Non-Riemannian geometry of M-theory

We construct a background for M-theory that is moduli free. This background is then shown to be related to a topological phase of the $\mathrm{E}_{8(8)}$ exceptional field theory (ExFT). The key ingredient in the construction is the embedding of non-Riemannian geometry in ExFT. This allows one to describe non-relativistic geometries, such as Newton-Cartan or Gomis-Ooguri-type limits, using the ExFT framework originally developed to describe maximal supergravity. This generalises previous work by Morand and Park in the context of double field theory.


Introduction
General relativity describes the geometry of gravity in terms of a dynamical (pseudo-)Riemannian metric. String theory and M-theory provide a route towards a quantum mechanical understanding of gravity. At low energies, the classical geometry of string theory/M-theory is again described by a metric, whose dynamics is governed by a supergravity theory in which the metric is accompanied by a collection of scalars and p-form gauge fields, plus fermions.
The presence of duality in these theories means that they exhibit a (hidden) symmetry which mixes metric and form field components. Inspired in large part by a desire to capture and explain this symmetry more fundamentally, and to find new notions of intrinsically "stringy" or "M-theoretic" geometry treating all the massless states of the theory on a more egalitarian footing, reformulations of the dynamics of supergravity have been found in which the geometry and the fields living in the geometry are united and covariance under the duality groups of string/M-theory is made manifest.
The starting point for these theories is to observe that the bosonic degrees of freedom of supergravity in a certain (n+d)-dimensional split can be recombined into multiplets of the groups O(d, d) (when n + d = 10 [18] with the original construction applicable to n = 0 [1,19]) or E d(d) (when n + d = 11, and so far allowing for d = 2, . . . , 9 [7-9, 20-24]). Then the full dynamics and local symmetries of 10- In this way, for instance, ExFT admits inequivalent solutions of the section condition giving either 11-dimensional supergravity or the 10-dimensional type IIB supergravity [7,25]. One can think of the usual supergravity theories as following from the single unifying ExFT formulation on solving the consistency conditions of the latter.
A more ambitious interpretation of the geometry of DFT/ExFT is to allow for solutions of the section condition, or parametrisations of the fields, which do not reduce to conventional supergravity.
A number of avenues have been explored, often involving notions of "non-geometry" in one form or another (for lots on non-geometry, see the review [26]). This includes the possibility of relaxing the section condition in order to carry out Scherk-Schwarz type reductions where the twist matrices may depend on dual coordinates. This leads to lower-dimensional gauged supergravities including those that are not possible to lift to ten-or eleven-dimensional geometries [27] as well as, in ten dimensions, the massive IIA theory [28,29] and the so-called "generalised supergravity". The latter was remarkably only discovered as a type IIB string background less than three years ago [30] (from examining in close detail the relationship between kappa symmetry and the string beta function) and was almost immediately shown to have a simple DFT and ExFT picture [31,32]. Furthermore, the DFT/ExFT geometry also provides a home for explicit solutions corresponding to strongly nonperturbative states known as exotic branes -whose existence is predicted by U-duality [33,34] that have no global description in supergravity [35][36][37][38]. In some instances it is possible to describe non-geometry by using alternative spacetime parametrisations of the DFT/ExFT fields, for instance in terms of a metric and bivector [39,40], rather than a metric and two-form.
The examples of the previous paragraph are still based on the idea that there is some spacetime description involving (possibly only locally) a Riemannian metric and some set of forms or bivectors. The novelty in the exotic backgrounds arises from global data as we "glue" patches using E d(d) transformations, rather than traditional diffeomorphisms, but locally there is a supergravity description of some sort though perhaps gauged or "generalised" due to a Scherk-Schwarz twist.
These non-geometric aspects of DFT/ExFT are of course of crucial importance. There is, however, a further generalisation we can make that will be the subject of this paper. This is the, perhaps rather unexpected, observation that DFT/ExFT also accommodates descriptions of non-Riemannian geometry. These are backgrounds where there is not an invertible spacetime metric but instead a non-relativistic geometry, or even no intrinsic geometric structure at all. Examples of such geometries go back to Newton-Cartan geometry [41,42], and include the non-relativistic limit of string theory studied by Gomis and Ooguri [43] and Danielsson, Guijosa and Kruczenski [44,45].
The exploration of non-Riemannian string theory geometries using DFT was pioneered in [46][47][48][49], leading to a classification of allowed non-Riemannian backgrounds [48]. The starting point of these papers was the realisation that backgrounds defined by a generalised metric without a conventional spacetime interpretation [46] could still be studied using the doubled sigma model [11,50], which describes a string whose target space is the doubled geometry of DFT. This was connected to the non-relativistic Gomis-Ooguri string in [47], including an analysis of the spectrum using spacetime DFT techniques. The paper [48] then offered a classification of non-Riemannian parametrisations of O(d, d) generalised metrics, as well as a study of particle and string actions in these backgrounds.
In this paper, we will study examples of M-theory non-Riemannian geometry in the context of exceptional field theory, 1 showing that they provide connections to non-relativistic and topological theories within a framework originally intended to describe maximal supergravity.

Non-Riemannian backgrounds in O(d, d): a first encounter
To the set scene, let us give a brief review of the ideas appearing in the non-Riemannian O(d, d) backgrounds studied in [46][47][48][49]. We will focus on "generalised metric", which may be interpreted In the usual DFT formulation, one then solves the compatibility condition (1.1) by parametrising H as follows: so that subsequently it is possible to interpret g as the spacetime metric and B 2 the Kalb-Ramond two-form of a d dimensional space. This generalised metric is also encountered in the usual string worldsheet theory, where it appears in the Hamiltonian form of the string worldsheet action: Here, if we integrate out the momenta P i we recover the usual Polyakov string action in conformal gauge, with background metric and B-field. This Hamiltonian action is actually very closely related to the doubled approach to the string sigma model, in which the target space has doubled coordinates X M = (X i ,X i ) andX ′ i = P i . However, to illustrate how the generalised metric describes "non-Riemannian" backgrounds, we will continue here to think in terms of the Hamiltonian of the usual string, rather than in DFT.
One of the advantages of the Hamiltonian picture is that it frequently allows one to take limits which would be singular in the Lagrangian formulation. Here, this manifests itself in the ability to choose H M N to have a degenerate bottom right d × d block. This is the block that would normally be interpreted as describing the inverse spacetime metric. However, the non-degeneracy of this block is compensated within H M N such that the whole generalised metric remains well-defined. An example of such a situation in the case d = 2 (suppressing the other target space coordinates) is given by The momenta P i cannot be integrated out but, rather, act to impose chirality conditions: The generalised metric is that which you obtain in the non-relativistic limit of string theory due to Gomis and Ooguri [43].
A more striking example is to set H M N = η M N (for any d). This choice clearly solves (1.1) but is not expressible as parameterised in (1.3). In the classification of [48], this has the special position of being "maximally non-Riemannian". The resulting sigma model is described by an entirely chiral theory, which appears to be related to recent work on certain chiral or "twisted" string theories with unusual properties [52][53][54][55][56] or a beta-gamma system [57]. The full quantum consistency of this model and its possible supersymmetrisation is a fascinating topic for future work. Viewed as an admissible background in DFT, this case has some very interesting properties: for instance, arbitrary variations δH M N evaluated on this background are automatically projected to zero [49]. Thus the maximally non-Riemannian space has no moduli!

Outline of this paper
The goal of this paper is to study the M-theory theory analogues of such backgrounds via exceptional field theory. This will involve re-analysing the consistency conditions on the generalised metrics of ExFT. Although in the string theory case, we have access to sigma models (either the Hamiltonian form presented above or the truly doubled sigma models such as [11,50]) with which to explore the non-Riemannian background, the nature of E d(d) covariant worldvolume theories for M-branes (if such theories exist) is mysterious [58] (though see [59][60][61] for some ExFT inspired approaches).
Nevertheless, some of the first examples of E d(d) generalised metrics (for d = 4, i.e. SL(5)) were found by studying the M2 worldvolume theory [3,62] (in fact they appear also in the M2 Hamiltonian).
We therefore expect, or hope, that the geometry of ExFT more generally tells us something about the structure of M-theory backgrounds beyond the conventional geometry. As examples, we will study the Gomis-Ooguri limit for M2 branes, and also some possible embeddings of Newton-Cartan geometry.
We will also focus on the generalisation of the DFT maximally non-Riemannian background H M N = η M N . This depends on the presence of an invariant tensor in the symmetric product of R 1 ⊗ R 1 . For the exceptional generalised diffeomorphism groups E d(d) , the representation theory precludes the existence of such a tensor in all finite dimensional cases except that of E 8 (8) . In this case, R 1 = 248 is the adjoint (also the fundamental), and we therefore can define M M N to be proportional to the Killing form κ M N . We propose to view the E 8(8) ExFT on the background M M N = −κ M N as the definition of the topological "phase" introduced in [63] and used to study three-dimensional superconformal field theories [64]. In that work, an ad hoc truncation was taken by setting M M N = 0 in order to preserve the full E 8(8) symmetry. By extending the set of allowed backgrounds to include non-Riemannian generalised metrics, we obtain a non-singular definition of the topological theory. It is satisfying that this realises an old idea that there should be some underlying topological phase of gravity and that the geometry should emerge through spontaneous breaking. What is interesting about the proposal using DFT and ExFT is that the topological phase exists not with a vanishing metric, as originally envisaged, but via a moduli-free maximally non-Riemannian metric.
We therefore see that both DFT and ExFT allow for the description of theories beyond the supergravities which inspired them by incorporating non-relativistic theories of gravity, topological three-dimensional theories, and more. We hope that this paper will stimulate further interest in these uses of DFT/ExFT and provide a starting point to study features of theories with non-Riemannian or non-relativistic geometries. What is appealing is that although one can think of many issues with such geometries, at least in DFT case the sigma model makes sense and so it appears to be a truly stringy geometry. (For the ExFT non-Riemannian backgrounds encountered in this paper, we can use the exceptional sigma model of [59,60], which describes a string action whose target space is the exceptional extended geometry of ExFT, reducing to descriptions of F1 strings and D1 branes in ten dimensions.) With this motivation, we should stress that, looking ahead, a full exploration of the quantum consistency of these sigma models is crucial. For now we focus on the classical presence of such solutions in these theories.

