Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th 2011

2 Уважаемые коллеги для меня болшая честь поступить сегодня перед Вами и читать доклад в знамеиитом Институте им. Стеклова. 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.

3 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 i and n+1 vector fields A The matrix N encodes together with the metric h ab Special Geometry

Special Kahler Geometry symplectic section

Special Geometry identities

The matrix N

When the special manifold is a symmetric coset.. Symplectic embedding

9 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 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 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 13 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 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 A 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

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 2(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

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 1 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

1 2 3

Tits Satake Projection: an example The D 3 » A 3 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 D 3 » A 3 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 D 3 » A 3 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 D 3 » A 3 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. It is the root system B 2 » C 2 of the Lie Algebra Sp(4,R) » SO(2,3)

Tits Satake Projection: an example The root system B 2 » C 2 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