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I'm reading this book The supersymmetric method in random matrix theory and applications to QCD. In page 302-303, the author calculate the following integral $$ Z=\int d\psi \, dH \, P(H)\exp\left(-\sum_{kl} i\phi^*_k(z-H)_{kl} \phi_l+i\chi^*_k (z+J-H)_{kl}\chi_l\right) $$ where $H$ is a $N\times N$ Hermitian matrix, $\phi_i$ and $\chi_i$ are respectively the commuting and anti-commuting (Grassmann) variables, the measurement $d\psi={-2\pi i}^{-1}\prod_{j=1}^N d\phi_j \, d\phi^*_j \, d\chi_j \, d\chi^*_j$. The distribution is $$ P(H)=e^{-\frac{N}{2} \operatorname{Tr}H^\dagger H}. $$ Integrating over the matrix $H$ results in $$ Z=\int d\psi \, \exp\left(-\frac{1}{2N} \operatorname{Str} \begin{pmatrix}\sum_j \phi^*_j\phi_j & \sum_j \chi^*_j \phi_j \\ \sum_j \chi_j \phi^*_j & \sum_j \chi^*_j \chi_j \end{pmatrix}^2-i\sum_j (\phi^*_jz \phi_j + \chi^*_j (z+J) \phi_j ) \right). $$ where the superetrace $\operatorname{Str} \begin{pmatrix}a & b \\ c & d\end{pmatrix}=\operatorname{Tr}a-\operatorname{Tr}d$. I'm not familiar with matrix integral and functional integral. How can I reach to the last equation?

I didn't realize that there's a very useful and introduction-level paper by Alexander D. Mirlin before I post my question, and this question will be closed.

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    $\begingroup$ the key reference is Efetov Supersymmetry in Disorder and Chaos --- this is quite sophisticated machinery, if you are unfamiliar with the basics I think you will first want to do some background reading. $\endgroup$ Commented Nov 2, 2022 at 16:39
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    $\begingroup$ I do not have the book, but if it is like the lecture notes of the same name, published as arxiv.org/abs/hep-th/0410211, then just read the earlier sections on bosonic and Grassman integrals; he seems to explain the basics therein. $\endgroup$ Commented Nov 2, 2022 at 21:45
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    $\begingroup$ @CarloBeenakker. I try to perform this integral by decomposing $dH$ into $dH_{11}dH_{12}\cdots dH_{NN}$, but I find it's not consistent with the answer in text book, even for integral over diagonal element $dH_{11}dH_{22}\cdots dH_{NN}$. Could you help me to check my derivations? $\endgroup$ Commented Nov 3, 2022 at 14:17
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    $\begingroup$ Are you using the fact that Grassmann variables anti-commute? $\endgroup$ Commented Nov 3, 2022 at 14:49
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    $\begingroup$ @TomDickens I've checked the Gaussian integral $\int \exp(-ax^2+bx)dx\propto \exp(b^2/4a)$, and I think it still holds for a Grassmann variable $b$. $\endgroup$ Commented Nov 4, 2022 at 4:03

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