Effective action for non-geometric fluxes duality covariant actions

Journal of High Energy Physics, Jul 2017

The (heterotic) double field theories and the exceptional field theories are manifestly duality covariant formulations, describing low-energy limit of various super-string and M-theory compactifications. These field theories are known to be reduced to the standard descriptions by introducing appropriately parameterized generalized metric and by applying suitably chosen section conditions. In this paper, we apply these formulations to non-geometric backgrounds. We introduce different parameterizations for the generalized metric in terms of the dual fields which are pertinent to non-geometric fluxes. Under certain simplifying assumptions, we construct new effective action for non-geometric backgrounds. We then study the non-geometric backgrounds sourced by exotic branes and find their U -duality monodromy matrices. The charge of exotic branes obtained from these monodromy matrices agrees with the charge obtained from the non-geometric flux integral.

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Effective action for non-geometric fluxes duality covariant actions

JHE Effective action for non-geometric fluxes duality Kanghoon Lee 0 1 4 Soo-Jong Rey 0 1 2 4 Yuho Sakatani 0 1 3 4 0 Seoul National University , Seoul 08826 , Korea 1 Institute for Basic Sciences , Daejeon 34047 , Korea 2 School of Physics & Astronomy and Center for Theoretical Physics 3 Department of Physics, Kyoto Prefectural University of Medicine 4 Fields, Gravity & Strings , CTPU The (heterotic) double field theories and the exceptional field theories are manifestly duality covariant formulations, describing low-energy limit of various superstring and M-theory compactifications. These field theories are known to be reduced to the standard descriptions by introducing appropriately parameterized generalized metric and by applying suitably chosen section conditions. In this paper, we apply these formulations to non-geometric backgrounds. We introduce different parameterizations for the generalized metric in terms of the dual fields which are pertinent to non-geometric fluxes. Under certain simplifying assumptions, we construct new effective action for non-geometric backgrounds. We then study the non-geometric backgrounds sourced by exotic branes and find their U -duality monodromy matrices. The charge of exotic branes obtained from these monodromy matrices agrees with the charge obtained from the non-geometric flux integral. Flux compactifications; String Duality - HJEP07(21)5 1 Introduction 2 General framework 2.1 2.2 2.3 2.4 2.5 Parameterization of Lie algebra The generalized metric Example: Double Field Theory Example: Einstein gravity Effective action for non-geometric fluxes 3 Non-geometric fluxes in EFT: M-theory 3.1 Parameterization of the generalized vielbein 3.2 3.3 Eleven-dimensional effective action Reduction to the type IIA theory 4 Non-geometric fluxes in EFT: type IIB section 4.1 Parameterizations of the generalized vielbein 3.1.1 3.1.2 3.1.3 3.1.4 4.1.1 4.1.2 4.1.3 4.1.4 6.1.1 6.1.2 6.1.3 6.2.1 6.2.2 6.3.1 Exotic branes in the M-theory 6.3 Exotic branes in the type IIB theory – i – p7−p-brane 164-brane 7 Discussion A Notations A.1 Ed(d) algebras: M-theory section A.2 Ed(d) algebras: type IIB section B Calculation of the EFT action B.1 Redefinitions of coordinates B.2 External part B.3 Internal (potential) part B.4 Summary C Double-vielbein formalism for gauged DFT C.1 Parameterization from defining properties of double-vielbein C.2 Connection and curvature C.3 Nongeometric fluxes and action D Exotic branes HJEP07(21)5 I do not wish, at this stage, to examine the logical justification of this form of argumentation; for the present, I am considering it as a practice, which we can observe in the habits of men and animals. Bertrand Russell, ‘Philosophy’. 1 Introduction Recently, a significant progress has been achieved for novel formulations of supergravity in which duality symmetries in string and M-theory compactification are manifest. They include the double field theory (DFT) [1–7], the exceptional field theory (EFT) [8–26] (see also [27–34] for closely related attempts) as well as the generalized geometry [35– 40]. One important advantage of these formulations is that they can treat wide variety of spacetimes, such as non-geometric backgrounds [41–44], that are not globally describable backgrounds arise quite naturally in superstring theories. Backgrounds sourced by exotic branes [47–53] are concrete examples. As an application of DFT and related formulations such as the β-supergravity [54–61], a background of a particular exotic brane, so-called a 522-brane, was studied in [45, 46, 62–72] and the exotic 522-brane was identified with a magnetic source of the non-geometric Q-flux [64, 70, 72]. – 1 – One reason why the exotic 522-brane received special attention is that the non-geometric Q-flux, which is intrinsic to the 522-brane background, is related to a T -duality monodromy, and the much developed DFTs efficiently describe such background. It is known that backgrounds of other exotic branes possess other non-geometric fluxes that are related to the Q-flux via U -duality transformations [51, 73]. In order to describe such non-geometric backgrounds, variants of the β-supergravity, which can describe the background of an exotic p-brane (called a p7−p-brane) or a 164-brane, was proposed in [74]. There, each of these 3 exotic branes was identified as the magnetic sources of a non-geometric P -flux [75–77] or a non-geometric Q-flux associated with a 6-vector, βm1···m6 [74]. However, the reformulation of [74] is applicable only to a limited situation; coexistence of different non-geometric fluxes U -duality covariant formulation of the supergravity, is a more suitable formulation, and indeed, backgrounds of the exotic 53-brane, 522-brane, and the 523-brane were studied in SL(5) EFT [78, 79]. One of the main purposes of this paper is to systematically identify the non-geometric fluxes in Ed(d) EFT for the cases of 4 ≤ d ≤ 7. The goal of this paper is to develop effective actions for a certain class of non-geometric flux backgrounds in Type II string and M-theories. Our starting point is the duality covariant action in an extended field theory, such as the manifestly U -duality covariant EFT. Since the U -duality orbit is of infinite order, there are in practice infinitely many possible parameterization of the U -duality group. The key idea is to identify the most effective parameterization for a given set of non-geometric flux background. Note that our non-geometric parameterization is efficient for backgrounds with only non-geometric fluxes. For backgrounds with both geometric and non-geometric fluxes, such as the truly non-geometric backgrounds of [80], a more general treatment will be required.1 Our construction can be extended to non-geometric flux backgrounds in heterotic string