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June 19, 2015

Asymptotic Safety and the Gribov Ambiguity

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

  1. Everything I am about to say was known to Iz Singer in 1978. Though, as with the corresponding result for nonabelian gauge theory, the import seems to be largely unappreciated by physicists working on the subject.
  2. I would like to thank Valentin Zakharevich, a very bright young grad student in our Math Department for a discussion on this subject, which clarified things greatly for me.

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 GG. For definiteness, we’ll talk about the Euclidean path integral, and take M=S 4M= S^4. Fix a principal GG-bundle, PMP\to M. We would like to integrate over all connections, AA, on PP, 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 PP, 𝒢\mathcal{G} the (infinite dimensional) group of gauge transformations (automorphisms of PP which project to the identity on MM), and =𝒜/𝒢\mathcal{B}=\mathcal{A}/\mathcal{G}, the gauge equivalence classes of connections.

“Really,” what we would like to do is integrate over \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 PP, and parametrize any other connection A=A¯+Q A= \overline{A}+Q with QQ a 𝔤\mathfrak{g}-valued 1-form on MM. Background field gauge is

(1)D A¯*Q=0D_{\overline{A}}* Q = 0

which picks out a linear subspace 𝒬𝒜\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\mathcal{A}\to \mathcal{B} and integrating over \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\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 \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 0𝒜\mathcal{A}_0\subset \mathcal{A}, on which 𝒢\mathcal{G} acts freely. We then have 𝒜 0 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 0\mathcal{A}_0\to \mathcal{B}_0, then we would have established 𝒜 0 0×𝒢 \mathcal{A}_0\cong \mathcal{B}_0\times \mathcal{G} But Singer proves that

  1. π k(𝒜 0)=0,k>0\pi_k(\mathcal{A}_0)=0,\,\forall k\gt 0. But
  2. π k(𝒢)0\pi_k(\mathcal{G})\neq 0 for some k>0k\gt 0.

Hence 𝒜 0 0×𝒢 \mathcal{A}_0\ncong \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 NN is the number of lattice sites. We can choose not to fix a gauge and instead divide our answers by Vol(G) NVol(G)^N, 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 \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 \mathcal{B}, we can always choose a gauge (any principal 𝒢\mathcal{G}-bundle is locally-trivial). On different patches of \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.


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 4M=S^4. Let ℳℯ𝓉\mathcal{Met} be the space of all metrics on MM, 𝒟𝒾𝒻𝒻\mathcal{Diff} the group of diffeomorphism, and =ℳℯ𝓉/𝒟𝒾𝒻𝒻\mathcal{B}= \mathcal{Met}/\mathcal{Diff} the space of metrics on MM modulo diffeomorphisms.

Pick a (fixed, but arbitrary) fiducial metric, g¯\overline{g}, on S 4S^4. Any metric, gg, can be written as g μν=g¯ μν+h μν g_{\mu\nu} = \overline{g}_{\mu\nu}+ h_{\mu\nu} They use background field gauge,

(2)¯ μh μν12¯ ν(h μ μ )=0\overline{\nabla}^\mu h_{\mu\nu}-\tfrac{1}{2}\overline{\nabla}_\nu(\tensor{h}{^\mu_\mu}) = 0

where ¯\overline{\nabla} 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 𝒬ℳℯ𝓉\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 \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\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 0\mathcal{Met}_0\to \mathcal{B}_0. Back in his 1978 paper, Singer already noted that

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

Of course, none of this matters perturbatively. When hh is small, i.e. for gg 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 hh 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.

Note: Since someone will surely ask, let me explain the situation in the Polyakov string. There, the gauge group isn’t 𝒟𝒾𝒻𝒻\mathcal{Diff}, but rather the larger group, 𝒢=𝒟𝒾𝒻𝒻Weyl\mathcal{G}= \mathcal{Diff}\ltimes \text{Weyl}. And we only do a partial gauge-fixing: we don’t demand a metric, but rather only a Weyl equivalence-class of metrics. That is, we demand a section of ℳℯ𝓉/Weylℳℯ𝓉/𝒢\mathcal{Met}/\text{Weyl} \to \mathcal{Met}/\mathcal{G}. And that can be done: in d=2d=2, every metric is diffeomorphic to a Weyl-rescaling of a constant-curvature metric.

1 To get the right measure on 𝒬\mathcal{Q}, we need to use the Fadeev-Popov trick. But, as long as 𝒬\mathcal{Q} is transverse to the gauge orbits, that’s all fine, and the prescription can be found in any textbook.

2 For more general choice of MM, we would also have to require H 2(M,)=0H^2(M,\mathbb{Z})=0.

3 When dim(M)>1dim(M)\gt 1, ℳℯ𝓉 0(M)\mathcal{Met}_0(M) is dense in ℳℯ𝓉(M)\mathcal{Met}(M). But for dim(M)=1dim(M)=1, ℳℯ𝓉 0=\mathcal{Met}_0=\emptyset. In that case, we actually can choose a global section of ℳℯ𝓉(S 1)ℳℯ𝓉(S 1)/𝒟𝒾𝒻𝒻(S 1)\mathcal{Met}(S^1) \to \mathcal{Met}(S^1)/\mathcal{Diff}(S^1).

Posted by distler at June 19, 2015 3:11 AM

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Re: Asymptotic Safety and the Gribov Ambiguity

It’s actually not clear whether there’s a Gribov problem for quantum gravity. This has ben investigated most carefully for the canonical path integral, where the relevant integration variables are the three-metric mod diffeos and the corresponding momentum. Here, Fischer and Moncrief showed back in 1996 that for many spatial topologies, the bundle is, in fact, trivial, and there is no Gribov problem – see Gen. Rel. Grav. 28 (1996) 221.

