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

Quantum (Non-commutative) Toric Geometry: Foundations