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Theory of Superdualities in D=2

Pietro Frè Talk at SQS 09 DUBNA. Theory of Superdualities in D=2. arXiv:0906.2510 Theory of Superdualities and the Orthosymplectic Supergroup Authors : Pietro Fré , Pietro Antonio Grassi , Luca Sommovigo , Mario Trigiante. Duality & Dualities.

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Theory of Superdualities in D=2

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  1. Pietro Frè Talk at SQS 09 DUBNA TheoryofSuperdualities in D=2 arXiv:0906.2510 Theory of Superdualities and the Orthosymplectic Supergroup Authors: Pietro Fré, Pietro Antonio Grassi, Luca Sommovigo, Mario Trigiante

  2. Duality & Dualities • There are dualitysymmetriesoffieldequations + Bianchi identities • There are activedualitiesthattransformonelagrangianintoanother. • In D=4 allBosedualities are symplecticSp(2n,R) • In D=2 allBosedualities are pseudorthogonal SO(m,m) • In D=2 we can constructsuperdualitiesofOsp(m,m|4n) applyingtoBose/Fermi -models

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

  4. The symplectictransformations Thesymplectic embedding

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

  6. The D=2 bosonic-model There are fields of two kinds Generalized electric/magnetic duality rotations are performed on the twisted scalars  Peccei-Quin symmetries ! + c

  7. The pseudo-orthogonalembedding Embedding of the group implies Embedding of the coset representative

  8. The SO(m,m) periodmatrix This is the pseudorthogonal generalization of the Gaillard-Zumino formula

  9. and under G the periodmatrix.. 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. D=2 Bose/Fermi -models barred index= fermion unbarred= boson If supercoset manifold

  11. Extensionof G/H (super) symmetries

  12. Super Dualities:

  13. The orthosymplecticembedding:catch of the method….. Each block A,B,C,D is by itself a supermatrix

  14. The super GZ formula The subalgebra is diagonally embedded in the chosen basis

  15. Summaryofwhatwelearnt so far.. • Wehaveseenthat the D=2 -modelswithtwistedscalars can beextendedto the Bose/Fermi case • The catch is the orthosymplecticembedding • In the Bose case wehaveinterestingcasesofmodelscomingfromdimensionalreduction • In thesemodels the twistedscalars can betypicallyeliminatedby a suitableduality • In this way onediscoversbiggersymmetries • Can weextendthismechanismalsoto the Bose/Fermi case??

  16. There are twotypesofdimensionalreductions D=4 ! D=3 ! D=2 • The two reductions are: • Ehlers • Maztner Missner • The resulting lagrangians are related by a duality transformation

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

  18. D=4 D=3 CONFORMAL GAUGE D=2 Liouville field SL(2,R)/O(2) - model Matzner&Misner reduction of pure gravity DIFFERENT SL(2,R) fields non locally related NO DUALIZATION OF VECTORS !!

  19. D=4 D=2 General Matzner&Misner reduction (P.F. Trigiante, Rulik e Gargiulo 2005)

  20. Universal, comes from Gravity Comes from vectors in D=4 Symplectic metric in d=2 Symplectic metric in 2n dim Structure of the Duality Algebra in D=3(P.F. Trigiante, Rulik and Gargiulo 2005)

  21. So what? • 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. D=4 D=3 D=2 N=8 E7(7) E8(8) E9(9) SO*(12) E7(-5) N=6 E7 N=5 SU(1,5) E6(-14) E6 N=4 SL(2,R)£SO(6,n) SO(8,n+2) SO(8,n+2) N=3 SU(3,n) £ U(1)Z SU(4,n+1)

  23. A superexample + twisted superscalars

  24. The embeddingwhichsolves the problem…. Analogue of G4 Analogue of SL(2,R) (Ehlers) The Ehlers G3supergroup

  25. Conclusions • 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?

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