I am trying to learn about some categorical aspects of topological quantum field theories. For concreteness, I am considering Chern Simons theory with gauge group G in three dimensions. As I understand, I can describe the theory as a functor F on a cobordism category where there are 0, 1 and 2 dimensional objects (manifolds) and morphisms between objects. Let me denote some C-linear n-categories as Cn, such that the functor F leads to following assignments:
F(S3)=C−1=:ZS3,F(S2)=C0=:HS2,F(S1)=C1,F({pt})=C2.
where
Sn is the
n-dimensional sphere,
HS2 is a vector space (with an inner product), and
ZS3 is a complex number, this question does not concern the 2-category
F({pt}). The 1-category
F(S1) is a category of some representations of the affine Lie group
ˆG.
Question: How do I compute the number ZS3 given only the 1-category F(S1)?
As a clarification of what I am asking let me explain how I would proceed for the simpler problem of computing ZS3 given the 0-category F(S2). Given HS2, I can construct S3 as S3=ˉD3⊔S2ˉD3 where I am gluing two closed three dimensional discs along their S2 boundaries and my understanding is that (I would like to know if this is incorrect) F assigns to ˉD3 an element of F(∂ˉD3)=F(S2)=HS2, i.e. F(ˉD3) is a vector and the result of the gluing is that F(S3)=⟨F(ˉD3),F(ˉD3)⟩ where ⟨−,−⟩ is an inner product (which can be written as a path integral over S3).
I would like to know what is the analogous procedure if I start from one dimension bellow. If the answer is standard or trivial I will be happy with a reference, and if it helps anyone willing to write an answer, I am more familiar with the physics side of the subject than the math side. Thanks in advance for any help.
Edit: (some attempts) I can try to construct F(S2) from the knowledge of F(S1) by gluing two ˉD2's along their boundary S1. F(ˉD2) should be an object of the 1-category F(S1), so F(ˉD2) is a ˆG-module, therefore the image of two disjoint ˉD2's (with one "incoming" and one "outgoing" boundary) under F should be F(ˉD2⊔(ˉD2)∗)=F(ˉD2)⊗F(ˉD2)∗.
Finally from the physical point of view the following seems reasonable:
F(S2)=F(ˉD2⊔S1(ˉD2)∗)=(F(ˉD2)⊗F(ˉD2)∗)ˆG.
Then I can proceed along the line described earlier for the simpler situation of having the data of
F(S2).
This kind of answers my original question. For the sake of being able to do computation I have two followup questions:
Followup question 1: How exactly do I decide which element of F(S2) to assign to F(ˉD3) (without using a path integral definition of a wave functional)? Is it part of the axioms or is there a derivation?
Followup question 2: In (1), if I choose a basis {ei} for F(ˉD2) (and a dual basis {e∗i} for F(ˉD2)∗), then F(S2) seems to be the one dimensional space Span(∑iei⊗e∗i), if F(ˉD2) is infinite dimensional then is there a problem with normalization for the vectors in F(S2)?
Perhaps I am making some silly mistakes, sorry for the lengthy post!
This post imported from StackExchange MathOverflow at 2017-01-11 10:39 (UTC), posted by SE-user Nafiz Ishtiaque Ornob