§ 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: . Can associate a hilbert space of states .
- Space of wave functions on field space.
- Axioms of hilbert space: (1) if there is no space, the hilbert space for it is the complex numbers. (2) If we re-orient the space, the hilbert space becomes the dual . (3) Hilbert space over different parts is the tensor product: .
- We want arbitrary spacetime topology. We start at space , and we end at a space . The space is given positive orientation to mark "beginning" and is given negative orientation to mark "end". We will have a time-evolution operator .
- We have a composition law of gluing: Going from to and then from to is the same as going from to . .
- If we start and end at empty space, then we get a linear map which is a linear map , which is a fancy way to talk about a complex number (scaling)
- If we start with an empty set and end at , then we get a function . But this is the same as picking a state, for example, [everything else is determined by this choice ].
- If a manifold has two sections and , we can glue to to get the trace.
- Quantum mechanics is
0 + 1 TQFT (!)
- TQFT of 1+1 dimensions.
- Take a circle: . Let be finite dimensional.
- A half-sphere has a circle as boundary. So it's like . This is the ket .
- This is quite a lot like a string diagram...
- Frobenius algebra
- Video: IAS PiTP 2015