Generalised metrics and diffeomorphisms
The local symmetries of general relativity, double field theory and exceptional field theory can all be treated in same manner, by defining (generalised) diffeomorphisms associated to a group G. For general relativity, this group is G = GL(d), for DFT, it is G = O(d, d), and for ExFT, it is E d(d) .
We work with coordinates (X µ , Y M ), where µ = 1, . . . , n and Y M transform in what we call the R 1 representation of G. In DFT and ExFT, we will call the X µ coordinates "external" and the Y M "internal" or "extended", mimicking the language we would use if we reduced to an n-dimensional theory (however no compactification is assumed or needed to formulate these theories). The R 1 representation is the d-dimensional fundamental of GL(d) in the case of general relativity, the 2ddimensional fundamental in the case of O(d, d), and for E d(d) the representations are listed in table 1 (the rule is that R 1 is the representation whose highest weight is the fundamental weight associated to the rightmost node on the Dynkin diagram).
We define (generalised) diffeomorphisms associated to the transformation of the coordinates where P adj M K N L denotes the projector from R 1 ⊗R 1 onto the adjoint representation, α is a constant which depends on the group under consideration (see table 1) and λ V denotes the weight of V M . It  [65], (generalised) vectors are valued in R 1 , and the section condition in R 2 . The intrinsic weight is given by ω = −1/(n − 2) in ExFT and ω = 0 in DFT.
is often useful to expand the projector to obtain an equivalent form of the generalised Lie derivative: which makes apparent how the structure differs from the ordinary Lie derivative (which is given by the first two terms). The modification involves the so-called Y -tensor, which is constructed out of group invariants [6] , and also a constant ω which can be thought of as an intrinsic weight. When G = GL(d), clearly Y M N KL = 0 and ω = 0.
The crucial distinction between the GL(d) Lie derivative of usual Riemannian geometry and the generalised Lie derivative is that though the former leads to a closed symmetry algebra (closed under the Lie bracket), the algebra of generalised Lie derivatives turns out to be obstructed. The root cause of this obstruction is the dependence of the fields and gauge parameters on the coordinates Y M . One way to guarantee closure is then to impose the section condition on the coordinate dependence of all fields and gauge parameters, which can be realised as the condition Here the derivatives may act on separate quantities or on a single quantity. Solutions to the section condition will break the (global and local) G-symmetry and amount to a choice of d coordinates Y i from amongst the Y M which are taken to be physical. This is how one reduces the formulation with manifest G-covariance to the standard geometric description. When further isometries are present such that ∂ M = 0 then the global group G remains a symmetry and is identified with a usual duality group. Essentially, duality arises from the ambiguity in identifying the physical spacetime inside the extended space when there are isometries.
The geometry of general relativity is, of course, described by a metric. Similarly the generalised, or "extended", geometry of DFT/ExFT will be described by a generalised metric. We define this to be a symmetric matrix, M M N , which is an element of G and so preserves the appropriate invariant tensors. The generalised Lie derivative of the generalised metric follows from (2.1) or (2.2) using the Leibniz property. It takes the form: in which the following projector appears: 4) or in terms of the adjoint projector, Note that as the Y -tensor, or equivalently the adjoint projector, is a group invariant it is preserved by the simultaneous action of M and M −1 on all four indices, which can be used to check that P M N KL is actually symmetric in both its upper and lower pairs of indices. We can think of equation We can calculate the trace of the projector to compute the number of independent components of the generalised metric, i.e. the dimension of the coset G/H in which it lives. In general, we find: Evidently, in general relativity we have α = 1, and the terms in (2.4) involving ω and the Y -tensor do not appear. Hence we find P M N M N = 1 2 d(d+ 1) which is the number of independent components of a symmetric matrix and also the dimension of the coset GL(d)/SO(d).
In DFT and ExFT the situation is rather more interesting. Part of the trace (2.6) is independent of the generalised metric and follows from representation theory as the Y-tensor can be related to the projector onto the R 2 representation [6]. For d = 4 to d = 6 it is directly proportional to this projector, and we find that its trace is Y M N M N = 2(d − 1)dim R 2 . For d = 7, an additional term appears in the Y-tensor involving the antisymmetric invariant of E 7(7) (i.e. a projector onto also the trivial representation) and in this case The crucial information about the coset then appears in the very final term in (2.6), which we may single out and define as One finds, as summarised in (and for E 7 (7) and E 6(6) in section 2.5).  [48], which we will review from the perspective of the projector P M N KL in section 3, and will appear below in an interesting context for the E 8(8) ExFT.

The action and equations of motion
Let us now discuss the dynamics of the generalised metric. Its equations of motion follow from the ExFT action, which is constructed using the requirement of invariance under the local symmetries of ExFT. These include not only generalised diffeomorphisms but also external diffeomorphisms associated to transformations of the coordinates X µ , and various generalised gauge transformations of gauge fields that also appear in the theory.
The projector then plays a vital role in the equations of motion for the generalised metric. (Here we are thinking only of the bosonic part of the action: if we include fermions then we will have to use a projector onto the variation of the generalised vielbein. We will comment more on this later.) In fact, it was in this context that the projector was first written down in [66]  The reason for this is that one must insist that the variations of the generalised metric δM M N are still compatible with G and so we impose this by a projector. In the standard formulation of ExFT, the actions do not explicitly impose this and so one needs to include these projectors by hand though it is equivalent to just calculating the variations of the action subject to G-compatibility.
Now, recalling that the projector depends on M M N , we might consider whether it is possible to find a generalised metric such that the projector vanishes:

The E 8(8) ExFT and its topological phase
Generalised Diffeomorphisms and the Action The E 8 (8) ExFT [9] is based on an extended geometry parametrised by 248 coordinates Y M valued therefore in the adjoint of E 8 (8) . Denoting its generators as T M , we define structure constants We freely raise and lower all indices using κ M N and its inverse κ M N .
The generalised Lie derivative of an adjoint vector of weight λ is explicitly given by in which we have used the projector onto the adjoint representation (P 248 ) M K N L defined by Alternatively, one can write the part of this transformation involving Λ M in the form (2.2) involving the Y-tensor, given here by (2.14) A special feature of the E 8 (8) ExFT is that it includes additional gauge transformations which appear alongside the conventional generalised Lie derivative. Under this extra gauge symmetry, generalised vectors transform as where the gauge parameter Σ M is not an arbitrary covector but is constrained as part of the section condition of the E 8 (8) ExFT. This section condition applies to any two quantities F M , F ′ M which are said to be "covariantly constrained" meaning that they vanish when their tensor product is projected These quantities include derivatives, ∂ M , as usual, the gauge parameters Σ M , and a number of other gauge parameters and field [9]. This section condition guarantees closure of the algebra of the combined action of generalised diffeomorphisms and constrained Σ M transformations, which we denote by The inclusion of the Σ M transformations is in fact necessary for closure: the algebra based on the ordinary generalised Lie derivative (2.12) alone cannot be made to close on its own. The underlying physical reason for the extra gauge transformation (2.15) is the appearance of dual graviton degrees of freedom in the generalised metric of the E 8 (8) ExFT, which we discuss briefly in section 2.4.
For further details on these subtleties, we refer the reader to the original paper [9] or the recent review [67].
We proceed to discuss the field content of the theory. This consists of the generalised metric, M M N , an external metric, g µν , and a pair of gauge fields (A µ M , B µM ), with B µM covariantly con-strained as in (2.16). These gauge fields have field strengths (F µν M , G µνM ) whose precise forms can be found in [9]. All these fields depend on the three-dimensional coordinates X µ as well as the 248-dimensional coordinates Y M , subject to the section condition. The gauge field A µ M can be thought of as serving as a gauge field for generalised diffeomorphisms while B µM is a gauge field for the constrained Σ M transformations. We define an improved derivative D µ ≡ ∂ µ − L (Aµ,Bµ) which is used in place of ∂ µ . The action for the E 8 (8) ExFT is constructed in [9] and is given by whereR[g] is the usual Ricci scalar for the metric g µν , except constructed in terms of D µ instead of ∂ µ . The two terms at the end are: which is usually referred to as the "potential", taking the point of view of the external threedimensional space, and the Chern-Simons term: written here in a manifestly gauge invariant form using the usual construction of an auxiliary space Σ 4 whose boundary ∂Σ 4 is the physical three-dimensional space, and where ∧ denotes the usual product with respect to the external indices, µ, ν, . . . .

