Modular forms, deformation of punctured spheres, and extensions of symmetric tensor representations

Let~$X=\text{Po}\mathrm{/}\mathrm{\Gamma }$ be an~$n$-punctured sphere, $n>3$. We introduce and study~$n-3$ deformation operators on the space of modular forms~$M_{\ast }(\mathrm{\Gamma })$ based on the classical theory of uniformizing differential equations and accessory parameters. When restricting to modular functions, we recover a construction in Teichm"uller theory related to the deformation of the complex structure of~$X$. We describe the deformation operators in terms of derivations with respect to Eichler integrals of weight-four cusp forms, and in terms of vector-valued modular forms attached to extensions of symmetric tensor representations.