Quantum (non-commutative) toric geometry: Foundations

Ludmil Katzarkov, Ernesto Lupercio, Laurent Meersseman, Alberto Verjovsky

Research output: Contribution to journalArticlepeer-review


In this paper, we will introduce Quantum Toric Varieties which are (non-commutative) generalizations of ordinary toric varieties where all the tori of the classical theory are replaced by quantum tori. Quantum toric geometry is the non-commutative version of the classical theory; it generalizes non-trivially most of the theorems and properties of toric geometry. By considering quantum toric varieties as (non-algebraic) stacks, we show that their category is equivalent to a certain category of quantum fans. We develop a Quantum Geometric Invariant Theory (QGIT) type construction of Quantum Toric Varieties. Unlike classical toric varieties, quantum toric varieties admit moduli and we define their moduli spaces, prove that these spaces are orbifolds and, in favorable cases, up to homotopy, they admit a complex structure.

Original languageEnglish (US)
Article number107945
JournalAdvances in Mathematics
StatePublished - Nov 19 2021


  • Irrational fans and polytopes
  • Mirror symmetry
  • Moduli spaces
  • Non-commutative geometry
  • Quantum integrable systems
  • Toric geometry

ASJC Scopus subject areas

  • Mathematics(all)


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