Modularity of Schur index, modular differential equations, and high-temperature asymptotics

In this paper we analytically explore the modularity of the flavored Schur index of 4d $\mathcal{N} = 2$ SCFTs. We focus on the $A_1$ theories of class-$\mathcal{S}$ and $\mathcal{N} = 4$ theories with $SU(N)$ gauge group. We work out the modular orbit of the flavored index and defect index, compute the dimension of the space spanned by the orbit, and provide complete basis for computing modular transformation matrices. The dimension obtained from the flavored analysis predicts the minimal order of the unflavored modular differential equation satisfied by the unflavored Schur index. With the help of modularity, we also study analytically the high-temperature asymptotics of the Schur index. In the high-temperature limit $\tau \to +i0$, we identified the (defect) Schur index of the genus-zero $A_1$ theories of class-$\mathcal{S}$ with the $S^3$-partition function of the $SU(2) \times U(1)^n$ star-shape quiver (with Wilson line insertion). In the identification, we observe an interesting relation between the linear-independence of defect indices and the convergence of the Wilson line partition functions.