University Habilitation Theoretical Physics Annaba University University Habilitation

Badis Ydri Ph.D.,2001 Theoretical Physics Syracuse University Annaba University New York

Outline CV Badis Ydri Publications Ph.D.,2001 Doctoral Syracuse University New York Thesis

Habilitation Dissertation

Future Directions January 19, 2011 Outline

University Habilitation Theoretical Physics Annaba University

Badis Ydri Curriculum Vitae. Ph.D.,2001 Syracuse Publications. University New York Doctoral Thesis:

Outline Fuzzy Physics. CV Habilitation Dissertation: Publications Noncommutative Gauge Theory and Emergent Geometry Doctoral Thesis From Yang-Mills Matrix Models. Habilitation Future Directions. Dissertation

Future Directions Education-Pre Doctoral

University Habilitation Theoretical Physics Annaba University

Badis Ydri Baccalaureat in Mathematics (June 1989), Moubarek El Ph.D.,2001 Syracuse Mili High School, Annaba. University New York Diploma of Higher Studies (D.E.S) in Theoretical Physics

Outline (June 1993), First Class Honor,Constantine University,

CV Algeria. Publications Diploma of Advanced Studies (D.E.A)1 in Theoretical Doctoral Thesis Physics (October 1994),Constantine University, Algeria.

Habilitation Dissertation

Future Directions

1The first year of the Magister degree. Education-Doctoral

University Habilitation Theoretical Physics 2 Annaba ICTP Diploma (August 1996), First Class Honor, University International Centre for Theoretical Physics,Trieste, Italy. Badis Ydri Ph.D.,2001 Dissertation:The 2-Dimensional O(N) Bosonization. Syracuse University Supervisor:K.S.Narain. New York Master of Science (M.S.) in Physics (December 2000), Outline Syracuse University,New York. CV Doctor of Philosophy (Ph.D.) in Theoretical Physics Publications (September 2001),Syracuse University, Syracuse, NY. Doctoral Thesis Thesis:Fuzzy Physics . Habilitation Supervisor:A.P.Balachandran. Dissertation

Future 2002 Syracuse University Doctoral Prize. Directions

2Equivalent to a Master of Science. Post Doctoral

University Habilitation Theoretical Physics Annaba University Hamilton Postdoctoral Fellow (2001-2004), School of

Badis Ydri Theoretical Physics, Dublin Institute for Advanced Ph.D.,2001 Syracuse Studies, Ireland. University 3 New York Marie-Curie Postdoctoral Fellow (2006-2008) , Institut fur Physik, Humboldt-Universitat zu Berlin, Germany. Outline

CV Theoretical Physics Faculty (2008-Current), Annaba

Publications University.

Doctoral Thesis DIAS Research Associate (2009-Current), School of

Habilitation Theoretical Physics, Dublin Institute for Advanced Dissertation Studies, Ireland. Future Directions

3Funded by The European Commission. Publications

University Habilitation Theoretical Physics Total Number of arXiv E-prints: 27. Annaba University Referred Journal Publications:21. Badis Ydri Ph.D.,2001 Published Doctoral Research Work:5. Syracuse University Published Post Doctoral Research Work:16. New York Journals: Outline Physical Review Letters (PRL) - 1 CV Communications in Mathematical Physics (CMP) - 1 Publications Journal of High Energy Physics (JHEP) - 5 Doctoral Thesis Physical Review D (PRD) - 3

Habilitation Nuclear Physics B (NPB) - 5 Dissertation Modern Physics Letters A (MPLA) - 3 Future Journal of Geometry and Physics (JGP) - 1 Directions International Journal of Modern Physics A (IJMPA) - 2 Doctoral Thesis

University Habilitation Theoretical Physics Annaba University Title:Fuzzy Physics. Badis Ydri Submitted: September 2001 . Defended: October 2001. Ph.D.,2001 Syracuse University Advisor: Professor A.P.Balachandran. New York Based on: Outline “Monopoles and solitons in fuzzy physics,” CMP 208, CV 787 (2000). Publications Doctoral “The fermion doubling problem and noncommutative Thesis geometry,” MPLA 15, 1279 (2000). Habilitation Dissertation “Fuzzy CP2,” JGP 42, 28 (2002). Future Directions Abstract of ”Fuzzy Physics”

