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Applications of Geometric Topology to Theoretical Physics

+ 7 like - 0 dislike
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Geometric topology is the study of manifolds, maps between manifolds, and embeddings of manifolds in one another. Included in this sub-branch of Pure Mathematics; knot theory, homotopy, manifold theory, surgery theory, and other topics are developed in extensive detail. Do you happen to know of any applications of the techniques and/or theorems from geometric topology to theoretical physics? I'm guessing that most applications are in topological quantum field theory. Does anyone know of some specific (I'm asking for technical details) uses of say, whitney tricks, casson handles, or anything from surgery theory?

If you cannot give a full response, references to relevant literature would work as well.

Thanks

This post imported from StackExchange Physics at 2014-04-01 12:42 (UCT), posted by SE-user Samuel Reid
asked Dec 29, 2011 in Mathematics by Samuel Reid (80 points) [ no revision ]
Community-wiki?

This post imported from StackExchange Physics at 2014-04-01 12:42 (UCT), posted by SE-user András Bátkai
Have you heard about topological quantum computing? It is very closely linked to braid theory, which has given rise to a quantum algorithm for approximating the Jones polynomial.

This post imported from StackExchange Physics at 2014-04-01 12:42 (UCT), posted by SE-user Joe Fitzsimons
@AndrásBátkai: I'm not sure I agree that this should be CW. Let's wait to see if any one has a strong opinion either way.

This post imported from StackExchange Physics at 2014-04-01 12:42 (UCT), posted by SE-user Joe Fitzsimons

4 Answers

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(sorry I don't have enough reputation to make a comment): This question is very broad/vague, as indeed algebraic/differential topology (symplectic geometry of course) is completely used in theoretical physics, in particular for Topological QFTs. From a physicist's perspective, start with Nakahara's Geometry, Topology, and Physics. Surgery, cobordism, and the likes are used in Knot Theory, which Witten has been studying for String Theory and other TQFT's.

And I believe the 'coolest' topic to start with is the Gauss-Bonnet Theorem, since it appears in the action functional.

This post imported from StackExchange Physics at 2014-04-01 12:42 (UCT), posted by SE-user Chris Gerig
answered Dec 29, 2011 by Chris Gerig (540 points) [ no revision ]
String theory is not a TQFT. Perhaps you mean topological string theory

This post imported from StackExchange Physics at 2014-04-01 12:42 (UCT), posted by SE-user Squark
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In 2 dimensions, TQFTs are given by Frobenius algebras. This fact can be seen by evaluating the TQFT functor on basic building blocks of 2d manifolds: pairs of pants and discs. These give the multiplication and trace on the Frobenius algebra.

Going up in dimension, every closed 3-manifold can be obtained by a surgery of $S^3$ along a link. This allowed Reshetikhin and Turaev to define invariants of 3-manifolds with links given a modular tensor category. It turned out these invariants organize into a 3-2-1 TQFT, which gives Witten's Chern-Simons TQFT when the modular tensor category is $U_q(\mathfrak{sl}_N)$ ($q$ is a root of unity).

More generally, the proof of the cobordism hypothesis due to Lurie (classifying fully extended TQFTs) uses Morse theory to build $n$-manifolds from $(n-1)$-manifolds with handles attached (inductive construction of the category of cobordisms by generators and relations).

Similar ideas (cutting and gluing) have been applied to many areas. For example, Eliashberg et al. developed symplectic field theory, which, in particular, allows one to compute Gromov-Witten invariants of closed symplectic manifolds by reducing them to simpler objects.

This post imported from StackExchange Physics at 2014-04-01 12:42 (UCT), posted by SE-user Pavel Safronov
answered Dec 30, 2011 by anonymous [ no revision ]
Excellent response, the links to both of those papers on arXiv will be very helpful to me and I'm interested to learn about SFT.

This post imported from StackExchange Physics at 2014-04-01 12:42 (UCT), posted by SE-user Samuel Reid
+ 5 like - 0 dislike

there are a sequence of articles that use 4-manifold results (exotic smooth structures on R^4, casson handles, slicing knots, akbulut corks, etc) to make assertions in physics from gravity to dark energy. You can start here http://arxiv.org/abs/1112.4882 and click.

This post imported from StackExchange Physics at 2014-04-01 12:42 (UCT), posted by SE-user Paul
answered Dec 31, 2011 by anonymous [ no revision ]
+ 2 like - 0 dislike

If you like Casson Handles, you'll love exotic smoothness. Basically, with an infinite tower of Casson handles one can construct a manifold which is homeomorphic but not diffeomorphic to the "usual" $\mathbb{R}^4$. The idea is to push the Morse points (which would show a topological change) "out to infinity" so that topological equivalence is maintained but there is now no diffeomorphism between the Casson handle and a subset of $\mathbb{R}^4$. Exotic $\mathbb{R}^4$, in turn, can be used to model dark matter and has numerous applications in quantum gravity.

A good physics review can be found in the text of Asselmeyer-Maluga and Brans, and a good mathematical reference is Scorpan, The Wild World of 4-Manifolds. Current research includes archive papers (most of which have been publishsed somewhere): 9610009, 1101.3168, 1112.4882.

This post imported from StackExchange Physics at 2014-04-01 12:42 (UCT), posted by SE-user levitopher
answered Jan 12, 2012 by levitopher (160 points) [ no revision ]
Funny! I just rented that book by Scorpan out from the library today and came home and saw you posted this. Looks like I'm meant to read it. :) Thanks for the post.

This post imported from StackExchange Physics at 2014-04-01 12:42 (UCT), posted by SE-user Samuel Reid

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