Abstract
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 \({n\bar{n}\ne 0}\) 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.
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1 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, \((n,{\bar{n}})\). The outcome – which we shall review in Sect. 2 – is that only the case of (0, 0) corresponds to conventional supergravity based on Riemannian geometry. Other generic cases of \({(n,{\bar{n}})\ne (0,0)}\) 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 \(\bar{n}\) dimensions respectively.
The non-Riemannian property is a point-wise or local statement [8,9,10,11,12] and differs from the global notion of ‘non-geometry’ [13,14,15,16,17] which is also well described by DFT [18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]. Possible examples of non-Riemannian geometries include Newton–Cartan geometry [33,34,35] as (1, 0), stringy Newton–Cartan [36] as (1, 1), (wonderland) Carroll geometry [37, 38] as \(({D-1},0)\), and non-relativistic Gomis–Ooguri string theory [39] as (1, 1). These are of continuous interest, e.g. [40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58]. Further, the fully \(\mathbf {O}(D,D)\) symmetric vacua of Double Field Theory turn out to be ‘maximally’ non-Riemannian, being of either (D, 0) or (0, D) type, compelling string to be completely chiral or anti-chiral. A remarkable insight from [11] is that, the ordinary Riemannian spacetime arises after spontaneous symmetry breaking of these fully \(\mathbf {O}(D,D)\) symmetric vacua while identifying the Riemannian metric, \(g_{\mu \nu }\), as a Nambu–Goldstone boson.
In this work we attempt to explore the dynamics of the generic \((n,{\bar{n}})\) sector in Double Field Theory. We analyze with care the relevant variational principle and recognize a nontrivial subtlety: when \(n{\bar{n}}\ne 0\), the resulting Euler–Lagrangian equations of motion depend whether the variations of the action keep the values of \((n,{\bar{n}})\) fixed or not. This rather unexpected subtle discrepancy contrasts DFT with the traditional approaches to the various non-Riemannian gravities.
The organization of the present paper is as follows.
In the remaining of this Introduction, to put the present work into context and set up notation, we describe DFT as the \(\mathbf {O}(D,D)\) completion of General Relativity along with a relevant doubled string action.
In Sect. 2, we review the \((n,{\bar{n}})\) classification of the non-Riemannian DFT geometries from [1].
In Sect. 3, we revisit the variational principle in DFT and confirm that the known Euler–Lagrangian equations, or ‘Einstein Double Field Equations’ (1.3) are still valid for non-Riemannian sectors.
In Sect. 4, now keeping \((n,{\bar{n}})\) fixed, we reanalyze the variational principle and show that the full Einstein Double Equations are not necessarily implied when \(n{\bar{n}}\ne 0\). We explain the discrepancy, and further propose a non-Riemannian differential tool kit as a ‘bookkeeping device’ to expound the equations.
We conclude in Sect. 5, followed by Appendix A and B.
1.1 Double Field Theory as the \(\mathbf {O}(D,D)\) completion of General Relativity
While the initial motivation of Double Field Theory was to reformulate supergravity in an \(\mathbf {O}(D,D)\) manifest manner [2,3,4,5,6,7] ([59,60,61] for reviews), through subsequent further developments [62,63,64,65], DFT has evolved and can be now identified as a pure gravitational theory that string theory seems to predict foremostFootnote 1 and may differ from General Relativity as it is capable of describing non-Riemannian geometries [1]. Specifically, DFT is the string theory based, \(\mathbf {O}(D,D)\) completion of General Relativity (GR): taking the \(\mathbf {O}(D,D)\) symmetry as the first principle, DFT geometrises not merely the Riemannian metric but the whole massless NS-NS sector of closed string as the fundamental gravitational multiplet, hence ‘completing’ GR. Further, the \(\mathbf {O}(D,D)\) symmetry principle fixes its coupling to other superstring sectors (R-R [68,69,70,71], R-NS [72], and heterotic Yang-Mills [73,74,75]). Having said that, regardless of supersymmetry, it can also couple to various matter fields which may appear in lower dimensional effective field theories [72, 76, 77], just as GR does so. In particular, supersymmetric extensions have been completed to the full (i.e. quartic) order in fermions for \(D=10\) cases powered by ‘1.5 formalism’ [78, 79], and the pure Standard Model without any extra physical degrees of freedom can easily couple to DFT in an \(\mathbf {O}(D,D)\) symmetric manner [80].
Schematically, governed by the \(\mathbf {O}(D,D)\) symmetry principle, DFT may couple to generic matter fields, say collectively \(\Upsilon \), which should be also in \(\mathbf {O}(D,D)\) representations:
Here, d is the \(\mathbf {O}(D,D)\) singlet DFT-dilaton, \(S_{\scriptscriptstyle {{(0)}}}\) is the DFT scalar curvature, and \({\nabla }_{A}\Upsilon \) denotes the covariant derivative of a matter field. To manifest the \(\mathbf {O}(D,D)\) symmetry, the action is equipped with an \(\mathbf {O}(D,D)\) invariant metric,
which, with its inverse \({\mathcal {J}}^{AB}\), is going to be always used to lower and raise the \(\mathbf {O}(D,D)\) vector indices (Latin capital letters). It splits the doubled coordinates into two parts, \(x^{A}=(\tilde{x}_{\mu },x^{\nu })\) and \(\partial _{A}=(\tilde{\partial }^{\mu },\partial _{\nu })\). Note that the doubling of the coordinates is crucial to manifest the \(\mathbf {O}(D,D)\) symmetry in DFT. Like GR, the General Covariance (DFT-diffeomorphisms) of the action (1.1) naturally gives rise to the definitions of the \(\mathbf {O}(D,D)\) completions of the Einstein curvature, \(G_{AB}\) [81] and also the Energy-Momentum tensor, \(T_{AB}\) [65], of which the former and the latter are respectively off-shell and on-shell conserved. Equating the two, they comprise the \(\mathbf {O}(D,D)\) completion of the Einstein field equations, or the Einstein Double Field Equations (EDFEs) [65, 82],
We summarize the basic geometrical notation of DFT in Table 1,Footnote 2 while the DFT-diffeomorphisms are generated by the so-called generalized Lie derivative [3, 7]: acting on a tensor density with weight \(\omega _{\scriptscriptstyle {T}}\),
In particular, being a scalar density with weight one (\({\omega _{{\scriptscriptstyle {T}}}=1}\)), the exponentiation \(e^{-2d}\) is the integral measure of DFT.
It is noteworthy and relevant to this work that, all the geometrical notation of the covariant derivative, \({\nabla }_{A}\), and the curvatures, \(S_{\scriptscriptstyle {{(0)}}},G_{AB}\), can be constructed strictly in terms of \(\mathbf {O}(D,D)\) covariant field variables, notably the \(\mathbf {O}(D,D)\) invariant DFT-dilaton, d, and the \(\mathbf {O}(D,D)\) covariant DFT-metric, \({\mathcal {H}}_{AB}\) (“generalized metric”), or more powerfully \(\mathbf {O}(D,D)\) covariant DFT-vielbeins, without necessarily referring to conventional, undoubled \(\mathbf {O}(D,D)\) breaking supergravity variables. Similarly, a doubled string action can be constructed in terms of \(\mathbf {O}(D,D)\) covariant objects as we review below.
1.2 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 \(\Phi _{r}\), and their products like \(\Phi _{s}\Phi _{t}\), the \(\mathbf {O}(D,D)\) invariant Laplacian should vanish
We remind the reader that the \(\mathbf {O}(D,D)\) indices are raised with \({\mathcal {J}}^{AB}\). 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, \(\Delta ^{A}\), is derivative-index-valued, meaning that its superscript index should be identifiable as a derivative index, for example \(\Delta ^{A}=\Phi _{s}\partial ^{A}\Phi _{t\,}\). This insight on the section condition may suggest that the doubled coordinates of DFT are in fact gauged by an equivalence relation,
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, \(\mathrm{d}x^{A}\), 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,
It is also a singlet of the coordinate gauge symmetry (1.8): \(\delta x^{A}=\Delta ^{A},\, \delta {\mathcal {A}}^{A}=\mathrm{d}\Delta ^{A},\, \delta (\mathrm{D}x^{A})=0\). The gauged one-form then naturally allows to construct a perfectly symmetric doubled string action [8, 84],
which enjoys symmetries like global \(\mathbf {O}(D,D)\), target spacetime DFT-diffeomorphisms, worldsheet diffeomorphisms, Weyl symmetry, and the coordinate gauge symmetry.Footnote 3 All the background information is encoded in the DFT-metric, \({\mathcal {H}}_{AB}\,\).
