Consistent Kaluza-Klein Truncations
[1.25ex] Exceptional Field Theory
Olaf Hohm and Henning Samtleben
Center for Theoretical Physics
Massachusetts Institute of Technology
Cambridge, MA 02139, USA
firstname.lastname@example.org Université de Lyon, Laboratoire de Physique, UMR 5672, CNRS
École Normale Supérieure de Lyon
46, allée d’Italie, F-69364 Lyon cedex 07, France
We present the generalized Scherk-Schwarz reduction ansatz for the full supersymmetric exceptional field theory in terms of group valued twist matrices subject to consistency equations. With this ansatz the field equations precisely reduce to those of lower-dimensional gauged supergravity parametrized by an embedding tensor. We explicitly construct a family of twist matrices as solutions of the consistency equations. They induce gauged supergravities with gauge groups SO and CSO. Geometrically, they describe compactifications on internal spaces given by spheres and (warped) hyperboloides , thus extending the applicability of generalized Scherk-Schwarz reductions beyond homogeneous spaces. Together with the dictionary that relates exceptional field theory to and IIB supergravity, respectively, the construction defines an entire new family of consistent truncations of the original theories. These include not only compactifications on spheres of different dimensions (such as AdS), but also various hyperboloid compactifications giving rise to a higher-dimensional embedding of supergravities with non-compact and non-semisimple gauge groups.
The consistent Kaluza-Klein truncation of higher-dimensional supergravity to lower-dimensional theories is an important and in general surprisingly difficult problem. Here consistent truncation means that any solution of the lower-dimensional theory can be embedded into a solution of the original, higher-dimensional theory. This requires that all coordinate dependence of the internal space is consistently factored out. Due to the non-linearity of the supergravity equations of motion this is a highly non-trivial and often impossible challenge for compactifications on curved backgrounds. Only very few examples are known in which such consistency cannot be attributed to an underlying symmetry argument. The simplest class of consistent truncations are the toroidal compactifications in which the internal coordinate dependence is completely dropped, extrapolating the original ideas of Kaluza and Klein [Kaluza:1921tu, Klein:1926tv] to higher dimensions. Consistency simply follows from the fact that all retained massless fields are singlets under the resulting gauge group. In the context of eleven-dimensional supergravity [Cremmer:1978km] such reductions give rise to the maximal ungauged (thus abelian) supergravities in lower dimensions [Cremmer:1979up].
More involved examples are sphere compactifications, the prime example being the compactification of eleven-dimensional supergravity on AdS, leading to maximal gauged supergravity in four dimensions [deWit:1982ig]. The required consistency conditions are so non-trivial that in the early days of Kaluza-Klein supergravity this shed serious doubt on the possible consistency of sphere compactifications. For AdS this consistency was, however, established in [deWit:1986iy], with recent improvements in [Nicolai:2011cy, deWit:2013ija, Godazgar:2013dma, Godazgar:2013pfa], employing an invariant reformulation of the original eleven-dimensional theory [deWit:1986mz]. Other consistent sphere reductions have been constructed in [Nastase:1999kf, Cvetic:2000dm, Cvetic:2000nc], including the compactification on AdS.
An important generalization of the usual compactification scheme was put forward by Scherk and Schwarz [Scherk:1979zr], relating the internal dimensions to the manifold of a Lie group. More recently, the advances in the understanding of the duality symmetries underlying string and M-theory have nourished the idea to identify generalized geometric (and possibly non-geometric) compactifications as generalized Scherk-Schwarz reductions in some extended geometry [Kaloper:1999yr, Hull:2004in, Dabholkar:2005ve, Hull:2007jy, DallAgata:2007sr, Hull:2009sg]. In the framework of double field theory [Siegel:1993th, Hull:2009mi, Hull:2009zb, Hohm:2010jy, Hohm:2010pp, Hohm:2010xe], which makes the T-duality of string theory manifest, generalized Scherk-Schwarz-type compactifications of an extended spacetime have been discussed in [Hohm:2011cp, Aldazabal:2011nj, Geissbuhler:2011mx, Hassler:2014sba], see also [Grana:2012rr, Hohm:2011ex] for reductions to deformations of double field theory. In the M-theory case, analogous ideas have been investigated in [Berman:2012uy, Musaev:2013rq, Aldazabal:2013mya, Berman:2013cli, Lee:2014mla, Baron:2014yua] in the duality covariant formulation of the internal sector of supergravity [Berman:2010is, Berman:2011jh, Coimbra:2011ky, Coimbra:2012af].
