Classification of non-Riemannian doubled-yet

Classification of non-Riemannian doubled-yet-gauged spacetime

K EVIN M ORAND] [

] Departamento

AND

J EONG -H YUCK PARK†∗

de Ciencias F´ısicas, Universidad Andres Bello

Republica 220, Santiago de Chile [ Departamento

de F´ısica, Universidad T´ecnica Federico Santa Mar´ıa

Centro Cient´ıfico-Tecnolo´gico de Valpara´ıso, Casilla 110-V, Valpara´ıso, Chile † Department ∗ Center

of Physics, Sogang University, 35 Baekbeom-ro, Mapo-gu, Seoul 04107, Korea

for Theoretical Physics of the Universe, Institute for Basic Science (IBS), Seoul 08826, Korea [email protected]

[email protected]

Abstract Assuming O(D, D) covariant fields as the ‘fundamental’ variables, Double Field Theory can accommodate novel geometries where Riemannian metric cannot be defined, even locally. Here we present a complete classification of such non-Riemannian spacetimes in terms of two non-negative integers, (n, n ¯ ), 0 ≤ n+n ¯ ≤ D. Upon these backgrounds, strings become chiral and anti-chiral over n and n ¯ directions respectively, while particles and closed strings are frozen over the n+¯ n directions. In particular, we identify (0, 0) as Riemannian manifolds, (1, 0) as non-relativistic spacetime, (1, 1) as Gomis-Ooguri nonrelativistic string, (D−1, 0) as ultra-relativistic Carroll geometry, and (D, 0) as Siegel’s chiral string. Combined with a covariant Kaluza-Klein ansatz which we further spell, (0, 1) leads to Newton-Cartan gravity. Alternative to the conventional string compactifications on small manifolds, non-Riemannian spacetime such as D = 10, (3, 3) may open a new scheme of the dimensional reduction from ten to four.

Contents 1

Introduction

1

2

General results

5

2.1

Classification of the DFT-metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.2

Particle and string on (n, n ¯ ) doubled-yet-gauged spacetime . . . . . . . . . . . . . . . . . .

8

2.3

DFT-vielbeins for (n, n ¯ ) doubled-yet-gauged spacetime . . . . . . . . . . . . . . . . . . . . 12

2.4

Kaluza-Klein ansatz for DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3

Applications

17

3.1

Maximally non-Riemannian (D, 0) : Siegel’s chiral string . . . . . . . . . . . . . . . . . . . 17

3.2

D = 10, (3, 3) : Non-Riemannian dimensional reduction from ten to four . . . . . . . . . . 18

3.3

(1, 1) : Non-relativistic limit a` la Gomis-Ooguri . . . . . . . . . . . . . . . . . . . . . . . . 19

3.4

(D − 1, 0) : Ultra-relativistic or Carroll . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.5

Least non-Riemannian (1, 0) or (0, 1) : Non-relativistic or Newton-Cartan . . . . . . . . . . 22

3.6

Embedding (0, 1) into ambient (0, 0) Kaluza-Klein ansatz : Carroll or Newton-Cartan . . . . 24

A Derivation of the most general form of the DFT-metric, Eq.(2.2)

1

30

Introduction

Ever since Einstein formulated his theory of gravity, i.e. General Relativity (GR), by employing differential geometry a` la Riemann, the Riemannian metric, gµν , has been privileged to be the only geometric and thus gravitational field. All other fields are meant to be ‘extra matters’. On the other hand, string theory suggests us to put a two-form gauge potential, Bµν , and a scalar dilaton, φ, on an equal footing along with the metric. Forming the massless sector of closed strings, this triplet of objects is ubiquitous in all string theories. Further, a genuine stringy symmetry, T-duality, can mix the three of them [1, 2], thus hinting at the existence of Stringy Gravity which should take the entire closed string massless sector as geometric and gravitational. After series of pioneering works on ‘doubled sigma models’ [3–8] and ‘double field theory’ (DFT) [9–13] (cf. [14–16] for reviews), such an idea of Stringy Gravity has materialized.1 1

Strictly speaking, string theory predicts not General Relativity but its own gravity, i.e. Stringy Gravity.

1

The word ‘double’ above refers to the fact that doubled (D +D)-dimensional coordinates are used for the description of D-dimensional physical spacetime. While, such a usage was historically first made in the case of a torus background – by introducing a dual coordinate conjugate to the string winding momentum – the doubled coordinates are far more general and can be applied to any compact or non-compact spacetime, and to not only string but also particle theories. Stringy Gravity of our interest adopts the doubled-yet-gauged coordinate system [17] which meets two properties. Firstly, an O(D, D) group is a priori postulated, having the invariant constant “metric”, 



 0 1  . JAB =    1 0

(1.1)

Along with its inverse, J AB , the invariant metric can be used to freely raise and lower the O(D, D) vector indices (capital letters, A, B, . . .). Secondly, the doubled coordinates are gauged by an equivalence relation, xA ∼ xA + ∆A (x) ,

(1.2)

where ∆A is an arbitrary ‘derivative-index-valued’ vector. This means that its superscript index must be identifiable as that of a derivative, ∂ A = J AB ∂B . For example, with arbitrary functions, Φ1 , Φ2 belonging to the theory, ∆A = Φ1 ∂ A Φ2 . The equivalence relation can be realized by requiring that all the fields or functions in Stringy Gravity – such as Φ1 , Φ2 , physical fields, local symmetry parameters, and their arbitrary derivatives – should be invariant under the coordinate gauge symmetry shift, ⇐⇒

Φ(x + ∆) = Φ(x)

∆A ∂A = 0 .

(1.3)

In this way, a single physical point is not represented by a point, as in ordinary Riemannian geometry, but as a gauge orbit in the doubled coordinate system. The above coordinate gauge symmetry invariance is equivalent to the so called ‘section condition’ in DFT, ∂A ∂ A = 0 .

(1.4)

With respect to the off block-diagonal form of the O(D, D) metric (1.1), the doubled coordinates split into two parts: xA = (˜ xµ , xν ) and ∂A = (∂˜µ , ∂ν ), such that ∂A ∂ A = 2∂µ ∂˜µ . The general solution to the section condition is then given by ∂˜µ ≡ 0, up to O(D, D) rotations [10, 11]. 2

Diffeomorphism covariance in doubled-yet-gauged spacetime reads δxA = ξ A ,

δ∂A = −∂A ξ B ∂B = (∂ B ξA − ∂A ξ B )∂B ,

(1.5)

and for a covariant tensor (or tensor density with weight ω), Pn

δTA1 ···An = −ω∂B ξ B TA1 ···An +

i=1 (∂B ξAi

− ∂Ai ξB )TA1 ···Ai−1 B Ai+1 ···An .

(1.6)

The latter corresponds to the passive counterpart of the “generalized Lie derivative”, Lˆξ , a` la Siegel [10]. The whole massless sector of closed strings, or stringy gravitons, can be represented by a unit-weighted scalar density, e−2d , and a symmetric projector, PA B PB C = PA C .

PAB = PBA ,

(1.7)

The complementary, orthogonal projector, P¯AB = JAB − PAB , satisfies, from (1.7), P P¯ = 0, P¯ 2 = P¯ . Covariant derivatives, ∇A = ∂A + ΓA , scalar curvature S(0) and “Ricci-like” curvature PA C P¯B D SCD are then expressed in terms of {PAB , P¯AB , d} and their derivatives or equivalently in terms of the stringy analogue of the Christoffel symbol, ΓABC (2.17) [18].2 The difference of the two projectors sets a symmetric O(D, D) element, known as the DFT-metric (or “generalized metric”), HAB = HBA = PAB − P¯AB

satisfying

HA C HB D JCD = JAB .

(1.8)

These O(D, D) covariant defining properties of the stringy gravitational fields can be conveniently solved by the conventional variables,   HM N =  

 g µν Bκρ

g ρν

−g µσ Bσλ gκλ −

Bκρ g ρσ Bσλ

 . 

(1.9)

However, this is not the most general solution: counter examples have been reported where the upper left D × D block of HAB is degenerate [20–22], and have been shown to provide a natural geometry for the 2

However, a fully covariant four-indexed “Riemann-like” curvature has been argued not to exist [18, 19]. This absence is, in a

way, consistent with the fact that there exists no locally inertial frame for extended object, i.e. string, where the stringy Christoffel connection (2.17) might vanish completely: Equivalence Principle holds for particles not strings.

3

non-relativistic string theory a` la Gomis and Ooguri [23].3 Namely, by assuming the O(D, D) covariant variables as the fundamental fields, DFT or Stringy Gravity becomes more general than GR: it encompasses ‘non-Riemannian’ spacetime where the Riemannian metric, gµν , cannot be defined, even locally. This includes various ‘singular’ limits of the Riemannian metric of which the inverse, g µν , becomes degenerate (c.f. T-fold, ‘non-geometries’ or ‘waves’ in global senses [6–8, 26–32]). Scope of the paper It is the purpose of the present paper to classify completely DFT backgrounds, by deriving the most general solution to the defining property of the stringy gravitational field, or (1.8). Our classification is given in terms of two non-negative integers, (n, n ¯ ), 0 ≤ n + n ¯ ≤ D. Except for the (0, 0) case, these are generically non-Riemannian. Since various DFTs and the relevant doubled sigma models have been constructed, strictly in terms of the O(D, D) covariant fields without referring to the Riemannian ones {gµν , Bµν , φ},4 our result can be readily and unambiguously applicable to these models which include, e.g. coupling to the Standard Model [33], higher spin [34], fluctuation or Noether analyses [21, 35–37], the doubled-yet-gauged Green-Schwarz superstring action [22] and the maximally supersymmetric D = 10 DFT [38]. In particular, this last example, with the Killing spinor equations therein, may lead to a new scheme of the dimensional reduction from ten to four, by assuming the six-dimensional internal space to be non-Riemannian, alternative to the traditional string compactification on ‘small’ Riemannian manifolds [39]. Further applications can be found in the holographic correspondences between Newton-Cartan gravity and condensed matter physics, e.g. [40, 41]. Organization of the paper The rest of the paper is structured as follows. In section 2, we classify the DFT-metric and spell the corresponding DFT-vielbeins. We discuss the dynamics of point particle and string upon these backgrounds. We also spell a path integral definition of the proper length in doubled-yet-gauged spacetime as well as a covariant Kaluza-Klein ansatz for DFT. In section 3, we discuss various applications, such as Gomis-Ooguri non-relativistic string, non-relativistic and ultra-relativistic geometries, Siegel’s chiral string and NewtonCartan gravity. Appendix contains the technical derivation of the main result.

