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. 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 OVERVIEW

3. OPEN PROBLEMS: 1.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 CENTRAL CHARGES CANONICAL QUANTIZATION 2+1 NCSYM THEORY MOTIVATION

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. 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 RESULTS

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

7. CLASSICALLY: STRING-LIKE SPIKES: H= M, N=1,..,9 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: NOT STRING-LIKE SPIKES (1) QUANTUM: BOSONIC PURELY DISCRETE SPECTRUM (2) FERMIONIC PURELY DISCRETE SPECTRUM (3) & TOPOLOGICAL CONDITION CENTRAL CHARGE CONDITION:

9. The only degrees of freedom to quantize are (X, A). 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

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

11. SPECTRAL PROPERTIES: SU(N) LEVEL CLASSICALLY: NO-STRING-LIKE SPIKES QUANTUM LEVEL: BOSONIC SECTOR QUANTUM LEVEL FERMIONIC SECTOR MPGM+ A. Restuccia 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 DgDg Gaussian measure on l2 Well-defined Matrix regularization

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

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

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

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

17. LARGE N LIMIT ? MATRIX REGULARIZATION IN COMPACTIFIED SPACES? TOPOLOGICAL INFORMATION? PROBLEM OF CLOSED HARMONIC FORMS De Wit+Peeters+Plefka D=11 SUPERMEMBRANES10D SYM H = SU(N) MATRIX REGULARIZATION DIMEN. REDUCCION 0+1 MATRIX MODELS