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Polchinski 12.3.22

  1. May 22, 2014 #1
    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
  2. jcsd
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