# 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.

## 1 Answer

In general you cannot, but in your special case it works out.

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