Why Cooper pairs live in AdS2: a spectral analysis of the Yukawa-SYK model

Abstract

We establish the spectral foundation of the geometric formulation of Cooper pairing in the Yukawa--Sachdev--Ye--Kitaev model. Starting from the large-$N$ bilocal effective action, we derive the Gaussian fluctuation kernel in the even-frequency spin-singlet Cooper channel. Following the construction of Maldacena and Stanford, we determine the kernel eigenvalue $k\left(h\right)$ analytically for arbitrary conformal weight $h$ and resolve the pairing fluctuations into the continuous and discrete sectors of the associated de-Sitter space Laplacian. We show that the superconducting instability and the universal low-energy fluctuations near it reside entirely in the continuous scattering sector, while the discrete modes remain non-critical. Expanding the kernel about the lowest continuum mode yields a Klein--Gordon action on ${\rm dS}_2$, restricted to the continuous spectral subspace. This is precisely the subspace on which the inverse Radon transform to ${\rm AdS}_2$ is well defined. The projected bilocal Cooper-pair field can therefore be mapped onto a scalar matter field propagating in ${\rm AdS}_2$, without requiring any further spectral restriction on the bulk theory. Our results provide the missing microscopic justification for the projection implicit in earlier holographic formulations and clarify how a local bulk field emerges from the bilocal pairing theory.