Gauge $\times$ Gauge $=$ Gravity on Homogeneous Spaces using Tensor Convolutions

A definition of a convolution of tensor fields on group manifolds is given, which is then generalised to generic homogeneous spaces. This is applied to the product of gauge fields in the context of `gravity $=$ gauge $\times$ gauge'. In particular, it is shown that the linear Becchi-Rouet-Stora-Tyutin (BRST) gauge transformations of two Yang-Mills gauge fields generate the linear BRST diffeomorphism transformations of the graviton. This facilitates the definition of the `gauge $\times$ gauge' convolution product on, for example, the static Einstein universe, and more generally for ultrastatic spacetimes with compact spatial slices.


Introduction
We introduce a convolution of tensor fields on group manifolds, which to the best of our knowledge has not been treated in the literature to date. This is generalised to homogeneous spaces, extending the special case of S 2 given in [1]. These convolutions are then applied to the notion of 'gravity = gauge × gauge'. In particular, it is shown that the symmetric convolution of two gauge potentials yields a graviton. The gauge symmetry BRST transformations generate via the convolution the diffeomorphism BRST transformations of the graviton, to linear order. This allows us to apply the field theoretic 'gravity = gauge × gauge' construction of [1][2][3][4][5][6][7][8][9] on curved spacetime backgrounds, such as the D = 4 spacetime dimensional Einstein universe. We also note that the convolution of tensor fields on homogeneous spaces is of intrisic interest and may have broader applications, cf. for example [10][11][12] and the references therein. Gravity = gauge × gauge Let us briefly review the theme of 'gravity = gauge × gauge'. For details and more complete references see the reviews [13,14]. From the Kawai-Lewellen-Tye (KLT) relations of string theory [15], which relate closed string tree amplitudes to the sums of products of open string tree amplitudes, we learn that the tree amplitudes of perturbatively quantised N = 0 supergravity (Einstein-Hilbert gravity, coupled to a dilaton ϕ and a Kalb-Ramond 2-form B) are the 'square' of Yang-Mills amplitudes. While the KLT relations are intrinsically tree level, it was shown in certain examples that this relation could be extended to low loop orders [16,17]. This programme was dramatically advanced with the Bern-Carrasco-Johansson colour/kinematic (CK) duality conjecture [18]: the gluon amplitudes can be cast in a form such that their 'kinematic numerators' (Lorentz invariant polynomials of momenta and polarisation tensors) obey the same relations as their 'colour numerators' (polynomials of the gauge group structure constants). CK duality has been shown to hold at tree level from a number of points of view [19][20][21][22][23][24][25][26][27]. It remains conjectural at loop level where it quickly becomes difficult to test [28,29], although there are numerous highly non-trivial examples [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46].
If CK duality is satisfied by a Yang-Mills amplitude integrand, then its double-copy is an amplitude integrand of N = 0 supergravity [30,47]. This suggests a fundamental relationship between gauge theory and gravity, at least perturbatively, and reveals new features and puzzles regarding the properties of perturbative quantum gravity [35,37,44]. This motivates some natural questions. Does CK duality and/or the double-copy hold to all orders in perturbation theory; is the double-copy special to amplitudes; can it be taken beyond perturbation theory; are there applications beyond the computation of gravity amplitudes; what are implications for quantum gravity? There are many approaches that one might take in addressing these challenges. For instance, there is an ambitwistor string approach to CK duality and the double-copy, related to the scattering equation formalism for the double-copy [48][49][50][51]. Beyond perturbation theory, for particular (e.g. Kerr-Schild) spacetimes there is a non-perturbative classical double-copy of Yang-Mills solutions [52][53][54][55][56], with an elegant variation relating the square of Yang-Mills field strengths to the Weyl tensor [55,[57][58][59][60]. As to applications, there is a vigorous and promising programme to bend amplitudes and the double-copy to the problem of classical black hole scattering in the context of gravity wave astronomy [61][62][63][64][65][66]. CK duality and the double-copy even find applications in fluid dynamics [67]. In [68], a double copy for asymptotic symmetries led to the discovery of a new symmetry for self-dual YM at null infinity, identified as the single copy of gravitational superrotations.
Two ideas that are central to our present discussion are (i) that the double-copy can be applied off-mass-shell to the fields of two gauge theories [2,47] and (ii) that this should be extended to all the fields of the BRST complex, including the longitudinal and ghost modes [4]. Regarding (i), it was shown that CK duality for physical gluon tree-level amplitudes can be made manifest order-by-order at the Lagrangian level [47,69]. One can then doublecopy the CK dual Lagrangian itself, yielding a theory that produces the correct tree level scattering amplitudes for N = 0 supergravity, as was shown to six points in [47]. In [2,4] an a priori off-shell convolution product of the fields, including those of the BRST complex, of two independent gauge theories was introduced. For two pure BRST Yang-Mills theories in flat space this yields the BRST complex and linear BRST transformations of perturbative N = 0 supergravity [4]. Combined with the Lagrangian double-copy [47] the pure Einstein-Hilbert action to cubic order was derived from that of Yang-Mills theory using the convolution product [6] 1 . The chief advantages of the convolution, including the extra BRST fields, are that (i) the ghost sector allows the dilaton to be truncated without imposing further constraints on the graviton [6,71] and (ii) the Yang-Mills gauge choice determines the N = 0 supergravity gauge choice to linear order, removing the ambiguities inherent in, for example, the double-copy of gauge dependent solutions. In [72,73] it was shown that tree-level CK duality holds for amplitudes involving all states of the BRST Fock space, including the unphysical longitudinal gluon and ghost modes, and that this can be made manifest in a purely cubic Yang-Mills BRST-Lagrangian 2 . It was then shown that its Lagrangian double-copy yields a BRST-Lagrangian that is pertubatively quantum equivalent to N = 0 supergravity to all orders, tree and loop [72,73]. A direct corollary is that all tree and loop Yang-Mills amplitude integrands can be written in a form that doublecopies correctly, i.e. yields a bona fide N = 0 supergravity amplitude. Let us emphasise that key to all of the preceding discussion was the derivation of the linear BRST operator of the double copy theory from the BRST operators of the gauge theory factors [4].

