D = 3 Unification of Curious Supergravities

We consider the dimensional reduction to D = 3 of four maximal-rank supergravities 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 supergravity with 7 chiral multiplets spanning the seven-disk manifold. Recently, this latter theory provided cosmological models for alpha-attractors, which are based on the disk geometry with possible restrictions on the parameter alpha. 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.

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) to D = 3; 3) dimensional reduction of 4 curious supergravities in D = 4 to D = 3. In particular, the resulting N = 2, D = 3 supergravity has the scalar manifold given by the eight-disk manifold which can be regarded as the unification of S-, T -and U -dualities of the N = 1, D = 4 corresponding theory mentioned above, augmented by the disk manifold SL(2,R) Ehlers pertaining to the D = 4 Ehlers group SL(2, R) Ehlers .
The paper is organized as follows. In Sec. 2 we recall the embedding of [SL(2, R)] ⊗8 into E 8 (8) . In Sec. 3 we give an interpretation of the four curious supergravities in terms of sequential reductions of M -theory 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 Sec. 4 we consider the so-called "Ehlers path", by compactifying these theories from D = 4 to D = 3. Finally, Sec. 5 contains some concluding remarks.

E 8(8) and the Eight-Disk Manifold
Almost all exceptional Lie algebras E enjoy a rank-preserving (generally non-maximal nor symmetric) embedding of the type This holds for E = e 8 , e 7 , f 4 , g 2 , with r = 8, 7, 4, 2, respectively. The unique exception 1 is provided by the rank-6 exceptional algebra e 6 , which embeds only [sl (2)] ⊕4 , and not [sl (2)] ⊕6 . In the following treatment, we will focus on the maximally non-compact (i.e., split) real form e 8(8) of e 8 , considering it at the Lie group level ( More specifically, starting from 2 E 8(8) we will analyze two paths yielding the same N = 2, D = 3 supergravity theory 3 , coupled to 8 matter multiplets, whose scalars coordinatize the completely factorized rank 4 -8 Hodge-Kähler symmetric, eight-disk manifold (1.8).

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) whereas the quantum one (in a stringy sense) reads , spanned by g IJ = g (IJ) and B IJ = B [IJ] . : namely a compactification retaining both geometric (g IJ , A µIJ ; ) and non-geometric (g µI , A IJK ) moduli of T 8 , down to maximal supergravity in D = 3 [13] (I, J, K = 1, ..., 8, and µ = 0, 1, 2); note that the 128 bosonic massless degrees of freedom can be organized in SO (8) irreprs. as follows : where the 1-form A µIJ = A µ[IJ] (playing the role of the "M -theoretical B-field") gets then dualized to scalar fields A IJ in D = 3. The next step corresponds to the first, maximal and symmetric embedding (3.1), which amounts to retaining only the geometric moduli of T 8 (i.e., to setting g µI = 0 = A IJK in the bosonic sector), thus giving rise upon compactification to half-maximal supergravity coupled to n = 8 matter multiplets in D = 3 : .
2. The chain of embeddings (4.1)-(4.4) has been used in [16] (also cfr. [12]) to study the tripartite entanglement of seven qubits inside E 7 . Moreover, it was recently exploited in [6] 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 : SO(4)×SO (4) ; since this latter is the c-map [18] of the corresponding vector-multiplets' projective special Kähler manifold , this model is self-mirror (also cfr. e.g. [20]) : . (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 [21] (also cfr. [22]); see, in particular, the discussion around Eq. (6.145) therein.