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

• 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

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

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

• 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

BFSS/IKKT CONJECTURE: D0 OR D-1 ACTION IS TAKEN AS THE FUNDAMENTALSYMMETRIES: EX. BMN MODEL ETC..

MATRIX MODELS

HEXACT

HEXACTCOMPACT.

HREGULARIZEDHREGULARIZEDCOMPACT

??

SYMMETRIES

?

ORIGINAL POINT OF VIEW: Halpern, Hoppe, De Wit, Hoppe, Nicolai , de wit, Peeters, Plefka etc..

OUR RESULTSFOLLOW ORIGINALPOINT OF VIEW: HREGULARIZED

M2

HEXACTM2

RECENT RESULT

HEXACTM2 = HNCSYM

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

THE SUPERMEMBRANE WITH CENTRAL CHARGESMartin,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:

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

SU(N) REGULARIZATION OF THE M2

GAUGE FIXING CONDITION:

CONSTRAINS:

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

Boulton+MPGM+Martin+Restuccia

Eigenvalues of V(X)

Large N for semiclassical aproximation

Semiclassical quantization of M2 braneDuff, Inami, Pope, Sezgin,Stelle

Semiclassical description of the regularized M2 brane

Dg Gaussian measure on l2

Well-defined

Matrix regularization

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

Spectrum of the exact Bosonic Hamiltonian II

NON-COMPACT INFINITE DIMENSIONAL LAPLACIAN

WITH

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

SQCD & M-Theory InterpretationIR UV

CONFINEMENT QUARK-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

LARGE N LIMIT ?

MATRIX REGULARIZATION IN COMPACTIFIED SPACES?

TOPOLOGICAL INFORMATION?

PROBLEM OF CLOSEDHARMONIC FORMS

De Wit+Peeters+Plefka

D=11 SUPERMEMBRANES 10D SYM

H =SU(N)

MATR

IX R

EGU

LARIZ

ATIO

N DIM

EN. R

EDUCC

ION 0

+1

MATRIX MODELS