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LIE SUPERALGEBRAS AND PHYSICAL MODELS My. Brahim SEDRA Ibn Tofail University Faculty of sciences, Physics Department, LHESIR, Kenitra WORKSHOP DE RABAT 6-8 JUIN 2013 1 Acknowledgements For invitation to present a talk. 2 1. Opening A brief comment about Supersymmetry is requested ! Before that: What is the contexte? 3 Structure de l’atome Mécanique Quantique Electron Interaction électromagnétique Noyau -10 10 4 m Strucure du noyau Proton Interaction forte Neutron -14 10 5 m Structure des nucléons Proton : 2 quarks up 1 quark down Interaction forte Neutron : 1 quark up 2 quarks down -15 10 6 m What happens at these very small scales of the matter? Classical physics is no longer valuable Quantum physics: Major Properties: - 7 Spin Incertainty (Heisenberg Principles) Duality: particles/waves Also quantum physics is not enough!! Quantum field theory! Mixture of quantum physics with relativity ! … String theory, 8 that's the contexte … 9 Susy: What is it? In nature there are bosons and fermions Bosons: particles having integer value of the spin Fermions: particles having half integer value of the spin Susy: a mechanism that associates to each boson a fermion. Susy is broken at the present scale of the univers ! Susy assumes that in nature (universe) the number of bosonic states should be the same as the number of fermionic states. By virtue of susy, bosons and fermions should have the same MASSE. 10 Susy theory assumes also that the super partner of the electron is a boson called the selectron: m (e)=m(se). However, there is no experimental (or observational) indication about the existence of the selectron. Interpretation ! The difficulty to observe the selectron can have two causes: 1. The selectron is very heavy ! or 1. There should exists an unknown mechanism that makes a screen on it. Thus, the observation of the selectron requests higher technology . 11 C/C: Since the masse of partners is not equilibrate, the susy is broken. Comment 1 Boson de Higgs : C’est une particule soupçonnée être à l'origine de l’attribution des masses à toutes les particules de l'univers physique 12 Comment 2 On 4 July 2012: CERN has announced in a conference that a new bosonic particle has been identified . Probably it’s the Higgs! The CERN is not yet completely assured about it! (Des études complémentaires seront nécessaires pour déterminer si cette particule possède l'ensemble des caractéristiques prévues pour le 13 boson de Higgs). 2. What’s about Lie superalgebras? The importance of LSA in physics deals, among other, with the connection with supersymmetry (briefly described before). In constructing supersymmetric integrable models, the request of integrability implies several solutions for the Cartan matrix Kij. In contrast to the standard (bosonic) LA, we don't have a unique Cartan matrix in the LSA. 14 Definition and Properties A Lie superalgeb ra (LSA) L is a Z 2 graded vector space L L0 L1 over the field (R or C) with a (supr) bracket [, } given by a, b ab 1 a b ba where a is the degree of a : 0 if a is even, aL0 a deg a 1 if a is odd, aL1 15 The superbracket is shown to satisfy: a) The supersymmetry: a, b 1deg adeg b b, a a) The super Jacobi Identity 1 a c a, b, c 1 c b c, a, b 1 b a b, c, a 0 Remark: The restriction of L to the even part L 0 gives a standard Lie algebra with a, b a, b satisfying the antisymmetry and the Jacobi identity. 