  Superconductivity results from an instability of the Fermi surface -- contour
of \textit{poles} of the single particle propagator -- to an infinitesimally
small attraction between electrons. Here, we instead discuss the analogous
problem on a model \textit{Luttinger} surface, or contour of \textit{zeros} of
the Green function. At zero temperature ($\beta \rightarrow \infty$) and a
critical interaction strength ($u_{c\infty}$) characterized by the residue of
self-energy pole, we find that the pair susceptibility diverges leading to a
superconducting instability. We evaluate the pair fluctuation partition
function and find that the spectral density in the normal state has an
interaction-driven, power-law $\frac{1}{\sqrt{\omega}}$ type, van-Hove
singularity (vHS) indicating non-Fermi liquid (NFL) physics. Crucially, in the
strong coupling limit ($\beta u \gg 1$), the leading order fluctuation free
energy terms in the normal state of this NFL-SC transition resemble the
equivalent $\left(O(1)\right)$ terms of the Sachdev-Ye-Kitaev (SYK) model. This
free energy contribution takes a simple form $-\beta F = \beta u_{c\infty} -
\gamma~\text{ln}\left(\beta u_{c \infty}\right)$ where $\gamma$ is a constant
equal to $\frac{1}{2}$. Weak impurity scattering ($\tau \gg \beta^{-1}$) leaves
the low-energy spectral density unaffected, but leads to an interaction-driven
enhancement of superconductivity. Our results shed light on the role played by
order-parameter fluctuations in providing the key missing link between Mott
physics and strongly coupled toy-models exhibiting gravity duals.

Strongly Correlated Electrons;Statistical Mechanics;Superconductivity;High Energy Physics - Theory