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.
Answer
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.
No comments:
Post a Comment