Vertex Operators and Moduli Spaces of Sheaves

Abstract

The Nekrasov partition function in supersymmetric quantum gauge theory is mathematically formulated as an equivariant integral over certain moduli spaces of sheaves on a complex surface. In ``Seiberg-Witten Theory and Random Partitions'', Nekrasov and Okounkov studied these integrals using the representation theory of ``vertex operators'' and the infinite wedge representation. Many of these operators arise naturally from correspondences on the moduli spaces, such as Nakajima's Heisenberg operators, and Grojnowski's vertex operators. In this paper, we build a new vertex operator out of the Chern class of a vector bundle on a pair of moduli spaces. This operator has the advantage that it connects to the partition function by definition. It also incorporates the canonical class of the surface, whereas many other studies assume that the class vanishes. When the moduli space is the Hilbert scheme, we present an explicit expression in the Nakajima operators, and the resulting combinatorial identities. We then apply the vertex operator to the above integrals. In agreeable cases, the commutation properties of the vertex operator result in modularity properties of the partition function and related correlation functions. We present examples in which the integrals are completely determined by their modularity, and their first few values.

48 pages, 2 figures

References in corpus (5)

• On the Cobordism Class of the Hilbert Scheme of a Surface
• Jack polynomials and Hilbert schemes of points on surfaces
• Quantum cohomology of the Hilbert scheme of points in the plane
• Random partitions and instanton counting
• Framed Rank r Torsion-free Sheaves on CP^2 and Representations of the Affine Lie Algebra \hat{gl(r)}

Cited by in corpus (9)

• Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra
• Vertex operators and 2-representations of quantum affine algebras
• A survey of Heisenberg categorification via graphical calculus
• Vertex Operators, Grassmannians, and Hilbert Schemes
• K-Theory of Hilbert Schemes as a Formal Quantum Field Theory
• Restriction theorems for $\mu$-(semi)stable framed sheaves
• A projection formula for the ind-Grassmannian
• On Okounkov's conjecture connecting Hilbert schemes of points and multiple q-zeta values
• $K$-theory of moduli spaces of sheaves and large Grassmannians