theories. Heterotic string exhibits O(D, D + 16) or O(D, D + dim G) duality group, where G is the heterotic Yang-Mills group, E8 × E8 or SO(32), and the corresponding heterotic DFT [1, 2, 81] provides a duality manifest description of the effective field theory. Again, the key idea is to identify the most effective parameterization. Through the non-geometric parameterization of heterotic generalized vielbein, we construct heterotic Q-flux which includes Chern-Simons like term and an additional non-geometric bi-vector flux associated with the heterotic Yang-Mills field strength. The corresponding non-geometric effective action can be constructed from O(D, D + dim G) gauged DFT [82–84]. If we take the maximal Abelian reduction of heterotic Yang-Mills gauge symmetry, G = U(1)16, the nongeometric gauged DFT reduces to the non-geometric parameterization by Blumenhagen and Sun [85]. This paper is organized as follows. In section 2, after reviewing some elements of Lie algebra, we explain the general construction of the generalized metric or vielbein. In 1Note that the section condition or the strong constraint in DFT/EFT can be relaxed through the generalized Scherk-Schwarz reduction [82], which provides all the fluxes in the maximal and half-maximal izations of the generalized vielbein; the conventional geometric parameterization and the dual non-geometric parameterization. Using the two different parameterizations, we write down two different eleven-dimensional effective actions. We also consider the dimensional reduction to the type IIA theory, and obtain the non-geometric fluxes in the type IIA theory. EFT in terms of the type IIB theory is discussed in section 4 and ten-dimensional action for the non-geometric fluxes in the type IIB theory is obtained. In section 5, we find a parameterization of heterotic DFT relevant for non-geometric fluxes. In section 6, the relation between the non-geometric fluxes and exotic branes are discussed. Discussions and future directions are given in section 7. We relegated much of technical details to the Aij (i, j = 1, . . . , rank g) that has the structure Aii = 2 , Aij ∈ Z≤0 (i 6= j) , Aij = 0 ⇔ Aji = 0 , det Aij > 0 , (2.1) where Z≤0 denotes non-positive integers. In g, consider the Chevalley basis generators {Hi, Ei, Fi}, which obey the properties [Hi, Hi] = 0 , [Hi, Ej ] = Aji Ej , [Hi, Fj ] = −Aji Fj , [Ei, Fj ] = δij Hi , | 1−{Azji } [Ei, [· · · , [Ei, Ej ] · · · ]] = 0 , | 1−{Azji } [Fi, [· · · , [Fi, Fj ] · · · ]] = 0 . It is known that the generators {Hi, Ei, Fi}, together with the commutators of Ei or Fi , [Ei1 , [· · · , [Eik−1 , Eik ] · · · ]] and [Fi1 , [· · · , [Fik−1 , Fik ] · · · ]] , 2Here, we suppose g is a split real Lie algebra, considering the applications to DFT, g = o(d, d), and EFT, g = ed(d). Application to a non-split case is considered in section 5. (2.2) (2.3) – 3 – form a complete set of basis of g . In the Chevalley basis, the generators Hi for i = 1, . . . , rank g form the Cartan subalgebra h, the generator Ei is associated with the positive simple root αi ∈ h∗ with αi(Hj ) = Aij , and the generator Fi is associated with the negative simple root −αi . We denote the space of positive root by Δ+ and the space of negative root by Δ−, respectively. For an arbitrary positive root α ∈ Δ+, we can construct the associated generator as k-tuple left-commutator HJEP07(21)5 Eα ≡ [Ei1 , [· · · , [Eik−1 , Eik ] · · · ]] . For the corresponding negative root −α ∈ Δ−, we also construct the associated generator as k-tuple right-commutator Fα ≡ [[· · · [Fik , Fik−1 ], · · · ], Fi1 ] . Denote the space spanned by Eα and Fα (α ∈ Δ+) as n+ and n−, respectively. Then, we obtain the triangular decomposition by decomposing the Lie algebra g as α = k X αin , n=1 g = n − ⊕ h ⊕ n+ . The second method is known as the Cartan decomposition. Define the Cartan involution θ by θ(Hi) = −Hi , θ(Ei) = −Fi , θ(Fi) = −Ei . From the distributive property that θ([s, t]) = [θ(s), θ(t)] for s, t ∈ g, it follows that θ(Eα) = −Fα , θ(Fα) = −Eα for every α ∈ Δ+ . Redefining the generators as we can diagonalize the Cartan involution as Sα ≡ Eα + Fα and Jα ≡ Eα − Fα for every α ∈ Δ+ , θ(Hi) = −Hi , θ(Sα) = −Sα , θ(Jα) = +Jα , and classify the generators according to the parity under the involution θ: k = {s ∈ g | θ(s) = +s} = span(Jα) and p = {s ∈ g | θ(s) = −s} = span(Hi, Sα) . We are thus decomposing the Lie algebra g as obtaining the Cartan decomposition. Since the number of the positive roots is (dim g − rank g)/2, we have dim k = dim g − rank g 2 dim g + rank g 2 . g = k ⊕ p , – 4 – (2.4) (2.5) (2.6) (2.7) (2.8) (2.9) (2.10) (2.11) (2.12) (2.13) G = Ed(d) D = dim l1 SL(5) 10 SO(5, 5) E6(6) 27 E7(7) 56 Although the commutator in p is not closed (since it has the odd parity under θ), the Lie commutators in k yields a subalgebra, sometimes called the Cartan-involution-invariant subalgebra, which coincides with the maximal compact subalgebra of g . The third method is known as the Iwasawa decomposition, the decomposition we shall be using in the present paper. There are two possible types of Iwasawa decomposition. The positive decomposition is defined by [h, ZM ] = − X(ρh)M N ZN N (h ∈ g) . θ(h), Z M = − X(ρh)M N Z N (h ∈ g) . N ≡ −θ ZM , we obtain from (2.17) the following commutator: Here, ρh is the matrix realization for the element h ∈ g in the l1-representation. Defining dual matrix realization (ρ¯h)M N ≡ δMK (ρh)K L δLN . We then obtain To render the position of indices consistent, we introduce the fundamental forms, δMN and δMN , whose components are equal to δMN (and are not generalized tensors), and define the h, Z M = −(ρ¯θ(h))M N Z N (h ∈ g) . g = k ⊕ h ⊕ n+ , g = n − ⊕ h ⊕ k , where b+ ≡ h ⊕ n+ is referred to as the positive Borel subalgebra. The negative decomposition is defined by where b − ≡ n − ⊕ h is referred to as the negative Borel subalgebra. Associated to the Lie algebra g, we construct the corresponding Lie group G as the exponential map. We can realize group element g ∈ G in any of the above decomposition of g. In particular, we can straightforwardly extend the definition of the Cartan involution θ to an arbitrary group element g ∈ G, and then define an anti-involution ♯ by g♯ ≡ θ(g−1) , (ab)♯ = b♯a♯ where g, a, b ∈ G . In section 3 and 4, we take the Lie algebra g = ed(d) and its Lie group G = Ed(d) as the duality symmetry (summarized in table 1). Suppose that the (generalized) momenta ZM (M = 1, . . . , D ≡ dim l1), which generate abelian translations ([ZM , ZN ] = 0) in the extended space X of the U -duality action, are in the fundamental representation l1 of the Lie group G [28], – 5 – (2.14) (2.15) (2.16) (2.17) (2.18) (2.19) ([h, v], w¯) + (v, [h, w¯]) = 0 (h ∈ g) , (Adg · v, Adg · w¯) = (v, w¯) , (2.20) (2.21) where Adg · v ≡ g v g−1 (g ∈ G) . We will normalize the abelian generators ZM such that the scalar product becomes the identity matrix, ZM , Z N = δMN , We also introduce a natural G-invariant scalar product (v, w¯) ≡ vM w¯M for an element v of the representation l1 spanned by AM and w¯ of the dual representation ¯l1 spanned by ∈ G, v = vM ZM , and w¯ = w¯M Z , we have M (eρh )M K (eρ¯θ(h) )N K = δMN . (2.22) (2.24) (2.25) (2.26) (2.27) (eρh )M N = (e−ρ¯θ(h) )N M = eρ¯h♯ N M = δNL eρh♯ L K δKM , where we defined h ♯ ≡ −θ(h) for h ∈ g , and used the dual representation (ρ¯h)N M = δNL (ρh)LK δKM in the last equality. This relation shows that the anti-involution ♯ defined in (2.16), sometimes called the generalized transpose, acts as the matrix transpose in the matrix realization of Lie algebra g. 2.2 The generalized metric hv, wi = hw, vi = vM wN δMN . – 6 – identities, ZM , θ(ZN ) = − ZM , Z symmetric and positive-definite, We next study the geometry of extended space X associated with the duality transformation group G. We shall define the generalized metric MMN of X and explain how to parameterize MMN in terms of appropriate physical fields (see [11, 86]). We first define a bilinear form hv, wi ≡ − v, θ(w) = −vM wN ZM , θ(ZN ) , for generalized vectors, v = vM ZM and w = wM ZM , in the l1-representation. From the N = −δMN = −δMN , we see that the metric (2.26) is However, as δMN is not a generalized tensor, this metric is not G-invariant. Indeed, for general element h ∈ g, we find that the adjoint action [h, v], w + v, [h, w] = − [h, v], θ(w) − v, [θ(h), θ(w)] = − [h − θ(h), v], θ(w) is nonzero. However, it is invariant under the maximal compact subgroup, K, of G, since h = θ(h) for h ∈ k. Starting from this (constant) positive-definite metric and a group element g ∈ G, we now define the generalized metric from the generalized bilinear form M(v, w) ≡ MMN vM wN ≡ hAdg−1 · v, Adg−1 · wi . (2.28) and w¯M ≡ δMN wN , we have The generalized bilinear form is positive-definite by construction, and it is defined to be G-invariant. We assume that the generalized metric MMN varies over the spacetime, so the group element g ∈ G should be spacetime-dependent as well. Denoting g = e−h (h ∈ g) θ Adg−1 · w = e[θ(h), · ] ZM w¯M = (eρ¯h )M N Z N w¯M , and the inner product in (2.28) becomes M(v, w) = vM (eρh )M K (eρ¯θ(h) )N L w¯N (ZK , ZL) = vM (eρh )M K (eρ¯h )N K w¯N = vM (eρh )M K (eρh♯ )K L δLN wN . Introducing the generalized vielbein as EM metric in the conventional form, A ≡ (eρh )M A, we can express the generalized MMN = EM A EN B δAB . Here, the indices A, B run over 1, . . . , D = dim l1, which play the same role as the original indices M, N but are interpreted as “flat indices.” As h · , · i is K-invariant, two generalized metrics constructed from g ∈ G and g · k (k ∈ K), respectively, have the same structure. Thus, the generalized metric can be parameterized by a coset representative of G/K, and so the number of the independent parameters is given by dim(G/K) = dim G − dim K. For an explicit construction of the generalized metric, we find it convenient to use the Iwasawa decomposition (2.14) and parameterize the representative g ∈ G/H, where H is the Cartan subgroup, in terms of functions, hi(x) and Aα(x), associated with generators of the positive Borel subalgebra b+, and the K equivalence class: g(x) = ePi hi(x) Hi ePα∈Δ+ Aα(x) Eα k(x) ∼ ePi hi(x) Hi ePα∈Δ+ Aα(x) Eα , (2.32) Here, k(x) denotes an element in the compact subgroup K and x refers to the coordinate system adopted. We can then obtain the generalized metric from the following generalized vielbein: EM A(x) = ehi(x) ρHi ePα∈Δ+ Aα(x) ρEα M Note that the generalized metric MMN is invariant under the anti-involution g → g (i.e. symmetric), while the generalized vielbein is not. Using the above decomposition, ♯ we have g♯(x) = k♯(x) ePα∈Δ+ Aα(x) Fα ehi(x) Hi with certain functions ehi(x), Aeα(x) and ek(x) ∈ K, whose relation to hi(x), Aα(x), and k(x) is in general complicated. This expression for g♯ corresponds to the alternative Iwasawa decomposition (2.15), so we can obtain the generalized vielbein in terms of the functions associated with the generators of negative Borel subalgebra b : − HJEP07(21)5 (2.34) (2.35) (2.36) (2.37) (2.38) The key idea of this paper is that the above replacement g → g♯, which does not change the generalized metric, generally corresponds to the replacement from the conventional geometric parameterization of the generalized metric MMN to the dual “non-geometric” parameterization of it. A transformation between the conventional and the dual parameterization is sometimes referred to as the exotic duality transformation [62, 74, 87]. In this paper, we will show that the exotic duality transformation is identifiable with the generalized transpose. It remains to confirm the tensorial property of the generalized metric. As we mentioned above, the flat bilinear form hv, wi = vM wN δMN was not G-invariant. However, the generalized bilinear form M(v, w) = hAdg−1 · v, Adg−1 · wi is invariant under G. This constrains the transformation rule for the group element g (i.e. the generalized vielbein). It then follows that, as the K-invariance of δMN , the transformation rule of the generalized vielbein generally has the following form: EM A → gM N EN B kBA for g ∈ G, k ∈ K . 2.3 Example: Double Field Theory Before presenting our new results, we first illustrate the above general consideration for the DFT. In this case, the T-duality group is G = O(d, d). We can decompose the generators of g = o(d, d) into representations of the GL(d), the gl(d)-generators Kab, Rab = R[ab], Rab = R[ab] (a, b = 1, . . . , d), which obey the following commutation relations: [Kab, Kcd] = δbc Kad − δda Kcb , [Kab, Rcd] = δbc Rad + δbd Rca , [Rab, Rcd] = 4 δ[[ca Kb]d] , [Kab, Rcd] = −δca Rbd − δda Rcb . The Cartan subalgebra h is generated by the diagonal components of Kab: Ha ≡ Kaa (no summation). The Cartan involution is given by EM A(x) = ePi ehi(x) ρHi ePα∈Δ+ Aeα(x) ρFα M θ(Kab) = −Kba θ(Rab) = −Rab , and – 8 – In particular, the (anti)chiral combinations, Ma±b ≡ (Jab ± Tab)/2 , satisfy the algebra for o(d) × o(d): [Ma±b, Mc±d] = 2 δ[da Ma±]c − 2 δ[ca Mb±]d , [Ma+b, Mc−d] = 0 . b), Rab} and {Ha, Kab (a > b), Rab}, respectively. The positive and negative Borel subalgebras, b+ and b−, are spanned by {Ha, Kab (a < In DFT, we take the fundamental (i.e. vector) representation, whose matrix realization is given by the matrices, δac δdb 0 0 −δda δbc ! 0 2 δacdb! 