What constrains ghost fields $c^+$ and $b_{++}$ as $z$ going to infinity

Question

In Green's superstring theory, Chapter 3, from equation 3.3.13 to 3.3.18, the author says that the singularity behavior of the ghost field $c^+$ must be no faster than $z^2$ as $z$ goes to infinity. And $b_{++}$ is required to approach $0$ as $z$ goes to infinity.

May I ask what is the reason for such constraints?

This post imported from StackExchange Physics at 2014-04-14 16:22 (UCT), posted by SE-user Han Yan

Comments

With a sphere topology, you have 2 patches of the sphere related by $u= \frac{1}{z}$. Holomorphic vectors $v^z$, and holomorphic quadratic differential $\omega_{zz}$ must be holomorphic at $z=0$ and $u=0$

This post imported from StackExchange Physics at 2014-04-14 16:22 (UCT), posted by SE-user Trimok

Answer 1

The behaviour of a vector field $v^z$ at $z=\infty$ can be described by first moving to coordinates that are well-defined in that coordinate patch. Defining $w = \frac{1}{z}$, we find $$ c^w = \frac{dw}{dz} c^z = - z^{-2} c^z $$ Now, $c^w$ must be well-defined at $w=0$. This implies that $z^{-2} c^z$ must be well-defined as $z \to \infty$. Thus $c^z$ cannot go faster than $z^2$ as $z \to \infty$.

Performing a similar analysis for $b_{zz}$, we find $$ b_{ww} = \left( \frac{dw}{dz} \right)^{-2} b_{zz} = z^4 b_{zz} $$ Now, since $z^4 b_{zz}$ has to be well-defined as $z\to\infty$, it must be that $b_{zz} \sim z^{-4}$ at large $z$.

This post imported from StackExchange Physics at 2014-04-14 16:22 (UCT), posted by SE-user Prahar

Comments

Thanks a lot, that's a very clear answer!

This post imported from StackExchange Physics at 2014-04-14 16:22 (UCT), posted by SE-user Han Yan