U-gravity : ${\mathbf{SL}(N)}$

We construct a duality manifest gravitational theory for the special linear group, ${\mathbf{SL}(N)}$ with $N{\neq 4}$. The spacetime is formally extended, to have the dimension $\textstyle{\frac{1}{2}} N(N-1)$, yet is `gauged'. Consequently the theory is subject to a section condition. We introduce a semi-covariant derivative and a semi-covariant `Riemann' curvature, both of which can be completely covariantized after symmetrizing or contracting the ${\mathbf{SL}(N)}$ vector indices properly. Fully covariant scalar and `Ricci' curvatures then constitute the action and the `Einstein' equation of motion. For $N\geq 5$, the section condition admits duality inequivalent two solutions, one $(N-1)$-dimensional and the other three-dimensional. In each case, the theory can describe not only Riemannian but also non-Riemannian backgrounds.

Unlike the Lorentz symmetry unification of space and time, the duality manifest extension of the spacetime calls for the existence of seemingly unphysical 'dual' spacetime. One simple prescription to eliminate this unphysical feature is to let all the fields be independent of the dual coordinates, e.g. [6,7] and 'Generalized Geometry' [8][9][10][44][45][46][47][48]. More covariant method is to enforce so called a section condition on all the functions defined on the extended spacetime. The section condition is a differential constraint and can be solved by a certain hyper-subspace, called 'section', on which the theory is restricted to live. Duality then rotates the section in the extended spacetime. Especially, acting on an isometry direction, it may produce a new solution while the section can remain unrotated. This is the very geometric insight that has motivated [6,7] or Double Field Theory (DFT) [49][50][51][52]. Fixing the section explicitly and parametrizing the DFT variables by Riemannian ones, DFT may locally reduce to Generalized Geometry. Then, like T-fold [53][54][55], by combining diffeomorphism and O(D, D) rotation as for a transition function, DFT may acquire nontrivial global aspects of non-geometry [56][57][58][59][60].
Further, once formulated in terms of genuine O(D, D) covariant variables, DFT does not merely repackage Generalized Geometry or known supergravities, but can also describe non-Riemannian backgrounds where the notion of Riemannian metric ceases to exist even locally [61]. In a somewhat abstract level, the DFT-metric can be defined simply as a 'symmetric O(D, D) element', with which (bosonic) DFT and a doubled string world-sheet action [61] still make sense. For most ("non-degenerate") cases the DFT-metric can be parametrized by Riemannian metric, g µν and Kalb-Ramond B-field, which allows DFT to describe an ordinary Riemannian gravity. But, for the other ("degenerate") cases the DFT-metric may not admit any Riemannian interpretation, even locally! An extreme example is the DFT vacuum solution where the DFT-metric coincides with the O(D, D) invariant constant metric. The doubled string action then reduces to a chiral sigma model [61], similar to [62].
As demonstrated in Refs. [57,61], imposing the section condition is, in fact, equivalent to postulating an equivalence relation on the doubled coordinate space. That is to say, spacetime is doubled yet gauged.
Accordingly, each equivalence class or gauge orbit represents a single physical point, and diffeomorphism symmetry means an invariance under arbitrary reparametrizations of the gauge orbits. This allows more than one finite transformation rule of diffeomorphism [56][57][58]. The idea has been pushed further to construct a completely covariant string world-sheet action on doubled-yet-gauged spacetime [61], where the coordinate gauge symmetry is realized literally as one of the local symmetries of the action. In a way, understanding the section condition by gauged spacetime agrees with the lesson learned from the 20th century that 'local symmetry dictates fundamental physics'. Table 1: Dynkin diagrams for A N −1 , D N −1 , E N −1 and E N .
• Complete covariantizations of them dictated by a projection operator. Especially when N = 5, the constructed theory of U-gravity reduces to our preceding research of 'SL (5) U-geometry' [30] (c.f. [22]). The present paper generalizes our previous work to an arbitrary special linear group, SL(N ) with N = 4, and also contains some novel findings, such as the semi-covariant Riemann curvature and an eight-index projection operator.
In the next section we spell out all the essential elements that constitute the SL(N ) U-gravity. Exposition will follow in section 3. We conclude with outlook in the final section. 2 Alternative approaches include [71][72][73][74][75][76][77].