Curved backgrounds
Everything till now has assumed perturbation theory around a flat spacetime (and gluon) background. It is natural to ask what of the double-copy survives in curved spaces, or at least in some suitable class of curved spaces. There are various possibilities. One could consider a flat gluon background on a curved spacetime, a curved gluon background on flat spacetime or both non-trivial gluon and spacetime backgrounds. For example, CK duality and the double-copy in a curved 'sandwich' plane-wave gluon background was considered in [74,75]. The Kerr-Schild version of the double copy also admits a formulation on curved backgrounds [76,77]. There have also been generalisations to AdS and conformal correlators [78][79][80][81][82].
On the other hand, in [1] a convolution product for the BRST complexes of two gauge theories on a spatial sphere (with trivial gluon background) was introduced. Trivially including a time dimension, this provided a convolution product in the D = 3 Einstein universe. It was shown that the linear BRST transformations correctly double-copy to those of perturbative N = 0 supergravity on an D = 3 Einstein universe background.
It was noted in [1] that the convolution of tensor fields on S 2 relied on rather generic properties of homogeneous spaces. Encouraged by this observation, in the present contribution we consider the convolution product for tensor fields on general Riemannian homogeneous spaces. Trivially including a time dimension facilitates the double-copy of the BRST transformations on a broad class of spacetimes such as the D = 4 Einstein universe.
Structure We proceed as follows. In subsection 2.1 we recall the essentials of the BRST formalism in the context of Yang-Mills theory and N = 0 supergravity. For complete details, see [73]. In subsection 2.2 we summarise what is required of the convolution product with respect to the goal of generating the linear diffeomorphism BRST transformations from the product of the Yang-Mills BRST transformations. Specialising to ultra-static spacetimes, reduces the problem to defining a convolution on Riemannian homogeneous manifolds, as discussed in subsection 2.3.
Our construction of a tensor field convolution on homogeneous spaces relies upon a simpler convolution on group manifolds G, which may be regarded as special class of homogeneous manifolds G ∼ = (G × G)/G. Accordingly, we first formulate in section 3 the tensor convolution on compact Lie groups. The features of group manifolds essential to the formulation of our tensor field convolution are reviewed in subsection 3.1. The convolution is introduced in subsection 3.2 and it properties under differentiation are determined in subsection 3.3. The convolution of functions on group manifolds is straightforward and well-known. It essentially relies on the existence of a G-bi-invariant measure (the Haar measure). The obvious obstruction to extending this construction to tensor fields is the need to compare tensors at different points on G. However, a basic property of Lie groups is that the left and right multiplication diffeomorphisms generate a set of left and right invariant basis vectors on T g G fo all g ∈ G, starting from some choice of basis of vectors at the identity T e G. Using this observation we introduce maps that generate left-and right-invariant vector fields from any vector at any point. This allows us to translate any vector field in a unique manner, facilitating a convolution of tensor fields as defined in subsection 3.2. Having introduced the convolution, its properties are developed. In particular, it is shown in subsection 3.3 that the symmetrised covariant derivative has the same properties as the flat space derivative acting on the flat space convolution when act on the convolution of functions with 1-forms. This implies that the BRST transformations of the graviton follow from those of the Yang-Mills gauge potentials to linear order. Finally, the generalisation of the familiar Convolution Theorem is given in subsection 3.4 Having treated the special case of Lie groups, we turn our attention to Riemannian homogeneous manifolds M ∼ = G/H. In subsection 4.1 we review the basics of homogeneous spaces. The group manifold convolution is then lifted to one defined on any Riemannian homogeneous space in subsection 4.2. The basic idea is to regard G as the fibre bundle H → G → M , using the projection π : G → M to 'pullback' the convolution of p-form fields on M to the convolution defined on G and then projecting back down to M . The action of the symmetrised covariant derivative on the symmetrised convolution is considered in subsubsection 4.2.2 and shown to have the same properties as the group manifold case, implying that the graviton BRST transformations are generated correctly.

BRST Review
The 'square' of pure Yang-Mills theory ought to correspond to the universal Neveu-Schwarz sector of the α ′ → 0 limit of closed string theories, N . Aside from the metric g and cosmological constant Λ, we have the dilaton ϕ and the Kalb-Ramond (KR) 2-form B with field strength H = dB. This is sometimes referred to as N = 0 supergravity, for short. For a Minkowski background this follows from the relationship between the tree-level BCJ double-copy and the KLT relations of string theory [15].
We shall be concerned with relating the linearised BRST transformations of Yang-Mills theory to those N = 0 supergravity. So, let us briefly review the linearised BRST formalism here. Of course, for free gauge and gravity theories, the ghosts decouple and there is no need to pass through BRST. However, it is nonetheless important in the context of squaring Yang-Mills theory [2,4,6,72,73].
Yang-Mills theory The BRST complex consists of the ghost number gh = 0 gauge potential A, its gh = 1 ghost c, and the trivial pair of the gh = 0 Nakanishi-Lautrup auxiliary field b and gh = −1 antighostc. The linearised gh = 1 off-shell nilquadratic Q 2 YM = 0 BRST transformations are given by The physical states are contained in the cohomology of Q YM . For a given gauge-fixing condition, the linearised Q YM -invariant BRST action can be written S lin YM, BRST = tr where Ψ A is the ghost number −1 gauge-fixing fermion with where g and φ 0 are a background metric and dilaton solving the Einstein and scalar equations of motion and for notational convenience we consider κ as our parameter. In fact, we shall consider arbitrary background metrics through the inclusion of arbitrary sources, but they will be treated only implicitly. Then where L FP is the Fierz-Pauli action quadratic in h µν and L[g, φ 0 , Λ, ϕ, h] is linear in the fluctuations ϕ and h. The Fierz-Pauli action has a gauge symmetry (the residue of diffeomorphism invariance upon taking the limit (2.10)), δh = ∇ξ, (2.11) or in components where ∇ µ is the covariant derivative with respect to the Levi-Civita connection of the background metric g.
Here we have introduced the coordinate independent symmetrised derivative ∇. For any function f it is defined by ∇f := df ∈ T * M , where d is the exterior derivative, and for any ω ∈ T * M we define it as for any X, Y ∈ T M . Note that the symmetrised covariant derivative ∇ has the same symbol as the usual covariant derivative ∇ X with respect to X ∈ T M (X µ ∇ µ in components), but any ambiguity between the two can be resolved by the fact that the symmetrised covariant derivative does not take a subscript argument. We work in Einstein frame so the dilaton is a scalar. The Fierz-Pauli BRST complex consists of the ghost number gh = 0 gauge potential h, the 1-form diffeomorphism gh = 1 ghost ξ, and its accompanying 1-form trivial pair of the gh = 0 Nakanishi-Lautrup auxiliary field π and gh = −1 diffeomorphism antighostξ. The Kalb-Ramond 2-form has a reducible gauge symmetry. In addition to the ghost number gh = 0 gauge potential B, there is the 1-form gh = 1 ghost Λ and the scalar gh = 2 ghost-for-ghost λ, and their accompanying 1-form W,Λ and 0-form w,λ trivial pairs and a final gh = 0 ghost η.
The linearised gh = 1 off-shell nilquadratic BRST transformations are given by and Q N =0 φ = 0. For the Kalb-Ramond sector see for example [4]. The physical states are contained in the cohomology of Q N =0 . We focus on the Fierz-Pauli sector. For a given gauge-fixing condition, the linearised Q N =0 -invariant BRST action can be written where Ψ h is the ghost number −1 diffeo-gauge-fixing fermion for gauge-fixing function G[h, ϕ] with Gaussian width ζ ∈ R. A typical choice of gaugefixing function is de Donder gauges G[A] = div(h − 1 2 gtrh), where the trace is taken with respect to the the background metric g. This yields which upon eliminating π gives with

