BPS blowup surface defects and Hurwitz chiral ring expansions

In this thesis we study a class of BPS surface defects of $SU(2)$ supersymmetric gauge theories defined on a blown-up geometry. We show that in the Nekrasov-Shatashvili limit the partition function in presence of these defects is in general a $\Tau$-function satisfying some Painlevé Hirota bilinear equation. Using a topological version of operator/state correspondence we compute the expansion of the $\Tau$-function in an integer basis, given in terms of the moduli of the quantum Seiberg-Witten curve. In the four dimensional case we study the modular properties of these solutions and show that they do directly lead to BCOV holomorphic anomaly equations for the corresponding topological string partition function. The resulting $\mathcal{T}$-functions are holomorphic and modular and as such they provide a natural non-perturbative completion of topological strings partition functions. In the five-dimensional setting, we discuss a UV completion of a class of Argyres-Douglas (AD) theories in the $\Omega$-background in terms of a renormalisation group flow from five dimensional $\mathcal{N}=1$ superconformal field theories (SCFT) on $S^1$. This is obtained via analysing these theories in the light of ($q$-)Painlev\'e/gauge theory correspondence, which allows to compute the five dimensional BPS partition functions as an expansion in the circular Wilson loop vev with integer $q$-polynomials coefficients. We discuss in detail the phase diagram of the four dimensional limits, pinpointing the special AD loci. Explicit computations are reported for $\tilde E_1$ SCFT and its limit to H$_0=(A_1,A_2)$ AD theory.