Pietro FrèTalk at SQS 09 DUBNA
arXiv:0906.2510 Theory of Superdualities and the Orthosymplectic Supergroup Authors: Pietro Fré, Pietro Antonio Grassi, Luca Sommovigo, Mario Trigiante
There are duality symmetries of field equations + Bianchi identities
There are active dualities that transform one lagrangian into another.
In D=4 all Bose dualities are symplectic Sp(2n,R)
In D=2 all Bose dualities are pseudorthogonal SO(m,m)
In D=2 we can construct superdualities of Osp(m,m|4n) applying to Bose/Fermi -models
The general form of a bosonic D=4 supergravity Lagrangian
For N>2 obligatoryFor N<3 possible
The symplectic embedding
?
It is the Cayley matrix which by conjugation realizes the isomorphism
The Gaillard Zumino Master Formula
There are fields of two kinds
Peccei-Quin symmetries ! + c
Generalized electric/magnetic duality rotations are performed on the twisted scalars
Embedding of the coset representative
Embedding of thegroup implies
This is the pseudorthogonal generalization of the Gaillard-Zumino formula
transforms with fractional linear transformations
NOW ARISES THE QUESTION:CAN WE EXTEND ALL THIS IN PRESENCE OF FERMIONS?
THE ANSWER IS YES!WE HAVE TO USE ORTHOSYMPLECTIC EMBEDDINGS AND WE ARRIVE AT ORTHOSYMPLECTIC FRACTIONAL LINEAR TRANSFORMATIONSWITH SUPERMATRICES
barred index= fermionunbarred= boson
If supercoset manifold
Each block A,B,C,D is by itself a supermatrix
The subalgebra
is diagonally embedded in the chosen basis
We have seen that the D=2 -models with twisted scalars can be extended to the Bose/Fermi case
The catch is the orthosymplectic embedding In the Bose case we have interesting cases
of models coming from dimensional reduction
In these models the twisted scalars can be typically eliminated by a suitable duality
In this way one discovers bigger symmetries
Can we extend this mechanism also to the Bose/Fermi case??
The two reductions are: Ehlers Maztner Missner
The resulting lagrangians are related by a duality transformation
CONFORMAL GAUGE DUALIZATION OF VECTORS TO SCALARS
D=4
D=3
D=2
Liouville field SL(2,R)/O(2) - model
+
D=4
D=3
D=2
CONFORMAL GAUGE
NO DUALIZATION OF VECTORS !!
Liouville field SL(2,R)/O(2) - model
DIFFERENT SL(2,R) fields non locally related
D=4
D=2
Universal,
comes
from Gravity
Comes from vectors in D=4
Symplectic metric in d=2 Symplectic metric in 2n dim
The twisted scalars of MM lagrangian come from the vector fields in D=4.
The Ehlers lagrangian is obtained by dualizing the twisted scalars to normal scalars.
The reason why the Lie algebra is enlarged is because there exist Lie algebras which whose adjoint decomposes as the adjoint of the D=4 algebra plus the representation of the vectors
N=8 E8(8)
N=6 E7(-5)
N=5 E6(-14)
N=4
SO(8,n+2)
N=3
SU(4,n+1)
D=4
E7(7)
SO*(12)
SU(1,5)
SL(2,R)£SO(6,n)
SU(3,n) £ U(1)Z
E9(9)
E7
E6
SO(8,n+2)
D=3 D=2
+ twisted superscalars
Analogue of G4
Analogue of SL(2,R) (Ehlers)
The Ehlers G3 supergroup
The fermionic dualities introduced by Berkovits and Maldacena and other can all be encoded as particular cases of the present orthosymplectic scheme.
The enlargement mechanism can be applied to physical interesting cases?
Are there hidden supersymmetric extension of the known dualities groups of supergravity?