Recently, an old post of mine about the Asymptotic Safety program for quantizing gravity received a flurry of new comments. Inadvertently, one of the pseudonymous commenters pointed out yet another problem with the program, which deserves a post all its own.

Before launching in, I should say that

Yang-Mills Theory

Let’s start by reviewing Singer’s explication of the Gribov ambiguity.

Say we want to do the path integral for Yang-Mills Theory, with compact semi-simple gauge group G G . For definiteness, we’ll talk about the Euclidean path integral, and take M = S 4 M= S^4 . Fix a principal G G -bundle, P → M P\to M . We would like to integrate over all connections, A A , on P P , modulo gauge transformations, with a weight given by e − S YM ( A ) e^{-S_{\text{YM}}(A)} . Let 𝒜 \mathcal{A} be the space of all connections on P P , 𝒢 \mathcal{G} the (infinite dimensional) group of gauge transformations (automorphisms of P P which project to the identity on M M ), and ℬ = 𝒜 / 𝒢 \mathcal{B}=\mathcal{A}/\mathcal{G} , the gauge equivalence classes of connections.

“Really,” what we would like to do is integrate over ℬ \mathcal{B} . In practice, what we actually do is fix a gauge and integrate over actual connections (rather than equivalence classes thereof). We could, for instance, choose background field gauge. Pick a fiducial connection, A ¯ \overline{A} , on P P , and parametrize any other connection A = A ¯ + Q A= \overline{A}+Q with Q Q a 𝔤 \mathfrak{g} -valued 1-form on M M . Background field gauge is

(1) D A ¯ * Q = 0 D_{\overline{A}}* Q = 0

which picks out a linear subspace 𝒬 ⊂ 𝒜 \mathcal{Q}\subset\mathcal{A} . The hope is that this subspace is transverse to the orbits of 𝒢 \mathcal{G} , and intersects each orbit precisely once. If so, then we can do the path integral by integrating1 over 𝒬 \mathcal{Q} . That is, 𝒬 \mathcal{Q} is the image of a global section of the principal 𝒢 \mathcal{G} -bundle, 𝒜 → ℬ \mathcal{A}\to \mathcal{B} and integrating over ℬ \mathcal{B} is equivalent to integrating over its image, 𝒬 \mathcal{Q} .

What Gribov found (in a Coulomb-type gauge) is that 𝒬 \mathcal{Q} intersects a given gauge orbit more than once. Singer explained that this is not some accident of Coulomb gauge. The bundle 𝒜 → ℬ \mathcal{A}\to \mathcal{B} is nontrivial and no global gauge choice (section) exists.

A small technical point: 𝒢 \mathcal{G} doesn’t act freely on 𝒜 \mathcal{A} . Except for the case2 G = SU ( 2 ) G=SU(2) , there are reducible connections, which are fixed by a subgroup of 𝒢 \mathcal{G} . Because of the presence of reducible connections, we should interpret ℬ \mathcal{B} as a stack. However, to prove the nontriviality, we don’t need to venture into the stacky world; it suffices to consider the irreducible connections, 𝒜 0 ⊂ 𝒜 \mathcal{A}_0\subset \mathcal{A} , on which 𝒢 \mathcal{G} acts freely. We then have 𝒜 0 → ℬ 0 \mathcal{A}_0\to \mathcal{B}_0 of which 𝒢 \mathcal{G} acts freely on the fibers. If we were able to find a global section of 𝒜 0 → ℬ 0 \mathcal{A}_0\to \mathcal{B}_0 , then we would have established 𝒜 0 ≅ ℬ 0 × 𝒢 \mathcal{A}_0\cong \mathcal{B}_0\times \mathcal{G} But Singer proves that

π k ( 𝒜 0 ) = 0 , ∀ k > 0 \pi_k(\mathcal{A}_0)=0,\,\forall k\gt 0 . But π k ( 𝒢 ) ≠ 0 \pi_k(\mathcal{G})

eq 0 for some k > 0 k\gt 0 .

Hence 𝒜 0 ≇ ℬ 0 × 𝒢 \mathcal{A}_0

cong \mathcal{B}_0\times \mathcal{G} and no global gauge choice is possible.

What does this mean for Yang-Mills Theory?

