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

[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

 

[Urs Schreiber]

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

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.

 

[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>

 

[Urs Schreiber]

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

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. ;-)

 

[Giulio Prisco]

Beware that in algebraic topology special dimensions of manifolds appear all the time, in all kinds of theorems... But one needs to beware of being too naive about these numbers.

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).

 

[Urs Schreiber]

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... ;-) ;-)

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

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

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

 

[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-dimensional paths between two spacetime events dominate the path integral, then we'll always find that the universe behaves 4-dimensionally.

I would add the magic word "yet" to your last line.

 

[Urs Schreiber]

If 4-dimensional paths between two spacetime events dominate the path integral, then we'll always find that the universe behaves 4-dimensionally.

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?

 

[arivero]

... 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).

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?

 

[Giulio Prisco]

What makes your cardinaliy argument ignore exotic smooth structure on spaces other than Euclidean space?

Good point. This is a simple starting argument, of course it should be detailed and generalized.
Aren't non Euclidean spaces locally equivalent to Euclidean spaces? If so, shouldn't the argument carry over to non Euclidean spaces?

 

[torsten]

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

I will start only with a little overview what was done in the last 6 or 7 years. The informations above are a little bit outdated...
But at first let me state that nearly all 4-dimensional manifolds admit exotic differential structures. For compact 4-manifolds there are countable infinite many whereas for non-compact there are uncountable many. Only "simple" examples like the 4-sphere or S²×S² are not known to admit such structures but all specialist are sure there are many.
Exotic space like R⁴ or S³×ℝ have very special properties which one would expect for a spacetime in quantum gravity like there is a foam-like structure (better fractal structure) at small distances, there is no global splitting in space and time, there are topology changes of the space, one has tree-like splittings of substructures (surfaces etc.), exotic structures are not an effect of continuous spaces it goes also over to the triangulation (and therefore to a discrete structure).
All these properties have an impact on physics. Here I can only give a short overview:
- exotic smoothness produces counterexamples to the censorship conjecture (see Etesi's paper)
- it can produce the cosmological constant (see my recent paper)
- used in cosmology one gets inflationayr behavior (see the old paper but I'm writing a better approach now)
- it gives restriction on possible spacetimes in cosmology
- the fractal structure behaves like a quantum state (see for a general approach including quantization)
- this result can be used to get an idea of quantum gravity by using smooth manifolds (I called it Smooth quantum gravity)
- exotic smoothness dominates the path integral (see here but also other papers of Duston)
You mentioned also our approach to understand matter. This part of our work is not so strongly connected to exotic smoothness like the others above. We showed that complements of knots behave like fermions.
It is only a short overview but maybe helpful.

 

[Giulio Prisco]

I will start only with a little overview what was done in the last 6 or 7 years...

Thanks Torsten, this is most helpful!

very special properties which one would expect for a spacetime in quantum gravity like there is a foam-like structure (better fractal structure) at small distances...

This is a point where I wanted to get indeed. Intuitively, a "broken" or fractal spacetime seems an appropriate mathematical arena for quantum physics. Feynman (in "Quantum Mechanics and Path Integrals') and others noted this before the term "fractal" was invented.

But since I don't have an intuitive picture of exotic smoothness, I am unable to see the connection with fractals. I can follow the mathematics of differential geometry/topology, but it doesn't help much without an intuitive visual picture. I can think of standard 4D spacetime with a mental picture of a 2D curved surface embedded in 3D space, and that helps (within limits), but how should I imagine an exotic 4D spacetime?

 

[torsten]

This is a point where I wanted to get indeed. Intuitively, a "broken" or fractal spacetime seems an appropriate mathematical arena for quantum physics. Feynman (in "Quantum Mechanics and Path Integrals') and others noted this before the term "fractal" was invented.

But since I don't have an intuitive picture of exotic smoothness, I am unable to see the connection with fractals. I can follow the mathematics of differential geometry/topology, but it doesn't help much without an intuitive visual picture. I can think of standard 4D spacetime with a mental picture of a 2D curved surface embedded in 3D space, and that helps (within limits), but how should I imagine an exotic 4D spacetime?

Now I found some rest to answer this question: how does this space looks like?
For simplicity I will concentrate on exotic versions of S³×ℝ. For the exotic space, there is no splitting into space and time globally. Therefore, there is no global, smooth S³ for a fixed time. If one wants to describe it with a local time then one get the picture that the space undergoes a series of topology changes. Here is where this fractal nature come into play: one will obtain a wild 3-sphere. The equator of this 3-sphere, which is a 2-sphere, is known to be Alexanders horned sphere. Here are some pictures:

In most cases, the corresponding strcuture are more complex but I think you will get an impression how it looks

 

[Giulio Prisco]

Thanks Torsten! Following your explanation, I now have some intuitive grasp of exotic smoothness: Topologically spherical 2-dimensional cross-sections of exotic 4-dimensional manifolds can be fractals like Alexander's horned sphere, which doesn't look like a sphere at all though topologically it's a sphere.

Something like that, more or less. Or not? Of course I understand that this mental image is limited, but a limited mental image is much better than nothing and permits trying to think visually.

The possibility of fractals geodesics opens the door (intuitively, and I would like to make this intuition more precise) to quantum physics.

there is no global, smooth S3 for a fixed time. If one wants to describe it with a local time

Doesn't this mean that special relativity also follows?

