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

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

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 Tags cft, representation, vertex operator, virasoro