Tensor Gaussian Integrals

Question

I wish to compute the most general tensor Gaussian integral of the form

$$\int [\mathcal{D}A] \exp(-A^{\mu_1\mu_2\cdots \mu_s}M_{\mu_1\mu_2\cdots\mu_s\nu_1\nu_2\cdots\nu_s}A^{\nu_1\nu_2\cdots \nu_s})$$

My guess is that the answer should be $$\det(M)^{-1/2}$$ where $\det{M}$ is defined here but I am unable to prove it. Can anybody provide a rigorous way to do such an integral?

Answer 1

Just rename indices, $\mu_1\mu_2\cdots \mu_s\to i$ (that is, $11\cdots 11\to 1, \ 11\cdots12\to2$, etc, $nn\cdots nn\to n^s$. Then your integral is $$ \int e^{-A_i M_{ij} A_j}=\underset{ij}{\det}(M_{ij})^{-1/2} $$

This is also $$ \underset{ij}{\det}(M_{ij})\epsilon_{j_1\cdots j_N}=\epsilon^{i_1\cdots i_N}M_{i_1j_1}M_{i_2,j_2}\cdots M_{i_N,j_N} $$ which you can "unfold" into your vector indices, $i_k\to \mu_{1,k}\mu_{2,k}\cdots\mu_{s,k}$, if you so desire.

This is how computers deal with higher-dimensional arrays, after all.