How is cohomology theory used in quantum field theory? Quantum field theory uses a large amount of mathematics and I was wondering about some applications of cohomology theory in QFT, I understand it has applications in string theory but I was wondering about possible applications in more traditional QFT.
 A: In the BRST quantisation of quantum field theories, cohomology does show up. The conserved charge associated to BRST quantisation $Q_B$ implies that physical states must be BRST-invariant via $Q_B |\psi \rangle=0$. It's easy to prove that $Q_B$ is nilpotent: $Q_B^2=0$ such that one can add a null state to any physical state without changing its BRST invariance: $|\psi'\rangle = |\psi\rangle + Q_B |\chi\rangle$ where $|\chi\rangle$ is not physical, but $Q_B |\chi\rangle$ is clearly exact and closed.
The equivalence classes under this equivalence relation are precisely the physical states:
\begin{equation}
\mathcal{H}_{\text{BRST}} = \mathcal{H}_{\text{closed}}/\mathcal{H}_{\text{exact}}
\end{equation}
$Q_B$ actually raised the ghost number by 1 and forms a cochain complex.
This is also used in string theory, but I didn't make any reference to it.
I don't know a lot about this, hence I may be wrong.
A: Also not an expert, but let me throw a couple things out there.  Nakahara's book "Geometry, Topology, and Physics" is excellent.  He has an entire chapter on de Rham cohomology, along with applications to harmonic forms.  My vague recollection in terms of the physics application of de Rham cohomology is to understand when closed forms are exact (all exact forms are closed), which--again, if memory serves--is equivalent to asking when one can express a force or a field in terms of a potential in a path-independent way; basically, if your manifold is topologically trivial, then you're able to write down the potential.  But if your space is topologically non-trivial, then you can get things like the Aharonov-Bohm effect.
A: Cohomology can be found lurking behind many condensed matter systems. More specifically, cohomology is the mathematical origin behind the Integer (Anomalous or conventional) and Fractional Quantum Hall effects in topological matter, such as topological insulators or Weyl semimetals.
Such systems can be studied by either using Quantum Mechanics, where we use single-particle states to calculate the relevant topological invariants, mostly the (first) Chern number, or by using Quantum Field theory through which we can study more carefully the transport properties of the system of interest via quantities such as correlation functions.
In the field-theoretical picture, Chern-Simons theories are very common. They are prevalent in research on the Fractional QHE, but in the past years they have become more and more commonplace in the study of the Integer QHE too.
If you want to see how quickly the Chern-Simons action can lead to the quantisation of the Hall conductivity with almost none of the assumption one has to do when tackling the problem from a quantum mechanical perspective, you can check out the first pages of Tong's basic but very informative notes:
https://www.damtp.cam.ac.uk/user/tong/qhe/five.pdf
Note that even though Tong does not mention the word cohomology, the "winding numbers" that he defines are indeed closely related (and sometimes equal) to the Chern number.
Edit:
You can find more on this topic and how comohology plays a role in the following notes by kauffmann et al: https://arxiv.org/pdf/1501.02874.pdf
