Physics Letters B 793 (2019) 65–69

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Physics Letters B

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Kaluza–Klein reduction on a maximally non-Riemannian space is moduli-free ∗ Kyoungho Cho, Kevin Morand, Jeong-Hyuck Park

Department of Physics, Sogang University, 35 Baekbeom-ro, Mapo-gu, Seoul 04107, Republic of Korea a r t i c l e i n f o a b s t r a c t

Article history: We propose a novel Kaluza–Klein scheme which assumes the internal space to be maximally non- Received 2 March 2019 Riemannian, meaning that no Riemannian metric can be defined for any subspace. Its description is Received in revised form 4 April 2019 only possible through Double Field Theory but not within supergravity. We spell out the corresponding Accepted 16 April 2019 Scherk–Schwarz twistable Kaluza–Klein ansatz, and point out that the internal space prevents rigidly Available online 18 April 2019 any graviscalar moduli. Plugging the same ansatz into higher-dimensional pure Double Field Theory and Editor: N. Lambert also to a known doubled-yet-gauged string action, we recover heterotic supergravity as well as heterotic worldsheet action. In this way, we show that 1) supergravity and Yang–Mills theory can be unified into higher-dimensional pure Double Field Theory, free of moduli, and 2) heterotic string theory may have a higher-dimensional non-Riemannian origin. © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction closed-string massless sector, conventionally represented by the three fields, {gμν, Bμν, φ}. They transform into one another un- Kaluza–Klein theory attempts to unify General Relativity and der T-duality and hence form a (reducible) O(D, D) multiplet [3,4]. electromagnetism into higher-dimensional pure gravity. Yet, the Furthermore, within the framework of Double Field Theory (DFT) (aesthetically unpleasing) cylindrical extra dimension brings along initiated in [5–9], O(D, D) T-duality becomes the manifest princi- an unwanted additional massless scalar field, i.e. radion or gravis- pal symmetry and the effective action itself is to be identified as an calar modulus, which is not observed in nature: it would spoil the integral of a stringy scalar curvature beyond Riemann. The whole Equivalence Principle by appearing on the right-hand side of the closed-string massless NS-NS sector may then be viewed as stringy geodesic equation. This moduli stabilization problem is essentially graviton fields consisting of the DFT-dilaton, d, and the DFT-metric, rooted in the fact that there is no natural scale in pure gravity HAB [10]. The latter satisfies two defining properties: which would fix or stabilize the radius of the cylinder. The prob- C D lem persists in modern string compactifications, in view of the HAB = HBA , HA HB JCD = JAB , (1) arbitrary size and shape of an internal space (mani/coni/orbifold,   compact or not). Turning on fluxes or non-perturbative correc- where J = 01 is the O(D, D) invariant constant metric which AB 10 tions might promise to solve the problem, but such scenarios re- can freely raise and lower the O(D, D) vector indices, A, B, ··· quire generically subtle analyses regarding both the validity of the (capital letters). A pair of