# A Physical meaning of "exotic smoothness" in (and only in) 4D

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1. Jan 9, 2018

### Giulio Prisco

I see that this has been discussed before, but the old threads are closed.

As Carl Brans and others note, it seems too big a coincidence to ignore.

Why is exotic smoothness "good" (in the sense that it permits richer physics or something like that)?

Exotic Smoothness and Physics,arXiv

"there are an infinity of differentiable structures on topological R4, no two of which are equivalent, i.e., diffeomorphic, to each other... The fact that R4Θ’s arise only in the physically significant case of dimension four makes the result even more intriguing to physicists..."

Exotic Smoothness and Physics: Differential Topology and Spacetime Models, book

2. Jan 9, 2018

### Urs Schreiber

The mathematical phenomenon of exotic $\mathbb{R}^4$s keeps seeming suggestive, but despite many pages filled with speculations over the decades, apparently there is still no concrete idea.

Often Witten 85 is cited as arguing that exotic smooth manifolds arise as "gravitational instantons". But Witten argues this only for the exotic spheres (the crux of the argument is on p. 12 of Witten 85) notably the exotic 7-sphere (as a fiber space for KK-compactification of 11d-supergravity to 4d).

In stark contrast to $\mathbb{R}^4$, it is open whether the 4-sphere $S^4$ admits any exotic smooth structure at all. If not, then, in view of how the whole theory of Yang-Mills instantons requires $\mathbb{R}^4$ to be thought of as one-point compactified to $S^4$ (here), the possible relevance of exotic $\mathbb{R}^4$s to 4d physics might evaporate.

Hence one should issue a warning here on basing dimensional numerology on few data points.

There are other, maybe more substantive arguments that serve to pick out special spacetime dimensions, notably dimension 4. For instance twistors work exactly only in dimensions 3 and 4 and 6 and to some extent in dimension 10 (here). This is the same numbers in which the Green-Schwarz superstring works. And one may trace this back to "first principles".

A detailed discussion of what is special about spacetime dimensions 3 and 4 and 6 and 10 from the point of view of spinors is in Mathematical Quantum Field Theory - 2. Spacetime.

3. Jan 9, 2018

### Giulio Prisco

Thanks Urs, I'll take a look at the material you provided. But allow me to indulge in some more dimensional numerology.

Some concrete ideas of how exotic R4 topology could be relevant to physics is exactly what I am looking for: What does it mean?

I am just starting to read up about this, but I see that there are speculations that exotic smoothness could represent a promising way to achieve the "geometrization of matter" that Einstein was looking for, or an entry point for quantum physics in general relativity.

<Mod note: profanity removed>

Last edited by a moderator: Jan 9, 2018
4. Jan 9, 2018

### Urs Schreiber

Yes. What I was trying to say is that many people have asked this question over the decades, and nothing tangible has emerged yet, it seems.

Beware that in algebraic topology special dimensions of manifolds appear all the time, in all kinds of theorems, just think of the Hopf invariant one theorem or the Kervaire invariant one theorem. Lots of numbers here, and lots of mathematical structures also seen in physics. But one needs to beware of being too naive about these numbers.

The algebraic topologist Jack Morava once made a half-joking observation about extracting the critical dimensions 10 and 26 of bosonic- and of super-string theory, respectively from the stable homotopy groups of spheres (I am taking the liberty of paraphasing from a private email by Jack Morava):

The first systematic family of $p$-torsion elements in the stable homotopy groups of spheres is the family $\alpha_k$, whose first element $\alpha_1$ lives in degree $2p-3$, e.g. at the prime two the first element $\alpha_1$ is the element $\eta$ related to the Möbius strip. These elements have nice geometric constructions, via work of Adams, Sullivan, Quillen and others on the J-homomorphism.

The next systematic family of $p$-torsion elements is called $\beta_k$; they are pretty well-understood through work of Miller, Ravenel and Wilson back in the 70s. The degree of $\beta_k$ is something like $2k(p^2-1) +$ a linear function of $p$.

The article
• Oka, Toda, "3-primary β-family in stable homotopy". Hiroshima Math. J. 5 (1975), no. 3, 447–460.
is one of the first papers on the subject; they show that there are certain elements $\beta_k$ in degree $16k - 6$ at the prime $p = 3$.

Morava's observation, intended at the time as a joke, was that the first of these elements is in dimension 10 and the second is in dimension 26; and that if there is a good theory of everything, it obviously should involve those elements somehow.

If you ever need an example of a really sophisticated joke, this is one. ;-)

Last edited: Jan 13, 2018
5. Jan 9, 2018

### Giulio Prisco

Yes, I guess if one wants to show that any number is special there's always a way.

Let me just add one more speculation (trigger warning, anthropic reasoning ahead):

If we consider non-equivalent differentiable structures on RN as different possible mathematical universes, since there is only a countable infinity of such things for N not equal to 4 (one for each N), but an uncountable infinity for N equal to 4, then the probability that our mathematical universe has 4 dimension is 1... ;-) ;-)

(I guess this half-joke could be formulated much more rigorously).

Last edited: Jan 9, 2018
6. Jan 13, 2018

### Urs Schreiber

That's a common thought. See Aaron Bergman's comment to the reply here.

No, the problem with these arguments is that nobody knows how to make them rigorous.

7. Jan 15, 2018 at 10:09 AM

### Giulio Prisco

Thanks Urs.

Bergman's comments is: "Oh, plenty of us know about the exotic structures on R4. It's a not entirely uncommon joke that the uncountable number of them dominates the path integral, and that's the reason why we live in 4 dimensions. The problem with such jokes and other things along these lines is that's it's been pretty much impossible to turn them into an actual theory."

I wasn't thinking of formulating this argument in terms of the path integral, but now that I think of it, it sounds like a cool idea. If 4-dimentional paths between two spacetime events dominate the path integral, then we'll always find that the universe behaves 4-dimensionally.

8. Jan 15, 2018 at 12:54 PM

### Urs Schreiber

You are using the words "spacetime events" in a funny way here, but I know what you really mean. Some questioons you should ask yourself:

What's the path integral? What's the difference between "we'll find" and "we'll always find"? What makes your cardinaliy argument ignore exotic smooth structure on spaces other than Euclidean space?

9. Jan 15, 2018 at 6:58 PM

### arivero

7-dim spaces are also nice to discuss, as 7+4=11. Now, are all the interesting exotic 7 spheres always some fibration of S4 by S3? Should that mean that KK compactification produces SO(5)xSO(4)? always?

Last edited: Jan 15, 2018 at 7:34 PM