D = 3 unification of curious supergravities
Received: November
Published for SISSA by Springer
London SW 0 1 6 7 8 9 10
AZ 0 1 6 7 8 9 10
U.K. 0 1 6 7 8 9 10
0 1 6 7 8 9 10
Open Access 0 1 6 7 8 9 10
c The Authors. 0 1 6 7 8 9 10
0 CH1211 Geneva , Switzerland
1 Woodstock Road, Radcliffe Observatory Quarter , Oxford, OX2 6GG, U.K
2 INFN  Laboratori Nazionali di Frascati
3 Mathematical Institute University of Oxford , Andrew Wiles Building
4 Theoretical Physics Department , CERN
5 Theoretical Physics, Blackett Laboratory, Imperial College London
6 Via Marzolo 8, I35131 Padova , Italy
7 Universita` di Padova and INFN , Sez. di Padova
8 Via Panisperna 89A, I00184, Roma , Italy
9 Mani L. Bhaumik Institute for Theoretical Physics , UCLA
10 Via Enrico Fermi 40 , I00044 Frascati , Italy
We consider the dimensional reduction to D = 3 of four maximalrank supergravities which preserve minimal supersymmetry in D = 11, 7, 5 and 4. Such “curious” theories were investigated some time ago, and the fourdimensional one corresponds to an N = 1 supergravity with 7 chiral multiplets spanning the sevendisk manifold. Recently, this latter theory provided cosmological models for αattractors, 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.
Supergravity Models; Extended Supersymmetry; MTheory; Differential and

3 unification of curious supergravities
1 Introduction 2 3 4
E8(8) and the eightdisk manifold
The M theory path
The Ehlers path
living in the sevendisk manifold
SL(2, R) ⊗7
This theory, proposed in [1] has some peculiar properties. It is the smallest member of a
a scalar manifold of (maximal) rank (0, 4, 6, 7), respectively, and endowed with a minimal
tively. Such theories couple naturally to supermembranes and admit these membranes as
solutions. In [7] the sevendisk manifold (1.1) was considered as providing possible
restricon the embeddings of the single onedisk into (1.1).
a 7manifold
with independent
Betti numbers
loosely associate Betti numbers with any supergravity with ns fields of spin s, whether or
not manifolds with these Betti numbers actually exist. We may then define a generalized
mirror transformation [1]
ρ X7 := X (−1)k+1 (k + 1) bk = 7b0 − 5b1 + 3b2 − b3,
defined by Joyce [2, 3]). Generalised selfmirror theories are here defined to be those for
selfmirror stu model [4, 5].) Similarly, we may define a generalized mirror transformation
c X6 := X (−1)k ck = 2c0 − 2c1 + 2c2 − c3
Given the unusual properties and possible cosmological applications of these curious
1) compactifications of M theory in terms of toroidal moduli;
SL(2, R) ⊗8
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.
an interpretation of the four curious supergravities in terms of sequential reductions of M
theory on an eightmanifold 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 socalled “Ehlers path”, by
Almost all exceptional Lie algebras E enjoy a rankpreserving (generally nonmaximal nor
symmetric) embedding of the type
E ⊃ [sl(2)]⊕r , r := rank(E ).
[sl(2)]⊕6.
In the following treatment, we will focus on the maximally noncompact (i.e., split)
context of D = 3 supergravity theories.
the completely factorized rank48 HodgeK¨ahler symmetric, eightdisk manifold (1.8).
The M theory path
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:
E8(8) ⊃ SO(8, 8)
⊃ SO(4, 4) × SO(4, 4)
⊃ [SO(2, 2)]⊗4 ∼= [SL(2, R)]⊗8 .
Each step of this chain has an interpretation in terms of truncations of the massless
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
Md := R+ × SL(d, R)
, spanned by gIJ = g(IJ),
1It should be here pointed that e6 stands on its own among exceptional Lie algebras for at least another
reason: it is the unique exceptional Lie algebra which does not embed maximally its principal (Kostant’s)
sl(2)P [8] algebra. Indeed, while all Lie algebras maximally embed sl(2)P (e8 and e7 actually maximally
embed three and two sl(2)’s , respectively), e6 embeds its sl(2)P through the chain of maximal embeddings
e6 ⊃ f4 ⊃ sl(2)P (in other words, e6 ”inherits” the sl(2)P of f4).