2 Review of [1]: classification of the non-Riemannian DFT geometries
The section condition can be generically solved, up to \(\mathbf {O}(D,D)\) rotations, by enforcing the tilde coordinate independency: \(\tilde{\partial }^{\mu }\equiv 0~\Rightarrow ~ \partial _{A}\partial ^{A}=2\partial _{\mu }\tilde{\partial }^{\mu }\equiv 0\). Choosing \(\Delta ^{A}=c_{\mu }\partial ^{A}x^{\mu }=(c_{\mu },0)\) for (1.8) and similarly \({\mathcal {A}}^{A}=A_{\mu }\partial ^{A}x^{\mu }=(A_{\mu },0)\), we note that the tilde coordinates are indeed gauged: \((\tilde{x}_{\mu },x^{\nu })\,\sim \,(\tilde{x}_{\mu }+c_{\mu },x^{\nu })\), \(\mathrm{D}x^{A}=(\mathrm{d}\tilde{x}_{\mu }-A_{\mu },\mathrm{d}x^{\nu })\). 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],
reduces DFT to supergravity. In this case, the single expression of the EDFEs (1.3) unifies all the equations of motion of the three fields, \(\{g_{\mu },B_{\mu \nu },\phi \}\). Further, after Gaussian integration of the auxiliary gauge potential, \(A_{\mu }\), the doubled-yet-gauged string action (1.11) reproduces the standard undoubled string action.
Yet, this is not the full story. The above parametrization (2.1) is merely one particular solution to the defining relations of the DFT-metric:
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 \(\mathbf {O}(D,D)\) invariant metric,
such that \({\mathcal {H}}_{A}{}^{B}=\delta _{A}{}^{B}\). This is a vacuum solution to DFT, or to the ‘matter-free’ EDFEs, \(G_{AB}=0\) (1.3), which is very special in several aspects. Firstly, compared with (2.1), there cannot be any associated Riemannian metric \(g_{\mu \nu }\) and hence it does not allow any conventional or Riemannian interpretation at all. It is maximally non-Riemannian. Secondly, it is fully \(\mathbf {O}(D,D)\) symmetric, being one of the two most symmetric vacua of DFT, \({\mathcal {H}}_{AB}=\pm {\mathcal {J}}_{AB}\). Thirdly, it is moduli-free since it does not admit any infinitesimal fluctuation, \(\delta {\mathcal {H}}_{AB}=0\) [75].Footnote 4 And lastly but not leastly, upon this background, the auxiliary gauge potential, \(A_{\mu }\), 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–Goldstone modes of the perfectly \(\mathbf {O}(D,D)\) symmetric vacuum (2.3).
Given the Riemannian and maximally non-Riemannian backgrounds, (2.1) v.s. (2.3), one may wonder 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, \((n,{\bar{n}})\),
where \(i,j=1,2,\ldots , n\), \(\bar{\imath },\bar{\jmath }=1,2,\ldots , {\bar{n}}\) and \(0\le n+{\bar{n}}\le D\).
- (i):
While the B-field is skew-symmetric as usual, \(H^{\mu \nu }\) and \(K_{\mu \nu }\) are symmetric tensors whose kernels are spanned by linearly independent vectors, \(\big \{X^{i}_{\mu },{\bar{X}}^{\bar{\imath }}_{\nu }\big \}\) and \(\big \{Y_{j}^{\mu },{\bar{Y}}^{\nu }_{\bar{\jmath }}\big \}\), respectively,
$$\begin{aligned} H^{\mu \nu }X^{i}_{\nu }= & {} 0,\quad \quad H^{\mu \nu }{\bar{X}}^{\bar{\imath }}_{\nu }=0,~\quad ~\quad \nonumber \\ K_{\mu \nu }Y_{j}^{\nu }= & {} 0,\quad \quad K_{\mu \nu }{\bar{Y}}_{\bar{\jmath }}^{\nu }=0. \end{aligned}$$(2.6)- (ii):
A completeness relation must be satisfied
$$\begin{aligned} H^{\mu \rho }K_{\rho \nu } +Y_{i}^{\mu }X^{i}_{\nu }+{\bar{Y}}_{\bar{\imath }}^{\mu }{\bar{X}}^{\bar{\imath }}_{\nu } =\delta ^{\mu }{}_{\nu }. \end{aligned}$$(2.7)
From the linear independency of the zero-eigenvectors, \(\left\{ X^{i}_{\mu }, {\bar{X}}^{\bar{\imath }}_{\nu }\right\} \), orthogonal/algebraic relations follow
Intriguingly, the B-field (hence ‘Courant algebra’) is universally present regardless of the values of \((n,{\bar{n}})\), and contributes to the DFT-metric through an \(\mathbf {O}(D,D)\) adjoint action:
where \(\mathring{{\mathcal {H}}}\) corresponds to the ‘B-field-free’ DFT-metric,
and \({\mathcal {B}}\) is an \(\mathbf {O}(D,D)\) element containing the B-field,
It is also worth while to note the ‘vielbeins’ or ‘square-roots’ of \(K_{\mu \nu }\) and \(H^{\mu \nu }:\)
where a, b are \((D-n-{\bar{n}})\)-dimensional indices subject to a flat metric, say \(\eta _{ab}\,\), whose signature is arbitrary. Essentially, \(\big \{K_{\mu }{}^{a},X_{\mu }^{i},{\bar{X}}_{\mu }^{\bar{\imath }}\big \}\) form a \(D\times D\) invertible square matrix whose inverse is given by \(\big \{H^{\mu }{}_{a},Y^{\mu }_{i},{\bar{Y}}^{\mu }_{\bar{\imath }}\big \}\) as
In fact, the analysis of the DFT-vielbeins corresponding to the \((n,{\bar{n}})\) DFT-metric (2.5) carried out in [1] shows that the local Lorentz symmetry group, i.e. spin group is
Here (t, s) is the arbitrary signature of \(\eta _{ab}\) or the nontrivial signature of \(H^{\mu \nu }\) and \(K_{\mu \nu }\) satisfying \(t+s+n+{\bar{n}}=D\). Of course, once the spin group of any given theory is specified, it is fixed once and for all. Thus, each sum, \(t+n\), \(s+n\), \(s+{\bar{n}}\), and \(t+{\bar{n}}\), 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 \(\mathbf {Spin}(1,9)\times \mathbf {Spin}(9,1)\), 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 \(\mathbf {Spin}(\hat{t},\hat{s})\times \mathbf {Spin}(\hat{s},\hat{t})\) with \(\hat{t}+\hat{s}=10\). The allowed non-Riemannian geometries will be then (n, n) types with \({n={\bar{n}}}\) running from zero to \({\text {min}}(\hat{t},\hat{s})\) [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 \((n,{\bar{n}})\) non-Riemannian geometries.
Crucially, the \((n,{\bar{n}})\) parametrization of the DFT-metric (2.5) possesses two local symmetries, namely \(\mathbf {GL}(n)\times \mathbf {GL}({\bar{n}})\) rotations and Milne-shift transformations. The \(\mathbf {GL}(n)\times \mathbf {GL}({\bar{n}})\) symmetry rotates the \(i,j,\ldots \) and \(\bar{\imath },\bar{\jmath },\ldots \) indices of the component fields: with infinitesimal local parameters, \(w_{i}{}^{j}\) and \({\bar{w}}_{\bar{\imath }}{}^{\bar{\jmath }}\),
The Milne-shift symmetry generalizes the so-called ‘Galilean boost’ in the Newtonian gravity literature [40, 41]. It acts with infinitesimal local parameters, \(V_{\mu i}\) and \({\bar{V}}_{\mu \bar{\imath }}\),Footnote 5
Remarkably, both transformations, (2.15) and (2.16), leave the DFT-metric invariant,
as the two local symmetries are actually parts of the underlying local Lorentz symmetries (2.14).
Upon the \((n,{\bar{n}})\) background, the doubled-yet-gauged worldsheet string action (1.11) reduces to
which should be supplemented by the chiral and anti-chiral constraints over the n and \({\bar{n}}\) directions,
These constraints are prescribed by the integrated-out auxiliary gauge potential \({\mathcal {A}}^{A}\) (1.10).
