Remarks on the non-Riemannian sector in Double Field Theory

Taking $\mathbf{O}(D,D)$ covariant field variables as its truly fundamental constituents, Double Field Theory can accommodate not only conventional supergravity but also non-Riemannian gravities that may be classified by two non-negative integers, $(n,\bar{n})$. Such non-Riemannian backgrounds render a propagating string chiral and anti-chiral over $n$ and $\bar{n}$ dimensions respectively. Examples include, but are not limited to, Newton--Cartan, Carroll, or Gomis--Ooguri. Here we analyze the variational principle with care for a generic $(n,\bar{n})$ non-Riemannian sector. We recognize a nontrivial subtlety for ${n\bar{n}\neq 0}$ that infinitesimal variations generically include those which change $(n,\bar{n})$. This seems to suggest that the various non-Riemannian gravities should better be identified as different solution sectors of Double Field Theory rather than viewed as independent theories. Separate verification of our results as string worldsheet beta-functions may enlarge the scope of the string landscape far beyond Riemann.


Introduction
This paper is a sequel to [1] which proposed to classify all the possible geometries of Double Field Theory (DFT) [2][3][4][5][6][7] by two non-negative integers, (n,n). The outcome -which we shall review in section 2is that only the case of (0, 0) corresponds to conventional supergravity based on Riemannian geometry.
Other generic cases of (n,n) = (0, 0) do not admit any invertible Riemannian metric and hence are non-Riemannian by nature. Strings propagating on these backgrounds become chiral and anti-chiral over n and n dimensions respectively.
It is noteworthy and relevant to this work that, all the geometrical notation of the covariant derivative,

Doubled but at the same time gauged string action
One of the characteristics of DFT is the imposition of the 'section condition': acting on arbitrary functions in DFT, say Φ r , and their products like Φ s Φ t , the O(D, D) invariant Laplacian should vanish We remind the reader that the O(D, D) indices are raised with J AB . Upon imposing the section condition, the generalized Lie derivative (1.4) is closed by commutators [3,7], The section condition is mathematically equivalent to the following translational invariance [8,83], where the shift parameter, ∆ A , is derivative-index-valued, meaning that its superscript index should be identifiable as a derivative index, for example ∆ A = Φ s ∂ A Φ t . This insight on the section condition may suggest that the doubled coordinates of DFT are in fact gauged by an equivalence relation, The expression of SAB in Table 1 is newly derived from [63] using ΓACDΓ CBD = Γ BCD ΓCAD = 1 2 ΓACDΓ BCD and ΓCADΓ DBC = ΓCADΓ CBD − 1 2 ΓACDΓ BCD which hold due to the symmetric properties, Γ [ABC] = 0 and Γ A(BC) = 0.
Each gauge orbit, i.e. equivalence class, represents a single physical point. As a matter of fact in DFT, the usual infinitesimal one-form of coordinates, dx A , is not DFT-diffeomorphism covariant, However, if we gauge the one-form by introducing a derivative-index-valued connection, we can have a DFT-diffeomorphism covariant one-form, provided that the gauge potential transforms appropriately, (1.10) It is also a singlet of the coordinate gauge symmetry (1.8): The gauged one-form then naturally allows to construct a perfectly symmetric doubled string action [84], [8],

