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We prove a blow-up formula for the generating series of virtual$\chi_y$-genera for moduli spaces of sheaves on projective surfaces, which isrelated to a conjectured formula for topological $\chi_y$-genera of G\"ottsche.Our formula is a refinement of one by Vafa-Witten relating to S-duality. Weprove the formula simultaneously in the setting of Gieseker stable sheaves onpolarised surfaces and also in the setting of framed sheaves on $\mathbb{P}^2$.The proof is based on the blow-up algorithm of Nakajima-Yoshioka for framedsheaves on $\mathbb{P}^2$, which has recently been extend to the setting ofGieseker $H$-stable sheaves on $H$-polarised surfaces by Kuhn-Tanaka.

The Blowup Formula for the Instanton Part of Vafa-Witten Invariants on Projective Surfaces