0 0 (ρKcd )AB = , (ρRcd )AB = , (ρRcd )AB = 0 −2 δcadb 0 0 ! , (2.41) where δacdb ≡ δ [[ac δbd]] (see appendix A for our conventions). The commutators with the generalized momenta ZM = (Pm, Pem) are given by (2.39) (2.40) (2.42) (2.43) N (2.44) (2.45) (2.46) and the Cartan-involution-invariant subgroup is generated by and Note that the variable ZM defined by [Kab, Pc] = −δca Pb , [Rab, Pc] = −2 δc[a Peb] , [Rab, Pc] = 0 , [Kab, Pec] = δbc Pea , [Rab, Pec] = 0 , [Rab, Pec] = 2 δ[ca Pb] . ZM ≡ ηMN ZN and ηMN ≡ 0 δm! n δmn 0 , M is in the same representation as Z . We thus see that the O(d, d)-invariance of (ZM , Z ) = δMN is equivalent to the O(d, d)-invariance of another metric, ((ZM , ZN )) ≡ ηMN , which is commonly used in DFT. We also have the K = O(d) × O(d)-invariant metric δAB. We define the generalized vielbein in the gauge of positive Borel subalgebra by EM A(x) = ePa ha(x) ρKaa ePa<b hab(x) ρKab e 2 1 Pa,b Bab(x) ρRab M in d-dimensions, and (e−T) is the inverse of the transpose of the vielbein. This generalized vielbein yields the conventional generalized metric in DFT: Upon the anti-involution, g → g♯, the generalized vielbein takes the lower-triangular HJEP07(21)5 and β( 2 )(x) = 0 −βab(x) 0 0 ! , (2.49) where ema(x) is a lower-triangular matrix. In this case, the generalized metric becomes where Gmn ≡ ema enb δab . form, parameterized by where , MMN Gmn − Bmk Gkl Bln Bmk Gkn! δmk Bmk 0 δm k −Gmk Bkn ! Gmn Gkl 0 ! 0 Gkl δ l n −Bln δnl 0 ! , EM A(x) = E(x) eβ( 2 )(x) M A , = = = = e ema(x) 0 0 e (e−T)ma(x) MMN ! Gemn (2.47) (2.48) (2.50) (2.51) studied in [88–92]. where Gemn ≡ eema eenb δab . These dual variables were first introduced and extensively 2.4 Example: Einstein gravity It is illuminating to compare the above results for DFT with the case of pure Einstein gravity. In Einstein gravity, the generators Rab and Rab are absent, the Cartan-involutioninvariant subgroup is simply generated by the local Lorentz O(d) rotations, and there is no important difference between the gauges of positive and negative Borel subalgebras. Indeed, as it is well-known, when we consider decomposing the spacetime into space and time, there are two natural parameterizations into upper or lower triangular decomposition: Arnowitt-Deser-Misner [93]: (gmn) = Landau-Lifschitz [94]: (gmn) = 1 N k! k 0 δi 1 −gi δik 0 ! −N 2 0 ! 0 hkl These two parameterizations are related simply by a usual local Lorentz transformation. In comparison, the situation is different in the DFT case. In order to relate two parameterizations (2.47) and (2.50), we need to use a non-trivial O(d) × O(d) subgroup of the T -duality group. In general, the parameterization (2.47) is suited for the conventional geometric backgrounds, while (2.50) is suited for non-geometric backgrounds, such as T folds. As such, we will refer to the latter, negative Borel subalgebra parameterization as the non-geometric parameterization. By definition, the actions of the extended field theories are independent of the explicit parameterization of the generalized metric. However, once we parameterize the generalized metric in terms of appropriate physical fields, we can straightforwardly construct the effective actions appropriate for describing dynamics of these field excitations. As is well known in DFT [7] or EFT [11], parameterizing the generalized metric in terms of the conventional supergravity fields, we can derive the conventional supergravity action from DFT or EFT action. For example, if we choose the conventional, geometric parameterization and impose the section constraint ∂em = 0, we find that the DFT action is reduced to HJEP07(21)5 (2.52) (2.53) (2.54) L = e−2φ 1 R(G) + 4 |dφ|2 − 2 |H(3)|2 , where φ is the conventional string dilaton field defined by the T-duality invariant dilaton ≡ | | of DFT, e−2d G 1/2 e−2φ, and the three-form H(3) strength for the Kalb-Ramond two-form potential B( 2 ). ≡ dB( 2 ), called the H-flux, is the field On the other hand, if we choose the dual, non-geometric parameterization (2.50), we reduce the DFT action to the so-called β-supergravity [54–56, 60, 61]. Although the full expression is complicated, with the simplifying assumption that indices of βmn contracted with ∂m always vanishes and the constraint ∂em = 0, the DFT action is reduced to the form Le = e−2φe R(Ge) + 4 |dφe| − 2 | 2 1 Q( 1,2 ) 2 | 1/2 e−2φe . Further, we defined Here, the tilde signifies the non-geometric parameterization, and φe is the dual dilaton field | Q( 1,2 ) 2 | ≡ 2 Gem1n1 Gem2n2 Gem3n3 Qm1 m2m3 Qn1 n2n3 , Qkmn ≡ ∂kβmn . The mixed-symmetry tensor,3 Qkmn, is called the non-geometric Q-flux. In this paper, we further generalize the β-supergravity starting from the (heterotic) DFT or EFT. 3 Non-geometric fluxes in EFT: M-theory In this section, we consider the eleven-dimensional supergravity of M-theory compactified on a d-torus, Td, equivalently, the ten-dimensional type IIA supergravity compactified on a (d − 1)-torus, Td−1. This theory possesses the U -duality transformation symmetry, and 3This behaves as a tensor only under the simplifying assumption [55]. Ed(d) Kd D αn SL(5) SO(5) 10 3 E6(6) Sp( 4 ) 27 6 E7(7) SU(8) 56 12 various noncompact dimensions, 4 ≤ n ≤ 7. the EFT provides the manifestly U -duality covariant formulation. To construct the EFT, we consider an exceptional spacetime with the following generalized coordinates: (XI ) = (xµ , Y M ) (µ, ν = 0, . . . , n − 1, M = 1, . . . , D) , (3.1) where n ≡ (11 − d) is the dimension of the uncompactified, external spacetime and D is the dimension of a fundamental representation of the exceptional group Ed(d) whose value for each n is shown in table 2. In this paper, we consider the cases of noncompact dimensions n = 4, 5, 6, 7, equivalently, cases of compact dimensions d = 7, 6, 5, 4. The EFT actions for n = 4, 5, 6, 7 are presented in [14, 15, 22, 23] (see also [24] for n = 9, [21] for n = 8, and [18] for n = 3). For simplicity, we focus on the following parts of the action, which are the relevant parts for our purposes: dnx dDY LEFT where LEFT = LEH + Lscalar + Lpot , Z SEFT = LEH = eR , 1/2, R is the Ricci scalar of the external metric gµν , and αn is Here, e abbreviates |det gµν | into account by Lpot. the integer shown in table 2. Note that the potential part in the EFT action is fully taken In the EFT, to render the gauge algebra closed, we will impose the section condition of the form, Y MN P Q ∂M (· · · ) ∂N (· · · ) = 0, where Y MN P Q for each EFT is given in [12, 13].4 As is well-known, there are two natural routes to solve for the section conditions: the M-theory section or the type IIB section [14, 98], where all background fields and gauge parameters depend only on d coordinates xi or d − 1 coordinates xm, respectively. In this section, we study the M-theory section and parameterize the generalized metric in terms 4The section condition of DFT can be relaxed in the flux formulation [82, 95, 96] or in the approach of [97], and the section condition of EFT may be also relaxed in these approaches. of the conventional/dual fields in eleven dimensions. We relegate the parameterization in the type IIB section to section 4. In the M-theory section, we decompose the internal D-dimensional coordinates Y M into some representations of SL(d). Explicitly, for each n, we introduce the following coordinates [11, 28]: n = 7 : (Y M ) = (xi, yij ) n = 6 : (Y M ) = (xi, yij , yi1···i5 ) n = 5 : (Y M ) = (xi, yij , yi1···i5 ) n = 4 : (Y M ) = (xi, yij , yi1···i5 , zi) (i, j = 7, 8, 9, M) , (i, j = 6, . . . , 9, M) , (i, j = 5, . . . , 9, M) , (i, j = 4, . . . , 9, M) , where the conventional M-theory circle direction, denoted by xM, is