Abstract
We briefly review the manifold incarnations and facets of the notion that gravity, in certain regards, can be re-conceived as the product of two gauge theories. We begin with a short history of the relationship between gauge and gravity theories in the context of “\(\hbox {gravity} = \hbox {gauge} \times \hbox {gauge}\)”. This is followed by modern approaches to gauge/gravity scattering amplitude relations, focussing on the Bern–Carrasco–Johansson colour-kinematic duality and double-copy construction of gravitational amplitudes from gauge amplitudes. This includes a partial characterisation of what gravity theories admit a gauge theory squared origin. We then consider classical and off-shell perspectives on “\(\hbox {gravity} =\hbox {gauge} \times \hbox {gauge}\)”. First we review field theoretic approaches to understanding the colour-kinematic duality and the double-copy prescription, including kinematic algebras and colour-kinematic duality and double-copy manifesting Lagrangians. We then consider classical double-copy-like methods for constructing gravitational solutions from gauge theory solutions. To close, we consider a purely field theoretic take on “\(\hbox {gravity} =\hbox {gauge} \times \hbox {gauge}\)”. This framework centres on a convolutive product of gauge theories, at the level of spacetime fields themselves, that yields gravitational theories.
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Notes
For a review of Kaluza–Klein theory see [11].
To be absolutely clear, here we are considering the Yang–Mills theory of the Poincaré group in the strict sense that it is treated as an internal symmetry, in direct analogy to, say, the \(\mathrm{SU}(3)\) of the Standard Model. By “gauged Poincaré” we really do mean a principal bundle P(M, G), for G the Poincaré group. Of course, the actual role of the global Poincaré group in a relativistic quantum field theory on \({\mathbb {R}}^{1,d}\) is as a spacetime symmetry. We can also promote this spacetime Poincaré group to be local, as was done in [13]. In this case, the local translations and rotations yield independent general coordinate and local Lorentz transformations. Demanding invariance then leads one, almost inevitably, to general relativity (with matter induced torsion) without invoking, a priori, any Riemannian geometry [13]. While certainly elegant, this seems to us to be gilding the lily somewhat.
An excellent account can be found in §7 of the classic reference [35].
In the stringy nomenclature we have the NS–NS (\(\mathbf {8}_v\times \mathbf {8}_v\)), NS-R (\(\mathbf {8}_v\times \mathbf {8}_c\)), R–NS (\(\mathbf {8}_s\times \mathbf {8}_v\)), and R–R (\(\mathbf {8}_s\times \mathbf {8}_c\)) sectors, where NS and R refer to Nevau–Schwarz and Ramond.
On the one hand, this is exactly what one would expect from our stringy intuition based on world-sheet supersymmetry, which is treated democratically in both the open string factors and their closed string product. It is, however, a little counter-intuitive from the spacetime “\(\hbox {gravity} = \hbox {gauge} \times \hbox {gauge}\)” perspective, where we take internal tensor products of spacetime little group representations, but external tensor products of the supersymmetry transformations, despite the fact that they anti-commute to give translations. It all works out, as we shall see.
With possibly distinct gauge groups. The construction did not rely on any particular properties of, or relations amongst, the gauge groups.
See [103] for possible explanations at three loops that nonetheless fail at four loops.
This is only unambiguous for matrix Lie algebras, but the general case is given by the obvious interpretation for an arbitrary Lie bracket \([~, ]:\mathfrak {g}\times \mathfrak {g}\rightarrow \mathfrak {g}\).
Proceed by induction: clearly true for \(n=3\). At n-points each graph has \(2n-3\) edges to which one can add an edge to give \(2n-3\) new \((n+1)\)-point graphs. Hence, assuming there are \((2m-5)!!\) graphs at n-points, there are \((2n-3)(2n-5)!!=(2(n+1)-5)!!\) graphs at \((n+1)\)-points.
As an example, for the s-channel we have
$$\begin{aligned} n_s=4\big ( \varepsilon _{1}\cdot p_{2} \varepsilon _{2 \lambda }-\varepsilon _{2}\cdot p_{1} \varepsilon _{1 \lambda } +\tfrac{1}{2}\varepsilon _{1}\cdot \varepsilon _{2} p_{12\lambda }\big )\big ( \varepsilon _{3}\cdot p_{4} \varepsilon _{4}^{\lambda }-\varepsilon _{4}\cdot p_{3} \varepsilon _{1}^{\lambda }+\tfrac{1}{2}\varepsilon _{3}\cdot \varepsilon _{4} p_{34}^{\lambda }\big ). \end{aligned}$$Of course, fixing the helicities and making some sensible choices for the polarisations, this can be significantly simplified.
We shall see momentarily that BCJ duality implies that this \((n-2)!\)-dimensional basis is over complete, but let us not put the cart before the horse.
One could demand that it is a Moore–Penrose pseudo-inverse, which always exists and is unique, picking out one particular solution.
The \(c_i\) should not be explicitly evaluated under the integral in case they accidentally vanish before being replaced by the loop-momenta dependent kinematic factors.
We are measuring complexity in terms of how simple it is to fully characterise the space of theories of interest: \({\mathcal {N}}=8\) supergravity is unique and determined by supersymmetry alone, as simple as can be, whereas the space of non-supersymmetric theories containing Einstein–Hilbert gravity is wild, even with reasonable consistency conditions imposed.