.8) and similarly
, we note that the tilde coordinates are indeed gauged: . With respect to this choice of the section, the well-known parametrization of the DFT-metric and the DFT-dilaton in terms of the conventional massless NS-NS field variables [88,89], DFT and the doubled-yet-gauged string action work well, provided these conditions are fulfilled. For example, instead of (2.1), we may let the DFT-metric coincide with the O(D, D) invariant metric, This is a vacuum solution to DFT, or to the 'matter-free' EDFEs, which is very special in several aspects. Firstly, compared with (2.1), there cannot be any associated Riemannian metric g µν and hence it does not allow any conventional or Riemannian interpretation at all. It is maximally non-Riemannian. Secondly, it is fully O(D, D) symmetric, being one of the two most symmetric vacua of DFT, H AB = ±J AB . Thirdly, it is moduli-free since it does not admit any infinitesimal fluctuation, δH AB = 0 [75]. 4 And lastly but not leastly, upon this background, the auxiliary gauge potential, A µ , appears linearly rather than quadratically in the doubled-yet-gauged string action (1.11). Consequently it serves as a Lagrange multiplier to prescribe that all the untilde target spacetime coordinates should be chiral [8] (c.f. [90,91]), An intriguing insight from [11] is then that, the usual supergravity fields in (2.1) would be the Nambu- about the existence of more generic non-Riemannian geometries (c.f. [8,10] for other examples and also [22] for 'timelike' duality rotations). This question was answered in [1]: the most general solutions to the defining properties of the DFT-metric (2.2) can be classified by two non-negative integers, (n,n), where i, j = 1, 2, · · · , n,ī, = 1, 2, · · · ,n and 0 ≤ n +n ≤ D. 4 Put HA B = δA B in (3.5).
(i) While the B-field is skew-symmetric as usual, H µν and K µν are symmetric tensors whose kernels are spanned by linearly independent vectors, X i µ ,Xī ν and Y µ j ,Ȳ ν  , respectively, (ii) A completeness relation must be satisfied From the linear independency of the zero-eigenvectors, X i µ ,Xī ν , orthogonal/algebraic relations follow Intriguingly, the B-field (hence 'Courant algebra') is universally present regardless of the values of (n,n), and contributes to the DFT-metric through an O(D, D) adjoint action: whereH corresponds to the 'B-field-free' DFT-metric, and B is an O(D, D) element containing the B-field, It is also worth while to note the 'vielbeins' or 'square-roots' of K µν and H µν : where a, b are (D − n −n)-dimensional indices subject to a flat metric, say η ab , whose signature is arbitrary. Essentially, K µ a , X i µ ,Xī µ form a D × D invertible square matrix whose inverse is given by In fact, the analysis of the DFT-vielbeins corresponding to the (n,n) DFT-metric (2.5) carried out in [1] shows that the local Lorentz symmetry group, i.e. spin group is Spin(t + n, s + n) × Spin(s +n, t +n) . (2.14) Here (t, s) is the arbitrary signature of η ab or the nontrivial signature of H µν and K µν satisfying t + s + n + n = D. Of course, once the spin group of any given theory is specified, it is fixed once and for all. Thus, each sum, t + n, s + n, s +n, and t +n, should be constant. For example, the Minkowskian D = 10 maximally supersymmetric DFT [85] and the doubled-yet-gauged Green-Schwarz superstring action [79], both having the local Lorentz group of Spin(1, 9) × Spin(9, 1), can accommodate (0, 0) Riemannian and (1, 1) non-Riemannian sectors only (see [12] for examples of supersymmetric non-Riemannian backgrounds).
Nevertheless, we may readily relax the Majorana-Weyl condition therein [79,85] and impose the Weyl condition only on spinors, such that the local Lorentz group can take any of Spin(t,ŝ) × Spin(ŝ,t) witĥ t +ŝ = 10. The allowed non-Riemannian geometries will be then (n, n) types with n =n running from zero to min(t,ŝ) [1]. On the other hand, bosonic DFT does not care about spin groups and hence should be free from such constraints. It can admit more generic (n,n) non-Riemannian geometries.
Upon the (n,n) background, the doubled-yet-gauged worldsheet string action (1.11) reduces to 1 2πα which should be supplemented by the chiral and anti-chiral constraints over the n andn directions, These constraints are prescribed by the integrated-out auxiliary gauge potential A A (1.10).