University Habilitation Theoretical Physics Annaba University

Badis Ydri Regularization of quantum field theories (QFT’s) can be Ph.D.,2001 Syracuse achieved by quantizing the underlying manifold (spacetime or University New York spatial slice) thereby replacing it by a non-commutative matrix model or a “fuzzy manifold”. Such discretization by Outline quantization is remarkably successful in preserving symmetries CV

Publications and topological features, and altogether overcoming the

Doctoral fermion-doubling problem. In this thesis, the fuzzification of Thesis coadjoint orbits and their QFT’s are put forward. Habilitation Dissertation

Future Directions Quantum Mechanics

University Habilitation Theoretical Physics Annaba Phase Space Quantization: University Badis Ydri x p canonical variables Ph.D.,2001 i , j = Syracuse −→ University xˆi , pˆj = hermitian operators : [ˆxi , pˆj ]= i~δij . New York

Outline The quantum phase space is a noncommutative space. It CV is also a fuzzy space since points are replaced with cells Publications due to Heisenberg uncertainty principle Doctoral Thesis 1~ Habilitation ∆x∆p . Dissertation ≥2 Future The commutative limit is the quasiclassical limit ~ 0. Directions −→ Noncommutative Geometry

University Habilitation Theoretical Spacetime Quantization: Physics Annaba University xi = coordinates −→ Badis Ydri Ph.D.,2001 xˆi = hermitian operators : [ˆxi , xˆj ]= iθij . Syracuse University 4 New York The quantized spacetime is a noncommutative space and

Outline points are fuzzy. CV The solution is given by the Heisenberg algebra. There are Publications no finite-dimensional matrix representations. In d = 2 we Doctoral have Thesis

Habilitation 1 + 1 + + Dissertation xˆ1 = (a + a ) , xˆ2 = (a a ) : [a, a ]= θ. Future √2 i√2 − Directions These are Moyal-Weyl noncommutative spaces.

4The signature is assumed to be Euclidean. Fuzzy Sphere

University Habilitation + Theoretical Quantum Mechanics: The Heisenberg algebra a, a is a Physics { } contraction of the angular momentum algebra J1, J2, J3 . Annaba { } University The angular momentum operators are N N matrices Badis Ydri N−1 × Ph.D.,2001 corresponding to spin j = 2 . Define Syracuse University New York 2 xa = Ja. 2 Outline √N 1 − CV

Publications We find

Doctoral 2 2 2 2 Thesis [xa, x ]= iǫ xc , x + x + x = 1. b 2 abc 1 2 3 Habilitation √N 1 Dissertation − Future This is a round sphere which is noncommutative: A Fuzzy Directions Sphere. In the limit N we recover the commutative −→ ∞ sphere. Fuzzy Spaces

University Habilitation Theoretical Physics A fuzzy space is a noncommutative space which can be Annaba University represented by finite-dimensional matrices. Badis Ydri Seminal Examples: The fuzzy sphere and its Cartesian Ph.D.,2001 Syracuse products. University New York Generalization: fuzzy CPn and fuzzy coadjoint orbits. Outline Field theory on a fuzzy space is automatically UV finite. CV A fuzzy space provides therefore a natural nonperturbative Publications regularization of field theory. Doctoral Thesis Advantages compared to lattice: symmetry and topology Habilitation Dissertation can be captured exactly on fuzzy spaces. Future Disadvantages: non-local effects and exotic phase Directions structure which are absent in commutative physics. Main Results of “Fuzzy Physics“

University Habilitation Theoretical Physics Annaba University Systematic construction of discrete topological Badis Ydri Ph.D.,2001 configurations such as monopoles and solitons with correct Syracuse University winding numbers. The main tools used are fuzzy physics, New York noncommutative geometry, K-theory and projective Outline modules. CV Resolution of the fermion doubling problem based on the Publications fuzzy sphere and construction of a fuzzy Ginsparg-Wilson Doctoral Thesis Algebra. Habilitation 2 Dissertation Construction of fuzzy CP and its Dirac operator.