The Goal: Diffeomorphism BRST from Yang-Mills BRST
We would like to define a product ' * ' of fields (i.e. sections of bundles) and set 3 Note that since h ∈ Sym(M ), where Sym(M ) denotes symmetric (0, 2)-tensor fields on M , the product of two 1-forms, A * Ã, must output a symmetric tensor. In order to recover the symmetries of linearised gravity, (2.14a) and (2.14b), using the symmetries of the two gauge fields, we require where Q denotes the double-copy transformation that acts either as Q YM or Q N =0 , depending on what fields it acts on.
Since QA = dc = ∇c, it suffices if the product * satisfies for any scalars f, g, and any 1-form ω. Assuming this we find and as desired. Note, in the last derivation we have used Qc = Qc = 0 and the anticommutativity of Q and the ghost fields. One might also ask for the derivative rule for any scalar f and 1-form ω. Such a rule would enable the recovery of the dilaton transformation [1]. Below we will see that this rule is more complicated than those in (2.22), and hence we leave it as an open question for future investigations. We also note that for the Kalb-Ramond 2-form we have where d is the exterior derivative. The requirement analogous to the function and 1-form case of (2.22) is then Surprisingly, this also proves to be less straightforward and is left for future work. Equation (2.22) can be seen as the analog of the familiar Leibniz failure property of convolutions on pseudo-Euclidean spaces ∂(X * Y ) = ∂X * Y = X * ∂Y , but for the symmetrized covariant derivative ∇ on M . (2.22) is the main goal, and the rest of the paper will be dedicated to constructing a definition of * satisfying (2.22) on a wide class of spacetime geometries.

From Spacetime to Space
Specifically, we shall be concerned with D = d + 1 ultrastatic spacetimesM = R × M , where the Riemannian geometry on M is held fixed over time. This essentially reduces the problem to defining a spatial product on the d-dimensional Riemannian space M . Hence we shall be concerned only with d-dimensional objects, and for notational convenience all (d + 1)-dimensional objects will be hatted henceforth (this is the first and last time we shall use this notation, for reasons we make clear momentarily). Similarly, we will hat the product onM , * → * , and the symmetrised covariant derivative ∇ →∇. Any occurrence of the un-hatted ' * ' will denote an, as of yet, undefined product over the spatial manifold M , and any occurrence of the un-hatted '∇' will refer to the symmetrised covariant derivative for the Levi-Civita connection on M .
Given the ultrastatic nature ofM we can make the convenient time-space split of any Here ω 0 is a function overM , and as such can vary in space as well as time. ω 1 is spatially-directed 1-form, i.e. ω 1 (∂ t ) = 0, which again can vary in space and time. In this way we can equivalently think ofω as the pair Here Ω p R (M ) denotes the space of 1-parameter families of p-forms. We will write ω i [·] for the i = 0, 1 component of this pair. Note that one can go backwards and constructω given such a pair, and hence this mapping between Ω 1 (M ) and Ω 0 is a bijection. We can connstruct similar bijections for all (0, p)-tensor fields. For functions is it trivial. Given anyf ∈ Ω 0 (M ), the corresponding 1-parameter family of functions on M , denoted x) for any t ∈ R and x ∈ M . The other relevant case for our purposes is the bijection for Sym(M ). For anyα ∈ Sym(M ), the time-space split can be written in the formα where ∨ denotes the symmetric tensor product, i.e. α∨β := α⊗β +β ⊗α. Here α 0 ∈ Ω 0 (M ) is a function onM , α 1 ∈ Ω 1 (M ) is a spatially-directed 1-form onM , i.e. α 1 (∂ t ) = 0, and α 2 ∈ Sym(M ) is a spatially-directed symmetric (0, 2)-tensor onM , i.e. α 2 (∂ t , X) = α 2 (X, ∂ t ) = 0 for any vector X ∈ TM . It is then clear thatα is equivalent to the triple where Sym R (M ) denotes the space of 1-parameter families of symmetric (0, 2)-tensor fields on M . We similarly write α i [·], (i = 0, 1, 2) for the corresponding component of the triple.
We are now ready to define the spacetime product * in terms of some, as of yet, undefined spatial product * . For a pair of functionsf ,ĝ ∈ Ω 0 (M ) we first find the corresponding The spacetime product,f * ĝ ∈ Ω 0 (M ), can then be defined through a specification of the corresponding 1-parameter family (f * ĝ)[·] ∈ Ω 0 R (M ), which, for any t ∈ R, we define as where, by assumption, * is defined between functions on M , and hence the integrand on the RHS is well-defined. Given the bijection between Ω 0 (M ) and Ω 0 R (M ), this specification of the 1-parameter family uniquely defines the productf * ĝ ∈ Ω 0 (M ). Modulo the definition of * , the above equation should be familiar to the reader as the usual Euclidean convolution of functions.
Given two 1-formsα,β ∈ Ω 1 (M ), we wish to go further and define the spacetime productsf * α ∈ Ω 1 (M ) andα * β ∈ Sym(M ). Both can be uniquely defined by specifying the corresponding tuples; a pair Ω 0 for the latter. We first compute the components α i [·] and β j [·], where i, j = 0, 1. We then define the components of the respective tuples, for all t ∈ R, as where the brackets denote the usual symmetrised sum: T (ab) = 1 2 (T ab + T ba ). Note that the * products in the integrands on the RHS's are between tensor fields on M , and hence are well-defined by assumption. One can of course extend the above definition to higher rank tensors, but the expressions becomes more complicated and will not be needed for our purposes.
Given the above definition of * , we can now show that it satisfies (2.22) for∇, assuming the spatial product * satisfies (2.22) for the spatial symmetrised covariant derivative ∇. To see this we first need to know how∇ acts on a functionf ∈ Ω 0 (M ) and a 1-formω ∈ Ω 1 (M ) in terms of the components f [·] ∈ Ω 0 R (M ) and ω i [·] ∈ Ω i R (M ) (i = 0, 1). Recall that∇f =df , whered is the exterior derivative onM . Asdf is a 1-form on M , it can be specified by a pair Ω 0 , for all t ∈ R, gives the correct specification, where d is the exterior derivative on M . Note that we may write (df ) 1 , as ∇ ≡ d on Ω 0 (M ). If we define the pair of operators D 0 := ∂ t and D 1 := ∇, we can write the components of∇f succinctly as Since∇ω is a symmetric (0, 2)-tensor field, it is uniquely specified by a triple Ω 0 for t ∈ R and i, j = 0, 1, gives the correct specification of the components.
(2.22) can now be verified by direct computation. For two functions we have We then note that the D i can be moved onto either argument of the * product in the integrand. For i = 1 (D 1 = ∇) this is true by linearity of the integral and by our assumption that * satisfies (2.22) for ∇.
, as it is the only part of the integrand that explicitly depends on t, and we have assumed bilinearity of the * product. To show that it can be moved onto f [t ′ ] one can run the same argument again following the change of integration variables, We can now write where any time derivative (for i = 0) on the RHS should be understood as a derivative with respect to the time variable of the respective function, e.g.
which verifies the desired derivative rule for a product of functions. The∇ derivative rule for the * product of a function and a 1-form onM can be verified in a similar manner. Again, one only requires * to be bilinear and to satisfy the derivative rule for ∇. The question, then, is whether such a * product on M can be constructed. Explicitly, we want (2.22) for functions and 1-formsm but on M instead ofM . From [1,2,4], the expectation is that * should correspond to some kind of convolution. On a generic spatial manifold, M , this is a difficult question, but on compact homogeneous spaces we are able to give a well defined convolution satisfying (2.22). This is what we shall develop in the remaining sections.

