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?