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\author{Gabriel C. Drummond-Cole}
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\title{RTG Seminar\\December 4, 2006\\
Alexander Kirillov, Jr.}
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Should I begin?  I was asked to give a brief overview of TQFTs in preparation for Segal's talk on Thursday.  I don't know precisely what he's going to talk about.  I will say some things that will be relevant and some things that will be irrelevant.  I hope to keep the ratio between the two reasonable.

Let me start by saying something about quantum field theory.  For starters, what is classical field theory?
 The classical theory is to have a manifold $N^d$ then you consider objects on this space, so you consider fields $\Ff(N^d)$ which are geometric objects, maybe sections of a bundle, tensors, usually some geometric object.
 
 There should be some equations of motion which explain how your fields change in time.  Solving them is usually not all that easy, but that's not what I'm gonig to talk about.
 
In quantum theory, you describe the configuration space not by a manifold but by a vector space, a state space, and a state in the system is a vector in the state space $\Hh.$  For example, this could be $L^2(F(N^d)),$ although it would usually not be.  For evolution you have an operator $A_t:\Hh\to\Hh,$ the time $t$ evolution.

How do you do this?  You write 
$$\langle \varphi_1|A_T|\varphi_0\rangle=\int_{F\in \Ff(N\times [0,T])}\varphi_0(F|_{N_0})\varphi_1(F|_{N_1})e^{\frac{i}{h}S(F)}DF$$  
This defines $A_T.$
The only real ingredient of the theory is $S(F),$ called the action, which is usually an integral whose value is based on the field and its derivatives, $S(F)=\int L(F,F',\ldots).$ It's very infrequent that you can actually make an explicit calculation.  In the whole tube, you don't care whether space time is a cylinder.  So first, you can generalize given any $d+1$-manifold $M$ and any vectors $\varphi_i\in \Hh(\boundary M)_i$ for each boundary component.

We can define an operator $A_M$ such that $$A_M(\varphi_1,\ldots,\varphi_n)=\int_F \prod \varphi_iF|_{\boundary M_i}e^{\frac{i}{h}S(F)}DF$$

Let me do another thing that is still immediate from this definition.  You know how $L^\infty(C^\infty(N))$ behaves under the disjoint union of $N.$  It becomes $\oplus$ in $C^\infty$ and then $L^2$ makes this into $\otimes.$

The first result is that if this thing can be defined accurately, well, you want the structure to respect composition.  If you wriet this, you get $A_M$ is that integral,.

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