Comment 1.
Matching with the content of the non-Riemannian component fields, 20) and the undoubled string worldsheet action resulting from (1.11), one can identify the original Newton-Cartan [33][34][35] as (1, 0), Stringy Newton-Cartan [36] as (1, 1), Carroll [37,38] as (D−1, 0), and Gomis-Ooguri [39] as (1, 1): see [1,11,57] for the details of the identifications. Further, the isometry of the (1, 1) 5 Through exponentiations, finite Milne-shift transformations can be achieved, which turn out to get truncated at finite orders, flat DFT-metric matches with the non-relativistic symmetry algebra such as Bargmann algebra [10], while the notion of T-duality persists to make sense in the non-relativistic string theory [47]. These all seem to suggest that DFT may be the home, i.e. the unifying framework, to describe various known as well as yetunknown non-Riemannian gravities. 6 Having said that there are also a few novel ingredients from DFT, such as the local GL(n) × GL(n) symmetry (2.15), the notion of 'Milne-shift covariance' as we shall discuss below (2.24), (2.26), and the very existence of the DFT-dilaton of which the exponentiation, e −2d , gives the integral measure in DFT being a scalar density with weight one, It is worth while to generalize the decomposition (2.9) to an arbitrary DFT tensor, Under diffeomorphisms, while the DFT tensor T A 1 ···An is surely subject to the generalized Lie derivative (1.4), the circled quantity,T A 1 ···An , is now governed by the undoubled ordinary Lie derivative which can be conveniently obtained as the truncation of the generalized Lie derivative by choosing the section, ∂ µ ≡ 0, and setting the parameter, ξ A = (0, ξ µ ) asξ ν ≡ 0: (2.23) Further, by construction, a DFT tensor is Milne-shift invariant. Yet, the circled one is Milne-shift covariant in the following manner, Explicitly, for a DFT vector, [76,93]) (2.25) That is to say, the circled quantities,T A 1 ···An ,V A , are 'B-field free', subject to the ordinary Lie derivative, For consistency, we also note for the O(D, D) invariant metric, Here we revisit with care the variational principle for a general DFT action coupled to matter (1.1) especially around non-Riemannian backgrounds. While the variations of the matter fields lead to their own Euler-Lagrange equations of motion, the variations of the DFT-metric and the DFT-dilaton give [65] δˆ1 Here G AB and T AB are respectively the stringy or O(D, D) completions of the Einstein curvature [81] and the Energy-Momentum tensor [65], as summarized in Table 1. The above result is easy to obtain once we neglect a boundary contribution arising from a total derivative [63]: and take into account a well-known identity which the infinitesimal variation of the DFT-metric should satisfy [7,62,94],  Table 1. , as the two variations, δH AB and δd, give the projected part and the trace part separately,  .12). We shall confirm that the full Einstein Double Field Equations are still valid for non-Riemannian sectors, either trivially due to projection properties or nontrivially from the genuine variational principle.

Variations of the DFT-metric around a generic (n,n) background
Here we shall identify the most general form of the infinitesimal fluctuations around a generic (n,n) DFTmetric (2.5). The fluctuations must respect the defining properties of the DFT-metric (2.2) and hence satisfy That is to say, the trace of the DFT-metric, H A A = 2(n −n), is invariant under continuous deformations.
Without loss of generality, like (2.9), we put With this ansatz, the former condition in (3.5) is met and the latter gives We need to solve these three constraints. For this, we utilize the completeness relation (2.13), and decompose each of {α, β, γ} into mutually orthogonal pieces, where, since α, β are symmetric, We remind the readers that, using the (D − n −n)-dimensional flat metric, η ab , we freely raise or lower the indices, a, b. Now, with the decomposition (3.9), it is straightforward to see that (3.8) implies Thus, the independent degrees of freedom for the fluctuations consist of (3.12) In total, as counted sequently as

Einstein Double Field Equations still hold for non-Riemannian sectors
Now, we proceed to organize the variation of the action induced by that of the (n,n) DFT-metric (3.1) in terms of the independent degrees of freedom for the fluctuations (3.12). 7 The only required property of the DFT-vielbeins is VApVB p +VApVBp = JAB. See [75] for a related discussion.
We apply the prescription (2.22) and write a pair of circled 'B-field-free' symmetric projectors, We also introduce a shorthand notation for the Einstein Double Field Equations, Hereafter, hatted quantities contain generically the H-flux, but, like the circled ones, there is no apparent bare B-field in them.
It is now straightforward to compute the variation in (3.1), In total, as counted sequently as, there is D 2 − (n −n) 2 number of independent on-shell relations, or EDFEs, in consistent with (3.13).
Up to the completeness relations (2.7), (2.13), and the identities (3.17), the first and the seventh in (3.21), the first and the eighth, the third and the fifth, the third and the sixth, the second and the last, the fourth and the last imply respectively, Finally, the first and the last, the second and the fifth, the third and the last, the fourth and the fifth give In this way, all the components of (P EP ) AB vanish and the full EDFEs persist to be valid universally for arbitrary (n,n) backgrounds.
Comment. From (3.17), off-shell relations hold among the components of the EDFEs, such that the full EDFEs are satisfied if 4 What if we keep (n,n) fixed once and for all ?
As it is a outstandingly hard problem to construct an action principle for non-Riemannian gravity (c.f. [45,46,48] for recent proposals), we may ask if the DFT action restricted to a fixed (n,n) sector might serve as the desired target spacetime gravitational action, c.f. (4.21). In this section, seeking for the answer to this question, we reanalyze the variational principle of DFT, crucially keeping (n,n) fixed. To our surprise, we obtain a subtle discrepancy with the previous section where the most general variations of the DFTmetric were analyzed. We shall see that, when the values of (n,n) are kept fixed and nn = 0, not all the components of the EDFEs (3.26) are implied by the variational principle.

