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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?

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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.

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  • $\begingroup$ Kirwin's paper seems beautiful. Thanks. I guess you meant to write "Mapping Class Groups do not have Kazhdan's Property (T)", as the article title says ;) $\endgroup$
    – Manuel Bärenz
    Commented Aug 29, 2014 at 15:37
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    $\begingroup$ Maybe one should amplify that such definitions (ncatlab.org/nlab/show/coherent+state+in+geometric+quantization) hence depend on having "pre-quantum" data (often called "classical" data). If you instead have a quantizED field theory without the information of how it came about from quantizATION, say a TQFT or ETQFT given by (just) a functor/n-functor, then these definitions of coherent state won't apply. $\endgroup$
    – Urs Schreiber
    Commented Aug 29, 2014 at 20:28
  • $\begingroup$ @Urs, is a good intuition for classical data something similar to a fibre functor (say, in Tannaka duality)? $\endgroup$
    – Manuel Bärenz
    Commented Dec 11, 2014 at 19:01

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