Badis Ydri Ph.D.,2001. January 19, 2011

Transcription

1 January 19, 2011

2 Curriculum Vitae.. : Fuzzy. : Noncommutative Gauge Theory and Emergent Geometry From Yang-Mills Matrix Models..

3 Education-Pre Baccalaureat in Mathematics (June 1989), Moubarek El Mili High School,. Diploma of Higher Studies (D.E.S) in (June 1993), First Class Honor,Constantine, Algeria. Diploma of Advanced Studies (D.E.A) 1 in (October 1994),Constantine, Algeria. 1 The first year of the Magister degree.

4 Education- ICTP Diploma 2 (August 1996), First Class Honor, International Centre for,trieste, Italy. :The 2-Dimensional O(N) Bosonization. Supervisor:K.S.Narain. Master of Science (M.S.) in (December 2000),,. Doctor of Philosophy (Ph.D.) in (September 2001),,, NY. :Fuzzy. Supervisor:A.P.Balachandran Prize. 2 Equivalent to a Master of Science.

5 Post Hamilton Postdoctoral Fellow ( ), School of, Dublin Institute for Advanced Studies, Ireland. Marie-Curie Postdoctoral Fellow ( ) 3, Institut fur Physik, Humboldt-Universitat zu Berlin, Germany. Faculty (2008-Current),. DIAS Research Associate (2009-Current), School of, Dublin Institute for Advanced Studies, Ireland. 3 Funded by The European Commission.

6 Total Number of arxiv E-prints: 27. Referred Journal :21. Published Research Work:5. Published Post Research Work:16. Journals: Physical Review Letters (PRL) - 1 Communications in Mathematical (CMP) - 1 Journal of High Energy (JHEP) - 5 Physical Review D (PRD) - 3 Nuclear B (NPB) - 5 Modern Letters A (MPLA) - 3 Journal of Geometry and (JGP) - 1 International Journal of Modern A (IJMPA) - 2

7 Title:Fuzzy. Submitted: September Defended: October Advisor: Professor A.P.Balachandran. Based on: Monopoles and solitons in fuzzy physics, CMP 208, 787 (2000). The fermion doubling problem and noncommutative geometry, MPLA 15, 1279 (2000). Fuzzy CP 2, JGP 42, 28 (2002).

8 Abstract of Fuzzy Regularization of quantum field theories (QFT s) can be achieved by quantizing the underlying manifold (spacetime or spatial slice) thereby replacing it by a non-commutative matrix model or a fuzzy manifold. Such discretization by quantization is remarkably successful in preserving symmetries and topological features, and altogether overcoming the fermion-doubling problem. In this thesis, the fuzzification of coadjoint orbits and their QFT s are put forward.

9 Quantum Mechanics Phase Space Quantization: x i,p j = canonical variables ˆx i, ˆp j = hermitian operators : [ˆx i, ˆp j ] = i δ ij. The quantum phase space is a noncommutative space. It is also a fuzzy space since points are replaced with cells due to Heisenberg uncertainty principle x p 1 2. The commutative limit is the quasiclassical limit 0.

10 Noncommutative Geometry Spacetime Quantization: x i = coordinates ˆx i = hermitian operators : [ˆx i, ˆx j ] = iθ ij. The quantized spacetime 4 is a noncommutative space and points are fuzzy. The solution is given by the Heisenberg algebra. There are no finite-dimensional matrix representations. In d = 2 we have ˆx 1 = 1 2 (a + a + ), ˆx 2 = 1 i 2 (a a+ ) : [a,a + ] = θ. These are Moyal-Weyl noncommutative spaces. 4 The signature is assumed to be Euclidean.

11 Fuzzy Sphere Quantum Mechanics: The Heisenberg algebra {a,a + } is a contraction of the angular momentum algebra {J 1,J 2,J 3 }. The angular momentum operators are N N matrices corresponding to spin j = N 1 2. Define We find [x a,x b ] = x a = 2 N 2 1 J a. 2 N 2 1 iǫ abcx c, x x2 2 + x2 3 = 1. This is a round sphere which is noncommutative: A Fuzzy Sphere. In the limit N we recover the commutative sphere.