I know much less about the full four-dimensional case. But it’s a plausible conjecture that for a manifold with the topology Rx(one of the manifolds considered by Fischer and Moncrief), there is no four-dimensional Gribov problem.

(One minor subtlety: this all breaks if you include diffeos that are not in the identity component as “gauge” symmetries. But there are reasonable arguments that you shouldn’t.

Posted by: Steve Carlip on June 19, 2015 3:17 PM | Permalink | Reply to this

Re: Asymptotic Safety and the Gribov Ambiguity

Well, among their conditions are

  1. M 3M_3 should not admit any Riemannian metric with a continuous isometry group.
  2. All of the higher homotopy groups of M 3M_3 should vanish: π i(M 3)=0,i2\pi_i(M_3)=0,\, i\geq 2.

Both of these conditions are badly-violated by M 3=S 3M_3=S^3.

And, certainly, the analogous conditions (if there were a similar theorem for 4-manifolds) would be badly violated by M=S 4M=S^4, which is the manifold used by the Asymptotic Safety people.

I was careful to restrict my statements about the gravity case to M=S nM=S^n. As you are pointing out, the topology of 𝒟𝒾𝒻𝒻(M)\mathcal{Diff}(M) depends rather sensitively on the topology of MM.

Posted by: Jacques Distler on June 19, 2015 3:59 PM | Permalink | PGP Sig | Reply to this

Re: Asymptotic Safety and the Gribov Ambiguity

I recently found this comment on Peter Woit’s blog:

“This and other threads of late have gotten me to thinking… It’s no secret that we cannot test string/multiverse theory and will not be able to anytime soon. It’s doubtful we’ll be able to in our lifetimes. But… what if we could? Would anything be different? Suppose we actually did have a working solar system-sized collider or similar device that allowed us to probe Planck-scale distances and energies. This is all speculative of course, but it’s reasonable to assume we’d find new particles and/or symmetries, a viable inflaton candidate perhaps, and more. What would we do with these discoveries? It seems to me that even if all this comes to pass we’ll still be faced with a fundamental problem.

Historically, successful theories have been able to make predictions because they offered fundamental paradigm shifts that led to them via their formalism. General relativity predicted things like gravitational lensing and the perihelion shift of Mercury’s orbit because it postulated a fundamentally different sort of space-time than classical mechanics did, and new field dynamics to go with it. The SM proposed actual underlying symmetries in the universe unlike those of its predecessors, and it was those that led to predictions of otherwise unexpected new particles, including the Higgs. All viable theories come in two parts: 1) A paradigm that proposes some new physical entity, state, or behavior; and 2) A mathematical framework that describes it. Successful theories are validated not by their ability to mathematically describe observation, but by their ability to verify their underlying paradigms within the frameworks they propose.

So here we are… our amazing super-duper-Planck-collider has scraped the universe right up to Planck scale energies and filled our databases with new particles, fields, etc. I suspect that regardless of what we find, all of that data will fit quite nicely with at least one possible string vacuum state. We will only have discovered which of the the 10^500 theoretically possible string vacua describes our universe. This raises a dilemma. For string/M-theory, the new paradigm is string/brane objects embedded in extra compactified dimensions. The mathematical framework that formalizes it has proven to be powerful but as we’ve seen, fluid enough to describe anything, including virtually anything our super-duper-Planck-collider manages to find. So the question is… how does the paradigm get verified here? How do we know that our “theory” of reality is an explanation of new physics in our universe and not simply an arcane but beautiful mathematical description of it?

The only way around this dilemma I can imagine would be to verify a multiverse… that is, to somehow directly observe other regions with different string vacuum states. But that is impossible, in principle as well as in practice. Unless I’m missing something, it seems to me that regardless of how elegant or workable the string/M-theory formalism is or how well it describes our observations, if it does not provide a way to verify its underlying paradigm it is nothing more than an arcane but lovely mathematical framework. It describes everything, but explains nothing… even if we do have Planck-scale observations.


Posted by: Interesting on August 13, 2015 3:11 PM | Permalink | Reply to this

Not over here.

My “thought” is that, if you want to have a discussion about stuff on Peter Woit’s blog, then you should have that discussion on Peter Woit’s blog.

He has a comment section over there; please use it.

Posted by: Jacques Distler on August 13, 2015 3:16 PM | Permalink | PGP Sig | Reply to this

Re: Asymptotic Safety and the Gribov Ambiguity

What do you think of the physics blog posts from Lubos Motl? Should I trust them?

Posted by: Jacob on August 15, 2015 12:04 AM | Permalink | Reply to this

Re: Asymptotic Safety and the Gribov Ambiguity

What are the latest interesting developments in string theory? Any promising avenues you’d like to talk about?

Posted by: Jesse on August 19, 2015 1:38 PM | Permalink | Reply to this

Re: Asymptotic Safety and the Gribov Ambiguity

What happens if the LHC doesn’t find SUSY? What would it mean?

Posted by: Detective Holder on August 20, 2015 10:54 PM | Permalink | Reply to this

Re: Asymptotic Safety and the Gribov Ambiguity

What do you think of this paper?

Posted by: thyme on October 2, 2015 2:30 PM | Permalink | Reply to this

Re: Asymptotic Safety and the Gribov Ambiguity

Hello, please have a look at this. Finally, a prediction for M-theory:

So I guess if no such gluino is found then, according to Kane, M-theory must be wrong?

Posted by: M-theory on November 23, 2015 10:18 AM | Permalink | Reply to this

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