A {\rm O}(p,q) tensors and identities

enhanced double field theory

Olaf Hohm, Edvard T. Musaev, Henning Samtleben
Simons Center for Geometry and Physics, Stony Brook University,

Stony Brook, NY 11794-3636, USA

ohohm@scgp.stonybrook.edu

Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut)

Am Mühlenberg 1, DE-14476 Potsdam, Germany

edvard.musaev@aei.mpg.de

Kazan Federal University, Institute of Physics, General Relativity Department,

Kremlevskaya 16a, 420111, Kazan, Russia

Univ Lyon, ENS de Lyon, Univ Claude Bernard Lyon 1, CNRS, Laboratoire de Physique, F-69342 Lyon, France

henning.samtleben@ens-lyon.fr

Abstract

Double field theory yields a formulation of the low-energy effective action of bosonic string theory and half-maximal supergravities that is covariant under the T-duality group O emerging on a torus . Upon reduction to three spacetime dimensions and dualisation of vector fields into scalars, the symmetry group is enhanced to O. We construct an enhanced double field theory with internal coordinates in the adjoint representation of O. Its section constraints admit two inequivalent solutions, encoding in particular the embedding of chiral and non-chiral theories, respectively. As an application we define consistent generalized Scherk-Schwarz reductions using a novel notion of generalized parallelization. This allows us to prove the consistency of the truncations of , and , supergravity on AdS .

## 1 Introduction

The T-duality property of closed string theory implies the emergence of an symmetry upon reduction of the low-energy effective actions on a torus . This holds for bosonic string theory but also for the maximal and half-maximal supergravities in and their lower-dimensional descendants. The O invariance is a ‘hidden’ symmetry from the point of view of conventional (super-)gravity in that it cannot be explained in terms of the symmetries present before compactification. Double field theory (DFT) is the framework that makes O manifest before reduction by working on a suitably generalized, doubled space [1, 2, 3, 4]. DFT can be defined for the universal NS sector consisting of metric, -field and dilaton, including bosonic string theory in and minimal supergravity in , but also for type II string theory [5, 6].

The group O is the universal duality symmetry arising for toroidal compactification of any string theory, but for special theories or backgrounds this symmetry may be enhanced further. For instance, for half-maximal supergravity coupled to vector multiplets (or heterotic string theory with ) the symmetry is enhanced to O, for which there is a DFT formulation [1, 7, 8]. Moreover, compactifications of half-maximal supergravity to also exhibit an SL duality, for which a DFT formulation has been obtained recently [9]. The case of interest for the present paper is the compactification to three spacetime dimensions. In this case, supergravity yields an O symmetry that, however, is enhanced to O for half-maximal and to E for maximal supergravity. Similarly, heterotic string theory exhibits an enhanced O duality [10], while the T-duality group of bosonic string theory on is enhanced to O. More generally, a string theory compactified on to three spacetime dimensions exhibits an O symmetry. This comes about because vector fields in three dimensions can be dualized into scalars which join the universal scalars to combine into a larger coset model [11, 12].

Our goal in this paper is to define an ‘enhanced double field theory’ that makes the larger duality group O manifest before compactification by working on a suitable extended internal space. More generally, we will define the theory for any pseudo-orthogonal group O. In this we closely follow the construction of the maximal E exceptional field theory [13] and the SL covariant formulation of Einstein gravity [14]. Concretely, we generalize the formulation of [15] to an enhanced double field theory, with external and (extended) internal coordinates, but the internal coordinates now live in the adjoint representation of O.1 The coordinates thus read with fundamental indices , subject to section constraints that generalize the level-matching constraint of DFT. A novel feature of this theory compared to the original DFT is that the section constraint has inequivalent solutions. As a consequence, we can embed in particular both the chiral and non-chiral theories in .

