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The following query is based on a reading of section 2.2 of a paper by Graña and Polchinski. The idea is to begin with the D3 brane action of the form

$$ ds^2 = Z^{-1/2}\eta_{\mu\nu}dx^\mu dx^\nu + Z^{1/2}dx^m dx^n$$

where $\mu, \nu = 0, 1, 2, 3$ and $m, n = 4, 5, \ldots, 9$ are indices along the longitudinal and transverse directions. Also, $\eta_{\mu\nu} = diag(-1, 0, 0, 0)$ and $Z$ is a harmonic function (in the paper it is taken as $Z = R^4/r^4$ where $R^4 = 4\pi g N\alpha'^2$).

Now, type IIB superstring theory has two fermonic superpartners of the NS$\otimes$NS and R$\otimes$R fields, namely the dilatino and the gravitino, the supersymmetry transformations of which are given in terms of a spinor parameter $\epsilon$ in equations (2.1) and (2.2) of the paper. The authors further assert that for bosonic backgrounds, and for constant $\tau = C + i e^{\Phi}$ where $C$ is the axion and $\Phi$ is the dilation, the dilatino variation is trivially zero. I understand this.

But when they set $\delta \psi_M = 0$ ($M = 0, 1, \ldots, 9$) they seem to go from

$$\delta \psi_M = \frac{1}{\kappa}D_M \epsilon + \frac{i}{480}\gamma^{M_1 \ldots M_5}F_{M_1\ldots M_5}\epsilon$$

to

$$k\delta \psi_M = \partial_\mu \epsilon - \frac{1}{8}\gamma_\mu \gamma_w (1 - \Gamma^4)\epsilon$$

where $\Gamma^4 = i \gamma^{0123}$, $w_m = \partial_m \ln Z$ and $\gamma_w = \gamma^m w_m$.

I am not sure how they arrive at this equation. What happened to the $i/480$ term?

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  • $\begingroup$ Divide through your second equation by $\kappa$, with $\kappa=8\pi^{7/2}\alpha'^2g$, get $i$ from $\Gamma^4$ and express $\gamma_\omega$ in terms of $\omega_m$. $\endgroup$ – Demosthene Apr 29 '15 at 13:22
  • $\begingroup$ What happened to the product of 5 gamma matrices? $\endgroup$ – leastaction Apr 29 '15 at 16:19

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