What is the Lie group of gravity?

Question

If the lie group of the three gauge forces are $SU(3)×SU(2)×U (1)$, then what is the symmetry group of gravity? $SL(2,C)$?

Just a newbie in Lie groups.

Answer 1

Not a complete answer, but some general notes:

About gauge groups:

• In the widest a sense, a (classical) gauge theory is a (classical, Lagrangian) field theory whose Lagrangian is degenerate, so the Hamiltonian formalism in the usual sense does not exist. When the covariant phase space is constructed, the canonical 2-form is a degenerate (pre-)symplectic form.

• The gauge group of such a theory consists of all field transformations that lie along the null directions of the presymplectic form. The true phase space of the theory is obtained by quotienting out this group from the group of all phase space flows. Note that the gauge group defined this way is pretty much always an infinite dimensional "Lie group" (in a very formal sense).

• We can without loss of generality assume that a field theory consists of a spacetime manifold $M$ and a fibre bundle $\pi:E\rightarrow M$ over spacetime (although I am sure everything is also valid at least with slight modifications if $\pi$ is a fibred manifold that isn't locally trivial), and a field is a (sufficiently differentiable) cross-section $\psi \in\Gamma(E)$ of this fibre bundle. Then a good candidate for the totality of all field transformations is the automorphism group $\mathrm{Aut}(E)$ of the fibre bundle.

• The (formally infinite dimensional Lie-) group $\mathrm{Aut}(E)$ has a subgroup $\mathrm{Aut}_V(E)$ of vertical automorphisms that move points only within fibres but do not change basepoints. $\mathrm{Aut}_V(E)$ is a normal subgroup of $\mathrm{Aut}(E)$, and we have $$ \mathrm{Aut}(E)/\mathrm{Aut}_V(E)\le\mathrm{Diff}(M), $$ where $\mathrm{Diff}(M)$ is the diffeomorphism group of $M$. Note that in general this quotient group is not the full diffeomorphism group, but on the "infinitesimal level", we have $\mathfrak{aut}(E)=\mathfrak X_{\mathrm{Pr}}(E)$ and $\mathfrak{aut}_V(E)=\mathfrak X_{V}(E)$ and then $$ \mathfrak X_{\mathrm{Pr}}(E)/\mathfrak X_V(E)=\mathfrak X(M), $$ where $\mathfrak X_{\mathrm{Pr}}(E)$ is the Lie algebra of all (smooth) $\pi$-projectable vector fields on $E$, and $\mathfrak X_V(E)$ is the Lie algebra of all $\pi$-vertical vector fields on $E$.

• Usually the whole of $\mathrm{Aut}(E)$ is not the gauge group of the theory, but there is an infinite dimensional subgroup $\mathcal G\le\mathrm{Aut}(E)$ under which the theory is invariant, and that is the gauge group.

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About structure groups:

• It is common that the configuration bundle $\pi:E\rightarrow M$ has a $G$-structure, and then there is a principal $G$-bundle $\pi_P:P\rightarrow M$ such that $E\rightarrow M$ is a $P$-associated bundle. If the field theory in question is compatible with this $G$-structure then we say that $G$ is the structure group of the theory.

• Let $\mathrm {Aut}(P)$ denote those automorphisms of the principal fibre bundle $P\rightarrow M$, which are $G$-equivariant, of these $\mathrm{Aut}_V(P)$ those that are also vertical. As it is well-known from the theory of associated bundles, each automorphism $\phi\in\mathrm{Aut}(P)$ can act on any associated bundle as automorphisms. Those automorphisms of $E$ which are obtained from $P$-automorphisms are called $G$-automorphisms of $E$, and will denote them as $\mathrm{Aut}_G(E)$, while the vertical ones (which are induced from elements of $\mathrm{Aut}_V(P)$) will be denoted as $\mathrm{Aut}_{G,V}(E)$.