One of the conceptually most intriguing aspects of double and exceptional field theories with three external dimensions is that they require the inclusion of ‘dual graviton’ degrees of freedom. Indeed, in dimensional reduction the three-dimensional vector fields need to be dualized into scalars in order to realize the enhanced symmetry, and these vectors include the Kaluza-Klein vector fields originating from the higher-dimensional metric. Thus, their duals would be part of a higher-dimensional dual graviton, whose existence within a more or less conventional field theory is excluded by strong no-go theorems [16]. This is reflected in the observation that the generalized Lie derivatives supposed to unify the internal diffeomorphisms and tensor gauge transformations do not define a consistent gauge algebra for duality groups associated to three dimensions such as O [17]. Within exceptional field theory this obstacle shows up in the gauge transformations of the tensor hierarchy in any dimension , among the gauge symmetries associated to the -forms [18, 19, 20, 21]. Nevertheless, consistent double and exceptional field theories can be defined upon including an additional gauge symmetry (subject to somewhat unusual constraints) and its associated gauge potential. Three external dimensions are special because the need for additional gauge symmetries is apparent already at the level of the ‘scalar’ fields, and the additional gauge potential features among the ‘vectors’ participating in the gauging and the needed Chern-Simons action.

Concretely, the internal (generalized) diffeomorphisms parameterized by have to be augmented by new gauge symmetries with parameters that are subject to ‘extended sections constraints’ requiring that they behave like a derivative in that, e.g., . Nevertheless, this additional gauge parameter cannot be reduced to the derivative of a (singlet) gauge parameter, nor can the associated gauge vector be eliminated in terms of (derivatives of) the other gauge fields. In the present paper we will confirm that precisely the same construction applies to enhanced DFT with duality group O. Moreover, we use the opportunity to clarify the properties of these enhanced gauge symmetries by showing that on the space of ‘doubled’ gauge parameters one has a generalized Dorfman product that shares all properties familiar from, say, DFT. In particular, we will show that the Chern-Simons action can be naturally defined in terms of a similarly ‘doubled’ gauge field .

As one of our main applications we will use the O DFT to define consistent generalized Scherk-Schwarz compactifications as in [22, 23], employing a novel notion of generalized parallelization. For a generalized Scherk-Schwarz reduction, the compactification data are entirely encoded in a group matrix (‘twist matrix’) and a singlet , both depending only on the internal coordinates . For duality group O the twist matrix can be decomposed into fundamental matrices , and we define a ‘doubled’ twist matrix as for the gauge parameters and gauge fields:

 U¯M¯N ≡ (ρ−1UK[¯MUL¯N],−14ρ−1(∂KLUP¯M)UP¯N). (1.1)

Although at the level of elementary gauge fields and parameters the additional (covariantly constrained) components cannot be eliminated in terms of (derivatives of) the other fields, for the Scherk-Schwarz ansatz the corresponding component can be written in terms of derivatives of the twist matrix. Note that with its indices being carried by a derivative, the above form is manifestly consistent with the constraint. We will show that a twist matrix gives rise to a consistent compactification provided the doubled tensor (1.1) satisfies the following algebra with respect to the (generalized) Dorfman product :

 U¯M¯N ∘ U¯K¯L = −X¯M¯N,¯K¯L¯P¯QU¯P¯Q, (1.2)

where the are constant and define the embedding tensor of gauged supergravity. For the ‘geometric component’ this relation encodes the familiar Lie algebra of Killing vector fields. The above defines a notion of generalized parallelizability. Writing the compactification ansatz in terms of the twist matrix, for instance for the ‘doubled’ gauge vector as , we will show that the -matrices and hence the -dependence factors out homogeneously, thus proving consistency of the compactification. We will thereby prove the consistency of a large class of compactifications to three dimensions, including the truncations of , and , supergravity on AdS.

This paper is organized as follows. In sec. 2 we introduce the generalized diffeomorphisms, the generalized Dorfman product and the associated gauge vectors. Based on this, we construct in sec. 3 the complete enhanced DFT, and discuss its relation, upon solving the section constraint, to (super-)gravity theories in various dimensions. In sec. 4 we discuss generalized Scherk-Schwarz compactifications in terms of generalized parallelizability and analyze the ‘twist equations’ (1.2). These results are then applied in sec. 5 in order to establish the consistency of various Kaluza-Klein truncations to three dimensions. We conclude in sec. 6 with a general outlook on further applications and generalizations. Appendix A collects some identities, and in appendix B we give for completeness the details of the generalized Dorfman product for (doubled) vectors in the case of E.

