# Spectral Properties of Supermembrane and Multibrane Theories A. Restuccia Simon Bolivar University, Caracas,Venezuela In collaboration with, L. Boulton.

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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)

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

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.

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)

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

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:

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.

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

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

OPEN POINT: Nonperturbative quantization of supersymmetric theories In 1988 11D 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,  2001- 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

H  = -  + + V F. Properties: 1- It is essentially self adjoint. 2- H  is a relatively bounded perturbation of -  +. 3- 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.B817:155-166,2009.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

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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