How to express Dirac delta-function in functional form? In the paper written by Dmitry Bagrets, Alexander Altland and AlexKamenev (Sachdev–Ye–Kitaev model as Liouville quantum mechanics: http://www.sciencedirect.com/science/article/pii/S0550321316302206?via%3Dihub , http://dx.doi.org/10.1016/j.nuclphysb.2016.08.002 , open access, arXiv:1607.00694), they introduced an identity, equation (5):
$$1=\int {D}G\delta(NG_{\tau,\tau^{\prime}}+A_{\tau,\tau^{\prime}})=\int DGD\Sigma e^{\frac{1}{2}\int d\tau\int d\tau^{\prime}\Sigma_{\tau,\tau^{\prime}}(NG_{\tau,\tau^{\prime}}+A_{\tau,\tau^{\prime}})} .\tag{5}$$
They said, we need $G_{\tau,\tau^{\prime}}=-\frac{1}{N}A_{\tau,\tau^{\prime}}$, and $\Sigma_{\tau,\tau^{\prime}}$ is the Lagrange multiplier (real field). Does this mean
$$\int D\Sigma e^{\Sigma(NG+A)}=\delta(NG+A)~?$$
I do not understand the upper equation, because in my mind we have
$$\int \frac{dx}{2\pi}e^{iax}=\delta(a).$$
And why is $\Sigma_{\tau,\tau^{\prime}}$ called Lagrange multiplier?
 A: Indeed,
$$\int D\Sigma \,e^{i \int_{\tau,\tau'}\Sigma_{\tau,\tau'}G_{\tau,\tau'} }=\int D\Sigma \,e^{i S[\Sigma]}=\prod_{\tau,\tau'}\delta(G_{\tau,\tau'})$$
is just a generalization of 
$$\int dx\, e^{i a x}=\delta (a).$$
Note that sometimes the $i$ in the path integral is omitted. This is under the assumption that the analytical continuation $\Sigma\to -i\Sigma$ is allowed (this can be sometimes justified in perturbation theory, but it might also not be allowed non-perturbatively). This is usually done to render the minimum of the action (and thus the Lagrange multiplier, see below) real.
$\Sigma$ is sometimes called a Lagrange multiplier (somewhat abusively), because a mean-field (or equivalently a stationary phase) approximation of $\int D\Sigma \,e^{i S[\Sigma]}$ is given by the extremization of $S[\Sigma]$, that is, for $\delta S/\delta \Sigma=0$ (in addition to other mean-field terms coming from the integration over $G$ and other fields).
In this approximation, $\Sigma$ exactly plays the role of a Lagrange multiplier.
