In the quantum harmonic oscillator, there exists a family of states called *coherent states* which form an overcomplete set of states. They are regarded as "the states most resembling classical states", in that their parameter satisfies the classical equation of motion of the harmonic oscillator. Multiple generalisations exist to diverse situations and have proven useful.


In TQFTs, we have Hilbert spaces and evolution operators as well, so do we also have coherent states of some sort? What are they being used for?


Bonus: Are there categorified versions of these in extended TQFTS?



Does the notion of a "coherent state" exist in TQFTs? (ETQFTs?)
Yes they do. In the geometric quantization approach to the 3d Chern-Simons TQFT, the vector space assigned to a closed surface is a space of holomorphic sections of a certain line bundle. In this context, we have the notion of "coherent state".


If you're worried that your notion of coherent state (coming from the harmonic oscillator) doesn't correspond to this mathematical notion of coherent state (coming from holomorphic sections of line bundles) - don't. These two notions of "coherent state" are compatible; see for instance Kirwin's paper [Coherent states in geometric quantization](http://arxiv.org/abs/math/0502026/%22Coherent%20states%20in%20geometric%20quantization%22) and references therein.


In fact, these coherent states play an important role in TQFT: Andersen used them to prove that the mapping class groups of surfaces do not have property (T). See [this paper](http://arxiv.org/abs/0706.2184).


