I am reading the following notes https://munsal.files.wordpress.com/2014/10/marino-lectures2014.pdf. On section 4.3 the euclidean Yang-Mills theory is considered. It is said that renormalizability and gauge invariance allow the following Lagrangian form

$$

\mathcal{L}_{\theta}=\mathcal{L}_{YM}-i\theta{}q(x)

$$

where

$$

q(x)=\frac{1}{32\pi^2}F_{\mu\nu}^a\tilde{F}^{a\mu\nu}

$$

where we have the usual YM lagrangian and the theta term. In equation (4.54) he defines what he calls the partition function in the presence of the $\theta$ angle as

$$

Z(\theta)=\int[\mathcal{D}A]e^{-\int{}d^4x\mathcal{L}}

$$

he then says that he will write this expression above as an exponential

$$

Z(\theta)=\int[\mathcal{D}A]e^{-\int{}d^4x\mathcal{L}}=e^{-VE_V(\theta)}

$$

where $V$ is the volume of the space. I need more motivation dor this step. Why can we write this as an exponential? He then says that the ground state energy density is given by

$$

\lim_{V\to\infty}E_V(\theta)=E(\theta)

$$

it is completely far from obvious why this should be the ground state energy. What is behind this?