one of the internal coordinates xi. The section condition is satisfied when all fields are functions only of xi, the physical coordinates on the d-torus. So, ∂/∂yij = ∂/∂yi1···i5 = ∂/∂zi = 0. Parameterization of the generalized vielbein We now examine parameterization of the generalized metric (or vielbein) in the M-theory section of the EFT. The generalized metric in the SL(5) EFT was first obtained in [8] (which in turn is based on the earlier work [99]) as Subsequently, the same generalized metric (up to an overall factor) was presented in [11] in the context of E11 program [27, 28], and its extensions to Ed(d) EFT with 5 ≤ d ≤ 7 were also presented (see also [100, 101] for d = 4, 5). The parameterization given in [11] was obtained by choosing the positive (or upper-triangular) Borel gauge. If we instead choose the negative (or lower triangular) Borel gauge, we can parameterize the generalized metric using the so-called dual Ω-fields (the explicit form of Ω-fields for SL(5) EFT is given in [78, 79], which we repeat below). As the Ω-fields are related to the non-geometric fluxes, we refer to the latter as non-geometric parameterization. In the rest of this subsection, we present two parameterizations of the generalized vielbein, i.e., the conventional parameterization and the non-geometric parameterization, for 4 ≤ d ≤ 7 (or 4 ≤ n ≤ 7). Using these parameterizations, we define the non-geometric fluxes in M-theory and construct the eleven-dimensional effective actions that are useful for describing these non-geometric fluxes. 3.1.1 For the g = sl(5) Lie algebra, we decompose the 24 generators to5 [11] 5We relegate their commutators in appendix A.1. (3.3) HJEP07(21)5 (3.4) where Kab are the gl( 4 ) generators and Ra1a2a3 and Ra1a2a3 are the generators that transform as totally antisymmetric under gl( 4 ). So, we are decomposing 24 generators into 16 + 4 + 4 generators. Using this decomposition, a group element g of G = SL(5) can be parameterized as This element can always be rewritten in the form of positive Borel gauge: where k ∈ H = SO(5) . (3.7) It turns out that the SO(5) element k does not contribute to the generalized metric. Disregarding it, the number of independent parameters are 10 + 4, which is equal to the dimension of the coset space G/H = SL(5)/SO(5). We can identify the parameters, eib ≡ (eh)ib ∈ GL( 4 )/SO( 4 ) and Aa1a2a3 , as the vielbein and the 3-form potential on the 4-torus, respectively. Note that the left index of the matrix (eh) is changed from a to i in order to interpret it as the curved index. From the formulas (2.31) and (2.33) and the matrix representations (A.12)–(A.16), the generalized vielbein and the metric become [8] MMN ≡ |G| 5 MMN , 1 EM A 1 ≡ |G| 10 EM A , |G| ≡ det Gij , Gij ≡ eia ej b δab , (EM A) ≡ Eb eA(3) =  e a i − √12 Aia1a2 0 eia1i12a2   , (MMN ) = EM A EN B δAB =  where (EbM A) ≡ A(3) 1 ≡ 3! eia 0 0 eia1i12a2 ! , Aabc ρRabc =  Gij + 12 Aikl Aklj  , 0 − √12 Aab1b2 0 0 ! , eia1i12a2 ≡ (e−T)i1 [a1 (e−T)i2 a2] , δAB ≡ δab 0 0 δa1a2, b1b2 ! Gi1···in, j1···jn ≡ δki11······iknn Gk1j1 · · · Gknjn , δa1a2, b1b2 ≡ δca11ca22 δc1b1 δc2b2 , and the indices are changed using the vielbein (e.g. Aia1a2 ≡ eic Aca1a2 ) and raised or lowered using the metric Gij and its inverse. See appendix A for further details of our conventions. – 14 – (3.8) (3.9) If we do not choose the Borel gauge, we can generally parameterize the SL(5) generalized metric as [78, 79] (EM A) ≡ Eb eA(3) eΩ(3) =  Choosing Ωijk = 0 or Aijk = 0, we obtain two alternative parameterizations for the Ωc1c2c3 ρRc1c2c3 = 0   where we defined the Ω-matrix: generalized metric, (3.10)  , (3.11) (3.12) (3.13) (3.14) (3.15) (3.16) (3.17) The first expression is the conventional, geometric parameterization, while the second expression is the non-geometric parameterization. From these two parameterizations, we obtain the following relation between the standard fields and the dual fields: Geij = | | G 1/9 | | E 1/9 Eij , Ωij1j2 = (E−1)ik Gj1k1 Gj2k2 Akk1k2 , where Further, associated to the two parameterizations, the external metric is also expressed in two alternative ways: Eij ≡ Gij + Aikl Aklj . 1 2 gµν = |G| n−2 gµν = |Ge| n−2 gµν . e 1 1 We confirm that gµν and Gij are components of the conventional metric in the elevendimensional supergravity, denoted by Gµˆνˆ (µ,ˆ νˆ = 0, . . . , 9, M). 3.1.2 n = 6: G = SO(5, 5) The generalized metric or vielbein generally has the overall factor, MM N ≡ | | G n−2 MM N , 1 that comes from the second term in the right-hand-side of (A.12). In the following, we focus on the parameterizations of MM N and EM A In the present case of G = SO(5, 5), we can similarly parameterize the generalized vielbein as [8] 0 0             ,   0 0 0   .  (3.19) (3.20) (3.21) (3.22) (3.23) (3.24) or as where we defined (EM A ) ≡ Eb e Ω(3) =  E b ≡  0 eia1i12a2 0 0       5 √ 5! 0 0 eia1·1····i·5a5 1 − √ 2 Ωi1i2a Ω[i1i2i3 Ωi4i5]a A(3) Ω (3) e a i 0    1 3! 1 3!       e a i 0 e a i e   ,     0  0 0    1 − √ 2 Aia1a2 5 √ 5! Ai[a1a2  , We can again redundantly parameterize the generalized vielbein as EM A ≡ Eb e A(3) Ω(3) e In the case G = E6(6), we can parameterize the generalized vielbein as [8] or as EM Eb ≡  0 0       ,  ≡ − 6! Ac1c2c3 ρRc1c2c3 =  0 Ac1···c6 ρRc1···c6 =  0 Ωc1c2c3 ρRc1c2c3 = − √12 Ωa1a2b Ωc1···c6 ρRc1···c6 =   0 0        ,  0 0 0 0 0 0 0 − √15! Ωa1···a5b 0 0 0  0  0  0  ,  0 0 0     ,   (3.26) (3.27) (3.28) (3.29) (3.30) (3.31) (3.32) (3.33) We remark that the normalization of the 6-form is different from that used in [8] by a factor 2. Note also that, in the middle expression of the last line, the minus sign is introduced in order to make the exotic duality, Ac1···c6 ↔ Ωc1···c6 , coincides with the matrix transpose. Stated differently, the negative sign comes from the fact that the Cartan involution (A.17) for Rc1···c6 appears with the positive sign, θ(Rc1···c6 ) = +Rc1···c6 . 3.1.4 In the E7(7) case, we can parameterize the generalized vielbein as [11] (EM A) ≡ Eb eA(6) eA(3) or (EM A) ≡ Eb eΩ(6) eΩ(3) , eia  0 Eb ≡  0  0 eia1i12a2 0 0 0 0 0 0 e a1···a5 i1···i5 0 0 0 |e|−1 eia     ,   − √15! Ωa1···a5b 0 0 0 0 0 0 0 0 0 0 0  0  .     , (3.34) (3.35) 0  0  0   ,  0 (3.36) (3.37) ≡ − 6! Ωc1···c6 ρRc1···c6 =  1 Ac1c2c3 ρRc1c2c3 =  00 0 0 0 0 0 0 0 0 1 4 1 4 35 2 6!√22 δ[ab1 ǫb2]c1···c6 Ωc1···c6 R(G) ≡ R(g) + gµν 1 4 ∂µ Gij ∂ν Gij + 1 4 ∂µ ln |G| ∂ν ln |G| + R(G) + Gij ∂igµν ∂j gµν + ∂i ln |g| ∂j ln |g| Fi1···i7 ≡ 7 ∂[i1 Ai2···i7] + A[i1i2i3 Fi4i5i6i7] , Fµ, k 1···k6 ≡ ∂µ Ak1···k6 − 10 A[k1k2k3| ∂µ A|k4k5k6] . We remark that the parameterizations for Ed(d) with 4 ≤ d ≤ 6 are obtainable by a truncation of those for E7(7). 