  The Spectral Form Factor (SFF) is a convenient tool for the characterization
of eigenvalue statistics of systems with discrete spectra, and thus serves as a
proxy for quantum chaoticity. This work presents an analytical calculation of
the SFF of the Chern-Simons Matrix Model (CSMM), which was first introduced to
describe the intermediate level statistics of disordered electrons at the
mobility edge. The CSMM is characterized by a parameter $ 0 \leq q\leq 1$,
where the Circular Unitary Ensemble (CUE) is recovered for $q\to 0$. The CSMM
was later found as a matrix model description of $U(N)$ Chern-Simons theory on
$S^3$, which is dual to a topological string theory characterized by string
coupling $g_s=-\log q$. The spectral form factor is proportional to a colored
HOMFLY invariant of a $(2n,2)$-torus link with its two components carrying the
fundamental and antifundamental representations, respectively. We check that
taking $N \to \infty$ whilst keeping $q<1$ reduces the connected SFF to an
exact linear ramp of unit slope, confirming the main result from
arXiv:2012.11703 for the specific case of the CSMM. We then consider the `t
Hooft limit, where $N \to \infty$ and $q \to 1^-$ such that $y = q^N $ remains
finite. As we take $q\to 1^-$, this constitutes the opposite extreme of the CUE
limit. In the `t Hooft limit, the connected SFF turns into a remarkable
sequence of polynomials which, as far as the authors are aware, have not
appeared in the literature thus far. A gap opens in the spectrum and, after
unfolding by a constant rescaling, the connected SFF approximates a linear ramp
of unit slope for all $y$ except $y \approx 1$, where the connected SFF goes to
zero. We thus find that, although the CSMM was introduced to describe
intermediate statistics and the `t Hooft limit is the opposite limit of the
CUE, we still recover Wigner-Dyson universality for all $y$ except $y\approx
1$.