Variational principle with fixed (n,n)
We start with (3.1) which gives the variation of the general DFT action induced by the DFT-metric. With fixed (n,n), the variation of the DFT-metric therein should comprise the variations of the (n,n) component fields: The variational principle implies either from the second line of (4.6), or alternatively from the third line of (4.6), Although (4.7) and (4.8) appear seemingly different, they are -as should be-equivalent. In fact, they are both equivalent to which are, from (3.25), further equivalent to more concise ones, Appendix A carries our proof.
The surprise which is manifest in (4.9) is that, when nn = 0 the variational principle with fixed (n,n) does not imply the full EDFEs (3.26): it does not constrain Y ρ i (P EP ) ρσȲ σ ı . However, as we have shown in the previous section, within the DFT frame they should vanish on-shell, Y ρ i (P EP ) ρσȲ σ ı = 0, and the full EDFEs should hold. We shall continue to discuss and conclude in the final section 5.
To identify the significance of the α iī parameter, we focus on the induced transformation of H µν , Geometrically the deformation of 2Y Without loss of generality, utilizing the completeness relation, decompose the zero-eigenvector, 17) substitute this ansatz into (4.16), and acquire the conditions the coefficients should satisfy, This shows that there are in total (n − rank [α iī ]) + (n − rank [α iī ]) = n +n − 2 × rank [α iī ] number of zero-eigenvectors. Moreover, from the invariance, δH A A = 0 (3.6), we note that the deformation by the α iī parameter actually changes the type of the 'non-Riemannianity' as This essentially explains why α iī vanishes in (4.13) where the (n,n) component field variables are varied with fixed values of (n,n), or fixed 'non-Riemannianity'. It is intriguing to note that the deformation makes the DFT-metric always less non-Riemannian. 8

Non-Riemannian differential geometry as bookkeeping device
This subsection is the last one before Conclusion, and is somewhat out of context. At first reading, readers may glimpse (4.21) in comparison with (4.20), and skip to the final section 5.
While the various (n,n) non-Riemannian geometries are universally well described by DFT through O(D, D) covariant tensors -as summarized in Table 1 Here in this last subsection, we propose an undoubled non-Riemannian differential tool kit, such as covariant derivative and curvature, for an arbitrary (n,n) sector. It descends from the DFT geometry, or the so-called "semi-covariant formalism" [63], and generalizes the standard Riemannian geometry underlying In particular, it enables us to extend the Riemannian expression of (4.20) in a way 'continuously' to the generic (n,n) non-Riemannian case, (4.21) We commence our explanation. First of all, D µ is our proposed 'upper-indexed' covariant derivative: which preserves both the undoubled diffeomorphisms (2.23) and the GL(n)×GL(n) local symmetries (2.15) as is equipped with proper connections: for undoubled ordinary diffeomorphisms, and for GL(n) × GL(n) rotations, We also denote a diffeomorphism-only preserving covariant derivative by and write for (4.22) and (4.24), Taking care of both spacetime and GL(n)×GL(n) indices, D µ acts on general tensor densities in a standard manner: On the other hand, D µ cares only the spacetime indices and ignores any GL(n) × GL(n) indices, For example, we have explicitly (4.29) It is instructive to see that the far right resulting expressions in (4.29) are clearly covariant under both diffeomorphisms and GL(n) × GL(n) local rotations, as the ρ, σ indices therein are skew-symmetrized and also contracted with H µρ , (KH) ν σ . However, without the GL(n) × GL(n) connections, we note and this breaks the GL(n) × GL(n) local symmetry.
Further, for the DFT-dilaton we should have where we have explicitly Because H µν and K ρσ are generically degenerate, the conventional relation (2.1) between the DFT-dilaton, d, and the string dilaton, φ, cannot hold. We stick to use the DFT-dilaton all the way. 9 The connections do the job as they transform properly under the diffeomorphisms (2.23), (2.25) and the GL(n) × GL(n) local rotations (2.15), (4.33) In particular, X i µ Ω µν λ ,Xī µ Ω µν λ , and H ρ[λ Ω µ]ν ρ are covariant tensors which might be viewed as "torsions". Finally, we define an upper-indexed Ricci curvature, which is diffeomorphism and GL(n) × GL(n) covariant, as it comes from the following commutator relation that is clearly also covariant, A scalar curvature follows naturally, which debuted in (4.21).
Our covariant derivative is "compatible" with the (n,n) component fields in a generalized fashion: (4.37) 9 We tend to believe that the conventional string dilaton, φ, is an artifact of the (0, 0) Riemannian geometry and the DFT-dilaton, d, is more fundamental as being an O(D, D) singlet.
the hatted new connection becomes Milne-shift covariant as well, in the sense of (2.16), (2.25), (2.26), where H λµν is a diffeomorphism covariant, GL(n) × GL(n) invariant, and Milne-shift invariant H-flux, The GL(n) × GL(n) connections (4.26) are inert to the addition of the H-flux-valued-torsion (4.38) as After all, in terms of a hatted covariant derivative, we can dismantle the DFT curvatures into a H-flux-free (circled) term and evidently H-flux-valued ones: where, as it should be obvious from our notation, we setŜ AB := (B −1 ) A C (B −1 ) B D S CD , and the circled quantities are all H-flux free: from Table 1 or [63,65], and, with (3.16), (4.45) While we organize the H-flux-valued parts in terms of the hatted covariant derivative, like (4.41), we have The only nontrivial distinction lies in As advertised in (4.21), we may further dismantleS (0) as well as (PSP ) µν into more elementary modules:S Comment 1. It is worth while to note and rewrite the 'kinetic term' of the DFT-dilaton in (4.21), Consequently, the proposed covariant derivative (4.25) and Ricci curvature (4.34) reduce to the standard covariant derivative and Ricci curvature in Riemannian geometry, The actual computation of the variations of the action, even with (n,n) fixed, are still powered by the semicovariant formalism, specifically (3.2).