3 4

cf. [24, 25] for U-duality analogues. Yet, there are quite a few works in the literature which do not meet this criterion, relying explicitly on the Riemannian variables.

Our results are thus not applicable therein.

4

2 2.1

General results Classification of the DFT-metric

As recalled in the introduction, the DFT-metric is by definition a symmetric O(D, D) element, satisfying the following relation: HAB = HBA ,

(2.1)

HA B HB C = δA C .

Our main result consists in providing a full classification for DFT-metrics by solving the above defining properties: the most general solution is characterized by two non-negative integers, (n, n ¯ ), 0 ≤ n + n ¯ ≤ D, and assumes the following form:  HAB

 = 

H µν

−H µσ Bσλ +

¯ κ¯ı Y¯¯ıν Bκρ H ρν + Xκi Yiν − X

Yiµ Xλi

−

¯ ¯ı Y¯¯ıµ X λ

ρ i B ¯ ¯ı ¯ρ Kκλ − Bκρ H ρσ Bσλ + 2X(κ λ)ρ Yi − 2X(κ Bλ)ρ Y¯ı

where i, j = 1, 2, · · · , n and ¯ı, ¯ = 1, 2, · · · , n ¯ . The variables,

   (2.2) 

n o ¯ ¯ı , Y¯ ν , H µν , Kµν , Bµν , Xµi , Yjν , X µ ¯

must meet the following properties: – H µν and Kµν are symmetric tensors H µν = H νµ ,

(2.3)

Kµν = Kνµ ,

  ¯ ν¯ı and Y µ , Y¯¯ν , respectively whose kernels are spanned by Xµi , X j H µν Xνi = 0 ,

¯ ν¯ı = 0 ; H µν X

Kµν Yjν = 0 ,

Kµν Y¯¯ν = 0 ;

(2.4)

– a completeness relation, ¯ ¯ı = δ µ ν ; H µρ Kρν + Yiµ Xνi + Y¯¯ıµ X ν

(2.5)

Bµν = −Bνµ .

(2.6)

– the skew-symmetry of B-field,

5

While the derivation is carried out in Appendix, some comments are in order. From (2.4), (2.5) and the  ¯ ν¯ı , orthonormal as well as algebraic relations follow linear independency of Xµi , X Yiµ Xµj = δi j ,

¯ µ¯ = δ¯ı ¯ , Y¯¯ıµ X

¯ µ¯ = Y¯¯ıµ Xµj = 0 , Yiµ X

H ρµ Kµν H νσ = H ρσ ,

Kρµ H µν Kνσ = Kρσ . (2.7)

With the choice of the section,

∂˜µ

≡ 0, the doubled-yet-gauged diffeomorphisms (1.5), (1.6), or the gener-

alized Lie derivative of the DFT-metric, cf. (A.4), imply that the variables transform covariantly as δXµi = Lξ Xµi , δH µν

= Lξ

H µν

,

¯ ¯ı = Lξ X ¯ ¯ı , δX µ µ

δYjν = Lξ Yjν ,

δKµν = Lξ Kµν ,

δ Y¯¯ν = Lξ Y¯¯ν ,

(2.8)

δBµν = Lξ Bµν + ∂µ ξ˜ν − ∂ν ξ˜µ ,

where Lξ denotes the ordinary, i.e. undoubled, Lie derivative with the local parameter, ξ ν , being part of the doubled vector field, ξ A = (ξ˜µ , ξ ν ). Our (n, n ¯ )-classification of the DFT-metric having the explicit parametrization (2.2) is particularly useful for the choice of the section, ∂˜µ ≡ 0. For example, the action for a massless scalar field reads (c.f. [43]) ˆ ˆ −2d AB e H ∂A Φ∂B Φ ≡ dD x e−2d H µν ∂µ Φ∂ν Φ .

(2.9)

section

For couplings to generic tensors or Yang-Mills fields, we refer to [18, 21, 36, 44, 45]. However, if the (n, n ¯) DFT-metric (2.2) admits an isometry direction, there appears arbitrariness in choosing the section. In this case, our parametrization (2.2) may be modified, see e.g. [32, 46]. Clearly, constant (n, n ¯ ) DFT-metric (2.2) and DFT-dilaton, d, solve the equations of motion of DFT. Thus, our (n, n ¯ ) classification also accounts for non-Riemannian ‘flat’ backgrounds. It is worthwhile to note that the characteristic value, (n, n ¯ ), may change point-wise in a given doubled-yet-gauged curved spacetime, typically at a “Riemannian singular point”. Further, O(D, D) rotations (along isometry directions) can also change the value of (n, n ¯ ), for example, the (0, 0) fundamental string background a` la Dabholkar et al. [47] can be mapped to (1, 1) by certain O(D, D) rotations [20] (cf. [24]). However, the trace of a DFT-metric, HA A = 2(n − n ¯) ,

(2.10)

remains invariant under O(D, D) rotations, and further point-wise if we fix the underlying spin group (2.50).

6

It is instructive to note that the B-field contributes to the DFT-metric by an O(D, D) conjugation, 





H  1 0   HAB =    ¯ ¯ı (Y¯¯ı )T B 1 X i (Yi )T − X

Yi

(X i )T

−

¯ ¯ı )T Y¯¯ı (X

K



  1 −B   ,   0 1

(2.11)

such that the contribution is ‘Abelian’, in the following sense: 









1 0   1 0  1 0     = .      B1 1 B2 1 B1 + B2 1

(2.12)

Further, the precise expression of the (n, n ¯ ) DFT-metric (2.2) as well as the fundamental algebraic relations, (2.4), (2.5), (2.6) are invariant under several transformations. Firstly under obvious GL(n) × GL(¯ n) rotations, ¯ µ¯ı , Y¯¯ıν Xµi , Yiµ , X




7−→



¯ ¯¯ı , R ¯ −1¯ı ¯ Y¯¯ν ¯ µ¯ R Xµj Rj i , R−1 i j Yjν , X



;

(2.13)

secondly under the transformation of only the B-field having two arbitrary skew-symmetric local parameters, mij = −mji , m ¯ ¯ı¯ = −m ¯ ¯¯ı , Bµν

7−→

¯ ¯ı X ¯ ¯ ¯ ¯ı¯ ; Bµν + Xµi Xνj mij + X µ νm

(2.14)

and lastly under the following somewhat less trivial transformations of {Yiµ , Y¯¯ıµ , Kµν , Bµν }, Yiµ

7−→

Yiµ + H µν Vνi ,

Y¯¯ıµ

7−→

Y¯¯ıµ + H µν V¯ν¯ı ,

Kµν

7−→

j i K ρσ ρσ ¯ i ρσ ¯ ¯ı ¯ ¯ı ¯ ¯ ¯ ¯  ) , Kµν − 2X(µ ν)ρ H Vσi − 2X(µ Kν)ρ H Vσ¯ı + (Xµ Vρi + Xµ Vρ¯ı )H (Xν Vσj + Xν Vσ¯

Bµν

7−→

i K H ρσ V + 2X ρσ ¯ ¯ ¯ı Kν]ρ H ρσ V¯σ¯ı + 2X i X ¯ ¯ı Bµν − 2X[µ σi ν]ρ [µ [µ ν] Vρi H Vσ¯ı ,

(2.15)

7

where Vµi and V¯µ¯ı are arbitrary local parameters. In fact, the latter two transformations, (2.14) and (2.15), can be unified into Yiµ

7−→

Yiµ + H µν Vνi ,

Y¯¯ıµ

7−→

Y¯¯ıµ + H µν V¯ν¯ı ,

Kµν

7−→

j i K ρσ ρσ ¯ i ρσ ¯ ¯ı ¯ ¯ı ¯ ¯ ¯ ¯  ) , Kµν − 2X(µ ν)ρ H Vσi − 2X(µ Kν)ρ H Vσ¯ı + (Xµ Vρi + Xµ Vρ¯ı )H (Xν Vσj + Xν Vσ¯

Bµν

7−→


ρ¯ i V i ¯ ¯ı ρσ ¯ ¯ ¯ı ¯ ¯ρ Bµν − 2X[µ ν]i + 2X[µ Vν]¯ı + 2X[µ Xν] Yi Vρ¯ı + Y¯ı Vρi + Vρi H Vσ¯ı . (2.16) H µν ,

H µν V

Note that in (2.15) the local parameters appear only through the contractions with i.e νi and µν ¯ H Vνi . On the other hand in (2.16), the B-field transformation contains orthogonal contributions. Substi¯ µ¯ into (2.16) reproduces (2.14). Alternatively, if we replace Vµi ¯ ¯ı¯X tuting Vµi = − 1 mij Xµj and V¯µ¯ı = 1 m 2

2

and V¯µ¯ı in (2.16) by Kµν H νρ Vρi and Kµν H νρ V¯ρ¯ı , we recover (2.15). The dynamics of the DFT-metric and the DFT-dilaton is dictated by the Euler-Lagrange equations of DFT. The expression of the (n, n ¯ ) DFT-metric (2.2) may be then inserted into the known stringy extension of the Christoffel symbol to lead to covariant derivatives and curvatures [18]. Yet, the trace (2.10) of the (n, n ¯ ) DFT-metric can be nontrivial, and this calls for some revision on the previous result: ΓCAB = 2 P ∂C P P¯ −4




[AB]

1


+ 2 P¯[A D P¯B] E − P[A D PB] E ∂D PEC (2.17) D

PM M −1

PC[A PB] +

¯

1

¯

P P P¯M M −1 C[A B]

D



∂D d +

(P ∂ E P P¯ )[ED]




,

which now allows generic values for the traces of the projectors, P¯M M = D − n + n ¯.