In this paper, we realize this scenario in full exceptional field theory (EFT) [Hohm:2013pua, Hohm:2013vpa, Hohm:2013uia, Hohm:2014fxa, Godazgar:2014nqa], which is the manifestly U-duality covariant formulation of the untruncated ten- and eleven-dimensional supergravities. The theory is formulated on a generalized spacetime coordinatized by , where we refer to as ‘external’ spacetime coordinates, while the describe some generalized ‘internal’ space with labeling the fundamental representation of the Lie groups in the exceptional series E, . The fields generically include an external metric , an internal (generalized) metric and various higher -forms, in particular Kaluza-Klein-like vectors in the fundamental representation and possibly 2-forms and higher forms. With respect to the internal space, all fields are subject to covariant section constraints on the extended derivatives which imply that fields depend only on a subset of coordinates. There are at least two inequivalent solutions to these constraints: for one the theory is on-shell equivalent to 11-dimensional supergravity, for the other to type IIB, in analogy to type II double field theory [Hohm:2011zr].
The recent ideas of realizing non-trivial (and possibly non-geometric) compactifications as generalized Scherk-Schwarz compactifications are based on an ansatz for the generalized metric of the form
in terms of group-valued twist matrices which capture the -dependence of the fields. With this ansatz, the -dependence in the corresponding part of the field equations consistently factors out, provided the twist matrices satisfy a particular set of first order differential equations which in full generality take the form
Here, is a -dependent scale factor, is the number of external space-time dimensions, and are constant so-called embedding tensors that encode the gauging of supergravity, and denotes projection onto a particular subrepresentation. With this ansatz, the scalar action for reproduces the scalar potential of gauged supergravity for , with the twist matrix encoding the embedding tensor which parametrizes the lower-dimensional theory [Berman:2012uy, Musaev:2013rq, Aldazabal:2013mya].
In this paper, we extend this scheme to the full exceptional field theory with the following main results
We extend the ansatz (1.1) to the field content of the full exceptional field theory, i.e. to the external metric, vector and -forms. In particular, we find that consistency of the reduction ansatz requires a particular form of the ‘covariantly constrained’ compensating gauge fields, which are novel fields required in exceptional field theory for a proper description of the degrees of freedom dual to those of the higher-dimensional metric. E.g. in exceptional field theory, most of the remaining fields reduce covariantly,
with the twist matrix in the corresponding representation and the scale factor taking care of the weight under generalized diffeomorphisms. In contrast, the constrained compensator field which in the E case corresponds to a 2-form in the fundamental representation is subject to a non-standard Scherk-Schwarz ansatz that reads
relating this field to the 2-forms present in gauged supergravity. The ansatz (1) for encodes the embedding of all four-dimensional vector fields and their magnetic duals. As such, it includes the recent results of [Godazgar:2013pfa] for the sphere compactification, but remains valid for a much larger class of compactifications, in particular, for the hyperboloids which we explicitly construct in this paper. The reduction ansatz for the fermionic fields in the formulation of [Godazgar:2014nqa] is remarkably simple, their -dependence is entirely captured by a suitable power of the scale factor .
We show that with the ansatz (1.1)–(1.4), the field equations of exceptional field theory precisely reduce to the field equations of the lower-dimensional gauged supergravity. Via (1.2), the twist matrix encodes the embedding tensor , which specifies the field equations of the lower-dimensional gauged supergravity [deWit:2002vt, deWit:2007mt, LeDiffon:2011wt]. In case , the lower-dimensional field equations include a gauging of the trombone scaling symmetry which in particular acts as a conformal rescaling on the metric [LeDiffon:2011wt]. These equations do not admit a lower-dimensional action. Yet, also in this case the generalized Scherk-Schwarz ansatz defines a consistent truncation and we reproduce in particular the exact scalar contributions to the lower-dimensional field equations. For , the reduction is also consistent on the level of the action and we reproduce the full action of gauged supergravity defined by an embedding tensor .