Conclusion
The very gravitational theory string theory predicts may be the Double Field Theory with non-Riemannian surprises, rather than General Relativity based on Riemannian geometry. The underlying mathematical structure of DFT unifies supergravity with various non-Riemannian gravities including (stringy) Newton-Cartan geometry, ultra-relativistic Carroll geometry, and non-relativistic Gomis-Ooguri string theory. The non-Riemannian geometries of DFT can be classified by two non-negative integers, (n,n) [1].
We have analyzed with care the variational principle. We have shown that the most general infinitesimal variations of an arbitrary (n,n) DFT-metric have D 2 − (n −n) 2 number of degrees of freedom, which matches with the dimension of the underlying coset [11], O(t+n,s+n)×O(s+n,t+n) (3.14). Through action principle, these variations imply the full Einstein Double Field Equations (3.22), (3.24). However, nn number of them change the value of (n,n), i.e. the type of non-Riemannianity (4.19). Consequently, if we keep (n,n) fixed once and for all, the variational principle gets restricted and fails to reproduce the full EDFEs: the specific part, Y µ i (P EP ) µνȲ ν ı , does not have to vanish on-shell (4.9). 10 The EDFEs are supposed to arise as the string worldsheet beta-functions [98,99]. For the doubled-yetgauged string action (1.11) upon an arbitrarily chosen (n,n) background, the (n,n)-changing variations of the DFT-metric would correspond to marginal deformations. We must stress that these deformations could not be realized by merely varying the background component fields with fixed (n,n) (4.13), c.f. [52,54,56].
Nevertheless, it is natural to expect that nn number of Y µ i (P EP ) µνȲ ν ı arise as the corresponding betafunctions too. That is to say, at least for nn = 0, the quantum consistency with the worldsheet string theory seems to forbid us to fix (n,n) rigidly. We conclude that the various non-Riemannian gravities should be identified as different solution sectors of Double Field Theory rather than viewed as independent theories.
Quantum consistency of the non-Riemannian geometries calls for thorough investigation, which may enlarge the scope of the string theory landscape far beyond Riemann. 10 As can be seen from (4.44), Y ρ i (P EP )ρσȲ σ ı contains a second order derivative of the DFT-dilaton along the Y µ i andȲī ν directions, i.e. Y µ iȲ ν ı ∂µ∂ν d .
Consequently, with the completeness relation (2.7), the identities from (3.17), and (A.2), we note Similarly we get with (A.3), B Derivation of the non-Riemannian differential tool kit from DFT The non-Riemannian differential geometry we have proposed in section 4.3, in particular the hatted Ω connection (4.38), descends from the known covariant derivatives in the DFT semi-covariant formalism [63]: In order to convert these into undoubled ordinary covariant quantities -or to get rid of the bare B-field in them-we multiply B −1 as in (2.22) and write Here we set∇ andΓ ABC is a naturally induced -or 'twisted' [101], c.f. Alternative combination of (B.1), rather than (B.5), can give different type of covariant derivatives, However, these can act only on one-form fields, and appear not so useful.