Timeline for What are the uses of the homotopy groups of spheres?
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| Mar 23, 2013 at 4:07 | comment | added | Tarun Chitra | Note that this propagator is usually formulated in terms of a sum over Euler characters (since the intuition behind this limit is that a sum over Feynman diagrams turns into a sum over closed connected surfaces). The authors themselves liken this to a 'sum over high-dimensional spheres.' This is likely ill-posed, but one might be able to view this process as the construction of an infinite sequence of $\mathsf{SU}(N)$-structures that asymptotically has some sort of $S^{\infty}$-bundle-like structure that resembles this whole process. I would love to learn if someone has already done this! | |
| Mar 23, 2013 at 4:02 | comment | added | Tarun Chitra | I'm not sure if this ends up having a connection to the homotopy groups on spheres, but part of the AdS/CFT correspondence is the idea (due to t'Hooft) that String Theories can be viewed as "large-N" Gauge Theories with gauge group $\mathsf{SU}(N)$ [0]. As mentioned in reference [0], the idea is to turn YM fields into a set of $\mathsf{SU}(N)$ fields and thehn take a "limit" (probably ill-defined) of $N\uparrow \infty$. In the paper they show that the propagator of theory depends on 'large-N sphere topology of spheres.' [0] Section 1.2 of hep-th:9905111 | |
| Apr 29, 2010 at 3:28 | comment | added | The Mathemagician | There's also a great introduction in Frankel's book. | |
| Apr 29, 2010 at 3:26 | comment | added | The Mathemagician | My friend's question that inspired the original question was inspired by the applications to physics. | |
| Apr 28, 2010 at 22:58 | comment | added | Sean Tilson | @jc: Hopkins hints at an invariant of manifolds that is not known or constructed. I am not sure how exactly this works, but the reference i am thinking of is his talk at santa barbara online.kitp.ucsb.edu/online/mp03/hopkins4 He is working by analogy in that other invariants of manifolds correspond to certain elements in the homotopy of other spectra, such as ko. I dont really understand this story very well, but I would like to. | |
| Apr 28, 2010 at 15:08 | comment | added | j.c. | I would be amazed (and delighted) to learn of any applications in physics of nontrivial homotopy groups of spheres π_m(S^k) where m or k get very large. | |
| Apr 28, 2010 at 13:56 | history | edited | Paul Siegel | CC BY-SA 2.5 |
added 4 characters in body
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| Apr 28, 2010 at 13:49 | history | answered | Paul Siegel | CC BY-SA 2.5 |