The consistency of the generalized Scherk-Schwarz ansatz being guaranteed by the differential equations (1.2), it remains an equally important task to actually solve these equations. For conventional Scherk-Schwarz compactifications the existence of proper twist matrices is guaranteed by Lie’s second theorem, but to our knowledge there is no corresponding theorem in this generalized context. In certain cases, the existence of solutions can be inferred from additional structures on the internal manifold, such as the Killing spinors underlying the original construction of [deWit:1986iy] and then [Godazgar:2013dma], or the generalized parallelizability underlying certain coset spaces, such as the round spheres [Lee:2014mla]. In this paper, we explicitly construct a family of twist matrices as solutions of (1.2), that via the generalized Scherk-Schwarz ansatz give rise to gauged supergravities with gauge groups SO and CSO. Geometrically, they describe compactifications on internal spaces given by (warped) hyperboloides (as first conjectured in [Hull:1988jw]), thus extending the applicability of generalized Scherk-Schwarz reductions beyond homogeneous spaces. Our construction is based on the embedding of the linear group into the EFT group with the internal coordinates decomposing according to
i.e. carrying the antisymmetric representation . We then construct a family of -valued twist matrices, parametrized by non-negative integers with , satisfying the version of the consistency conditions (1.2). They depend on a subset of coordinates , embedded into (1.5) as , such that the section constraint of exceptional field theory is identically satisfied. Upon embedding into , these twist matrices turn out to solve the full version of consistency conditions (1.2), provided the number of external dimensions is related to as
With the three principal integer solutions , we thus obtain solutions of the consistency conditions (1.2) within , , and EFT. Their coordinate dependence is such that the reduction ansatz explicitly satisfies the EFT section constraints.
Combining these explicit solutions to the consistency equations (1.2) with the generalized Scherk-Schwark ansatz (1.1)–(1.4), we thus define consistent truncations of the full exceptional field theory to lower-dimensional supergravities with gauge groups and . Together with the dictionary that relates exceptional field theory to and IIB supergravity, respectively, (which is independent of the particular choice of the twist matrix ), the construction thus gives rise to an entire family of consistent truncations in the original theories, including spheres of various dimensions and warped hyperboloids.111 It depends on the embedding (1.5) of , if the coordinate dependence of the twist matrix falls into the class of eleven-dimensional (IIA) or IIB solutions of the exceptional field theory. This defines in which higher-dimensional theory the construction gives rise to consistent truncations. Unsurprisingly, this is IIA for , and IIB for . Specifically, we compute the internal metric induced by our twist matrices via the Scherk-Schwarz ansatz (1.1), and find
with the further split of coordinates , , and , and the combinations , . This space is conformally equivalent to the direct product of flat directions and the hyperboloid . The three integer solutions to (1.6) in particular capture the compactifications around the three maximally supersymmetric solutions AdS, AdS, AdS. We stress that in the general case however the metric (1.7) will not be part of a solution of the higher-dimensional field equations. This is equivalent to the fact that the lower-dimensional supergravities in general do not have a critical point at the origin of the scalar potential, as explicitly verified in [Hull:1988jw] for the supergravities. Nevertheless, in all cases the generalized Scherk-Schwarz ansatz continues to describe the consistent truncation of the higher-dimensional supergravity to the field content and the dynamics of a lower-dimensional maximally supersymmetric supergravity. The construction thus enriches the class of known consistent truncations not only by the long-standing AdS, but also by various hyperboloid compactifications giving rise to non-compact and non-semisimple gauge groups.
