I actually have an integral which is used to prove Atiyah-Singer index theorem for spin complex in a path integral fashion. The integral I need to evaluate is following (in simplified form)
$\int \prod_{n=1}^\infty da_n^\mu exp{\frac{1}{4}}\sum_{n=1}^\infty[(2n \pi)^2 (a_n^\mu)^2 + a_n^{\mu} a_n^{\nu} R_{\mu\alpha} R_{\nu}^{\alpha}/M^4]$ where $a_n^\mu$ are fourier coefficients and $R_{\mu\nu}$ is the famous curvature tensor with it's two indices contracted (Ricci tensor),
the result to which should be
$\int \prod_{n=1}^\infty det |1-\frac{R_{\mu}^{\alpha} R_{\nu}^{\alpha}}{M^4 (2n\pi)^2}|^{-1/2}$.
Now I am having difficulty to verify this result. I have tried to diagonalize the Ricci tensor in terms of the eigenvalues $x_i$ and $-x_i$ and hence the product of two tensor in $R_{\mu\alpha} R_{\nu}^{\alpha}$ will give only diagonal terms which can be combined with the first term to give something similar to $[1-\frac{R_{\nu}^{\alpha} R_{\nu}^{\alpha}}{M^4 (2n\pi)^2}](2n \pi)^2 (a_n^\mu)^2$ and then I can perform Gaussian integration (did not work) to get similar but not same result.
Can anyone help me out?