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N =1 2+1 NCSYM & Supermembrane with Central Charges .

N =1 2+1 NCSYM & Supermembrane with Central Charges. Lyonell Boulton, M.P. Garcia del Moral, Alvaro Restuccia. Hep-th/0609054. TORINO U. & POLITECNICO TORINO & U. ALESSANDRIA. RTN WORKSHOP NAPOLI 2006. OVERVIEW. MOTIVATION D=11 SUPERMEMBRANE ( M2 )

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N =1 2+1 NCSYM & Supermembrane with Central Charges .

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  1. N=1 2+1 NCSYM & Supermembrane with Central Charges . Lyonell Boulton, M.P. Garcia del Moral, Alvaro Restuccia Hep-th/0609054 TORINO U. & POLITECNICO TORINO & U. ALESSANDRIA RTN WORKSHOP NAPOLI 2006

  2. OVERVIEW • MOTIVATION • D=11 SUPERMEMBRANE (M2) • N=1 SUPERMEMBRANE WITH CENTRAL CHARGES (M2) • N=1 M2 vs 2+1 NCSYM THEORY • DISCRETNESS OF THE BOSONIC SPECTRUM AT EXACT LEVEL • CENTER, CONFINEMENT, TRANSITION PHASE. • INTERPRETATION IN TERMS OF M-THEORY & N=1 SQCD • CONCLUSIONS

  3. MOTIVATION • OPEN PROBLEMS: • NONPERTURBATIVE QUANTIZATION OF STRING THEORY: • QUANTIZATION OF M-THEORY: M2 (SUPERMEMBRANE), M5, • Attempts: QUANTIZATION OF M2 • 2. NONPERTURBATIVE QUANTIZATION OF YANG-MILLS THEORIES TOWARDS A COMPLETE DESCRIPTION OF QCD. • Attempts: SPIN CHAINS, TWISTORS, GAUGE-GRAVITY, • LARGE N MATRIX MODELS.. THE CANONICAL QUANTIZATION OF THE SUPERMEMBRANE CENTRALCHARGES CANONICAL QUANTIZATION 2+1 NCSYM THEORY

  4. RESULTS • At exact level, the first results of the spectral properties of • N=1 2+1 NCYM that can live in 4D: • - purely discrete with eingenvalues of finite multiplicity • - mass gap. • At exat level, the first results of the spectral properties of • the N=1 supermembrane with central charges, 4D: • - purely discrete with eigenvalues of finite multiplicity • - mass gap • It represents a nonperturbative quantization of a sector of • M-theory

  5. RESULTS • We have identified the center of the group as a • mechanism for confinement in both theories, • at exact and regularized level. • Interpretation in terms of SQCD: • - The N=1 NCSYM or the Supermembrane with central • charges are the IR phase of the theory. • - Through a breaking of the center due a • topological transition the theory enters in a • - UV phase that corresponds to the compactified • N=4 Supermembrane as a many body object interpreted • in terms of a quark-gluon plasma

  6. MATRIX MODELS ORIGINAL POINT OF VIEW: Halpern, Hoppe, De Wit, Hoppe, Nicolai , de wit, Peeters, Plefka etc.. COMPACT. M2 H H H EXACT EXACT EXACT ? ? ? SYMMETRIES COMPACT H H REGULARIZED REGULARIZED M2 NCSYM H =H EXACT RECENT RESULT OUR RESULTS FOLLOW ORIGINAL POINT OF VIEW: M2 H REGULARIZED BFSS/IKKT CONJECTURE: D0 OR D-1 ACTION IS TAKEN AS THE FUNDAMENTAL SYMMETRIES: EX. BMN MODEL ETC..

  7. H= M, N=1,..,9 CLASSICALLY: STRING-LIKE SPIKES: QUANTUM: BOSONIC FERMIONIC PURELY DISCRETE CONTINUUM!! 2º QUANTIZED THEORY!!!: MANY BODY OBJECT OF D0´S OLD PROBLEMS OF M2 QUANTIZATION (L.C.G)De Wit+Hoppe+Nicolai; De wit+Marquard+ Nicolai, De wit+Luscher+Nicolai, D.wit+Peeters+Plefka, MPGM+Navarro+Perez+Restuccia

  8. THE SUPERMEMBRANE WITH CENTRAL CHARGES Martin,Ovalle,Restuccia;MPGM,Restuccia(1); Boulton,MPGM.,Restuccia(3), Boulton,MPGM.,Martin, Restuccia,Boulton(2),MPGM+R, Bellorin,Restuccia,97-06 SU(N) Spectrum: CLASSICALLY: NOTSTRING-LIKE SPIKES (1) QUANTUM: BOSONIC PURELY DISCRETE SPECTRUM (2) FERMIONIC PURELY DISCRETE SPECTRUM (3) TOPOLOGICAL CONDITION CENTRAL CHARGE CONDITION: &

  9. With the decomposition allowed by fixed central charges: A N=1 2+1 symplectic NCSYM coupled to scalars proceeding from NCSYM 10D reduccion=N=1 Supermembrane with Z The only degrees of freedom to quantize are (X, A).

  10. SU(N) REGULARIZATION OF THE M2 GAUGE FIXING CONDITION: CONSTRAINS:

  11. SPECTRAL PROPERTIES: SU(N) LEVEL CLASSICALLY: NO-STRING-LIKE SPIKES MPGM+ A. Restuccia QUANTUM LEVEL: BOSONIC SECTOR Boulton+MPGM+Martin+Restuccia QUANTUM LEVEL FERMIONIC SECTOR Boulton+MPGM+Martin+Restuccia Eigenvalues of V(X)

  12. Large N for semiclassical aproximation Semiclassical quantization of M2 brane Duff, Inami, Pope, Sezgin,Stelle Semiclassical description of the regularized M2 brane Matrix regularization Gaussian measure on l2 Dg Well-defined

  13. Spectrum of the exact bosonic hamiltonian I Su(N) proof: Exact proof: Compact Phase Space Infinite dimensional Phase Space Configuration Space (X,A) Banach space Potential Defining

  14. Spectrum of the exact Bosonic Hamiltonian II WITH NON-COMPACT INFINITE DIMENSIONAL LAPLACIAN

  15. The center as a mechanism of confinement Symmetries at exact level: NON-COMMUTATIVE THEORY FIXING THE HARMONIC SECTOR Symmetries at SU(N) level: CENTER OF MASS TERMS Mass terms

  16. SQCD & M-Theory Interpretation IR UV Topological transition Many-body object Dirac monopoles, Mass terms m(Z) Z(2) string , glueballs CONFINEMENT QUARK-GLUON PLASMA N=4 Compactified Supermembrane (N=1) Supermembrane with Z= (N=1) NCYM

  17. MATRIX MODELS D=11 SUPERMEMBRANES 10D SYM MATRIX REGULARIZATION DIMEN. REDUCCION 0+1 SU(N) H= LARGE N LIMIT ? PROBLEM OF CLOSED HARMONIC FORMS MATRIX REGULARIZATION IN COMPACTIFIED SPACES? TOPOLOGICAL INFORMATION? De Wit+Peeters+Plefka

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