Future Directions From “Fuzzy Physics“: Fermion Doubling

University Habilitation Lattice Regularization: The eigenvalues of the Laplacian Theoretical Physics (kinetic energy) of a Dirac fermion on a square lattice are Annaba University 2 2 2 Badis Ydri λ = (sin ak1 + sin ak2)/a . Ph.D.,2001 Syracuse 2 2 University They approach k1 + k2 for small momenta k 0 and New York −→ for large momenta k π/a. −→ ± Outline We have therefore 22 fermion species instead of just CV one:Fermion Doubling. Publications Fuzzy Regularization: The eigenvalues of the Laplacian of Doctoral Thesis a Dirac fermion on a fuzzy sphere are Habilitation Dissertation λ = (j + 1/2)2 , j = 1/2, 3/2, ..., N 1/2. Future − Directions This is precisely the spectrum of the Laplacian on the ordinary sphere only cutoff. Habilitation Dissertation

University Habilitation Noncommutative Gauge Theory and Emergent Theoretical Title: Physics Geometry From Yang-Mills Matrix Models Annaba . University Based on: Badis Ydri Ph.D.,2001 “Geometry in transition: A model of emergent Syracuse University geometry,” PRL 100, 201601 (2008). New York

Outline “Monte Carlo simulation of a noncommutative gauge

CV theory on the fuzzy sphere,” JHEP 0611, 016 (2006). Publications “A gauge-invariant UV-IR mixing and the Doctoral Thesis corresponding phase transition for U(1) fields on the Habilitation fuzzy sphere,” NPB 704, 111 (2005). Dissertation Future “Quantum equivalence of noncommutative and Directions Yang-Mills gauge theories in 2D and matrix theory,” PRD 75, 105008 (2007). Abstract

University Habilitation Theoretical Physics Annaba We find for pure matrix models with global SO(3) symmetry University an exotic line of discontinuous transitions with a jump in the Badis Ydri Ph.D.,2001 energy, characteristic of a 1st order transition, yet with Syracuse University divergent critical fluctuations and a divergent specific heat with New York critical exponent α = 1/2. The low temperature phase (small

Outline values of the gauge coupling constant) is a geometrical one CV with gauge fields fluctuating on a round sphere. As the Publications temperature increased the sphere evaporates in a transition to Doctoral a pure matrix phase with no background geometrical structure. Thesis

Habilitation These models present an appealing picture of a geometrical Dissertation phase emerging as the system cools and suggests a scenario for Future Directions the emergence of geometry in the early universe. IKKT Matrix Models

University Habilitation The IKKT Yang-Mills matrix model in d = 10 (II B matrix Theoretical Physics model) is postulated to give a constructive definition of Annaba University type II B superstring theory. Badis Ydri Ph.D.,2001 It is obtained from the dimensional reduction of 10 d Syracuse − University supersymmetric Yang-Mills theory to zero dimension (a New York point). Outline Quantum Mechanics: BFSS and BMN models give a CV constructive definition of M-theory. Publications The IKKT exists also in d = 4, 6. The bosonic truncation Doctoral Thesis exists also in d = 3. Habilitation Dissertation Yang-Mills theories on noncommutative tori can be Future obtained as effective field theories of IKKT matrix models. Directions Mass deformed IKKT Yang-Mills matrix models in various dimensions admit the fuzzy sphere as a solution. The IKKT Matrix Model in d =4

University Habilitation Theoretical Physics Annaba University The action is

Badis Ydri Ph.D.,2001 N 2 + Syracuse S0 = Tr[Xµ, Xν] + Trθ i[X4, ..]+ σa[Xa, ..]
θ. University − 4 New York

Outline The most general supersymmetric SO(3) mass deformation is CV given by Publications Doctoral 2iNα 2α2N 2α Thesis 2 + S1 = ǫabc TrXaXbXc + TrXa + Trθ θ. Habilitation 3 9 3 Dissertation

