in 2D.
Euler characteristic of
The two-point correlator:
determines a scalar product on the manifold.
The triple correlator
defines the structure of theoperator algebra A associated with
quantum gravity in two dimensions?
1) Matrix models of gravity (Parisi, Izikson, Zuber,…)
: the partition is an integral in the space of N x N
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”
: Deligne-Mumford compactification
: line bundles over
X : smooth projective variety
: moduli space of stable curves on X of genus g and
degree with m marked points
Gromov-Witten invariants of genus g
total Gromov-Witten potential
Witten’s conjecture: the models 1) and 2) of quantum gravity are
= log of the -function of a solution of the KdV hierarchy
(Witten): The generating functions of GWI can be written as a
hierarchy of systems of n evolutionary PDEs for the dependent
and the hamiltonian densities of the flows given by
WDVV equations (1990)
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.
(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
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
Def. 7. is a shift operator:
Def. 8. The positive part of the operator
is defined by:
Def. 9.The residue is
Def 11. The flows of the extended hierarchy are given by:
Remark. We have two different fractional powers of the Lax operator:
Logaritm of L. Let us introduce the dressing operators
The logarithm of L is defined by
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
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:
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
Lemma 3. (symmetry property of the omega function)
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:
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
correspondong tau function satisfes the Virasoro constraints
Here is a collection of formal power
series in .
2. The system of Virasoro constrants is satisfied.
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:
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.
In particular, it represents a natural geometrical setting for the
study of differential-difference systems of Toda type.
Toda hierarches associated to the orbit spaces of other
extended affine Weyl groups.
GW invariants orbifold and integrable hierarchies.
FM and Drinfeld-Sokolov hierarchies.