  Let $\rho$ be compactly supported on $D \subset \mathbb R^2$. Endow $\mathbb
R^2$ with the metric $e^{\rho}(dx_1^2 + dx_2^2)$. As $\delta \to 0$ the set of
Brownian loops centered in $D$ with length at least $\delta$ has measure
$$\frac{\text{area}(D)}{2\pi \delta} + \frac{1}{48\pi}(\rho,\rho)_{\nabla}+
o(1).$$ When $\rho$ is smooth, this follows from the classical Polyakov-Alvarez
formula. We show that the above also holds if $\rho$ is not smooth, e.g. if
$\rho$ is only Lipschitz. This fact can alternatively be expressed in terms of
heat kernel traces, eigenvalue asymptotics, or zeta regularized determinants.
Variants of this statement apply to more general non-smooth manifolds on which
one considers all loops (not only those centered in a domain $D$).
  We also show that the $o(1)$ error is uniform for any family of $\rho$
satisfying certain conditions. This implies that if we weight a measure $\nu$
on this family by the ($\delta$-truncated) Brownian loop soup partition
function, and take the vague $\delta \to 0$ limit, we obtain a measure whose
Radon-Nikodym derivative with respect to $\nu$ is $\exp\bigl(
\frac{1}{48\pi}(\rho,\rho)_{\nabla}\bigr)$. When the measure is a certain
regularized Liouville quantum gravity measure, a companion work [APPS20] shows
that this weighting has the effect of changing the so-called central charge of
the surface.
