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 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 have property (T). See this paper.

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 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.