Comment 1. Matching with the content of the non-Riemannian component fields,
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) 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 yet-unknown non-Riemannian gravities.Footnote 6 Having said that there are also a few novel ingredients from DFT, such as the local \(\mathbf {GL}(n)\times \mathbf {GL}({\bar{n}})\) 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,
Comment 2. It is worth while to generalize the decomposition (2.9) to an arbitrary DFT tensor,
Under diffeomorphisms, while the DFT tensor \(T_{A_{1}\cdots A_{n}}\) is surely subject to the generalized Lie derivative (1.4), the circled quantity, \(\mathring{T}_{A_{1}\cdots A_{n}}\), 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, \(\tilde{\partial }^{\mu }\equiv 0\), and setting the parameter, \(\xi ^{A}=(0,\xi ^{\mu })\) as \(\tilde{\xi }_{\nu }\equiv 0\):
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, \(V_{A}={\mathcal {B}}_{A}{}^{B}\mathring{V}_{B}\), we have (c.f. [76, 93])
That is to say, the circled quantities, \(\mathring{T}_{A_{1}\cdots A_{n}}\), \(\mathring{V}_{A}\), are ‘B-field free’, subject to the ordinary Lie derivative, and Milne-shift covariant rather than invariant. More specifically, the undoubled lower Greek indices are Milne-shift covariant, while the upper ones are invariant: from (2.10), (2.16), (2.25),
For consistency, we also note for the \(\mathbf {O}(D,D)\) invariant metric,
3 Variational principle around non-Riemannian backgrounds
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]
Here \(G_{AB}\) and \(T_{AB}\) are respectively the stringy or \(\mathbf {O}(D,D)\) 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],
Equation (3.2) holds due to the nice variational property of the semi-covariant curvature, \(\delta S_{AB}={\nabla }_{[A}\delta \Gamma _{C]B}{}^{C}+{\nabla }_{[B}\delta \Gamma _{C]A}{}^{C}\), and the compatibility of the derivative, \({{\nabla }_{A}{\mathcal {J}}_{BC}}{=0}\), \({{\nabla }_{A}{\mathcal {H}}_{BC}=0}\), \({{\nabla }_{A}d=0}\), see Table 1. Equation (3.3) holds because the DFT-metric is constrained to be a symmetric \(\mathbf {O}(D,D)\) element (2.2), see also (3.5) below. This is the reason why in the variation of the action (3.1) \(\delta {\mathcal {H}}_{AB}\) is contracted with a projected quantity, i.e. \((PG{\bar{P}})^{AB}-8\pi G(PT{\bar{P}})^{AB}\). Equation (3.1) is then supposed to give the EDFEs, \(G_{AB}=8\pi GT_{AB}\) (1.3) [65], as the two variations, \(\delta {\mathcal {H}}_{AB}\) and \(\delta d\), give the projected part and the trace part separately,
which comprise the full EDFEs. While there is no issue on the equation of motion of the DFT-dilaton, i.e. the trace part in (3.4), there might be some ambiguity on the DFT-metric variation especially around a non-Riemannian background. For example, let us take one of the two maximally non-Riemannian, fully \(\mathbf {O}(D,D)\) symmetric vacua, as \({{\mathcal {H}}_{AB}={\mathcal {J}}_{AB}}\). Because it does not allow any infinitesimal variation or moduli, \({\delta {\mathcal {H}}_{AB}=0}\) [75], the induced variation of the action is null and therefore it should not generate any nontrivial Euler–Lagrange equation of motion. Nevertheless, in this case the ‘barred’ projector vanishes automatically, \({{\bar{P}}_{AB}=0}\), and the projected part of the EDFEs in (3.4) is satisfied rather trivially. It appears that we have a slightly puzzling situation for the non-Riemannian background, \({{\mathcal {H}}_{AB}={\mathcal {J}}_{AB}}\): it allows no infinitesimal variation \(\delta {\mathcal {H}}_{AB}=0\) and hence one may expect that the variation of the action should be trivial and there should be no nontrivial Euler–Lagrange equation of motion of the DFT-metric. This is all true, but nevertheless the full EDFEs are still valid! (though in a trivial manner as \({\bar{P}}=0\)).
Below, through Sects. 3.1 and 3.2, we shall rigorously revisit the variational principle of DFT around a generic non-Riemannian background. Basically, we are questioning whether it is really safe from (3.3) to put \(\delta {\mathcal {H}}_{AB}=2P_{(A}{}^{C}{\bar{P}}_{B)}{}^{D}{\mathcal {M}}_{CD}\) and read off the Euler–Lagrange equation of motion of the DFT-metric as if \({\mathcal {M}}_{CD}\) is a generic symmetric matrix. To answer this, we shall directly identify the truly independent degrees of freedom in the infinitesimal fluctuations of an arbitrary \((n,{\bar{n}})\) non-Riemannian DFT-metric, as (3.12). 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.
3.1 Variations of the DFT-metric around a generic \((n,{\bar{n}})\) background
Here we shall identify the most general form of the infinitesimal fluctuations around a generic \((n,{\bar{n}})\) DFT-metric (2.5). The fluctuations must respect the defining properties of the DFT-metric (2.2) and hence satisfy
It follows that \(\delta {\mathcal {H}}_{A}{}^{B}=-{\mathcal {H}}_{A}{}^{C}\delta {\mathcal {H}}_{C}{}^{D}{\mathcal {H}}_{D}{}^{B}\), and hence equivalent (3.3) holds. In particular, \(\delta {\mathcal {H}}_{A}{}^{B}\) is traceless,
That is to say, the trace of the DFT-metric, \({\mathcal {H}}_{A}{}^{A}=2(n-{\bar{n}})\), 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 \(\{\alpha ,\beta ,\gamma \}\) into mutually orthogonal pieces,
where, since \(\alpha ,\beta \) are symmetric,
We remind the readers that, using the \((D-n-{\bar{n}})\)-dimensional flat metric, \(\eta _{ab}\), 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
In total, as counted sequently as
there are \(D^{2}-(n-{\bar{n}})^{2}\) number of degrees of freedom which matches precisely the dimension of the underlying coset [11],
Furthermore, if we employ the DFT-vielbeins,Footnote 7 \(V_{Ap}, {\bar{V}}_{A{\bar{p}}}\), the projected part of the EDFEs (3.4) is equivalent to
As the local Lorentz vector indices p and \({\bar{p}}\) run from one to \({D+n-{\bar{n}}}\) and \({D-n+{\bar{n}}}\) respectively, there are in total \((D+n-{\bar{n}})\times (D-n+{\bar{n}})=D^{2}-(n-{\bar{n}})^{2}\) number of components in (3.15) which coincides with the total number of independent fluctuations of the \((n,{\bar{n}})\) DFT-metric (3.13). As the number of the equations and the fluctuations are the same, we may well expect that the former should be implied by the variational principle generated by the latter. Below, we confirm this directly through explicit computation, without using the DFT-vielbeins.
3.2 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 \((n,{\bar{n}})\) DFT-metric (3.1) in terms of the independent degrees of freedom for the fluctuations (3.12).
We apply the prescription (2.22) and write a pair of circled ‘B-field-free’ symmetric projectors,
which satisfy \(\mathring{P}_{A}{}^{B}+\mathring{{\bar{P}}}_{A}{}^{B}=\delta _{A}{}^{B}\), \(\,\mathring{P}_{A}{}^{B}\mathring{{\bar{P}}}_{B}{}^{C}=0\), and useful identities,
We also introduce a shorthand notation for the Einstein Double Field Equations,
Hereafter, hatted quantities contain generically the \(\mathbb {H}\)-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),
Each term is independent and thus, from the variational principle, should vanish individually on-shell,
In total, as counted sequently as,
there is \({D^{2}-(n-{\bar{n}})^{2}}\) 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 \((\mathring{P}\widehat{E}\mathring{{\bar{P}}})_{AB}\) vanish and the full EDFEs persist to be valid universally for arbitrary \((n,{\bar{n}})\) 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 \((n,{\bar{n}})\) 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 \((n,{\bar{n}})\) 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 \((n,{\bar{n}})\) fixed. To our surprise, we obtain a subtle discrepancy with the previous section where the most general variations of the DFT-metric were analyzed. We shall see that, when the values of \((n,{\bar{n}})\) are kept fixed and \(n{\bar{n}}\ne 0\), not all the components of the EDFEs (3.26) are implied by the variational principle.
4.1 Variational principle with fixed \((n,{\bar{n}})\)
We start with (3.1) which gives the variation of the general DFT action induced by the DFT-metric. With fixed \((n,{\bar{n}})\), the variation of the DFT-metric therein should comprise the variations of the \((n,{\bar{n}})\) component fields:
Further, from their defining relations, (2.6), (2.7), the variations of the \((n,{\bar{n}})\) component fields are not entirely independent. They must meet
From (2.12), we also note
which imply in particular,
It is then evident from (4.2), (4.3), and (4.4) that we have freedom to choose either \(\{\delta K_{\mu }{}^{a},\delta X^{i}_{\rho },\delta {\bar{X}}^{\bar{\imath }}_{\sigma }\}\) or \(\{\delta H^{\mu }{}_{a},\delta Y_{i}^{\rho },\delta {\bar{Y}}_{\bar{\imath }}^{\sigma }\}\) as independent variations. Each of them has (formally) \(D^{2}\) number of degrees of freedom.
Now, we substitute (4.1) into (3.1), and utilize (4.2), (4.3), (4.4), (4.5) to obtain
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 \(n{\bar{n}}\ne 0\) the variational principle with fixed \((n,{\bar{n}})\) does not imply the full EDFEs (3.26): it does not constrain \(\,Y_{i}^{\rho }(\mathring{P}\widehat{E}\mathring{{\bar{P}}})_{\rho \sigma }{\bar{Y}}^{\sigma }_{\bar{\imath }}\). However, as we have shown in the previous section, within the DFT frame they should vanish on-shell, \(\,Y_{i}^{\rho }(\mathring{P}\widehat{E}\mathring{{\bar{P}}})_{\rho \sigma }{\bar{Y}}^{\sigma }_{\bar{\imath }}=0\), and the full EDFEs should hold. We shall continue to discuss and conclude in the final Sect. 5.
4.2 Difference between keeping \((n,{\bar{n}})\) fixed or not
In order to understand the discrepancy in the resulting Euler–Lagrangian equations, (3.24) vs. (4.9), here we investigate how the infinitesimal variations of the component fields of the \((n,{\bar{n}})\) DFT-metric (4.1),
contribute actually to the \(\alpha ,\beta ,\gamma \) variables defined in the generic variation of the DFT-metric (3.7),
With (3.9), one can identify the contributions thoroughly:
and
This is consistent with the general result of (3.11). However, one surprise is that \(\alpha ^{i\bar{\imath }}\) must be trivial when the \((n,{\bar{n}})\) component fields (4.11) are varied while keeping \((n,{\bar{n}})\) fixed.