## 2 O(p,q) generalized diffeomorphisms and tensor hierarchy

In this section we introduce the covariant generalized Lie derivatives that define generalized diffeomorphisms. Their structure follows [14, 13] and is conceptually different from theories with external dimension : they are defined with respect to a pair of gauge parameters, one of which is subject to further constraints. We clarify their algebraic structure by defining a generalized Dorfman product on the space of such pairs. This significantly simplifies the subsequent constructions, including the tensor hierarchy and the definition of the Chern-Simons action.

### 2.1 Generalized diffeomorphisms

We begin by spelling out our conventions for the group . Its fundamental representations is indicated by indices . Hence, objects living in the adjoint representation, like the coordinates , are labelled by index pairs:

 YM ≡ Y[MN] ≡ YMN. (2.1)

The structure constants are given by

 fMN,KLPQ = 8δ[P[MηN][KδQ]L], (2.2)

with the invariant metric , which we use in the following to raise and lower indices. In addition, for we use two more invariant tensors:

 sPQ,MNKL = 8δ(K[PηQ][MδL)N], (2.3)

which is symmetric under exchange of with , and

 APQRSKLMN ≡ δKLMNPQRS ≡ 124(δKPδLQδMRδNS±permutations), (2.4)

which is totally antisymmetric in the lower and upper sets of indices.

We can now define section constraints for the derivatives dual to the adjoint coordinates (2.1) in analogy to other double and exceptional field theories. In terms of the above defined tensors, we impose

 sMNKLPQ∂MN⊗∂KL = 0fMNKLPQ∂MN⊗∂KL = 0, AMNKLPQRS∂MN⊗∂KL = 0,ηMKηNL∂MN⊗∂KL = 0. (2.5)

Writing out the invariant tensors in terms of and Kronecker deltas it is easy to see that the section constraints are equivalent to

 ∂[MN⊗∂KL] = 0 = ηNK∂MN⊗∂KL, (2.6)

which is the form we will use from now on. We recall that as for higher-dimensional DFTs and ExFTs these constraints are meant to hold for arbitrary functions and their products, so that for instance for fields we impose and . The constraints simplify when the second-order differential operator acts on a single object as follows

 0 = ∂M[N∂PQ]A⇒∂MN∂PQA = −2∂M[P∂Q]NA. (2.7)

This can be verified by using that partial derivatives commute, .

We now turn to the definition of generalized Lie derivatives acting on arbitrary tensors. For a tensor in the adjoint representation it is defined as

 L(Λ,Σ)VMN ≡ ΛKL∂KLVMN+2(p+q−2)PPQRSMNKL∂PQΛRSVKL+λ∂KLΛKLVMN (2.8) −ΣPQfPQ,MNKLVKL,

where is the projector to the adjoint representation, explicitly given in (A.3), and we have also allowed for an arbitrary density weight . While these terms capture the generic structure of generalized diffeomorphisms [24, 25] the last term describes a local adjoint transformation with parameter which, subject to constraints, will be seen momentarily to be necessary for consistency. Its presence is typical for ExFTs with three external dimensions [14, 13]. The projector can be written in terms of the above invariant tensors, so that one obtains for the generalized Lie derivative

 L(Λ,Σ)VMN= (2.9) =ΛKL∂KLVMN−VKL∂KLΛMN+(λ−1)∂PQΛPQVMN +(6APQMNRSKL+116sPQ,MNUVsRS,KLUV+116fPQ,MNUVfRS,KLUV)∂PQΛRSVKL −ΣPQfPQ,MNKLVKL.

Let us emphasize that in the following we will always refer to as the density weight of a field, as opposed to the ‘effective weight’ emerging in the first line of (2.9).