2E8(8) belongs to the socalled exceptional En(n)sequence [9–11] of symmetries of maximal supergravities
in 11 − n dimensions.
4The rank of a manifold is defined as the maximal dimension (in R) of a flat (i.e., with vanishing Riemann
tensor), totally geodesic submanifold (see e.g. §6, page 209 of [13]).
whereas the quantum one (in a stringy sense) reads
Md :=
SO(d) × SO(d)
, spanned by gIJ = g(IJ) and BIJ = B[IJ].
The first, starting step of the M theoretical path (3.1)–(3.3) corresponds to:5
T 8 (geom−+→nongeom) N = 16, D = 3 :
SO(8) irreprs. as follows:
dualized to scalar fields AIJ in D = 3.
The next step corresponds to the first, maximal and symmetric embedding (3.1), which
the bosonic sector), thus giving rise upon compactification to halfmaximal supergravity
coupled to n = 8 matter multiplets in D = 3:
T 8 −(g→eom) N = 8, D = 3, n = 8 :
SO(8) × SO(8)
The subsequent maximal and symmetric embedding (3.2) corresponds to a
compactii′, j′ = 5, . . . , 8):
thus giving rise to the following N = 4, D = 3 supergravity model:
T 4×T−4→(geom) N = 4, D = 3, n = 8 :
SO(4) × SO(4)
SO(4) × SO(4)
The last step is given by the maximal and symmetric embedding (3.3), corresponding
to a compactification on T 2 × T 2 × T 2 × T 2 retaining only the related geometric moduli
the eightdisk manifold (1.8):
T 2×T 2×T−2→×T 2 (geom) N = 2, D = 3 :
SL(2, R) ⊗8
Some comments are in order.
5”B” and “F ” denote the number of bosonic and fermionic massless degrees of freedom throughout.
1. All symmetric scalar manifolds in (3.6), (3.8), (3.10) and (3.12) have rank 8, as
a consequence of the fact that all embeddings of the chain (3.1)–(3.3) are
rankthe four curious supergravities, studied in [1] and mentioned in section 1. These
numbers), R (reals) denote the four Hurwitz division algebras),with scalar manifolds
of rank 0, 4, 6, 7 respectively. As observed in [1], such N
= 8, 4, 2, 1, D = 4
cuplane, and hence they admit a division algebraic interpretation consistent with the
socalled “blackhole/qubit” correspondence (cfr. e.g. [16] for an introduction and a
view of the curious supergravities.
The Ehlers path
starts with the socalled Ehlers embedding (cfr. e.g. [17], and refs. therein) for maximal
and rankpreserving embeddings which has already been considered in [7, 14, 18]:
E8(8) ⊃ E7(7) × SL(2, R)Ehlers
⊃ SO(6, 6) × SL(2, R)Ehlers × SL(2, R)
⊃ SO(4, 4) × [SL(2, R)]⊗2 × SL(2, R)Ehlers × SL(2, R)
⊃ [SL(2, R)]⊗8
tion; cfr. table XVIII of [1]).
the study of MaxwellEinstein systems coupled to nonlinear sigma models ([19], thereby
including the cmap [20, 21] relating projective special K¨ahler manifolds to quaternionic
manifolds), for N = 1 the dimensional reduction reads
(B, F )=(16, 16) :
SL(2, R) ⊗7
SL(2, R) ⊗8
N =1,D=4,nc=7,nv=0
N =2,D=3,n=8
coupled only to 7 chiral multiplets, with no vectors at all. Therefore, under (spacelike)
enlarged only by a further factor manifold SL(2,R)Ehlers , spanned by the axiodilaton given
by the S1radius of compactification and by the dualization of the corresponding
KaluzaKlein vector. In other words, the added SL(2,R)Ehlers manifold pertains to the two degrees
any degree of freedom): as mentioned in section 1, the sevendisk manifold (1.1) [1, 7] gets
Some observations are:
1. All symmetric scalar manifolds in (4.6), (4.7) and (4.8) have rank 7, as a consequence
of the fact that all embeddings of the chain (4.1)–(4.4) are rankpreserving.
