frobeniusmanifolds and integrablehierarchiesof todatypea 2-dimensional frobenius manifolds m qh *(cp...
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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.