Future Directions The IKKT (Ishibashi, Kawai, Kitazawa, and Tsuchiya) Matrix Model in d =4

University Habilitation The dynamical variables are four hermitian N N matrices Xµ Theoretical × Physics together with a 4 dimensional Majorana spinor. This has Annaba − = 1 supersymmetry. In d = 4 the determinant of the Dirac University N Badis Ydri operator is positive definite. The action is Ph.D.,2001 Syracuse University N 2 + New York S0 = Tr[Xµ, Xν] + Trθ i[X4, ..]+ σa[Xa, ..]
θ. − 4 Outline

CV The ground state is given by commuting matrices. The most

Publications general supersymmetric SO(3) mass deformation is given by

Doctoral Thesis 2 2iNα 2α N 2 2α + Habilitation S1 = ǫabc TrXaXbXc + TrXa + Trθ θ. Dissertation 3 9 3

Future Directions The cubic (Chern-Simons) term is due to Myers effect. This is the essential ingredient in the phenomena of condensation of geometry at low temperature. The ground state is a noncommutative gauge theory on a fuzzy sphere

University Habilitation The minimum energy configuration of the bosonic action is Theoretical Physics Annaba University X4 = 0 , Xa = φ(α)Ja. Badis Ydri Ph.D.,2001 Syracuse The Ja are angular momentum operators with spin (N 1)/2. University − New York This corresponds to a fuzzy sphere configuration. We consider fluctuations around the ground state given by Outline CV X4 =Φ4 , Xa = φ(α)(Ja + Aa). Publications U Doctoral We obtain a noncommutative (1) gauge theory on a fuzzy Thesis sphere with two adjoint scalar fields given by Φ4 and the Habilitation Dissertation normal scalar field Φ defined by

Future Directions 1 2 2 Φ= (Xa φ c2) Aana , N . φ2√N2 1 − −→ −→∞ − Theory: perturbative and large N expansions(quenched model in d = 3)

University Habilitation First result: We can show the existence of a UV-IR mixing Theoretical Physics problem. Annaba Second result: In the limit N the path integral is University −→∞ dominated by the configuration X = αφJ and the one-loop Badis Ydri a a Ph.D.,2001 becomes dominant. The free energy is Syracuse University New York 3 4 4 1 4 1 3 Veff = logα ˜ +α ˜  φ φ  + logαφ ˜ , α˜ = α√N Outline 4 8 − 6 CV There is a solution φ of the equation of motion only for values Publications ofα ˜ above the value Doctoral Thesis 8 3 Habilitation α˜∗ = ( ) 4 . Dissertation 3 Future Directions Below this value the minimum configurations are commuting matrices and the background spherical geometry evaporates. We have a phase transition atα ˜∗ . Nonperturbative Monte Carlo results-Latent heat

University 4 Habilitation The inverse temperature is defined by β =α ˜ . The energy Theoretical jumps from the value 5/12 at low temperataure to the value Physics Annaba 3/4 at high temperature. There is latent heat. This is a first University order transition.The high temperature is highly interacting. Badis Ydri Ph.D.,2001 Every matrix contributes 1/4 to the energy. Syracuse University New York

2 Outline 1 m = 0 N=16 N=24 CV N=32 0.5 N=48 Publications exact

Doctoral 2 Thesis 0

Habilitation /N Dissertation -0.5

Future Directions -1

-1.5 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 α∼ Monte Carlo-The radius of the sphere (order parameter)

University Habilitation This is defined by Theoretical Physics 1 1 2 Annaba = TrDa , Xa = αDa. University r Nc2

Badis Ydri The sphere expands then evaporates. Ph.D.,2001 Syracuse University New York 6 N=16 N=12 Outline 5 N=10 theory CV 4 Publications

Doctoral r 3 Thesis

Habilitation 2 Dissertation 1 Future Directions 0 -2 0 2 4 6 8 10 12 α∼ Monte Carlo-The specific heat