 

[torsten]

Thanks Torsten! Following your explanation, I now have some intuitive grasp of exotic smoothness: Topologically spherical 2-dimensional cross-sections of exotic 4-dimensional manifolds can be fractals like Alexander's horned sphere, which doesn't look like a sphere at all though topologically it's a sphere.

Something like that, more or less. Or not? Of course I understand that this mental image is limited, but a limited mental image is much better than nothing and permits trying to think visually.

The possibility of fractals geodesics opens the door (intuitively, and I would like to make this intuition more precise) to quantum physics.

Doesn't this mean that special relativity also follows?

No special relativity is not effected. This local splitting is equivalent to the existence of a Lorentz structure at a manifold (a globally non-vanishing vector field).
But more is true in the exotic context: exotic smoothness admit locally hyperbolic structures and the isometry of the 3D hyperbolic space is the Lorentz group.

 

[mitchell porter]

The Donaldson invariants, that distinguish the different smooth structures of a 4-manifold, can apparently be computed in a "topologically twisted" form of super-Yang-Mills theory now known as Donaldson-Witten theory.

Maybe one could construct a kind of quantum gravity in which spacetime can tunnel from one smoothness structure to another.

edit: Of course! Twisting an Akbulut cork is the primordial example of changing the smooth structure. So if that could be represented as a form of quantum tunneling... Though if I read @torsten right, this couldn't be represented as a process in 4d spacetime, since it would necessarily cut across the time-slices? Perhaps it could be represented as tunneling, in a formal fifth dimension, between Euclidean "vacua", which can then be made Lorentzian via the usual procedure of analytic continuation.

 

[Giulio Prisco]

The Donaldson invariants, that distinguish the different smooth structures of a 4-manifold, can apparently be computed in a "topologically twisted" form of super-Yang-Mills theory now known as Donaldson-Witten theory.

Maybe one could construct a kind of quantum gravity in which spacetime can tunnel from one smoothness structure to another.

edit: Of course! Twisting an Akbulut cork is the primordial example of changing the smooth structure. So if that could be represented as a form of quantum tunneling... Though if I read @torsten right, this couldn't be represented as a process in 4d spacetime, since it would necessarily cut across the time-slices? Perhaps it could be represented as tunneling, in a formal fifth dimension, between Euclidean "vacua", which can then be made Lorentzian via the usual procedure of analytic continuation.

Interesting! What is the best reference work to read?

 

[mitchell porter]

Interesting! What is the best reference work to read?

I haven't forgotten this question, but I also haven't figured out (to my own satisfaction) how to think about this topic. I have a bunch of unanswered questions of my own:

Witten argued, from a field-theoretic perspective, that certain 10d gravitational instantons are exotic spheres; what do they correspond to in a proper stringy (worldsheet) description?

The "Donaldson-Witten" topological field theory which is used to detect the possibility of exotic smooth structures on a 4-manifold, I believe has a relationship to Ashtekar's formulation of general relativity, considered as a topological field theory. What are the implications of that?

There's a 2009 paper which actually incorporates exotic smooth structures into a gravitational path integral. There's also a 2010 paper about "topspin networks", an extension of spin networks meant to contain topological information too, which implies that the "piecewise linear" structure they describe, can be upgraded to describe smooth structure.

Also, veteran PF contributor @MTd2 was very interested in this topic.

 

[mitchell porter]

There is now (January 2025) a paper that has derived differences in physical properties from a difference in smooth structure.

"Physics on manifolds with exotic differential structures" (Chiapi-Ngamako and Paranjape)

It's a bit complicated. They start with the 7-sphere, which is the manifold on which provably distinct differentiable structures were first constructed - 28 of them altogether. They represent them as different fiberings of S^4 with copies of S^3. They suppose a gravitational metric on the S^7 and treat the S^3 as infinitesimal, i.e. Kaluza Klein. You end up with general relativity coupled to an SO(4) gauge field on S^4, where SO(4) is the isometry group of S^3.

In constructing a smooth structure on a manifold, you cover the manifold with overlapping coordinate patches and then have smooth transition functions in the overlap, explaining how to convert from one set of coordinates to the other. For these 7-dimensional smooth structures, there are two coordinate patches on each S^3, covering northern and southern hemispheres, and then the full smooth structure describes how these two S^3 patches twist around throughout the S^3 bundle over the S^4. When we do the Kaluza-Klein reduction to just the S^4, these twistings correspond to something different in the construction of the SO(4) gauge field and/or its configurations - I haven't understood this part, but there are parameters "h" and "l" which specify the smooth structure, which now also enter into the gauge field.

Then we look at the energy levels of Dirac spinors for different values of "h" and "l", and find that the energies are different (examples provided in equations 64 and 65). The authors remark that the map between different smooth structures, must have at least one non-differentiable point. This reminds me of how point impurities in condensed matter can alter the electron energy bands. I don't know how accurate this analogy is, but it's what I've got for now. :-)

On page 4, the authors briefly reflect on "what part of physical reality depends only on the notion of continuity, and not on the notion of differentiability". In their conclusions on page 13, the authors speculate that choice of smooth structure can make a physical difference in "condensed matter physics and Kaluza-Klein supergravity".