Frobenius manifolds

1
Frobenius manifolds
and
Integrable hierarchies of Toda type
joint work with B. Dubrovin
Piergiulio Tempesta
SISSA - Trieste
Gallipoli, June 28, 2006
2
Topological field theories (WDVV equations) 1990
Integrable hierarchies of PDEs (60)
Witten, Kontsevich (1990-92)
Frobenius manifolds (Dubrovin, 1992)
Manin, Kontsevich (1994)
Gromov-Witten invariants (1990)
Singularity theory (K. Saito, 1983)
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
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
Wittens 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. 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
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. 7. is a shift operator
Def. 8. 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 km1.

• G.Carlet, B. Dubrovin, J. Zhang, Russ. Math.
  Surv. (2003)

• B Dubrovin, J. Zhang, CMP (2004)

• q 0,

• q 0,

• q 1,

dove
13

• 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
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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)
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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)
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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
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.
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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
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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.

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.