Instantons and vortices on noncommutative toric varieties

Lucio S. Cirio*, Giovanni Landi, Richard J. Szabo

*Corresponding author for this work

Research output: Contribution to journalLiterature reviewpeer-review

4 Citations (Scopus)
66 Downloads (Pure)


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 quantized 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.

Original languageEnglish
Article number1430008
Number of pages51
JournalReviews in Mathematical Physics
Issue number9
Publication statusPublished - Oct 2014


  • Noncommutative algebraic geometry
  • noncommutative gauge theory
  • instantons
  • vortices


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