Pietro Frè

Pietro Frè

University of Torino and Italian Embassy in Moscow

Based on common work with Aleksander S. Sorin

& Mario Trigiante

MGU June 8th 2011

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Уважаемые коллеги для меня болшая честь поступить сегодня перед Вами ичитать доклад в знамеиитом Московском Государственном Университете им. Ломоносова.

Я Вам очень блогадарен за приглашение и хотел бы передать найлучшее пожелания Итальянского посольство и Господина Посла.

Элсперимент Антарес один прекрасный премер международного сотрудночества в котором участвуют ученые многих стран чтобы открыть самые глубокие таины вселенной, особинно чтобы определить кто является космическоами ускорителями космических излучений .

Сильнами кандидатами в этой области являются черные диры и иммено а них сегодна буду я говорить. Но я хотел сразу извиняться перед Вами и просить прошениее за то что мои черные дыры очень маленкие и обсолютно не грандиозные как ускорители.

Мои черные дыры являются только сложные матаматические

решения уравниении Эинстеина в рамках суперправитации. Они маленькие маленькие, больше похожи на элемептарные частицы которые никто еще не наблюдал но являют чудом математики.

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Our goal is just to find and classify all spherical symmetric solutions of Supergravity with a static metric of Black Hole type

The solution of this problem is found by reformulating it into the context of a very rich mathematical framework which involves:

1. The Geometry of COSET MANIFOLDS

2. The theory of Liouville Integrable systems constructed on Borel- type subalgebras of SEMISIMPLE LIE ALGEBRAS

3. The addressing of a very topical issue in conyemporary ADVANCED LIE ALGEBRA THEORY namely:

1. THE CLASSIFICATION OF ORBITS OF NILPOTENT

OPERATORS

The N=2 Supergravity Theory

We have gravity and

n vector multiplets

2 n scalars yielding n complex

sca ^{lars z}

and n+1 vector fields A

The matrix N

encodes together with the metric

h

Special Geometry

Special Kahler Geometry

symplectic section

Special Geometry identities

The matrix N _

When the special manifold is a symmetric coset ..

Symplectic embedding

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The main point

Dimensional Reduction to D=3

D=4 SUGRA with SK_n D=3 -model on Q_4n+4

4n + 4 coordinates

Gravity scalars From vector fields

Metric of the target manifold

THE C-MAP

Space red. / Time red.

Cosmol. / Black Holes

SUGRA BH.s = one-dimensional Lagrangian model

Evolution parameter

Time-like geodesic = non-extremal Black Hole Null-like geodesic = extremal Black Hole

Space-like geodesic = naked singularity

A Lagrangian model can always be turned into a Hamiltonian one by means of standard procedures.

SO BLACK-HOLE PROBLEM = DYNAMICAL SYSTEM

FOR SK_n = symmetric coset space THIS DYNAMICAL SYSTEM is LIOUVILLE INTEGRABLE, always!

When homogeneous symmetric manifolds

C-MAP

General Form of the Lie algebra decomposition

Relation between

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One just changes the sign of the scalars coming from W

part in:

Examples

The solvable parametrization

There is a fascinating theorem which provides an identification of the geometry of moduli spaces with Lie algebras for (almost) all supergravity theories.

THEOREM:All non compact (symmetric) coset manifolds are metrically equivalent to a solvable group manifold

•There are precise rules to construct Solv(U/H)

•Essentially Solv(U/H) is made by

•the non-compact Cartan generators H_i 2 CSA  K and

•those positive root step operators E^ which are not orthogonal to the non compact Cartan subalgebra CSA  K

Splitting the Lie algebra U into the maximal compact subalgebra H plus the orthogonal complement K

The simplest example G ₂₍₂₎

One vector multiplet

Poincaré metric

Symplectic section

Matrix N_

OXIDATION 1

The metric

where Taub-NUT charge

The electromagnetic charges

From the -model viewpoint all these first integrals of the motion

Extremality parameter

OXIDATION 2

The electromagnetic field-strenghts

U, a,  » z, Z

Coset repres. in D=4

Ehlers SL(2,R) gen. in (2,W)

Element of

From coset rep. to Lax equation

From coset representative

decomposition

R-matrix Lax equation

Integration algorithm

Initial conditions

Building block

Found by Fre & Sorin 2009 - 2010

Key property of integration algorithm

Hence all LAX evolutions occur within distinct orbits of H*

Fundamental Problem: classification of

ORBITS

The role of H*

Max. comp. subgroup

Different real form of H

COSMOL.

