Instantons on the Blown-up Surface and the Affine Vertex Algebra

Vafa-Witten observed that Yoshioka’s blow-up formula for the Euler characteristics of rank $r$ instantons on an algebraic surface coincides with the character of the Wess-Zumino-Witten model for $\mathrm{S}\mathrm{U}(r)$ at level $1$, and raised the question of finding a rational conformal field theory explanation for this striking coincidence. In this work, we provide an answer to this question by constructing and analyzing the affine ${\mathrm{g}\mathrm{l}}_r$ action on various cohomology theories, including the Grothendieck group of coherent sheaves, Hochschild homology, Chow groups, and Hodge cohomology, of the moduli space of stable sheaves on a blown-up surface. A key ingredient in our proof is a representation-theoretic reformulation of the theory of Grassmannians of Tor-amplitude $[0,1]$-perfect complexes studied by the first-named author in terms of the spin representation of the finite-dimensional Clifford algebra. This may be viewed as a finite analog of the question of Vafa-Witten via the Boson-Fermion correspondence.