Constitution of SL(N ) U-gravity
Essential elements that constitute SL(N ) U-gravity are as follows.
• Extended-yet-gauged spacetime. The spacetime is formally extended, being 1 2 N (N −1)-dimensional. The coordinates carry a pair of anti-symmetric SL(N ) vector indices, and hence so does the derivative, However, the spacetime is gauged: the coordinate space is equipped with an equivalence relation, and ϕ are arbitrary -but not necessarily covariant-functions in the theory of U-gravity. As usual, ǫ c 1 c 2 ···c N denotes the totally anti-symmetric Levi-Civita symbol with ǫ 12···N ≡ 1. Apparently, the above equivalence relation makes sense for N ≥ 5. For N = 2, 3, the spacetime is not to be gauged.
• Section condition. The invariance under the coordinate gauge symmetry (2.4) is, in fact, equivalent to a section condition, Acting on arbitrary functions, Φ, Φ ′ , and their products, the section condition leads to • Diffeomorphism. Diffeomorphism symmetry in SL(N ) U-gravity is generated by a generalized Lie derivative, Here we let the tensor density, T a 1 a 2 ···ap b 1 b 2 ···bq , carry the 'total' weight, 1 2 p − 1 2 q + ω, such that each upper or lower index contributes to the total weight by + 1 2 or − 1 2 respectively, while ω corresponds to a possible 'extra' weight.
In particular, the generalized Lie derivative of the Kronecker delta symbol is trivial, 9) and the commutator of the generalized Lie derivatives is closed by a generalized bracket [25], It is a somewhat surprising result of us that the above definition of the generalized Lie derivativeincluding the total weight-is independent of the rank of the duality group, or N , and thus is identical to the known one in [25,78] for the case of N = 5. • Semi-covariant derivative and semi-covariant Riemann curvature. We define a semi-covariant derivative, (2.14) The semi-covariant derivative obeys the Leibniz rule and annihilates the Kronecker delta symbol, • Connection. The connection of the semi-covariant derivative and the semi-covariant Riemann curvature is given by where we set This connection is the unique solution to the following five constraints: 3 The first relation (2.20) is equivalent to the U-metric compatibility condition, The second condition (2.21) is natural from the skew-symmetric nature of the coordinates, x (ab) = 0 and hence ∂ (ab) = ∇ (ab) = 0. The next two constraints, (2.22) and (2.23), make the semi-covariant derivative compatible with the generalized Lie derivative as well as with the generalized bracket, The last formula (2.24) is a projection condition which we impose intentionally in order to ensure the uniqueness. 3 See [64,65] for the analogous constraints in DFT.
• Projection operator. The eight-index projection operator, used in (2.24), is explicitly, (2.28) Crucially, the projection operator dictates the anomalous terms in the diffeomorphic transformations of the semi-covariant derivative and the semi-covariant Riemann curvature, Ω abcd = P abcd klm n ∂ kl ∂ me X ne . (2.29) • Complete covariantizations. Both the semi-covariant derivative and the semi-covariant Riemann curvature can be fully covariantized by (anti-)symmetrizing or contracting the SL(N ) vector indices properly [30], (2.31) • Action. The action of SL(N ) U-gravity is given by the fully covariant scalar curvature, where the integral is taken over a section, Σ.
• The Einstein equation of motion. The equation of motion corresponds to the vanishing of the 'Einstein' tensor, Diffeomorphism symmetry of the action implies a conservation relation, • Two inequivalent sections. Up to SL(N ) duality rotations, there exist two inequivalent solutions to the section condition, which we denote here by Σ N −1 and Σ 3 .
More than one solution to a section condition has been also reported in EFT [37,38].

Exposition
In this section we provide detailed exposition of the main results listed in section 2. All the mathematical analyses are parallel to those in the DFT-geometry of [57,61,64,65,80].