Generalised metric and projector
Conventionally, we view the generalised metric as being an element of and then this coset is parametrised in terms of a spacetime metric and p-form fields. Instead, following the intuition from the DFT approach of [48] where the generalised metric was defined as a symmetric two index object obeying the O(d, d) compatibility condition (1.1), we will define the E 8 (8) generalised metric by the properties that are needed in [9] to ensure the invariance of the action (2.18). Thus we define the E 8(8) generalised metric to be the symmetric two index object that obeys the constraints: One can check that the conventional coset parametrisation of M M K obeys these constraints but new results will follow from a solution to these constraints that does not obey the coset parametrisation.
The full generalised Lie derivative (including the additional transformations involving Σ M ) of the generalised metric takes the form with the projector given simply by The trace is where A is a spinor index corresponding to the 128 of SO (16), and I the 16-dimensional vector representation, with E M IJ = −E M JI in the 120 of SO (16). The generalised metric is then given Now, ordinary three-dimensional gravity is topological, so this action naively resembles that of a topological gravity theory plus a Chern-Simons term, though matters are complicated by the dependence on the coordinates Y M and the modified partial derivative used in the construction of the Ricci scalar. Remarkably, however, the entire truncation (2.28) including the external metric and the gauge fields is indeed a novel sort of topological theory. This was shown in [63] where the theory described by the action (2.28) was reformulated as a Chern-Simons theory based on a "Leibniz algebra" 3 incorporating both the three-dimensional Poincaré symmetry and the generalised diffeo-

Why is E 8(8) special?
Let's briefly discuss some of the physics behind why the E 8 (8) ExFT is special, and how this can be tied in to the existence of the solution M M N = −κ M N .

Gauge transformations and dual graviton
We saw above that the generalised Lie derivative of In fact, these dual graviton degrees of freedom occur in all exceptional field theories [67], at some point in the tensor hierarchy of generalised p-form gauge fields, specifically when one encounters a representation R p equal toR 1 . In order to remove them, it becomes necessary to include additional gauge transformations, with gauge parameters Σ M ∈R 1 , which are covariantly constrained to obey the section constraint. As one goes from E d(d) to E d+1(d+1) , these dual graviton components appear at one degree lower in the tensor hierarchy: thus in the E 6(6) ExFT [7] they appear in the two-form B µνM , for the E 7(7) ExFT [8] they appear in the one-form A µ M , and then for E 8(8) they appear inevitably in the generalised metric itself.

Extended Coordinates and Exotic Branes
As well as the appearance of the dual graviton in the generalised metric, the other novel physical feature which becomes relevant for the first time in the E 8 (8) ExFT is the appearance of extended coordinates which can be interpreted as conjugate to winding modes of exotic branes. (Of course, in the construction of ExFT no compactification, toroidal or otherwise is assumed, so here we speak of brane windings firstly as a useful mnemonic for classifying and interpreting the extended coordinates and secondly to highlight the essential string or M-theoretic context.) We can see this by decomposing the adjoint of E 8 (8) under GL(8), such that where the subscript denotes the charge with respect to GL(1) ⊂ GL (8). Thus, the 248 coordinates split as with i = 1, . . . , 8. We can associate the dual coordinates with windings of various branes. It is convenient to introduce a "generalised momenta" P M , conjugate to the coordinates Y M (this momentum interpretation follows from basic intuition about duality, and specifically in an ExFT context, can be seen for instance at the level of solutions [66] or particle actions [69]), which decomposes into GL(8) tensors carrying antisymmetric indices as follows: (2.32) (Note we dualise lower indices into upper or vice versa using η i 1 ...i 8 in some cases relative to (2.31).) This decomposition corresponds to physical momenta p i then windings of the M2, M5, KK6, 5 3 , 2 6 and 0 (1,7) branes [33]. The latter three are exotic, highly non-perturbative states which must exist in M-theory for U-duality to exist as a symmetry, and which do not exist as wrapped states in dimensions greater than three. A comprehensive modern discussion, and argument that "exotic" states are actually generic in string theory, can be found in [34].
The key point for us is that the E 8 (8) ExFT is the first ExFT for which the branes more exotic than the KK6 provide dual coordinates and that the way in which they appear is to fill up the 248 representation, which is self-conjugate. Hence every GL(8) representation appears alongside its conjugate (apart from the KK6 states in the (63 ⊕ 1) 0 which is the GL(8) adjoint and so again self-conjugate). This means that there is a natural pairing between the representations. As a result, the Killing form in this basis (see appendix B for more) is given by: The What we see from this discussion and from the dual graviton is that in the E 8(8) ExFT, dual degrees of freedom for both the form fields and the graviton appear within both the extended coordinates and the generalised metric of the theory (whereas in lower rank ExFTs, the dual graviton degrees of freedom do not enter the generalised metric, and we have fewer exotic winding coordinates). The non-perturbative self-duality of M-theory therefore enters this theory in a fundamental way. We can only speculate about the connection between these clues and the presence of the topological phase in the non-Riemannian background.
Reduction: from E 8 (8) to E 7 (7) We can also ask what happens to this background if we descend to a lower rank ExFT. We can consider the reduction from E 8 (8) to E 7(7) via the breaking of the adjoint of E 8 (8) to E 7(7) × SL (2) where Y ab is symmetric in ab. The E 8(8) Killing form is: Here we see the appearance of the E 7(7) invariant Killing form κ αβ and antisymmetric tensor Ω M N , as well as the SL (2) invariant ε ab .
To give an ExFT interpretation, we pick the Then we might attempt to understand the generalised metric block decomposition ExFT, and a component g ♯♯ of the external metric of the same (with some additional fields in certain cases, including E 8 (8) to E 7 (7) , related to the treatment of dual graviton components). The precise details of this reduction have not, however, been worked out, much less any modifications that will likely be necessary in reducing (7) theory which we expect to lack a generalised metric (as there is no symmetric invariant for it to equal, put differently after unpacking (2.35) we should find that the "normal" definition of the E 7(7) generalised metric will vanish). We can infer that the decomposition (2.35) could be viewed as being a non-Riemannian reduction itself. Detailed understanding of this possibility is beyond the scope of the present paper.
Reduction: from E 8 (8)  to the E 8 (8) ExFT rather than to an ordinary DFT.
which again likely would not correspond to a standard reduction from the O (8,8) to the O(7, 7) theory.
We may note that the decomposition of the 248 of E 8 (8) to O(7, 7) produces two fundamental representations of the latter. Returning to the brane picture, these are associated to momentum/string winding and to windings of the 1 6 4 and 0 We may also note the section condition of the O(8, 8) theory also admits solutions corresponding to six-dimensional supergravity [70]. The interpretation of non-Riemannian backgrounds in this context may also be interesting to understand more completely in the future.