PM M = D + n − n ¯,

2.2

(2.18)

Particle and string on (n, n ¯ ) doubled-yet-gauged spacetime

While the notion of doubled-yet-gauged spacetime might sound somewhat mysterious, it is possible to define proper length and hence to show that it is a ‘metric space’. To do so, we first note that the usual infinitesimal one-form, dxA , is neither diffeomorphism covariant (1.5), (1.6), δ(dxA ) = dxB ∂B V A 6= dxB (∂B V A − ∂ A VB ) , 8

(2.19)

nor coordinate gauge symmetry invariant (1.3), since d∆A = dxB ∂B ∆A 6= 0 .

(2.20)

Thus, the naive contraction with the DFT-metric, dxA dxB HAB , cannot give any sensible definition of proper length in doubled-yet-gauged spacetime. To cure the problem, we need to gauge dxA explicitly, introducing a connection, AA , which should satisfy the same property as the coordinate gauge symmetry generator, ∆A (1.3), DxA := dxA − AA ,

AA ∂ A = 0 ,

AA AA = 0 .

(2.21)

Provided the connection transforms appropriately, DxA becomes a well-behaved i.e. covariant vector [20], δxA = ξ A ,

δAA = ∂ A ξB (dxB − AB ) =⇒

δ(DxA ) = (∂B ξ A − ∂ A ξB )DxB ;

δxA

δAA

δ(DxA )

(2.22) =

∆A ,

=

d∆A

=⇒

= 0.

We propose then to define the proper distance in doubled-yet-gauged spacetime by path integral [48],  ˆ  ˆ 2p A B Dx Dx HAB . (2.23) ||x1 , x2 || := − ln DA exp − 1

By letting ∂˜µ ≡ 0 and therefore AA ≡ (Aµ , 0), we may solve the constraints and write DxA ≡ (d˜ xµ − Aµ , dxν ) .

(2.24)

That is to say, only the half of the doubled coordinates, i.e. x ˜µ directions, are gauged. Furthermore, with the Riemannian DFT-metric (1.9), we get [20] DxA DxB HAB ≡ dxµ dxν gµν + (d˜ xµ − Aµ + dxρ Bρµ ) (d˜ xν − Aν + dxσ Bσν ) g µν .

(2.25)

Thus, after integrating out the auxiliary connection, our proposal (2.23) reduces – at least classically – to ´ 2p the conventional, i.e. Riemannian proper distance, ||xµ1 , xµ2 || = 1 dxµ dxν gµν . Being independent of A ≡ ||xµ , xµ ||, indeed the formula (2.23) measures the distance the gauged x ˜µ coordinates, i.e. xA 1 , x2 1 2 between two ‘gauge orbits’.

9

The exponent in (2.23) sets immediately the action for a point particle in doubled-yet-gauged spacetime, or its square-root free einbein formulation [42], ˆ Sparticle = dτ e−1 Dτ xA Dτ xB HAB (x) − 41 m2 e .

(2.26)

It also easily extends to (Nambu-Goto type) area and volume, which in turn provides the doubled-yet-gauged string action [20] (c.f. [8] and also [22] for the extension to the Green-Schwarz superstring), ˆ √ 1 Lstring = − 12 −hhαβ Dα xA Dβ xB HAB (x) − αβ Dα xA AβA . Sstring = 4πα0 d2 σ Lstring ,

(2.27)

These two actions are fully covariant under O(D, D) rotations, coordinate gauge symmetry (1.3), targetspacetime diffeomorphisms (1.6), world-volume diffeomorphisms and Weyl symmetry in the string case. Besides the constraint imposed by the auxiliary potential, AA , the equation of motion of the former particle action can be spelled in terms of the stringy Christoffel connection (2.17), d e dτ (e−1 HAB Dτ xB ) + 2ΓABC (P¯ Dτ x)B (P Dτ x)C = 0 .

(2.28)

On the other hand, for a string propagating on the (0, 0) Riemannian background, the auxiliary potential, AA , implies the self-duality (i.e. chirality) over the entire doubled spacetime [20], Dα xA +

√1 α jβ HA B Dβ xB −h

= 0,

(2.29)

and the Euler-Lagrangian equation of xA gets simplified to give the stringy geodesic equation, √

√ 1 ∂α ( −h

−hHAB Dα xB ) + ΓABC (P¯ Dα x)B (P Dα x)B = 0 ,

(2.30)

which extends (2.28), yet with a different numerical factor in front of the connection, 2 versus 1. For a generic non-Riemannian background, the analysis is more subtle which we investigate hereafter. We substitute the generic (n, n ¯ ) DFT-metric (2.2) into the covariant actions, and move from doubled formalism to undoubled one. One useful identity which generalizes (2.25) from Riemannian (0, 0) to a generic (n, n ¯ ) case is, with Dα xA = (∂α x ˜µ − Aαµ , ∂α xν ), D α x M D β x N HM N


= ∂α xµ ∂β xν Kµν + (Dα x ˜µ − Bµκ ∂α xκ ) Dβ x ˜ν − Bνλ ∂β xλ H µν     ¯ µ¯ı ∂(α xµ Dβ) x +2Xµi ∂(α xµ Dβ) x ˜ν − Bνρ ∂β) xρ Yiν − 2X ˜ν − Bνρ ∂β) xρ Y¯¯ıν , (2.31) 10

which reads more explicitly for particles, Dτ xM Dτ xN HM N

= x˙ µ x˙ ν Kµν + x ˜˙ µ − Aτ µ − Bµκ x˙ κ


x ˜˙ ν − Aτ ν − Bνλ x˙ λ H µν






¯ ¯ı x˙ µ x +2Xµi x˙ µ x ˜˙ ν − Aτ ν − Bνρ x˙ ρ Yiν − 2X ˜˙ ν − Aτ ν − Bνρ x˙ ρ Y¯¯ıν . µ (2.32) Note that, in accordance with the completeness relation (2.5), the auxiliary vector potential decomposes as
¯ ¯ı Y¯¯ıρ Aαρ . Aαµ = Kµν (H νρ Aαρ ) + Xµi (Yiρ Aαρ ) + X µ

(2.33)

– Particle dynamics. Integrating out H µν Aτ ν gives the on-shell relation, H µν Aτ ν ≡ H µν x ˜˙ ν − Bνλ x˙ λ





H µν Dτ x ˜ν − Bνλ x˙ λ ≡ 0 ,

or equivalently

(2.34)

which implies that the ‘dual’ conjugate momenta are trivial along D − n − n ¯ number of x ˜µ directions. On the other hand, integrating out the remaining components, Yiρ Aτ ρ and Y¯¯ıρ Aτ ρ , we acquire constraints on the xµ coordinates, Xµi x˙ µ ≡ 0 ,

¯ µ¯ı x˙ µ ≡ 0 . X

(2.35)

Namely, the particle freezes over n + n ¯ directions on the physical section formed by xµ coordinates. – String dynamics. For string, combining the useful identity (2.31) with the topological term in the action (2.27), we can reduce the world-sheet Lagrangian, 1 4πα0 Lstring

=

1 0 2πα0 Lstring

,

√ L0string = − 12 −hhαβ ∂α xµ ∂β xν Kµν + 21 αβ ∂α xµ ∂β xν Bµν + 21 αβ ∂α x ˜µ ∂β xµ h  √ − 12 −hhαγ Xµi ∂α xµ +

√1 α β ∂β xµ −h

i

(∂γ x ˜ν − Aγν − Bνρ ∂γ xρ ) Yiν

h  √ ¯ ¯ı ∂α xµ − + 12 −hhαγ X µ

√1 α β ∂β xµ −h

i

(∂γ x ˜ν − Aγν − Bνρ ∂γ xρ ) Y¯¯ıν

√ − 14 −hhαβ (Cαµ − Aαµ ) (Cβν − Aβν ) H µν , 11

(2.36)

where for shorthand notation we set Cαµ := ∂α x ˜µ − Bµν ∂α xν +

√1 α β Kµν ∂β xν −h

.

(2.37)

Now, integrating out H µν Aαν we obtain the on-shell relation, H µν A

αν

≡

H µν C

αν

H µν

or equivalently



Dα x ˜µ − Bµν ∂α

xν

+

√1 α β Kµν ∂β xν −h



≡ 0, (2.38)

and integrating out Xµi



Yiν Aαν ,

∂α

xµ

+

Y¯¯ıν Aαν , we obtain chiral constraints,

√1 α β ∂β xµ −h



≡ 0,

 ¯ ı ¯ Xµ ∂α xµ −

√1 α β ∂β xµ −h



≡ 0.

(2.39)

Namely, string becomes chiral over n directions and anti-chiral over n ¯ directions on the section coordinatized by xµ . The chirality further implies that closed strings which meet the periodic boundary condition are also frozen, or localized, over the n + n ¯ directions, similarly to the particle case (2.35).

2.3

DFT-vielbeins for (n, n ¯ ) doubled-yet-gauged spacetime

In order to couple to fermions [33, 49] or R-R sector [50], as well as for supersymmetrizations [38, 51, 52], it is necessary to introduce a pair of DFT-vielbeins, VAp and V¯A¯p , from which one can construct the projectors, PAB = 12 (JAB + HAB ) = VAp VBq η pq ,

P¯AB = 21 (JAB − HAB ) = V¯A¯p V¯B q¯η¯p¯q¯ ,

(2.40)

where ηpq and η¯p¯q¯ are the two constant metrics of twofold local Lorentz symmetries for two distinct locally inertial frames, one for the left and the other for the right closed string modes [53].5 To ensure the symmetric, orthogonal and completeness properties of the projectors (1.7), the DFT-vielbeins must satisfy their own defining properties: VAp V A q = ηpq ,

V¯A¯p V¯ A q¯ = η¯p¯q¯ ,

VAp V¯ A¯p = 0 ,

VAp VB p + V¯A¯p V¯B p¯ = JAB .

