D = 3 unification of curious supergravities

Journal of High Energy Physics, Jan 2017

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 \( \mathcal{N}=1 \) 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.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

http://link.springer.com/content/pdf/10.1007%2FJHEP01%282017%29023.pdf

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 CH-1211 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, I-35131 Padova , Italy 7 Universita` di Padova and INFN , Sez. di Padova 8 Via Panisperna 89A, I-00184, Roma , Italy 9 Mani L. Bhaumik Institute for Theoretical Physics , UCLA 10 Via Enrico Fermi 40 , I-00044 Frascati , Italy 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 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. Supergravity Models; Extended Supersymmetry; M-Theory; Differential and - 3 unification of curious supergravities 1 Introduction 2 3 4 E8(8) and the eight-disk manifold The M -theory path The Ehlers path living in the seven-disk 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 seven-disk manifold (1.1) was considered as providing possible restricon the embeddings of the single one-disk into (1.1). a 7-manifold 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 self-mirror theories are here defined to be those for self-mirror 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 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 Almost all exceptional Lie algebras E enjoy a rank-preserving (generally non-maximal 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 non-compact (i.e., split) context of D = 3 supergravity theories. the completely factorized rank4-8 Hodge-K¨ahler symmetric, eight-disk 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 so-called 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−+→non-geom) 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 half-maximal 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 eight-disk 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 so-called “black-hole/qubit” correspondence (cfr. e.g. [16] for an introduction and a view of the curious supergravities. The Ehlers path starts with the so-called Ehlers embedding (cfr. e.g. [17], and refs. therein) for maximal and rank-preserving 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 Maxwell-Einstein systems coupled to non-linear sigma models ([19], thereby including the c-map [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 axio-dilaton given by the S1-radius 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 seven-disk 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 rank-preserving. 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 half-maximal 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 c-map [20] of the corresponding vector, this model is self-mirror (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, self-mirror 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, self-mirror stu model, (B,F )=(32,32) N =1,D=4,nc=7,nv=0, (B,F )=(16,16) metry in the gravity theory with non-linear 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 non-geometric 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 rank-7 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 rank-8 Hodge-K¨ahler symmetric (from the T 2-factors of the 8-dimensional internal manifold), the SL(2, R)’s of S-duality compactified on a suitable 7-dimensional manifold with G2-structure. 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 full-fledged symmetry of the fully factorised rank-7 Hodge-K¨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 INFN-CSN4-GSS. 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 (CC-BY 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 7-manifolds with holonomy G2. I, [3] D. Joyce, Compact Riemannian 7-manifolds with holonomy G2. II, Nucl. Phys. B 459 (1996) 125 [hep-th/9508094] [INSPIRE]. string triality, Phys. Rev. D 54 (1996) 6293 [hep-th/9608059] [INSPIRE]. arXiv:1610.04163 [INSPIRE]. group, Amer. J. Math. 81 (1959) 973. AMS-SIAM 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, Four-dimensional string-string-string 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, Seven-disk manifold, α-attractors and B-modes, [8] B. Kostant, The three-dimensional sub-group 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, Kac-Moody symmetry of gravitation and supergravity theories, invited talk at physics, Chicago U.S.A. July 6-16 1982 [INSPIRE]. [12] B. de Wit, A.K. Tollsten and H. Nicolai, Locally supersymmetric D = 3 nonlinear σ-models, Nucl. Phys. B 392 (1993) 3 [hep-th/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, Three-dimensional supergravity theories, Nucl. Phys. B 228 (1983) 145 [INSPIRE]. [16] L. Borsten, M.J. Duff and P. L´evay, The black-hole/qubit correspondence: an up-to-date review, Class. Quant. Grav. 29 (2012) 224008 [arXiv:1206.3166] [INSPIRE]. [17] S. Ferrara, A. Marrani, M. Trigiante, A. Marrani and M. Trigiante, Super-Ehlers 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 [quant-ph/0609227] [INSPIRE]. [19] P. Breitenlohner, D. Maison and G.W. Gibbons, Four-dimensional black holes from Kaluza-Klein 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 Calabi-Yau spaces, Nucl. Phys. B 332 (1990) 317 [INSPIRE]. [22] S. Ferrara and A. Marrani, N = 8 non-BPS 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 four-dimensions, JHEP 03 (2002) 025 [hep-th/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


This is a preview of a remote PDF: http://link.springer.com/content/pdf/10.1007%2FJHEP01%282017%29023.pdf

M. J. Duff, S. Ferrara, A. Marrani. D = 3 unification of curious supergravities, Journal of High Energy Physics, 2017, DOI: 10.1007/JHEP01(2017)023