Vertex operator - state mapping in Polchinski's book

Question

In Polchinski's textbook String Theory, section 2.8, the author argues that the unit operator $1$ corresponds to the vacuum state, and $\partial X^\mu$ is holomorphic inside couture $Q$ in figure 2.6(b), so operators $\alpha_m^\mu$ with $m>=0$ vanishes.

I am a bit confused about why $\partial X^\mu$ has no pole inside the contour. Before this section $\partial X^\mu$ always has the singularity part ($1/z^m$). Therefore would it be possible for you to give a more mathematical argument what condition requires $\partial X^\mu$ having no poles in this case?

Thanks a lot for your time!

This post imported from StackExchange Physics at 2014-04-14 16:20 (UCT), posted by SE-user Han Yan

Answer 1

The main point is that the operator-state correspondence maps all the annihilation operators to zero, so that an operator-valued Laurent series in $z$ and $\bar{z}$ maps to a ket-state-valued power series in $z$ and $\bar{z}$.

This post imported from StackExchange Physics at 2014-04-14 16:20 (UCT), posted by SE-user Qmechanic