Definitions
Consider a compact Lie group G and the associated Lie algebra g ∼ = T e G. For any g ∈ G we denote the corresponding right and left action as R g and L g respectively, i.e. R g g ′ = g ′ g and L g g ′ = gg ′ . Note that R g and L g are diffeomorphisms of G.
For any function f ∈ Ω 0 (G), the associated pull-backs, R * g f, L * g f ∈ Ω 0 (G), are defined as For any vector, X g ∈ T g G, we define the push-forward, R g ′ * : T g G → T gg ′ G, as that which satisfies for any function f ∈ Ω 0 (G). Similarly, for L g ′ * : For any 1-form ω ∈ Ω 1 (G) we define the pull-back associated with R g as for any X ∈ T G, and similarly for the pull-back L * g . We can now define the subspace of vector fields, RG ⊂ T G, that are right-invariant.
for any g, g ′ ∈ G. Similarly, we define LG ⊂ T G as the subspace of left-invariant vector fields which satisfy Using the pull-back we can similarly define right/left-invariant (0, p)-tensor fields. Given that G is compact, there is a unique (up to a constant rescaling) bi-invariant (right-and left-invariant) metric on G, cf. for example [83].
To any x ∈ g ∼ = T e G we can associate a left-invariant vector field X ∈ LG by pushing x forward from e to any g ∈ G with L g * . That is, X g = L g * x. This sets up an isomorphism between g and LG (and similarly for RG).
Recall that for any X, Y ∈ T G, and any push- . This implies that for any two vector fields X, Y ∈ LG, the Lie bracket, [X, Y ], is also leftinvariant. The Lie bracket, and the isomorphism between g and LG, then defines the usual Lie bracket on g.
Consider some orthonormal (with respect to the bi-invariant metric) basis of the Lie algebra, {e a } (a = 1, ..., dim(G)). In this basis the structure constants, f c ab , satisfy For any g ∈ G we have the map Ad g : g → g, given by Ad g := L g * R g −1 * . We can then decompose the vector Ad g e a in terms of the basis {e a } to get where the components of this decomposition, (Ad g ) b a , are the adjoint representation of G in this basis.
Given the basis {e a } we can form the corresponding orthonormal right-and leftinvariant bases over G, denoted by r a and l a respectively. At any g ∈ G these basis vectors are given by Given some coordinates, x µ , the basis vector field l a can be decomposed as l a (x) = l µ a (x)∂ µ , and similarly for r a . In this way, l a and r a can be thought of as vielbeins on G, with respective components l µ a (x) and r µ a (x). One can then verify that the left and right bases are related to each other via when evaluated at some point g ∈ G. The bi-invariance of the metric ensures that these bases remain orthonormal, and the fact that LG is closed under the Lie bracket implies that We can use these left-and right-invariant bases to decompose any vector field X ∈ T G: where, for each index a, the components X a R and X a L are functions on G. We denote by r a and l a the dual basis 1-forms. With tensor products one can then expand any tensor field using r a , l a , r a , and l a .
Given a vector at a point g ∈ G, say X g ∈ T g G, we can generate the corresponding left/right-invariant vector field by using the push-forward to move it to every other point g ′ ∈ G. We define maps, that generate left-and right-invariant vector fields out of vectors at g 4 . Explicitly, if we evaluate the fields l g (X) and r g (X) at any point g ′ ∈ G we have The right-and left-invariant basis fields clearly satisfy r g (r a ) = r a and l g (l a ) = l a . For a 1-form ω ∈ Ω 1 (G), we define the maps l g (·) and r g (·) in an analogous way, but with the pull-back instead: We can then extend the definition of these maps to all (0, p)-tensor fields using the pull-back in the obvious way. Note that, for any function f ∈ Ω 0 (G), l g (f ) and r g (f ) are constant functions on G, and at any point g ′ ∈ G they take the value f (g), i.e. (l g (f )) (g ′ ) = (r g (f )) (g ′ ) = f (g).
Given any diffeomorphism f : G → G, and any h ∈ Ω 0 (G) and ω ∈ Ω 1 (G), the pull-back f * satisfies where we recall that the pull-back of a function is defined as f * h := h • f . If we write ω = (ω L ) a l a , we then have that In this basis L * g simply acts as the pull-back on the component functions. Similarly we have One can then verify that (3.20)

