Download Supersymmetry

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
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Acknowledgements
For invitation to present a talk.
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1. Opening
A brief comment about
Supersymmetry
is requested !
Before that: What is the contexte?
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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:
-
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Spin
Incertainty (Heisenberg Principles)
Duality: particles/waves
Also quantum physics is not enough!!
 Quantum field theory!
Mixture of quantum physics with relativity !
…
 String theory,
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that's the contexte
…
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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.
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 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 .
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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
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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
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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.
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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, aL0
a  deg a  
1 if a is odd, aL1
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The superbracket is shown to satisfy:
a) The supersymmetry:
a, b   1deg adeg 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.
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Superbracket and Physics !
Consider:
a, b   1
deg adeg b
b, a
and let B and F be Fermionic and Bosonic operators respectively
such that



with
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B deg B0
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
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defines a LSA of
dimension (1  2r )
3. What is new in LSA?
Two type of
Simple Roots
FERMIONIC
(ODD)
DIFFERENT DYNKIN DIAGRAMM !!
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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
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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),
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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)
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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.
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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
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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
.......
•
•
•
•
•
•
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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
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