Chapter 3 Quantum Anomaly in 1+1d QFT

In this chapter we learn about quantum anomaly of symmetry in 1+1-dimensional QFT. A QFT is a quantum mechanics equipped with a continuous locality (while a lattice system possess a discrete locality).12 Here, the locality means that we have the notion of “space”, and the observables are contained in the space. This locality (plus some assumption) prohibits the projective phase that we discussed so far, but then there is possibly more subtle anomaly about how much local the composition law of the symmetry is. It is not a very traditional way but we see this phenomenon through the action of the symmetry on the Hilbert spaces of QFT on an interval with different boundary conditions. The quantum anomaly in this dimensions for a finite group \(G\) can be captured by the 3rd group cohomology.

  1. Most things in this chapter are supposed to be applied to both of QFTs and lattice systems. It is just that KO is more familiar with QFTs and also the lecture is for high-energy students.↩︎