# Question on the form of a vertex operator in a proof

 P: 1 Ok, never mind - I decided to find the solution in a different way.. This is a little too specialized anyway. (Is there a way to delete the thread?) Hi, I am reading paper [1] and I found that formula (33), $$\psi(xy)\psi^*(y)=\frac 1{x^{1/2}-x^{-1/2}}\exp\left(\sum_n\frac{(xy)^n-y^n}{n}\alpha_{-n}\right)\exp\left(\sum_n\frac{y^{-n}-(xy)^{-n}}n\alpha_n\right)$$ is almost in accordance to its alleged source [2, Theorem 14.10], except for the factor at the front, namely, $$\frac1{x^{1/2}-x^{-1/2}}.$$ Does anyone know where that comes from? Probably this comes from the shift of coordinates that happens when Eskin and Okounkov use half-integers for the indices in the infinite wedge representation, instead of the usual whole integers. But I have not found the way to fully justify the term using this. I'd really appreciate a hint! Thanks! Schure [1] A. Eskin and A. Okounkov, Pillowcases and quasimodular forms, http://arxiv.org/pdf/math/0505545.pdf [2] Kac, Infinite dimensional Lie algebras, 3rd edition