(2.41)

Essentially, with HAB = VAp VB p − V¯A¯p V¯B p¯, DFT-vielbeins diagonalize JAB and HAB simultaneously, with the eigenvalues, (η, +¯ η ) and (η, −¯ η ). 5

Ensuring the twofold spin groups in the maximally supersymmetric DFT [38] and the doubled-yet-gauged Green-Schwarz

superstring action [22], the conventional IIA and IIB theories are unified into a single theory that is chiral with respect to both spin groups. The distinction of IIA and IIB then refers to two different types of (Riemannian) ‘solutions’ rather than ‘theories’.

12

The main result of this subsection is the construction of the DFT-vielbeins, VAp and V¯A¯p , for the general (n, n ¯ ) DFT-metric (2.2). They are given by 2D × (D + n − n ¯ ) and 2D × (D − n + n ¯ ) matrices respectively,     µ µ Yp Y¯p¯     ¯A¯p = √1  . , (2.42) V VAp = √12      2 Xν q ηqp + Bνσ Yp σ X¯ν q¯η¯q¯p¯ + Bνσ Y¯p¯σ ¯ ), (D + n − n ¯ ) × D, D × (D − n + n ¯ ) and Here Xµ p , Yp µ , X¯ν q¯ and Y¯q¯ν are respectively D × (D + n − n (D − n + n ¯ ) × D matrices, such that 1 ≤ p ≤ D + n − n ¯ and 1 ≤ p¯ ≤ D − n + n ¯ . Explicitly, with the smaller range of indices, 1 ≤ a, a ¯ ≤ D−n − n ¯ and 1 ≤ i ≤ n,     ha µ  Xµ p := kµ a X i Xµj ,  µ µ :=  Y  Yµ p    i  , X¯µ p¯ := k¯µ a¯ X ¯ µ¯ ¯ µ¯ı X  Yjµ

1 ≤ ¯ı ≤ n ¯ as before, the matrices read        ,   

Y¯p¯µ

    :=    

¯ a¯ µ  h   µ  , ¯ Y¯ı     Y¯¯µ

(2.43)

¯ a¯ µ , k¯ν ¯b } are two sets of the “square-roots” of H µν and Kµν , where {ha µ , kν b } and {h ¯

¯

¯ a¯ µ h ¯ ¯ν , H µν = η ab ha µ hb ν = −¯ η a¯b h b

Kµν = kµ a kν b ηab = −k¯µ a¯ k¯ν b η¯a¯¯b .

(2.44)

The ‘total’ twofold local Lorentz symmetry group is clearly Spin(t + n, s + n) × Spin(s + n ¯, t + n ¯ ), with t+s+n+n ¯ = D, where (t, s) is the signature of H µν and Kµν . The corresponding constant metrics are ηpq and η¯p¯q¯ respectively, while ηab and η¯a¯¯b are (t + s) × (t + s) sub-blocks of them, of which the signatures are numerically opposite to each other [18],  

ηpq

 ηab 0    =  0 −δij    0 0 

0 0 +δij

 η¯a¯¯b 0 0    η¯p¯q¯ =  0 +δ¯ı¯ 0    0 0 −δ¯ı¯

    ,   

ηab = diag(− · · − −} + · · + +}) , | − ·{z | + ·{z t

s

(2.45)

     ,   

η¯a¯¯b = diag(+ · · + +} − · · − −}) . | + ·{z | − ·{z t

13

s

¯ a¯ µ , k¯ν ¯b }, in accordance with (2.4), (2.5): There are defining properties of {ha µ , kν b } and {h ha µ Xµi = 0 ,

¯ ¯ı = 0 , ha µ X µ

Yiµ kµ a = 0 ,

Y¯¯ıµ kµ a = 0 ,

¯ a¯ µ X i = 0 , h µ

¯ a¯ µ X ¯ ¯ı = 0 , h µ

Yiµ k¯µ a¯ = 0 ,

Y¯¯ıµ k¯µ a¯ = 0 ,

¯ µ¯ı Y¯¯ıν = δµ ν , kµ a ha ν + Xµi Yiν + X

ha µ kµ b = δa b ,

¯ a¯ ν + X i Y ν + X ¯ ¯ı Y¯ ν = δµ ν , k¯µ a¯ h µ i µ ¯ı

¯ a¯ µ k¯µ¯b = δa¯¯b . h

(2.46)

It follows that ¯ ¯ı Y¯ ν , Xµ p Yp ν = δµ ν + Xµi Yiν − X µ ¯ı  

Yp λ Xλ q

 δa b 0 0    =  0 δi k δi l    0 δj k δj l

¯ ¯ı Y¯ ν , X¯µ p¯Y¯p¯ν = δµ ν − Xµi Yiν + X µ ¯ı   ¯

 δa¯ b 0 0    Y¯p¯λ X¯λ q¯ =  0 δ¯ı k¯ δ¯ı ¯l    ¯ ¯ 0 δ¯k δ¯l

    ,   

    ,   

(2.47)

and   PAB =  

 1 2H

X i (Yi )T + 21 (K + B)H

Yi 1 2 (K

(X i )T

+

1 2 H(K

− B)

+ B)H(K − B) + BYi (X i )T − X i (Yi )T B

 , 

  P¯AB =  

 ¯ ¯ı )T + Y¯¯ı (X

− 21 H ¯ ¯ı (Y¯¯ı )T + 1 (K − B)H X 2

1 2 H(K

+ B)

¯ ¯ı )T − X ¯ ¯ı (Y¯¯ı )T B − 12 (K − B)H(K + B) + B Y¯¯ı (X

where the superscript T converts column vectors to row ones. As expected, PAB ¯ ¯ı , Y¯¯} and {X i , Yj }. free of the barred and unbarred variables, {X

14

 , 

(2.48) ¯ and PAB are respectively

In a parallel manner to (2.11), the B-field contributes to the DFT–vielbeins through O(D, D) multiplications,  VM p =

√1 2







T  1 0  Y    ,    B 1 Xη

V¯M p¯ =

√1 2





T  1 0   Y¯    .    ¯ B 1 X η¯

(2.49)

For consistency, the trace of the DFT-metric reads HA A = ηp p − η¯p¯p¯ = (t + s + 2n) − (t + s + 2¯ n) = (D + n − n ¯ ) − (D − n + n ¯ ) = 2(n − n ¯ ) . (2.50) The symmetry of the DFT-metric (2.16) extends to DFT-vielbeins: Yiµ

7−→

Yiµ + H µν Vνi ,

Y¯¯ıµ

7−→

Y¯¯ıµ + H µν V¯ν¯ı ,

kµ a

7−→

¯ µ¯ı η ab hb ν V¯ν¯ı , kµ a − Xµi η ab hb ν Vνi − X

k¯µ a¯

7−→

¯¯ ν ¯ ¯ν V¯ν¯ı , ¯ ¯ı η¯a¯¯b h k¯µ a¯ − Xµi η¯a¯b h ¯b Vνi − X µ b

Bµν

7−→


ρ¯ i V i ¯ ¯ı ρσ ¯ ¯ ¯ı ¯ ¯ρ Bµν − 2X[µ ν]i + 2X[µ Vν]¯ı + 2X[µ Xν] Yi Vρ¯ı + Y¯ı Vρi + Vρi H Vσ¯ı .

(2.51)

As seen from the doubled-yet-gauged actions for particle and string (2.32), (2.36), as well as the coupling to a scalar field (2.9), it is not the full signatures of the spin group, Spin(t + n, s + n) × Spin(s + n ¯, t + n ¯) ,

(2.52)

but the signature of Kµν and H µν , i.e. (t, s), that matters for unitarity. The choice of t = 1 then amounts to the usual Minkowskian spacetime.

2.4

Kaluza-Klein ansatz for DFT

The ordinary Kaluza-Klein ansatz for Riemannian metric can be ‘block-diagonalized’, 

 0

T

 g + aga gˆ =   T

ga



 0

ag   g  = exp [ˆ a]    g 0

0   T  exp a ˆ  g

15

 where

 0 a ˆµˆ νˆ =   0

 aµ0 ν 0

 . 

(2.53)

ˆ ˆ ˆ, In a similar fashion, we propose the Kaluza-Klein ansatz for DFT-metric, H MN 



i h h i  H0 0  ˆT , ˆ  ˆ = exp W  exp W H   0 H

(2.54)

ˆ = D0 + D, such that for which we decompose D  ˆ D) ˆ → O(D0 , D0 ) × O(D, D) , O(D,

 Jˆ =  

 J0 0

0  ,  J

(2.55)

and set an off-block-diagonal so(D, D) element,   0 ˆ = W  ¯ W

 −W   ∈ so(D, D) ,  0

¯ M M 0 := W M 0 M = JM N WN 0 N J 0N 0 M 0 . W

(2.56)

Further, we impose a constraint on the 2D0 × 2D matrix, WM 0 N , ¯W =0 W

or explicitly

0

(2.57)

WL0 M W L N = 0 ,

which sets half of its components trivial. At least for the Riemannian, i.e. (0, 0) case, this constraint makes the counting of the degrees of freedom consistent: gµν and Bµν have D2 degrees of freedom, while WM 0 N has 2D0 D degrees, such that ˆ 2 = (D0 + D)2 = D02 + D2 + 2D0 D , D

(2.58)

ˆ and {H0 , H, W }. Essentially, gˆµ0 ν and B ˆµ0 ν constitute WM 0 N . matching the degrees of freedom between H ˆ 3 = 0 and Explicitly, we have W   (1 − ˆ= H 

 1 0 ¯ 2 W W )H (1

−

1 ¯ T 2WW)

+

W HW T

¯ H0 (1 − 1 W W ¯ )T −HW T + W 2

−W H + (1 −

1 0 ¯T ¯ 2 W W )H W

¯ H0 W ¯T H+W

16

 , 

(2.59)

0 which is classified by four non-negative integers: (n, n ¯ ) for HAB and (n0 , n ¯ 0 ) for HA 0 B 0 , with the total trace, ˆ A 0 0 ˆ H ˆ = 2(n + n − n ¯−n ¯ ). A

Especially, in the maximally non-Riemannian case of H0 = J 0 , i.e. (n0 , n ¯ 0 ) = (D0 , 0), the above expression dramatically simplifies   ˆ= H 

J0

− 2W P¯ W T 2P¯ W T

 ¯ 2W P  .  H

(2.60)

Intriguingly, the resulting field content, HAB , P¯AB WA0 B , coincides with the ansatz for heterotic DFT proposed by Hohm, Sen and Zwiebach [54]. We leave it as a future work to explore the tantalizing connection between heterotic string and non-Riemannian doubled-yet-gauged spacetime, possibly using the ScherkSchwartz reduction scheme in DFT [52, 55–62].