The usual cosets
In this subsection, we demonstrate for the cases of E 6(6) and E 7(7) how to compute the quantity r defined in (2.7) which appears in the trace (2.6) of the projector P M N M N , and which encodes the vital information about the coset in which the generalised metric is valued. We will not study these particular ExFTs elsewhere in this paper. However, we think that the form of the verification that r = 0 may be instructive for future generalisations of non-Riemannian parametrisations to these groups.
For the d = 6 E 6(6) ExFT, R 1 is the fundamental 27. The Y-tensor is given by with d M N K and d M N K the symmetric cubic invariants of E 6(6) . We use the USp (8) construction of [71] in which the generalised metric is formed from the generalised vielbein V M ij carrying an antisymmetric pair of eight-dimensional indices i, j, which transform in the fundamental of USp (8).
These indices are raised and lowered with the symplectic form Ω ij such that Finally, the totally symmetric invariant d M N K is given in terms of the symplectic form as We can then calculate which vanishes by (2.39).
For E 7 (7) , [8], R 1 is the fundamental 56. Denote the generators acting on the fundamental by t αM N , and the antisymmetric invariant by HereMMN represents the flat generalised metric. As where Eᾱ α is the adjoint representation of the vielbein (which we do not need) and tᾱMN corresponds to the E 7(7) generator in the SU(8) basis. In this basis VM = (V AB , V AB ) and Vᾱ = (V A B , V ABCD ), Ω AB CD = δ CD AB and (see e.g. appendix of [73]) the components of tᾱMN are then (2.47) We then want to compute which is automatically zero forᾱ = ABCD and forᾱ = A B turns out to vanish on evaluat-  [48] fits into this picture. Then, we will review the explicit details of these parametrisations and look at some examples which will inspire us in our later study of the SL(5) ExFT. The O(D, D) compatibility condition implies the existence of two projectors

Generalised metric and coset projectors
where h AB andhĀB are respectively (D + y) × (D + y) and (D − y) × (D − y) diagonal matrices of signatures (p, q) and (p,q). This is quite general; we will see how different choices of signature allow for different coset descriptions and constrains (p, q) and (p,q). Constructing a vielbein for the full generalised metric, where the 2D × 2D flat metric, is of signature (p +p, q +q) we can check that then has signature (p +q, q +p). . (3.8) The denominator agrees with the generalised Lorentz factors established in [48]. Note that this coset has dimension d 2 + 2d(n +n) + 4nn = D 2 − (n −n) 2 . There are thus (n −n) 2 fewer components than would ordinarily be present.
Note that the explicit parametrisation that will be used in the subsequent subsection does not make this component counting manifest, as it uses variables which are written in a D-dimensionally  [48]. Taking the section condition solution, ∂ i = 0,∂ i = 0, they found that the most general parametrisation of the generalised metric is given by Here both H ij and K ij are symmetric D × D matrices which may be non-invertible, with {X,X} spanning the kernel of H ij and {Y,Ȳ } spanning the kernel of K ij . Both kernels have dimensions n +n, and we index the zero vectors by a = 1, . . . , n andā = 1, . . .n. Explicitly, We have some completeness relations which are necessary for the invertibility of H M N , namely where σ u v is conjugate to diag(δ a b , −δā b ). Thus X,X and Y,Ȳ diagonalise σ u v . Finally, note there is also a shift symmetry preserving the parametrisation (3.9), involving arbitrary parameters b ia ,b iā : which we can view as eliminating some components of the B-field in the non-Riemannian geometry.
A variety of interesting example have been considered in [48].  [48].) In general, the non-Riemannian background (3.9) can be studied using the doubled sigma model, and it was shown in [48] that the zero vectors X i a pick out n string target space coordinates which become chiral, while theX iā lead ton antichiral directions.
The paper [48] also introduced generalised vielbeins as follows. Let d denote the rank of H ij and K ij , such that D = d + n +n. Suppose that H ij and K ij have signature (t, s, n +n), and define the flat matrices which obey and similarly for the barred quantities. Now define out of which we construct

Examples: Gomis-Ooguri limit and timelike duality
Here we review two closely linked examples of DFT non-Riemannian geometry.

Gomis-Ooguri
The original idea of Gomis-Ooguri [43] is to consider the string sigma model in a special background for which one take a certain scaling limit leading to a description of string theory in a non-relativistic background geometry. This limit can be taken starting with the flat background which, for G → ∞, is non-singular but non-Riemannian. We have 22) and the preferred basis of zero vectors is

Non-Riemannian geometry from timelike duality
We can also obtain a non-Riemannian generalised metric by acting with T-duality on the supergravity solution corresponding to a fundamental string solution. This appeared in the DFT context first in [46], although the timelike dual of the F1 solution was studied long ago in [74]. The F1 solution is: Normally one takes c = −1 such that the B-field vanishes at infinity: in general it should lie in the range 0 > c > −2 [75]. After constructing the doubled generalised metric and dilaton using the usual parametrisation (1.3), we can T-dualise in both the t, z directions, giving the T-dual generalised metric (the part of the generalised metric describing the transverse space with coordinates x 8 is trivial so we do not write it) while the generalised dilaton is invariant and is e −2d = 1. Defining the corresponding spacetime geometry is • c = −1 corresponds to the usual asymptotically flat F1 solution, and gives rise here to a dual solution which is automatically asymptotically flat withH = 2 − H being given bỹ The ADM mass is minus that of the F1, and the solution can be interpreted as in [76] as describing a negative tension F1, or negative F1 for short. There is a singularity atH = 0, which can be thought of as marking the position of a "bubble" surrounding the negative tension brane, inside which the spacetime signature flips and we should use an exotic variant of string theory, of the type investigated by Hull [77,78], to describe its physics. As the generalised metric is non-singular atH = 0, the DFT description is perfectly well-defined (see [79,80] for some discussion of such exotic theories in DFT). Indeed, we see that in this case the generalised metric is which is in fact exactly of the non-Riemannian type appearing in the Gomis-Ooguri limit, and so is described by the same parametrisation with µ = 1 and (X, Y ) interchanged with (X,Ȳ ) (we could also change the sign of the original B-field). This maybe provides an interesting interpretation of the singularity in the background of a negative brane: the string theory becomes non-relativistic at the special pointH = 0.
• c = 0 corresponds toH = 0 and the spacetime is singular. We see though that the generalised metric (3.25) is well-defined, and given bỹ which clearly describes the same type of non-Riemannian background as the Gomis-Ooguri limit, identifying H = +2µ.

Example: Newton-Cartan non-relativistic geometry from null duality
As well as the Gomis-Ooguri limit, the Morand-Park classification can be used to describe various non-relativistic backgrounds. Here we will discuss an example which did not in fact appear in [48], namely a version of non-relativistic Newton-Cartan geometry which can be obtained from a Lorentzian geometry by a null T-duality. This was used in [81,82] to obtain the Polyakov action for a string in the Newton-Cartan geometry (see also the description of T-duality in the "string Newton-Cartan" background of [83], which showed that a non-relativistic string was T-dual to a Lorentzian background with a null isometry). This procedure naturally lives in the doubled formalism. Note that this will therefore again be an example of a (1, 1) non-Riemannian generalised metric (whereas the non-relativistic examples in [48] had n =n): one of the advantages of starting with a (0, 0) generalised metric and dualising is that we know what happens to the generalised dilaton and can therefore uplift to ExFT later on. Here we follow the notation and conventions of [81,82]. We start with the metric for a d + 1 dimensional Lorentzian spacetime with a null isometry, which can always be put in the form where u denotes the null direction, and the d dimensional matrix h µν has rank d − 1. The fields τ µ , m ν and h µν together describe a torsional Newton-Cartan geometry and transform under Galilean local symmetries in a particular way that need not concern us here. We also introduce a vector v µ and a rank d − 1 matrix h µν such that It is convenient to also definē which are invariant under Galilean boosts and rotations, but not under U(1) gauge transformations of m µ , δm µ = ∂ µ λ. In fact, the completeness relation holds with these variables, h µρh ρν −v µ τ ν = δ µ ν . We can then compute the inverses of g ij andh µν : We embed the Lorentzian background in the generalised metric H M N = diag (g ij , g ij ). We exchange the direction u for a dual directionũ using the analogue of a Buscher transformation (on the components of the generalised metric, this amounts to swapping the u and u indices). The dual generalised metric is This does not admit a Riemannian parametrisation. Instead, it is again of type (1, 1), with: Observe that generalised diffeomorphisms include gauge transformations δB ij = 2∂ [i λ j] , which provide the U(1) transformations of m µ on noting we have ∂ i = (∂ µ , ∂ũ) and ∂ũ = 0 (i.e. we do not depend on the direction dual to the original isometry direction). This is as expected, as prior to dualising these transformations were part of the diffeomorphism symmetry of the original metric.
The other DFT field which is present is the generalised dilaton, which is invariant under O(D, D) and so given by e −2d = | det g|, where This provides a measure factor for the Newton-Cartan geometry.
It is straightforward to check that inserting the above non-Riemannian generalised metric (3.35) in a doubled sigma model and integrating out the dual coordinates reproduces the string action of [82] (one can use for instance the general result of [48]).

Riemannian backgrounds and exotic supergravities in SL(5) ExFT
We will now focus on the SL(5) ExFT [4,22,25], a good testing ground as it is simple enough to allow one to realise various constructions very explicitly, and simultaneously complex enough to be interesting. Already at the level of Riemannian parametrisations, the SL(5) ExFT describes not only the conventional 10-and 11-dimensional supergravities, but exotic variants [25], with all information about the nature of the spacetime theory is encoded in the generalised metric via the choice of parametrisation.

