1. Pietro Frè Talk at SQS 09 DUBNA arXiv:0906.2510arXiv:0906.2510 Theory of Superdualities and the Orthosymplectic Supergroup Authors: Pietro Fré, Pietro Antonio Grassi, Luca Sommovigo, Mario TrigiantePietro FréPietro Antonio GrassiLuca Sommovigo Mario Trigiante

2.  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

3. The general form of a bosonic D=4 supergravity Lagrangian For N>2 obligatory For N<3 possible

4. The symplectic embedding

5. WHAT IS THE MATRIX ? It is the Cayley matrix which by conjugation realizes the isomorphism The Gaillard Zumino Master Formula

6. There are fields of two kinds Peccei-Quin symmetries   !   + c  Generalized electric/magnetic duality rotations are performed on the twisted scalars  

7. Embedding of the coset representative Embedding of the group implies

8. This is the pseudorthogonal generalization of the Gaillard-Zumino formula

9. 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 TRANSFORMATIONS WITH SUPERMATRICES

10. barred index= fermion unbarred= boson If supercoset manifold

13. Each block A,B,C,D is by itself a supermatrix

14. The subalgebra is diagonally embedded in the chosen basis

15.  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??

16.  The two reductions are:  Ehlers  Maztner Missner  The resulting lagrangians are related by a duality transformation

17. CONFORMAL GAUGE DUALIZATION OF VECTORS TO SCALARS D=4 D=3 D=2 Liouville field SL(2,R)/O(2)  - model +

18. 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

19. D=4 D=2

20. Universal, comes from Gravity Comes from vectors in D=4 Symplectic metric in d=2Symplectic metric in 2n dim

21.  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

22. N=8 E 8(8) N=6 E 7(-5) N=5 E 6(- 14) N=4 SO(8,n+2) N=3 SU(4,n+1) D=4 E 7(7) SO*(12) SU(1,5) SL(2,R)£SO(6,n) SU(3,n) £ U(1) Z E 9(9) E7E7 E6E6 SO(8,n+2)  D=3D=2

23. + twisted superscalars

24. Analogue of G 4 Analogue of SL(2,R) (Ehlers ) The Ehlers G 3 supergroup

25.  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?