  Starting from H. Fr\"ohlich's second-quantized Hamiltonian for a
$d$-dimensional electron gas in interaction with lattice phonons describing the
quantum vibrations of a metal, we present a rigorous mathematical derivation of
the superconducting state, following the principles laid out originally in 1957
by J. Bardeen, L. Cooper and J. Schrieffer. As in the series of papers written
on the subject in the 90es, of which the present paper is a continuation, the
representation of ions as a uniform charge background allows for a
$(1+d)$-dimensional fermionic quantum-field theoretic reformulation of the
model at equilibrium. For simplicity, we restrict in this article to $d=2$
dimensions and zero temperature, and disregard effects due to electromagnetic
interactions. Under these assumptions, we prove transition from a Fermi liquid
state to a superconducting state made up of Cooper pairs of electrons at an
energy level $\Gamma_{\phi}\sim \hbar\omega_D e^{-\pi/m\lambda}$ equal to the
mass gap, expressed in terms of the Debye frequency $\omega_D$, electron mass
$m$ and coupling constant $\lambda$. The dynamical $U(1)$-symmetry breaking
produces at energies lower than the energy gap $\Gamma_{\phi}$ a Goldstone
boson, a non-massive particle described by an effective $(2+1)$-dimensional
non-linear sigma-model, whose parameters and correlations are computed. The
proof relies on a mixture of general concepts and tools (multi-scale cluster
expansions, Ward identities), adapted to this quantum many-body problem with
its extended infra-red singularity located on the Fermi circle, and a specific
$1/N$-expansion giving the leading diagrams at intermediate energies. Ladder
diagrams are proved to provide the leading behavior in the infra-red limit, in
agreement with mean-field theory predictions.
