Theory of Superdualities in D=2

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?