Exotic branes in Exceptional Field Theory: the SL(5) duality group

Published for SISSA by Springer Received: November 9, 2017 Revised: June 7, 2018 Accepted: July 22, 2018 Published: August 7, 2018 Ilya Bakhmatov,a,b David S. Berman,c Axel Kleinschmidt,d Edvard T. Musaevd,b and Ray Otsukic a Asia Pacific Center for Theoretical Physics, Postech, Pohang, 37673 Korea b Institute of Physics, Kazan Federal University, Kremlevskaya 16a, Kazan, 420111 Russia c Queen Mary University of London, Centre for Research in String Theory, School of Physics and Astronomy, Mile End Road, London, E1 4NS England d Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut) Am Mühlenberg 1, Potsdam, DE-14476 Germany E-mail: , , , , Abstract: We study how exotic branes, i.e. branes whose tensions are proportional to gs−α , with α > 2, are realised in Exceptional Field Theory (EFT). The generalised torsion of the Weitzenböck connection of the SL(5) EFT which, in the language of gauged supergravity describes the embedding tensor, is shown to classify the exotic branes whose magnetic fluxes can fit into four internal dimensions. By analysing the weight diagrams of the corresponding representations of SL(5) we determine the U-duality orbits relating geometric and nongeometric fluxes. As a further application of the formalism we consider the Kaluza-Klein monopole of 11D supergravity and rotate it into the exotic 6(3,1) -brane. Keywords: M-Theory, p-branes, String Duality ArXiv ePrint: 1710.09740 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP08(2018)021 JHEP08(2018)021 Exotic branes in Exceptional Field Theory: the SL(5) duality group Contents 1 2 Exotic branes in Type II and M-theory 2.1 gs−3 -branes from EFT 2.2 Fluxes of SL(5) EFT 5 6 10 3 Orbits of the SL(5) U-duality group 3.1 Root system of SL(5) 3.2 Orbits inside the 15 representation 3.3 Orbits inside the 40 representation 11 12 13 15 4 Uplift of SUGRA solutions into the SL(5) EFT 4.1 SL(5) theory 4.2 KK6 monopole 4.3 U-duality transformation of KK6-monopole 4.4 Generalised torsion and fluxes 18 18 21 22 24 5 E11 embedding and U-duality orbits 26 6 Discussion 29 7 Conclusions 30 A Conventions A.1 Indices A.2 Metrics, vielbeins and metric determinants 32 32 32 1 Introduction In the seminal paper of Cremmer and Julia it was shown that D = 11 supergravity compactified on a torus Tn results in a d = (11 − n)-dimensional theory that exhibits a hidden symmetry described by the exceptional series En(n) (the split real subgroup of the complexified group) [1–3]. Subsequent to this [4–6] suggested the idea that the compactified d-dimensional theory can be reformulated to make T-dualities manifest. Then a series of works suggested [7–9] that one could make U-dualities manifest as a exceptional field theory (EFT) living on a larger space where the U-duality symmetries appear linearly realised, i.e. as symmetries acting on the coordinates of the extended space that transforms under a particular representation Rn of En(n) . Such a space is parametrized by dim Rn coordinates, n of which –1– JHEP08(2018)021 1 Introduction n En Kn Rn 3 SL(2) × SL(3) SO(3) × SO(2) 3⊕2 4 SL(5) SO(5) 10 M2 5 SO(5, 5) SO(5) × SO(5) 15 M2, M5 6 E6(6) USp(8) 27 M2, M5 7 E7(7) SU(8)/Z2 56 M2, M5, KK6 8 E8(8) Spin(16)/Z2 248 M2, M5, KK6, Exotic Windings correspond to the usual Riemannian coordinates of spacetime, whilst the other dim Rn − n coordinates correspond to wrapping modes of the extended objects of M-theory, i.e. M2-, M5-branes, KK-monopole etc. This coordinate representation Rn differs for each group En(n) and depends on the field content of the resulting theory [10, 11]. Table 1 collects a selection of these. A simpler case comes from compactifying D = 10 type II supergravity on a d-dimensional torus, where the duality group is just the O(d, d) of T-duality. Then the dimension of the extended space becomes doubled RO(d,d) = 2d and the extended coordinates comes from string winding modes. This is what is known as Double Field Theory (DFT) [12–15]. In both DFT and EFT, the geometry of the extended space becomes closely tied to the duality group through the local symmetries of the theory and the EFT action is invariant under the so-called generalised Lie derivative which is described in terms of the projector P to the adjoint representation of the duality group as δ Λ V M = L Λ V M = Λ N ∂ N V M + α n PM N K L ∂ K Λ L V N + λ n ∂ N Λ N V M , (1.1) where the indices M, N = 1, . . . , dim Rn ; the prefactor αn depends on the duality group and λn is weight of the generalised vector V M . The derivative here is taken with respect to the coordinate Ŷ M on the extended space. To close the algebra of such transformations one must impose a condition on the extended space, which may be written as: Y M N KL ∂M f ∂N g = 0. (1.2) This was first constructed for the specific case of SL(5) in [10] and for the general case of Ed,d in [16]. Here f, g are any functions on the extended space and the Y -tensor Y M N KL is constructed from group invariants as described in [11, 16] and further satisfy M N N Y M N KL = −αn PK M L N + βn δK δL + δLM δK , R) Y (M N KL Y R)L P Q − Y (M N P Q δK = 0, Y MP KQ Y QN for P L = (2 − αn )Y d ≤ 5, MN M N M N KL + (Dβn + αn )βn δK δL + (αn − 1)δL δK . (1.3) –2– JHEP08(2018)021 Table 1. A selection of EFT symmetry groups En(n) , along with their maximal compact subgroups Kn , coordinate representation Rn and contributing windings of extended objects. The exotic objects will be discussed in more detail below. where the six coordinates ycd describe membrane winding modes and the four coordinates xa are the usual spacetime coordinates. The section condition simply reads mnklp ∂mn ⊗ ∂kl = 0. (1.5) In section 4, the geometry of the SL(5) theory is described in greater detail. The central object in this construction is the so-called generalised metric MM N which contains all the scalar degrees of freedom and is an element of the coset space SL(5)/ SO(5) (or more generally, of En(n) /Kn where Kn is the maximal compact subgroup of En(n) ). In the series of papers [16, 18–27] it was shown that one may consider full (supersymmetric) d+dim Rn -dimensional theories that descend to the maximal D = 11, half-maximal D = 10 Type IIA/B or ungauged d-dimensional supergravity theories. In addition, upon a generalised Scherk-Schwarz reduction, these reproduce gauged d-dimensional supergravity with all gaugings expressed in terms of the generalised vielbein. The gaugings are controlled by the so called embedding tensor whose origin in EFT is the generalised torsion of the generalised Weitzenböck connection [28–31]. The role of the Weitzenböck connection and the generalised torsion in exceptional field theories was first discussed in [16] and later for DFT in [32]. From the perspective of the lower dimensional gauged supergravity, all components of the embedding tensor may be divided into two groups: geo (...truncated)