To identify the significance of the \(\alpha ^{i\bar{\imath }}\) parameter, we focus on the induced transformation of \(H^{\mu \nu }\),
Geometrically the deformation of \(2Y^{(\mu }_{i}{\bar{Y}}^{\nu )}_{\bar{\imath }}\alpha ^{i\bar{\imath }}\) is ‘orthogonal’ to \(H^{\mu \nu }\), and thus we expect it should reduce the kernel of \(H^{\mu \nu }\). To verify this explicitly, we solve for the eigenvectors of \(H^{\prime \mu \nu }\) with zero eigenvalue,
Without loss of generality, utilizing the completeness relation, \(K_{\mu a}H^{\nu a}+X_{\mu }^{i}Y^{\nu }_{i}+{\bar{X}}_{\mu }^{\bar{\imath }}{\bar{Y}}^{\nu }_{\bar{\imath }}=\delta _{\mu }{}^{\nu }\), we decompose the zero-eigenvector,
substitute this ansatz into (4.16), and acquire the conditions the coefficients should satisfy,
This shows that there are in total \((n-\hbox {rank}\,[\alpha ^{i\bar{\imath }}])+({\bar{n}}-\hbox {rank}\,[\alpha ^{i\bar{\imath }}])=n+{\bar{n}}-2\times \hbox {rank}\,[\alpha ^{i\bar{\imath }}]\) number of zero-eigenvectors. Moreover, from the invariance, \({\delta {\mathcal {H}}_{A}{}^{A}=0}\) (3.6), we note that the deformation by the \(\alpha ^{i\bar{\imath }}\) parameter actually changes the type of the ‘non-Riemannianity’ as
This essentially explains why \(\alpha ^{i\bar{\imath }}\) vanishes in (4.13) where the \((n,{\bar{n}})\) component field variables are varied with fixed values of \((n,{\bar{n}})\), or fixed ‘non-Riemannianity’. It is intriguing to note that the deformation makes the DFT-metric always less non-Riemannian.Footnote 8
4.3 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 Sect. 5.
While the various \((n,{\bar{n}})\) non-Riemannian geometries are universally well described by DFT through \(\mathbf {O}(D,D)\) covariant tensors – as summarized in Table 1 – it may be desirable in practical computations to break the manifest \(\mathbf {O}(D,D)\) symmetry spontaneously by fixing the section, \(\tilde{\partial }^{\mu }\equiv 0\), and dismantle the \(\mathbf {O}(D,D)\) covariant tensors or curvatures into smaller modules which should be still covariant under undoubled ordinary diffeomorphisms, B-field gauge symmetry, and \(\mathbf {GL}(n)\times \mathbf {GL}({\bar{n}})\) local rotations. We remind the readers that in the case of the (0, 0) Riemannian sector, the \(\mathbf {O}(D,D)\) singlet DFT scalar curvature reduces to four modules (c.f. [27, 95, 96]):
Here in this last section, we propose an undoubled non-Riemannian differential tool kit, such as covariant derivative and curvature, for an arbitrary \((n,{\bar{n}})\) sector. It descends from the DFT geometry, or the so-called “semi-covariant formalism” [63], and generalizes the standard Riemannian geometry underlying (4.20) in a consistent manner. It breaks the manifest \(\mathbf {O}(D,D)\) symmetry spontaneously, but preserves the ordinary diffeomorphisms, B-field gauge symmetry, and the \(\mathbf {GL}(n)\times \mathbf {GL}({\bar{n}})\) local symmetries as desired. In particular, it enables us to extend the Riemannian expression of (4.20) in a way ‘continuously’ to the generic \((n,{\bar{n}})\) non-Riemannian case,
We commence our explanation. First of all, \(\mathbb {D}^{\mu }\) is our proposed ‘upper-indexed’ covariant derivative:
which preserves both the undoubled diffeomorphisms (2.23) and the \(\mathbf {GL}(n)\times \mathbf {GL}({\bar{n}})\) local symmetries (2.15) as is equipped with proper connections: for undoubled ordinary diffeomorphisms,
and for \(\mathbf {GL}(n)\times \mathbf {GL}({\bar{n}})\) 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 \(\mathbf {GL}(n)\times \mathbf {GL}({\bar{n}})\) indices, \(\mathbb {D}^{\mu }\) acts on general tensor densities in a standard manner:
On the other hand, \({\mathfrak {D}}^{\mu }\) cares only the spacetime indices and ignores any \(\mathbf {GL}(n)\times \mathbf {GL}({\bar{n}})\) indices,
For example, we have explicitly
It is instructive to see that the far right resulting expressions in (4.29) are clearly covariant under both diffeomorphisms and \(\mathbf {GL}(n)\times \mathbf {GL}({\bar{n}})\) local rotations, as the \(\rho \), \(\sigma \) indices therein are skew-symmetrized and also contracted with \(H^{\mu \rho }\), \((KH)_{\nu }{}^{\sigma }\). However, without the \(\mathbf {GL}(n)\times \mathbf {GL}({\bar{n}})\) connections, we note
and this breaks the \(\mathbf {GL}(n)\times \mathbf {GL}({\bar{n}})\) local symmetry.
Further, for the DFT-dilaton we should have
where we have explicitly
Because \(H^{\mu \nu }\) and \(K_{\rho \sigma }\) are generically degenerate, the conventional relation (2.1) between the DFT-dilaton, d, and the string dilaton, \(\phi \), cannot hold. We stick to use the DFT-dilaton all the way.Footnote 9
The connections do the job as they transform properly under the diffeomorphisms (2.23), (2.25) and the \(\mathbf {GL}(n)\times \mathbf {GL}({\bar{n}})\) local rotations (2.15),
In particular, \(X^{i}_{\mu }\Omega ^{\mu \nu }{}_{\lambda }\), \({\bar{X}}^{\bar{\imath }}_{\mu }\Omega ^{\mu \nu }{}_{\lambda }\), and \(H^{\rho [\lambda }\Omega ^{\mu ]\nu }{}_{\rho }\) are covariant tensors which might be viewed as “torsions”.
Finally, we define an upper-indexed Ricci curvature,
which is diffeomorphism and \(\mathbf {GL}(n)\times \mathbf {GL}({\bar{n}})\) 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 \((n,{\bar{n}})\) component fields in a generalized fashion:
Another characteristic is that, if we add one more torsion linear in the \(\mathbb {H}\)-flux to the \(\Omega \)-connection,
the hatted new connection becomes Milne-shift covariant as well, in the sense of (2.16), (2.25), (2.26),
where \(\widehat{\mathbb {H}}^{\lambda \mu \nu }\) is a diffeomorphism covariant, \(\mathbf {GL}(n)\times \mathbf {GL}({\bar{n}})\) invariant, and Milne-shift invariant \(\mathbb {H}\)-flux,
The \(\mathbf {GL}(n)\times \mathbf {GL}({\bar{n}})\) connections (4.26) are inert to the addition of the \(\mathbb {H}\)-flux-valued-torsion (4.38) as
while they transform under the Mine-shift as \(\delta _{\scriptscriptstyle {\mathrm{M}}}\Upsilon ^{\mu i}{}_{j}= -2H^{\rho \sigma }V_{\rho j}\mathbb {D}^{\mu }X^{i}_{\sigma \,}\), \(\,\delta _{\scriptscriptstyle {\mathrm{M}}}{{\bar{\Upsilon }}}^{\mu \bar{\imath }}{}_{\bar{\jmath }}=-2H^{\rho \sigma }{\bar{V}}_{\rho \bar{\jmath }}\mathbb {D}^{\mu }{\bar{X}}^{\bar{\imath }}_{\sigma \,}\).
After all, in terms of a hatted covariant derivative,
we can dismantle the DFT curvatures into a \(\mathbb {H}\)-flux-free (circled) term and evidently \(\mathbb {H}\)-flux-valued ones:
where, as it should be obvious from our notation, we set \({\hat{S}}_{AB}:=({\mathcal {B}}^{-1})_{A}{}^{C}({\mathcal {B}}^{-1})_{B}{}^{D}S_{CD}\), and the circled quantities are all \(\mathbb {H}\)-flux free: from Table 1 or [63, 65],
and, with (3.16),
While we organize the \(\mathbb {H}\)-flux-valued parts in terms of the hatted covariant derivative, like (4.41), we have
The only nontrivial distinction lies in
Since \(e^{-2d}\widehat{\mathbb {H}}^{\lambda \mu \nu }\) carries a unit weight, its contraction with the ordinary derivative, \(\partial _{\lambda }(e^{-2d}\widehat{\mathbb {H}}^{\lambda \mu \nu })\), is also by itself diffeomorphism covariant. In this way, every single term in (4.43) is symmetric under both undoubled diffeomorphisms and \(\mathbf {GL}(n)\times \mathbf {GL}({\bar{n}})\) local rotations. On the other hand, as we have singled out the \(\mathbb {H}\)-flux-valued terms from the \(\mathbb {H}\)-flux-free parts, each individual term is not necessarily Milne-shift covariant.