But not supergravity, the analog in this case is \({\mathcal {N}}=7\) supersymmetry implies \({\mathcal {N}}=8\).
\({\mathcal {N}}=2\) supergravity coupled to a single vector multiplet with non-compact global symmetry group \(\mathrm{SL}(2, {\mathbb {R}})\), under which the two Maxwell field strengths and their duals transform as the \(\mathbf {4}\) [246, 247]. Can be regarded as the “symmetrisation” of the STU model [248], where the there complex scalars S, T, U are identified.
But see final point of this list.
This may be trivially true, for example a cubic theory of scalars transforming in the adjoint of a global symmetry.
As it stands this can of course only be established in general at tree-level, with supporting evidence from case-by-case examples of loop-level amplitudes. Our present analysis is explicitly tree-level only.
Of course, for tree-level amplitudes they may be consistently truncated to leave only the graviton scattering amplitudes of Einstein–Hilbert gravity.
This follows from the requirement that the spectator is the convolutive pseudo-inverse of the \(\phi ^3\) field: \(\phi \cdot \Phi \cdot \phi = \phi \) implies \([\Phi ]=(3D+2)/2\).
We stop at \(D=3\), which has \(E_{8(8)}\) U-duality, the largest finite dimensional exceptional Lie algebra. One can continue to \(D=2,1,0\), invoking the infinite dimensional extended algebras \(E_{9(9)}, E_{10(10)}, E_{11(11)}\) [313,314,315,316]. Although we will not discuss theses cases here, it would be interesting to investigate if they can be understood from the perspective of Yang–Mills squared.
It might be elucidating to not identify the spacetime symmetries. For example, one can reformulate the \({\mathcal {N}}=0\) supergravity action so as to have manifest left and right Lorentz symmetries [281, 284]. Nonetheless, from the point of view of spectrum matching and global symmetries we should identify the spacetime little groups.
It is tempting to spectulate that this modification is related to the spectator scalar of the field product.
We must retain the homogenous assumption. In fact, as far as we are aware there are as yet no examples of double-copy constructible supergravity theories without homogenous scalar manifolds, despite there ubiquity for \({\mathcal {N}}\le 2\).
As far as we are aware the first instance in this context.
A quadratic norm on a vector space V over a field \({\mathbb {R}}\) is a map \({\mathbf {n}}:V\rightarrow {\mathbb {R}}\) such that: (1) \({\mathbf {n}}(\lambda a)=\lambda ^2{\mathbf {n}}(a), \lambda \in {\mathbb {R}}, a\in V\) and (2) \( \langle a, b\rangle :={\mathbf {n}}(a+b)-{\mathbf {n}}(a)-{\mathbf {n}}(b) \) is bilinear.
One way to understand this is in terms of Jordan algebras. Points in \({\mathbb {O}}\mathbb {P}^2\) are bijectively identified with trace 1 projectors in \(\mathfrak {J}_{3}^{{\mathbb {O}}}\), the Jordan algebra of \(3\times 3\) octonionic Hermitian matrices. However, for \(m>3\), \(m\times m\) octonionic Hermitian matrices do not form a Jordan algebra.
Non-associativity, however, implies that the line through the origin containing the point (a, b) is not given by \(\{(\alpha a, \alpha b) | \alpha \in {\mathbb {O}}\}\), unless \(x=1\) or \(y=1\). This obstacle is easily avoided as all non-zero octonions have an inverse; (a, b) is equivalent to \((b^{-1}a, 1)\) or \((1, a^{-1}b)\) for \(b\not =0\) or \(a\not =0\), giving two charts with a smooth transition function on their overlap. See [195].
Note, the additional factors are given by intermediate algebras: \(\mathfrak {tri}({\mathbb {A}})\ominus \mathfrak {int} ({\mathbb {A}})=\emptyset , \mathfrak {u}(1), \mathfrak {sp}(1), \emptyset \) for \({\mathbb {A}}={\mathbb {R}}, {\mathbb {C}}, {\mathbb {H}}, {\mathbb {O}}\) [196].
The objects \(1_{\pm }\) act as projection operators dividing \({\mathbb {C}}\otimes {\mathbb {C}}\) into two 2-dimensional subspaces, on which \(i_{\pm }\) act as complex structures, so that \({\mathbb {C}}\otimes {\mathbb {C}}\cong {\mathbb {C}}\oplus {\mathbb {C}}\).
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Acknowledgements
I gratefully acknowledge T. Adamo, Z. Bern, B. Cerchiai, M. Chiodaroli, S. Deser, S. Ferrara, Y. Geyer, A. Hodges, C. M. Hull, H. Johansson, B. Julia, A. Luna, L. Mason, G. Mogull, R. Monteiro, A. Ochirov, D. O’Connell, R. Roiban, O. Schlotterer, W. Siegel, K. Stelle and C. White and for many helpful conversations and correspondences. I would especially like to thank my collaborators A. Anastansiou, M. J. Duff, M. J. Hughes, A. Marrani, S. Nagy and M. Zoccali. This work was supported by a Schrödinger Fellowship.
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Borsten, L. Gravity as the square of gauge theory: a review. Riv. Nuovo Cim. 43, 97–186 (2020). https://doi.org/10.1007/s40766-020-00003-6
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DOI: https://doi.org/10.1007/s40766-020-00003-6