Equivalence between the coordinate gauge symmetry and the section condition
Here, following a parallel argument in DFT [61], we show the equivalence between the coordinate gauge symmetry invariance (2.4), and the section condition (2.7), Note that, in (3.1) we put a continuous real parameter, s, in order to control the shift.
First of all, from the standard series expansion of Φ(x + s∆) in s, it is clear that the strong constraint, (3.2), implies the invariance (3.1). The converse is also true: taking derivative at s = 0, we get This shows that the invariance (3.1) indeed implies the strong constraint (3.2). Further, from the strong constraint, it follows that the following N (N −1) Since any nilpotent matrix is traceless 5 , we have which leads to the weak constraint (3.3), In this way, the strong constraint (3.2) implies the weak constraint (3.3), and is actually equivalent to the coordinate gauge symmetry invariance (3.1). This completes our proof.

Projection operator
The eight-index projection operator (2.27), satisfies the 'projection' property, The verification of this identity requires straightforward yet tedious computations, which can be simplified by noting symmetric properties, and 'trace' properties, P abs sklmn = 0 . It is also useful to note (3.13) As we shall see below, the projection operator plays crucial roles in U-gravity. 6 Compared to the ordinary Riemannian geometry, the existence of a projection operator and its key role appear to be novel distinct features of the extended-yet-gauged spacetime geometries, such as DFT-geometry in [64][65][66][67][68][69][70] and the present SL(N ) U-gravity.

Compatibility of the semi-covariant derivative
Here we discuss the compatibilities of the semi-covariant derivative, firstly with the generalized Lie derivative, secondly with the generalized bracket, and lastly with the U-metric. 6 The construction of the projection operator (3.8) is one of the major improvements made in this paper compared to the previous work on SL(5) U-geometry [30]. An operator therein, called J abcd klmn , is consistently related to the projection operator by J abcd Specifically, we start by postulating the generalized Lie derivative and the semi-covariant derivative to take the following forms, Here, α(p, q, ω) andᾱ(p, q, ω) are yet-undetermined total weights which may depend on p, q, ω, i.e. the numbers of upper, lower indices and the extra weight. Below, in section 3.3.1, by demanding the compatibility with the generalized Lie derivative, we shall fix the dependency and derive the final expression, which is linear in p, q, ω and remarkably independent of N . This result will, in particular, ensure that both the generalized Lie derivative and the semi-covariant derivative annihilate the Kronecker delta symbol, This implies that the duality invariant integral measure with unit extra weight is It is instructive to note that, irrespective of the choice of α(p, q, ω), upon the section condition, the commutator of the generalized Lie derivative is closed by a generalized bracket [25], Further, it is obvious from this expression that the generalized bracket satisfies up to the section condition, (3.21)

Compatibility with the generalized Lie derivative
If we replace all the ordinary derivatives by semi-covariant derivatives in the definition of the generalized Lie derivative expressed in (3.14), we get where we set for the parameter, X ab ,β ≡ᾱ(2, 0, 0) .

(3.23)
The compatibility of the semi-covariant derivative with the generalized Lie derivative means that the right hand side of (3.22) should vanish algebraically. In order to achieve this, it is required that the four-index quantity, Γ [abc] d , should be, at least, related to the two-index quantities, Γ eab e and Γ abe e . There is one unique such an ansatz which is self-consistent, 7 Note that the left and right hand sides of this ansatz share the same anti-symmetric properties, and also that the contractions of the two indices, one lower and the other upper (for example c and d ), agree. 7 The division by N − 2 in (3.24) needs not cause any alarm to exclude the case of N = 2, since after all we shall have Assuming the ansatz (3.24), the expression (3.22) reduces to (3.25) Now, each line above should vanish separately. More precisely, with the skew-symmetry, we should require

