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Black Holes Tits -Satake Universality classes and Nilpotent Orbits

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

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Black Holes Tits -Satake Universality classes and Nilpotent Orbits

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  1. Pietro Frè Universityof Torino and ItalianEmbassy in Moscow BlackHolesTits-SatakeUniversalityclassesand NilpotentOrbits StekhlovInstituteJune 30th 2011 Based on common work withAleksander S. Sorin & Mario Trigiante

  2. A welldefinedmathematicalproblem Our goal is just tofind and classifyallsphericalsymmetricsolutionsofSupergravitywith a staticmetricofBlackHoletype • The solutionofthisproblemisfoundbyreformulatingitinto the contextof a veryrichmathematicalframeworkwhichinvolves: • The GeometryofCOSET MANIFOLDS • The theoryofLiouvilleIntegrablesystemsconstructed on Borel-typesubalgebrasofSEMISIMPLE LIE ALGEBRAS • The addressingof a verytopicalissue in conyemporary ADVANCED LIE ALGEBRA THEORY namely: • THE CLASSIFICATION OF ORBITS OF NILPOTENT OPERATORS

  3. The N=2 SupergravityTheory 2 nscalarsyielding n complexscalarszi and n+1 vectorfieldsA Wehavegravity and nvectormultiplets The matrixNencodestogetherwith the metrichab SpecialGeometry

  4. SpecialKahlerGeometry symplecticsection

  5. SpecialGeometryidentities

  6. The matrix N

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

  8. The mainpoint

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

  10. 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!

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

  12. Relation between One just changes the signof the scalarscomingfrom W(2,R) part in: Examples

  13. The solvable parametrization Thereis a fascinatingtheoremwhichprovidesanidentificationof the geometry of moduli spaceswithLiealgebrasfor (almost) allsupergravitytheories. THEOREM:Allnon compact (symmetric) cosetmanifolds are metricallyequivalentto a solvablegroupmanifold Splitting the Lie algebra Uinto the maximal compact subalgebraH plus the orthogonalcomplementK • There are precise rulestoconstructSolv(U/H) • EssentiallySolv(U/H)ismadeby • the non-compactCartangeneratorsHi2 CSA  K and • those positive rootstepoperatorsEwhich are notorthogonalto the non compact CartansubalgebraCSA  K

  14. The simplest example G2(2) One vector multiplet Poincaré metric Symplecticsection Matrix N

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

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

  17. Fromcoset rep. toLaxequation Fromcosetrepresentative decomposition R-matrix Laxequation

  18. Integrationalgorithm Initialconditions Building block FoundbyFre & Sorin 2009 - 2010

  19. Key propertyofintegrationalgorithm Henceall LAX evolutionsoccurwithindistinctorbitsofH* FundamentalProblem: classificationof ORBITS

  20. The roleofH* Max. comp. subgroup COSMOL. BLACK HOLES Differentrealformof H In oursimple G2(2)model

  21. The algebraicstructureofLax For the simplestmodel ,the Laxoperator, is in the representation of We can constructinvariants and tensorswithpowersof L

  22. Invariants & Tensors QuadraticTensor

  23. Tensors 2 QUADRATIC BIVECTOR

  24. Tensors 3 Hencewe are abletoconstructquartictensors ALL TENSORS, QUADRATIC and QUARTIC are symmetric Theirsignaturesclassifyorbits, both regular and nilpotent!

  25. Tensorclassificationoforbits How do wegettothisclassification? The answeris the following: bychoosing a newCartansubalgebra inside H* and recalculating the stepoperatorsassociatedwithroots in the newCartanWeylbasis!

  26. Relation betweenold and newCartanWeylbases

  27. Hencewe can easilyfindnilpotentorbits Everyorbitpossesses a representativeof the form Genericnilpotency 7. Thenimposereductionofnilpotency

  28. The general pattern

  29. The method of standard triplets

  30. Angular momenta

  31. 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

  32. TitsSatakeTheory • Toeach non maximallynon-compactrealformU (non split) of a Lie algebra ofrank r1isassociated a uniquesubalgebraUTS½ Uwhichismaximallysplit. • UTShasrank r2 < r1 • The CartansubalgebraCTS½ UTSis the non compact part of the full cartansubalgebra

  33. 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

  34. Two type of roots 1 2 3

  35. The Dynkin diagram is 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) INGREDIENT 3 We consider the real section SO(2,4)

  36. The D3» A3 root system contains 12 roots: Complex Lie algebra SO(6,C) The Dynkin diagram is We consider the real section SO(2,4) Tits Satake Projection: an example 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

  37. The D3» A3 root system contains 12 roots: Complex Lie algebra SO(6,C) The Dynkin diagram is We consider the real section SO(2,4) Tits Satake Projection: an example 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. The D3» A3 root system contains 12 roots: Complex Lie algebra SO(6,C) The Dynkin diagram is We consider the real section SO(2,4) Tits Satake Projection: an example 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

  39. 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)

  40. 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.

  41. Universality Classes

  42. Oneexample Tits-Satake Projection SO(4,5)

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

  44. Спосибо за внимание Thank you for your attention

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