3.2 Eleven-dimensional effective action The eleven-dimensional effective action is obtained by solving the section condition such that the eleven-dimensional coordinates are given by (xµˆ) ≡ (xµ , xi); see appendix B for the detailed derivation. For instance, consider the E7(7) EFT in the geometric parameterization. The action becomes L = |G| 2 1 where R(G) − g µν 1 − 2 · 4! (3.38) 1 1  0  0  0    1 βeα, m1m2m3n ≡ 2 ǫm1m2m3np1p2 βαp1p2 , For the external part, we focus on the following two-derivative terms: LEH = e R(g) and e 4αn Recalling the relation, gµν = |G| n−12 gµν , the first term is given by LEH = |g| 12 |G|1/2 R(g) + 2 (n − 1) ∂µ e gµν ∂ν ln |G|− n−12 + |g| 21 |G| n−2 n/2 n − 1 1 = |G| 2 R(g) + 4(n − 2) n − 1 4(n − 2) g µν ∂µ ln |G| ∂ν ln |G| g µν ∂µ ln |G| ∂ν ln |G| , where we defined |G| 2 = |g| 12 |G| 2 and neglected the total derivative term at the second 1 1 equality. For the scalar part, Lscalar, noting that the matrix V has a block-wise upper/lower triangular form with constant diagonal elements, we obtain γ Gµ,ˆ m1···m4 ≡ ∂µˆDm1···m4 − 3 ǫγδ B[m1m2| ∂µˆB|δm3m4] , Gµ,ˆ m1···m6 ≡ ∂µˆBm1···m6 − 15 B[βm1m2 ∂µˆDm3···m6] + 15 ǫγδ B[m1m2 Bm3m4| ∂µˆB|δm5m6] . β β β γ e 4αn g µν ∂µ McMN ∂νMc MN g µν Mc MN McP Q ωµM P ωµN Q , (B.85) e − 2αn where the first term simply becomes 4αn g = µ McM N ∂ν Mc M N   | | g 4 4 1 µν g 4 ∂µ Gij ∂ν Gij − 4(n − 2) g µν ∂µ Gmn ∂ν Gmn − 4(n − 2) g µν 1 1 ∂µ mαβ ∂ν mαβ ∂µ ln |G| ∂ν ln |G| . ∂µ ln |G| ∂ν ln |G| +Lscalar (mat) (M-theory) ∂µ ln |G| ∂ν ln |G| | | where we used, e gµν • SL(5), SO(5, 5) (geometric): • E6, E7 (geometric): L(smcaalat)r = − | | G 21 gµν | | L(smcaalat)r = − 2 · 3! 1 |G| 2 | | R(g) + + 1 µν g 4 1 µν g 4 1 µν g 4 ∂µ Gij ∂ν Gij + ∂µ Gmn ∂ν Gmn + 1 µν g 4 1 µν g 4 ∂µ mαβ ∂ν mαβ + Lscalar (mat) g µν M M N McP Q ωµM c P ωνN Q , We can calculate the explicit form of Lscalar as follows: = |G| 21 gµν and Mc M N (mat) McP Q = M M N McP Q . c g µν Gi1i2i3, j1j2j3 ∂µ Ai1i2i3 ∂ν Aj1j2j3 , (B.89) Gei1i2i3, j1j2j3 Sµ i1i2i3 Sν j1j2j3 , L(smcaalat)r = −|Ge| 2 egµν 1 µν mαβ Gm1m2, n1n2 ∂µ Bmα1m2 ∂ν Bnβ1n2 , (B.93) | | G 2 µν 2 mαβ Gm1m2, n1n2 2! ∂µ Bmα1m2 ∂ν Bnβ1n2 + Gm1···m4, n1···n4 4! Gµ, m1···m4 Gν,n1···n4 , (B.94) mαβ Gm1m2, n1n2 ∂µ Bmα1m2 ∂ν Bnβ1n2 + + 2! Gm1···m4, n1···n4 4! mαβ Gm1···m6, n1···n6 6! Gµ, m1···m4 Gν,n1···n4 α β Gµ, m1···m6 Gν,n1···n6 , (B.95) • E7 (geometric): L(smcaalat)r = − 1 | | G 2 µν g 2 • SL(5) (non-geometric): • E7 (non-geometric): L(smcaalat)r = − 1 µν |e| g L(smcaalat)r = − 2 · 2! e µν 1 mαβ Gem1m2, n1n2 Qα,µ m1m2 Qβ,ν n1n2 , (B.96) • SO(5, 5), E6 (non-geometric): L(smcaalat)r = − 1 µν mαβ Gem1m2, n1n2 Qα,µ e 2! m1m2 Qβ,ν n1n2 4! + Gem1···m4, n1···n4 P m1···m4 Pν n1···n4 , µ (B.97) mαβ Gem1m2, n1n2 Qα,µ e 2! m1m2 Qβ,ν n1n2 4! + Gem1···m4, n1···n4 P m1···m4 Pν n1···n4 µ + e mαβ Gem1···m6, n1···n6 Qα,µ 6! m1···m6 Qβ,ν n1···n6 . (B.98) Internal (potential) part The internal part, or the potential part, consists of three terms Here, we choose the canonical section, (∂M ) = (∂i , 0, . . . , 0), where the index i represents i in the M-theory section or m in the type IIB section. In this case, the first and the third terms can be obtained as follows: Lpot ≡ L(p1o)t + L(p2o)t + L(p3o)t , ( 2 ) (3) (1) Lpot ≡ e Lpot ≡ −e 2 M 1 4αn 1 M . (B.99) (B.100) (B.101) (B.102) 4(n − 2)2 Gij ∂i ln |G| ∂j ln |G| − 2(n − 2) n Gij Gkl ∂i ln |G| ∂lGjk + + 4 2 1 Gij ∂i ln |g| ∂j ln |g| + 1 Gij ∂i ln |G| ∂j ln |g| + 1 On the other hand, as we show later, the second term L(p2o)t can be written as L(p2o)t = −e 2 Mc 1 MN ∂N Mc KL ∂LMcMK + ΔL(p2o)t , 1 2αn 1 − 2αn 1 4 Gmn Gpq ∂m ln |G| ∂pGnq + Gmn ∂mmαβ ∂nmαβ . (B.104) (B.105) (M-theory) (type IIB)  =      1 1 |G| 2                     1 1 n − 2 1 1 n − 2 Gij ∂kGil ∂j Gkl + Gmn ∂pGmq ∂nGpq + where ΔL(p2o)t does not include derivatives of metric. Gij Gkl ∂i ln |G| ∂kGjl + ΔL(p2o)t 1 2(n − 2)2 Gij ∂i ln |G| ∂j ln |G| 1 2(n − 2)2 Gmn ∂m ln |G| ∂n ln |G| where we used the formula R(G) = 1 Gij ∂iGkl ∂jGkl 1 Gij ∂kGil∂jGkl + ∂mmαβ ∂nmαβ M MN McP Q ωmM c P ωnN Q 1 Gij Gkl ∂i ln |G| ∂kGjl − 1 ∂i |G| 21 Gij Gkl ∂jGkl − ∂kGlj 1 Gij Gkl ∂i ln |G| ∂kGjl + 1 Gij ∂i ln |g| ∂j ln |G| + 1 1 ∂i |G| 21 Gij Gkl ∂jGkl − ∂kGlj , , − 2αn +Gmn 4 4 − 2 − 1 |G| 2 We thus obtain the potential as Thus, comparing this with (B.105), we obtain |G| 2 R(G) + Gij ∂igµν ∂j gµν + Gij ∂i ln |g| ∂j ln |g| − 2αn 1 |G| 2 2 e 1 and dropped the boundary term. Calculation of L(p2o)t. of ΔL(p2o)t. case, noticing VM i = δi M = (V −1)M i and (V T)iM = δi M = (V −T)iM , we obtain First, let us calculate L(p2o)t in the case of the conventional parameterization. In this Here, we show equation (B.105) and determine the explicit form L(p2o)t = − 2 e e e McP Q ωijP ωklQ = |G| 2 Gil Gjk McP Q ωijP ωklQ . We next calculate L(p2o)t in the non-geometric parameterization. In this case, we use the simplifying assumption [54] that requires any derivatives contracted with the dual potentials vanishes (e.g. βmn ∂m = 0). In our notation, it can be expressed as · · · VM i ∂i = · · · δM i ∂i , ∂i · · · VM i = ∂i · · · δM i V = V or V −1 , (B.110) where the ellipsis represent arbitrary tensors or derivatives. Using the simplifying assumption, we obtain L(p2o)t = − 2 M Mi ∂iM Kj ∂jMMK e e = − 2 MN ∂N Mc KL ∂LMcMK . (B.111) where, in the third equality, we used the simplifying assumption and McP i = δj ji P Mc , and in the fourth equality, we used (V −T)M j = δjM and Mkl = Mckl which are generally satisfied in the non-geometric parameterization. Comparing (B.111) with (B.105), we obtain ΔL(p2o)t = 0 in the non-geometric parameterization. Summary of the potential Lpot. To summarize, we obtained 1 |G| 2 R(G) + Gmn ∂m ln |g| ∂n ln |g| 1 4 4 1 4 1 4 + Gmn ∂mmαβ ∂nmαβ + Lpot (mat) Lpot 1 (mat) = −|G| 2 1 Gi1···i4, j1···j4 Fi1···i4 Fj1···j4 + 2 · 7! 1 Gi1···i7, j1···j7 Fi1···i7 Fj1···j7 , • geometric parameterization: Lpot (mat) = − 2 1 |G| 2 1 αn • non-geometric parameterization: Lpot Geij McMN McP Q ωiM P ωjN Q . More explicit form of L(pmotat) in each case is given as follows: • SL(5), SO(5, 5), E6 (geometric): Lpot (mat) = − 2 · 4! |G| 2 1 Gi1···i4, j1···j4 Fi1···i4 Fj1···j4 , Gij McMN ωiM P ωjN Q − Gil Gjk ωijP ωklQ McP Q , (B.113) . (mat) Lpot 1 = − 2 · 3! e Gij Gei1i2i3, j1j2j3 Sii1i2i3 Sj j1j2j3 , (B.117) • E6, E7 (non-geometric): (mat) mαβ Gm1m2m3, n1n2n3 Hmα1m2m3 Hnβ1n2n3 , (B.119) (mat) Lpot mαβ Gm1m2m3, n1n2n3 3! Hmα1m2m3 Hnβ1n2n3 + Gm1···m5, n1···n5 5! Gm1···m5 Gn1···n5 , (B.120) • E6(6), E7(7) (geometric): • SL(5) (non-geometric): (mat) Lpot 1 = − 2 · 2! e • SO(5, 5), E6(6) (non-geometric): (mat) 1 Gmn • E7(7) (non-geometric): (mat) = − e Gmn 1 m1···m6 Qβ, nn1···n6 . (B.123) mαβ Gmn Gem1m2, n1n2 Qα, m e m1m2 Qβ, nn1n2 , (B.121) − Gem1···m4, n1···n4 Pm 4! m1···m4 Pnn1···n4 , (B.122) mαβ Gem1m2, n1n2 Qα, m e m1m2 Qβ, nn1n2 mαβ Gem1m2, n1n2 Qα, m e m1m2 Qβ, nn1n2 − Gem1···m4, n1···n4 Pm 4! + e mαβ Gem1···m6, n1···n6 Qα, m 6! m1···m4 Pnn1···n4 2! 2! Fi1···i4 ≡ 4 ∂[i1Ai2i3i4] , Hmα1m2m3 ≡ 3 ∂[m1Bmα2m3] , B.4 Summary δ Gm1···m5 ≡ 5 ∂[m1Dm2···m5] − 15 ǫγδ B[m1m2 ∂m3Bm4m5] γ = 5 ∂[m1Cm2···m5] + 30 H[1m1m2m3 Cm4m5] . 