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 E7. Moreover, it was recently exploited
3. The maximal and symmetric embedding (4.2) corresponds to the truncation of
maxfourth line of (4.5).
N =8,D=4, (B,F )=(128,128)
SO(6) × SO(6)
N =4,D=4,n=6, (B,F )=(64,64)
4. The subsequent step (4.3) corresponds to the truncation of halfmaximal D = 4
multiplets’ projective special K¨ahler manifold h SL(2,R) i⊗3
to 4 hypermultiplets, whose quaternionic scalars coordinatize the symmetric scalar
manifold SO(4)×SO(4) ; since this latter is the cmap [20] of the corresponding
vector, this model is selfmirror
(also cfr. e.g. [22]):
SO(6) × SO(6)
N =4,D=4,n=6, (B,F )=(64,64)
SL(2, R) ⊗3
SO(4) × SO(4)
N =2,D=4,nv=3,nH =4, selfmirror stu model, (B,F )=(32,32)
SL(2, R) ⊗3
SO(4) × SO(4)
SL(2, R) ⊗7
N =2,D=4,nv=3,nH =4, selfmirror stu model, (B,F )=(32,32)
N =1,D=4,nc=7,nv=0, (B,F )=(16,16)
metry in the gravity theory with nonlinear sigma model given by (1.8). Besides the
must be performed. This last step is particularly challenging for the consistency with
cfr. [24]); see, in particular, the discussion around eq. (6.145) therein.
curious supergravities (3.6), (3.8), (3.10) and (3.12) with symmetric scalar manifolds of
(maximal) rank 8 in D = 3:
geometric and nongeometric moduli of T 8, and then geometric moduli of T 8, of
T 4 × T 4, and of T 2 × T 2 × T 2 × T 2. This is given by the M theoretical path (3.1)–
(3.3) discussed in section 3.
point 3 of section 3.
2. Toroidal compactification of the four curious supergravities [1] (defined in 11, 7, 5, 4
supergravities with rank7 scalar manifolds (after dualization; cfr. table XVIII of [1]).
This is given by the Ehlers path (4.1)–(4.4) discussed in section 4.
By comparing the two paths (3.1)–(3.3) and (4.1)–(4.4), it is evident that they exhibit
different and features.
The M theoretical path (3.1)–(3.3) is deeply rooted in M theory, and it makes
“octality”, pertaining to the symmetry of the fully factorised rank8 HodgeK¨ahler symmetric
(from the T 2factors of the 8dimensional internal manifold), the SL(2, R)’s of Sduality
compactified on a suitable 7dimensional manifold with G2structure.
unified, and they stand on the same footing.
On the other hand, the Ehlers path (4.1)–(4.4), makes only “septality”, pertaining to
the fullfledged symmetry of the fully factorised rank7 HodgeK¨ahler symmetric coset in
U  dualities get unified.
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 E11 interpretation [25] of these four
16, 8, 4, 2 theories are associated with the 7, 3, 1, 0 quadrangles of the Fano plane and the
dual Fano plane.7
We are grateful to Renata Kallosh, for useful discussions and related collaboration. MJD
is grateful to the Leverhulme Trust for an Emeritus Fellowship and to Philip Candelas
for hospitality at the Mathematical Institute , Oxford. This work was supported by the
STFC under rolling grant ST/G000743/1. The work of SF is supported in part by CERN
TH Dept. and INFNCSN4GSS. AM wishes to thank the CERN Theory Division, for
the kind hospitality during the realization of this work. MJD and SF acknowledge the
hospitality of the GGI institute in Firenze, where this work was completed during the
workshop ‘Supergravity: What Next?’.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
[arXiv:1010.3173] [INSPIRE].
J. Diff. Geom. 43 (1996) 291.
J. Diff. Geom. 43 (1996) 329.
[2] D. Joyce, Compact Riemannian 7manifolds with holonomy G2. I,
[3] D. Joyce, Compact Riemannian 7manifolds with holonomy G2. II,
Nucl. Phys. B 459 (1996) 125 [hepth/9508094] [INSPIRE].
string triality, Phys. Rev. D 54 (1996) 6293 [hepth/9608059] [INSPIRE].
arXiv:1610.04163 [INSPIRE].
group, Amer. J. Math. 81 (1959) 973.
