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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 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.
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