Spacetime decompositions
In general, in order to match exceptional field theory with standard supergravity, it is convenient to start with an intelligent decomposition of the fields of the latter. For instance, the 11-or 10dimensional Einstein frame metricĝμν can be decomposed in the following manner (corresponding to a partial fixing of Lorentz symmetry): splitting the 11-or 10-dimensional indexμ = (µ, i), where µ is an n-dimensional index, let where ω is the intrinsic weight appearing in the generalised Lie derivative (listed in Table 1). For SL (5), ω = −1/5. The ExFT formalism will work regardless of the signatures of the blocks g µν and φ ij . We will denote the signature of metrics by (t, s). Let φ ij be a d-dimensional metric with Alternatively we can evaluate (3.38) by replacing hµν withhµν so that det g = − dethh µν τµτν, hence and there are no extra signs in the contractions between ǫ with indices up and those with indices down.
As well as the metric, it can be convenient to redefine the components of the gauge fields which carry the external µ, ν indices, making use of the field A µ i . The details are not important in the present paper. The reader can consult the appendices of [84] for details adapted to the SL(5) case.

The SL(5) ExFT
For SL (5), the representation R 1 is the antisymmetric 10-dimensional representation; we will write an R 1 index M as an antisymmetic pair of five-dimensional indices a, b, so that V M ≡ V ab = −V ba .
We will contract indices with a factor of 1/2, The generalised Lie derivative is defined by giving the Y-tensor, which is Y M N KL = η aa ′ bb ′ e η cc ′ dd ′ e , and the section condition is η abcde ∂ bc ∂ de = 0.
when written out in terms of the spacetime metric,ĝμν (as in (4.1)), and spacetime coordinates, where the ellipsis denotes terms involving dual coordinates. Pullbacks of the expression (4.3) are used to construct particle and string actions with target space the extended geometry of ExFT, and the relative sign between the two terms is fixed by the appropriate notion of gauge covariance under the ExFT gauge symmetries [60,69]. As it is M M N that appears in (4.3), we stress that it is the parametrisation of this version of the generalised metric which must be considered fundamental, though we will almost always write down explicit expressions using the more compact notation of the little metric m ab . (Note we can also express m ab via m ab = 1 6 η aM N η bP Q M M P M N Q .) The gauge fields of the SL(5) ExFT appearing in the action are a one-form A µ M , two-form, B µνa with field strength H µνρa , and three-form, C µνρ a , whose field strength J µνρσ a appears in the Chern-Simons term but does not have a kinetic term. The equation of motion for C µνρ a accordingly amounts to a duality relation relating it to the degrees of freedom in the other gauge fields. The action is defined by and the Chern-Simons term is described in [22].

M-theory parametrisations
The M-theory solution of the section condition is based on splitting a = (i, 5), where i is a fourdimensional index, and choosing the physical coordinates to be Y i ≡ Y i5 and the dual coordinates to be Y ij , with the section condition solution then provided by ∂ i = 0, ∂ ij = 0. Generalised diffeomorphisms are generated by Λ ab = (Λ i5 , Λ ij ). The vector Λ i is then found to generate fourdimensional diffeomorphisms, while Λ ij = 1 2 η ijkl λ kl produces gauge transformations of the threeform. This allows us to parametrise the generalised metric in terms of the internal spacetime metric, φ ij , and the internal components of the three-form, C ijk . It is convenient to turn C ijk into a vector by defining v i ≡ 1 3! ǫ ijkl C jkl . Then we have: This parametrisation incorporates two sign factors. The first of these is (−1) t , which depends on the number of timelike directions t in φ ij . This appears in order that the generalised metric parametrise the correct coset SL(5)/SO(2, 3) rather than SL(5)/SO (5), and ensures that the determinant is +1. Such timelike variants of the classic G/H cosets were analysed in [65]. The second sign factor is denoted by λ, and controls the sign of the kinetic term of the three-form, providing an ExFT parametrisation for exotic variants of 11-dimensional supergravity related to timelike dualities [77,78]. See also [25] for an earlier discussion of such parametrisations in the SL (5) ExFT. The parametrisation of the big generalised metric that we use corresponds to Studying the gauge transformations of the ExFT gauge fields in this solution of the section condition, we find that the obvious components of the 11-dimensional three-form can be identified with certain components of the ExFT gauge fields, schematically A µ ij = 1 2 η ijkl C µkl , B µνi = C µνi , C µνρ = C µνρ (see the appendices of [84] for more precise relationships). Apart from the obvious identification A µ i = A µ i , the other components of the gauge fields are related to the dual 11-dimensional sixform, and can be eliminated from the ExFT action using duality relations. As a result, one finds by explicit calculation that the ExFT action is equivalent to that of 11-dimensional supergravity: In general we see that λ = +1 corresponds to the usual relative sign between the Ricci scalar and  (2,9) and containing M2 branes whose worldvolume has Euclidean signature [76][77][78].
We can summarise some of the sign choices appearing in the little generalised metric (4.7), with reference to figure 1: • The signature of φ ij is (0, 4) and λ = +1 so that the signature of m ab is (0, 5), and if the external metric has signature (1, 6) this describes the usual 11-dimensional SUGRA.
• The signature of φ ij is (1, 3) and λ = +1 so that the signature of m ab is (2, 3), and if the external metric has signature (0, 7) this describes the usual 11-dimensional SUGRA.
• The signature of φ ij is (2, 2), and λ = −1 so that the signature of m ab is (2, 3), and if the external metric has signature (0, 7) this describes the unusual 11-dimensional SUGRA with signature (2,9) and wrong sign kinetic term, the low energy limit of the M * theory (see diagram 1).
• Other choices can correspond to ExFT descriptions of other exotic variants of M-theory.

IIB parametrisations
For the IIB solution of the section condition we split a = (i, α) where i a three-dimensional index, and α is a two-dimensional index associated to the unbroken SL(2) S-duality symmetry of IIB. The physical coordinates are then the three coordinates Y ij . It can be convenient to view the i index as being naturally down, i.e. Y M = (Y ij , Y i α , Y αβ ), such that the physical coordinates can be defined to have the usual index position via Y i = η ijk Y jk .
The generalised diffeomorphism parameter Λ ab = (η ijk Λ k , Λ i α , Λ αβ ) now produces three-dimensional diffeomorphisms generated by Λ i , gauge transformations Λ i α of the two-form doublet, and gauge transformations Λ αβ ≡ ε αβ 1 3! η ijk λ ijk of the four-form singlet. The generalised metric can be parametrised in terms of the internal metric, φ ij , the two twoforms (C ij , B ij ) = C ij α (which we again write as vectors, v iα ≡ 1 2 ǫ ijk C jk α ), and a two-by-two matrix, H αβ , containing the dilaton Φ and RR zero-form C 0 . We write Again, we allow for a general distribution of sign factors when the coset is SL(5)/SO (2,3). Here the signs σ i = ± dictate whether the parametrisation corresponds to a set of variants of type IIB, We also identify the gauge fields such that (schematically) A µ αβ = ε αβ 1 3! η ijk C µijk and similarly for the higher rank fields. Then the SL(5) ExFT dynamics are equivalent to those following from the type pseudo-IIB action 6 of the form which matches the Einstein frame action exactly for the type IIB σ F σ D supergravities [76]. We see that the choice of signs σ F , σ D will determines which kinetic terms come with the wrong sign. When σ F = −1, the NSNS B-field does, while when σ D = −1 the RR two-form does.
We can summarise some of the sign choices appearing in the little generalised metric (4.11), with reference to figure 1: • The signature of φ ij is (0, 3), then we can describe either usual IIB ++ , in the SL(5)/SO (5) description, with the external metric of signature (1, 6), or also IIB −− , in the SL(5)/SO (2,3) description.
• The signature of φ ij is (1, 2), then we can describe IIB ++ (with Euclidean external metric) as the (−1) t factor flips the signature of the upper three-by-three block: this way we describe the usual Lorentzian supergravity (in [25] this was viewed as working with a mostly minus signature in the spacetime picture). We can also describe IIB +− or IIB −+ .
• Other choices can correspond to ExFT descriptions of other exotic variants of IIB supergravity.

Non-Riemannian backgrounds in SL(5) ExFT
We will now generate and describe non-Riemannian parametrisations of the SL(5) theory. We will first play with the same simple examples as worked in the DFT case: namely, a Gomis-Ooguri scaling limit, and a U-duality of the worldvolume directions of an M2 brane solution. We will then demonstrate how to think more systematically about such parametrisations in this ExFT.

Gomis-Ooguri
Consider the flat background where η αβ = diag (−1, 1, 1) and G and µ are tunable constants: we will take the G → ∞ limit keeping µ fixed. The relative scaling between the worldvolume coordinates z α = (t, z 1 , z 2 ) and the transverse coordinates x 8 is the same as in the M2 example considered in [43], however we need to rescale the overall metric by a factor of G 2/3 in order to obtain a finite generalised metric in the limit. We pick ExFT physical coordinates Y i5 = (z α , w) where w denotes any one of the x 8 directions. We take η t12w = +1. The only non-vanishing ExFT fields are the external metric and generalised metric (using the parametrisation (4.7) with, here, λ = t = +1): The G → ∞ limit is well-defined and non-singular at the level of the generalised metric, leading to should be proportional to the spacetime metric, is degenerate. We have therefore very easily generated an example of a non-Riemannian parametrisation in exceptional field theory.

