Chiral algebra, Wilson lines, and mixed Hodge structure of Coulomb branch

We find an intriguing relation between the chiral algebra and the mixed Hodge structure of the Coulomb branch of four dimensional N = 2 superconformal field theories. We identify the space of irreducible characters of the N = 4 $SU(N)$ chiral algebra V[TSU(N)] by analytically computing the Wilson line Schur index, and imposing modular invariance. We further establish a map from the V[TSU(N)] characters to the characters of the Tp, N chiral algebra. We extract the pure part of the mixed Hodge polynomial PHc of the Coulomb branch compactified on a circle, and prove that PHc encodes the representation theory of V[TSU(N)]. We expect this to be a new entry of the 4D mirror symmetry framework.