BLACK HOLES

In our simple G

model

The algebraic structure of Lax

For the simplest model ,the Lax operator, is in the representation

of

We can construct invariants and tensors with powers of L

Invariants & Tensors

Quadratic Tensor

Tensors 2

BIVECTOR

QUADRATIC

Tensors 3

Hence we are able to construct quartic tensors

ALL TENSORS, QUADRATIC and QUARTIC are symmetric

Their signatures classify orbits, both regular and nilpotent!

Tensor classification of orbits

How do we get to this classification? The answer is the following: by choosing a new Cartan subalgebra inside H* and recalculating the step operators

associated with roots in the new Cartan Weyl basis!

Relation between old and new Cartan Weyl bases

Hence we can easily find nilpotent orbits

Every orbit possesses a representative of the form

Generic nilpotency 7. Then imposereduction of nilpotency

The general pattern

 To each non maximally non-compact real

form U (non split) of a Lie algebra of rank r ₁ is associated a unique subalgebra U _TS ½ U which is maximally split.

 U _TS has rank r ₂ < r ₁

 The Cartan subalgebra C _TS ½ U TS is the non

compact part of the full cartan subalgebra

root system of rank r

Projection

Several roots of the higher system have the same projection.

These are painted copies of the same wall.

The Billiard dynamics occurs in the rank r2 system

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2

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Tits Satake Projection: an example

The D₃ » A3 root system contains 12 roots:

Complex Lie algebra

SO(6,C)

We consider the real section SO(2,4) The Dynkin diagram is

Let us distinguish the roots that have

a non-zero z-component,

from those that have a vanishing

z-component

INGREDIENT 3

Tits Satake Projection: an example

The D₃ » A3 root system contains 12 roots:

Complex Lie algebra

SO(6,C)

We consider the real section SO(2,4) The Dynkin diagram is

Let us distinguish the roots that have

a non-zero z-component,

from those that have a vanishing

z-component

Now let us project all the root vectors onto the

plane z = 0

Tits Satake Projection: an example

The D₃ » A3 root system contains 12 roots:

Complex Lie algebra

SO(6,C)

We consider the real section SO(2,4) The Dynkin diagram is

Let us distinguish the roots that have

a non-zero z-component,

from those that have a vanishing

z-component

Now let us project all the root vectors onto the

plane z = 0

Tits Satake Projection: an example

The D₃ » A3 root system contains 12 roots:

Complex Lie algebra

SO(6,C)

We consider the real section SO(2,4) The Dynkin diagram is

The projection creates new vectors

in the plane z = 0 They are images of more than one root in the original system

Let us now consider the system of 2-dimensional vectors obtained from the projection

Tits Satake Projection: an example

This system of vectors is actually

a new root system in rank r = 2.

₁

2

₁ ₂ 2 ₁ ₂

It is the root system B₂ » C2 of the Lie Algebra Sp(4,R) » SO(2,3)

Tits Satake Projection: an example

The root system B₂ » C2

of the Lie Algebra Sp(4,R) » SO(2,3)

so(2,3) is actually a subalgebra of so(2,4).

It is called the

Tits Satake subalgebra The Tits Satake

algebra is maximally split. Its rank is equal to the non compact rank of the original algebra.

Universality Classes

One example

Tits-Satake Projection SO(4,5)

The orbits are the same for all members of the

universality class (still unpublished result)

Спосибо за внимание

Thank you for your attention