Let us stress that throughout this paper we impose the strong version of the section constraint, which implies that locally the fields (i.e. the twist matrix and scale factor ) depend only on the coordinates of the usual supergravities.11footnotemark: 1 This is indispensable in order to deduce that the consistent truncations from exceptional field theory induce a consistent truncation of the original supergravities. On the other hand, it puts additional constraints on the solutions of (1.2), which makes the actual construction of such solutions a more difficult task. Although naively, one might have thought that for a given embedding tensor a simple exponentiation of provides a candidate for a proper twist matrix, the failure of Jacobi identities of the ‘structure constants’ , and the non-trivial projection in (1.2) put a first obstacle to the naive extrapolation of the Lie algebra structures underlying the standard Scherk-Schwarz ansatz. On top of this, an object like in general violates the section constraint, since in general will have a rank higher than is permitted by the six/seven coordinates among the that are consistent with the section constraint. From this point of view, the standard sphere compactifications take a highly ‘non-geometric’ form. While we do not expect to encode in this construction genuinely non-geometric compactifications (unless global issues of the type addressed in double field theory in [Hohm:2013bwa] become important), we expect that a proper understanding of highly non-trivial compatifications like for spheres and hyperboloides will be a first step in developing a proper conceptual framework for non-geometric compatifications, which so far are out of reach. It should be evident that the advantage even of the strongly constrained exceptional field theory formulations is a dramatic technical simplification of, for instance, the issues related to consistency proofs, allowing to resolve old and new open questions. In fact, with the full EFTs at hand we can potentially provide a long list of examples of consistent truncations that were previously considered unlikely, such as hyperboloides, warped spheres, compactifications with massive multiplets, etc. Of course, eventually one would like to also include in a consistent framework truly non-geometric compactifications, pointing to a possible relaxation of the strong form of the section constraint.
The rest of this paper is organized as follows. In section 2, we give a brief review of the EFT. Although in this paper most detailed technical discussions will be presented for the EFT the analogous constructions go through for all other EFTs. In section LABEL:sec:reduction we describe the generalized Scherk-Schwarz ansatz for the full field content of the theory. We show that it defines a consistent truncation of the EFT which reduces to the complete set of field equations of lower-dimensional gauged supergravity with embedding tensor , , even in presence of a trombone gauging . For , the reduction is also consistent on the level of the action. In section LABEL:sec:sphere, we construct twist matrices as explicit solutions of the consistency conditions (1.2). We define a family of twist matrices and show that upon proper embedding into the exceptional groups they solve equations (1.2). The lower-dimensional theories have gauge groups and , respectively. The construction provides the consistent reduction ansaetze for compactifications around spheres and (warped) hyperboloides . Discussion and outlook are given in section LABEL:sec:conclusions.
2 E exceptional field theory
We start by giving a brief review of the E-covariant exceptional field theory, constructed in Refs. [Hohm:2013pua, Hohm:2013uia, Godazgar:2014nqa] (to which we refer for details) . All fields in this theory depend on the four external variables , , and the 56 internal variables , , transforming in the fundamental representation of , however the latter dependence is strongly restricted by the section condition [Berman:2010is, Coimbra:2011ky, Berman:2012vc]
where the notation should indicate that both derivative operators may act on different fields. Here, is the symplectic invariant matrix which we use for lowering and raising of fundamental indices according to , . The tensor is the representation matrix of in the fundamental representation, . These constraints admit (at least) two inequivalent solutions, in which the fields depend on a subset of seven or six of the internal variables. The resulting theories are the full supergravity and the type IIB theory, respectively.
2.1 Bosonic field equations
The bosonic field content of the E-covariant exceptional field theory is given by
The field is the vierbein, from which the external (four-dimensional) metric is obtained as . The scalar fields are described by the symmetric matrix constructed as from an valued 56-bein, parametrizing the coset space . Vectors and 2-forms transform in the fundamental and adjoint representation of , respectively. The 2-forms in the fundamental representation describe a covariantly constrained tensor field, i.e. satisfy algebraic equations analogous to (2.1)
Their presence is necessary for consistency of the hierarchy of non-abelian gauge transformations and can be inferred directly from the properties of the Jacobiator of generalized diffeomorphisms [Hohm:2013uia]. In turn, after solving the section constraint these fields ensure the correct and duality covariant description of those degrees of freedom that are on-shell dual to the higher-dimensional gravitational degrees of freedom.
The bosonic exceptional field theory is invariant under generalized diffeomorphisms in the internal coordinates, acting via [Coimbra:2011ky]
on arbitrary tensors of weight . The weights of the various bosonic fields of the theory are given by eμαMMNAμMBμναBμνMλ:12012112