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A vertex operator is a linear map associating every state to a operator-valued distributions(quantum field) on a algebra curve,which is also called operator-state correspondence.
Chose a local complex coordinate, we can locally expand quantum fields as operator valued formal laurent series,this process is called mode expansion(?), the coefficients are called fourier coefficients. I confuse the terminology fourier coefficients,why people give them this name,does mode expansion relate to fourier transformation?

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If you expand a meromorphic function in a Laurent series about $z=0$ and now take $z$ on the unit circle in the complex plane, so that $z= e^{i\theta}$, then the Laurent series is a Fourier series.

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