symmetric projection matrices are then effective-field-theory approximation and calculational control [1] defined, (cf. [2] and references therein for related current controversies). In this paper we propose a novel Kaluza–Klein scheme to unify 1 ¯ 1 P AB = (JAB + HAB), P AB = (JAB − HAB), (2) Stringy Gravity and Yang–Mills (including Maxwell), which pos- 2 2 tulates a non-Riemannian internal space and consequently does while their ‘square roots’ give twofold DFT-vielbeins, not suffer from any moduli problem. By Stringy Gravity, we mean p q ¯ ¯ p¯ ¯ q¯ P = V V η , P = V V η¯ ¯ ¯ , (3) the string theory effective action of the NS-NS (or purely bosonic) AB A B pq AB A B pq which satisfy their own defining properties,

A ¯ ¯ A ¯ ¯ A * Corresponding author. V ApV q = ηpq , V Ap¯ V q¯ = ηp¯ q¯ , V ApV q¯ = 0 , (4) E-mail addresses: [email protected] (K. Cho), [email protected] (K. Morand), [email protected] (J.-H. Park). or equivalently [11] https://doi.org/10.1016/j.physletb.2019.04.042 0370-2693/© 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 66 K. Cho et al. / Physics Letters B 793 (2019) 65–69

p q ¯ p¯ ¯ q¯ ˆ  V V η + V V η¯ ¯ ¯ = J . (5) A = A A =  (10) A B pq A B pq AB z (y , x ), ∂Aˆ (∂A ,∂A ). ¯ Here ηpq and ηp¯ q¯ are local Lorentz invariant metrics characterizing In (9), Wˆ is an off-block-diagonal so(Dˆ , Dˆ ) element which satisfies the twofold spin groups which have dimensions Tr(P) and Tr(P¯ ) ˆ ˆ T separately. They are distinguished by unbarred and barred small W Jˆ + Jˆ W = 0 , (11) letter indices. and takes the form, The above stringy graviton fields constitute the diffeomorphic  − c  DFT-Christoffel symbols [12–14], ˆ 0 W c A A AB  B W = , W  := W  = J J   W , (12)     W 0 A A A B B ¯ ¯ D ¯ E D E  = 2 P ∂ P P + 2 P[ P ] − P[ P ] ∂ P CAB C [AB] A B A B D EC    ×  A D ¯ ¯ D   where the 2D 2D block, W A , should meet [24] − P C[A P B] + P C[A P B] + E ¯ 4 F − ¯ F ∂Dd (P ∂ P P )[ED] ,   P F 1 P F −1 A AB A W A W = 0 , W A ∂A = 0 . (13) and, consequently with ∇ := ∂ +  , twofold local Lorentz spin A A A This condition sets half of the components to be trivial, truncates connections [15–17],   the exponential, = B ∇ = B + C   Apq V p A V Bq V p ∂A V Bq AB V Cq , ˆ ˆ 1 ˆ 2   (6) exp W = 1 + W + W , ¯ ¯ B ¯ ¯ B ¯ C ¯ 2 Ap¯ q¯ = V p¯ ∇A V Bq¯ = V p¯ ∂A V Bq¯ + AB V Cq¯ . and makes the above ansatz consistent with the ordinary Kaluza– These connections form covariant curvatures, Ricci and scalar [13, Klein ansatz in supergravity. Moreover, crucially for the pur- 17], of which the latter can be constructed as pose of the present work, the ansatz (9)can accommodate non- Riemannian geometry in which the Riemannian metric cannot be S = (P AC P BD − P¯ AC P¯ BD)S , (0) ABCD defined, see [14]for classification and [25–27]for earlier exam- G = F Ap Bq + 1 Apq ples including the attainment of the Gomis-Ooguri non-relativistic (0) ABpqV V 2 Apq , (7) ¯ ¯ ¯ ¯ string [28,29]. G¯ = F¯ ¯ ¯ ¯ Ap ¯ Bq + 1 ¯ ¯ ¯ ¯ Apq (0) ABpq V V 2 Apq , Henceforth, we focus on a specific internal space of which   = + E − ↔ the DFT-metric is fully O(D , D ) symmetric and maximally non- where, with RCDAB ∂A BCD AC BED (A B),    Riemannian, namely (D , 0) type as classified in [14]: := 1 + − E S ABCD 2 R ABCD RCDAB  ABECD ,   H   ≡ J   . (14) r r A B A B FABpq := ∇A Bpq −∇B Apq + Ap Brq − Bp Arq , In general, from the defining relations (1), the infinitesimal varia- F¯ := ∇ ¯ −∇ ¯ + ¯ r¯ ¯ − ¯ r¯ ¯ ABp¯ q¯ A B p¯ q¯ B Ap¯ q¯ Ap¯ Br¯q¯ B p¯ Ar¯q¯ . tion of any DFT-metric should satisfy While S coincides exactly with the well-known expression of        (0) δH = P δH P¯ + P¯ δH P . (15) scalar curvature written in terms of d, HAB [18], Meanwhile, the particular choice of the internal space (14)im- = 1 HAB H HCD + 1 HAB C H D H   ¯  S(0) 8 ∂A CD∂B 2 ∂ AD∂ BC plies P = J and P = 0. Thus, the fluctuation must be trivial: no AB AB AB graviscalar modulus can be generated and the non-Riemannian in- −∂A ∂B H + 4H (∂A ∂Bd − ∂Ad∂Bd) + 4∂AH ∂Bd , ternal space is rigid, the other two accommodate the vielbeins and read [19]  δH   = 0 . (16) +G = 1 + A − A + 1 A Bp A B (0) S(0) 2∂A ∂ d 2∂Ad∂ d ∂A V Bp∂ V , 2 2   × ¯ (8) In fact, (14)sets the “twofold” internal spin group to be O(D , D ) −G¯ = 1 − A + A − 1 ¯ ¯ A ¯ B p (0) 2 S(0) 2∂A ∂ d 2∂Ad∂ d 2 ∂A V B p∂ V . O(0, 0), such that the coset structures of the internal and the am- H Hˆ Clearly their differences would vanish upon the section condition, bient DFT-metrics, , , are ‘trivial’ and ‘heterotic’ respectively, if A H ∂A ∂ ≡ 0, but they provide precisely the known ‘missing’ pieces is Riemannian (27), in the Scherk–Schwarz reduction of DFT while relaxing the section   condition on the internal space, as previously added by hand in O(D , D ) O(Dˆ , Dˆ ) = 1 , . [20–23]. Hence, the above vielbein formalism generates these cru-   ×  ˆ G −G¯ O(D , D ) O(0, 0) O(D + 1, D − 1) × O(D − 1, 1) cial terms in a natural manner. In particular, (0) (0) matches the action adopted in [22]. Below we focus on computing +G and (0) (17) −G¯ separately in higher dimensions with the Scherk–Schwarz (0)  twisted non-Riemannian Kaluza–Klein ansatz. The latter has dimension D2 + 2DD , which matches the total de- grees of the external DFT-metric, H (27), and the gravivector,  AB A 2. Moduli-free non-Riemannian Kaluza–Klein ansatz W A (13)[30], cf. [31]. The corresponding DFT-vielbeins are [14] ˆ =  + With D D D, the DFT Kaluza–Klein ansatz [14]breaks ˆ ˆ   ×      O(D, D) into O(D , D ) O(D, D) and takes the form: V   0 ˆ ˆ A p ¯ˆ ˆ 0   V ˆ = exp W V ˆ = exp W       Apˆ , Aˆ p¯ ¯ , H 0 J 0 0 V V ¯ Hˆ = exp Wˆ exp Wˆ T , Jˆ = . (9) Ap Ap 0 H 0 J (18) In our notation, hatted, primed, and unprimed symbols refer to  the ambient, internal, and external spaces respectively. In partic- where V A p is now an invertible ‘square’ matrix,  ular, the ambient doubled coordinates split into the internal and   p  V   V  = J   . external ones as A p B A B K. Cho et al. / Physics Letters B 793 (2019) 65–69 67

3. Reduction to heterotic DFT In each transformation above, the first line with ξ A is the (ex- ˙ ternal) diffeomorphic DFT Lie derivative and the second with A Before inserting the ansatz (9), (18)into the ambient scalar cur- is the (internal) Yang–Mills gauge symmetry. In fact, every single Gˆ G¯ˆ vatures (8), or (0), (0) (all hatted), we perform a Scherk–Schwarz term in (23)is (external) diffeomorphism-invariant [39]. Gˆ twist over the internal space. Following [19], we introduce a twist- It is worth while to note the only