# Polchinski 12.3.22

1. May 22, 2014

### Illuminatum

Forming the supersymmetric string using superfields and superspace, Polchinski claims that the function
$$G \sim \ln{\left|z_{1} - z_{2} - \theta_{1}\theta_{2}\right|^{2}}$$
satisfies the equation
$$D\bar{D}G = \delta^{2}(z_{1} - z_{2})\delta^{2}\left(\theta_{1} - \theta_{2}\right)$$
which would be the Green's function for the kinetic term in the action he presents
$$\int DX \cdot \bar{D}X.$$
where $$D = \partial_{\theta} + \theta\partial_{z}$$

I've tried showing that this is indeed true but can't quite get the correct form. To begin I simply expanded the Green's function in terms of its fermionic parts and believe it is
$$G \sim \ln|z_{1}-z_{2}|^{2} - \frac{\theta_{1}\theta_{2}}{z_{1} - z_{2}} - \frac{\bar{\theta_{1}}\bar{\theta_{2}}}{\bar{z_{1}} - \bar{z_{2}}}$$
because higher order terms vanish because of anti-commutation or cancel directly.

Then writing
$$\delta^{2}(\theta_{1} - \theta_{2}) = (\theta_{1} - \theta_{2})(\bar{\theta_{1}} - \bar{\theta_{2}})$$
I find myself missing the final $\theta_{2}\bar{\theta_{2}}$ term. I'm making use here of the relationship
$$\partial_{z}\frac{1}{\bar{z} - \bar{z'}} \sim \delta^{2}(z - z')$$
and extending it for the super derivative to
$$D\frac{1}{\bar{z} - \bar{z'}} \rightarrow \theta \partial_{z}\frac{1}{\bar{z} - \bar{z'}} \sim \theta \delta^{2}(z - z')$$
which may be the cause of the issue.

Could anyone provide an explanation of how / why the relationship holds in the fashion that Polchinski suggests?

Many many thanks for your time

Last edited: May 22, 2014