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Supergravity Black Holes and Nilpotent Orbits

Supergravity Black Holes and Nilpotent Orbits. Pietro Fré Torino University , Italy And Embassy of ITALY in the Russian Federation. Talk given at the Sternberg Institute MGU, Moscow March 16th 2011 Based on work in collaboration with A.S. Sorin & M. Trigiante.

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Supergravity Black Holes and Nilpotent Orbits

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  1. SupergravityBlackHolesand NilpotentOrbits Pietro Fré Torino University, Italy And Embassyof ITALY in the RussianFederation Talk given at the SternbergInstitute MGU, Moscow March 16th 2011 Based on work in collaboration with A.S. Sorin & M. Trigiante

  2. where G is Newton’s constant Newtonian Gravity Two bodies of mass M and m at a distance R attract each other with a force From this formula we work out the escape velocity (namely the minimal velocity that a body must have in order to be able to escape from the surface of a star having radius R and mass M)

  3. Laplace 1796 In hisExpositiondu System de Monde, Laplace foresaw the possibilitythat a celestial body withradius Rmighthave a mass Mso big that the correspondentescapevelocityislargerthan the speedof light: In this case the celestial bodywouldbeinvisible..... Indeed no light-signalcould emerge fromit and reachus: BLACK HOLE.

  4. TrueastrophysicalBlackHoles are not the mainconcern in this talk Supermassive Black Holes (106 solar masses) are the hidden engines of Galaxies and Quasars Black holes are revealed by the observation of their accretion disk and of the flares orthogonal to the accretion disk plane. The BlackHoleswe deal with are solitonsofStringTheory…..! No whoknows…?

  5. KARL SCHWARZSCHILD 1873 – 1916 (He was born in Frankurt am Mein in a well to do Jewish family) Very young determined orbits of binary stars Since 1900 Director of the Astronomical Observatory of Gottingen (the hottest point of the world for Physics and Mathematics at that time and in subsequent years) Famous scientist and member of the Prussian Academy of Sciences in 1914 he enrolled as a volounteer in the German Army and went to war first on the western and then on the eastern front against Russia. At the front in 1916 he wrote two papers. One containing quantization rules discovered by him independentely from Sommerfeld. The second containing Schwarzschild solution of GR. At the front he had learnt GR two months before reading Einstein’s paper. Einstein wrote to Schwarschild : ….I did not expect that one could formulate the exact solution of the problem in such a simple way…. Few months later Schwarschild died from an infection taken at the front 1916

  6. Fundamentalsolution: the Schwarzschildmetric (1916) Using standard polarcoordinates plus the time coordinatet Is the mostgeneralstatic and sphericalsymmetricmetric

  7. The Schwarzschild metric has a singularity at the Schwarzschild radius It took about 50 years before its true interpretation was found. Kruskal In the meantime another solution was found This is the Reissner Nordstrom solution which describes a spherical object with both mass m and charge q

  8. The Reissner Nordstrom solution Hans Jacob Reissner was a German aeronautical engineer with a passion for mathematical physics. He solved Einstein equations with an electric field in 1916. Later he emigrated to the USA and was professor in Illinois and in Brooklin Gunnar Nordstrom was a finnish theoretical physicist who worked in the Netherlands at Leiden in Ehrenfest’s Institute. In 1918 he solved Eisntein’s equation for a spherical charged body extending Reissner’s solution for a point charge. Hans Jacob Reissner (1874-1967) & wife in 1908 Gunnar Nordstrom (1881-1923) THIS EARLY SOLUTION HAS AN IMPORTANT PROPERTY which is the tip of an iceberg of knowledge….when m=q….

  9. The first instanceofextremalBlackHoles The Reissner Nordstrom metric: Has two “horizons” at There is a true singularity that becomes “naked” if It was conjectured a principle named COSMIC CENSORSHIP It is intimately related to supersymmetry

  10. ExtremalReissnerNordstromsolutions Harmonic function The largest part of new developments in BH physics is concerned with generalizations of this solution and with their deep relations with Supergravity and Superstrings

  11. Historicalintroduction 1995-96 Discoveryof the attractionmechanism in BPS BH.sbyFerrara & Kallosh 1995-96 Discoveryof the statisticalinterpretationof the area of the horizonascountingofstringmicrostatesbyStrominger. 1997-1999 Extensivestudyof BPS solutionsofsupergravityof BH typebymeansof FIRST ORDER EQUATIONS, followingfrompreservationof SUSY NEW WAVE of interest mid 2000s: also non BPS BlackHoleshave the attractionmechanism! Alsotherewe can find a fake-superpotential! D=3 approach and NOW INTEGRABILITY!

  12. 1995-1998 Ferrara, Kallosh, Strominger Renata Kallosh graduatedfrom MSU in 1966, prof. in Stanford Sergio Ferrara born in Rome, oneof the threefoundersofSupergravity, staff @CERN, prof. UCLA The mainpioneersof the newBlackHoleseason Andrew Strominger Harvard Professor

  13. What is a BPS black hole? • To explain this idea we have to introduce a few basic facts about the supersymmetry algebra…..

  14. Extended SUSY algebra in D=4 NORMAL FORM of CENTRAL CHARGES

  15. Rewritingof the algebra BogomolnyBound Reducedsupercharges

  16. BPS states = short susymultiplets Fieldtheorydescription

  17. A lessontaughtby RN BlackHoles For m=|q| ForextremalBlackHoles the area of the horizondependsonly on the charges

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

  19. SpecialKahlerGeometry symplecticsection

  20. SpecialGeometryidentities

  21. The matrix N

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

  23. 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

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

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

  26. 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!

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

  28. 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)

  29. The Quartic Invariant The vector of electric and magnetic charges of SL(2,R) Quarticsymplectic invariant

  30. Attraction mechanism & Entropy Potential Central charges of supersym.

  31. Critical Points of the Potential (Ferrara et al) THREE TYPES of Critical Points Special Geometry Invariants

  32. Invariants at Fixed Points At BPS attractor points At BPS attractor points of type I At BPS attractor points of type II Identityeverywhere

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

  34. Integrationalgorithm Initialconditions Building block

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

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

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

  38. Invariants & Tensors QuadraticTensor

  39. Tensors 2 QUADRATIC BIVECTOR

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

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

  42. Relation betweenold and newCartanWeylbases

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

  44. The general pattern

  45. EXAMPLE : NON BPS attractorwith 2 charges InitialLax p,q charges

  46. Initialcosetrepresentative

  47. Solution At the horizon

  48. The attractionmechanism in picture

  49. LiouvilleIntegrability 1° The Poissonianstructure Borelsubalgebra PoissonBracket Evolutionequations

  50. LiouvilleIntegrability 2° KostantDecomposition ForanyLie algebra elementholdstrue Howtofindcommutinghamiltonians?

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