difference between 2 (0) and  A − G¯ˆ ing matrix, U A˙ (y), which depends on the internal coordinates 2 (0): the former contains a DFT-cosmological constant term [13],   − ˙ ˙ ˙ only and is an O(D , D ) element satisfying 1 2d ˙ ˙ ˙ ABC 3 e f ABC f , but the latter does not. As anticipated in [36,40],   A B  ˙ our result (23)is manifestly symmetric for O(D, D) as well as U ˙ U ˙ J   = J ˙ ˙ (19)   A B A B AB , any subgroup of O(D , D ) which stabilizes the structure constant, ˙    J ˙ ˙ J   f ˙ ˙ ˙ . where AB coincides with A B numerically: both are O(D , D ) ABC invariant constant metrics. Essentially the twist converts all the Finally, if we adopt the well-known Riemannian parametriza- primed indices to dotted ones, tion of the DFT-metric,  − −  ˙    ˙  1 − 1 √ A = A A A  = ˙ A  A (20) g g B −2d −2φ W A (z) W A (x)U A˙ (y), V p (z) V p (x)U A˙ (y), H = − − , e = −ge , (27) AB Bg 1 g − Bg 1 B  ˙ ˙ p ˙ where now V ˙  V ˙ = J ˙ ˙ . We further put Ap B AB A and solve both the external section condition, ∂A ∂ = 0, and (13), ˙ ˙ ˆ ¯ = ˜μ ≡ A ≡ A d(z) = λ(y) + d(x), V Ap(x), V Ap¯ (x), (21) by letting ∂A (∂ , ∂ν) (0, ∂ν) and W A (0, Wν ), our main result (23)reproducesthe heterotic supergravity action [41,42], up such that the external fields are independent of the internal coor- ¯ to total derivatives, dinates: ∂A d = 0, ∂A V Bp = 0, ∂A V B p¯ = 0. Finally, we impose the standard section condition on the exter- nal space, ∂ ∂ A = 0, and the twistability conditions on the internal − A −2e 2dG¯ space separately [22], [19]: (0)

 √   1 A˙ B E˙ −2φ μ 1 λμν 1 μνC˙  − U   U ˙ = 0 f ˙ ˙ f ˙ ˙ ˙ = 0 (22) = − + − ˜ ˜ − ˙ ∂A λ 2 A ∂B A , [AB C]D E , ge R 4∂μφ∂ φ 12 Hλμν H 4 FμνC F ,     =  A B where f A˙ B˙ C˙ 3 ∂B U[A˙ U B˙ U C˙ ]A which we require to be con- stant. (28) After straightforward yet tedious computation – assisted by a where computer algebra system [32] and through intermediate expres- ˜ sions like (4.14) in [19]– we obtain our main result: with the Hλμν = 3∂[λ Bμν] − ωλμν , (29) above Scherk–Schwarz twisted Kaluza–Klein ansatz substituted, the higher-dimensional scalar curvatures (8)reduce precisely to which is invariant under the Yang–Mills gauge symmetry (26), or (cf. [22,33–38]) specifically in components, ˆ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ + Gˆ =− G¯ + 1 ˙ ˙ ˙ ABC A = A + A ˙ ˙ B C = [ A ] ˙ 2 (0) 2 (0) 3 f ABC f , δWμ ∂μ f BC Wμ ,δBμν W μ ∂ν A . ˆ ˙ −2G¯ = S − 1 HACHBDF A F ˙ However, we have not pinned down the gauge group to be either (0) (0) 4 AB CDA (23) E8 × E8 or SO(32). Further, as the starting scalar curvatures (8) − 1 HADHBEHCF 12 ωABCωDEF are at most two-derivative, our final result (23)lacks the higher- 1 derivative gravitational Chern–Simons term [43], cf. [44–46]. + HADHBEHCF H[ G H ] 2 ωABC D ∂E F G , where we set Yang–Mills and Chern–Simons terms, 4. Reduction to heterotic string ˙ ˙ ˙ ˙ ˙ ˙ F C = ∂ W C − ∂ W C + f ˙ ˙ C W A W B , AB A B B A AB A B Henceforth we discuss the worldsheet aspect of the DFT back- ˙ ˙ ˙ ˙ (24) = A + A B C ground (9)involving the maximally non-Riemannian internal ωABC 3W[A ∂B W C]A˙ f A˙ B˙ C˙ W A W B W C , space. The analysis of the string propagating on the most gen- C˙ of which the O(D, D) indices are totally skew-symmetric, F AB = eral