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Spectral Properties of Spectral Properties of SupermembraneSupermembrane
and Multibrane Theories 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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MOTIVATIONOTIVATION
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 ValueMolchanov’s Mean ValueQ: 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 PosednessLocal 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 XH = - + Di 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 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
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H = - + + VF .
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 L2 norm.
4- H H strongly in the generalized sense .
5- H has a compact resolvent
2. The minimum eigenvalue 0 when 0 belongs to H1 ( 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. Lundholm, M. Trzetrzelewski, Nucl.Phys.B817:155-166,2009.
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 = - + VB + VF 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.