3

Applications

The case of (0, 0) admits a well defined Riemannian metric and hence corresponds to Riemannian geometry, or to “Generalized Geometry” [63–68] when equipped with the pair of DFT-vielbeins. In this section, we discuss various applications of other (n, n ¯ ) backgrounds and identify the corresponding geometries.

3.1

Maximally non-Riemannian (D, 0) : Siegel’s chiral string

In the maximally non-Riemannian case of (D, 0), with i = 1, 2, · · · , D, we can view Xµi as a nondegenerate D × D square matrix. Then from (2.7) and   Xλj Yjµ Xµi = Xλi ,

(3.1)

we conclude that Xλj Yjµ is actually an identity, Xλj Yjµ = δλ µ .

(3.2)

Thus, in the case of (D, 0), we have JAB = HAB = PAB , 17

P¯AB = 0 .

(3.3)

The corresponding DFT-vielbein, VAp (2.42) and the Spin(D, D) metric are also 2D ×2D square matrices,  VAp =

√1 2

  







 −δij ηpq =   0

1 1  ,  −1 1

0 +δij

 . 

(3.4)

On the other hand, V¯A¯p is trivial. The resulting string action is completely chiral on the D-dimensional section (2.36) [20], ˆ Sstring =

1 4πα0

d2 σ αβ ∂α x ˜µ ∂β xµ ,

∂ α xµ +

√1 α β ∂β xµ −h

= 0.

(3.5)

From the conventional (0, 0) set-up, noting the sign difference, JAB = VA p VB q ηpq + V¯A p¯V¯B q¯η¯p¯q¯ ,

HAB = VA p VB q ηpq − V¯A p¯V¯B q¯η¯p¯q¯ ,

(3.6)

we may regard the substitution of the O(D, D) invariant metric, JAB , into the DFT-metric, HAB , inside the doubled-yet-gauged string action (2.27) as the flipping of the spin group signature, η¯p¯q¯

−→

(3.7)

−¯ ηp¯q¯ ,

such that ηpq and −¯ ηp¯q¯ assume not opposite (2.45) but rather identical signatures. That is to say, there is no right modes: only left modes exist. This is consistent with (3.5), and realizes the chiral string theory a` la Siegel [69]6 in a rather geometric set-up.

3.2 D = 10, (3, 3) : Non-Riemannian dimensional reduction from ten to four If we set n = n ¯ , then the DFT-metric is traceless and the two spin groups become commonly D-dimensional, Spin(t + n, s + n) × Spin(s + n, t + n)

where

t + s + 2n = D .

(3.8)

Thus, the maximally supersymmetric D = 10 DFT [38] and the doubled-yet-gauged Green-Schwarz superstring [22], both of which assume the Minkowskian Spin group, Spin(1, 9)×Spin(9, 1), can accommodate 6

See also [70, 71].

18

(0, 0) and (1, 1). However, the theories constructed in [22, 38] can be readily generalized to an arbitrary signature, Spin(tˆ, sˆ) × Spin(ˆ s, tˆ), with tˆ+ sˆ = 10, by relaxing the Majorana condition on the spinors and employing their charge conjugations only, without involving the complex Dirac conjugations. In this case, the
theory can describe (n, n) non-Riemannian doubled-yet-gauged spacetime with n = 0, 1, 2, · · · , min tˆ, sˆ . An interesting choice then appears to be Spin(4, 6) × Spin(6, 4). Such a choice can encompass sixdimensional (3, 3) non-Riemannian ‘internal’ spacetime, while maintaining the ordinary four-dimensional Minkowskian ‘external’ spacetime. As analyzed in subsection 2.2, point particles and closed strings freeze on the (3, 3) internal spacetime and this may imply a natural dimensional reduction of string theory from ten to four, alternative to the conventional compactification on ‘small’ Riemannian manifolds, e.g. CY3 . The latter will be of interest to analyze the Killing spinor equations in [38] for the D = 10 (3, 3) DFTvielbeins (2.42). Certainly, constant ‘flat’ backgrounds are maximally supersymmetric.

3.3 (1, 1) : Non-relativistic limit a` la Gomis-Ooguri In this subsection, we identify (1, 1) as the non-relativistic limit a` la Gomis-Ooguri [23]. We start by considering a generic Riemannian metric which depends explicitly on the speed of light, c, gµν = −c2 Tµ Tν (1 − S ρ S σ Φρσ ) + 2cT(µ Φν)ρ S ρ + Φµν ,

(3.9)

where Tµ and S ν are orthogonal time-like and space-like vectors, Tµ S µ = 0 .

(3.10)

Essentially, (3.9) is the ‘covariantized’ form of the ordinary Kaluza-Klein ansatz for the Riemannian metric (2.53) as gµν = (δµ ρ + cTµ S ρ )(δν σ + cTν S σ )(−c2 Tρ Tσ + Φρσ ) ,

[exp(cT ·S)]µ ν = δµ ν + cTµ S ν .

(3.11)

The inverse of the metric is then given by g µν = Υµν − S µ S ν + 2c N (µ S ν) − c12 N µ N ν = (δ µ ρ − cS µ Tρ )(δ ν σ − cS ν Tσ )(− c12 N ρ N σ + Υρσ ) , (3.12) where the variables, N ν and Υµν , meet by construction,7 Tµ N µ = 1 , 7

Tµ Υµν = 0 ,

N µ Φµν = 0 ,

Tµ N ν + Φµρ Υρν = δµ ν .

In subsections 3.4 and 3.5, {Tµ , N ν , Υµν , Φµν } will be identified as either Carroll or Newton-Cartan variables.

19

(3.13)

Now, we introduce an ansatz for the B-field in a similar manner, 0 Bµν = 2cT[µ Bν] + Bµν ,

(3.14)

and require that the Riemannian DFT-metric (1.9) should be non-singular in the non-relativistic, large c limit. In (3.14), without loss of generality we may set Bµ to be orthogonal to N ν , i.e. N µ Bµ = 0. Further, 0 Bµν denotes the zeroth order in c which is arbitrary and should survive once the limit is taken, as expected

from the ‘Abelian’ nature of the B-field from (2.11) and (2.12). Clearly in the limit of c → ∞, the inverse, g µν , is regular. We only need to ensure both g −1 B and g − Bg −1 B to be non-singular. The former implies (Υµν − S µ S ν ) Bµ = 0 ,

0 lim g µρ Bρν = (Υµρ − S µ S ρ )Bρν + S µ Bν − (S ρ Bρ )N µ Tν .

(3.15)

c→∞

In turn, Bg −1 B cannot be quadratically singular, and hence for the regularity of g − Bg −1 B, the leading power of g must be first order in c, i.e. the apparent second order term in (3.9) must be trivial, S ρ S σ Φρσ = 1 .

(3.16)

Therefore, the nontrivial cancellation of diverging terms inside g − Bg −1 B takes place at the first order, which reads h i c × (Φµρ S ρ − Bρ S ρ Bµ ) Tν + (Φνρ S ρ − Bρ S ρ Bν ) Tµ = 0 .

(3.17)

Contraction of the quantity inside the square bracket with N ν gives Bρ S ρ Bµ = Φµρ S ρ .

(3.18)

Bρ S ρ = ±1 .

(3.19)

Hence from (3.16) and (3.18), we obtain It follows that g − Bg −1 B is non-singular as 0 0 lim (gµν − Bµρ g ρσ Bσν ) = Φµν − Bµ Bν − Bµρ (Υρσ − S ρ S σ )Bσν

c→∞

(3.20)

0 0 ∓(Tµ N σ − Φµρ S ρ S σ )Bσν ± Bµσ (N σ Tν − S σ S ρ Φρν ) .

After all, the DFT-metric becomes completely regular,   SS T

Υ−  1 0   HAB =   
± ΦSS T − T N T B0 1



 ±

SS T Φ Φ−

20

−

NTT

ΦSS T Φ




  1 −B   ,   0 1 0

(3.21)

which can be easily and precisely identified as the (1, 1) type of the classification (2.2) as Υµν − S µ S ν ≡ H µν ,

Φµν − Φµρ S ρ Φνσ S σ ≡ Kµν ,



 ¯ν . Tµ , Φνρ S ρ ≡ Xµ , X

(3.22)

As demonstrated in [20], constant flat background belonging to this type generates the Gomis-Ooguri nonrelativistic string [23] (see also [22] for its Green-Schwarz superstring extension). Thus, a generic (1, 1) DFT-metric provides a curved spacetime generalization of the non-relativistic string.

3.4 (D − 1, 0) : Ultra-relativistic or Carroll The Riemannian metric (3.9) in the previous section defines the proper length. Rescaling the metric by an overall factor of c−2 , it becomes the Riemannian metric for the proper time: gµν = −Tµ Tν (1 − S ρ S σ Φρσ ) + 2c T(µ Φν)ρ S ρ + g µν

=

c2 (Υµν

−

SµSν )

+

2cN (µ S ν)

−

N µN ν

1 Φ c2 µν

, (3.23)

,

where the variables should satisfy (3.10) and (3.13), which we recall here Tµ S µ = 0 ,

Tµ N µ = 1 ,

Tµ Υµν = 0 ,

N µ Φµν = 0 ,

Tµ N ν + Φµρ Υρν = δµ ν .

