# Summing over topologies

Sometimes in quantum cosmology, when we are thinking about 'wave functions of the universe' we have in mind some sort of formal path integral, where we include not just the variations in the dynamical fields (metric and so forth), but also possibly some sort of prescription for summing over all possible topologies. At least, that seems to be some sort of heurestic guess often encountered in the literature (Hawking et al).

Now, in String theory, admittedly a different context but there is a rather well defined notion of how this works (basically as a generalization of Feynman graphs over Riemann surfaces).

However I don't understand exactly how this is supposed to work in quantum cosmology exactly (say in the Wheeler-De Witt context). First of all there seems to be a massive amount of overcounting already at the metric level (where presumably one needs to mod out by all the diffeomorphisms), but how exactly does one deal with the topologies? Do they only include connected topologies? Is there even a mathematical formalism on how to even approach this problem? Any good papers that deal with this in a comprehensive way?

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Such an excellent question! One pessimistic answer is that the sum over metrics is only a semi-classical statement, correct is some saddle point approximation to something more accurate. We then likely need to know all the details of such microscopic description before making sense of quantum cosmology, including such questions as which saddle points we include in the path integral in the semi-classical limit. I'd be curious about less pessimistic statements though... –  user566 Jan 16 '11 at 2:25

it's an important question but the available answers don't fully match this importance. However, there are several facts that are known:

1. First of all, you mentioned the redundancies coming from diffeomorphisms. Of course, any path integral calculation has to be divided by the diffeomorphism group, i.e. to guarantee that each class of geometries (or, more generally, configurations) that are diff-equivalent (or, more generally, gauge equivalent: diff symmetry is a kind of gauge symmetry) are only counted once. For gauge symmetries, the BRST quantization with the Faddeev-Popov ghosts (originally introduced by Feynman) is a poweful tool to do so. In supergravity, its extension with antighosts etc. has to be used and is known as the BV formalism which is mathematically heavier and "more nonlinear" than the BRST formalism, so only dozens of people in the world can really use the BV formalism.

2. In the world sheet description of perturbative string theory, the summation over topologies is well-defined and produces Feynman diagrams of increasing order in the string coupling constant, as the number of "handles" increases or the Euler character of the manifold decreases. The reason why this expansion in terms of the metric works is that the metric in the world sheet theory is a fundamental, exact degree of freedom - so we're dealing with the correct theory. The metric is valid up to arbitrarily short distances on the world sheet because the world sheet theory is scale-invariant (at least at short distances) so if the metric is OK at one scale, it has to be right at any scale.

3. In quantum gravity in 3+1 dimensions or higher (and I will skip the special 2+1-dimensional case which has no physical polarizations of the graviton, which makes it look like things work "automatically" but they actually don't, even in this case, if one is accurate enough), the metric tensor is not a fundamental object that is perfectly well-defined at any length scale. Instead, near the Planck scale (and sometimes lower scale such as the string scale), it is supplemented by infinitely many massive fields that the metric can mix with and create non-trivial topological configurations that "knit" both the metric and other degrees of freedom etc. So the probable main reason why no known functioning description of quantum gravity in terms of a path integral exists for the metric in 3+1 dimensions or higher is that it is a wrong problem.

Quantum gravity is an extremely constrained problem and the "quantization of the metric" without any extra degrees of freedom and any extra powerful arrangements simply cannot produce a consistent theory, to all orders. At least, about 50 years of the efforts to "map the landscape of possibilities" how to define a theory of quantum gravity suggest that treating the metric tensor as the right and only degree of freedom in quantum gravity is a wrong way to proceed. So the quantization of the metric is only consistent as an approximation that only tells us how gravity works up a certain order in the expansion in the "strength of the gravitational field" - with the order measured by powers of $\hbar$. However, any topology change is inevitably a non-perturbative effect which is much more inaccessible than a 2-loop or 50-loop contribution. The quantization of general relativity can't even produce the right, unambiguous 2-loop contributions to the effective action and dynamics (because of well-known non-renormalizable divergences). So of course, it is also unable to produce different-topology-contributions to the effective action because they're equivalent to infinitely-many-loops contributions.

In supergravities, the non-renormalizable loop divergences may cancel at 2 loops and may appear at $k$ loops where $k$ is (much) greater than 2. In $N=8$ supergravity in $d=4$, all the perturbative divergences probably cancel. But the non-perturbative physics is still inconsistent - and the spontaneous topology change is always a non-perturbative effect. There are many ways to see that the supergravity Lagrangian cannot know about the nonperturbative effects. For example, its exact continuous exceptional noncompact symmetry is incorrect; it must be broken to its discrete group by quantum effects such as the Dirac quantization rule that guarantees that various electric and magnetic charges have to be quantized and belong to inverse lattices (with respect to one another). So even $N=8$ supergravity without anything else is unable to calculate the impact of topology-changing histories.