In the following we will show that the generalized Lie derivatives form a closed algebra, which in turn fixes the coefficient in front of the projector in (2.8). More precisely, the for do not close separately, but closure follows upon including a ‘covariantly constrained’ parameter satisfying the same constraints as the derivatives :

 Σ[MN⊗∂KL] = 0 = ηNKΣMN⊗∂KL,etc.. (2.10)

Indeed, defining the gauge variations of a generic tensor field by the generalized Lie derivative, , and provided the section conditions (2.6) are satisfied, one finds for the gauge algebra

 [δ(Λ1,Σ1),δ(Λ2,Σ2)] = δ[(Λ2,Σ2),(Λ1,Σ1)],[(Λ2,Σ2),(Λ1,Σ1)] ≡ (Λ12,Σ12), (2.11)

with the effective parameters

 Λ12MN = 2Λ[2KL∂KLΛ1]MN−6AMNKLPQRSΛ[2PQ∂KLΛ1]RS −116(sPQRSUVsMNKLUV+fPQRSUVfMNKLUV)Λ[2PQ∂KLΛ1]RS, Σ12MN = −2Σ[2|MN∂KLΛ1]KL+2Λ[2KL∂KLΣ1]MN−2Σ[2|KL∂MNΛ1]KL (2.12) −18fRSKLPQΛ[2RS∂MN∂PQΛ1]KL.

In order to prove the above closure result it is convenient (and sufficient) to work with the Lie derivative acting on an object in the fundamental representation of , i.e., a vector , for which we write

 L(Λ,Σ)VM = ΛKL∂KLVM+KMN(Λ,Σ)VN+λ∂KLΛKLVM, (2.13)

where we defined

 KMN(Λ,Σ) = 4(∂[MKΛN]K+ΣMN). (2.14)

The action of the generalized Lie derivative on a tensor with an arbitrary number of fundamental indices is then defined straightforwardly, with a term for each index. In particular, one may verify that this definition reproduces the above form of the generalized Lie derivative acting on an adjoint vector .

Closure of the gauge transformations given by the generalized Lie derivatives (2.13) can now be proved by a direct computation. Specifically, one may quickly verify that closure is equivalent to the following condition on :

 KMN(Λ12,Σ12) = ΛKL2∂KLKMN(Λ1,Σ1)+KMK(Λ2,Σ2)KKN(Λ1,Σ1)−(1↔2), (2.15)

where and , given in (2.12), can be simplified by writing out the invariant tensors in terms of (2.2)–(2.4):

 ΛMN12 =2Λ[2KL∂KLΛ1]MN−6Λ[2[MN∂KLΛ1]KL]−4Λ[2K[M∂LN]Λ1]KL,Σ12MN =−2Σ[2|MN∂KLΛ1]KL+2Λ[2KL∂KLΣ1]MN−2Σ[2|KL∂MNΛ1]KL−Λ[2PK∂MN∂PLΛ1]KL. (2.16)

As a help for the reader and an illustration of the use of the section constraints (2.6) and the analogous constraints (2.10) on , we display the relevant terms involving . It is easy to see that, as a consequence of the constraints, they are linear in and vanish by use of the first constraint in (2.10) in the form

 0 ≡ 6Σ[MK∂NP] = 2ΣM[K∂|N|P]+2ΣN[P∂|M|K]−ΣMN∂KP−ΣKP∂MN. (2.17)

We will next discuss the transformation rules for partial derivatives of tensor fields. Let us compute the variation of the partial derivative of a vector of weight :

 δΛ,Σ(∂MNVK) =∂MN(ΛPQ∂PQVK+KKP(Λ,Σ)VP+λ∂PQΛPQVK) =ΛPQ∂PQ∂MNVK+∂MNΛPQ∂PQVK+KKP∂MNVP+λ∂PQΛPQ∂MNVK+∂MNKKPVP+λ∂MN∂PQΛPQVK. (2.18)

In order to compare this with the expression for a generalized Lie derivative, we use the section constraint as in (2.17), which yields

 ∂MNΛPQ∂PQVK = 2K[MP∂|P|N]VK−∂PQΛPQ∂MNVK. (2.19)

Thus, using this in (2.18), we have shown

 δΛ,Σ(∂MNVK) = L[λ−1](Λ,Σ)(∂MNVK)+∂MNKKPVP+λ∂MN∂PQΛPQVK, (2.20)

where the notation in the first term indicates that the generalized Lie derivative acts now with weight . [We will use similar notations in the following whenever it is convenient.] The additional terms involving second derivatives of the gauge parameter are referred to as non-covariant variations. The non-covariant variations for the (first) partial derivatives of arbitrary tensors take the analogous form, with a term for each index and one term proportional to involving (which, of course, vanishes for zero density weight).

