# Dirac operator partial integration

When you have an action with bosonic $X$ and fermionic $\psi$ (Majorana) fields and perform a SUSY transformation $\epsilon$ (the constant, infinitesimal parameter of transformation, a real, anticommuting spinor) can you do normal partial integration on the Dirac operator just like you're used doing with a derivative? For example: \begin{align} S &= \int \mathrm d^{2}\sigma\; \bar{\epsilon}[\left(\gamma^{\alpha}\partial_{\alpha}\gamma^{\beta}\partial_{\beta}\right)X(\sigma)]\psi(\sigma)\\ &= \int \mathrm d^{2}\sigma\; \bar{\epsilon}[{\not}\partial{\not}\partial X(\sigma)]\psi(\sigma)\\ &= -\int \mathrm d^{2}\sigma\; \bar{\epsilon}[{\not}\partial X(\sigma)][{\not}\partial\psi(\sigma)] + boundary \end{align}

This would make calculations much easier because I'm lost with all matrix multiplications and properties of the Gamma matrices...I wanted to check because I couldn't find any information about this.

You should be aware of what the objects in your expression actually are, and how they relate to each other. $X$ is a bosonic field and as such does not feel the presence of gamma matrices at all. Your first line could be rewritten as $$S = \int \mathrm d^2 \sigma \, \bar \epsilon \gamma^\alpha \gamma^\beta \psi(\sigma) \, \partial_\alpha \partial_\beta X(\sigma)$$ The expression $\bar \epsilon \gamma^\alpha \gamma^\beta \psi$ does not have any free spinor indices! Now, you can use partial integration to get $$S = -\int \mathrm d^2 \sigma \partial_\beta \left(\bar \epsilon \gamma^\alpha \gamma^\beta \psi(\sigma) \right) \partial_\alpha X(\sigma)$$ which is $$S = -\int \mathrm d^2 \sigma \bar \epsilon \gamma^\alpha \not \partial \psi(\sigma) \, \partial_\beta X$$ and this is your last line.
• The special case that there are actually only two spinors. If you have an expression like $S = \int \mathrm d^dx \bar \psi_1 \not \partial \psi_2 \, \bar \psi_3 \not \partial \psi_4$ with four spinors $\psi_i$ you can only move the partial derivative, but not the gamma matrix between $\psi_1$ and $\psi_2$ to $\psi_3$ and $\psi_4$. To move gamma matrices around with more than two spinors in your expression you need the so-called Fierz Identities (Wiki only has a special case, these exist for a wide variety of frameworks). Nov 12, 2014 at 7:22