35 2 Fi1···i7 ≡ 7 ∂[i1Ai2···i7] + A[i1i2i3 Fi4i5i6i7] , (B.124) In this appendix section, we evaluated several external terms in the EFT action, 4αn and the potential part, Lpot. Combining these, we obtain 1 L =|G| 2 R(g) + 4 1 gµν ∂µ Gij ∂νGij + 4 1 gµν ∂µ ln |G| ∂ν ln |G| + R(G) + 4 1 Gij ∂i ln |g| ∂j ln |g| + L(smcaalat)r + Lpot (mat) 1 ≡ |G| 2 R(G) + L(smcaalat)r + L(pmotat) . For example, for the E7(7) EFT in the geometric parameterization, this becomes Gi1···i4, j1···j4 Fi1···i4 Fj1···j4 + Gi1···i7, j1···j7 Fi1···i7 Fj1···j7 . (B.129) C C.1 Double-vielbein formalism for gauged DFT Parameterization from defining properties of double-vielbein The previous result from the Iwasawa decomposition provides the upper or lower triangular parameterization of the generalized vielbein. However, the triangulation breaks the full local structure group into the diagonal subgroup. If we decompose O(1, D−1 + dim G) as O(D − 1, 1) × O(dim G), then we choose the diagonal gauge-fixing by identifying the two local Lorentz groups, O(D − 1, 1) × O(1, D − 1) O(D − 1, 1)D . (C.1) Here, we shall construct the geometric parameterization and the non-geometric parameterization directly from the defining conditions of double-vielbein. This approach does not require any gauge-fixing condition and ensures manifest O(1, D − 1) × O(1, D−1 + dim G) covariance. Analogous to the ordinary O(D, D) case, double-vielbein for O(D, D + dim G) gauged DFT satisfies the following defining properties [127], VMcpV Mcq = ηpq , VMcpV¯ Mc q¯ˆ = 0 , V¯Mcpˆ¯ V¯ Mc q¯ = η¯ˆpˆ¯q¯ˆ , VMcpVNb p + V¯Mcp¯ˆ V Nb p¯ˆ = JˆMcNb , (C.2) (B.125) (B.126) (B.127) (B.128) where ηmn and η¯ˆpˆ¯q¯ˆ are O(1, D − 1) and O(D − 1, 1 + dim G) metric, respectively. The double-vielbein is then decomposed as Mc V m = VM m! Vαm and V¯ m¯ˆ = Mc V¯M m¯ V¯M a¯! V¯αm¯ V¯αa¯ . Note that the usual geometric parameterization is obtained by assuming that the upperhalf blocks of VM m and V¯M m¯ are non-degenerate and by identifying them as a pair of conventional vielbeins [127]. However, the non-degeneracy assumption can be relaxed in a consistent manner. Suppose that the upper-half blocks of V m and V¯ m¯ are given by V µm = (e−1)µm + β′µν eνm and V¯ µ m¯ = (e¯−1)µm + β′µν e¯νm , where eµ m and e¯µ m¯ are two copies of the D-dimensional vielbein corresponding to the same metric gµν eµmeνnηmn = −e¯µ m¯ e¯ν n¯η¯m¯ n¯ = gµν , and β′ is an arbitrary tensor. Then, V µm and V¯ µ m¯ are not guaranteed to be non-degenerate. Substituting the previous decomposition ansatz (C.3) and (C.4) into the defining properties (C.2), we find the most general parameterization that satisfy all the algebraic constraints (C.2) for VMˆ Vαm = √ m 1 1 2 2 V¯M m¯ = √ V¯αm¯ = √ 1 1 2 2 eµ m + Bµν′ (e−1)νm − β′νρ eρm ! (e−1)µm − β′µν eνm καβ(AT)βµ (e−1)µm − β′µν eνm − καβ(A˜T)βµ eµ m , e¯µ m¯ + Bµν′ (e¯−1)ν m¯ − β′νρ e¯ρ m¯ ! − β′µν e¯νm¯ (AT)αµ (e¯−1)µ m¯ − β′µν e¯νm¯ − (A˜T)αµ e¯µ m¯ , , , −Aµ a¯ + Bµν′ A˜νa¯! A˜µα (φT)αa¯ , V¯αa¯ = φa¯α + φa¯β(A˜T)βµ Aµα . and for V¯ m¯ˆ Mc Here, Bµν′ and β′µν are defined as in which Bµν and βµν are antisymmetric tensors. Bµν′ = Bµν + 1 α′Aµ α(AT)αν , β′µν = βµν 1 α′A˜µα (A˜T)αν , 2 − 2 (C.3) (C.4) (C.5) (C.6) (C.7) (C.8) However, if we assume that each blocks of VM m and V¯M m¯ are non-degenerate, this solution is over-parameterized. The physical degrees of freedom are determined by the coset O(D, D + dim G) O(D−1, 1) × O(1, D−1 + dim G) , and the associated number of degrees of freedom is given by 1 2 1 1 (2D + G)(2D + G − 1) − 2 D(D − 1) − 2 (D + G)(D + G − 1) = D2 + DG , where G denotes dim G. The D2 components arise from the gµν , Bµν or g˜µν , βµν , and to make up the parameterization. The geometric parameterization, which is for the conventional heterotic supergravity [128], is obtained by turning off βµν and A˜µ a¯, eµ m + Bµν′ (e−1)νm! e¯µ m¯ + Bµν′ (e¯−1)ν m¯! (e¯−1)µ m¯ Aµ α(φT)αa¯! 0 , , Vαm = √ (AT)αµ (e−1)µm , , V¯αm¯ = √ (AT)αµ (e¯−1)µ m¯ , 1 2 1 2 1 V¯αa¯ = √α′ (φa¯)α . Under the non-degeneracy assumption, one can show through a field redefinition that the geometric parameterization is essentially the same as the most general solution (C.6) and (C.7). On the other hand, if we assume that some of components of V µm or V¯ µ m¯ are vanishing, we can define an another class of non-geometric background, which cannot be related by field redefinition from geometric parameterization [129]. Using the relation the projection operators and double-vielbein: PMcNb = VMcmηmn(V T)nNˆ and PMcNb = V¯ m¯η¯m¯ n¯(V¯ T n¯ ¯ Mc ) Nb +V¯Mca¯ κa¯¯b(V¯ T ¯b , (C.13) ) Nb we construct a geometric parameterization for the projection operators as and and P = 1 2  g + α′A κ At + B′g−1(B′)t Aκ + B′g−1Aκ 1 + B′g−1 κAt + κ Atg−1(B′)t 1 + g−1(B′)t κ Atg−1A κ g−1A κ κ Atg−1  ,  g−1 P¯ = 1 2  −g − α′A κ At − B′g−1(B′)t −κAt − κAtg−1(B′)t 1 − g−1(B′)t −Aκ − B′g−1Aκ −κAtg−1Aκ − α2′ κ −g−1Aκ 1 − B′g−1 −κAtg−1  .  −g−1 (C.9) (C.10) (C.11) (C.12) (C.14) (C.15) 0 1 eµ m 2 (e−1)µm − β′µν eν m 1 Vαm = − √2 καβ(A˜T)βµ eµ m , ¯ m¯ = √ VM 1 − β′µν e¯ν m¯ √α′A˜µα (φT)αa¯ e¯µ m¯ , 1 , V¯α m¯ = − √2 καβ(A˜T)βµ e¯µ m¯ , V¯αa¯ = √α′ 1 φa¯α . The corresponding projection operators are constructed as P = −A˜κ + β′g˜A˜κ 1 − g˜β′T −κA˜T + κA˜Tg˜β′T g˜−1 + β′g˜β′T + α′A˜κA˜T P¯ = −κA˜Tg˜A˜κ − α2′ κ A˜κ − β′g˜A˜κ 1 + g˜β′T κA˜T − κA˜Tg˜β′T −g˜−1 − β′g˜β′T − α′A˜κA˜T by H = P − P¯ takes the form: Here, we used Kαβ = −(ta¯)αTκa¯¯bt¯bβ. In this parameterization, it follows that the projection operators satisfy the complete relation, J = P + P¯ and that the generalized metric defined H =  g + B′g−1(B′)t + AκAt κAt + κAtg−1(B′)t Aκ + B′g−1Aκ κAtg−1Aκ + α1′ κ g−1Aκ B′g−1  κAtg−1 .  g−1 Consider next the non-geometric parameterization. As for the geometric parameterization, it is simply given by turning off Bµν and Aµ a¯ while keeping β and A˜ in (C.6) and (C.7): HJEP07(21)5  ,   .   .  (C.16) (C.17) (C.18) (C.19) (C.20) (C.21) and and satisfied and that the generalized metric H = P − P¯ is expressed by Once again, in this parameterization, it follows that the complete relation J = P + P¯ is H = −κA˜Tg˜ κA˜Tg˜A˜κ + α1′ κ −A˜κ + β′g˜A˜κ −g˜β′T −κA˜T + κA˜Tg˜β′T g˜−1 + β′g˜β′T + α′A˜κA˜T One notes that this result is consistent with the parameterization in terms of the Iwasawa decomposition given in (5.34). We should remark that, ultimately, the double-vielbein formalism is imperative. For the bosonic case, the geometric parameterization and the non-geometric parameterization of double-vielbein, (C.11) and (C.18), respectively, are equivalent to the previous result constructed by coset representative, as they should. Even though these two approaches are consistent for the bosonic case, for introducing supersymmetry, the double-vielbein formalism is the most adequate approach [128, 130, 131]. Connection and curvature The gauge symmetry for gauged DFT is given by a twisted generalized Lie derivative which is defined by (LˆX V )McNb = (Lˆ0X V )McNb − f McPbQbXPbV QbNb − fNbPb QbXPbV Mc , Q b The Lˆ0X is the ordinary generalized Lie derivative defined in the un-gauged DFT by (Lˆ0X V )McNb = XPb∂ V Mc Pb Nb + (∂McX Pb − ∂PbXMc)V PbNb + (∂Nb XPb − ∂PbXNb )V Mc , Pb Lˆ0X d = XMc∂Mcd − 2 ∂McXMc , 1 where fMcNbPb are the structure constants for Yang-Mills