27)
Γ c(ab) c = 0 . There is a good reason for the contraction, Γ abc c , to be nontrivial: as we shall discuss more in section 3.3.3, the compatibility of the semi-covariant derivative with the U-metric, and hence with its determinant, implies for some value 8 of ω * , For this to hold, Γ abc c should not vanish in general. Thus, Eq.(3.29) tells us In particular, for the special case of p = 2, q = 0, ω = 0, this reduces to where, like (3.23), we set β ≡ α(2, 0, 0). Hence, we have eitherβ = 2 N or β = 1 2 . Ifβ = 2 N , Eq.(3.31) would get simplified to giveᾱ(p, q, ω) = p−q N . However, this is not a desired result. In order to meet the compatibility with the U-metric determinant (3.30),ᾱ(p, q, ω) must depend nontrivially on ω rather than being independent of it. Therefore, we should choose β = 1 2 . Now, rather than trying to look for the most general solution, for simplicity we focus on the case of β =β = 1 2 and search for a 'linear' solution. Then, Eq.(3.31) implies a more generic equality, α =ᾱ, and naturally we are lead to the final expression for the total weight, i.e. (3.15

Compatibility with the generalized bracket
If we replace all the ordinary derivatives by semi-covariant derivatives in the definition of the generalized bracket (3.20), we get, in a similar fashion to (3.22), which further reduces, with the ansatz (3.24), to (3.36) In order to meet the compatibility, each line should vanish separately. Hence, we require which in fact coincide with (3.27) and (3.28). Thus, puttingβ ≡ 1 2 , we re-derive (3.33) and, from (3.24), we arrive at the same conditions as before for the connection (3.34),

Compatibility with the U-metric
Having fixed the total weight to be 1 2 ( 1 2 p − 1 2 q + ω) as (3.15), the compatibility of the semi-covariant derivative with the U-metric reads Thus, the metric compatibility (3.39) is equivalent to It is useful to note (3.42)

Determining the connection uniquely
Here, we derive the connection (2.18), We start by recalling the five conditions for the connection, The first condition (3.46) is equivalent to the metric compatibility, ∇ ab M cd = 0, as discussed in sec- less' conditions. These are all -including the projection condition-analogous to the DFT-geometry of [65].
While the first condition, (3.46), fixes the symmetric part of the connection, the remaining ones should determine the anti-symmetric part, and also from (3.47), (3.49), We then only need to determine 56) which satisfies, by construction, symmetric properties, 57) and contributes to the connection through Now, all the constraints except the last one (3.50), boil down to On the other hand, the last projection condition (3.50) fixes Y abcd uniquely, Following the method in [30], the uniqueness can be also verified directly. First, it is straightforward to check that the connection given in and hence, the verification of the uniqueness,

Semi-covariant derivative and its complete covariantization
The infinitesimal diffeomorphic transformation of the U-metric, induces upon the section condition, and hence It is then straightforward to derive the variation of the connection under diffeomorphism, For consistency, this expression is compatible with all the properties of the connection, such as which can be easily verified using e.g. the projection property 'P(1 − P) = 0' (3.9) and an identity, Further, up to the section condition, we have It is crucial to note that the last term in (3.69), which we put hereafter 9 generates 'anomalous' terms in the variation of the semi-covariant derivative acting on a generic covariant tensor density, The second line is the anomalous part. Hence, the semi-covariant derivative of a generic covariant tensor density is not necessarily covariant. 10   These ensure that, for consistency, the followings are exceptionally, fully covariant.
i) The U-metric compatibility (3.46), ii) Scalar density with an arbitrary extra weight, iii) Kronecker delta symbol, The key characteristic of the semi-covariant derivative is that, by (anti-)symmetrizing or contracting the SL(N ) vector indices in an appropriate manner, it can generate completely covariant derivatives acting on a generic covariant tensor density, (2.30), Of course, from the U-metric compatibility, ∇ ab M cd = 0, the SL(N ) vector indices above can be freely raised or lowered without spoiling the full covariance. For example, the following is also fully covariant along with (3.80), Especially, for the case of q = 0, the divergence (3.84) reads explicitly, and hence, This is a useful relation for the discussion of the 'total derivative' or 'surface integral' for the action.
Successive applications of the above procedure to a scalar and a vector -or directly from (3.91)-lead to the following second-order covariant derivatives, which turn out to be all trivial due to (3.47), (3.48), (3.49) and the section condition. Similarly, for arbitrary a scalar and a vector, we have an identity, It is worth while to note, from (3.74), Further, from (2.24) and (3.75), we have which also implies with (2.21) and (3.75),

