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?