On a four-manifold underlying a complex smooth projective surface S, Tanaka-Thomas gave an algebro-geometric definition of the SU(r) Vafa-Witten partition function. When S has a non-zero holomorphic 2-form, the partition function is essentially topological with a rich modular structure. The algebro-geometric viewpoint provides new insights into Vafa-Witten theory such as (1) new mathematical verifications of S-duality, (2) K-theoretic refinement, (3) higher rank expressions with relations to the Rogers-Ramanujan continued fraction.