non-Riemannian background was carried out in [14], with the C˙ F[AB] , ωABC = ω[ABC], and further from (13), conclusion that string becomes generically (anti-)chiral. To review this and apply it to the current non-Riemannian background (9), F AB ˙ = 0 ABC = 0 (25) C ∂A , ω ∂A . we recall the doubled-yet-gauged string action [47], [25](cf. [27, The higher Dˆ -dimensional diffeomorphisms generated by the 48,49]), standard DFT Lie derivative give rise, with a twisted parameter, ˆ ˙  1 2 ˆ A = A A A Sstring =  d σ Lstring , (30) ξ ( U A˙ , ξ ), to the D-dimensional diffeomorphisms plus 4πα Yang–Mills gauge symmetry, cf. [19,22], which contains a generic DFT-metric, ˙ ˙   ˙ A = C A + B − B A δW A ξ ∂C W A ∂A ξ ∂ ξA W B ˆ ˆ ˆ 1 αβ A B ˆ αβ A ˙ ˙ ˙ ˙ Lstring =− −hh Dα z Dβ z H ˆ ˆ −  Dα z A ˆ . + A + A B C 2 AB β A ∂A f B˙ C˙ W A , ˆ ˆ C C C A = ˜ ν δHAB = ξ ∂C HAB + 2∂[AξC]H B + 2∂[B ξC]HA (26) With ambient doubled coordinates, z (zμˆ , z ), and an auxiliary ˆ  ˙   ˙  A A + A HC + A H C gauge potential, α , a covariant derivative is introduced, W[A ∂C] A˙ B W[B ∂C] A˙ A , Aˆ Aˆ Aˆ Aˆ = C − 1 C D z := ∂ z − A , A ∂ ˆ = 0 . (31) δd ξ ∂C d 2 ∂C ξ . α α α α A 68 K. Cho et al. / Physics Letters B 793 (2019) 65–69

While this action is completely covariant with respect to the 5. Conclusion desired symmetries like O(D, D) T-duality, Weyl symmetry, and worldsheet as well as target-spacetime diffeomorphisms, it also re- The maximally non-Riemannian DFT background specified by   alizes concretely the assertion that doubled coordinates in DFT are the DFT-metric, H   = J   (14), is singled out to be completely   A B A B O(D , D ) symmetric and rigid: it does not admit any linear fluc- actually gauged and each gauge orbit in the doubled coordinate  tuation, H   = 0(16), nor graviscalar moduli, and the coset system corresponds to a single physical point [50]. Specifically, δ A B = ˜μˆ ≡ structure is trivial (17). It is the most symmetric vacuum in DFT. with the choice of the section, ∂Aˆ (∂ , ∂νˆ ) (0, ∂νˆ ), the con- ˆ For the DFT Kaluza–Klein ansatz, (9)and (18), we set the inter- A A = dition on the auxiliary gauge potential (31)is solved by α nal space to be maximally non-Riemannian, performed a Scherk– A  ( αμˆ , 0). Therefore, half of the doubled coordinates, namely the Schwarz twist (20), (22), and computed the ambient higher (D + tilde coordinates, are gauged, +Gˆ −G¯ˆ D)-dimensional DFT scalar curvatures, (0) and (0) (8), which ˆ   lead to the O(D, D)-manifest formulation of the non-Abelian het- A = ˜ − A νˆ +Gˆ Dα z ∂α zμˆ αμˆ ,∂α z . erotic supergravity (23). Only the former, (0), contains a DFT- cosmological constant. In this way, supergravity and Yang-Mills the- Upon the Riemannian background (27), the auxiliary gauge poten- ory can be unified into a higher-dimensional pure Stringy Gravity, free of tial appears quadratically in the action (30). Then after integrating moduli. it out, one recovers the standard undoubled string action with gμν Plugging the same non-Riemannian Kaluza–Klein ansatz (9)into and B . On the other hand, upon a generic non-Riemannian DFT μν the doubled-yet-gauged string action (30)may reproduce the usual background which is characterized by two non-negative integers, ˆ heterotic string action, (n, n¯), the components of the auxiliary field, A A , appear linearly α   for n and n¯ directions, playing the role of Lagrange multipliers. 