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Superconductivity results from an instability of the Fermi surface -- contourof \textit{poles} of the single particle propagator -- to an infinitesimallysmall attraction between electrons. Here, we instead discuss the analogousproblem on a model \textit{Luttinger} surface, or contour of \textit{zeros} ofthe Green function. At zero temperature ($\beta \rightarrow \infty$) and acritical interaction strength ($u_{c\infty}$) characterized by the residue ofself-energy pole, we find that the pair susceptibility diverges leading to asuperconducting instability. We evaluate the pair fluctuation partitionfunction and find that the spectral density in the normal state has aninteraction-driven, power-law $\frac{1}{\sqrt{\omega}}$ type, van-Hovesingularity (vHS) indicating non-Fermi liquid (NFL) physics. Crucially, in thestrong coupling limit ($\beta u \gg 1$), the leading order fluctuation freeenergy terms in the normal state of this NFL-SC transition resemble theequivalent $\left(O(1)\right)$ terms of the Sachdev-Ye-Kitaev (SYK) model. Thisfree 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 constantequal to $\frac{1}{2}$. Weak impurity scattering ($\tau \gg \beta^{-1}$) leavesthe low-energy spectral density unaffected, but leads to an interaction-drivenenhancement of superconductivity. Our results shed light on the role played byorder-parameter fluctuations in providing the key missing link between Mottphysics and strongly coupled toy-models exhibiting gravity duals.

Pairing instability on a Luttinger surface: A non-Fermi liquid to superconductor transition and its Sachdev-Ye-Kitaev dual