12 Fuzzy Spaces A fuzzy space is a noncommutative space which can be represented by finite-dimensional matrices. Seminal Examples: The fuzzy sphere and its Cartesian products. Generalization: fuzzy CP n and fuzzy coadjoint orbits. Field theory on a fuzzy space is automatically UV finite. A fuzzy space provides therefore a natural nonperturbative regularization of field theory. Advantages compared to lattice: symmetry and topology can be captured exactly on fuzzy spaces. Disadvantages: non-local effects and exotic phase structure which are absent in commutative physics.

13 Main Results of Fuzzy Systematic construction of discrete topological configurations such as monopoles and solitons with correct winding numbers. The main tools used are fuzzy physics, noncommutative geometry, K-theory and projective modules. Resolution of the fermion doubling problem based on the fuzzy sphere and construction of a fuzzy Ginsparg-Wilson Algebra. Construction of fuzzy CP 2 and its Dirac operator.

14 From Fuzzy : Fermion Doubling Lattice Regularization: The eigenvalues of the Laplacian (kinetic energy) of a Dirac fermion on a square lattice are λ = (sin 2 ak 1 + sin 2 ak 2 )/a 2. They approach k1 2 + k2 2 for small momenta k 0 and for large momenta k ±π/a. We have therefore 2 2 fermion species instead of just one:fermion Doubling. Fuzzy Regularization: The eigenvalues of the Laplacian of a Dirac fermion on a fuzzy sphere are λ = (j + 1/2) 2, j = 1/2,3/2,...,N 1/2. This is precisely the spectrum of the Laplacian on the ordinary sphere only cutoff.

15 Title:Noncommutative Gauge Theory and Emergent Geometry From Yang-Mills Matrix Models. Based on: Geometry in transition: A model of emergent geometry, PRL 100, (2008). Monte Carlo simulation of a noncommutative gauge theory on the fuzzy sphere, JHEP 0611, 016 (2006). A gauge-invariant UV-IR mixing and the corresponding phase transition for U(1) fields on the fuzzy sphere, NPB 704, 111 (2005). Quantum equivalence of noncommutative and Yang-Mills gauge theories in 2D and matrix theory, PRD 75, (2007).

16 Abstract We find for pure matrix models with global SO(3) symmetry an exotic line of discontinuous transitions with a jump in the energy, characteristic of a 1st order transition, yet with divergent critical fluctuations and a divergent specific heat with critical exponent α = 1/2. The low temperature phase (small values of the gauge coupling constant) is a geometrical one with gauge fields fluctuating on a round sphere. As the temperature increased the sphere evaporates in a transition to a pure matrix phase with no background geometrical structure. These models present an appealing picture of a geometrical phase emerging as the system cools and suggests a scenario for the emergence of geometry in the early universe.

17 IKKT Matrix Models The IKKT Yang-Mills matrix model in d = 10 (II B matrix model) is postulated to give a constructive definition of type II B superstring theory. It is obtained from the dimensional reduction of 10 d supersymmetric Yang-Mills theory to zero dimension (a point). Quantum Mechanics: BFSS and BMN models give a constructive definition of M-theory. The IKKT exists also in d = 4,6. The bosonic truncation exists also in d = 3. Yang-Mills theories on noncommutative tori can be obtained as effective field theories of IKKT matrix models. Mass deformed IKKT Yang-Mills matrix models in various dimensions admit the fuzzy sphere as a solution.

18 The IKKT Matrix Model in d = 4 The action is S 0 = N 4 Tr[X µ,x ν ] 2 + Trθ +( i[x 4,..] + σ a [X a,..] ) θ. The most general supersymmetric SO(3) mass deformation is given by S 1 = 2iNα 3 ǫ abctrx a X b X c + 2α2 N TrXa α 3 Trθ+ θ.

19 The IKKT (Ishibashi, Kawai, Kitazawa, and Tsuchiya) Matrix Model in d = 4 The dynamical variables are four hermitian N N matrices X µ together with a 4 dimensional Majorana spinor. This has N = 1 supersymmetry. In d = 4 the determinant of the Dirac operator is positive definite. The action is S 0 = N 4 Tr[X µ,x ν ] 2 + Trθ +( i[x 4,..] + σ a [X a,..] ) θ. The ground state is given by commuting matrices. The most general supersymmetric SO(3) mass deformation is given by S 1 = 2iNα 3 ǫ abctrx a X b X c + 2α2 N TrXa α 3 Trθ+ θ. 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.