As advertised in (4.21), we may further dismantle \(\mathring{S}_{\scriptscriptstyle {{(0)}}}\) as well as \((\mathring{P}\mathring{S}\mathring{{\bar{P}}})^{\mu \nu }\) into more elementary modules:
From (3.26), vanishing of all the five quantities in (4.43) characterizes the \((n,{\bar{n}})\) vacuum geometry of DFT.
Comment 1. It is worth while to note
and rewrite the ‘kinetic term’ of the DFT-dilaton in (4.21),
Comment 2. Especially when \(n+{\bar{n}}=D\), i.e. in the maximally non-Riemannian cases, all the quantities like \(H^{\mu \nu },K_{\rho \sigma },\Omega ^{\lambda \mu }{}_{\nu }, \widehat{\mathbb {H}}^{\lambda \mu \nu },\mathbb {D}^{\mu }d, R^{\mu \nu },S_{\scriptscriptstyle {{(0)}}}, (\mathring{P}{\hat{S}}\mathring{{\bar{P}}})^{\mu \nu }\) are trivial except the term of interest, \(Y^{\mu }_{i}(\mathring{P}{\hat{S}}\mathring{{\bar{P}}})_{\mu \nu }{\bar{Y}}^{\mu }_{\bar{\imath }}\).
Comment 3. Restricted to the (0, 0) Riemannian case, we have \(K_{\mu \nu }=g_{\mu \nu \,}\), \(\,H^{\mu \nu }=g^{\mu \nu }\), \(\,K_{\mu \rho }H^{\rho \mu }=\delta _{\mu }{}^{\nu }\),
and the vectors, \(\{X_{\mu }^{i},{\bar{X}}_{\nu }^{\bar{\imath }},Y^{\rho }_{j},{\bar{Y}}^{\sigma }_{\bar{\jmath }}\}\), are trivially absent. Both the \(\Omega \) and \(\widehat{\Omega }\) connections (4.23, 4.38) coincide with nothing but the standard Christoffel symbols with one index raised by the Riemannian metric,
Consequently, the proposed covariant derivative (4.25) and Ricci curvature (4.34) reduce to the standard covariant derivative and Ricci curvature in Riemannian geometry,
Comment 4. Besides \((\mathring{P}\mathring{S}\mathring{{\bar{P}}})^{\mu \nu }\), we have not been able to dismantle other circled \(\mathbb {H}\)-flux-free DFT Ricci curvatures which carry at least one lower index. In addition to \({\mathfrak {D}}^{\mu }=H^{\mu \rho }\partial _{\rho }+\Omega ^{\mu }\), separate type of covariant derivatives containing \(Y^{\mu }_{i}\partial _{\mu }\) or \({\bar{Y}}^{\mu }_{\bar{\imath }}\partial _{\mu }\) might help, c.f. (B.13).
Comment 5. Appendix B sketches how we have arrived at the above proposal of the non-Riemannian differential tool kit starting from the semi-covariant formalism of DFT. In any case, our proposal is meant to provide a bookkeeping device to expound the EDFEs into smaller modules and to single out the \(\mathbb {H}\)-fluxes. The actual computation of the variations of the action, even with \((n,{\bar{n}})\) fixed, are still powered by the semi-covariant formalism, specifically (3.2).
5 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, \((n,{\bar{n}})\) [1].
We have analyzed with care the variational principle. We have shown that the most general infinitesimal variations of an arbitrary \((n,{\bar{n}})\) DFT-metric have \(D^{2}-(n-{\bar{n}})^{2}\) number of degrees of freedom, which matches with the dimension of the underlying coset [11], \( \frac{\mathbf {O}(D,D)}{\mathbf {O}(t+n,s+n)\times \mathbf {O}(s+{\bar{n}},t+{\bar{n}})}\) (3.14). Through action principle, these variations imply the full Einstein Double Field Equations (3.22), (3.24). However, \(n{\bar{n}}\) number of them change the value of \((n,{\bar{n}})\), i.e. the type of non-Riemannianity (4.19). Consequently, if we keep \((n,{\bar{n}})\) fixed once and for all, the variational principle gets restricted and fails to reproduce the full EDFEs: the specific part, \(Y^{\mu }_{i}(PE{\bar{P}})_{\mu \nu }{\bar{Y}}^{\nu }_{\bar{\imath }}\), does not have to vanish on-shell (4.9).Footnote 10
The EDFEs are supposed to arise as the string worldsheet beta-functions [97, 98]. For the doubled-yet-gauged string action (1.11) upon an arbitrarily chosen \((n,{\bar{n}})\) background, the \((n,{\bar{n}})\)-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 \((n,{\bar{n}})\) (4.13), c.f. [52, 54, 56]. Nevertheless, it is natural to expect that \(n{\bar{n}}\) number of \(Y^{\mu }_{i}(\mathring{P}\widehat{E}\mathring{{\bar{P}}})_{\mu \nu }{\bar{Y}}^{\nu }_{\bar{\imath }}\) arise as the corresponding beta-functions too. That is to say, at least for \(n{\bar{n}}\ne 0\), the quantum consistency with the worldsheet string theory seems to forbid us to fix \((n,{\bar{n}})\) 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.
Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors’ comment: This work is genuinely theoretical.]
Notes
The expression of \(S_{AB}\) in Table 1 is newly derived from [63] using \(\Gamma _{ACD}\Gamma ^{CBD}=\Gamma ^{BCD}\Gamma _{CAD} ={{\textstyle \frac{1}{2}}}\Gamma _{ACD}\Gamma ^{BCD\,}\) and \( \Gamma _{CAD}\Gamma ^{DBC}=\Gamma _{CAD}\Gamma ^{CBD} - {{\textstyle \frac{1}{2}}}\Gamma _{ACD}\Gamma ^{BCD\,}\) which hold due to the symmetric properties, \(\Gamma _{[ABC]}=0\) and \(\Gamma _{A(BC)}=0\).
Put \({\mathcal {H}}_{A}{}^{B}=\delta _{A}{}^{B}\) in (3.5).
Through exponentiations, finite Milne-shift transformations can be achieved, which turn out to get truncated at finite orders, for example \(e^{\delta _{\scriptscriptstyle {\mathrm{M}}}}Y_{i}^{\mu }=Y_{i}^{\mu }+H^{\mu \nu }V_{\nu i}\,\). See Eq.(2.16) of [1] for the full list.
The only required property of the DFT-vielbeins is \(\,V_{Ap}V_{B}{}^{p}+{\bar{V}}_{A{\bar{p}}}{\bar{V}}_{B}{}^{{\bar{p}}}={\mathcal {J}}_{AB}\). See [75] for a related discussion.
In a way, on the space of full DFT geometries, the (0, 0) Riemannian geometry corresponds to an open set as \(\det (H^{\mu \nu })\ne 0\), while the genuine non-Riemannian geometries form a closed set, \(\det (H^{\mu \nu })= 0\). Infinitesimally, it is impossible to leave an open set but possible to leave a closed set.
We tend to believe that the conventional string dilaton, \(\phi \), is an artifact of the (0, 0) Riemannian geometry and the DFT-dilaton, d, is more fundamental as being an \(\mathbf {O}(D,D)\) singlet.
As can be seen from (4.44), \(\,Y_{i}^{\rho }(\mathring{P}\widehat{E}\mathring{{\bar{P}}})_{\rho \sigma }{\bar{Y}}^{\sigma }_{\bar{\imath }}\) contains a second order derivative of the DFT-dilaton along the \(Y_{i}^{\mu }\) and \({\bar{Y}}_{\nu }^{\bar{\imath }}\) directions, i.e. \(\,Y^{\mu }_{i}{\bar{Y}}^{\nu }_{\bar{\imath }}\partial _{\mu }\partial _{\nu }d\,\).
While \({\nabla }_{C}V_{D}\) itself is not covariant, the projected ones in (B.1) are covariant, and hence the name, ‘semi-covariant formalism’.
Note \({\mathcal {B}}_{A}{}^{B}\partial _{B}=\partial _{A}\) as \(\tilde{\partial }^{\mu }\equiv 0\).
