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Consider a Brownian loop soup $\mathcal{L}_D^\theta$ with subcritical
intensity $\theta \in (0,1/2]$ in some 2D bounded simply connected domain. We
define and study the properties of a conformally invariant field $h_\theta$
naturally associated to $\mathcal{L}_D^\theta$. Informally, this field is a
signed version of the local time of $\mathcal{L}_D^\theta$ to the power
$1-\theta$. When $\theta=1/2$, $h_\theta$ is a Gaussian free field (GFF) in
$D$.

Our construction of $h_\theta$ relies on the multiplicative chaos
$\mathcal{M}_\gamma$ associated with $\mathcal{L}_D^\theta$, as introduced in
[ABJL23]. Assigning independent symmetric signs to each cluster, we restrict
$\mathcal{M}_\gamma$ to positive clusters. We prove that, when $\theta=1/2$,
the resulting measure $\mathcal{M}_\gamma^+$ corresponds to the exponential of
$\gamma$ times a GFF. At this intensity, the GFF can be recovered by
differentiating at $\gamma=0$ the measure $\mathcal{M}_\gamma^+$. When
$\theta<1/2$, we show that $\mathcal{M}_\gamma^+$ has a nondegenerate
fractional derivative at $\gamma=0$ defining a random generalised function
$h_\theta$.

We establish a result which is analoguous to the recent work [ALS23] in the
GFF case ($\theta=1/2$), but for $h_\theta$ with $\theta \in (0,1/2]$. Relying
on the companion article [JLQ23], we prove that each cluster of
$\mathcal{L}_D^\theta$ possesses a nondegenerate Minkowski content in some
non-explicit gauge function $r \mapsto r^2 |\log r|^{1-\theta+o(1)}$. We then
prove that $h_\theta$ agrees a.s.\ with the sum of the Minkowski content of
each cluster multiplied by its sign.

We further extend the couplings between CLE$_4$, SLE$_4$ and the GFF to
$h_\theta$ for $\theta\in(0,1/2]$. We show that the (non-nested) CLE$_\kappa$
loops form level lines for $h_\theta$ and that there exists a constant height
gap between the values of the field on either side of the CLE loops.