We close this subsection by discussing trivial gauge parameters or gauge symmetries of gauge symmetries, i.e., choices of whose generalized Lie derivative (2.8) gives zero on all fields as a consequence of the constraints. The simplest example is

 ΛMN = ∂KLχ[MNKL], (2.21)

with . Indeed, the transport term vanishes by the section constraint, and as a consequence of the section constraint in the form (2.7). There are more subtle trivial gauge parameters, involving both and , parameterized by an arbitrary :

 ΛMN = ∂[MKχN]K,ΣMN = −14∂MN∂KLχKL. (2.22)

Again, triviality follows from the section constraints, which immediately imply that transport (and density) terms vanish, as well as by a quick computation with (2.7). Note that can be symmetric, in which case the parameter vanishes. In particular, this contains as a special case the familiar DFT trivial parameter via . There is a more general trivial parameter for the latter:

 ΛMN = ΩMN,with ΩMN covariantly % constrained, (2.23)

by which we mean , etc. Finally, there is a trivial parameter that generalizes (2.22) for antisymmetric. Indeed, the E case suggests that , where is covariantly constrained in the first index, is trivial. Here we find that

 ΛMN = Ω[MKN]K,ΣMN = −18∂MNΩKLKL−18∂KLΩMNKLwith ΩKLPQ {covariantly constrained} (% ???) in the first index pair, (2.24)

is indeed trivial.

### 2.2 Generalized Dorfman structure

We will now introduce a new notation that allows us to exhibit an algebraic structure on the space of gauge parameters , that is analogous to the Dorfman product appearing for DFTs and ExFTs with external dimension . We introduce for the gauge parameters the pair notation

 Υ ≡ (Λ,Σ), (2.25)

and we treat the first entry as an adjoint vector of weight and the second entry as a co-adjoint vector of weight zero that is covariantly constrained according to (2.10).2

Our goal is to define a product for such doubled objects such that its antisymmetric part coincides with the gauge algebra structure introduced in the previous subsection and its symmetric part is a trivial gauge parameter. It turns out these conditions are satisfied for

 Υ1∘Υ2≡ (Λ1,Σ1)∘(Λ2,Σ2)≡ (L[1]Υ1ΛMN2,L[0]Υ1Σ2MN+14ΛKL2∂MNK(Υ1)KL), (2.26)

where we used the notation (2.14) for . Moreover, the Lie derivatives in here act as defined in the previous subsection, with carrying weight one and weight zero. Specifically, using that is constrained one computes

 LΥ1Σ2MN = L[0](Λ1,Σ1)Σ2MN = ΛKL1∂KLΣ2MN+∂MNΛKL1Σ2KL+∂KLΛKL1Σ2MN. (2.27)

Note that, curiously, the ‘anomalous’ terms in the component of (2.26) have the order of 1 and 2 such that we cannot think of this as a deformed Lie derivative of w.r.t. , because enters explicitly. This ordering turns out to be crucial for the following construction.

We first verify that the antisymmetrized product defines the expected bracket:

 [Υ1,Υ2] ≡ 12(Υ1∘Υ2−Υ2∘Υ1) ≡ [(Λ1,Σ1),(Λ2,Σ2)] ≡ (Λ[1,2],Σ[1,2]), (2.28)

where

 ΛMN[1,2] ≡Λ1KL∂KLΛ2MN−3Λ1[MN∂KLΛ2KL]−2Λ1K[M∂LN]Λ2KL+4Σ[M1KΛ|K|N]2−(1↔2),Σ[1,2]MN ≡12(ΛKL1∂KLΣ2MN+∂MNΛKL1Σ2KL+∂KLΛKL1Σ2MN−ΛKL1∂MNΣ2KL−Λ1PK∂MN∂PLΛ2KL−(1↔2)). (2.29)

This is not quite of the form (2.16), but is equivalent to it upon adding trivial gauge parameters. Indeed, the gauge algebra is only well-defined up to trivial gauge parameters, and adding a trivial parameter of the form (2.24), with

