# Classification of manifolds and smoothness structures

1. Jun 3, 2013

### tom.stoer

Except for the work of Torsten I am not aware of any paper which discusses these topics.

Some ideas:
1) all theories for QG I am aware of do either use manifolds and smootheness (string theory, geometrodynamics, shape dynamics, ...) or are constructed from them (LQG, CDT, ...)
2) in some theories the number of dimensions is fixed, whereas in other theories they become dynamical (spectral dimension)
3) we know that piecewise-wise linear and smooth manifolds are not necessarily the same
4) we know that for dim > 3 the classification of manifolds is not decidable, that there is no algorithm to distinguish between arbitrary manifolds, and that manifolds cannot be "listed"
5) we know that in dim = 4 there are in general uncountably many smoothness structures for a given (non-compact) manifold
5') however if we restrict the domain to manifolds which allow for a global foliation M4 = R * M3 then (3-5) become trivial
6) due to (4) we are not able to define a "sum over all manifolds" and due to (5) we are not able to define a "sum over all smoothness structures for a given manifold"
7) in all approaches I am aware of a kind of gauge fixing is required; therefore we need to know the diffeomorphism group / mapping class group and the fibre bundle topology (Gribov ambiguities) for local Lorentz invariance (for the connections and n-Beins)

Is there any paper discussing these topics and their relevance in the contex of QG?

Last edited: Jun 3, 2013
2. Jun 9, 2013

3. Jun 9, 2013

### Physics Monkey

I have no answers, but I think its a great question. Presumably we will realize (or already know?) that summing over manifolds is a bit too naive. For example, even in ordinary quantum mechanics in the path integral language most paths are not smooth.

It should be lattice models all the way down :)

4. Jun 9, 2013

### atyy

http://arxiv.org/abs/1112.5104
"If we try to join the two main messages of Einstein and Feynman we get the equation

Quantum Gravity = Random Geometry. (2)

There is quite a wide agreement on this equation among all the main schools on quantum gravity, although neither string theory nor LQG take it as their starting point. The problem is that geometry is rich. Especially in three or four dimensions there does not seem to be a unique way to put a canonical probability theory on it.

We can take inspiration from another outstanding idea of Feynman, namely to represent quantum histories by graphs. We feel that it is perhaps not sufficiently emphasized in the graph theory community that quantum field theory and Wick's theorem provide a canonical measure ..."

5. Jun 10, 2013

### tom.stoer

atty, I understand the basic idea: let geometry emerge from some per-geometric structure; but there are approaches like asymptotic safety relying on smooth manifolds all the way down; so what I am asking for are the mathematical structures we expect to be present in

$\sum_\text{top}\;\int_\text{diff}\;\ldots$

where "diff" are diffential structures for a specific topology "top"

All questions refer to this idea

6. Jun 10, 2013

### Haelfix

hey Tom.

It's a very good question, but ultimately that's the central problem of quantum gravity and the snag where everyone hits the proverbial brick wall for many of the same reasons you list. Namely it's not even obvious how to set up and partition the sum much less solve it.

The state of the art basically doesn't even address the question directly.
For instance a recent paper by Maloney and Castro sets up the sum in 2+1 Ads space, and you can already see the complications they run into. For instance, do you include nontrivial diffeomorphisms in the sum? How do you weigh the geometries etc. The authors make plausible choices, but it's still not obvious if those correspond to the real physics and there are multiple potential technical snags lurking. http://arxiv.org/abs/1111.1987

In a different vein, this type of calculation is easier to write down in perturbative string theory (sum of Riemann structures) and topological string theories but even there you quickly run into calculational complexities.

Even the masters of these types of game (eg Witten) doesn't have much to say in 4d.

I believe a lot of people are skeptical regarding whether it's even a sensible question. Eg This sum over geometries might only be a semi classical artifact valid only in very specific circumstances.

7. Jun 10, 2013

### tom.stoer

yes, of course

yes, why not?

I know; nevertheless it may be interesting to take it seriously.

The interesting thing is that if you define the counting AND sum over ALL dimensions (!) then the sum is naturally peaked at dim=4 due to the uncountably many smoothness structures for non-compact 4-manifolds.

8. Jun 10, 2013

### atyy

Does the question remain if we ask for a canonical formulation, instead of a path integral?

9. Jun 10, 2013

### tom.stoer

I am not sure.

The problem with the canonical formalism is that it requires a global foliation M4 = R * M3. Doing this changes (or restricts) the allowes 4-manifolds. For the remaining 3-manifolds some questions (3-5) become trivial or are at least solvable, as I said in (5')

However the key question is whether this restriction is physical (and therefore QG must be based on this restricted subset of 4-manifolds) or whether the restriction is unphysical (and the resulting theory is incomplete or wrong).

