There’s a nice article in Nature News about the solution to the Kervaire invariant problem mentioned here. It’s an excellent and accurate description of the result and its significance, except for the last paragraph, on “Brane science”, where the author can’t resist following the convention of appending some nonsensical hype about string theory:
Because the new approach involves looking at topological problems of a manifold from the perspective of a space that has one more dimension, it is analogous to the use of one-dimensional strings as the basis of zero-dimensional (point-like) fundamental particles. Similarly, it has become popular for cosmologists to study the behaviour of space-time from the perspective of higher-dimensional ‘branes’ that interact with one another. This is why studying the Kervaire invariant problem might offer useful mathematical techniques to fundamental physics.
Update: This news is now featured on the AMS web-site, together with the misleading hype about strings and branes:
Ball explains “although it looks at face value to be extremely abstruse, the mathematics involved in the solution might be relevant to quantum theory and string theory, not to mention brane theory, which has been invoked to explore some issues in Big Bang cosmology.”
For a framed 4k+2 manifold Kervaire defined a refined intersection pairing (on middle dimensional homology). However the refinement is non-trivial only if the exist certain elements in the stable homotopy groups of spheres that are represented by elements in Ext groups. Hill-Hopkins-Ravenel show these elements don’t exist for 4k + 2 != 2,6,14,30,62, or 126.
The case 4k + 2 = 6 is very relevant to string theory. See for example
Hopkin’s ICM address: math.AT/0212397
Loosely speaking, the existence of certain quadratic refinements is essential for string/M-theory to avoid potential anomalies.
Of course the new results are about k >> 1, so its not clear what the implications for physics will be.
There’s much to learn about tmf and index of Dirac operators on loop space and the new results are very exciting.
Claiming that the hopeful wish:
“studying the Kervaire invariant problem might offer useful mathematical techniques to fundamental physics.”
is “nonsensical hype” seems flippant at best.
newcomer to QFT,
That an old result about a topological invariant of 6d manifolds has some application to string/M-theory is not surprising, but the new Hopkins et. al result is about non-existence of similar topological invariants of manifolds with dimension 254 and higher. There simply is zero reason to believe that this has anything to do with string theory. The one possible connection to physics that Hopkins mentioned is not about string theory, but a speculative idea about a possible 4d QFT.
It is also just absurd to describe this kind of mathematical work as being based on the idea “analogous to the use of one-dimensional strings as the basis of zero-dimensional (point-like) fundamental particles”. This is not just nonsensical hype, but actively misleading.
Some elementary explanations by John Baez why this problem is string theory:
http://golem.ph.utexas.edu/category/2009/04/kervaire_invariant_one_problem.html
Hi,
I just found that your book “Not Even Wrong” was recently translated into Korean. Since it is not easy to translate ‘not even wrong’ into Korean, the title of the book is “The truth of the superstring theory” (in Korean). Congratulations.
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Austin,
You’re misreading Baez’s posting, and presumably the same misreading is what led to the Nature News article.
Baez explains that there are important relationships between level 2 cohomology theories (elliptic cohomology, tmf) and 2d QFT. This goes back to work by Witten, where he investigated index theory on the loop space of a manifold. Index theory on the manifold itself can be understood in terms of supersymmetric quantum mechanics, going to the loop space means one’s path integral is now an integral over world-sheets, so a 2d QFT (which is different than string theory, but maintaining that distinction seems to be a lost cause…).
The new work of Hopkins et. al involves a level 4 cohomology theory, which (speculatively) might have something to do with some sort of quantum field theory on 4d manifolds. Baez refers to such a theory as involving “3-branes”, just because for any n-dim QFT, one can call its n-1 dim boundary conditions “n-1 branes”.
So, what Baez was explaining was a speculative connection between this mathematics and 4d QFT. 4d QFT is not string theory.
If you want to make the argument that any QFT in any dimension d has dimension d-1 boundary conditions and people have called these “branes”, and “branes” also occur in work on string theory, so any QFT is really string theory, I suppose you can do that. But it’s extremely misleading….
Well, they are not exactly hyping string theory. Rather Nature is hyping the result by claiming a connection to string theory.
Typical blog post.
You are at a meeting where they are discussing blah blah blah or there is an article about blah blah blah. Various smart and accomplished people have done the following interesting thing which is being discussed. Unfortunately these smart and accomplished people are under the delusion that their work has some connection to string theory. Whereas I, PW know that is impossible because nothing interesting or useful can possibly be connected to string theory.
At some point I would think most of your readers would draw the obvious conclusion.
Jr,
The hype does go in both directions. There’s some of both here, with the author claiming that “the new approach” of Hopkins et. al. is somehow based on the idea of replacing points by strings (which is simply untrue).
H-I-G-G-S,
Do you know anything at all about the mathematics involved here and what its connections to 2d QFT and to string theory are? If you’d like to discuss this, please go ahead, it’s an interesting subject. If you don’t, you might think for a minute about whether using anonymous blog comments to attack people who do know something about this is a good idea or not. I don’t know who you are, but if you don’t have the excuse of being a high school student, I find your behavior on this blog both shameful and unprofessional.