What is a vague motivational intro to the relationship between topological quantum field theory, cohomological quantum field theory, and quantum field theory?
I am a beginner.
Here are the vague basic definitions (coming from a noob)-
A topological qft is a functor (infinity category style and satisfying some conditions) from cobordism categories to vector spaces. A two dimensional for instance is a frobenius algebra.
A cohomological field theory is; fix a vector space $V$, then it's a family of functors $V^n \to H^*(\mathcal{M}_{g,n})$ satisfying appropriate gluing conditions along natural operations of the moduli space.
A QFT on $X$ is a Hamiltonian and good operators on $L^2(Fields(X))$.
Does each of them have a forgetful functor to the previous one?
Until someone smarter comes along here is what I gathered. Nothing here is original except mistakes.
Nlab in enter link description here does a great job explaining all but CFT.
I will give a short summary here-
There are two approaches to QFT that are nonperturbative that are analogous of the two approaches; schrodinger and heisenberg pictures in the standard qm.
The functorial one which assigns a vector space of states which move along time, while operators don't change with time.
A $d$ dimensional functorial QFT is a functor $Bord^S_{d-1,d} \to Vect_k$ where $S$ means that we keep more information like riemannian structure when we do cobordisms. Physical intuition; For qft on spacetime, we'd take a $4$ dimensional qft that on each spaceslice (or other slices whatever you want!) returns the hilbert space of fields on that slice. While flowing along time gives a functor from one hilbert states to another.
The other approach is the algebraic one. For this one, we say states don't matter, qm is all about the noncommutative algebra of operators. Thus on spacetime we define a cosheaf $F$ which on a set returns the operators that live on it. The important condition is that if we have two space-separated sets $U,V$, then their operators must commute since they can't affect each other. I.e $F(U),F(V) \subset F(U \cup V)$ commute.
Now TQFT
Take the functorial approach above and hope that we can forget the Riemannian structure, so it becomes topological (And thus easier).
Most cool physics depends on the Riemannian structure, but whatever.
Here is a physical construction of a natural TQFT (i.e a functor, I will really explain how to assign every closed manifold a vector space). If we have a space $M$ with Hamiltonian $H$, then the ground states (i.e $0$ eigenvalues of $H$) acting on $L^2(M)$ are those that don't move with time, so are more topological as there is no dynamics happening. However in the Riemannian case, mostly our Hamiltonian is basically the Laplacian, and there are no nonconstant harmonic functions. The solution is to take the "derived" harmonic functions. Meaning if you consider instead of $L^2(M)$ the space $L^2(\Omega_M)$ which has operators $Q=d, Q^*=d^*, \Delta$, then $[Q,Q^*] = \Delta$, so the chain-complex with derivative $Q$ has an homotopy $Q^*$ showing $\Delta$ vanishes. Thus we got the derived vacuum states.
Now CQFT Sadly I still don't understand how this relates.