10. Jun 10, 2013

### atyy

I guess I wasn't thinking necessarily of canonical LQG - but more generally - like if the path integral form of the theory summing over dimensions and smoothness structures existed, presumably one could get from there to a Hilbert space and unitary evolution or some sort of canonical-like formulation?

11. Jun 10, 2013

### tom.stoer

I don't believe in the PI as a method to construct the canonical formalism rigorously. And I don't believe in the PI as fully equivalent formalism in general.

In the restricted domain the canonical formulation is the better starting point. In the unrestricted domain no canonical formulation will exist - neither constructed directly, nor indirectly via the PI. I think topology and smoothness structures a the weak point of all approaches to QG I know - except for thorsten's work which I still do not really understand.

Afaik in 4-dim. even piecewise linear structures (related to discrete approaches like CDT) can be devided into inequivalent equivalence classes.

12. Jun 10, 2013

### torsten

Tom,
these a really important questions.
I know only of one paper (unpublished) which adress them:

Quantum general relativity and the classification of smooth manifolds
by H. Pfeiffer
http://arxiv.org/abs/gr-qc/0404088

But let me also discuss some of your points:

Point 3) Is only true for manifolds of dimension 8 or higher. All manifolds of dimension 7 or smaller have equivalent Piecewise-linear (PL) and smooth structure. So the sum over PL is equivalent to a sum over smooth manifolds

Point 5') Is misleading: Is the global foliation smooth or only continuous? For instance (3-space) cross Line has always uncountable many smooth structures.

Point 4) the decidability depends only on the fundamental group (word problem). But if you assume causality then you have at least demand that spacetime is simply connected (trivial fundamental group). We try to calculate the "sum over smoothness structures" with partly success.

Point 7) I would add to gauge fixing: the group of large diffeomorphisms (not connected to the identity) can be very complicated beginning wth dim 3. In the "sum over structures" you have to include them.

If you want to read a non-technical summary of our approach. I took part of the FQXi essay contest this year. Here is the link:
http://fqxi.org/community/forum/topic/1780

13. Jun 10, 2013

### torsten

Here I completely agree with. In the last two years we analyzed more concrete examples like S^3xR and found some interesting features.
In a canonical approach, one starts with NxR with N a 3-manifold. This spacetime has to be choosen also smoothly, otherwise one contradicts the strong causality. But strong causality means that one has a unique Cauchy surface (some N for a time) and one has unique geodesics starting from the Cauchy surface and going to the future and to the past. These conditions are enough to show that one needs NxR smoothly. Every canonical formulation starts with these assumptions. But what does it mean? If I assume an open future (at least for a quantum system) then why I have to assume unique geodesics? Instead I would assume that there saddle points where I have branching geodesics. Surprisingly we found such structures in exotic smooth 4-manifolds.

14. Jun 10, 2013

### tom.stoer

Torsten, I do really appreciate the information you provide!

15. Jun 11, 2013

### TrickyDicky

Good point, this should give us some clue.
And then there are those approaches that point towards getting rid of formulations in terms of Lorentzian manifolds (a lá Connes non-commutative algebra) because they seem too limiting but I think those are not introducing gravity yet..

16. Jun 11, 2013

### tom.stoer

Connes still relies on a differentiable (spin-) manifold structure, otherwise he would not be able to introduce a Dirac operator; in that sense his approach does not solve any of the above mentioned problems, as far as I can see

17. Jun 11, 2013

### TrickyDicky

You are right it doesn't solve the OP problems.

18. Jun 11, 2013

### tom.stoer

But honestly, I didn't think about the (differential) topological properties of non-commutative geometry.

19. Jun 11, 2013

### Physics Monkey

Maybe I'm confused, but does Connes really require this? I thought one was, at some level, just working with an algebra of non-commutative degrees of freedom plus extra data specifying the Dirac operator. This algebra indirectly determines a "non-commutative manifold" in the sense that the usual commutative algebra of functions plus extra data also determines a conventional commutative manifold.

20. Jun 11, 2013

### marcus

In what I've read there is no smooth manifold required---instead there is a bunch of axioms that say (among other things) how the Dirac operator should behave.

In special cases/examples he can start with an algebra of functions on a manifold, so there is a manifold in the picture. But there does not have to be.

Tom may be thinking of a special case relevant to physics where the algebra is "almost commutative" in the sense that a big piece of it is constructed by taking the functions on a conventional manifold (and that part of the algebra is commutative) but then a little non-commutative piece is tacked on. The direct sum of the two is non-commutative. this construction comes up when they want to realize the standard particle model.

Last edited: Jun 11, 2013