  We consider $N$ bosons in a box in $\mathbb {R}^d$ with volume $N/\rho$ under
the influence of a mutually repellent pair potential. The particle density
$\rho\in (0,\infty)$ is kept fixed. Our main result is the identification of
the limiting free energy, $f(\beta,\rho)$, at positive temperature $1/\beta$,
in terms of an explicit variational formula, for any fixed $\rho$ if $\beta$ is
sufficiently small, and for any fixed $\beta$ if $\rho$ is sufficiently small.
The thermodynamic equilibrium is described by the symmetrized trace of
$e^{-\beta {\mathcal{H}}_N}$, where ${\mathcal{H}}_N$ denotes the corresponding
Hamilton operator. The well-known Feynman--Kac formula reformulates this trace
in terms of $N$ interacting Brownian bridges. Due to the symmetrization, the
bridges are organized in an ensemble of cycles of various lengths. The novelty
of our approach is a description in terms of a marked Poisson point process
whose marks are the cycles. This allows for an asymptotic analysis of the
system via a large-deviations analysis of the stationary empirical field. The
resulting variational formula ranges over random shift-invariant marked point
fields and optimizes the sum of the interaction and the relative entropy with
respect to the reference process. In our proof of the lower bound for the free
energy, we drop all interaction involving "infinitely long" cycles, and their
possible presence is signalled by a loss of mass of the "finitely long" cycles
in the variational formula. In the proof of the upper bound, we only keep the
mass on the "finitely long" cycles. We expect that the precise relationship
between these two bounds lies at the heart of Bose--Einstein condensation and
intend to analyze it further in future.
