Connecting topological strings and spectral theory via non-autonomous Toda equations

We consider the Topological String/Spectral theory duality on toric Calabi-Yau threefolds obtained from the resolution of the cone over the $Y^{N,0}$ singularity. Assuming Kyiv formula, we demonstrate this duality in a special regime thanks to an underlying connection between spectral determinants of quantum mirror curves and the non-autonomous (q)-Toda system. We further exploit this link to connect small and large time expansions in Toda equations. In particular we provide an explicit expression for their tau functions at large time in terms of a strong coupling version of irregular $W_N$ conformal blocks at $c=N-1$. These are related to a special class of multi-cut matrix models which describe the strong coupling regime of four dimensional, $\mathcal{N}=2$ $SU(N)$ super Yang-Mills.