I have a question about notation used in Information, Physics, and Computation for the $p$-spin glass model with $N$ Ising spins. The energy of the configuration $\sigma=\{\sigma_1,\cdots,\sigma_N\}$ is defined as $$E(\sigma)=-\sum_{i_1<i_2<\cdots <i_p} J_{i_1,\cdots,i_p}\sigma_{i_1}\cdots\sigma_{i_p},$$ with $\sigma_{i}\in \{\pm 1\}$. The couplings $J_{i_1,\cdots,i_p}$ with $1\leq i_1<\cdots<i_p\leq N$ are i.i.d Normal random variables with mean zero and variance $\dfrac{p!}{2N^{p-1}}$.
The following is the replica calculation by Mezard and Montanari:
So, I do not understand the notation in (8.29). I believe the $\delta$ in $\delta\left(Q_{ab}-\dfrac{1}{N}\sum_{i=1}^N\sigma_i^a\sigma_i^b\right)$ is the discrete time unit impulse. The notation for $\lambda_{ab}$ is not clear either. Is it a $0$-$1$ matrix like $Q_{ab}$? Could you also provide a brief explanation of $8.29$ and how it was used in $8.28$ to get $8.30$?
They are using the distributional definition of the Dirac Delta function in terms of its fourier transform:
$$\delta(x-a)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{it(x-a)}dt.$$
So $\lambda_{ab}$ are just variables living in $(-\infty,\infty)$.