Derivatives
We will mostly be concerned with the covariant derivative, ∇, specifically the torsionless connection that is compatible with the bi-invariant metric. Before discussing ∇ we briefly comment on a useful property of the exterior derivative, d : Using the bases (3.10) we can write for any f ∈ Ω 0 (G). For higher degree forms the action of d in the left/right-invariant basis is more complicated, but we will not need it in what follows. Moving on to the covariant derivative, for any pair of left-invariant (or right-invariant) basis vector fields l a , l b ∈ LG, ∇ satisfies For any two vector fields X, Y ∈ T G we can write them in the l a basis as Using the algebraic properties of ∇, we have where the covariant derivative of a function f ∈ Ω 0 (G), with respect to some vector field X ∈ T G, is defined as ∇ X f := X(f ). The covariant derivative of some 1-form ω ∈ Ω 1 (G) is defined as for any X, Y ∈ T G. By writing ω = (ω L ) a l a we find The definition of the covariant derivative can be extended to any tensor field in the usual way.
For any function f we define the symmetrised covariant derivative ∇f ∈ Ω 1 (G) as ∇f := df , and for any ω ∈ Ω 1 (G) we define it as for any X, Y ∈ T G. Note that the symmetrised covariant derivative has the same symbol as the usual covariant derivative. Any ambiguity between the two can be resolved by the fact that the symmetrised covariant derivative does not take a subscript argument, while the usual covariant derivative does.
Using (3.26), and the anti-symmetry of f c ab , we find We also have that where ∨ denotes the symmetrised tensor product, i.e. α ∨ β := α ⊗ β + β ⊗ α. We can therefore write where the last equality follows from the fact that there is no distinction between the properties of left and right fields. In other words, we could re-do the above derivation entirely with right-invariant fields, and we would arrive at the same result. The fact that the symmetrised covariant derivative has this simple form in terms of the left/right-invariant basis will be extremely convenient for our purposes.

Definition for Functions
Consider the group R with addition. For any x, x ′ ∈ R the group action is defined as x ′ · x := x + x ′ . The usual convolution of two functions on R is where on the far right we have written it in a form that looks more generalisable to any Lie group, provided the integration measure can be defined. In the above case the group is abelian, so we can equivalently write the argument off as For a general group, that may be non-abelian, there are then two obvious choices for the convolution of two functions.

33)
where dg is the Haar measure, normalised such that vol(G) = 1.
For compact groups the Haar measure is invariant under right and left action, and inversion.
Note that where, from line 1 to line 2, we have changed integration variables from g tog = g −1 g ′ , and used the fact that the integration measure is invariant.
We also note that in [84] they are able to define the above convolution for more than just smooth functions.

Definition for Tensor Fields
Generalising from functions to tensor fields, we want to write the right and left convolution as where S is a (0, p)-tensor field, T is a (0, q)-tensor field, and the convolution should be a (0, p + q)-tensor field. Of course this does not work, as we cannot take tensor products of S and T at different points in the integrand. Since we want the output to be a (0, p+q)-tensor at point g ′ , we need some way of 'shifting' S from g to g ′ , and T from g −1 g ′ , or g ′ g −1 , to g ′ . The right and left pull-back, and the maps l g and r g , enable us to do this in a way that is consistent with the convolution on functions.
First, we note that we can re-write the right and left convolutions of two functions, f,f ∈ Ω 0 (G), as (3.37) One can verify this by evaluating the above at any point g ′ ∈ G. Since the left convolution, * L , involves the map r g on the first argument, and L * g −1 on the second, we will henceforth refer to is as the right-left convolution, or the RL-convolution, and denote it by * RL . Likewise, the right convolution will be referred to as the left-right, or LR-convolution, and will be written as * LR . One can also define the LL-and RR-convolutions, using the respective maps, but since r g (f ) = l g (f ) for any function f ∈ Ω 0 (G), we have * LL = * RL and * RR = * LR . Notably, this will not be the case for tensor fields.
Given that r g , l g , and the pull-backs are defined for any (0, p)-tensor field, we immediately have our desired definitions for tensor fields: Definition 3.2. For any S ∈ T 0 p G and any T ∈ T 0 q G, the RLand LR-convolutions, Note, one can define the convolution on T p 0 G tensors using the left/right push-forwards and the corresponding l, r maps for vector fields (3.15).
In general, it is no longer true that ⊗ LL = ⊗ RL and ⊗ RR = ⊗ LR , as it was for the convolution between functions. Indeed, these identities are only valid for functions and for abelian groups where l a = r a . Our choice of the 'mixed' convolutions, ⊗ RL and ⊗ LR , will become more obvious when we discuss derivatives below.
In what follows we will sometimes write ∧ AB and ∨ AB instead of ⊗ AB , where A, B = R, L. In this case one should substitute ∧ and ∨ respectively for ⊗ in Def. 3.2.
Using the basis fields r a and l a we can decompose two 1-forms ω,ω ∈ Ω 1 (G) as and similarly, From this it is clear that the tensor convolutions we have defined amount to convolutions of the component functions in the right-and left-invariant bases. This generalises to convolutions between any (0, p)-tensor fields. We briefly note the following properties of the AB-convolutions: Commutativity -Writing the convolution in terms of the right-and left-invariant bases it is then clear that for any f ∈ Ω 0 (G) and any ω,ω ∈ Ω 1 (G). For any α ∈ Ω p (G), and any β ∈ Ω q (G), it is also clear that Associativity -In appendix A we prove the following associativity property: for any S ∈ T 0 p G, T ∈ T 0 q G, and U ∈ T 0 r G. Of particular importance for our purposes is how the convolution acts under derivatives, which we will now discuss.

Exterior Differentiation
In this section we will be concerned with the wedge convolution on differential forms and its action under the exterior derivative. First, we show the following Proof. To see this consider the RL-convolution between two functions, f * RLf . Using (3.21) the exterior derivative of the resulting function can be written as Consider any left-invariant basis vector l a . For any g ′ ∈ G we then have where the curve γ(λ) is a representative of l a at g ′ , i.e. γ(λ) = g ′ exp(λe a ) where exp : g → G is the exponential map. Continuing on we have where the last line follows from the left-invariance of l a . We have just shown that Similarly, one can show that Together with the commutativity of * AB we have and similarly r a (f * RLf ) = r a (f ) * RLf . (3.53) We can now show that and that Similarly, one can show that This property under the exterior derivative is why the mixed convolutions, ⊗ RL and ⊗ LR , are more useful for our purposes. Unfortunately, this sort of derivative property does not continue to higher degree forms. For example, for ω ∈ Ω 1 (G) we have whereas and so Similarly, one can show that The extra term on the far right in both expressions spoils the distributivity of d that we want for the convolution, and is the reason we cannot use our convolution, as defined, to recover the linearised BRST symmetries of the 2-form field. On the other hand, the symmetrised covariant derivative, ∇, does satisfy the desired distributivity property with respect to our convolution.