1 αβ αβ μ ν αβ μˆ − −hh gμν +  Bμν ∂αx ∂β x +  ∂α z˜ ˆ ∂β z , Consequently, string becomes chiral and anti-chiral over the n and 2 μ n¯ directions respectively [14].     Now, for the present non-Riemannian background (9), in accor- αβ √αβ μ with chiral internal coordinates, h + ∂β y = 0. This is dance with the decomposition of the ambient space (10), we put −h   A ν A ν indeed the case when the Yang–Mills sector is trivial. Therefore, y = (y˜ μ , y ) for the internal doubled coordinates, x = (x˜μ, x )   heterotic string theory may have a higher-dimensional origin with non- A ˜ ν for the external ones, and Wμ = (Wμν , Wμ ) for the gravivec-  Riemannian internal space. Aˆ A A tor. Further, for the ambient gauge potential, Aα = (Aα , Aα ), Our analyses have been focused on the bosonic sectors, both on  A we set the internal and the external ones as Aα = (Aαμ , 0) and the target-spacetime and on the world-sheet. Inclusion of fermions A Aα = (Aαμ, 0). With this preparation and (9), (27), the string is highly desired, cf. [16,17,27]. We also leave the exploration of the action (30) becomes quadratic in the combination of the gauge po- worldsheet aspect of the somewhat mysterious ‘relaxed’ section  − ˜ μ   condition designed for the Scherk–Schwarz twist [19–23]for future tential components, Aαμ Wμ Aαμ , and linear in Aαμ . Hence,  A = the former leads to a Gaussian integral and the latter plays the role work with nontrivial gravivector, W A 0, for the action (32). Up- of a Lagrange multiplier. After all, the doubled-yet-gauged string lift of the Standard Model of particle physics coupled to DFT [24]to higher dimensions (bottom-up) would be also of interest, as well 1 2 L action (30)reduces to 2πα d σ Het, with as applications to string compactifications (top-down), possibly on other types of non-Riemannian internal spaces [52]. √ 1 αβ μ ν 1 αβ μ ν LHet =− −hh ∂αx ∂β x gμν +  ∂αx ∂β x Bμν 2 2 Note added   + 1 αβ ˜  + μ  μ + λ ˜ μ 2  (∂α yμ ∂αx Wμμ )(∂β y ∂β x Wλ ) (32)  Motivated by the first version of this work on arXiv, Berman, + 1 αβ x˜ − W  y A xμ 2  (∂α μ μA ∂α )∂β . Blair, and Otsuki explored non-Riemannian geometries in M-   theory [53]. In particular, they pointed out that the maximally Furthermore, ∂ yμ + ∂ xμW˜ μ must be chiral: α α μ non-Riemannian E8 background naturally realizes the topological      (hence ‘moduli-free’) phase of Exceptional Field Theory [54]. αβ √1 αβ μ μ ˜ μ h +  ∂β y + ∂β x W = 0 . (33) −h μ Acknowledgements A˙ Especially when the gravivector, Wμ , is trivial, the second and third lines in (32)merge into a known topological term, We wish to thank Stephen Angus, Kanghoon Lee for use- 1 αβ ˜ μˆ ful comments, and Wonyoung Cho for valuable help at the  ∂α zμˆ ∂β z [47,51], while (33)gets simplified to make the in- 2 early stage of the project. This work was supported by the Na- ternal coordinates chiral: tional Research Foundation of Korea through the Grants, NRF-   2016R1D1A1B01015196 and NRF-2018H1D3A1A01030137 (Korea μ = μ + y (τ , σ ) y (0, τ σ ). (34) Research Fellowship Program). 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