Non-Riemannian geometry from U and S duality
Our next trick will involve dualising supergravity solutions. We will apply the following U-duality transformation: where n in i = 1. If the physical directions are indexed by i = 1, 2, 3, 4 and we want to do a Uduality acting on the 1, 2, 3 directions, then we let n 4 = 1 =n 4 . This U-duality then reduces (after reduction on the 3 direction, say) to a pair of Buscher T-dualities acting in the 1, 2 directions plus an interchange of the x 1 and −x 2 directions. The generalised metric transforms tom = U −T mU −1 .

U-duality between M2 and non-Riemannian background
The M2 solution, which in fact inspired the form of the Gomis-Ooguri background considered above, is One would normally have c = −1, such that the three-form vanishes at infinity, however we leave this constant general (corresponding to a large gauge transformation at infinity). We again pick ExFT physical coordinates Y i5 = (z α , w) with z α = (t, z 1 , z 2 ) and w denoting one of the x 8 directions.
The ExFT embedding of this solution (again, with λ = t = +1) is We dualise on the (isometric) (t, z 1 , z 2 ) directions. The SL(5) transformation (5.4) and transformed generalised metric are: The spacetime parametrisation that one should use form ab depends on the value ofH ≡ −2c−Hc 2 .
• IfH < 0, then the signature of the upper left 4 by 4 block is still (1, 3) so this admits the conventional geometric parametrisation. We find • IfH > 0, then the signature of the upper left 4 by 4 block is now (2,2). It seems we have a choice of whether to use the parametrisation with λ = 1 or λ = −1. The latter preserves the SO(8) invariance of the transverse directions, giving For c = −1 we haveH = 2 − H = 1 − h/r 6 . Then the solution (5.8) and (5.9) describes a negative M2 in the exotic M − 2,9 theory with two timelike directions. In the region outside the brane whereH > 0 and the spacetime is described by (5.9). Passing through the singularity atH = 0, the interior (H < 0) configuration (5.8) then has as usual flipped signature in the worldvolume directions (so in fact is described by M-theory with conventional signature). 7 The generalised metric is non-Riemannian atH = 0 when c = 0, and everywhere when c = 0, These are evidently non-Riemannian in the same manner as the Gomis-Ooguri example we considered previously.

S-duality between D(−1) and non-Riemannian background
We can also consider the generalised metric (5.6) of the embedding of the M2 in the SL(5) ExFT and this time interpret it in a type IIB solution of the section condition. The appropriate parametrisation corresponds to the type IIB +− theory (related by T-duality on the timelike direction to type IIA).
We find while the 10-d Einstein metric is flat. This can be interpreted as the solution for a D(−1) (normally this is obtained as a solution of the Euclideanised type IIB, a subtlety we will ignore. It seems natural to view the D(−1) here as having been obtained by timelike T-duality of a D0). We can relate this IIB parametrisation back to the M-theory one: the relationship is evidently given by T-dualising the D(−1) on all three physical coordinates to get a D2 which lifts to the M2 we started off by thinking about. 7 In [85] it is argued that the change of signature of spacetime should instead be viewed as a breakdown in the gauge fixing of the generalised vielbein, implying that it is impossible to everywhere describe the background in terms of a metric and a three-form. Instead, owing to the singularity atH = 0, we could introduce a parametrisation involving a dual metric and a trivector which will be globally defined on the dual geometry. Here we instead adopt the perspective that though the generalised metric is well-defined everywhere, the choice of parametrisation that one makes is discontinuous atH = 0. At the singularityH = 0, the geometry becomes non-Riemannian but on either side we have different Riemannian parametrisations in which spacetime has different signatures. Similar comments should apply to the DFT example considered previously, also.
The U-duality transformation in (5.7) now in fact corresponds to the S-duality that inverts the string coupling constant. The resulting configuration (5.7) can be simply described as: where σ F is determined by requiring e Φ be positive.
Let's consider the two cases of most interest: • When c = −1, we haveH = 2 − H as before, and the solution can be written everywhere as ForH > 0 we have σ F = −1, σ D = +1, so that the theory corresponds to IIB −+ , while for H < 0, we have σ F = +1, σ D = −1, so this is IIB +− . This therefore is a negative D(−1) in the IIB −+ theory. Note there is no signature flip (as we are dealing with a worldpoint).
• When c = 0, there is no parametrisation in terms of e Φ and C 0 , although the spacetime metric is well-defined. The "non-Riemannian" nature of the generalised metric is thus that the parametrisation of the part of the generalised metric, H αβ , describing the axio-dilaton of type IIB, is non-standard.

Non-Riemannian little metrics
We can begin to understand the presence of non-Riemannian parametrisations in this ExFT by looking anew at the generalised metric. We continue to use the little metric, and will work in an M-theory solution of the section condition. Then, a general parametrisation of the little metric is: assuming that it is symmetric and subject to the sole constraint that det m = 1, such that We will now attempt to solve this constraint to find allowed parametrisations of m ab .

Non-Riemannian solutions
Now, we suppose det k = 0. This takes us into the realm of non-Riemannian parametrisations, as evidently we will not be able to identify k ij as being proportional to the four-dimensional spacetime metric anymore. Define a vector Then the constraint (5.15) is equivalent to u i χ i = 1, and we have k ij u j = 0. Note that then k ij has rank 3, and we cannot satisfy (5.15) if it has lower rank. We introduce the almost inverseh ij such Then, the inverse metric is Explicitly,h ij = − 1 2 η iklm η jpqr χ m χ r k kp k lq . The parametrisations (5.14) and (5.19) together with the relations (5.18) completely encode the non-Riemannian parametrisation. To understand the quantities appearing, let us examine the transformation of the generalised metric under generalised diffeomorphisms [4] In the M-theory solution to the section condition, we let Λ ab = (Λ i5 , Λ ij ) such that where L Λ denotes the usual Lie derivative with respect to the parameter Λ i5 , with k ij , χ i ,φ, u i and h ij of weight −4/5, 1/5, 6/5, −1/5 and +4/5 respectively. Normally one views the transformations Λ ij as giving gauge transformations of the 11-dimensional three-form, with Λ ij = 1 2 η ijkl λ kl . We observe that though k ij and its zero vector To see how we could introduce a three-form into the non-Riemannian parametrisation trans-forming under the gauge transformations, suppose that we start with the degenerate k ij and its zero vector u i . If all we require of χ i is that it obeys u i χ i = 1, there is an ambiguity χ ∼ χ i + k ij b j for arbitrary vector b j . We can provide a partial fixing of this ambiguity by defining a particular covector X i such that u i X i = 1. Taking this as a reference, we can write and require that X i be wholly "geometric" in the sense that it is unchanged by the gauge transformations Λ ij . Then, we find that This can be factorised as m ab = (U T Vm U V ) ab where The relations in (5.18) are now: In this version of the parametrisation, as V i transforms under three-form gauge transformations as δ λ V i = 3η ijkl ∂ j λ kl , we can think of relating this to three-form as V i = 1 3! η ijkl C jkl . This form of the generalised metric (5.26) appears to contain the degrees of freedom we might expect for SL(5)/SO (2,3): in place of a Riemannian metric we instead have the degenerate k ij and X i , along with a three-form (encoded in V i ) and the extra scalar ϕ, subject to the constraint u i X i = 1. However, given the ambiguity in introducing V i , the situation is quite subtle. If we insist on using the parametrisation (5.26), then we must note that it is invariant under the shift symmetry: mirroring the shift symmetry (3.13) appearing in the O(D, D) non-Riemannian parametrisations of [48]. Note that for b i proportional to u i , X i is invariant while V i and ϕ transform. One can eliminate ϕ using the latter transformation, with b i = − 1 2 u i ϕ, or eliminate V i using b i = −V i . Furthermore, one can insert (5.26) into the SL(5) ExFT action (4.5) evaluated on the M-theory solution of the section condition to investigate see how ϕ and V i appear. We focus on the terms involving derivatives of generalised metric. In the "potential" V , we have: while the "kinetic term" involves: 8 Before discussing the interpretation of this, note that for these non-Riemannian backgrounds the time-like direction is in what is normally called the "internal space" and the external space is Euclidean (this must be the case as in the examples we looked at, we obtained the non-Riemannian background by U-dualising from a generalised metric in SL(5)/SO(2, 3), i.e. including the timelike direction in the extended space). Thus the names "potential" and "kinetic term" are misnomers inherited from the more usual situation where time is in the external space. The fact that V i and ϕ drop out of the action involving internal derivatives implies that there are no time derivatives for these fields and so their canonical momentum will vanish. This leads to (in the language of canonical quantisation) a first class constraint which we normally associate with a gauge symmetry.
The local symmetry is the shift symmetry we have identified and thus leads to the conclusion that there are fewer physical degrees of freedom than expected relative to the usual case. This should not be a surprise since having non-Riemannian directions will mean that forms will also have fewer degrees of freedom.
We have seen previously that, in the maximally non-Riemannian case, all the propagating degrees of freedom were projected out and that one should think of the coset as G/G. Now that we have some subset of non-Riemannian directions one will naturally have fewer degrees of freedom, as indicated by the presence of the shift symmetry and the first class constraint on V i . Constructing the coset description for each case with a different non-Riemannian structure is an invidious task that we leave for future work. The general story is that H enhances as more dimensions become non-Riemannian. One should also note that from the "kinetic terms" ie. the action containing derivatives of the external space on m ab there are additional constraints from the equations of 8 Here Dµ = ∂µ − LA µ with the generalised Lie derivative acting as explained in the text above.
motion, these will be second class. A full detailed analysis of the constraint structure is beyond this paper. What we wish to emphasise is that there is a reduction in the form degrees of freedom due to the non-Riemannian nature of the space which may be seen from the absence of time derivative terms for V i . The other (second class) constraints maybe viewed as providing a restriction on how a non-Riemannian space may be fibred over some other space. As such this is model provides a fascinating playground for studying different aspects of how non-Riemannian spaces are embedded in ExFT.
In spite of the above discussion, we can continue to make use of a SL(5)/SO(2, 3) description for the generalised vielbein. This is perfectly fine so long as we keep in mind the additional shift symmetry (5.29) that will ultimately lower the degrees of freedom in the coset by enhancing H.
Then, to construct the generalised vielbein, letā,b be flat five-dimensional indices, andī, be three-dimensional flat indices. The flat generalised metric can be taken to be where σ = ±1 and ηī is the three-dimensional Minkowskian metric. A family of generalised vielbein Eā a , satisfying m ab = Eā a Eb b ηāb, is provided by Here, α is an arbitrary non-zero constant and kī i k j ηī = k ij with kī i u i = 0.
We can also write down the big generalised metric, which will have the parametrisation where the sign factor will depend on the signature of k ij and on σ. Now, consider the example non-Riemannian generalised metric (5.3) we generated above by taking the Gomis-Ooguri limit (or equivalently those appearing after U-dualising the M2 supergravity solution, (5.10)). This embeds into the general form of the little metric (5.14) as a non-Riemannian parametrisation with: Alternatively, one could fix X i = χ i ,φ = 0 and write it in the form (5.26) in terms of the following dualised three-form V i = 0 0 0 µ , (5.36) which implies that we have C t12 = −µ, i.e. this three-form is only defined with legs in the threedimensional space orthogonal to the zero vector of k ij . This form of the solution is closest to the DFT non-Riemannian parametrisation (3.22) with a non-vanishing B-field.