Covariant Differentiation
Here we show that the symmetrised covariant derivative, defined in (3.27) and also denoted by ∇ (but with no subscript argument), satisfies the following For any f,f ∈ Ω 0 (G) and any ω ∈ Ω 1 (G),
Proof. Recall that on functions the symmetric covariant derivative acts as ∇f = df . From Prop. 3.1 we immediately have for any two functions f,f ∈ Ω 0 (G). For any 1-form ω ∈ Ω 1 (G) we have as f * RL (ω L ) a are the l a components of f ∨ RL ω. Comparing the previous two equations we have our desired derivative property: With a similar calculation one can verify the analogous property for ∨ LR .
Proof. Above we saw that The same can be shown for ∨ LR .

Analogue of the Convolution Theorem
It is worth pausing to comment on the analogue of the Convolution Theorem, given the above definition of a tensor field convolution on the compact Lie group G. The analogue of the Convolution Theorem is well understood for the convolution of two functions on G, and so the only generalisation we are providing here is an extension to tensor fields.
Recall that the Convolution Theorem on Euclidean space states that the Fourier coefficients of f * f (where f andf are functions) are given by the pointwise product (in Fourier space) of the Fourier coefficients of f andf , up to some normalisation factor.
On a general Lie group G the situation is slightly more complicated, even for the convolution of two functions. For any function, f ∈ Ω 0 (G), the Peter-Weyl Theorem [87] tells us that we can decompose it as The first sum is over equivalence classes, [ρ], of irreducible unitary representations of G, where the equivalence is up to isomorphism.Ĝ denotes the space of all such equivalence classes. d ρ is the dimension of the representation ρ, and f (ρ|i, j) denotes the corresponding 'Fourier coefficient' for the representation ρ and indices i, j = 1, ..., d ρ . Lastly, φ ρ ij (g) denotes the (i, j)-th matrix element of the ρ representation of g ∈ G, in some orthonormal basis of the vector space used in the representation, i.e. φ ρ ij (g) := v i , ρ(g)v j for some orthonormal basis {v i } i=1,...,dρ . One can then verify that where · * denotes complex conjugation. Now consider the convolution of two functions, f * ABf . The analogue of the Convolution Theorem on Euclidean space is then a relationship between the Fourier coefficients of f * ABf , i.e. (f * ABf )(ρ|i, j), and the Fourier coefficients f (ρ|i, j) andf (ρ|i, j). Using the identities in (3.70), we find that f * RLf evaluated at some g ∈ G simplifies to and hence This can be written more succinctly if we let f (ρ) denote the d ρ × d ρ 'Fourier matrix' with elements f (ρ|i, j). The above equation can then be written as where '.' denotes matrix multiplication. The comparison to the Convolution Theorem is now clear. Specifically, the Fourier matrix of the convolution is given by the pointwise 5 matrix product of the Fourier matrices of the input functions. If the group G is abelian, then the representations are all 1-dimensional, and this matrix product reduces to normal multiplication. In this case one recovers the direct analogue of the Convolution Theorem on Euclidean space. Similarly, for the LR-convolution one finds Now that we have established the analogue of the Convolution Theorem for functions on G, we turn our attention to the convolution of two tensor fields S ∈ T 0 p (G) and T ∈ T 0 q (G). After expanding S and T in terms of the right and left bases repsectively, we can use (3.42) to write S ⊗ RL T as then the Fourier matrix for the components (S ⊗ RL T ) a 1 ,...,bq is given by (3.76). Similarly, for the convolution S ⊗ LR T we can expand it as The Fourier matrices for the components (S ⊗ LR T ) a 1 ,...,bq are then given by

Homogeneous Spaces
The convolution of functions on Riemannian homogeneous spaces G/H is reasonably welldeveloped, cf. for example [85] and the references therein. Here we generalise to tensor fields.

Definitions
Consider a Riemannian manifold M on which a compact Lie group G acts transitively. Pick a base point η ∈ M and consider the subgroup H for which hη = η for all h ∈ H. Then M ∼ = G/H, and G can be thought of as a H-principal bundle over M with the projection π : G → M , g → π(g) := gη.
Given some X ∈ T G, the push-forward gives the vector π * X ∈ T M , which acts on a function f ∈ Ω 0 (M ), at the point gη ∈ M , as (4.1) For any vector X g ∈ T g G we can find a representative curve γ(λ), with γ(0) = g and such that γ(λ) is tangent to X g at g. Consider the special case where the vector X g has a representative curve of the form γ(λ) = gh(λ), where h(λ) ∈ H and where gh(0) = g e = g. The action of the vector π * X g ∈ T gη M on any function f ∈ Ω 0 (M ) is then and hence (π * X g ) gη = 0. The vector X g is then in the kernel of the push-forward π * . Furthermore, if X g ∈ ker(π * ) then L g ′ * X g and R h * X g are also in the kernel for any g ′ ∈ G and any h ∈ H. This means that the kernel of π * is left-invariant and R H -invariant. We call the kernel of π * the vertical subspace of T G, and denote it by VT G.
Using the bi-invariant metric on G we can find the subspace of T G that is orthogonal to VT G. This is the horizontal subspace, denoted by HT G. From the bi-invariance of the metric, and the left-and R H -invariance of VT G, one can see that HT G is also left-and R H -invariant. We can now write T G = HT G ⊕ VT G. For any X ∈ T G we have where H (V) is the horizontal (vertical) projector.
Recall that g ∼ = T e G. We now have the orthogonal decomposition g = m ⊕ h, where m = HT e G and h = VT e G. One can also show that h is the Lie algebra of the subgroup H.
Let us pick our basis, {e a }, of g, such that {e i } is a basis for m, where i = 1, ..., dim(m), and such that e A is a basis for h, where A = dim(m) + 1, ..., dim(G). As HT G and VT G are left-invariant, the left-invariant basis {l a } also splits in a similar way, i.e. {l i } is a basis of HT G, and {l A } is a basis of VT G. Any vector field X ∈ T G can then be written as For any ω ∈ Ω 1 (G) we can decompose it using the dual basis: and similarly for any (p, q)-tensor field. We note that for any h ∈ H. To see this, we first note that the vertical subspace is R H -invariant, which means that R h * l A (for any h ∈ H) can be expanded in terms of the l A basis only. From the right-invariance of the metric we know that {R h * l A } is an orthonormal set, and since each vector R h * l A lies in VT G, we know that {R h * l A } is an orthonormal basis of VT G. From the right-invariance of the metric we also know that R h * l i is orthogonal to each R h * l A , and hence R h * l i ∈ HT G. This means that R h * l i can be expanded in terms of the l i basis only. As the set {R h * l i } is orthonormal (from the right-invariance of the metric) we then have that {R h * l i } is a basis of HT G. The preceeding argument shows that R h * acts separately on the bases {l A } and {l i }, and rotates them amongst themselves. This means that R h * and H commute as desired.
Definition 4.1. Given some X ∈ T M , the horizontal lift, L(X) ∈ HT G, is the unique horizontal vector field which satisfies for any g ∈ G.
More simply, we can write π * L(X) = X. Note that this condition implies that L(X) is R H -invariant, as for any Y g ∈ T g G we have for any h ∈ H. We note some basic identities that will be useful later. For any X ∈ T M , Y g ∈ T g G, f ∈ Ω 0 (M ), and ω ∈ Ω 1 (M ), π * L(X) = X , (4.10) (π * ω)(L(X)) g = ω(X) gη . (4.13) We will also need the following map that averages along the fiber H: For any (0, p)-tensor field (including functions) we define the map, A H : T 0 p G → T 0 p G, that averages along the fiber H as where dh is the Haar measure for the compact subgroup H.

