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Spectral Properties of Supermembrane and Multibrane Theories A. Restuccia Simon Bolivar University, Caracas,Venezuela In collaboration with, L. Boulton (Heriot-Watt U.), M.P. Garcia del Moral (Oviedo U. Spain), I. Martin (Simon Bolivar U)

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PLAN OF THE TALK Motivation ◦ State of Art Bosonic Analysis. ◦ Molchanov Mean Value. ◦ New Results: The M2 analysis. Relation to Initial Value Problem. Multibrane Spectra. Supersymmetric Analysis ◦ M2 with topological constraints spectrum, ◦ ABJM, BLG spectrum ◦ D2-D0 matrix model. D=11 Supergravity from M-Theory: The Zero Eigenvalue Problem. Conclusions

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OTIVATION MOTIVATION OPEN PROBLEM : Nonperturbative quantization of StringTheory General Goal : Nonperturbative analysis of M-theory. ◦ In Particular: 1-Spectral analysis of M2, M5,p-branes, and multibrane theories ◦ 2-Stability properties of these theories. Strategy: In distinction with field approximation or ADS/CFT correspondence, we will follow a complementary approach: ◦ Operatorial analysis of matrix models of those theories.

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State of Art 11D Supermembrane was found to have continuous spectrum ( De Wit, Luscher, Nicolai (88)) No analogous to string quantization for the M2-brane. M2-brane was reinterpreted as a macroscopic object (i.e. interacting theory in terms of fundamental d.o.f carried by D0’s) BFSS-conjecture. No direct way to obtain M-theory formulation (96 Townsend, Witten) or nonperturbative complection of string theory. Very Succesful Avenues to deal with strongly coupled gauge theories in a decoupling limit of gravity- AdS/CFT. (Maldacena (00)) 11D Supermembrane with a topological condition has supersymmetric discrete spectrum and then allows for nonperturbative quantization. (Boulton, Garcia del Moral, Restuccia NPB03)

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Molchanov’s Mean Value Molchanov’s Mean Value Q: Is there a precise condition on the potential for the discreteness of spectrum of bosonic matrix models ? Barry Simon (83) and Lusher (87) gave different proofs on the discreteness of the bosonic M2 using theorems of sufficient conditions on the hamiltonian. A Necessary and Sufficient Condition

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Molchanov’s mean value: Membranes Molchanov’s mean value: Membranes M.P. Garcia Moral, L. Navarro, A. J. Perez, A. Restuccia. Nucl. Phys.B 2007 An exact condition for discreteness of bosonic membranes:

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Local Well Posedness Local Well Posedness Allen, Andersson & Restuccia, Comm. Math. Phys.2010 ( to appear) Theorem Q: Is there a relation between the discreteness condition and the initial value problem? A:The symbol of the elliptic operator contained in the hyperbolic structure of the field Eqn is the same that determined the discreteness of the spectrum.

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Multibrane bosonic spectra Discreteness of the Spectrum of Schrödinger Operators with Regular Potentials Let : below M.P. Garcia del Moral, I. Martin, L. Navarro, A.J. Perez, A. Restuccia. NPB 2010

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1.The BLG Potential: Scalar potential analysis The regularity condition Chern-Simon Contribution : Solve for F (A) in terms of X H = - + D i X D i X + V(x) - + V(x) H has discrete spectrum 2. The ABJM / ABJ Potential: Scalar potential analysis H has discrete spectrum Multibrane bosonic theories M.P. Garcia del Moral, I. Martin, L. Navarro, A.J. Perez, A. Restuccia. NPB 2010

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OPEN POINT: Nonperturbative quantization of supersymmetric theories In D Supersymmetric M2 has continuous spectrum.De Wit, Luscher, Nicolai NPB 1988 Semiclassical analysis a M2 + topological constraint has discrete Spectrum Duff, Inami, Pope,Sezgim, Stelle NPB Supersymmetric Models 0, Nonperturbative Analysis of the Supersymmetric M2+ Topological constraint : Purely Discrete Spectrum. - Boulton, Garcia del Moral Restuccia NPB 2003; NPB 2008; 2010 IN PREPARATION Spectral analysis of BLG, ABJM, ABJ: Continuous Spectrum, gap. Other Matrix Models?: D2-D0 Brane : Yang Mills + Topological Restriction Continuous Spectrum 0, NEW RESULTS: L. Boulton, M.P. Garcia del Moral, A. Restuccia In preparation. Physical Implications! 4D description: w/ Pena JHEP08, G2 compactification w/ Belhaj,,Garcia del Moral, Segui, JP. VeiroJPHA09 Non abelian description w/Garcia del Moral JHEP10

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H = - + + V F. Properties: 1- It is essentially self adjoint. 2- H is a relatively bounded perturbation of - is bounded from below by the square of the L 2 norm. 4- H H strongly in the generalized sense. 5- H has a compact resolvent 2.The minimum eigenvalue 0 when 0 belongs to H 1 ( R n ) 3. We consider now S n and the Embedding Theorem : The closure in the L2 norm of the unit H1 ball is compact. There exists a convergent subsequence such that 4. belongs to the domain of H and H = 0 A different Approach was followed in: J. Hoppe, D. LundholmJ. Hoppe, D. Lundholm, M. Trzetrzelewski, Nucl.Phys.B817: ,2009.M. Trzetrzelewski Eigenvalue Zero Problem of the 11D M2: Is 11D Supergravity really contained in M-theory)? OPEN PROBLEM (88) Boulton, Garcia del Moral, Martin, Restuccia. For H = - + V B + V F consider Deformation of the Bosonic potential 0

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CONCLUSIONS We obtain a Necessary and Sufficient condition for the discreteness of the spectrum of all bosonic polynomial matrix models : M2, M5, p-branes, ABJ/M,... We characterize the spectrum of the supermembrane with a topological condition: It is the only model with pure discrete spectrum. For the supersymmetric multibrane models (BLG, ABJ/M) the spectrum is continuous and has a mass gap. The existence of the eigenfunction with zero eigenvalue for the 11D supermembrane: A step forward.

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