D = 3 unification of curious supergravities

We consider the dimensional reduction to D = 3 of four maximal-rank super-gravities which preserve minimal supersymmetry in D = 11, 7, 5 and 4. Such “curious” theories were investigated some time ago, and the four-dimensional one corresponds to an N=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=1 $$\end{document} supergravity with 7 chiral multiplets spanning the seven-disk manifold. Recently, this latter theory provided cosmological models for α-attractors, which are based on the disk geometry with possible restrictions on the parameter α. A unified picture emerges in D = 3, where the Ehlers group of General Relativity merges with the S-, T- and U-dualities of the D = 4 parent theories.


Introduction
Among compactifications of D = 11 supergravity on a 7-manifold to D = 4, an interesting N = 1 theory emerges, whose spectrum consists of seven chiral (Wess-Zumino) multiplets living in the seven-disk manifold SL(2, R) U(1)

⊗7
. (1.1) This theory, proposed in [1] has some peculiar properties. It is the smallest member of a family of four "left curious" supergravities, defined in D = (11,7,5,4) dimensions, having a scalar manifold of (maximal) rank (0, 4, 6, 7), respectively, and endowed with a minimal number ν of supersymmetries in the corresponding dimensions, ν = (32, 16,8,4), respectively. Such theories couple naturally to supermembranes and admit these membranes as solutions. In [7] the seven-disk manifold (1.1) was considered as providing possible restrictions on the parameter α of the cosmological α-attractors models for inflation, depending on the embeddings of the single one-disk into (1.1). When compactified on a 7-manifold X 7 with independent Betti numbers (b 0 , b 1 , b 2 , b 3 ) = (b 7 , b 6 , b 5 , b 4 ), the number of fields of spin s = (2, 3/2, 1, 1/2, 0) in the resulting D = 4 supergravity is given by n s = (b 0 , b 0 + b 1 , b 1 + b 2 , b 2 + b 3 , 2b 3 ), and we may loosely associate Betti numbers with any supergravity with n s fields of spin s, whether or not manifolds with these Betti numbers actually exist. We may then define a generalized mirror transformation [1]
In the special case X 7 = X 6 × S 1 , ρ = χ and the two symmetries coincide. Given the unusual properties and possible cosmological applications of these curious supergravities, in the present note we give a D = 3 three-way unified picture in terms of 1) compactifications of M -theory in terms of toroidal moduli; 2) dimensional reduction of the four curious supergravities D = (11,7,5,4)  pertaining to the D = 4 Ehlers group SL(2, R) Ehlers .
The paper is organized as follows.
In section 2 we recall the embedding of [SL(2, R)] ⊗8 into E 8 (8) . In section 3 we give an interpretation of the four curious supergravities in terms of sequential reductions of Mtheory on an eight-manifold with only toroidal moduli of T 8 , T 4 ×T 4 , and T 2 ×T 2 ×T 2 ×T 2 (" M -theoretical path"). Then, in section 4 we consider the so-called "Ehlers path", by compactifying these theories from D = 4 to D = 3. Finally, section 5 contains some concluding remarks.

The M -theory path
The first path starts from M -theory (or, more appropriately, N = 1, D = 11 supergravity), and performs iterated compactifications on tori T 8 , T 4 × T 4 , and on T 2 × T 2 × T 2 × T 2 ; this corresponds to the following chain of maximal and symmetric embeddings: Each step of this chain has an interpretation in terms of truncations of the massless spectrum of M -theory dimensionally reduced to D = 3, such as to preserve N = 16, 8, 4, 2 local supersymmetries. As we discuss below, the last three are obtained keeping only the geometric moduli of the tori T 8 , T 4 × T 4 and T 2 × T 2 × T 2 × T 2 , respectively. It is worth here recalling that the classical moduli space of a d-dimensional torus is (I, J = 1, . . . , d)

The Ehlers path
The second path yielding the N = 2, D = 3 supergravity theory with scalar manifold (1.8) starts with the so-called Ehlers embedding (cfr. e.g. [17], and refs. therein) for maximal supergravity in D = 4 → D = 3, and then proceeds with a chain of maximal, symmetric and rank-preserving embeddings which has already been considered in [7,14,18]: Since this path, which we name Ehlers path, starts with a D = 4 → D = 3 dimensional reduction, it is immediate to realize that the D = 3 scalar manifolds given in (3.6), (3.8), (3.10) and (3.12) are nothing but the dimensional reduction of the D = 4 cosets of N = 8, 4, 2, 1 curious supergravities with rank-7 scalar manifolds (after dualization; cfr. table XVIII of [1]).

U(1)
, spanned by the axio-dilaton given by the S 1 -radius of compactification and by the dualization of the corresponding Kaluza-Klein vector. In other words, the added SL(2,R) Ehlers 2. The chain of embeddings (4.1)-(4.4) has been used in [18] (also cfr. [14]) to study the tripartite entanglement of seven qubits inside E 7 . Moreover, it was recently exploited in [7] in order to obtain the N = 1, D = 4 theory with 7 WZ multiplet given in the fourth line of (4.5).
3. The maximal and symmetric embedding (4.2) corresponds to the truncation of maximal D = 4 supergravity to half-maximal supergravity coupled to 6 matter (vector) multiplets: .
(4.8) This step is non-trivial for what concerns the retaining of an N = 1 local supersymmetry in the gravity theory with non-linear sigma model given by (1.8). Besides the necessary truncation of the N = 1 gravitino multiplet coming from the supersymmetric N = 2 → N = 1 reduction of the N = 2 gravity multiplet, one has to truncate all N = 1 vector multiplets coming from the supersymmetry reduction of the three N = 2 vector multiplets; furthermore, a truncation of half of the N = 1 chiral multiplets stemming from the supersymmetry reduction of the four N = 2 hypermultiplets must be performed. This last step is particularly challenging for the consistency with local N = 1 supersymmetry, which is however granted by the results 6 in [23] (also cfr. [24]); see, in particular, the discussion around eq. (6.145) therein.
However, notwithstanding the first step (4.1) which seems to single out the D = 4 Ehlers group SL(2, R) Ehlers , a complete equivalence between the two paths is reached at their final steps. It would be worth pursuing an E 11 interpretation [25] of these four maximal rank theories preserving minimal supersymmetry in D = 11, 7, 5, 4.