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Black Holes Tits -Satake Universality classes and Nilpotent Orbits

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Presentation on theme: "Black Holes Tits -Satake Universality classes and Nilpotent Orbits"— Presentation transcript:

1 Black Holes Tits -Satake Universality classes and Nilpotent Orbits
Pietro Frè University of Torino and Italian Embassy in Moscow Stekhlov Institute June 30th 2011 Based on common work with Aleksander S. Sorin & Mario Trigiante

2 Introduction 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 A well defined mathematical problem
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: The Geometry of COSET MANIFOLDS The theory of Liouville Integrable systems constructed on Borel-type subalgebras of SEMISIMPLE LIE ALGEBRAS The addressing of a very topical issue in conyemporary ADVANCED LIE ALGEBRA THEORY namely: THE CLASSIFICATION OF ORBITS OF NILPOTENT OPERATORS

4 The N=2 Supergravity Theory
2 n scalars yielding n complex scalars zi and n+1 vector fields A We have gravity and n vector multiplets The matrix N encodes together with the metric hab Special Geometry

5 Special Kahler Geometry
symplectic section

6 Special Geometry identities

7 The matrix N

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

9 The main point

10 Dimensional Reduction to D=3
THE C-MAP D=4 SUGRA with SKn D=3 -model on Q4n+4 Metric of the target manifold Space red. / Time red. Cosmol. / Black Holes 4n + 4 coordinates From vector fields Gravity scalars

11 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 SKn = symmetric coset space THIS DYNAMICAL SYSTEM is LIOUVILLE INTEGRABLE, always!

12 When homogeneous symmetric manifolds
C-MAP General Form of the Lie algebra decomposition

13 Relation between One just changes the sign of the scalars coming from W(2,R) part in: Examples

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

15 The simplest example G2(2)
One vector multiplet Poincaré metric Symplectic section Matrix N

16 OXIDATION 1 The metric Taub-NUT charge where
The electromagnetic charges From the -model viewpoint all these first integrals of the motion Extremality parameter

17 OXIDATION 2 U, a,  » z, ZA Element of
The electromagnetic field-strenghts U, a,  » z, ZA parameterize in the G/H case the coset representative Ehlers SL(2,R) gen. in (2,W) Coset repres. in D=4 Element of

18 From coset rep. to Lax equation
From coset representative decomposition R-matrix Lax equation

19 Integration algorithm
Initial conditions Building block Found by Fre & Sorin

20 Key property of integration algorithm
Hence all LAX evolutions occur within distinct orbits of H* Fundamental Problem: classification of ORBITS

21 The role of H* Max. comp. subgroup Different real form of H
COSMOL. BLACK HOLES Different real form of H In our simple G2(2) model

22 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

23 Invariants & Tensors Quadratic Tensor

24 Tensors 2 QUADRATIC BIVECTOR

25 Their signatures classify orbits, both regular and nilpotent!
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!

26 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!

27 Relation between old and new Cartan Weyl bases

28 Hence we can easily find nilpotent orbits
Every orbit possesses a representative of the form Generic nilpotency 7. Then imposereduction of nilpotency

29 The general pattern

30 The method of standard triplets

31 Angular momenta

32 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

33 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

34 Projection rank r2 < r1 root lattice root system of rank r1
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 Projection root system of rank r1 rank r2 < r1 root lattice

35 Two type of roots 1 2 3

36 Tits Satake Projection: an example
The D3 » A3 root system contains 12 roots: Let us distinguish the roots that have a non-zero z-component, from those that have a vanishing z-component Complex Lie algebra SO(6,C) The Dynkin diagram is INGREDIENT 3 We consider the real section SO(2,4)

37 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

38 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

39 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

40 Tits Satake Projection: an example
This system of vectors is actually a new root system in rank r = 2. It is the root system B2 » C2 of the Lie Algebra Sp(4,R) » SO(2,3)

41 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.

42

43 Universality Classes

44 One example Tits-Satake Projection SO(4,5)

45 The orbits are the same for all members of the universality class (still unpublished result)

46 Спосибо за внимание Thank you for your attention


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