A Universe of Sorts

Once again, watching a videos for shits and giggles.
Geometrically, we cut and paste topological indices / defects.
QFT in dimensions n+1 (n space, 1 time)
Manifold: $X^n$ . Can associate a hilbert space of states $H_x$ .
Space of wave functions on field space.
Axioms of hilbert space: (1) if there is no space, the hilbert space $H_\emptyset$ for it is the complex numbers. (2) If we re-orient the space, the hilbert space becomes the dual $H_{-X} = H_X^\star$ . (3) Hilbert space over different parts is the tensor product: $H_{X \cup Y} = H_X \otimes H_Y$ .
We want arbitrary spacetime topology. We start at space $X$ , and we end at a space $Y$ . The space $X$ is given positive orientation to mark "beginning" and $Y$ is given negative orientation to mark "end". We will have a time-evolution operator $\Phi: H_X \rightarrow H_Y$ .
We have a composition law of gluing: Going from $X$ to $Y$ and then from $Y$ to $Z$ is the same as going from $X$ to $Z$ . $\phi_{N \circ M} = \phi_N \circ \phi_M$ .
If we start and end at empty space, then we get a linear map $\Phi: H_\emptyset \rightarrow H_\emptyset$ which is a linear map $\Phi: \mathbb C \rightarrow \mathbb C$ , which is a fancy way to talk about a complex number (scaling)
If we start with an empty set and end at $Y$ , then we get a function $\Phi: H_\emptyset \rightarrow H_Y \simeq \mathbb C \rightarrow \mathbb Y$ . But this is the same as picking a state, for example, $\Phi(1) \in H_Y$ [everything else is determined by this choice ].
If a manifold has two sections $X$ and $-X$ , we can glue $X$ to $-X$ to get the trace.
Quantum mechanics is 0 + 1 TQFT (!)
TQFT of 1+1 dimensions.
Take a circle: $S^1 \rightarrow H$ . Let $H$ be finite dimensional.
A half-sphere has a circle as boundary. So it's like $H_\emptyset \rightarrow H_{S^1}$ . This is the ket $|0\rangle$ .
This is quite a lot like a string diagram...
Frobenius algebra
Video: IAS PiTP 2015

§ Intro to Topological Quantum Field Theory