The symplectic origin of conformal and Minkowski superspaces

Abstract

Supermanifolds provide a very natural ground to understand and handle supersymmetry from a geometric point of view; supersymmetry in $d=3,4,6$ and $10$ dimensions is also deeply related to the normed division algebras. In this paper we want to show the link between the conformal group and certain types of symplectic transformations over division algebras. Inspired by this observation we then propose a new\,realization of the real form of the 4 dimensional conformal and Minkowski superspaces we obtain, respectively, as a Lagrangian supermanifold over the twistor superspace $\mathbb{C}^{4|1}$ and a big cell inside it. The beauty of this approach is that it naturally generalizes to the 6 dimensional case (and possibly also to the 10 dimensional one) thus providing an elegant and uniform characterization of the conformal superspaces.Comment: 15 pages, references added, minor change

Klein and conformal superspaces, split algebras and spinor orbits

“…In [51], the compactified 3-dimensional Minkowski space M 2,1 was constructed, along with its N = 1 supersymmetric extension M 2,1|1 , in terms of a Lagrangian manifold over the twistor space R 4 , by exploiting the Lie group isomorphism Spin(3, 2) ∼ = Sp(4, R). Taking inspiration from the isomorphisms (1.2)-(1.5) and also relying on [41], in [19] a symplectic characterization of the 4-dimensional (compactified and real) Minkowski space M 3,1 and N = 1 Poincaré superspace M 3,1|1 was given, exploiting the Lie group isomorphism Spin(4, 2) ∼ = Sp(4, C). Therein, it was also argued the possibility to extend the approach also to the other critical dimensions D = 6 and 10, thus providing a uniform and elegant description of N = 1 Poincaré superspaces M q+1,1|1 in critical dimensions D = q + 2 in terms of the four normed division algebras A's.…”

“…The present paper is then devoted to the study of the and so(q/2 + 2, q/2 + 2) is the Lie algebra of the Klein-conformal group in D = q + 2. More in detail, in this paper we give an explicit proof and take advantage of the Lie group isomorphism Spin(2, 2) ∼ = SL(2, H s ) and Spin(3, 3) ∼ = Sp(4, C s ), by constructions similar to the ones made in [41] and [19]. While in our treatment the construction and the Lie group isomorphisms analogues of (1.7) and (1.10) are explicitly worked out in those cases, nothing 2 seemingly prevents us from putting forward the conjecture that our approach equally works well in the other critical dimensions with ultrahyperbolic signature, i.e.…”

“…Our approach to M 2,2 and its N = 1 super-extensions will follow closely the one of [19], which in turn developed a procedure exploited in [66,21], in which the complex 4-dimensional Minkowski (super)space was realized inside a complex flag (super)manifold, with the conformal group SL(4, C) acts naturally. It is here worth remarking that this is a more physics-oriented approach, in which superspaces come along with the supergroups describing their supersymmetries; this is to be contrasted to the approach e.g.…”

“…However, sharing the same approach as in [19] and essentially relying on [20] and [21], we would like to point out that in the present investigation we will strive to leave abstract subtleties pertaining to the formal machinery of functor of points on the background, though employing its descriptive power while dealing with T -points of a supergroup or with a superspace.…”

The Segre embedding of the quantum conformal superspace

“…For example, in Ref. [33] a symplectic realization of the chiral conformal superspace is proposed, which can be extended to the 6 and 10 dimensional cases by using matrix groups over quaternions and octonions. Also, a generalization to split signatures (n, n) has been considered recently in Refs.…”