slide1 n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Frobenius manifolds PowerPoint Presentation
Download Presentation
Frobenius manifolds

Loading in 2 Seconds...

play fullscreen
1 / 22

Frobenius manifolds - PowerPoint PPT Presentation


  • 155 Views
  • Uploaded on

Frobenius manifolds. and. Integrable hierarchies of Toda type. joint work with B. Dubrovin. Piergiulio Tempesta. SISSA - Trieste. Gallipoli, June 28, 2006. Topological field theories (WDVV equations) 1990. Witten, Kontsevich (1990-92). Integrable hierarchies of PDEs (’60).

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

Frobenius manifolds


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
slide1

Frobenius manifolds

and

Integrablehierarchies of

Toda type

joint work with B. Dubrovin

Piergiulio Tempesta

SISSA - Trieste

Gallipoli, June 28, 2006

slide2

Topological field

theories

(WDVV equations)

1990

Witten,Kontsevich

(1990-92)

Integrable hierarchies

of PDEs

(’60)

Frobenius manifolds

(Dubrovin, 1992)

Manin, Kontsevich (1994)

Singularity theory

(K. Saito, 1983)

Gromov-Witten invariants

(1990)

slide3

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 theoperator algebra A associated with

the model:

slide4

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

slide5

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

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

equivalent.

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

slide6

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:

slide7

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

slide8

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

slide9

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.
slide10

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

slide11

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

slide12

Example. Consider the case k=m=1.

  • G.Carlet, B. Dubrovin, J. Zhang, Russ. Math. Surv. (2003)
  • B Dubrovin, J. Zhang, CMP (2004)
  • q = 0,
  • q = 0,
  • q = 1,

dove

slide13

Objective: Toextend thetheory of Frobenius manifoldsto 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

slide14

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)

slide15

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 thetau 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)

slide16

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:

slide18

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:

slide19

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

slide20

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.

slide21

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

slide22

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.