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

1. ### Schure

1
[Closed] 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?)

[STRIKE]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[/STRIKE]

Last edited: Apr 23, 2012
2. ### Physics Monkey

1,359
Do stick around Schure, it's good to have some specialized discussion now and again. I find it a nice change of pace from the nth iteration of interpreting quantum mechanics, the twin paradox, or "is string theory science?"

There are some very knowledgeable people who hang out here.