§ Intro to topological quantum field theory

• 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