16 Superbracket and Physics ! Consider: a, b 1 deg adeg b b, a and let B and F be Fermionic and Bosonic operators respectively such that with 17 B deg B0 F deg F 1 B, B B B, F F F , F B Example Let L be the Heisenberg LSA defined as the set j j of raising and lowering operators b and b , j 1,2 ,...,r. These operators satisfy the following relations b b b j j j , b k jk , b k 0, , b k 0. .1, and b , 1 0, b , 1 0, 1, 1 0. j j Then 1 b j , b j , j 1,..., r 18 defines a LSA of dimension (1 2r ) 3. What is new in LSA? Two type of Simple Roots FERMIONIC (ODD) DIFFERENT DYNKIN DIAGRAMM !! 19 LSA with odd simple roots play an important role in Susy Integrable models. These integrable models are defined through a zero curvature condition D A , D A 0 20 The important result (Literature): There are classes of LSA whose Cartan matrices lead to integrable models in such way that the simple roots is chosen to be purely fermionic (odd). The constraint of integrability, leads to some explicit solutions of the Cartan matrix. As an example A(n|n-1)=sl(n+1|n), B(n|n)=osp(2n+1|2n) B(n-1|n)=osp(2n-1|2n), D(n+1|n)=osp(2n+2|2n) D(n|n) =osp(2n|2n), 21 D(2|1; a) LSA Sl(n+1/n) These are 1 2 2,3,...,2n 1 osp(1/2) 3/2 osp(3/2) 3/2, 2 osp(2n-1/2n), n≥2 1 3,4,7,8,11,12,...,4n 5,4n 4,4n 1 2 osp(2n+1/2n), n≥2 1 3,4,7,8,11,12,...,4n 1,4n 2 Osp(2/2)≈sl(2/1) 1, 3/2 Osp(2n/2n) , n≥2 1 3,4,7,8,11,12,...,4n 5,4n 4,4n 1, n 2 Osp(2n+2/2n) D(2/1,a) 22 SPIN OF CONSERVED CURRENTS 1 3,4,7,8,11,12,...,4n 1,4n , n 1 2 2 3/2, 3/2,2 4. How things work in physics? Integrable models are systems of non linear differential equations . Solving these equations is not an easy job. To avoid the non linearity, we use: Operators belonging to some The famous: Lax technique Lie algebra structure The principal idea of the LT : We start from a non linear diff. Equation with some fixed degrees of freedom. We assume the existence of a Lax pair, defined in some Lie algebra structure. If the Lax pair exists, the integrability is assured. 23 Bosonic case – To illustrate the previous arguments, let’s consider the following physical model: The 2d Conformal Liouville Field Theory S d 2 z exp( 2 ) where is a scalar bosonic field. The equation of motion is 2 exp( 2 ) 0 This is a n.l.d.eq. That can be solved by the following Lax pair: h ,e 2 e Az h e Az ex p 2 f with h , f 2 f su (2) e , f h We underline that the Lax pair satisfy the zero curvature condition Fzz Az Az Az , Az 0 Fermionic case – The super(symmetric) case consists in considering similar steps: The 2d super Liouville Field Theory S d zd D D exp( ) 2 2 where : - is superfield , - D, D are spinors, the super derivative s - the Grassmann variables As in the bosonic case, the Lax pair exists in this case in order to ensure the integrability of the model. The Lie symmetry is given by the Superalgebras Osp(1|2) osp (1 2) h, e2 , f 2 0 e1 , f1 1 Rank 1 dim 5 Dynkin Diagram D : (one simple root) Results MBS (and collaborators): http://inspirehep.net/search: M.B.Sedra. More on Lie superalgebras and Physical Models: MBS (Thèse de doctorat d’Etat 1995) Et references dedans 28 References 1. M.Scheunert, The theory of Lie superalgebras, Lecture Notes in math (1979); 2. J.Wess and J. Bagger, supersymmetry and supergravity, princeton series in physics, 1983, 3. H. Nohara and all, Toda field theories, CFT (1990, 1991) 4. M.B. Sedra, • ADSTP (2011), with K. Bilal, A. Boukili, M. Nach • CJP, (2009) with A. Boukili, A. Zemate ....... • • • • • • 29 Nucl.Phys. B513:709-722,1998 J.Math.Phys.37:3483-3490,1996. Mod.Phys.Lett.A9:3163-3174,1994, Mod.Phys.A9:1994. Class.Quant.Grav.10:1937-1946, 1993. J.Math.Phys.35, 3190,1993 Thanks 30