LIE SUPERALGEBRAS AND PHYSICAL MODELS

My. Brahim SEDRA

Ibn Tofail University Faculty of sciences, Physics Department, LHESIR,

Kenitra

1

WORKSHOP DE RABAT 6-8 JUIN 2013

Acknowledgements

2

For invitation to present a talk.

3

A brief comment about

Supersymmetry

is requested !

Before that: What is the contexte?

1. Opening

4

Structure de l’atome

Noyau

Electron

Interaction électromagnétique

10-10

m

MécaniqueQuantique

5

Strucure du noyau

10-14

m

Neutron

Proton

Interaction forte

6

Structure des nucléons

10-15

m

Proton :2 quarks up1 quark down

Neutron :1 quark up2 quarks down

Interaction forte

Classical physics is no longer valuable Quantum physics:

7

What happens at these very small scales of the matter?

Major Properties:

- Spin- Incertainty (Heisenberg Principles)- Duality: particles/waves

Quantum field theory!

8

Also quantum physics is not enough!!

Mixture of quantum physics with relativity !

… String theory,

9

that's the contexte

…

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 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: What is it?

Susy is broken at the present scale of the univers !

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.

The difficulty to observe the selectron can have two causes:1. The selectron is very heavy !or1. There should exists an unknown mechanism that makes a screen on

it.

Thus, the observation of the selectron requests higher technology .

C/C: Since the masse of partners is not equilibrate, the susy is broken.

11

Interpretation !

12

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

Comment 1

13

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 boson de Higgs).

Comment 2

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

2. What’s about Lie superalgebras?

Definition and Properties

:a of degree theis a where

1,

bygiven }[,bracket (supr) a with C)or (R field over the

LLL

space vector a is L (LSA) rasuperalgeb LieA 2

baabba

gradedZ

ba

10

15

0

1

L

L

a even, is a if 0

a odd, is a if 1

deg aa

The superbracket is shown to satisfy:a) The supersymmetry:

a) The super Jacobi Identity

Remark:The restriction of L to the even part gives a standard Lie algebra with satisfying the antisymmetry and the Jacobi identity.

abba ba ,1, degdeg

0,,1,,1,,1 acbbaccba abbcca

0L

baba ,,

16

Superbracket and Physics !Consider:

and let B and F be Fermionic and Bosonic operators respectively such that

with

abba ba ,1, degdeg

17

BFF

FFB

BBB

,

,

,

0deg

1deg

BB

FF

.21

,b and b operators lowering and raising of

set theas definedLSA Heisenberg thebe Let

jj

,...,r,j

LExample

.0b ,b

,0b ,b

,1.b ,b

kj

kj

kj

jk

)21(dimension

ofLSA a defines

,,1

Then

r

bb jj

r1,...,j

These operators satisfy the following relations

18

., , , b, , b

and

jj 0110101

19

3. What is new in LSA?

Two type of Simple Roots

DIFFERENT DYNKIN DIAGRAMM !!DIFFERENT DYNKIN DIAGRAMM !!

20

LSA with odd simple roots play an important role in Susy Integrable models.

These integrable models are defined through a zero curvature condition

0, ADAD

21

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

The important result (Literature):

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),

D(2|1; )

22

These are LSA SPIN OF CONSERVED CURRENTS

Sl(n+1/n)

osp(1/2) 3/2

osp(3/2) 3/2, 2

osp(2n-1/2n), n≥2

osp(2n+1/2n), n≥2

Osp(2/2)≈sl(2/1) 1, 3/2

Osp(2n/2n) , n≥2

12,...,3,22

1n

14,44,54,...,12,11,8,7,4,32

1 nnn

nn 4,14,...,12,11,8,7,4,32

1

nnnn ,14,44,54,...,12,11,8,7,4,32

1

Osp(2n+2/2n)

D(2/1,a) 3/2, 3/2,2

2

1,4,14,...,12,11,8,7,4,3

2

1

nnn

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:

The famous: Lax technique 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.

Operators belonging to some

Lie algebra structure

23

Bosonic case – To illustrate the previous arguments, let’s consider the

following physical model: The 2d Conformal Liouville Field Theory

The equation of motion is

This is a n.l.d.eq. That can be solved by the following Lax pair:

with

0)2exp(2

field. bosonicscalar a is where

)2exp(2

zdS

ehzA

fzA

2exp

eeh

ffh

hfe

su2,

2,

,

)2(

We underline that the Lax pair satisfy the zero curvature condition

0, zzzzzz AAAAF

Fermionic case – The super(symmetric) case consists in considering

similar steps: The 2d super Liouville Field Theory

variablesGrassmann the -

sderivativesuper thespinors, are DD, -

,superfield is -

:where

)exp(22

DDzddS

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)

root) simple (one

:

5dim

1

,,,)21( 111022

DDiagramDynkin

Rank

fefehosp

28

MBS (and collaborators):

http://inspirehep.net/search: M.B.Sedra.

Results

More on Lie superalgebras and Physical Models:MBS (Thèse de doctorat d’Etat 1995) Et references dedans

29

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.......

• 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

30

Thanks