Clearly, for any
where we have used the invariance of dh under right action by H. For any T ∈ T 0 p G, we can then see that A H T is R H -invariant. For any f ∈ Ω 0 (G), the function A H f is constant along the fibers of H, and its value along any fiber is the average of f along that fiber.
Given (4.6), we have where we recall that H projects onto the horizontal subspace.
Using π * we can push-forward any X ∈ T G to T M , and using π * we can pull-back any T ∈ T 0 p M . We can also lift any vector field X ∈ T M from T M to T G with L(·). We now define the analogous map that takes some T ∈ T 0 p G to a (0, p)-tensor field in T 0 p M : . For any f ∈ Ω 0 (G), the π H -projected function π H f ∈ Ω 0 (M ) is the unique function on M that satisfies The value of π H f at any point gη ∈ M is then the average of f along the fiber above gη. Similarly, we have: For any T ∈ T 0 p G, the π H -projected tensor field π H T ∈ T 0 p M is the unique tensor field on M that satisfies π * π H T = HA H T . (4.18) It is straightforward to verify the uniqueness of π H T using the isomorphism between T M and horizontal R H -invariant vector fields in T G.

Covariant Differentiation
Consider the Levi-Civita connection, ∇, on G. One can see that ∇ L(X) L(Y ) is R H -invariant for any X, Y ∈ T M as follows: where the first line follows from the fact that R h is an isometry on G, and the second line follows from the R H -invariance of L(X) and L(Y ). This allows us to make the following One can also show that π * ∇f = ∇π * f for any f ∈ Ω 0 (M ) as follows. For any X ∈ T G we have As ∇f = df , this is simply the statement that the pull-back commutes with d.
For any ω ∈ Ω 1 (G), and any horizontal vector fields X, Y ∈ HT G, we have the identity which can be seen as follows. Recall that X = X i L l i for any X ∈ HT G, and similarly Hω = (ω L ) i l i for any ω ∈ Ω 1 (G). We have where we have used (3.28) in line 1. For any ω ∈ Ω 1 (M ), the symmetrised covariant derivative on M also takes on a simple form. To see this we first note that, for any X, Y ∈ T M , 25) which means that, after symmetrising over any pair X, Y ∈ T M , we have We also have the following derivation: 27) which implies that π * ∇ω = ∇π * ω , (4.28) when acting on lifted vector fields L(T M ) ⊂ T G.

Definition
Consider a compact manifold M on which a compact Lie group G acts transitively. We make the following Definition 4.6. For any S ∈ T 0 p M and any T ∈ T 0 q M , we define the convolution S ⊗ AB T ∈ T 0 p+q M , for AB = RL, LR, as S ⊗ AB T := π H (π * S ⊗ AB π * T ) . where the convolution π * S ⊗ AB π * T on the RHS is the convolution defined on G in Def.

3.2.
In words, the convolution of two fields on M consists of first pulling them back to the group manifold G, convolving them on G, averaging over the fiber H, then projecting the result back down to M . Recall that the π H -projection involves averaging over the fiber and projecting onto the horizontal component. In fact, the averaging is not necessary as π * S ⊗ AB π * T is R Hinvariant. To see this, we focus on the ⊗ RL and ⊗ LR convolutions separately.
For any h ∈ H we have where from line 1 to 2 we used the fact that R * h r g = r g (as r g already creates a right-invariant field), and the fact that right and left pull-backs commute. From line 2 to 3 we used the fact that any pulled-back field, e.g. π * T , is R H -invariant.
For the ⊗ LR convolution we have using the fact that R * g R * g ′ = R * gg ′ . In order to proceed we need to determine how R * h and l g commute. Consider some ω ∈ Ω 1 (G), then R * h l g ω evaluated at g ′ ∈ G is where from line 1 to 2 we used the definition of l g in (3.16). In summary, R * h l g = l gh −1 R * h . This then means that where from line 1 to 2 we changed integration variables tog = gh −1 . Now that we have established the R H invariance of π * S ⊗ AB π * T , we get the following simplification of the convolution S ⊗ AB T on M , when acting on test vector fields X 1 , X 2 , ... ∈ T M : (S ⊗ AB T )(X 1 , ...) = (S ⊗ AB T )(π * L(X 1 ), ...) = (π * (S ⊗ AB T ))(L(X 1 ), ...) = (π * π H (π * S ⊗ AB π * T ))(L(X 1 ), ...) where we have used (4.10) in line 1, Def. (4.6) of the convolution on M from line 2 to 3, and Def. (4.4) of π H from line 3 to 4. To get the final line we used the fact that the horizontal projection H can be removed as the tensor is already acting on horizontal lifted vector fields, and the fact that A H acts trivially as π * S ⊗ AB π * T is R H -invariant.
In words, we have just shown that the convolution of two tensors, S and T on M , acts on vector fields on M as the convolution of the pulled-back tensors, π * S and π * T on G, on the corresponding lifted vector fields. For the convolution of a pair of functions, f ∨ ABf , this simplifies to the statement that the value of f ∨ ABf at the point gη ∈ M is equal to the value of π * f ∨ AB π * f at any point in the fiber above gη.

