Frobenius manifolds

Integrable hierarchies of Toda type

Piergiulio Tempesta

SISSA - Trieste

Gallipoli, June 28, 2006

joint work with B. Dubrovin

and

Topological fieldtheories

(WDVV equations)1990

Integrable 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 2D

arbitraryxgij 0 S

g

yx ......

Simplest example: the Einstein-Hilbert gravity in 2D.

xdgRS 2 Euler characteristic of

,..., xLS

• Consider a TFT in 2D on a manifold, with N primary fields: .,...,1 N

The two-point correlator:

determines a scalar product on the manifold.

The triple correlator

c

defines the structure of the operator algebra A associated with the model:

c

cc 1

Problem: how to formulate a coherent theory of quantum gravity in two dimensions?

1) Matrix models of gravity (Parisi, Izikson, Zuber,…)

Discretization: g polyhedron

2) Cohomological field theory (Witten, Kontsevich, Manin):

g

: moduli space of Riemann surfaces of genus g with s “marked points”

NZ : the partition is an integral in the space of N x N Hermitian matrices NN

N Z function of a solution of the KdV hierarchy.

g,sM

g,sM

sg,s xx ,...,, 1M

.022 ,0 ,0 sgsg (stability)

sg ,M : Deligne-Mumford compactification

sLL ,...,1 : line bundles over sg ,M

Fiber over gxi iTx *:

Witten’s conjecture: the models 1) and 2) of quantum gravity are equivalent.

0

22 ,,g

Xg

gX F tt

= log of the -function of a solution of the KdV hierarchy

Gromov-Witten invariants of genus g

total Gromov-Witten potential

Gromov-Witten theory

X : smooth projective variety

,,mgX : moduli space of stable curves on X of genus g and degree with m marked points

C;dim: * XHn

mp

mp

Xgpp

m

mvirt

mg

mncevcev LL 1

*11

*1,

...:,..., 1

,,

111

imgi xffXXev ,: ,,

n ,...,,1 21 basis

,00 ,...,1 gm

,

,

;

.....1

11

11

2gpp

pp

m XH

Xg mm

mmttm!

F

Z

,tXF

GWI and integrable hierarchies

(Witten): The generating functions of GWI can be written as a hierarchy of systems of n evolutionary PDEs for the dependentvariables

0,0,1

22

100

,

ttw

X

t

p

X

pp tth

,0,1

22

10,

,

t

and the hamiltonian densities of the flows given by

ttt

tF

ttt

tF

ttt

tF

ttt

tF

3333

WDVV equations (1990)

N ,..., 1 ,,,,,

0,0 0,0, pt,wtgtF

Crucial observation: ttt

tFtc

3

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:

, ,A

Azy,x, , z yx, z y,x

,

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:

MTv

Mv

FM1. The metric over M is flat. v ,

FM2. Let . Then the 4-tensor MTzyxzyxzyxc v ,, , , : ,,

must be symmetric in x,y,z,w. zyxcw ,,

FM3. vector field s.t. ME

yxyExyxEyxE ,,,

FM WDVV

F(t)

Bihamiltonian Structure

udxH 1 (Casimir for ) 1 ,

,1,0,1,211 , , jHuHu jj

jH1H : primary Hamiltonian; : descendent Hamiltonians

Tau function: (1983)

1

212

222 ,....,,log,...,,

j

jxj tx

ttxuuuh

dxuuuhH jxjj 22,....,

Dispersionless hierarchies and Frobenius manifolds

Frobenius manifold solution of WDVV eqs.

,, tctF

,...,Nuucu Xp

T p 1 ,,

an integrable hierarchy of quasilinear PDEs of the form

yxyuxu x ,

pTHxuup ,,,

dxxuhH pp 1,, xuhuxcuxh ,,

yxuuconstcyxxugyuxu xx

2,

Frobenius manifold

Dispersionless hierarchies

Topological fieldtheories

Full hierarchies

Witten, Kontsevich

Whithamaveraging

Tau 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:1) a tau structure;2) 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)

11 2 nqnqnq eeeq

Bigraded Extended Toda Hierarchy

xe Def. 7. is a shift operator: xfxf

• 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

Zl

ll xQQ̂

Q̂

is defined by:

0

ˆl

ll xQQ

Def. 9. The residue is 0ˆ QQres

G. Carlet, B. Dubrovin 2004

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.

• q = 0,

1

1

• q = 0, 2

• q = 1, 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.

1) 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 is

1 ,~ mkLAW L

k

1 ,~ mkLAW L

k

K. Saito, 1983 : flat structures in the space of parametersof the universal unfolding of singularities.nA

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 the tau function of the hierarchy. It is defined by

Tau structure

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

LAWM ~

The 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 reads

Theorem 5. The tau function admits the following genus expansion

where represents the tau function for the solution ddd uuuw 1010 ,, 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.

Main Theorem

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 2

2. The transformations

3. For a generic solution of the extended Toda hierarchy, thecorrespondong tau function satisfes the Virasoro constraints

1,0,1

mLm t

ct

Here is a collection of formal power series in .

pc ,c

Conjecture 1.

For any hierarchy of the family of bigraded extended TodaHierarchy, i.e.for any value of (k, m):

1. 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

1* CPQHMToda

12 ~

/ AWMToda C

Alternatively, it can be identified with the quantum cohomology of the complex projective line

ueuvF 2

2

1

Conjecture 2.

),(0

22 log,, mkMg

Xg

gX

TodaF

tt

The total Gromov-Witten potential for the weighted projective

mkCP ,1space is the logarithm of the tau function of a

particular solution to the bigraded extended Toda hierarchy.

GWI orbifold Integrable hierarchies

1,~

/2),( C -mkLAWM LmkToda

In the bigraded case:

mkCPQHM mkToda ,1*,

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 perspectives

GW 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.