(3.24)

Clearly, the expression of g µν in (3.23) indicates the possibility of taking a small c i.e. ultra-relativistic limit, as the inverse remains non-singular, yet degerate having the rank one, lim g µν = −N µ N ν .

(3.25)

c→0

In this subsection, we propose a (D − 1, 0) DFT-metric as the ultra-relativistic ‘completion’ of the above degerate inverse (3.25), 



 1 0   HAB =    B 1

 −N N T

ΥΦ

ΦΥ

−T T T



  1 −B   ,   0 1

(3.26)

where all the variables are from (3.24). It is easy to check this ansatz satisfies the defining properties of DFT-metric (2.1) and HA A = 2Υµν Φµν = 2(D − 1). Note the identification, H µν

≡

−N µ N ν

,

Kµν ≡ −Tµ Tν ,

D−1 X i=1

21

Xµi Yiν ≡ Φµρ Υρν .

(3.27)

From (2.35), particles freeze over almost all the directions except one, Φµν x˙ ν ≡ 0 .

(3.28)

This is in agreement with the ultra-relativistic limit of Riemannian geodesics a` la Bergshoeff et al. [72]. Namely, particles cannot move faster than light, and thus must freeze in the ultra-relativistic limit, c → 0. In fact, (D − 1, 0) forms Carroll structure [73, 74]: Φµν is known as a Carrollian metric, i.e. a rank (D − 1) covariant metric whose kernel is spanned by the Carroll vector, N ν , and Tµ is a principal connection. The Carrollian boost symmetry [74] is given, with an arbitrary local parameter, V µ , by Tµ

7−→

Tµ + Φµν V ν ,

Υµν

7−→

Υµν − 2N (µ Υν)ρ Φρσ V σ + N µ N ν Φρσ V ρ V σ ,

Bµν

7−→

Bµν + 2T[µ Φν]ρ V ρ ,

(3.29)

which leaves our (D − 1, 0) DFT-metric (3.26) invariant, and can be identified with the symmetry of the DFT-vielbein (2.51) for the case of (D − 1, 0).

3.5

Least non-Riemannian (1, 0) or (0, 1) : Non-relativistic or Newton-Cartan

The ordinary Kaluza-Klein ansatz (2.53) treats the two block-diagonal Riemannian metrics, g and g 0 , asymmetrically. Exchanging the two will lead to an alternative Kaluza-Klein ansatz. In this subsection, we consider such an alternative ansatz for the Riemannian metric (3.9), which reads gµν

= (δµ ρ − c−1 Φµκ U κ N ρ )(δν σ − c−1 Φνλ U λ N σ )(−c2 Tρ Tσ + Φρσ ) (3.30) =

−c2 Tµ Tν

+ 2cT(µ Φν)ρ

Uρ

+ Φµν − Φµρ

U ρΦ

νσ

Uσ

,

with the inverse, g µν

= (δ µ ρ + c−1 N µ U κ Φκρ )(δ ν σ + c−1 N ν U λ Φλσ )(−c−2 N ρ N σ + Υρσ ) (3.31) =

Υµν

+

2c−1 N (µ U ν)

−

c−2 N µ N ν

(1 −

U ρΦ

ρσ

Uσ

+ 2cTρ

U ρ)

.

Clearly the inverse of the Riemannian metric allows non-singular large c limit, lim g µν = Υµν ,

c→∞

22

(3.32)

of which the rank is D − 1. The DFT-metric which completes this degenerate inverse is then 





Υ  1 0   HAB =    B 1 ±T N T

±N T T Φ



  1 −B   ,   0 1

(3.33)

with HA A = ±2. Here the upper and lower signs correspond to (1, 0) and (0, 1) respectively. Satisfying (3.13) which we recall Tµ N µ = 1 ,

Tµ Υµν = 0 ,

N µ Φµν = 0 ,

Tµ N ν + Φµρ Υρν = δµ ν ,

(3.34)

{Tλ , Υµν } forms a Leibnizian structure (cf. e.g. [75, 76]): Tλ is the absolute clock and Υµν is a collection of absolute rulers with non-negative signature, i.e. ηab = δab from (2.44). Further, the vector, N µ , corresponds to a field of observers, and the covariant rank D − 1 metric, Φµν , provides the associated transverse metric. The transformation (2.15) reduces to, with U µ = Υµν Vν ∈ Ker(T ), Nµ

7−→

Nµ + Uµ ,

Φµν

7−→

Φµν − 2T(µ Φν)ρ U ρ + U ρ Φρσ U σ Tµ Tν ,

Bµν

7−→

Bµν ∓ 2T[µ Φν]ρ U ρ ,

(3.35)

which is sometimes referred to as Milne transformation or Galilean boost in the literature [77]. From (2.35), particles freeze over the time direction only, Tµ x˙ µ = 0 .

(3.36)

so that the observer x˙ µ is said to be space-like. This is naturally dual to the ultra-relativistic Carroll dynamics (3.28) where time flows but all spatial directions freeze. In order to account for the dynamics of time-like observers (for which time flows), one needs to introduce external forces, as done in the following subsection within the ambient framework of a null Kaluza-Klein reduction.

23

3.6

Embedding (0, 1) into ambient (0, 0) Kaluza-Klein ansatz : Carroll or Newton-Cartan

ˆ = 1 + D Kaluza-Klein ansatz (2.60) for a Riemannian ambient DFT-metric, We start by considering the D 0 0 i.e. (ˆ n, n ¯ˆ ) = (0, 0). As for the ‘internal’ space, we assume D0 = 1, (n0 , n ¯ 0 ) = (1, 0) with HA 0 B 0 ≡ JA0 B 0 . Then the ‘external’ DFT-metric, HAB , must be of the (n, n ¯ ) = (0, 1) type8 , i.e. the lower sign in (3.33), ˆ ˆ Aˆ = 2(ˆ ˆ which ensures H n−n ¯ ) = 2(n + n0 − n ¯−n ¯ 0 ) = 0. We let (˜ y , y) denote the primed coordinates, A (˜ x01 , x01 ), and write for the ambient doubled coordinates, ˆ Dτ xA = (Dτ y˜ , y˙ , Dτ xA ) = (y˜˙ − Aτ y˜ , y˙ , x ˜˙ µ − Aτ µ , x˙ ν ) .

(3.37)

We solve the constraint on WM 0 N (2.57) by putting Wµ0 N ≡ 0, such that for the present case of D0 = 1, simply we have
WM 0 N ≡ W N , 0 ,

(3.38) 0

where the O(D, D) vector, W N , carries no hidden index. By choosing this – instead of letting e.g. W µ N ˆ vanish – we ensure a null Killing vector, ξ A = (ξ˜µˆ , ξ νˆ ) (2.8), (A.4) with ξ µˆ ∂µˆ = ∂y , satisfying from (2.23),9

ˆ ln

  q ˆ ˆ ˆ ˆ ˆ A A B B = 0. DA exp − (ξ − A )(ξ − A )HAˆBˆ

The ambient DFT-metric (2.60) takes then the following form, 

ˆ ˆˆ H AB

 −2Wp¯W p¯    = 1    2V¯A¯p W p¯

1 0 0

(3.39)



2V¯B p¯W p¯     , 0    HAB

(3.40)

where, using the notations of section 2.3, we set a (D + 1)-dimensional Spin(s + 1, t + 1) vector,10   p¯ A ¯ p¯ , (3.41) W = W VA ≡ W a¯ , √1 (W+ + W− ) , √1 (W+ − W− ) 2

8

2

0 0 ¯ The alternative choice of (n0 , n ¯ 0 ) = (0, 1) obtained by setting HA 0 B 0 ≡ −JA0 B 0 will involve replacing P by −P in (2.60),

and accordingly the external DFT-metric, H, will need to be of (1, 0) type. 9 ˆ ˆν = −2∂[µˆ ξ˜νˆ] , and (3.39) means ξ µˆ ξ νˆ gˆµˆ In terms of ordinary Lie derivative, Lξ gˆµˆ ˆ ν = 0, Lξ Bµˆ ˆ ν = 0. 10 If we had chosen (n, n ¯ ) = (1, 0), from (2.42), the expression (3.41) would have reduced to ‘W p¯ = W a¯ ’ without W± .

24

such that, from (2.42), P¯ A

B

WB

=

V¯ A

p¯ p¯W

 =

¯ a W a¯ √1 k 2 µ¯

+ Tµ W− + Bµρ



¯¯ ρ √1 W b h ¯b 2

+ W+ N ρ



 ,

¯¯ ν √1 W b h ¯b 2

+ W+ N ν

Wp¯W p¯ = WA WB P¯ AB = Wa¯ W a¯ + 2W+ W− . (3.42) It is also convenient to define from (2.43), (2.45), √ √ Wµ := 2X¯µ p¯Wp¯ = 2k¯µ¯a W a¯ + 2W− Tµ .

(3.43)

Note the identification of the conventions, ¯ Φµν ≡ Kµν = −k¯µ a¯ k¯ν b η¯a¯¯b ,

¯

¯ a¯ µ h ¯ ¯ν . Υµν ≡ H µν = −¯ η a¯b h b

(3.44)

Now, with the lower sign choice of (3.33), plugging (3.40) into the master doubled-yet-gauged action for a point particle (2.26), we obtain in a similar fashion to (2.32), ˆ S = ˆ

ˆ ˆ ˆ 1 2 dτ e−1 Dτ xA Dτ xB H ˆB ˆ − 4m e A

i h
dτ e−1 2Dτ y˜ y˙ + 2Dτ xA V¯A¯p W p¯ − Dτ y˜Wp¯W p¯ + Dτ xA Dτ xB HAB − 41 m2 e   h i 1 −1 µ ν µ 2 2 ˆ x˙ x˙ Φµν + 2Dτ y˜Wµ x˙ − 4(Dτ y˜) W+ W− + 2yD ˙ τ y˜ − 4 m e   e , = dτ    −1 µ −1 µ a ¯ ν ¯ ¯ −2e (Tµ x˙ − 2Dτ y˜W+ ) Λ − e ha¯ Λµ h Λν

=

(3.45)

where we set for shorthand notation as well as for a convenient field redefinition to replace Aτ µ , Λµ := x ˜˙ µ − Aτ µ − Bµκ x˙ κ − Dτ y˜Wµ ,

Λ := Λµ N µ + 2Dτ y˜W− .