Differentiation
In this section we will be concerned with the symmetrised covariant derivative of the symmetrised convolution ∨ AB . For our purposes, we only need to consider the convolution S ∨ AB T , where S and T are either both functions, or one of them is a function and the other is a 1-form. The symmetrised covariant derivative of the convolution, ∇(S ∨ AB T ), is then either a 1-form or a symmetric (0, 2)-tensor respectively, and hence acts on either one or two vector fields. We write (∇(S ∨ AB T ))(X 1 , ...), where X 1 , ... ∈ T M , to cover both these cases.
We have where from line 2 to 3 we have used the fact that, for any function or 1-form U on M , π * ∇U = ∇π * U when acting on lifted vector fields (see (4.28)). From line 5 to 6 we have used (4.23) to remove the horizontal projection H. We also removed A H as it acts trivially on the R H -invariant π * S ∨ AB π * T . We can now utilise the derivative rules for the convolution on G (Prop. (3.2) and Prop. (3.3)) to write ∇(π * S ∨ AB π * T ) = ∇π * S ∨ AB π * T = π * S ∨ AB ∇π * T . (4.36) If S is a function, we can use the fact that π * and ∇ commute on functions to write ∇π * S ∨ AB π * T = π * ∇S ∨ AB π * T , (4.37) and hence If we take B = L, i.e. we are considering the ∨ RL convolution, then (L * g −1 ∇π * T )(L(X 1 ), ...) = (∇π * T )(L g −1 * L(X 1 ), ...) = (π * ∇T )(L g −1 * L(X 1 ), ...) To go from line 1 to 2 we have used the fact that the left push-forward of a lifted vector field is still a lifted vector field (that is, it is horizontal and R H -invariant), and hence we can apply (4.28) to commute ∇ and π * . This then means that If we take B = R, i.e. we are considering ∇(S ∨ LR T ) with S ∈ Ω 0 (M ) and T ∈ Ω 1 (M ), we cannot make the same simplification, as the right push-forward of a lifted vector field is not a lifted vector field. It is not clear then whether the derivative rule ∇(S ∨ LR T ) = S ∨ LR ∇T is true when S ∈ Ω 0 (M ) and T ∈ Ω 1 (M ). This potential asymmetry between the LR-and RL-convolutions may be expected for a homogeneous space, as the manifold M is identified with the left coset space {gH | g ∈ G}.

Discussion
We introduced convolution products for tensor fields on group manifolds and homogeneous spaces and demonstrated that for symmetric convolutions of scalars and 1-forms they satisfy the usual (non-Leibniz) derivative rule. This followed from the observation that group manifolds, and homogeneous spaces, come with a natural path-independent notion of transporting tensors in a manner compatible with the covariant derivative. This allowed us to apply the convolutions to the construction of double copy field dictionaries for the BRST complex of graviton theory, considered to linear order in perturbation theory on these background manifolds. The robustness under BRST transformations was shown to follow from the convolution properties. We extended the construction to static universe backgrounds by the addition of a time direction.
It was shown that the spin-2 gauge transformations of the free graviton are reproduced correctly via the convolution. However, one can also generate the dilaton and 2-form Kalb-Ramond field, in the latter case by taking the two Yang-Mills BRST complexes to be different. Moreover, p-forms are expected to appear in more general double-copy constructible theories, particularly in the presence of supersymmetry. We leave the construction of a convolution which can adequately describe these types of fields to future work. Encouragingly, this was shown to exist in the simple case of the 2-sphere in [1].
Another natural question is whether one can formulate the convolution on backgrounds of cosmological interest, such as de Sitter (perhaps making use of the fact that it is foliated by spheres in global coordinates, for which we already know how to define double copy dictionaries), or anti-de Sitter, which would allow us to explore potential links to holography. An important extension would be to construct the gravitational theories to higher orders perturbatively on these backgrounds, as was done for flat backgrounds in [6,72,73,86]. Here one could exploit recent developments extending color-kinematics duality to these spaces [80,81].

A Associativity
We first note the identities for any g, h ∈ G. From the definition of r g , we can also see that, for any T ∈ T 0 p G, for any g, h ∈ G. Similarly, That is, have the identities r g r h = r h , l g l h = l h . (A.4) We will also need the following identity: which can be seen by considering the action of both the left and right sides of the above equation on some (0, p)-tensor T . Take the LHS: and then the RHS: We are now ready to prove the associativity of ⊗ RL . Proof where we have used the fact that the pull-back distributes onto both sides of a tensor product. If we change integration variables from h toh = gh, we get A similar calculation can be done to prove the associativity of ⊗ LR .

B Levi-Civita connection on M
For clarity, in this appendix we denote the bi-invariant metric on G asg, and the corresponding Levi-Civita connection on G as∇. Recall that bi-invariance ofg means that L * gg = R * g ′g =g, for any g, g ′ ∈ G. g induces a metric, g, on the base space M in the following way. For any point gη ∈ M , and any X gη , Y gη ∈ T gη M , we find the horizontal lifts L(X) g , L(Y ) g ∈ T g G, and define g(X gη , Y gη ) :=g(L(X) g , L(Y ) g ) . (B.1) Crucially, the RHS is independent of the point g in the fiber above gη. This follows from the right-invariance ofg and the R H -invariance of the horizontal lifts.
where line 2 follows from the definition of g, and line 3 follows from (4.11), and line 4 follows from the fact thatX g andỸ g are already horizontal. Following the last line in (B.7), this implies that where line 2 follows from the left-invariance ofg, and line 3 follows from the definition of g.∇ is torsionless and compatible withg. Respectively, for any X, Y, Z ∈ T G. Using the above, and the definitions of g and ∇, we now show that Proposition B.2. ∇ is torsionless. That is, for any X, Y ∈ T M , Proof. We first note that, for any g ∈ G, any X 1 , ..., X p ∈ T M , and any T ∈ T 0 p M , π * T (X 1 , ..., X p ) gη = (π * T )(L(X 1 ), ..., L(X p )) g , (B. 19) where the LHS is the pull-back of the function T (X 1 , ..., X p ) on M . For any X, Y ∈ T M , we can then use (B.8) to write the pull-back of the function g(X, Y ) on M as π * g(X, Y ) gη =g(L(X), L(Y )) g . (B.20) In (B.18) we can then write X(g(Y, Z)) gη = π * L(X)(g(Y, Z)) gη = L(X) π * g(Y, Z) Putting it all together we find (∇ X g)(Y, Z) gη = X g(Y, Z) gη − g(∇ X Y, Z) gη − g(Y, ∇ X Z) gη = L(X) g(L(X), L(Y )) g