ℛ2 supergravity
JHE
R2 supergravity
Sergio Ferrara 0 1 3 5 6 7 8 9
Alex Kehagias 0 1 2 3 8 9
Massimo Porrati 0 1 3 4 7 8 9
0 Los Angeles , CA 90095-1547 , U.S.A
1 Via Enrico Fermi 40 , I-00044 Frascati , Italy
2 Physics Division, National Technical University of Athens
3 CH 1211 , Geneva 23 , Switzerland
4 CCPP, Department of Physics
5 INFN - Laboratori Nazionali di Frascati
6 Department of Physics and Astronomy, University of California
7 Physics Department, Theory Unit , CERN
8 Open Access , c The Authors
9 15780 Zografou , Athens , Greece
We formulate R2 pure supergravity as a scale invariant theory built only in terms of superfields describing the geometry of curved superspace. The standard supergravity duals are obtained in both “old” and “new” minimal formulations of auxiliary fields. These theories have massless fields in de Sitter space as they do in their non supersymmetric counterpart. Remarkably, the dual theory of R2 supergravity in the new minimal formulation is an extension of the Freedman model, describing a massless gauge field and a massless chiral multiplet in de Sitter space, with inverse radius proportional to the Fayet-Iliopoulos term. This model can be interpreted as the “de-Higgsed” phase of the dual companion theory of R + R2 supergravity.
Superspaces; Supergravity Models
1 Introduction 2 3 4
R2 supergravity in the old minimal formulation
Scale invariant supergravity
Scale invariant matter couplings
R2 supergravity in the new minimal formulation
Recently, various authors [1–3] considered pure R2 theories of gravity coupled to
matter. These theories are particularly interesting also in regard to cosmology because they
naturally accommodate for de Sitter universes.
While demanding conformal invariance
(Weyl local gauge symmetry) would require spin two ghosts arising from the Weyl square
term [4, 5], the rigid scale invariant R2 theory propagates only physical massless modes in
de Sitter space, in contrast with the R + R2 theory, which has in addition a Minkowski
phase with a massive scalar, the inflaton. An Einstein term is then obtained through
quantum effects as substantiated by the analysis of [1]. Further restrictions that follow from the
supersymmetric extensions of these theories are the aim of the present investigation. In
particular, in this note we implement the analysis of [1] by requiring that pure R2
supergravity be effectively derived solely in terms of the geometry of curved superspace. This
poses severe restrictions on the dual standard supergravity theory which, in fact, cannot be
an arbitrary scale invariant theory of supergravity. For example, we find that only certain
cases are possible among the ones worked out in [1]. Moreover, one conformally coupled
chiral superfield (that we call S) and another one that we call the chiral superfield T , are
not matter fields but have a pure gravitational origin. In fact, out of the three (unique)
suggested forms of superpotential in [1], only a certain linear combination of T S and S3
can arise; T 3/2 alone is also possible. This result parallels the same analysis made in the
R + R2 theory [6, 7]. Also, the T 3/2 theory, where only the T field is present, is not an
ory originally investigated in [8–13]. The latter has in fact an anti-de Sitter rather than
a de Sitter phase. All these theories have instabilities in some scalar direction and the
problem caused by these instabilities is similar to the one found in the context of R + R2
supergravity [13–16, 18–38], which was solved in [15]. Even more interesting is the analysis
in the new minimal formulation [39, 40]. Here the dual supergravity R + R2 theory is a
gauge theory in the Higgs phase [17–20]. The de-Higgsed phase corresponds to the pure
the gauge coupling goes to zero. This theory is in fact the extension of the Freedman
model [41], where a massless vector multiplet with a Fayet-Iliopoulos (FI) term gives rise
to a positive cosmological constant. Here there is an additional massless chiral field, dual
to the antisymmetric tensor auxiliary field that has become dynamical.
The paper is organized as follows. In section 2 we present the superconformal rules
needed for our analysis and we discuss the pure R2 in the old minimal formulation of the
In section 3 we describe pure R2 supergravity in the new minimal formulation; we conclude
in section 4.
R2 supergravity in the old minimal formulation
For our convenience we report here some rules of superconformal tensor calculus that will
be useful in order to go from the R2 theory to its standard supergravity form. These rules
are explained in [42], and also in [13, 21].
Superconformal fields are denoted by their Weyl weight w and chiral weight n. So we
will use the notation Xw,n and we will only consider scalar superfields. The basic operator
of a chiral superfield f of weight (0, 0) is
denotes, as usual, the standard D- and F-term density formulae of conformal supergravity,
for a real superfield O with scaling weight 2 and vanishing (...truncated)