Algebraic deformations of toric varieties II. Noncommutative instantons

TL;DR

This paper explores noncommutative deformations of toric varieties, developing a noncommutative twistor theory and ADHM construction to explicitly build and analyze instanton gauge bundles and their moduli spaces.

Contribution

It introduces a noncommutative twistor theory and a braided ADHM construction, providing new tools for studying instantons on noncommutative toric varieties.

Findings

Constructed a noncommutative four-sphere with twistor theory.

Developed a braided ADHM construction for moduli of sheaves.

Found that partition functions match supersymmetric gauge theory results.

Abstract

We continue our study of the noncommutative algebraic and differential geometry of a particular class of deformations of toric varieties, focusing on aspects pertinent to the construction and enumeration of noncommutative instantons on these varieties. We develop a noncommutative version of twistor theory, which introduces a new example of a noncommutative four-sphere. We develop a braided version of the ADHM construction and show that it parametrizes a certain moduli space of framed torsion free sheaves on a noncommutative projective plane. We use these constructions to explicitly build instanton gauge bundles with canonical connections on the noncommutative four-sphere that satisfy appropriate anti-selfduality equations. We construct projective moduli spaces for the torsion free sheaves and demonstrate that they are smooth. We define equivariant partition functions of these moduli spaces, finding that they coincide with the usual instanton partition functions for supersymmetric gauge theories on C^2.