References
K. Morand, J.H. Park, Classification of non-Riemannian doubled-yet-gauged spacetime. Eur. Phys. J. C 77(10), 685 (2017). Erratum: [Eur. Phys. J. C 78(11), 901 (2018) ] https://doi.org/10.1140/epjc/s10052-017-5257-z, https://doi.org/10.1140/epjc/s10052-018-6394-8, arXiv:1707.03713 [hep-th]
W. Siegel, Two vierbein formalism for string inspired axionic gravity. Phys. Rev. D 47, 5453 (1993). https://doi.org/10.1103/PhysRevD.47.5453. arXiv:hep-th/9302036
W. Siegel, Superspace duality in low-energy superstrings. Phys. Rev. D 48, 2826 (1993). https://doi.org/10.1103/PhysRevD.48.2826. arXiv:hep-th/9305073
C. Hull, B. Zwiebach, Double field theory. JHEP 0909, 099 (2009). https://doi.org/10.1088/1126-6708/2009/09/099. arXiv:0904.4664 [hep-th]
C. Hull, B. Zwiebach, The gauge algebra of double field theory and Courant brackets. JHEP 0909, 090 (2009). https://doi.org/10.1088/1126-6708/2009/09/090. arXiv:0908.1792 [hep-th]
O. Hohm, C. Hull, B. Zwiebach, Background independent action for double field theory. JHEP 1007, 016 (2010). https://doi.org/10.1007/JHEP07(2010)016. arXiv:1003.5027 [hep-th]
O. Hohm, C. Hull, B. Zwiebach, Generalized metric formulation of double field theory. JHEP 1008, 008 (2010). https://doi.org/10.1007/JHEP08(2010)008. arXiv:1006.4823 [hep-th]
K. Lee, J.H. Park, Covariant action for a string in doubled yet gauged spacetime. Nucl. Phys. B 880, 134 (2014). arXiv:1307.8377 [hep-th]
J.H. Park, Y. Suh, U-gravity: SL(N). JHEP 1406, 102 (2014). https://doi.org/10.1007/JHEP06(2014)102. arXiv:1402.5027 [hep-th]
S.M. Ko, C. Melby-Thompson, R. Meyer, J.H. Park, Dynamics of perturbations in double field theory & non-relativistic string theory. JHEP 1512, 144 (2015). https://doi.org/10.1007/JHEP12(2015)144. arXiv:1508.01121 [hep-th]
D.S. Berman, C.D.A. Blair, R. Otsuki, Non-Riemannian geometry of M-theory. JHEP 1907, 175 (2019). https://doi.org/10.1007/JHEP07(2019)175. arXiv:1902.01867 [hep-th]
Y. Sakatani, Type II DFT solutions from Poisson-Lie T-duality/plurality, PTEP 073B04 (2019). https://doi.org/10.1093/ptep/ptz071. arXiv:1903.12175 [hep-th]
N.A. Obers, B. Pioline, U duality and M theory. Phys. Rep. 318, 113 (1999). https://doi.org/10.1016/S0370-1573(99)00004-6. arXiv:hep-th/9809039
C.M. Hull, A geometry for non-geometric string backgrounds. JHEP 0510, 065 (2005). https://doi.org/10.1088/1126-6708/2005/10/065. arXiv:hep-th/0406102
C.M. Hull, R.A. Reid-Edwards, Flux compactifications of string theory on twisted tori. Fortsch. Phys. 57, 862 (2009). https://doi.org/10.1002/prop.200900076. arXiv:hep-th/0503114
G. Dall’Agata, N. Prezas, H. Samtleben, M. Trigiante, Gauged supergravities from twisted doubled tori and non-geometric string backgrounds. Nucl. Phys. B 799, 80 (2008). https://doi.org/10.1016/j.nuclphysb.2008.02.020. arXiv:0712.1026 [hep-th]
R. Blumenhagen, A. Deser, D. Lust, E. Plauschinn, F. Rennecke, Non-geometric fluxes, asymmetric strings and nonassociative geometry. J. Phys. A 44, 385401 (2011). https://doi.org/10.1088/1751-8113/44/38/385401. arXiv:1106.0316 [hep-th]
D. Andriot, M. Larfors, D. Lust, P. Patalong, A ten-dimensional action for non-geometric fluxes. JHEP 1109, 134 (2011). https://doi.org/10.1007/JHEP09(2011)134. arXiv:1106.4015 [hep-th]
D. Andriot, O. Hohm, M. Larfors, D. Lust, P. Patalong, Non-geometric fluxes in supergravity and double field theory. Fortsch. Phys. 60, 1150 (2012). https://doi.org/10.1002/prop.201200085. arXiv:1204.1979 [hep-th]
R. Blumenhagen, A. Deser, E. Plauschinn, F. Rennecke, Non-geometric strings, symplectic gravity and differential geometry of Lie algebroids. JHEP 1302, 122 (2013). https://doi.org/10.1007/JHEP02(2013)122. arXiv:1211.0030 [hep-th]
G. Dibitetto, J.J. Fernandez-Melgarejo, D. Marques, D. Roest, Duality orbits of non-geometric fluxes. Fortsch. Phys. 60, 1123 (2012). https://doi.org/10.1002/prop.201200078. arXiv:1203.6562 [hep-th]
E. Malek, Timelike U-dualities in generalised geometry. JHEP 1311, 185 (2013). https://doi.org/10.1007/JHEP11(2013)185. arXiv:1301.0543 [hep-th]
F. Hassler, D. Lust, Consistent compactification of double field theory on non-geometric flux backgrounds. JHEP 1405, 085 (2014). https://doi.org/10.1007/JHEP05(2014)085. arXiv:1401.5068 [hep-th]
M. Cederwall, T-duality and non-geometric solutions from double geometry. Fortsch. Phys. 62, 942 (2014). https://doi.org/10.1002/prop.201400069. arXiv:1409.4463 [hep-th]
I. Bakhmatov, A. Kleinschmidt, E.T. Musaev, Non-geometric branes are DFT monopoles. JHEP 1610, 076 (2016). https://doi.org/10.1007/JHEP10(2016)076. arXiv:1607.05450 [hep-th]
M.A. Heller, N. Ikeda, S. Watamura, Unified picture of non-geometric fluxes and T-duality in double field theory via graded symplectic manifolds. JHEP 1702, 078 (2017). https://doi.org/10.1007/JHEP02(2017)078. arXiv:1611.08346 [hep-th]
K. Lee, S.J. Rey, Y. Sakatani, Effective action for non-geometric fluxes duality covariant actions. JHEP 1707, 075 (2017). https://doi.org/10.1007/JHEP07(2017)075. arXiv:1612.08738 [hep-th]
A. Chatzistavrakidis, L. Jonke, F.S. Khoo, R.J. Szabo, Double field theory and membrane sigma-models. JHEP 1807, 015 (2018). https://doi.org/10.1007/JHEP07(2018)015. arXiv:1802.07003 [hep-th]
V.E. Marotta, R.J. Szabo, Para Hermitian geometry, dualities and generalized flux backgrounds. Fortsch. Phys. 67(3), 1800093 (2019). https://doi.org/10.1002/prop.201800093. arXiv:1810.03953 [hep-th]
E. Plauschinn, Non-geometric backgrounds in string theory. Phys. Rep. 798, 1 (2019). https://doi.org/10.1016/j.physrep.2018.12.002. arXiv:1811.11203 [hep-th]
A. Deser, C. Sämann, Extended Riemannian geometry III: global double field theory with nilmanifolds. JHEP 1905, 209 (2019). https://doi.org/10.1007/JHEP05(2019)209. arXiv:1812.00026 [hep-th]
D.S. Berman, E.T. Musaev, R. Otsuki, Exotic branes in M-theory. arXiv:1903.10247 [hep-th]
E. Cartan, Sur les variétés à connexion affine et la théorie de la relativité généralisée. (première partie). Ann. Sci. Ecole Norm. Sup. 40, 325 (1923)
H.P. Künzle, Galilei and Lorentz structures on space-time—comparison of the corresponding geometry and physics. Ann. Inst. H. Poincare Phys. Theor. 17, 337 (1972)
C. Duval, G. Burdet, H.P. Kunzle, M. Perrin, Bargmann structures and Newton-Cartan theory. Phys. Rev. D 31, 1841 (1985). https://doi.org/10.1103/PhysRevD.31.1841
R. Andringa, E. Bergshoeff, J. Gomis, M. de Roo, ’Stringy’ Newton-Cartan gravity. Class. Quantum Gravity 29, 235020 (2012). https://doi.org/10.1088/0264-9381/29/23/235020. arXiv:1206.5176 [hep-th]
M. Henneaux, Zero hamiltonian signature spacetimes. Bull. Soc. Math. de Belgique XXXI, 31–47 (1979)
C. Duval, G.W. Gibbons, P.A. Horvathy, P.M. Zhang, Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time. Class. Quantum Gravity 31, 085016 (2014). https://doi.org/10.1088/0264-9381/31/8/085016. arXiv:1402.0657 [gr-qc]
J. Gomis, H. Ooguri, Nonrelativistic closed string theory. J. Math. Phys. 42, 3127 (2001). https://doi.org/10.1063/1.1372697. arXiv:hep-th/0009181
E.A. Milne, A Newtonian expanding universe. Quat. J. Math. (Oxf. Ser.) 5, 64–72 (1934)
C. Duval, On Galileian isometries. Class. Quantum Gravity 10, 2217 (1993). https://doi.org/10.1088/0264-9381/10/11/006. arXiv:0903.1641 [math-ph]
M.H. Christensen, J. Hartong, N.A. Obers, B. Rollier, Torsional Newton-Cartan geometry and Lifshitz holography. Phys. Rev. D 89, 061901 (2014). https://doi.org/10.1103/PhysRevD.89.061901. arXiv:1311.4794 [hep-th]
