# Integral of $e^{-xy}$ is a Dirac Delta

I was looking at papers about the SYK model page 33 (equation 112), in which they write

$$\int\mathscr{D}\Sigma\,\mathscr{D}G~e^{-\frac{N}{2}\int\limits_{\left[0,\beta\right]^2} d\tau_1d\tau_2\Sigma\left(G-\frac{1}{N}\sum\limits_{I=1}^N\psi_i\left(\tau_1\right)\psi_i\left(\tau_2\right)\right)}=1.\tag{112}$$

This would follow from $$\int d\sigma\,dg\ e^{-\sigma g}=1.$$

Wolfram alpha confirms this as a Cauchy principle value.

How would this be proved?

1. The underlying integral $$\int_{\mathbb{R}} \mathrm{d}\sigma~ e^{-\sigma g}, \qquad g~\in~\mathbb{R},$$ in eq. (112) (using the so-called Euclidean formulation cf. footnote 36) is not well-defined.
2. Presumably it should be defined via a Wick rotation $$\int_{\mathbb{R}} \mathrm{d}\sigma~ e^{\color{red}{i}\sigma g}~=~2\pi \delta(g), \qquad g~\in~\mathbb{R},$$ (using the so-called Minkowskian formulation).
• But $\sigma$ is a field. Why is there a factor of $i$ under Wick rotation? – Chetan Vuppulury Jun 30 '18 at 12:26