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?



About vertex algebra, mode expansion
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.