X. Bekaert, K. Morand, Connections and dynamical trajectories in generalised Newton-Cartan gravity I. An intrinsic view. J. Math. Phys. 57(2), 022507 (2016). https://doi.org/10.1063/1.4937445. arXiv:1412.8212 [hep-th]
X. Bekaert, K. Morand, Connections and dynamical trajectories in generalised Newton-Cartan gravity II. An ambient perspective. J. Math. Phys. 59(7), 072503 (2018). https://doi.org/10.1063/1.5030328. [arXiv:1505.03739 [hep-th]]
E.A. Bergshoeff, J. Rosseel, Three-dimensional extended Bargmann supergravity. Phys. Rev. Lett. 116(25), 251601 (2016). https://doi.org/10.1103/PhysRevLett.116.251601. arXiv:1604.08042 [hep-th]
J. Hartong, Y. Lei, N.A. Obers, Nonrelativistic Chern-Simons theories and three-dimensional Hořava-Lifshitz gravity. Phys. Rev. D 94(6), 065027 (2016). https://doi.org/10.1103/PhysRevD.94.065027. arXiv:1604.08054 [hep-th]
E. Bergshoeff, J. Gomis, Z. Yan, Nonrelativistic string theory and T-duality. JHEP 1811, 133 (2018). https://doi.org/10.1007/JHEP11(2018)133. arXiv:1806.06071 [hep-th]
D. Hansen, J. Hartong, N.A. Obers, Action principle for Newtonian gravity. Phys. Rev. Lett. 122(6), 061106 (2019). https://doi.org/10.1103/PhysRevLett.122.061106. arXiv:1807.04765 [hep-th]
T. Harmark, J. Hartong, L. Menculini, N.A. Obers, Z. Yan, Strings with non-relativistic conformal symmetry and limits of the AdS/CFT correspondence. JHEP 1811, 190 (2018). https://doi.org/10.1007/JHEP11(2018)190. arXiv:1810.05560 [hep-th]
E.A. Bergshoeff, K.T. Grosvenor, C. Simsek, Z. Yan, An action for extended string Newton-Cartan gravity. JHEP 1901, 178 (2019). https://doi.org/10.1007/JHEP01(2019)178. arXiv:1810.09387 [hep-th]
K. Morand, Embedding Galilean and Carrollian geometries I. Gravitational waves. arXiv:1811.12681 [hep-th]
J. Gomis, J. Oh, Z. Yan, Nonrelativistic string theory in background fields. arXiv:1905.07315 [hep-th]
D. Hansen, J. Hartong, N.A. Obers, Non-relativistic expansion of the Einstein–Hilbert Lagrangian. arXiv:1905.13723 [gr-qc]
A.D. Gallegos, U. Gürsoy, N. Zinnato, Torsional Newton Cartan gravity from non-relativistic strings. arXiv:1906.01607 [hep-th]
T. Harmark, J. Hartong, L. Menculini, N.A. Obers, G. Oling, Relating non-relativistic string theories. arXiv:1907.01663 [hep-th]
E.A. Bergshoeff, J. Gomis, J. Rosseel, C. ŞimŞek, Z. Yan, String theory and string Newton-Cartan geometry. arXiv:1907.10668 [hep-th]
C.D.A. Blair, A worldsheet supersymmetric Newton-Cartan string. arXiv:1908.00074 [hep-th]
D. Pereñiguez, p-brane Newton–Cartan geometry. arXiv:1908.04801 [hep-th]
G. Aldazabal, D. Marques, C. Nunez, Double field theory: a pedagogical review. Class. Quantum Gravity 30, 163001 (2013). https://doi.org/10.1088/0264-9381/30/16/163001. arXiv:1305.1907 [hep-th]
D.S. Berman, D.C. Thompson, Duality symmetric string and M-theory. Phys. Rep. 566, 1 (2014). https://doi.org/10.1016/j.physrep.2014.11.007. arXiv:1306.2643 [hep-th]
O. Hohm, D. Lüst, B. Zwiebach, The spacetime of double field theory: review, remarks, and outlook. Fortsch. Phys. 61, 926 (2013). https://doi.org/10.1002/prop.201300024. arXiv:1309.2977 [hep-th]
I. Jeon, K. Lee, J.H. Park, Differential geometry with a projection: application to double field theory. JHEP 1104, 014 (2011). https://doi.org/10.1007/JHEP04(2011)014. arXiv:1011.1324 [hep-th]
I. Jeon, K. Lee, J.H. Park, Stringy differential geometry, beyond Riemann. Phys. Rev. D 84, 044022 (2011). https://doi.org/10.1103/PhysRevD.84.044022. arXiv:1105.6294 [hep-th]
O. Hohm, B. Zwiebach, On the Riemann tensor in double field theory. JHEP 1205, 126 (2012). https://doi.org/10.1007/JHEP05(2012)126. arXiv:1112.5296 [hep-th]
S. Angus, K. Cho, J.H. Park, Einstein double field equations. Eur. Phys. J. C 78(6), 500 (2018). https://doi.org/10.1140/epjc/s10052-018-5982-y. arXiv:1804.00964 [hep-th]
S.M. Ko, J.H. Park, M. Suh, The rotation curve of a point particle in stringy gravity. JCAP 1706(06), 002 (2017). https://doi.org/10.1088/1475-7516/2017/06/002. arXiv:1606.09307 [hep-th]
J.H. Park, Stringy gravity: solving the dark problems at ‘short’ distance. EPJ Web Conf. 168, 01010 (2018). https://doi.org/10.1051/epjconf/201816801010. arXiv:1707.08961 [hep-th]
A. Rocen, P. West, E11, generalised space-time and IIA string theory: the R-R sector. https://doi.org/10.1142/9789814412551_0020. arXiv:1012.2744 [hep-th]
O. Hohm, S.K. Kwak, B. Zwiebach, Unification of type II strings and T-duality. Phys. Rev. Lett. 107, 171603 (2011). https://doi.org/10.1103/PhysRevLett.107.171603. arXiv:1106.5452 [hep-th]
O. Hohm, S.K. Kwak, B. Zwiebach, Double field theory of type II strings. JHEP 1109, 013 (2011). https://doi.org/10.1007/JHEP09(2011)013. arXiv:1107.0008 [hep-th]
I. Jeon, K. Lee, J.H. Park, Ramond-Ramond cohomology and O(D, D) T-duality. JHEP 1209, 079 (2012). https://doi.org/10.1007/JHEP09(2012)079. arXiv:1206.3478 [hep-th]
I. Jeon, K. Lee, J.H. Park, Incorporation of fermions into double field theory. JHEP 1111, 025 (2011). https://doi.org/10.1007/JHEP11(2011)025. arXiv:1109.2035 [hep-th]
O. Hohm, S.K. Kwak, Double field theory formulation of heterotic strings. JHEP 1106, 096 (2011). https://doi.org/10.1007/JHEP06(2011)096. arXiv:1103.2136 [hep-th]
O. Hohm, A. Sen, B. Zwiebach, Heterotic effective action and duality symmetries revisited. JHEP 1502, 079 (2015). https://doi.org/10.1007/JHEP02(2015)079. arXiv:1411.5696 [hep-th]
K. Cho, K. Morand, J.H. Park, Kaluza-Klein reduction on a maximally non-Riemannian space is moduli-free. Phys. Lett. B 793, 65 (2019). https://doi.org/10.1016/j.physletb.2019.04.042. arXiv:1808.10605 [hep-th]
I. Jeon, K. Lee, J.H. Park, Double field formulation of Yang-Mills theory. Phys. Lett. B 701, 260 (2011). https://doi.org/10.1016/j.physletb.2011.05.051. arXiv:1102.0419 [hep-th]
X. Bekaert, J.H. Park, Higher spin double field theory: a proposal. JHEP 1607, 062 (2016). https://doi.org/10.1007/JHEP07(2016)062. arXiv:1605.00403 [hep-th]
I. Jeon, K. Lee, J.H. Park, Supersymmetric Double Field Theory: Stringy Reformulation of Supergravity. Phys. Rev. D 85, 081501 (2012), Erratum: [Phys. Rev. D 86 (2012) 089903]. https://doi.org/10.1103/PhysRevD.86.089903, https://doi.org/10.1103/PhysRevD.85.081501, https://doi.org/10.1103/PhysRevD.85.089908. arXiv:1112.0069 [hep-th]
I. Jeon, K. Lee, J.H. Park, Y. Suh, Stringy unification of type IIA and IIB supergravities under N = 2 D = 10 supersymmetric double field theory. Phys. Lett. B 723, 245 (2013). https://doi.org/10.1016/j.physletb.2013.05.016. arXiv:1210.5078 [hep-th]
K.S. Choi, J.H. Park, Standard model as a double field theory. Phys. Rev. Lett. 115(17), 171603 (2015). https://doi.org/10.1103/PhysRevLett.115.171603. arXiv:1506.05277 [hep-th]
J.H. Park, S.J. Rey, W. Rim, Y. Sakatani, \({ O}(D, D)\) covariant Noether currents and global charges in double field theory. JHEP 1511, 131 (2015). https://doi.org/10.1007/JHEP11(2015)131. arXiv:1507.07545 [hep-th]
J.H. Park, \({\bf O} (D,D)\) completion of the Einstein field equations. arXiv:1904.04705 [hep-th]
J.H. Park, Comments on double field theory and diffeomorphisms. JHEP 1306, 098 (2013). https://doi.org/10.1007/JHEP06(2013)098. arXiv:1304.5946 [hep-th]
C.M. Hull, Doubled geometry and T-folds. JHEP 0707, 080 (2007). arXiv:hep-th/0605149