If we’re working on the lattice, then 𝒢 = G N \mathcal{G}= G^N , where N N is the number of lattice sites. We can choose not to fix a gauge and instead divide our answers by Vol ( G ) N Vol(G)^N , which is finite. That is what is conventionally done.

, where is the number of lattice sites. We can choose not to fix a gauge and instead divide our answers by , which is finite. That is what is conventionally done. In perturbation theory, of course, you never see any of this, because you are just working locally on ℬ \mathcal{B} .

. If we’re working in the continuum, and we’re trying to do something non-perturbative, then we just have to work harder. Locally on ℬ \mathcal{B} , we can always choose a gauge (any principal 𝒢 \mathcal{G} -bundle is locally-trivial). On different patches of ℬ \mathcal{B} , we’ll have to choose different gauges, do the path integral on each patch, and then piece together our answers on patch overlaps using partitions of unity. This sounds like a pain, but it’s really no different from what anyone has to do when doing integration on manifolds.

Gravity

The Asymptotic Freedom people want to do the path-integral over metrics and search for a UV fixed point. As above, they work in Euclidean signature, with M = S 4 M=S^4 . Let ℳℯ𝓉 \mathcal{Met} be the space of all metrics on M M , 𝒟𝒾𝒻𝒻 \mathcal{Diff} the group of diffeomorphism, and ℬ = ℳℯ𝓉 / 𝒟𝒾𝒻𝒻 \mathcal{B}= \mathcal{Met}/\mathcal{Diff} the space of metrics on M M modulo diffeomorphisms.

Pick a (fixed, but arbitrary) fiducial metric, g ¯ \overline{g} , on S 4 S^4 . Any metric, g g , can be written as g μ ν = g ¯ μ ν + h μ ν g_{\mu

u} = \overline{g}_{\mu

u}+ h_{\mu

u} They use background field gauge,

(2) ∇ ¯ μ h μ ν − 1 2 ∇ ¯ ν ( h μ μ ) = 0 \overline{

abla}^\mu h_{\mu

u}-\tfrac{1}{2}\overline{

abla}_

u(\tensor{h}{^\mu_\mu}) = 0

where ∇ ¯ \overline{

abla} is the Levi-Cevita connection for g ¯ \overline{g} , and indices are raised and lowered using g ¯ \overline{g} . As before, (2) defines a subspace 𝒬 ⊂ ℳℯ𝓉 \mathcal{Q}\subset \mathcal{Met} . If it happens to be true that 𝒬 \mathcal{Q} is everywhere transverse to the orbits of 𝒟𝒾𝒻𝒻 \mathcal{Diff} and meets every 𝒟𝒾𝒻𝒻 \mathcal{Diff} orbit precisely once, then we can imagine doing the path integral over 𝒬 \mathcal{Q} instead of over ℬ \mathcal{B} .

In addition to the other problems with the asymptotic safety program (the most grievous of which is that the infrared regulator used to define Γ k ( g ¯ ) \Gamma_k(\overline{g}) is not BRST -invariant, which means that their prescription doesn’t even give the right path-integral measure locally on 𝒬 \mathcal{Q} ), the program is saddled with the same Gribov problem that we just discussed for gauge theory, namely that there is no global section of ℳℯ𝓉 → ℬ \mathcal{Met}\to\mathcal{B} , and hence no global choice of gauge, along the lines of (2).

As in the gauge theory case, let ℳℯ𝓉 0 \mathcal{Met}_0 be the metrics with no isometries3. 𝒟𝒾𝒻𝒻 \mathcal{Diff} acts freely on the fibers of ℳℯ𝓉 0 → ℬ 0 \mathcal{Met}_0\to \mathcal{B}_0 . Back in his 1978 paper, Singer already noted that

π k ( ℳℯ𝓉 0 ) = 0 , ∀ k > 0 \pi_k(\mathcal{Met}_0)=0,\,\forall k\gt 0 , but 𝒟𝒾𝒻𝒻 \mathcal{Diff} has quite complicated homotopy-type.

Of course, none of this matters perturbatively. When h h is small, i.e. for g g close to g ¯ \overline{g} , (2) is a perfectly good gauge choice. But the claim of the Asymptotic Safety people is that they are doing a non-perturbative computation of the β \beta -functional, and that h h is not assumed to be small. Just as in gauge theory, there is no global gauge choice (whether (2) or otherwise). And that should matter to their analysis.