University Habilitation The specific heat diverges at the discontinuity from the sphere Theoretical side and remains constant from the matrix side. This is a Physics Annaba second order behaviour with critical fluctuations only from one University side of the transition which is quite novel. Theoretically we find Badis Ydri Ph.D.,2001 the critical exponent 1/2. Syracuse University New York 2.2 Outline N=16 2 N=24 CV N=32 1.8 N=48 Publications theory Doctoral 1.6 Thesis v 1.4 C Habilitation Dissertation 1.2 Future 1 Directions 0.8 0.6 0 1 2 3 4 5 6 α∼ The matrix phase

University Habilitation The matrix phase is dominated by commuting matrices. The Theoretical eigenvalues distribution of X can be derived by assuming that Physics 3 Annaba the joint eigenvalues distribution of the the three commuting University matrices X , X and X is uniform inside a solid ball. We obtain Badis Ydri 1 2 3 Ph.D.,2001 3 Syracuse x R2 x2 R University ρ( )= 3 ( ) , = 2. (1) New York 4R −

Outline 35 CV N=16 N=12 30 Publications N=10 theory Doctoral 25 Thesis

/N 20 2

Habilitation a Dissertation 15 Tr X Future Directions 10

5

0 -0.5 0 0.5 1 1.5 2 2.5 3 The most general SO(3) symmetric quartic matrix − model is obtained by adding the potential

University Habilitation Theoretical Physics Annaba University

Badis Ydri 2 Ph.D.,2001 m 2 2 2 2 Syracuse V = N Tr(Xa ) α µTr(Xa ). University 2c2 − New York 2 Outline The value µ = m is of particular interest. In this case we are CV giving a large mass to the normal scalar field Φ. The matrix Publications phase persists. The nature of the transition changes as we Doctoral m2 m2 Thesis increase . The critical value in the limit is 4 2 −→ ∞ Habilitation α˜∗ = 8/m . The perturbative UV-IR mixing disappears in this Dissertation limit. Future Directions The phase diagram

University Habilitation Theoretical 2 Physics m 2 2 2 2 2 Annaba V = N Tr(Xa ) α m Tr(Xa ). University 2c2 −

Badis Ydri The phase diagram Ph.D.,2001 Syracuse University New York ∼ Ln α 1.5 α∼ s Outline Ln c theory CV 1 Fuzzy Sphere Publications

c Phase

0.5 Doctoral ∼ α , s

Thesis ∼ α 0 Matrix Phase

Habilitation Ln Dissertation -0.5 Future Directions -1

-8 -6 -4 -2 0 2 4 6 Ln m2 Some generalizations and other results

University Habilitation Theoretical Physics 2 2 Annaba The 4 dimensional complex projective spaces S S and University × CP2. There is condensation of geometry (not necessarily 4 Badis Ydri Ph.D.,2001 dimensional) at low temperatures. Syracuse University CPn CP3 New York Higher dimensional ( in particular) and more generally coadjoint orbits. Outline The bosonic truncation of mass deformed IKKT matrix CV

Publications models in 4 dimensions with SO(3) symmetry.The

Doctoral condensed geometry is still two dimensional. Thesis The supersymmetric mass deformed IKKT matrix model in Habilitation Dissertation 4 dimensions with SO(3) symmetry. The geometry is more Future stable and supersymmetry can be spontaneously broken. Directions Some current and future directions

University Habilitation Current Theoretical Physics Cohomological deformations which reduce IKKT models to Annaba University a single matrix model accessible to random matrix theory.

Badis Ydri The impact of supersymmetry on emergent geometry. Ph.D.,2001 The use of the Monte Carlo method to study exact Syracuse University supersymmetry via matrix models. New York Finding matrix models in which we have emergent 4 2 2 Outline dimensional geometries. A candidate is S S . × 4 CV The renormalizability and phase structure of nc φ using

Publications the fuzzy sphere as the matrix base.

Doctoral The Polchinski renormalization group equation for Thesis noncommutative field theories and matrix models. Habilitation Dissertation Future

Future The AdS/CFT correspondence and Yang-Mills matrix Directions models. Supersymmetric gauge theories in 4 dimensions and Instanton calculus.