(3.46)

¯ a¯ µ Λµ being integrated out as Note that the very last term in (3.45) is a perfect square which vanishes after h ¯ a¯ µ Λµ ≡ 0 . h

(3.47)

Since y is the coordinate for the isometry direction, it serves as a Lagrange multiplier: it forces the conjugate momentum of y, or p, to be constant, d dτ


e−1 Dτ y˜ ≡ 0

=⇒

2Dτ y˜ = ep 25

with constant p .

(3.48)

,

Integrating out Λ gives a constraint, EΛ := Tµ x˙ µ − epW+ ≡ 0 ,

(3.49)

such that the time is generically not frozen, c.f. (3.36). Further, integrating out the auxiliary field, Aτ y˜ inside Dτ y˜ determines the velocity, with (3.47), (3.48), (3.49), y˙ = epWp¯W p¯ − 2Dτ xA V¯A¯p W p¯ (3.50) = −2W+ Λ − Wµ

x˙ µ

+ 2epW+ W− .

The einbein imposes the Hamiltonian constraint, Ee := Φµν x˙ µ x˙ ν + e2 p2 W+ W− − 2epW+ Λ + 14 e2 m2 ≡ 0 .

(3.51)

From (3.50) and (3.51), it follows that: − py˙ = e−1 Φµν x˙ µ x˙ ν + pWµ x˙ µ − ep2 W+ W− + 14 m2 e .

(3.52)

That is to say, whenever p 6= 0, y˙ is completely fixed by the dynamics of the xµ coordinates. The auxiliary variable, Λ, is also fixed in the same manner. Making use of the world-line diffeomorphisms, we hereafter normalize the einbein: e ≡ 1,

(3.53)

such that τ coincides with the proper length. The equation of motion for xµ reads now

Eµ := Φµν x ¨ν + ∂ρ Φσµ − 12 ∂µ Φρσ x˙ ρ x˙ σ + Tµν Λ − 12 pWµν x˙ ν + 12 p2 ∂µ (W+ W− ) − pΛ∂µ W+ − Tµ Λ˙ , (3.54) where we defined for simplicity, the field strengths Tµν := ∂µ Tν − ∂ν Tµ ,

Wµν := ∂µ Wν − ∂ν Wµ .

(3.55)

Computing the contractions, N µ Eµ , x˙ µ Eµ , respectively, we obtain the time derivative of the auxiliary variable, Λ˙ = N µ

h

i

∂ρ Φσµ − 12 ∂µ Φρσ x˙ ρ x˙ σ + Tµν Λ − 21 pWµν x˙ ν + 12 p2 ∂µ (W+ W− ) − pΛ∂µ W+ , 26

(3.56)

and a consistency relation among the constraints (3.49), (3.51), ˙ Λ − 1 E˙e = 0 . x˙ µ Eµ + ΛE 2

(3.57)

While (3.54) determines partially the acceleration, x ¨µ , the time derivative of the constraint (3.49) can provide the missing component, E˙Λ = Tµ x ¨µ + ∂(µ Tν) x˙ µ x˙ ν − px˙ µ ∂µ W+ = 0 .

(3.58)

All together, the combination, Υλµ Eµ + N λ E˙Λ , fully determines the acceleration, i h λ µ ν x˙ x˙ + Υλµ Tµν Λ − p(N λ ∂ν W+ + 21 Υλµ Wµν ) x˙ ν + 12 p2 Υλµ ∂µ (W+ W− ) − pΛΥλµ ∂µ W+ = 0 , x ¨λ + γµν (3.59) where

λ γµν

denotes the following coefficients, λ γµν := N λ ∂(µ Tν) + 12 Υλρ (∂µ Φνρ + ∂ν Φρµ − ∂ρ Φµν ) .

(3.60)

We emphasize that the dynamics of the D-dimensional coordinates xµ as prescribed by (3.59) is independent of the Kaluza-Klein direction, y. Geometrically, this means that one can interpret xµ as coordinates on the quotient manifold of the ambient spacetime by the light-like direction along the vector field, ξ µˆ ∂µˆ = ∂y . In the special case where Tµν vanishes (i.e. the one-form, Tµ , is closed) and W+ is a (non-vanishing) constant, the expression (3.59) simplifies to λ µ ν x ¨λ + γµν x˙ x˙ = 12 pΥλµ [Wµν x˙ ν − p∂µ (W+ W− )] ,

(3.61)

of which the right-hand side can be interpreted as the Lorentz plus Coulomb forces. In this particular case, the coefficients (3.60) are the ones associated to the so-called ‘special Galilean connection’ for the field of observers, N µ , (cf. e.g. [76]). In accordance with the usual Riemannian ambient approach of [79–81] (cf. also [74, 82, 84, 85]), the resulting dynamical trajectories (3.61) can be interpreted as Newton-Cartan geodesics. These are of two different types, depending on the value of p : • p = 0 (Space-like observer). In this case, the constraint11 , Tµ x˙ µ = 0, holds as a consequence of (3.49) so that we recover the case 11

From the ambient perspective, the constraint, Tµ x˙ µ = 0, implies that the dynamics becomes restricted to a D-dimensional

hypersurface of the ambient manifold, transverse to the null isometry vector field, ξ µˆ ∂µˆ = ∂y . Such a light-like hypersurface is naturally endowed with a Carrollian metric structure [73], and the equations of motion (3.61) together with (3.50) and (3.56) can be naturally interpreted as geodesics associated to a suitable Carrollian connection induced by the ambient metric structure (cf. [73, 83] for details). The role of Carrollian time is then played by y and the ‘space-like’ directions are generically unfrozen, thus generalizing the Carrollian dynamics discussed in section 3.4.

27

investigated in section 3.5 for which time freezes. Geometrically, the observer trajectory is restricted to a (D − 1)-dimensional hypersurface (absolute space). The absolute spaces are Riemannian spaces (of Euclidean signature), since the degenerate metric Φµν becomes invertible on Ker T (cf. e.g. [76]). Equation (3.61) thus describes geodesics associated to the spatial Riemannian metric and the Hamilp 2 tonian constraint (3.51) can be solved as e = |m| Φµν x˙ µ x˙ ν ≡ 1.

• p 6= 0 (Time-like observer). In this case, τ is parametrized to ensure e =

1 ˙µ pW+ Tµ x

≡ 1 such that the observer x˙ µ is time-like.

Equation (3.61) can thus be reformulated as λ µ ν x ¨λ + γˆµν x˙ x˙ = 0 ,

(3.62)

λ are defined as where the coefficients γˆµν λ λ γˆµν := γµν + Υλρ T(µ Fν)ρ ,

with Fµν := ∂µ Aν − ∂ν Aµ and Aµ :=

1 2 W+

(3.63)

(Wµ − W− Tµ ).

The connection associated to the coefficients (3.63) is naturally interpreted as a Newtonian connection [78], i.e. a torsion-free connection compatible with the Leibnizian structure (Υµν , Tµ ) such that the associated field strength, Fµν , is closed. In summary, assuming the triviality of Tµν and W+ , the doubled-yet-gauged particle action (2.26) with the ambient (D+1)-dimensional Kaluza-Klein ansatz (2.60) reproduces the full content of Newtonian dynamics (unifying the space-like and time-like cases) on the D-dimensional manifold quotient along the light-like direction, y. In principle, the assumption regarding the triviality of the variables, Tµν and W+ , should be examined by considering the on-shell dynamics of the DFT-metric, i.e. the Euler-Lagrangian equations of DFT. In the present work, we have focused on the kinematical side of the DFT-metric and the subsequent particle and string dynamics on the background. We leave the investigation of the dynamical aspect of the (n, n ¯ ) DFTmetric for future work. From our perspective, the DFT action and its full equations of motion determine universally and unambiguously all the dynamics of the (n, n ¯ ) backgrounds, including (0, 0) Riemannian General Relativity and (0, 1) Newton-Cartan gravity, in a unifying manner.

28

Acknowledgements We would like to thank Xavier Bekaert and Daniel Waldram for helpful discussions as well as Soo-Jong Rey for supportive encouragement. K.M is grateful to Sogang University for hospitality while part of this work was completed. This work was supported by the National Research Foundation of Korea through the Grants 2015K1A3A1A21000302 and 2016R1D1A1B01015196, as well as by the Chilean Fondecyt Postdoc Project N◦ 3160325.

29

A

Derivation of the most general form of the DFT-metric, Eq.(2.2)

By definition (2.1), the DFT-metric is a symmetric O(D, D) element, such that it satisfies (A.1)

HL M HM N = δL N .

HM N = HN M ,

With respect to the O(D, D) metric (1.1) and the choice of the section, ∂˜µ ≡ 0, we decompose the DFTmetric,  HM N



µν  H  = Hκ ν

Hµ λ  .  Hκλ

(A.2)

The defining condition (A.1) reads explicitly, Hµν = Hνµ ,

Hµ ν = H ν µ ,

Hµν = Hνµ ,

(A.3) H(µ

ρ

Hν)ρ

= 0,

Hρ(µ

Hρ

ν)

Hµ

= 0,

ρ

Hρ

ν

+

Hµρ H

ρν

=

δµ

ν

.

