  We study on the metric graphs two types of scalar Gaussian free fields (GFF),
the usual one and the one twisted by a $\{-1,1\}$-valued gauge field. We show
that the latter can be obtained, up to an additional deterministic
transformation, by conditioning the first on a topological event. This event is
that all the sign clusters of the field should be trivial for the gauge field,
that is to say should not contain loops with holonomy $-1$. We also express the
probability of this topological event as a ratio of two determinants of
Laplacians to the power $1/2$, the usual Laplacian and the gauge-twisted
Laplacian. As an example, this gives on annular planar domains the probability
that no sign cluster of the metric graph GFF surrounds the inner hole of the
domain.
  Based on our result on the metric graph, and on previous works by Werner and
Cai-Ding on the clusters of the metric graph GFF in high dimension, we
formulate an intensity doubling conjecture. According to it, if the space
dimension is high enough, the cycles in the sign clusters of the metric graph
GFF converge in the scaling limit to a Brownian loop soup of intensity
parameter $\alpha = 1 = 2\times \dfrac{1}{2}$, which is the double of the
intensity parameter appearing in isomorphism theorems.
