I am reading the Section 15.9 of Weinberg's book "The Quantum Theory of Fields, vol. 2". Under a shift $\delta\Psi[\chi]$ in $\Psi[\chi]$, we have
$$ \begin{split} \delta Z&=i\int[d\chi]\,\exp\big(iI_{\Psi}[\chi]\big) \left(\frac{\delta_RS[\chi,\chi^{\ddagger}]}{\delta\chi_n^{\ddagger}} \right)_{\chi^{\ddagger}=\delta\Psi/\delta\chi}\left( \frac{\delta(\delta\Psi[\chi])}{\delta\chi^n}\right) \\ &=i\int[d\chi]\,\exp\big(iI_{\Psi}[\chi]\big)\left\{\frac{\delta_L} {\delta\chi^n}\left(\frac{\delta_RS}{\delta\chi_n^{\ddagger}} \delta\Psi\right)-\frac{\delta_R}{\delta\chi_n^{\ddagger}} \frac{\delta_LS}{\delta\chi^n}\delta\Psi\right\}_{\chi^{\ddagger} =\delta\Psi/\delta\chi} \\ &=\int[d\chi]\,\exp\big(iI_{\Psi}[\chi]\big)\left\{ \frac{\delta_RS[\chi,\chi^{\ddagger}]}{\delta\chi_n^{\ddagger}} \frac{\delta_LI_{\Psi}[\chi]}{\delta\chi^n}-i\Delta S[\chi,\chi^{\ddagger}]\right\}_{\chi^{\ddagger}=\delta\Psi/ \delta\chi}\delta\Psi[\chi] \end{split} $$
The last line is exactly the same to Eq. (15.9.33). Referring to the definition of antibracket
$$ (F,G)=\frac{\delta_RF}{\delta\chi^n}\frac{\delta_LG}{\delta\chi_n^{\ddagger}}- \frac{\delta_RF}{\delta\chi_n^{\ddagger}}\frac{\delta_LG}{\delta\chi^n} $$
we can see that the quantum master equation reads
$$ -(S,S)-2i\Delta S=0 $$
which has an extra minus sign. I are not sure whether this is a typo or not. Could someone help me to check this derivation?
Besides, I am also confused by $\delta_L$ and $\delta_R$. Any clarifications will be appreciated.
Many thanks in advance!