A quick glance at IIB parametrisations
We could approach the issue similarly in a IIB solution of the section condition, writing with i, j = 1, 2, 3 the spacetime indices and α, β = 1, 2 the SL(2) indices. We have the unit determinant constraint We shall not study this in detail here. It is clear that one can take k ij to be non-invertible, meaning there is no standard spacetime metric. Examples could be found by taking Gomis-Ooguri-type limits of the F1 and D1 SUGRA solutions. However, there are also "non-Riemannian" possibilities that do not involve assuming lack of invertibility of k ij and H αβ . Indeed, the example ( in which the three-by-three and two-by-two blocks are invertible, but, as we explained before, the factor H αβ does not admit the standard parametrisation in terms of the dilaton and RR zero form as in (4.11). Thus not only the geometry, as encoded in the relationship between k ij and the spacetime metric, but the information about the string coupling can be modified in a generic parametrisation of the generalised metric.
, such that the components of the usual DFT generalised metric are given by Let us carry this process out for the non-Riemannian parametrisation in the original form (5.14), that is, let us write where now a = (î, 5) in order to rewrite the four-dimensional indices asî = (i, 4). We assume that k 44 = 0 and, for simplicity, that k 44 > 0 so the reduction is (in some sense) spacelike. We can then write a Kaluza-Klein style decomposition for the degenerate matrix kî, Then det k = k 44 detk and so detk = 0. The unit determinant condition (5.15) on the generalised metric implies The null vector becomes withk ij u i = 0,χ i u i = 1. Using (5.40) we find explicit expressions for the generalised dilaton, the RR fields, and the spin generalised metric: This leads, via (5.42), to the components of the vector generalised metric: where ± corresponds to the choice of sign in relating the parametrisation of M ab,cd to that of the little metric. We note that H ijk jk Y k = 0 (5.49) for arbitrary vector Y k . Now,k ij has only one zero vector u i , so its kernel has dimension one, and thus its image has dimension two. This implies that H ij has a two-dimensional kernel. We therefore identify it directly with the degenerate matrix H ij of the DFT parametrisation. We also see that it can be shown that this corresponds to: In this case, the zero vectors of K ij are those Y i such thatX i Y i = 0, while those of H ij are those X i such that X i = k ij Y j . This parametrisation amounts to a special choice of B-field such that Regardless of the ambiguity in directly identifying the blocks as in (5.50) and (5.51), the expressions (5.48) allow us obtain the full DFT generalised metric without ambiguities. For instance, it is simple to check that one can reduce the Gomis-Ooguri little metric (5.35) to match the parametrisation (3.22) obtained in the DFT case (this requires taking the minus sign in the ± of (5.50), and also noting that as our four-dimensional convention was that η t12w = +1, the three-dimensional alternating symbol after reducing on the directionî = 2 say would be η ijk with η t1w = −1). Similarly one can reduce the background resulting from U-duality of the M2 to find the generalised metric

Embedding Newton-Cartan
We will now offer three variations on a theme of Newton-Cartan. Recall from section 3.4 that the Newton-Cartan geometry is described by degenerate matrices h µν , h µν and their zero vectors v µ and τ µ . Here the index µ is d dimensional and h µν and h µν have rank d − 1. We will describe below methods to embed a Newton-Cartan geometry in an SL(5) description for d = 4, 3 and 2. We will only discuss the generalised metric, i.e the "internal" sector of the ExFT. Including the external metric and other degrees of freedom would extend this to putative 11, 10 and 9-dimensional non-relativistic geometries. The latter case corresponds to the embedding of the Newton-Cartan geometry discussed in the context of string theory in section 3.4, so that our description here might be thought of as an M-theory uplift of this. The 10-dimensional case would be obtained in M-theory by U-dualising an 11-dimensional metric along a null isometry direction. The 11-dimensional case would perhaps be best thought of as needing ExFT for a higher dimensional interpretation, in place of a 12-dimensional Lorentzian geometry. We leave detailed inquiry into these possibilities, and their role in an potential non-relativistic duality web, for future work.

d = 4 Newton-Cartan: direct non-Riemannian and Riemannian embeddings
The structure of our non-Riemannian parametrisation, in terms of degenerate metric-like quantities with zero vectors is clearly very similar to that of the Newton-Cartan geometry. Indeed, given the Newton-Cartan structure h µν , h µν , v µ and τ µ (we will consider the gauge field m µ subsequently), with µ, ν = 1, . . . , 4, let us define first such that |g| 1/2 may serve as a density factor. (If we were to embed the Newton-Cartan geometry in a Lorentzian metric with a null isometry as in section 3.4, then g is indeed the determinant of this metric.) Then the following non-Riemannian SL(5) parametrisation describes this geometry: where we have to include a sign factor g |g| , in order that det m ab = 1. (Note the form m ab itself resembles that of a five-dimensional metric with a null isometry.) Curiously, we can include the Newton-Cartan gauge field m µ in the form of conjugation by the following SL(5) element (which seems natural from the point of view of the index structure of m µ ): Ordinarily, the factorisation involving U W would be associated with a trivector Ω µνρ defined by The presence of the trivector, analogous to the bivector which appears in O(d, d), is usually associated to non-geometry (for instance, it provides a potential for non-geometric fluxes [35]). This may signal here that we should really interpret this generalised metric in terms of a "dual" solution of the section condition. Furthermore,m ab now actually represents a Riemannian parametrisation as deth = −2Φe 2 = 0. We leave a detailed understanding of this geometry, and its local symmetries, for future work. The second approach mimics what we did in section 3.4. We start with the form of the Lorentzian metric (3.31) with null isometry, embed this into the SL(5) ExFT and then U-dualise. That means we are thinking of (3.31) as describing an 11-dimensional metric, which may provide a route to a 10-dimensional Newton-Cartan geometry. We focus on a four-dimensional part of the metric: where now µ is a three-dimensional index, andh µν = h µν − 2τ (µ m ν) as before. We insert this into the Lorentzian parametrisation of the little metric (4.7) with λ = 1 and (−1) t = −1. Then we U-dualise via (5.4) with n i = (n µ , 0). This corresponds to a U-dualisation on the null isometry direction u and two of the three-directions indexed by µ.