gauge group. The gauge parameter XMc consists of ordinary generalized Lie derivative part and a Yang-Mills gauge symmetry part in an O(D, D + dim G) covariant way. As for the covariant differential operator of the gauge transformations (C.22), we present a covariant derivative which can be applied to any arbitrary O(D, D + dim G), Spin(D − 1, 1) and Spin(1, D − 1 + dim G) representations as follows DˆMc := ∂ Mc + Γ Mc + Φ Mc + Φ¯ Mc . where Φ Mcmn and Φ¯ are constructed in gauged DFT [83] Mcm¯ˆ n¯ˆ are spin-connections and ΓMcNbPb is semi-covariant connection which (C.22) (C.23) (C.24) QbRbSb . (C.25) (C.26) (C.27) (C.28) ΓPbMcNb = Γ0PbMcNb + δP QbP McRbP Sb + δPbQbP¯ Nb Mc 2 RbP¯ Sb fQbRbSb − 3 P + P¯ PbMcNb Nb QbRbSbf where Γ0P MN is the connection for ordinary DFT [127], Γ0PbMcNb = 2(P ∂PbP P¯)[McNb] + 2(P¯[Mc QbP¯ R Nb] b − P[Mc QbPNb]Rb)∂QbPRbPb 4 − D − 1 PP [McP Nb]Qb + PPb[McPNb] Q b ∂Qbd + (P ∂RbP P¯ [RbQb] , and PPbMcNb QbRbSb and P¯PbMcNb QbRbSb are rank-six projection operators PPbMcNb SbQbRb :=PPbSbP[Mc PPbMcNb SbQbRb :=P¯ SbP¯[Mc Pb Rb] + Rb] + 2 2 D − 1 PPb[McPNb] ¯ D − 1 PPb[McP ¯ Nb] [QbP Rb]Sb , which are symmetric and traceless, PPbMcNbQbRbSb = PQbRbSbPbMcNb = PPb[McNb]Qb[RbSb] , PPbMcNbQbRbSb = P¯QbRbSbPbMcNb = P¯Pb[McNb]Qb[RbSb] , ¯ Here the superscript ‘0’ indicates a quantity defined in the un-gauged DFT. The spin-connections are defined by using the semi-covariant derivative Although these are not gauge covariant, we can project out to the tensor part Φ Mcmn = V Nb m∂McVNbn + ΓMcNbPbV Nb mV Pbn , Φ¯ Mcm¯ˆ n¯ˆ = V¯ Nb mˆ¯ ∂McV¯Nb n¯ˆ + ΓMcNbPb V¯ Nb m¯ˆ V Pbn¯ˆ . (C.29) Φp¯mn , Φ¯ pm¯ n¯ , Φ¯ [a¯¯bc¯] , Φa¯mn , Φ¯pm¯ a¯ , Φ¯ p¯ˆp¯ˆm¯ , Φ[pmn] , Φpa¯¯b , Φ¯ p¯ˆp¯ˆa¯ . Φppm , Φ¯ [p¯m¯ n¯] , Φ¯ [p¯m¯ a¯] , Φ¯ [p¯a¯¯b] , (C.30) These will be the building block that the formalism uses. Various covariant quantities can be generated by using these spin-connections and their derivatives [83]. The heterotic DFT action is given by the generalized curvature tensor from semicovariant curvature tensor S McNbPbQb as 1 2 SMcNbPbQb = R McNbPbQb + RPbQbMcNb − ΓRbMcNb ΓRbPbQb , (C.31) where R McNbPbQb is defined from the standard commutator of the covariant derivatives R McNbPbQb = ∂McΓNbPbQb − ∂Nb ΓMcPbQb + ΓMcPbRbΓ NbRbQb − ΓNbPbRbΓ McRbQb + fRbMcNb R Γ bPbQb . (C.32) jection operators Then, the generalized curvature scalar is defined by contraction of SMcNbPbQb with the proS := 2P McNb P PbQbS McPbNbQb = 2 2∂mΦnmn − ΦmmpΦnnp − 2 3 Φ[mnp]Φmnp − 2 1 Φp¯mnΦp¯mn − 2 1 Φa¯mnΦa¯mn (C.33) − fpmnΦpmn − fp¯mnΦp¯mn − fa¯mnΦa¯mn . C.3 Nongeometric fluxes and action There are several approaches for constructing differential geometry of the gauged DFT [65, 83]. Here, we follow the so called semi-covariant formalism [83] which is well-suited for To define non-geometric fluxes, we adopt the non-geometric parameterizations of double-vielbein obtained in (C.17) and (C.18), and substitute them to the definition of generalized spin connection (C.29). Not all of the components of generalized spin connection are involved for defining heterotic DFT action. The relevant components of generalized spin-connection should be invariant under the generalized diffeomorphism for gauged DFT 10See appendix C for the concise review of double-vielbein formalism for gauged DFTs. Φm¯ np = + √ Φa¯mn = − 2 Φ[mnp] = + √ 1 1 2 2 1 2 Φmmn = + √ (e−1)µm ωµmn − 2∂nφ , In the above expression, the components of generalized spin connection comprise three kinds of fluxes that were introduced in (5.40). Consider now the non-geometric action of heterotic DFT in terms of the non-geometric fluxes. The action is given by the generalized curvature scalar S, which is defined in (C.33) in terms of the generalized spin-connections: Shet = Z e−2d 2S , 1 F˜µν a¯eµm eνn , 1 1 ωm¯ np − 2 Qρµν e¯ρm¯ eµn eνp + Qρµν eρneµp e¯ν m¯ , ω[mnp] − 2 Qρµν eρ[me|µ |ne|ν|p] . or twisted generalized Lie derivative. They define the non-geometric fluxes where D Exotic branes them by S = 2∂mΦnmn − ΦmmpΦnnp − 2 3 Φ[mnp]Φmnp − 2 1 Φp¯mnΦp¯mn − 2 1 Φa¯mnΦa¯mn . By substituting (C.34) into this action, one can show that (C.36) is equivalent to the previous non-geometric heterotic action (5.40). A defect brane refers to a codimension-two configuration in type II string theory. Denote b(nd, c)(n1 · · · nb, m1 · · · , mc, ℓ1 · · · ℓd) , for the configuration wrapped or smeared over the 7-torus [45, 46, 51] and thus has the mass: Mb(nd, c) = 1 gsn ls Rn1 · · · Rnb l b s Rm1 · · · Rmc l c s 2 constant and bcn ≡ b(nd=0, c) and bn ≡ b(nd=0, c=0). Here, Ri is the compactification radius in the xi-direction and gs is the string coupling In this paper, we consider compactification on shrinking tori. As an example, consider a 522(34567, 89)-brane (6.49) in the E7(7) EFT. In this case, xm (m = 4, . . . , 9) are compactified on a six-torus and x3 direction is a noncompact direction. In this case, the “522(34567, 89)brane” is a one-dimensional extended object with the tension, T = 1 2πgs2 ls2 R4 · · · R9 l 5 s R8R9 l 2 s 2 (C.35) (C.36) (D.1) (D.2) (D.3) type IIA theory  01 = D0  (MP(n) = Rn−1) 10(n) = F1(n) 21(n1n2) = D2(n1n2)  41(n1 · · · n4) = D4  52(n1 · · · n5) = NS5  61(n1 · · · n6) = D6 512(n1 · · · n5, n6) = KKM 613(n1 · · · n5, n7) 522(n1 · · · n4) 433(n1 · · · n4, m1m2m3)  253(n1n2, m1 · · · m5)  164(n1, m1 · · · m6) 073(, 3 · · · 9) 0(1, 6)(, n1 · · · n6, m1)  4  P M-theory   P(M)  P(n1) M2 = 23 M5 = 56 KKM = 619  (MP(n) = Rn−1)    M2(nM)  M2(n1n2)  M5(n1 · · · n4M)  M5(n1 · · · n5) 512 215 3  53(n1 · · · n5, m1m2M)  53(n1 · · · n4M, m1m2m3) 6  26(n1n2, m1 · · · m5M)  26(n1M, m1 · · · m6) 0(118, 7)  0(1, 7)(, 3 · · · 9, M)  0(1, 7)(, n1 · · · n6M, m1)   KKM(n1 · · · n6, M) KKM(n1 · · · n5M, n6)  KKM(n1 · · · n6, n7) We will still call it a point-like 522(34567, 89)-brane as its mass becomes that of the usual 522(34567, 89)-brane after further compactifying the x3-direction. A list of defect branes in the type IIA theory compactified on a seven-torus is collected in table 3. As shown in the table, each defect brane of the type IIA theory can be regarded as a reduction of a defect brane of the M-theory compactified on an eight-torus. By using the relation ls = RM−1/2 l131/2 and gs = RM3/2 l1−13/2 , where RM is the radius in the M-theory direction and l11 is the Planck length in eleven dimensions, a b(nd, c)-brane in the type IIA theory can be identified with a defect b(˜d, c)-brane n in the M-theory with the mass, Mb(nd, c) = ˜ l11 ˜ n Rn1 · · · Rnb RMb ˜ n l11 2 Rℓ1 · · · Rℓd 3 , 2 b + 2c + 3d − n + 1 2 RMd 3 (D.4) (D.5) (D.6) HJEP07(21)5 Here, the indices ni, mi, ℓi run over 3, . . . , 9, M, where M represents the M-theory direction. Note that the subscript n˜ of b(n˜d, c) is usually suppressed. Open Access. 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Kanghoon Lee, Soo-Jong Rey, Yuho Sakatani. Effective action for non-geometric fluxes duality covariant actions, Journal of High Energy Physics, 2017, 75, DOI: 10.1007/JHEP07(2017)075