• Special attention is worth giving to the vertical $P$-automorphisms $\mathrm{Aut}_V(P)$. We can construct two associated bundles, $P_{\mathbf{Ad},G}$, which is given by letting $G$ act on itself on the left via conjugation $\mathbf{Ad}_g(h)=ghg^{-1}$, and the adjoint bundle $\mathrm{Ad}(P)\cong P_{\mathrm{Ad},\mathfrak{g}}$, which is constructed by letting $G$ act on its Lie algebra $\mathfrak g$ on the left via the adjoint representation $$\mathrm{Ad}_g(A)=\frac{d}{dt}(g\exp(tA)g^{-1})|_{t=0}.$$ Then $$ \mathrm{Aut}_V(P)=\Gamma(P_{\mathbf{Ad},G})\quad \mathfrak{aut}_V(P)=\Gamma(\mathrm{Ad}(P)). $$

• In a Yang-Mills type theory (with or without matter fields) formulated on a fixed background, the gauge group is $\mathcal G=\mathrm{Aut}_V(P)=\Gamma(P_{\mathbf{Ad},G})$ (or rather its image on the corresponding associated bundle). By construction (see the above bullet point), the set of all gauge transformations correspond to the smooth sections of a fibre bundle $P_{\mathbf{Ad},G}$ whose fibres are $G$. This gives a link between the gauge group and the structure group, since formally the gauge group is a fibration whose fibres are isomorphic to the structure group. Essentially, you have a different copy of $G$ over each point of $M$. Here I also note that the gauge group acts purely vertically.

• If the Yang-Mills type theory is coupled to gravity, then the gauge group in consideration is the full automorphism group $\mathrm{Aut}(P)$.

• Diffeomorphisms of $M$ in general cannot be lifted (uniquely) to automorphisms of $P$, since $P$ is not a natural bundle. On the other hand, on the infinitesimal level, we have seen that $\mathfrak X(M)=\mathfrak {diff}(M)=\mathfrak{aut}(P)/\mathfrak{aut}_V(P)$, and locally in a given trivialization $U\subseteq M$ of $P$, we actually have $\mathfrak{aut}(\pi^{-1}_P(U))=\mathfrak{aut}_V(\pi^{-1}_P(U))\oplus\mathfrak{diff}(U)$ (semi-direct sum), but this is trivialization-dependent so in this context diffemorphism-invariance must be interpreted together with gauge invariance as $\mathrm{Aut}(P)$-invariance, and there is no "covariant" way to split the two.

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About gravity:

• All the previous drivel has been to highlight that a structure group (which is $\mathrm{U}(1)\times \mathrm{SU}(2)\times\mathrm{SU}(3)$ in OP's example) can be related to a gauge group's vertical part, but not all gauge groups have to come from a structure group, and even if they do, the full gauge group has horizontal parts as well that never come from a gauge group.

• When considering either pure gravity, or gravity with matter fields that are completely tensorial, all fibre bundles involved are natural bundles, which admit functorial lifts of diffeomorphisms to fibre bundle automorphisms, and the gauge group is essentially $\mathrm{Diff}(M)$. Note that this does not act in a purely horizontal manner on any structure bundle (and horizontality doesn't even make sense globally), and the vertical part (as much as a vertical part can be covariantly separated, eg. not at all, and this remark must be understood heuristically) is always a $\mathrm{GL}(4,\mathbb R)$-automorphism.

• The latter however is a common aspect to all "generally covariant theories" and cannot really be interpreted in any way that would carry information specific to Einstein gravity. In particular, a diffeomorphism-induced fibre bundle automorphism is purely vertical if and only if it is the identity transformation, so in generally covariant theories, there are no purely vertical gauge transformations.

• The conclusion is that general relativity has a gauge group (it is a degenerate Lagrangian theory after all), which is the diffeomorphism group $\mathrm {Diff}(M)$, but has no (vertical) structure group in the Yang-Mills sense.

• There are alternative formulations of gravity, especially metric-affine theories like Einstein-Cartan theory that have gauge interpretations. In particular, EC theory can be interpreted as a gauge-natural (as opposed to natural) theory of a connection on a Poincaré-principal bundle. The translational part of the structure group is broken by a Higgs-like mechanism, which gives rise to a metric, while the unbroken Lorentz-part gives rise to the spin-connection. In this case for example the broken structure group is the Poincaré group and the unbroken structure group is the Lorentz group.

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Reference: Fatibene, Francaviglia: Natural and Gauge Natural Formalism for Classical Field Theories