D = 3 unification of curious supergravities

Published for SISSA by Springer Received: November 1, 2016 Accepted: December 24, 2016 Published: January 9, 2017 D = 3 unification of curious supergravities a Theoretical Physics, Blackett Laboratory, Imperial College London, London SW7 2AZ, U.K. b Mathematical Institute University of Oxford, Andrew Wiles Building, Woodstock Road, Radcliffe Observatory Quarter, Oxford, OX2 6GG, U.K. c Theoretical Physics Department, CERN, CH-1211 Geneva, Switzerland d INFN — Laboratori Nazionali di Frascati, Via Enrico Fermi 40, I-00044 Frascati, Italy e Department of Physics and Astronomy and Mani L. Bhaumik Institute for Theoretical Physics, UCLA, Los Angeles CA 90095-1547, U.S.A. f Museo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi”, Via Panisperna 89A, I-00184, Roma, Italy g Dipartimento di Fisica e Astronomia “Galileo Galilei”, Università di Padova and INFN, Sez. di Padova, Via Marzolo 8, I-35131 Padova, Italy E-mail: , , Abstract: 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. Keywords: Supergravity Models, Extended Supersymmetry, M-Theory, Differential and Algebraic Geometry ArXiv ePrint: 1610.08800 Open Access, c The Authors. Article funded by SCOAP3 . doi:10.1007/JHEP01(2017)023 JHEP01(2017)023 M.J. Duff,a,b S. Ferrarac,d,e and A. Marranif,g,c Contents 1 2 E8(8) and the eight-disk manifold 3 3 The M -theory path 3 4 The Ehlers path 5 5 Conclusion 7 1 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) ⊗7 . (1.1) U(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 (b0 , b1 , b2 , b3 ) = (b7 , b6 , b5 , b4 ), the number of fields of spin s = (2, 3/2, 1, 1/2, 0) in the resulting D = 4 supergravity is given by ns = (b0 , b0 + b1 , b1 + b2 , b2 + b3 , 2b3 ), and we may 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] (b0 , b1 , b2 , b3 ) → (b0 , b1 , b2 − ρ/2, b3 + ρ/2), under which ρ X 7
:= 7 X (−1)k+1 (k + 1) bk = 7b0 − 5b1 + 3b2 − b3 , (1.2) (1.3) k=0 changes sign: ρ → −ρ –1– (1.4) JHEP01(2017)023 1 Introduction (In the special case b1 = 0, ρ reversal reduces to the reflection symmetry of G2 manifolds defined by Joyce [2, 3]). Generalised self-mirror theories are here defined to be those for which ρ vanishes. Under further toroidal compactification to D = 4, the four curious supergravities have N = 8, 4, 2, 1 supersymmetries and Betti numbers (b0 , b1 , b2 , b3 ) = (1, N − 1, n, 3n − 5N + 12) and thus are all self-mirror. (The N = 2 theory is just the self-mirror stu model [4, 5].) Similarly, we may define a generalized mirror transformation for 6-manifolds X 6 [1] with independent Betti numbers (c0 , c1 , c2 , c3 ) = (c6 , c5 , c4 , c3 ) (1.5) 6 X
c X 6 := (−1)k ck = 2c0 − 2c1 + 2c2 − c3 (1.6) χ → −χ (1.7) under which k=0 changes sign: (In the special case c1 = 0, χ reversal reduces to ordinary mirror symmetry of CalabiYau [6]). Generalised self-mirror theories are here defined to be those for which χ vanishes. 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) 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 eightdisk manifold   SL(2, R) ⊗8 , (1.8) U(1) which can be regarded as the unification of S-, T - and U - dualities of the N = 1, D = Ehlers 4 corresponding theory mentioned above, augmented by the disk manifold SL(2,R) U(1) 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 E8(8) . In section 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 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. –2– JHEP01(2017)023 (c0 , c1 , c2 , c3 ) → (c0 , c1 , c2 − χ/2, c3 + χ) 2 E8(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 E ⊃ [sl(2)]⊕r , r := rank(E). (2.1) 3 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: E8(8) ⊃ SO(8, 8) ⊃ SO(4, 4) × SO(4, 4) ⊗8 ⊃ [SO(2, 2)]⊗4 ∼ = [SL(2, R)] . (3.1) (3.2) (3.3) 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 (...truncated)