Frobenius manifolds

Integrable hierarchies of

Toda type

Piergiulio Tempesta

SISSA - Trieste

Gallipoli, June 28, 2006

joint work with B. Dubrovin

and

Topological field

theories

(WDVV equations)

1990

Integrable hierarchies

of 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 = δ 0 =Sδ

( ) ( )g

yx ...... βαβα φφφφ ≡Σ

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

∫Σ

== xdgRS 2 Euler characteristic of Σ

[ ] ( )∫Σ

= ,..., xLS φφφ

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

The two-point correlator:

βααββα ηηφφ ==

determines a scalar product on the manifold.

The triple correlator

γβααβγ φφφ=c

defines the structure of theoperator algebra Aassociated withthe model:

γγαββα φφφ c=⋅

αβεγεγ

αβ η cc = ( ) ( ) 1−= αβαβ ηη

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

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

Discretization: gΣ polyhedron

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

gΣ

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

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

∞→N −≡∞ τZ function of a solution of the KdV hierarchy .

g,sM

g,sM

( ){ }sg,s xx ,...,, 1Σ=M

.022 ,0 ,0 <−−>≥ sgsg (stability )

sg ,M : Deligne-Mumford compactification

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

Fiber over gxi iTx Σ *:

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

( ) ( )∑≥

−=ℑ0

22 ,,g

Xg

gX F tt εεε

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

Gromov-Witten invariants of genus g

τ

total Gromov-Witten potential

Gromov-Witten theory

X : smooth projective variety

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

( )C;dim: * XHn =

( ) ( ) ( )[ ]

( ) ( ) ( )mp

mp

Xgpp

m

mvirt

mg

mncevcev LL 1

*11

*1,

...:,..., 1

,,

111∧∧∧= ∫ ααβαα φφφτφτ

β

( )imgi xffXXev α ,: ,, →β

nφφφ ,...,,1 21 = basis

( ) ( )βαα φτφτ

,00 ,...,1 gm

( )( ) ( )

βαααα

βφτφτ

,

,

;

.....1

11

11

2gpp

pp

m XH

Xg mm

mmttm!

F ∑ ∑∈

=Z

( )ε,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

,αααεεφτφτ

ttw

X

∂∂ℑ∂== t

( ) ( ) ( )p

X

pp tth

,0,1

22

10,

,ααα

εεφτφτ∂∂

ℑ∂== t

and the hamiltonian densitiesof the flows given by

( ) ( ) ( ) ( )µδα

λµλβγµδγ

λµλβα ηη

ttt

tF

ttt

tF

ttt

tF

ttt

tF

∂∂∂∂

∂∂∂∂=

∂∂∂∂

∂∂∂∂ 3333

WDVV equations (1990)

N ,..., 1 ,,,,, =µλδγβα

( ) ( )0,0 0,0, ===ℑ= >pt,wtgtF αα

α

Crucial observation: ( ) ( )γβααβγ ttt

tFtc

∂∂∂∂=

3

Frobenius manifold

Definition 1. A Frobenius algebrais a couplewhere A is an associative, commutative algebra with unity over A field k (k = R, C) and is a bilinear symmetric formnon 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 χ∈

[ ] [ ] [ ] yxyExyxEyxE ⋅=⋅−⋅−⋅ ,,,

FM WDVV

F(t)

Bihamiltonian Structure

∫=− udxH 1(Casimir for ){ }1 ,

{ } { } ,1,0,1,211 , , −==+ jHuHu jj

jH1−H : primary Hamiltonian; : descendent Hamiltonians

Tau function: (1983)

( )( ) ( )1

212

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

+

+

∂∂∂=

j

jxj tx

ttxuuuh

τε( )( )∫+= dxuuuhH j

xjj 22,....,

Dispersionless hierarchies and Frobenius manifolds

Frobenius manifold solution of WDVV eqs. ⇔( ) ( )( )αβ

γαβ η ,, tctF

∃( ) ( ) ,...,Nuucu Xp

T p 1 ,, =∂=∂ αγγαβ

βα

an integrable hierarchy of quasilinear PDEs of the form

( ) ( ){ } ( )yxyuxu x −∂= δηαββα ,

( ){ }pTHxuup ,,, α

ββα =∂

( )( )∫ += dxxuhH pp 1,, αα ( ) ( ) ( )xuhuxcuxh ,, αεεβγαγβ ∂=∂∂

( ) ( ){ } ( )( ) ( ) ( ) ( )yxuuconstcyxxugyuxu xx −∂+−∂= δδ γαβγ

αββα2,

Frobenius manifold

Dispersionless hierarchies

Topological field

theories

Full hierarchies

Witten, Kontsevich

Whitham

averaging

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 correspondencein the case of hierarchies of differential-difference equations.

Toda equation (1967)

( ) ( ) ( )11 2 −+ +−= nqnqnq eeeq&&

Bigraded Extended Toda Hierarchy

xe ∂=Λ εDef. 7. is a shift operator: ( ) ( )ε+=Λ xfxf

• Two parametric family of integrable hierarchies of differential-difference equations

• It is a Marsden-Weinstein reductionof the 2D Toda hierarchy.

Def. 8. The positive partof the operator ( )∑∈

Λ=Zl

ll xQQ̂

Q̂

is defined by: ( )∑≥

+ Λ=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 differentfractional powersof the Lax operator:

which satisfy:

Logaritm of L . Let us introduce the dressing operators

such that

The logarithm of Lis defined by

Example. Consider the case k=m=1.

• q = 0,

1=β

1=β

• q = 0, 2=β

• q = 1, 1=β

dove

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

• B Dubrovin, J. Zhang, CMP (2004)

Objective: To extend the theory of Frobenius manifoldsto 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 extendaffine 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 respectto 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):

βα ,

Def. 13For any solution of the bigraded extended Toda hierarchythere exists a function

called thetau function of the hierarchy. It is defined by

Tau structure

Lemma 2. The hamiltonian densities are related to the taustructure by

Lemma 3. (symmetry property of the omega function)

Lie symmetries and Virasoro algebras

Theorem 4. There exists an algebraof linear differential operatorsof the second order

associated with the Frobenius manifold . These operatorssatisfy 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 2δ

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 τεεε

Here is a collection of formal power series in .

( ) ( )εε α pc ,=cε

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 symmetriesgenerated by the action of theVirasoro operators.

2. The system of Virasoro constrants is satisfied.

Toda hierarchies and Gromov-Witteninvariants

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 cohomologyofthe complex projective line

ueuvF += 2

2

1

Conjecture 2.

( ) ( ) ),(0

22 log,, mkMg

Xg

gX

TodaF τεεε ==ℑ ∑

≥

− 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 manifoldsallows 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 otherextended 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.