Instantons and vortices on noncommutative toric varieties

TL;DR

This paper explores the quantization of toric varieties using noncommutative geometry, constructs instantons and vortices on these spaces, and finds that their partition functions match classical results, revealing new geometric structures.

Contribution

It introduces a new noncommutative four-sphere compatible with twistor theory and extends instanton and vortex moduli space computations to noncommutative settings.

Findings

Derived a unique noncommutative four-sphere compatible with twistor correspondence.

Extended instanton partition function computations to noncommutative gauge theories.

Constructed moduli spaces of noncommutative vortices with matching classical limits.

Abstract

We elaborate on the quantization of toric varieties by combining techniques from toric geometry, isospectral deformations and noncommutative geometry in braided monoidal categories, and the construction of instantons thereon by combining methods from noncommutative algebraic geometry and a quantised twistor theory. We classify the real structures on a toric noncommutative deformation of the Klein quadric and use this to derive a new noncommutative four-sphere which is the unique deformation compatible with the noncommutative twistor correspondence. We extend the computation of equivariant instanton partition functions to noncommutative gauge theories with both adjoint and fundamental matter fields, finding agreement with the classical results in all instances. We construct moduli spaces of noncommutative vortices from the moduli of invariant instantons, and derive corresponding equivariant partition functions which also agree with those of the classical limit.