20 The ground state is a noncommutative gauge theory on a fuzzy sphere The minimum energy configuration of the bosonic action is X 4 = 0, X a = φ(α)j a. The J a are angular momentum operators with spin (N 1)/2. This corresponds to a fuzzy sphere configuration. We consider fluctuations around the ground state given by X 4 = Φ 4, X a = φ(α)(j a + A a ). We obtain a noncommutative U(1) gauge theory on a fuzzy sphere with two adjoint scalar fields given by Φ 4 and the normal scalar field Φ defined by Φ = 1 φ 2 N 2 1 (X 2 a φ2 c 2 ) A a n a, N.

21 Theory: perturbative and large N expansions(quenched model in d = 3) First result: We can show the existence of a UV-IR mixing problem. Second result: In the limit N the path integral is dominated by the configuration X a = αφj a and the one-loop becomes dominant. The free energy is V eff = 3 4 log α4 + α 4[ 1 8 φ4 1 6 φ3] + log αφ, α = α N There is a solution φ of the equation of motion only for values of α above the value α = ( 8 3 ) 3 4. Below this value the minimum configurations are commuting matrices and the background spherical geometry evaporates. We have a phase transition at α.

22 Nonperturbative Monte Carlo results-latent heat The inverse temperature is defined by β = α 4. The energy jumps from the value 5/12 at low temperataure to the value 3/4 at high temperature. There is latent heat. This is a first order transition.the high temperature is highly interacting. Every matrix contributes 1/4 to the energy. <S>/N m 2 = 0 N=16 N=24 N=32 N=48 exact

23 Monte Carlo-The radius of the sphere (order parameter) This is defined by 1 r = 1 TrDa 2, X a = αd a. Nc 2 The sphere expands then evaporates. r N=16 N=12 N=10 theory α

24 Monte Carlo-The specific heat The specific heat diverges at the discontinuity from the sphere side and remains constant from the matrix side. This is a second order behaviour with critical fluctuations only from one side of the transition which is quite novel. ly we find the critical exponent 1/2. C v N=16 N=24 N=32 N=48 theory

25 The matrix phase The matrix phase is dominated by commuting matrices. The eigenvalues distribution of X 3 can be derived by assuming that the joint eigenvalues distribution of the the three commuting matrices X 1, X 2 and X 3 is uniform inside a solid ball. We obtain ρ(x) = 3 4R 3(R2 x 2 ), R = 2. (1) Tr X a 2 /N N=16 N=12 N=10 theory

26 The most general SO(3) symmetric quartic matrix model is obtained by adding the potential V = N [ m 2 Tr(Xa 2 2c )2 α 2 µtr(xa 2 )]. 2 The value µ = m 2 is of particular interest. In this case we are giving a large mass to the normal scalar field Φ. The matrix phase persists. The nature of the transition changes as we increase m 2. The critical value in the limit m 2 is α 4 = 8/m2. The perturbative UV-IR mixing disappears in this limit.

27 The phase diagram V = N [ m 2 2c 2 Tr(X 2 a )2 α 2 m 2 Tr(X 2 a )]. The phase diagram Ln α s, α c Matrix Phase Ln α s Ln α c theory Fuzzy Sphere Phase Ln m 2

28 Some generalizations and other results The 4 dimensional complex projective spaces S 2 S 2 and CP 2. There is condensation of geometry (not necessarily 4 dimensional) at low temperatures. Higher dimensional CP n (CP 3 in particular) and more generally coadjoint orbits. The bosonic truncation of mass deformed IKKT matrix models in 4 dimensions with SO(3) symmetry.the condensed geometry is still two dimensional. The supersymmetric mass deformed IKKT matrix model in 4 dimensions with SO(3) symmetry. The geometry is more stable and supersymmetry can be spontaneously broken.

29 Some current and future directions Current Cohomological deformations which reduce IKKT models to a single matrix model accessible to random matrix theory. The impact of supersymmetry on emergent geometry. The use of the Monte Carlo method to study exact supersymmetry via matrix models. Finding matrix models in which we have emergent 4 dimensional geometries. A candidate is S 2 S 2. The renormalizability and phase structure of nc φ 4 using the fuzzy sphere as the matrix base. The Polchinski renormalization group equation for noncommutative field theories and matrix models. The AdS/CFT correspondence and Yang-Mills matrix models. Supersymmetric gauge theories in 4 dimensions and Instanton calculus.