People have speculated about many phenomena that quantum gravity, if it could be expressed as a path integral, could produce because of the summation over topologies. In particular, one could generically sum over spacetimes that always have some tiny handles or wormholes: this picture of the typical contribution to the path integral, envisioned by Wheeler, is referred to as the quantum form. Moreover, splitting and reconnecting "baby universes" were also described in the literature and they were hypothesized to be able to re-adjust values of coupling constants.

But I won't mention any paper (with one exception below) because no paper of this kind that leads to any convincing calculation of anything exists. It has been pretty much established that a consistent theory of quantum gravity cannot be defined in the way you (and others) suggested. And many pictures that emerged from the only consistent theory of quantum gravity we have, string/M-theory, indicate that the picture of summing over many topologies is inadequate. In some vague sense, the new phenomena that have to exist in a functioning theory of quantum gravity make the short-distance spacetime more peaceful and de facto remove the contributions of nontrivial topologies.

Well, it can't be a 100% accurate description and I will mention one paper by Iqbal, Nekrasov, Okounkov, and Vafa, anyway.

http://arxiv.org/abs/hep-th/0312022

In topological string theory, which may be thought of as a truncation of the spectrum of the full string theory where only special "holomorphic" states are allowed (which removes most of the dynamical states from the bulk), it is true that the whole path integral - or partition function - may be written down as a summation over all topologies of Calabi-Yau manifolds. So at least in this limited context, well-defined observables - partition sums of a system that may also be defined as a melting crystal - may be calculated by summing over infinitely many topologies of 6-dimensional geometries (which is higher than 2-dimensional ones).

However, no corresponding calculation that would actually have an answer that has any relationship to the observations or other methods to justify that the answer is correct is known in any "physical" theories of quantum gravity with dynamical gravitons - in theories that actually contain the gravitational attraction (where the objects have a variable momentum). So at this moment, it surely looks like that the metric tensor is just an "effective" degree of freedom and this degree of freedom can be only used for problems at distances much longer than the Planck length. In particular, only calculations up to a fixed number of quantum loops - usually 0 or 1 (the classical limit and the semiclassical calculations) - can be done using this metric tensor.

Non-trivial topologies that don't involve any curvature with curvature radius comparable to or smaller than the Planck scale become irrelevant in this picture because they're suppressed by something like $\exp(-A/A_{Planck})$ where $A$ are the typical $(D-2)$-dimensional areas of the "handles" that modify the topology. Needless to say, because $A_{Planck}$ is a tiny area, the exponential is zero for all practical (and most of the impractical) purposes. The exponentially suppressed term is much smaller than many other terms that are already incalculable.

Some of the answers above may get modified in the future and the absence of such a path-integral description with new topologies may sometimes be blamed on our current ignorance. It is plausible that a spacetime-based path-integral approach with different topologies exists; it remains puzzling how such a description can ever be consistent with the holographic principle (which says that the entropy in a finite region is finite and bounded by the surface). However, it seems almost guaranteed that such a hypothetical description would also include many other - perhaps dynamically generated - degrees of freedom besides the metric. The non-existence of a "pure gravity" in more than 3+1 dimensions seems to be a claim with an extensive amount of circumstantial evidence - a claim independent of any particular description - and is unlikely to be due to our lack of imagination. It seems to be a by-product of the research of quantum gravity.

When research gets really profound, people sometimes find out that some of their basic assumptions - or "dogmas" - were really invalid. The possibility to define quantum gravity in terms of the "metric tensor only" seems to be an example of such a pre-scientific superstition.

Best wishes Lubos

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Thank you Lubos and Moshe. Based upon my search, I tend to agree with your characterization of the state of the art. Actually it is a little bit frustrating, b/c this subject seems to largely be lore or unpublished. Almost no calculations to be found, even in simple contexts. The topological string paper is interesting, and what I would call a well defined toy model, however it is unfortunately largely outside my ken (toric varieties... ouch) –  Columbia Jan 17 '11 at 3:19
On a more pedestrian level, regarding the diffeomorphism group part and the Fadeev Popov procedure. Do you know of an explicit calculation where the full gauge group (eg Diff(M) is explicitly modded out of some problem (even a toy problem, say in lower dimensions)? There too I see a lot of papers where the author talk about doing it, but almost no paper where the calculation is actually done. Technically, it is of course far more challenging than what we know how to calculate from Yang Mills –  Columbia Jan 17 '11 at 3:26
@Lubos "guarantee that each class of geometries (or, more generally, configurations) that are diff-equivalent (or, more generally, gauge equivalent: diff symmetry is a kind of gauge symmetry) are only counted once." I wish this statement made any sense to me: All we know is that diffeomorphisms are a symmetry group of GR, i don't understand this stubbornness to believe it is also a gauge group. We don't even know if the gauge group formalism should be applied to such monster as GR. When we deal with symmetry groups, we never think that we have to sum 'once', but actually [continued] –  lurscher Feb 4 '12 at 15:37
that we multiply the result obtained once with the quotient space. Sorry if i'm not making any sense to you, but i feel this is a very legitimate concern –  lurscher Feb 4 '12 at 15:39