 ΩMNKL = −4Σ1MNΛKL2−(1↔2), (2.30)

shows that the above indeed defines the gauge algebra bracket. Next we have to prove that the symmetric part of the product is trivial. We compute:

 12(Υ1∘Υ2+Υ2∘Υ1) = (3∂KL(Λ2[MNΛ1KL])+Ω[MKN]K+∂[MKχN]K,−18∂MNΩKLKL−18∂KLΩMNKL), (2.31)

where

 ΩMNKL≡ −4Σ1MNΛ2KL−2∂MNΛ1[KPΛ2L]P+(1↔2),χMN≡ 2Λ2(MKΛ1N)K. (2.32)

We infer that the result is indeed of the trivial form (2.21), (2.22) and (2.24).

Our next goal is to show that the product satisfies a certain Jacobi or Leibniz-type identity that will be instrumental for our subsequent construction. To this end it is convenient to extend the notion of generalized Lie derivative slightly so as to act on doubled objects of the same type as :

 LΥA ≡ Υ∘A. (2.33)

From the definition (2.26) of the product we infer that for the first component (the ’ component’) this reduces to the conventional generalized Lie derivative, but for the component there is an additional contribution due to the ‘anomalous’ term in (2.26). We will next prove, however, that these extended generalized Lie derivatives still close according to the same bracket:

 [LΥ1,LΥ2]A = L[Υ1,Υ2]A. (2.34)

Again, for the component this reduces to the closure of standard generalized Lie derivatives established in the previous subsection, but for the component one obtains additional contributions, so that after a brief computation

 [LΥ1,LΥ2]A = (L[Υ1,Υ2]AMN,L[Υ1,Υ2]BMN+14AKLLΥ1(∂MNK(Υ2)KL)−(1↔2)). (2.35)

On the other hand, the right-hand side of (2.34) equals

 L[Υ1,Υ2]A = (L[Υ1,Υ2]AMN,L[Υ1,Υ2]BMN+14AKL∂MNK([Υ1,Υ2])KL). (2.36)

In order to prove that the above two right-hand sides are actually identical we use

 ∂MN(LΥ1K(Υ2)KL) = LΥ1(∂MNK(Υ2)KL)+2∂MNK(Υ1)[KPK(Υ2)|P|L]. (2.37)

This follows as in (2.20), using that the Lie derivative acts on , defined in (2.14), as a tensor of zero density weight. With this one can quickly establish

 Extra open brace or missing close brace (2.38)

where

 XKL ≡ LΥ1K(Υ2)KL−K(Υ1)KPK(Υ2)PL−(1↔2)−K([Υ1,Υ2])KL. (2.39)

Using (2.15) it is easy to see that this is actually zero, completing the proof of (2.34).

We now derive a Leibniz identity for the product from the closure relation (2.34). We first note that for trivial the extended generalized Lie derivative (2.33) vanishes:

 Υtrivial⇒Υ∘A = 0. (2.40)

This holds by definition for the component and for the component follows from the fact that the entering the anomalous term of (2.26) is zero for trivial parameters. Thus, using that the symmetric part (2.31) of the product is trivial, the closure relation can also be written as

 [LΥ1,LΥ2]A = LΥ1∘Υ2A. (2.41)

Using (2.33) twice we can write this as

 Υ1∘(Υ2∘A)−Υ2∘(Υ1∘A) = (Υ1∘Υ2)∘A. (2.42)

Upon renaming the doubled objects entering here and reordering the equations, we have thus established the Leibniz identity

 A∘(B∘C) = (A∘B)∘C+B∘(A∘C). (2.43)

Let us finally note that formally all relations that hold for conventional Dorfman products are then also satisfied for the product defined here, except that the relevant objects are doubled in the sense of (2.25). In particular, the Jacobiator of the bracket (2.28) can then be proved to be trivial in precise analogy to the original DFT and ExFTs for E with .