Semi-covariant Riemann curvature and its complete covariantization
The commutator of the semi-covariant derivative leads to an expression, where R abcde f denotes the standard "field strength" of the connection, (3.100) The semi-covariant Riemann curvature can be rewritten, using the semi-covariant derivative, the fake curvature varies, (3.111) 12 Again, this is precisely analogous to the DFT-geometry, c.f. Eq. (27) in Ref. [65]. 13 Note that S ab and S are related to 'R ab ' and 'R' in [30] by factor two: S ab = 2R ab , S = 2R.
For later use, it is worth while to have an explicit expression of the completely covariant scalar curvature, where, as defined before (3.41), Hence, disregarding surface integral, arbitrary variation of the U-metric induces the following transformation of the U-gravity action (2.32),

Reductions
Here we discuss the reduction of SL(N ) U-gravity upon each section, Σ N −1 and Σ 3 separately.
Here φ, v α and g αβ denote a scalar, a vector and a Riemannian metric on Σ N −1 , such that v α = g αβ v β , v 2 = g αβ v α v β and g = det(g αβ ). The vector can be freely dualized to an (N −2)-form potential which may couple to an (N −3)-brane.
It is crucial to note that a nontrivial assumption has been implicitly made to write the ansatz (3.116), namely that the upper left (N − 1) × (N − 1) block of the U-metric is non-degenerate and hence we are allowed to write " g αβ / |g| " there. However, the rank of the (N − 1) × (N − 1) block can be N − 2 (but not less than that for the U-metric to be non-degenerate). The degenerate case then corresponds to a non-Riemannian background where the Riemannian metric ceases to exist.
Nevertheless, SL(N ) U-gravity has no problem with that. One example of such a non-Riemannian background is given by a U-metric of which the only nontrivial components are M 1N = M N 1 and Mαβ with 2 ≤α,β ≤ N − 1.
Writing (3.117), it has been assumed that the upper left 3 × 3 block of M ab is non-degenerate. But, in general, its rank can be less than 3. In fact, when N ≥ 6 the whole block can vanish: for example the only nontrivial components of the inverse of the U-metric can be, M µî = Mî µ and Mĩ wherê ı = 4, 5, 6 and 7 ≤ĩ, ≤ N . When N = 5, the rank of the 3 × 3 block is either 3 (non-degenerate) or at least 2 (degenerate).

Outlook
On the extended-yet-gauged spacetime, the usual differential one-form of the coordinate, dx ab , is not invariant under the coordinate gauge symmetry (2.3), and thus needs to be gauged, c.f. [30] Dx ab := dx ab − A ab , A ab ∂ ab ≡ 0 . Here a connection has been introduced which assumes the same 'value' as the coordinate gauge symmetry generator, or the shift (2.4). Essentially it gauges away the orthogonal directions to a chosen section.
The gauged one-form can be then used to construct an SL(N ) duality manifest world-volume action for an (N −3)-brane propagating in the extended-yet-gauged spacetime, as done for a string in [30] (c.f. [81][82][83]).
The notion of the cosmological 'constant' depends on the kind of differential geometry in use [65]. In SL(N ) U-gravity, the natural cosmological constant term reads Σ M 1 and the whole RR-sector is represented by a single Spin(1, 9) L × Spin(1, 9) R bi-spinorial object which is a priori O(10, 10) singlet. After diagonal gauge fixing of the doubled local Lorentz group, the single bi-spinorial object may decompose into various RR p-form potentials which are no longer O(10, 10) singlet but form an O(10, 10) spinor, to agree with [71,72]. On the other hand, in SL(11) U-gravity, the SL(11) group does not mix the RR nine-form potential with other RR p-form potentials, since only the nine-form potential enters the parametrization of the U-metric (2.39). This might shed light on the E 11 duality manifest reformulation of the maximal supergravity. But here we can only speculate.