AMSSIAM summer seminar on applications of group theory in physics and mathematical
[1] M.J. Duff and S. Ferrara, Four curious supergravities, Phys. Rev. D 83 (2011) 046007 [4] M.J. Duff, J.T. Liu and J. Rahmfeld, Fourdimensional stringstringstring triality, [5] K. Behrndt, R. Kallosh, J. Rahmfeld, M. Shmakova and W.K. Wong, STU black holes and [6] Y.S. Tung, Essays on mirror manifolds, Int. Press, Hong Kong (1992). [7] S. Ferrara and R. Kallosh, Sevendisk manifold, αattractors and Bmodes, [8] B. Kostant, The threedimensional subgroup and the Betti numbers of a complex simple Lie [9] B. Julia, Group disintegrations, in Superspace and supergravity, S.W. Hawking and M. Rocek eds., Cambridge Univ. Press, Cambridge U.K. (1981) [INSPIRE]. [10] E. Cremmer, Supergravities in 5 dimensions, in Superspace and supergravity, S.W. Hawking and M. Rocek eds., Cambridge Univ. Press, Cambridge U.K. (1981) [INSPIRE]. [11] B. Julia, KacMoody symmetry of gravitation and supergravity theories, invited talk at physics, Chicago U.S.A. July 616 1982 [INSPIRE]. [12] B. de Wit, A.K. Tollsten and H. Nicolai, Locally supersymmetric D = 3 nonlinear σmodels, Nucl. Phys. B 392 (1993) 3 [hepth/9208074] [INSPIRE]. [13] S. Helgason, Differential geometry, Lie groups and symmetric spaces, Academic Press, New York U.S.A. (1978). [14] L. Borsten, D. Dahanayake, M.J. Duff, H. Ebrahim and W. Rubens, Black holes, qubits and octonions, Phys. Rept. 471 (2009) 113 [arXiv:0809.4685] [INSPIRE]. [15] N. Marcus and J.H. Schwarz, Threedimensional supergravity theories, Nucl. Phys. B 228 (1983) 145 [INSPIRE]. [16] L. Borsten, M.J. Duff and P. L´evay, The blackhole/qubit correspondence: an uptodate review, Class. Quant. Grav. 29 (2012) 224008 [arXiv:1206.3166] [INSPIRE]. [17] S. Ferrara, A. Marrani, M. Trigiante, A. Marrani and M. Trigiante, SuperEhlers in any dimension, JHEP 11 (2012) 068 [arXiv:1206.1255] [INSPIRE]. [18] M.J. Duff and S. Ferrara, E7 and the tripartite entanglement of seven qubits, Phys. Rev. D 76 (2007) 025018 [quantph/0609227] [INSPIRE]. [19] P. Breitenlohner, D. Maison and G.W. Gibbons, Fourdimensional black holes from KaluzaKlein theories, Commun. Math. Phys. 120 (1988) 295 [INSPIRE]. [20] S. Cecotti, S. Ferrara and L. Girardello, Geometry of type II superstrings and the moduli of superconformal field theories, Int. J. Mod. Phys. A 4 (1989) 2475 [INSPIRE]. [21] S. Ferrara and S. Sabharwal, Quaternionic manifolds for type II superstring vacua of CalabiYau spaces, Nucl. Phys. B 332 (1990) 317 [INSPIRE]. [22] S. Ferrara and A. Marrani, N = 8 nonBPS attractors, fixed scalars and magic supergravities, Nucl. Phys. B 788 (2008) 63 [arXiv:0705.3866] [INSPIRE]. [23] L. Andrianopoli, R. D'Auria and S. Ferrara, Supersymmetry reduction of N extended supergravities in fourdimensions, JHEP 03 (2002) 025 [hepth/0110277] [INSPIRE]. [24] S. Ferrara, R. Kallosh and A. Marrani, Degeneration of groups of type E7 and minimal coupling in supergravity, JHEP 06 (2012) 074 [arXiv:1202.1290] [INSPIRE]. [25] P. West, A brief review of E theory, Int. J. Mod. Phys. A 31 (2016) 1630043