### 2.3 Gauge fields, tensor hierarchy, and Chern-Simons action

We will now introduce gauge fields that, roughly speaking, take values in the algebra given by the Dorfman product defined above. More precisely, we introduce gauge fields and and combine them into a pair or doubled object as above:

 Aμ ≡ (AμMN,BμMN). (2.44)

In particular, carries weight one and weight zero while being constrained according to (2.10), i.e.

 Bμ[MN⊗∂KL] = 0 = ηNKBμMN⊗∂KL,etc.. (2.45)

Their transformation rules receive inhomogeneous terms as to be expected for gauge fields. Indeed, in analogy to Yang-Mills theories we postulate the following gauge transformations w.r.t. doubled parameters (2.25)

 δΥAμ ≡ DμΥ, (2.46)

where we defined the covariant derivative

 Dμ = ∂μ − Aμ∘. (2.47)

It should be emphasized that the covariant derivative as written is only defined on doubled objects, which is indicated by the mathfrak notation. We can, however, define covariant derivatives for any field with a well-defined action of the generalized Lie derivatives in sec. 2.1. For a generic (undoubled) tensor field we define

 DμV ≡ ∂μV−L(Aμ,Bμ)V. (2.48)

For instance, for a vector of zero weight this reads explicitly

 DμVM ≡ Missing or unrecognized delimiter for \right (2.49) −4ηMLBμLKVK.

Despite not being a doubled object we can prove in an index-free fashion that the covariant derivative indeed transforms covariantly:

 δ(Λ,Σ)(DμV)= δΥ(∂μV−LAμV) = ∂μ(LΥV)−L∂μΥ−Aμ∘ΥV−LAμ(LΥV)= L∂μΥV+LΥ(∂μV)−L∂μΥV+LAμ∘ΥV−LAμ(LΥV)= LΥ(∂μV−LAμV)+([LΥ,LAμ]+LAμ∘Υ)V= LΥ(DμV), (2.50)

where we used (2.41) in the last step. This proves the covariance of the covariant derivative under combined tensor transformations given by generalized Lie derivatives and vector gauge transformations, whose component form is with (2.46) and (2.26) found to be

 δ(Λ,Σ)AμMN= DμΛMN,δ(Λ,Σ)BμMN= DμΣMN−ΛKL∂MNBμKL−ΛKL∂MN∂KPAμLP, (2.51)

which of course may also be verified with a direct component computation. This clarifies the seemingly ‘non-covariant’ terms in the gauge transformations of , first identified for the SL and E cases [14, 13], and explains them as a consequence of the necessary ‘anomalous’ terms of the Dorfman product.

Let us next discuss the gauge structure and invariant field strengths for the gauge vectors. With the Leibniz identity (2.43) it is straightforward to compute the commutator of two gauge transformations (2.46):

 [δΥ1,δΥ2]Aμ = δΥ1∘Υ2Aμ+2{Υ[1,DμΥ2]}, (2.52)

where we introduced the notation

 {A,B} ≡ A∘B + B∘A. (2.53)

We infer from (2.52) that the vector gauge transformations do not quite close, but the failure of closure involves the symmetrized product, which is trivial, c.f. (2.31). This implies that the extra terms can be absorbed into higher-form (here 2-form) gauge transformations, as is standard in the tensor hierarchy. Thus, the combined one- and two-form transformations close. Another way to see the need for 2-forms is by inspection of the naive field strength for the gauge vectors:

 Fμν ≡ ∂μAν − ∂νAμ − [Aμ,Aν] + ⋯, (2.54)

where the ellipsis denotes 2-form terms to be added momentarily. In components, writing , this reads

 FμνMN ≡ 2∂[μAν]MN−2A[μKL∂KLAν]MN+6A[μ[MN∂KLAν]KL]−4A[μK[M∂N]LAν]KL, GμνMN ≡ 2D[μBν]MN−A[μKP∂PQ∂MNAν]KQ. (2.55)

We consider now the general variation under an arbitrary , for which we compute

 δFμν = 2D[μδAν]+{A[μ,δAν]}+⋯. (2.56)

We do not quite obtain the expected identity with only the covariant curl of , but the additional terms are trivial and can hence be absorbed into the 2-forms. More precisely, 2-forms are introduced in precise correspondence with the trivial terms in the symmetrized product (2.31). We thus define the full field strength to be