On Blowup Formulae for the S-duality Conjecture of Vafa and Witten Ii: the Universal Functions

This is a continuation of our work [L-Q] on blowup formulae for the S-duality conjecture of Vafa and Witten. In [V-W], Vafa and Witten formulated some mathematical predictions about the Euler characteristics of instanton moduli spaces derived from the S-duality conjecture in physics. From these mathematical predictions, a blowup formula was proposed based upon the work of Yoshioka [Yos]. Roughly speaking, the blowup formula says that there exists a universal relation between the Euler characteristics of instanton moduli spaces for a smooth four manifold and the Euler characteristics of instanton moduli spaces for the blowup of the smooth four manifold. The universal relation is independent of the four manifold and related to some modular forms. In [L-Q], we verified this blowup formula for the gauge group SU(2) and its dual group SO(3) when the underlying four manifold is an algebraic surface. In fact, we proved a stronger blowup formula in [L-Q], i.e. a blowup formula for the virtual Hodge numbers of instanton moduli spaces. However, in [L-Q], we did not find a closed formula for the universal function which appears in this stronger blowup formula. Our goal of the present paper is to determine a closed formula for this universal function. To state the blowup formulae proved in [L-Q], we recall some standard definitions and notations. Let φ : X̃ → X be the blowing-up of an algebraic surface X at a point x0 ∈ X, and E be the exceptional divisor. For simplicity, we always assume that X is simply connected. Fix a divisor c1 on X, c̃1 = φ∗c1 − aE with a = 0 or 1, and an ample divisor H on X with odd (H · c1). For an integer n, let MH(c1, n) be the moduli space of Mumford-Takemoto H-stable rank-2 bundles with Chern classes c1 and n, MH(c1, n) be the moduli space of Gieseker H-semistable rank-2 torsion-free sheaves with Chern classes c1 and n, and MH(c1, n) be the Uhlenbeck compactification of MH(c1, n) from gauge theory [Uhl, Don, LiJ]. It is well-known that both the Gieseker moduli spaces and