(5.57)
This becomes non-Riemannian for n µ = τ µ and n µ = −v µ or n µ = −v µ . For instance, in the latter case we have The final approach we take is to start with the O(3, 3) description of the Newton-Cartan geometry, which has a concrete origin in string theory via the dualisation procedure, and uplift this. So, we start again with the Newton-Cartan variables h µν , h µν , τ µ , and v µ , where now µ is a two-dimensional index. The Newton-Cartan geometry is especially simple as h µν and h µν have rank 1, and can be written in general as where such that h µ v µ = 0 = h µ τ µ , h µ h µ = −1 and the completeness relation holds, h µρ h ρν − v µ τ ν = δ µ ν . Here our conventions are that ε 12 = 1, ε 12 = 1 and so ε µρ ε νρ = +δ ν µ (ε µν rather than η µν to denote the alternating symbol as the latter may be confused for a metric). We have chosen to parametrise h µν in terms of a positive function e 2 > 0, as required by going back to the Lorentzian metric from which the Newton-Cartan geometry can be obtained by null duality, which has det g = −e 2 , using the formula (3.38).
We can then reverse engineer the SL(5) generalised metric which reduces to the O(3, 3) Newton-Cartan generalised metric (3.35). Using (5.48), we find that the SL(5) ExFT uplift of the Newton- Note that τ µ τ ν = e 2 ε µρ ε νσ h ρσ . It is possible to find U-duality transformations that take it to a Riemannian background -we describe a couple of possibilities in appendix C.

Discussion
Exceptional Field Theory was constructed to reproduce ordinary supergravity but with the additional feature that it could manifestly include the different perspectives of U-duality related geometries in M-theory. Since its inception, some surprises have emerged, and the full spectrum of theories and backgrounds which can be accommodated in the ExFT (and DFT) framework is still being uncovered. ExFT (and DFT) can describe many variants of supergravity theories. By focusing on non-Riemannian parametrisations, we extend the range of DFT/ExFT further, finding that it can accommodate non-relativistic theories and theories seemingly without a standard dynamical gravity at all in the form of the maximally non-Riemannian solutions which gave either chiral string theory (in DFT) in a beta-gamma sense [57] or a topological three-dimensional theory (in E 8 (8) ExFT).
In this paper we have extended the results of [46][47][48][49] and initiated the study of non-Riemannian backgrounds appearing in exceptional field theory. We have shown how the E 8(8) ExFT contains a maximally non-Riemannian solution with no moduli. In this background the ExFT becomes the topological theory previously described by Hohm and Samtleben [63] as a truncation of ExFT.
For the SL(5) ExFT, we have shown how to parametrise the generalised metric in order to obtain theories of non-relativistic type, related to the Gomis-Ooguri scaling limit and to Newton-Cartan type geometries.
The next stage is to understand these maximally non-Riemannian backgrounds and more of their properties. One can perhaps think of there being a full "package" associated to DFT/ExFT which is by now well understood and incorporates for instance geometric notions (such as connections and curvatures), supersymmetry, Scherk-Schwarz compactifications, and aspects of the descriptions of strings and branes [87,88]. This package should be brought to bear on understanding theories of non-Riemannian geometry and their role in string and M-theory.
A natural step is to examine sigma models in these backgrounds and determine their quantum consistency. An immediate observation is that the doubled sigma model in such a background contains chiral theories [48], and thus the quantum consistency of the partition function is highly constrained [57]. The situation is similar for the exceptional sigma model of [59,60], which provides a way to describe strings and D1 branes in ExFT non-Riemannian backgrounds (likely reproducing and generalising the results of the very recent paper [89]). Furthermore one could study the general brane actions of [61] in such backgrounds. The closer study of sigma model descriptions will be reported in further work [90]. A key issue would be to determine the thermodynamics of DFT/ExFT and, as such, determine questions regarding the phase structure of these theories, and where the E 8(8) topological phase described here fits into M-theory. It is remarkable that ExFT is a theory that contains both topological and "metric" phases; it would be further revealing if one could view the "metric" phase as a spontaneously broken phase of the ExFT and thus the metric and other fields would be like Goldstone modes.
In general, the inclusion of non-Riemannian geometries in M-theory using ExFT is another example of the surprises of the formalism. The potential role for non-Riemannian geometries to fix the moduli of the Kaluza-Klein dimensions and its implications for string phenomenology needs extensive study but the basic idea that the hidden dimensions could have Newton-Cartan form or even be maximally non-Riemannian could have significant implications for how we see the string landscape.
generalised metric (5.40) with , (A.1) and the usual generalised dilaton e −2d = e −2Φ |g|. Here B i ≡ 1 2 η ijk B jk is the B-field, C ijk is the RR three-form and C i the RR one-form. The number t denotes the number of timelike directions present in g ij , which is the string frame metric. The sign factor η takes into account whether the reduction is spacelike (η = +1) or timelike (η = −1). Using the result (5.42), with the sign factor ± replaced by (−1) t ηλ, we find that The following sign factors are possible: • λ = +1, η = +1 corresponds to a spacelike reduction from conventional M-theory to conventional IIA, giving the usual DFT generalised metric • λ = +1, η = −1 corresponds to a timelike reduction from conventional M-theory to exotic IIA −+ , giving the generalised metric for the exotic DFT − constructed in [80].
• λ − 1, η = −1 corresponds to a timelike reduction from M − to IIA +− , giving the usual DFT generalised metric. Here though, the RR fields will have the wrong sign kinetic term

A.2 IIB parametrisation
We can start with the IIB parametrisation (4.11) and rewrite in terms of a string frame metric g ij = e Φ/2 φ ij . The sign factor η is identified with σ F . It is convenient to also raise all the spinorial indices in the decomposition of the little metric (5.40). This reflects the fact that the RR spinors in a IIB parametrisation of DFT have opposite chirality to those in a IIB parametrisation. Thus, we write and the usual generalised dilaton e −2d = e −2Φ |g|. Again B i ≡ 1 2 η ijk B jk is the B-field, while C ij is the RR two-form and C 0 the RR zero-form. The number t denotes the number of timelike directions present in g ij . We now get for the generalised metric (the coordinates are now given by , using that the ± factor is given by (−1) t in IIB parametrisations of the SL(5) ExFT, We thus see that when σ F = +1, we get the usual generalised metric of DFT, while when σ F = −1, we get that of DFT − , as expected to describe IIB −+ .
A.3 DFT non-Riemannian parametrisation from SL(5) non-Riemannian parametrisation In this subappendix, we explain how the equations (5.48), namely, which is not unique but is particularly natural from the reduction point of view as it is clearly of rank 1. Other choices of K ij can be obtained using shift symmetries. Having committed to this choice, we note that H i j K jk = 0 = H ij H j k . Now, we let Z i j ≡ X a i Y j a −Xā iȲ j a in terms of the preferred basis of zero vectors of K ij and H ij . As H i j = Z i j + B ik H kl we see that B ij H jk K kl = 0. We then need to solve ± (k 44 ) 1/2γk ij = B ik Z j k + B jk Z i k , (A. 10) with Z as in (A.9). Contracting with u i , the left-hand-side vanishes. Writing B ij = η ijk B k we can then show thatk ij B j = 0 and hence B i = βu i . Inserting this back into (A.10) it follows that β = − 1 2γ . Hence, we find .11) B A practical guide to the E 8 (8) algebra In this appendix, we give a comprehensive account of the E 8 (8) algebra, in order to compensate for some gaps in the literature and to fix our conventions for the Killing form.

B.1 Some projectors
In terms of the Killing form, the projector onto the adjoint takes on the form

B.2 The SL(9) basis
In the SL(9) basis, the algebra of E 8 (8) is [91]:  Note that [92] appears to be missing the commutators between the 3-index and 2-index operators.
We now fix the coefficients a, α, . . . , δ to convenient values. In the conventions of [93] (which we follow), the contraction of two E 8(8) adjoint indices is given by for which the identity matrix is given by 9 1 M N = diag δ m n , δ n 1 n 2 m 1 m 2 , δ m 1 m 2 m 3 n 1 n 2 n 3 , δ m 1 n 2 δ n 1 m 2 − Note that the coefficients a and α, . . . , δ are not actually entirely free; (a, α) are still constrained by the traceless condition of the SL(9) generators tmn. In particular, the traceless condition for the SL(9) generators is still satisfied tmm = t m m + t 9 9 ≡ 0 since t m m + t 9 9 = E m m − 1 α (8a + 1)N (B.38) 9 One may verify that the trace of this, with the above contraction convention, gives δ M M = 248 as required. 10 Note that one may fix α, . . . , δ to conform with the other common contraction convention which vanishes for our choice of (a, α). In fact, since E m m = 0 this means that E m n generate an SL(8) algebra, which is augmented to GL(8) by the final generator N . Actually, because we have treated N as an independent generator, it is obvious that α may be set consistently to any value (barring zero). One may find it more convenient to choose a = − 1 8 , fixing α, and treat the tracefull E m n as generators of GL (8). In this case, their trace N , is no longer an independent generator.
Additionally, (γ, δ) should be related since they descend from the same set of generators.