J.H. Park, Green-Schwarz superstring on doubled-yet-gauged spacetime. JHEP 1611, 005 (2016). arXiv:1609.04265 [hep-th]
A.S. Arvanitakis, C.D.A. Blair, Unifying type-II strings by exceptional groups. Phys. Rev. Lett. 120(21), 211601 (2018). arXiv:1712.07115 [hep-th]
A.S. Arvanitakis, C.D.A. Blair, The exceptional sigma model. JHEP 1804, 064 (2018). arXiv:1802.00442 [hep-th]
A. Giveon, E. Rabinovici, G. Veneziano, Duality in string background space. Nucl. Phys. B 322, 167 (1989). https://doi.org/10.1016/0550-3213(89)90489-6
M.J. Duff, Duality rotations in string theory. Nucl. Phys. B 335, 610 (1990). https://doi.org/10.1016/0550-3213(90)90520-N
W. Siegel, Amplitudes for left-handed strings. arXiv:1512.02569 [hep-th]
E. Casali, P. Tourkine, Windings of twisted strings. Phys. Rev. D 97(6), 061902 (2018). https://doi.org/10.1103/PhysRevD.97.061902. arXiv:1710.01241 [hep-th]
C.D.A. Blair, E. Malek, J.H. Park, M-theory and type IIB from a duality manifest action. JHEP 1401, 172 (2014). https://doi.org/10.1007/JHEP01(2014)172. arXiv:1311.5109 [hep-th]
C.M. Hull, Finite gauge transformations and geometry in double field theory. JHEP 1504, 109 (2015). https://doi.org/10.1007/JHEP04(2015)109. arXiv:1406.7794 [hep-th]
J. Berkeley, D.S. Berman, F.J. Rudolph, Strings and branes are waves. JHEP 1406, 006 (2014). https://doi.org/10.1007/JHEP06(2014)006. arXiv:1403.7198 [hep-th]
D. Andriot, A. Betz, \(\beta \)-supergravity: a ten-dimensional theory with non-geometric fluxes, and its geometric framework. JHEP 1312, 083 (2013). https://doi.org/10.1007/JHEP12(2013)083. arXiv:1306.4381 [hep-th]
C.D.A. Blair, E. Malek, Geometry and fluxes of SL(5) exceptional field theory. JHEP 1503, 144 (2015). https://doi.org/10.1007/JHEP03(2015)144. arXiv:1412.0635 [hep-th]
D.S. Berman, N.B. Copland, D.C. Thompson, Background field equations for the duality symmetric string. Nucl. Phys. B 791, 175 (2008). https://doi.org/10.1016/j.nuclphysb.2007.09.021. arXiv:0708.2267 [hep-th]
N.B. Copland, A double sigma model for double field theory. JHEP 1204, 044 (2012). https://doi.org/10.1007/JHEP04(2012)044. arXiv:1111.1828 [hep-th]
K. Peeters, Introducing Cadabra: A Symbolic computer algebra system for field theory problems. arXiv:hep-th/0701238
W. Cho, J.J. Fernández-Melgarejo, I. Jeon, J.H. Park, Supersymmetric gauged double field theory: systematic derivation by virtue of twist. JHEP 1508, 084 (2015). https://doi.org/10.1007/JHEP08(2015)084. arXiv:1505.01301 [hep-th]
D.S. Berman, K. Lee, Supersymmetry for gauged double field theory and generalised Scherk-Schwarz reductions. Nucl. Phys. B 881, 369 (2014). https://doi.org/10.1016/j.nuclphysb.2014.02.015. arXiv:1305.2747 [hep-th]
Acknowledgements
We would like to thank David Berman, Chris Blair, and Kevin Morand for helpful discussions. Parts of computations are assisted by a computer algebra system, Cadabra [99]. This work was supported by the National Research Foundation of Korea through the Grant, NRF-2016R1D1A1B01015196.
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Appendices
Appendix
Proof on the equivalence among (4.7), (4.8), (4.9), and (4.10)
Taking \(\left\{ \delta K_{\mu }{}^{a},\delta X^{i}_{\rho },\delta {\bar{X}}^{\bar{\imath }}_{\sigma }\right\} \) as independent variations, from the second line of (4.6), the variational principle implies (4.7) which we enumerate here:
Alternatively taking \(\left\{ \delta H^{\mu }{}_{a},\delta Y_{i}^{\rho },\delta {\bar{Y}}_{\bar{\imath }}^{\sigma }\right\} \) as independent variations, we acquire from the third line of (4.6),
Henceforth we show that Eqs. (A.1, A.2, A.3, A.4) and Eqs. (A.5, A.6, A.7, A.8) are all equivalent to (4.9) as well as to (4.10). The equivalence between (4.9) and (4.10) should be obvious from the off-shell relation (3.25), and therefore we focus on (4.9) which we recall for quick reference:
Proof
It is manifest that (A.9) implies both Eqs. (A.1, A.2, A.3, A.4) and Eqs. (A.5, A.6, A.7, A.8). Thus, we only need to show the converse. Equations (A.4) and (A.8) are common and give, combined with (3.17),
With these in mind we first focus on the former set of Eqs. (A.1, A.2, A.3, A.4), of which the first and the last imply
Consequently, with the completeness relation (2.7), the identities from (3.17), and (A.2), we note
Similarly we get with (A.3),
It follows that
Thus, Eqs. (A.1, A.2, A.3, A.4) are equivalent to (A.9).
We now turn to the latter set of equations (A.5, A.6, A.7, A.8). In a parallel manner to (A.12), (A.13), we note from (A.6), (A.7),
which imply with (A.5),
and hence in particular,
Then with (A.8) and from
we note
It follows then, with (A.8), (A.10),
Finally, as in (A.15), we have
Thus, Eqs. (A.5, A.6, A.7, A.8) are also equivalent to (A.9), and this completes our Proof. \(\square \)
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 \(\widehat{\Omega }\) connection (4.38), descends from the known covariant derivatives in the DFT semi-covariant formalism [63]: specifically,Footnote 11
In order to convert these into undoubled ordinary covariant quantities —or to get rid of the bare B-field in them— we multiply \({\mathcal {B}}^{-1}\) as in (2.22) and write
Here we set
and \(\hat{\Gamma }_{ABC}\) is a naturally induced – or ‘twisted’ [100], c.f. [101] – new connection,Footnote 12
The very last term on the second line involves certain six-indexed projectors formed by \(\mathring{P}_{A}{}^{B},\mathring{{\bar{P}}}_{A}{}^{B}\) (c.f. Eq. (17) of [63] and Eq. (2.26) of [65]), and is actually irrelevant as it is always projected out in the final results. Using the new connection (B.4) we can conveniently separate the B-field contributions and eventually acquire the results (4.43).
Now, remembering \(\mathring{P}^{\mu \nu }=-\mathring{{\bar{P}}}^{\mu \nu }={{\textstyle \frac{1}{2}}}H^{\mu \nu }\) (3.16) and \(\mathring{P}_{A}{}^{B}+\mathring{{\bar{P}}}_{A}{}^{B}=\delta _{A}{}^{B}\), we subtract the two quantities in (B.2), and acquire a desired covariant derivative, or \(\,\widehat{\mathfrak {D}}^{\mu }=H^{\mu \rho }\partial _{\rho }+\widehat{\Omega }^{\mu }\) (4.38):
where, with shorthand notation,
we set, extending (4.39),
With \(\partial _{A}=(0,\partial _{\mu })\) and \(\xi ^{A}=(0,\xi ^{\mu })\) (2.23), using Eq. (2.43) of [65], we get under diffeomorphisms,
Hence both \({\widehat{\Phi }}^{\lambda }{}_{\mu \nu }\) and \(\widehat{\mathbb {H}}^{\lambda \mu \nu }\) are diffeomorphism covariant (and surely \(\mathbf {GL}(n)\times \mathbf {GL}({\bar{n}})\) invariant) tensors.
Further, due to identities,
\(\widehat{\mathbb {H}}^{\lambda \mu \nu }\) and \({\widehat{\Phi }}^{\lambda }{}_{\mu \nu }\) are skew-symmetric,
and we may express \(\widehat{\Omega }^{\mu \nu }{}_{\lambda }\) in different ways,
In particular, when the circled vector, \(\mathring{V}_{A}=(0,\mathring{V}_{\mu })\), is derivative-index-valued as \({\mathring{V}^{\mu }\equiv 0}\), from (2.26) \(\mathring{V}_{\mu }\) becomes Milne-shift invariant and so does \(\widehat{\mathfrak {D}}^{\lambda }\mathring{V}_{\mu \,}\),
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.
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Cho, K., Park, JH. Remarks on the non-Riemannian sector in Double Field Theory. Eur. Phys. J. C 80, 101 (2020). https://doi.org/10.1140/epjc/s10052-020-7648-9
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DOI: https://doi.org/10.1140/epjc/s10052-020-7648-9