Black HolesTits -Satake Universality classesand Nilpotent OrbitsPietro FrèUniversity of Torino and Italian Embassy in Moscow

Based on common work with Aleksander S. Sorin & Mario Trigiante

Stekhlov Institute June 30th 2011

A well defined mathematical problem

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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 MANIFOLDS2. The theory of Liouville Integrable systems constructed on Borel-

type subalgebras of SEMISIMPLE LIE ALGEBRAS3. 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 andn vector multiplets

2 n scalars yielding n complex scalars zi

and n+1 vector fields AThe matrix N encodes together with the metric hab 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 SKn

D=3 -model on Q4n+4

4n + 4 coordinates

Gravity

From vector fields

scalars

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 HoleNull-like geodesic = extremal Black HoleSpace-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 SKn = symmetric coset space THIS DYNAMICAL SYSTEM is LIOUVILLE INTEGRABLE, always!

When homogeneous symmetric manifolds

C-MAPGeneral Form of the Lie algebra decomposition

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Relation between

One just changes the sign of the scalars coming from W(2,R) 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 Hi 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 G2(2)

One vector multiplet

Poincaré metric

Symplectic section

Matrix N

OXIDATION 1The 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, ZA parameterize in the G/H case the coset representative

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 G2(2) 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

The method of standard triplets

Angular momenta

Partitions

(j=3) The largest orbit NO5

(j=1, j=1/2, j=1/2) The orbit NO2

(j=1, j=1,j=0) Splits into NO3 and NO4 orbits

(j=1/2, j=1/2, j=0, j=0, j=0) The smallest orbit NO1

Tits Satake Theory

To each non maximally non-compact real form U (non split) of a Lie algebra of rank r1 is associated a unique subalgebra UTS ½ U which is maximally split.

UTS has rank r2 < r1

The Cartan subalgebra CTS ½ UTS is the non compact part of the full cartan subalgebra

root systemof rank r1

ProjectionSeveral 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

Two type of roots

1

2

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

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

1

2

1 2

2 1 2The 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