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

Abstract

We answer a long-standing question raised by Vafa--Witten on a relation between S-duality and conformal field theory, which related Yoshioka's blow-up formula and the WZW model for $\mathrm{SU}(r)$ at level $1$. Precisely, for the moduli space of Euler characteristics of rank $r$ instantons on the blow-up of an algebraic surface along a closed point, we construct the affine $\mathrm{gl}_r$-action on various cohomology theories, including the Grothendieck group of coherent sheaves, Hochschild homology groups, Chow groups and Hodge cohomology groups, and identifying the module as a basic representation. 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.