The generalized Lie derivative of the DFT-metric, c.f. (1.6), Lˆξ HAB = ξ C ∂C HAB + (∂A ξC − ∂C ξA )HC B + (∂B ξC − ∂C ξB )HA C ,

(A.4)

leads to δHµν = Lξ Hµν ,

δHκλ = Lξ Hκλ + (∂κ ξ˜ρ − ∂ρ ξ˜κ )Hρ λ − Hκ ρ (∂ρ ξ˜λ − ∂λ ξ˜ρ ) ,

δHµ λ = Lξ Hµ λ − Hµρ (∂ρ ξ˜λ − ∂λ ξ˜ρ ) ,

δHκ ν = Lξ Hκ ν + (∂κ ξ˜ρ − ∂ρ ξ˜κ )Hρν . (A.5)

Viewed as a D × D matrix, if Hµν is non-degenerate, we may identify it as the inverse of Riemannian metric. It is easy to see then that the remaining constraints are all solved by a skew-symmetric B-field, such that the most general DFT-metric in this case takes the well-known form (1.9). Henceforth, we look for the most general form of the DFT-metric, where Hµν is degenerate. Firstly, we focus on the case where the rank of Hµν is D−1, admitting only one zero-eigenvector, Xµ , Hµν ≡ H µν ,

H µν Xν = 0 . 30

(A.6)

From (A.3), Hµ ρ H ρν is skew-symmetric, and hence Xµ Hµ ρ H ρν = −Hν ρ H ρµ Xµ = 0 .

(A.7)

Without loss of generality then, introducing a skew-symmetric B-field,12 we may put Hµ ρ H ρν ≡ −H µρ Bρσ H σν ,

(A.8)

Bµν = −Bνµ .

It follows that, with some vector field, Y µ , Hµ ν takes the form, Hµ ν = −H µρ Bρν + Y µ Xν .

(A.9)

We proceed to put with a new symmetric variable, Kµν = Kνµ , Hµν ≡ Kµν − Bµρ H ρσ Bσν + 2X(µ Bν)ρ Y ρ .

(A.10)

H µρ Kρν + (Y ρ Xρ )Y µ Xν = δ µ ν .

(A.11)

Y µ Xµ = ±1 .

(A.12)

The last relation in (A.3) gives

Contracting this with Xµ shows Lastly we impose the skew-symmetric condition of Hµρ Hρ ν , which gives with (A.11), Kµρ Y ρ Xν + Kνρ Y ρ Xµ = 0 ,

(A.13)

and hence in particular, contracting with Y µ , we have Kνρ Y ρ = ∓(Y µ Kµρ Y ρ )Xν .

(A.14)

Substituting this back into (A.13), we conclude that Y µ Kµρ Y ρ must be trivial, and hence in fact from (A.14), Kνρ Y ρ = 0 .

(A.15)

We may perform a field redefinition, Y µ → ±Y µ , in order to remove the sign factor in the normalization of (A.12). After all, the most general form of the DFT-metric in the ‘least’ degenerate case takes the form,   HM N

12

 = 

H µν Bκρ H ρν ± Xκ Y ν

−H µσ Bσλ ± Y µ Xλ Kκλ − Bκρ H ρσ Bσλ ± 2X(κ Bλ)ρ Y ρ

 , 

The ambiguity in introducing the B-field through (A.8) amounts to the symmetry of the final result (2.16).

31

(A.16)

of which the variables must meet H µν Xν = 0 ,

Kµν Y ν = 0 ,

Y µ Xµ = 1 ,

H λµ Kµν + Y λ Xν = δ λ ν ,

Bµν = −Bνµ .

(A.17)

The above analysis can be straightforwardly extended to the most general degenerate cases, where there are N number of linearly independent zero-eigenvectors, Xµi , i = 1, 2, · · · , N , such that the rank of Hµν is D − N, Hµν ≡ H µν ,

(A.18)

H µν Xνi = 0 .

From H(µ ρ H ν)ρ = 0 ,

Hµ ρ H ρν Xνi = 0 ,

(A.19)

Xµi Hµ ρ H ρν = 0 ,

Eqs.(A.9) and (A.10) generalize, defining Yiµ and Mµν , to Hµ ν ≡ −H µρ Bρσ + Yiµ Xνi ,

ρ i B Hµν ≡ Mµν − Bµρ H ρσ Bσν + 2X(µ ν)ρ Yi ,

(A.20)

such that the DFT-metric assumes the following intermediate form,   HM N =  

H µν

−H µσ B

Bκρ H ρν + Xκi Yiν

σλ

+

Yiµ Xλi

Mκλ − Bκρ H ρσ Bσλ +

ρ i B 2X(κ λ)ρ Yi

  . 

(A.21)

In the above, the repeated index, i, is surely summed from one to N . The remaining constraints in (A.3) give H µρ Mρν + (Yiρ Xρj )Yjµ Xνi = δ µ ν ,

(A.22)

Mµρ Yiρ Xνi + Mνρ Yiρ Xµi = 0 .

(A.23)

Contraction of (A.22) with Xµk leads to Xνi (Yi ·X j Yj ·X k ) = Xνk ,

(A.24)

where we set Yi ·X j ≡ Yiµ Xµj . Since k = 1, 2, · · · , N is arbitrary and Xνk ’s are independent, the above result actually implies that Yi ·X j is an involutory N × N matrix, Yi ·X j Yj ·X k = δi k .

32

(A.25)

On the other hand, contraction of (A.23) with (Yj ·X k )Ykµ leads to Mνρ Yjρ = −(Yj ·X k )(Yk ·M ·Y i )Xνi ,

(A.26)

where we set Yk ·M ·Y i ≡ Ykρ Mρσ Yiσ for shorthand notation. Substituting into (A.23), we get (Yj ·X k )(Yk ·M ·Y i )Xµi Xνj + (Yj ·X k )(Yk ·M ·Y i )Xνi Xµj = 0 ,

(A.27)

which, after contraction with Ylµ Ymν and from (A.25), can be seen to be equivalent to (Yl ·X i )(Yi ·M ·Y m ) = −(Ym ·X i )(Yi ·M ·Y l ) .

(A.28)

It follows from (A.22), (A.25), (A.26) that (Yi ·X j )Yjµ Xνi and H µρ Mρν are mutually orthogonal and complementary (A.22) projection matrices, (Yi ·X j )Yjλ Xµi (Yk ·X l )Ylµ Xνk = (Yi ·X j )Yjλ Xνi , j

(Yi ·X )Yjλ Xµi H µσ Mσν = 0 ,

H λρ Mρµ H µσ Mσν = H λρ Mρν , (A.29) H λρ Mρµ (Yk ·X

l

)Ylµ Xνk

= 0.

Now, we may recast (A.26) into h n o i Mµν + Xµi Xνl (Yl ·X k )(Yk ·M ·Y j )Xρj Yiρ Ymν = 0 .

(A.30)

It is crucial to note, from the symmetric property, Yk ·M ·Y j = Yj ·M ·Y k , that the free indices, µ and ν, below are symmetric, n o n o Xµi Xνl (Yl ·X k )(Yk ·M ·Y j )Xρj Yiρ = Xνi Xµl (Yl ·X k )(Yk ·M ·Y j )Xρj Yiρ ,

(A.31)

and further from the skew-symmetric property (A.28), that the free indices, ν and ρ, below are skewsymmetric, Xνl (Yl ·X k )(Yk ·M ·Y j )Xρj = −Xρl (Yl ·X k )(Yk ·M ·Y j )Xνj .

(A.32)

Therefore, if we perform a field redefinition, Bµν

−→

Bµν + 21 Xµi (Yi ·X k )(Yk ·M ·Y j )Xνj ,

(A.33)

among the components of the DFT-metric spelled in (A.21), H µσ Bσλ , Bκρ H ρσ , and Bκρ H ρσ Bσλ remain ρ i B invariant, but Mκλ + 2X(κ λ)ρ Yi transforms as follows, ρ i B Mκλ + 2X(κ λ)ρ Yi

−→

n o j ρ i i B (Yj ·X k )(Yk ·M ·Y l )Xρl Yiρ + 2X(κ Mκλ + X(κ Xλ) λ)ρ Yi . 33

(A.34)

We then let n o j i Kκλ := Mκλ + X(κ Xλ) (Yj ·X k )(Yk ·M ·Y l )Xρl Yiρ ,

(A.35)

which satisfies nicely H µρ Kρν + (Yi ·X j )Yjµ Xνi = δ µ ν .

Kκλ Yiλ = 0 ,

(A.36)

Finally, we perform a similarity transformation, (Xµi , Yjν ) → (S i k Xµk , Ykν S −1k j ), which leaves Yiµ Xνi invariant but diagonalizes Yiρ Xρj with the eigenvalues of either +1 or −1. We then let N = n + n ¯ in order to denote the numbers of the +1 and −1 eigenvalues of Yiρ Xρj . If the corresponding eigenvalue is −1, we ¯ ¯ı , Y¯ ν ) := (X i , −Y ν ), which involves the change of the index from i further perform a field redefinition, (X µ

¯ı

µ

i

to ¯ı. In this way, we arrive at the most general form of the DFT-metric, (2.2), classified by two non-negative integers, n, n ¯. It is also worthwhile to decompose the B-field utilizing the completeness relation (2.5), i ¯ ¯ ¯µ¯ X ¯ ¯ − B ¯ν¯ X ¯ ¯ + X i X j bij + X ¯ ¯ı X ¯ ¯ Bµν = βµν + Bµj Xνj − Bνj Xµj + B , ν µ µ ν µ ν b¯ı¯ + 2X[µ Xν] bi¯

(A.37)

for which we set βµν := (KH)µ ρ (KH)ν σ Bρσ , bi¯ :=

Yiµ Y¯¯ν Bµν

,

bij := Yiµ Yjν Bµν , b¯ı¯ :=

Y¯¯ıµ Y¯¯ν Bµν

,

¯ µ¯bi¯ , Bµi := Bµν Yiν − Xµj bji + X ¯µ¯ı := B

Bµν Y¯¯ıν

−

¯ µ¯b¯¯ı − X

(A.38)

Xµj bj¯ı .

¯µ¯ı and βµν are completely orthogonal to the vectors, Y µ and Y¯¯µ , The variables, Bµi , B j Bµi Yjµ = 0 ,

Bµi Y¯¯µ = 0 ,

¯µˆı Y µ = 0 , B j

¯µˆı Y¯¯µ = 0 , B

34

βµν Yjµ = 0 ,

βµν Y¯¯µ = 0 .

(A.39)

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