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arxiv: math/0612139 · v3 · submitted 2006-12-06 · 🧮 math.AG · hep-th· math.CT· math.QA· math.RA

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Calabi-Yau algebras

Victor Ginzburg
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classification 🧮 math.AG hep-thmath.CTmath.QAmath.RA
keywords calabi-yaualgebrasalgebraexamplesmanifoldsnoncommutativepotentialrelated
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We introduce some new algebraic structures arising naturally in the geometry of Calabi-Yau manifolds and mirror symmetry. We give a universal construction of Calabi-Yau algebras in terms of a noncommutative symplectic DG algebra resolution. In dimension 3, the resolution is determined by a noncommutative potential. Representation varieties of the Calabi-Yau algebra are intimately related to the set of critical points, and to the sheaf of vanishing cycles of the potential. Numerical invariants, like ranks of cyclic homology groups, are expected to be given by `matrix integrals' over representation varieties. We discuss examples of Calabi-Yau algebras involving quivers, 3-dimensional McKay correspondence, crepant resolutions, Sklyanin algebras, hyperbolic 3-manifolds and Chern-Simons. Examples related to quantum Del Pezzo surfaces will be discussed in [EtGi].

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Cited by 2 Pith papers

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  1. Proper modules over Ginzburg dg algebras and compact Fukaya categories of plumbings

    math.SG 2026-05 unverdicted novelty 7.0

    Compact Fukaya categories of general plumbings are generated by proper modules over associated Ginzburg dg algebras and equivalent to proper modules over wrapped Fukaya categories and to microlocal sheaves.

  2. Some faithful algebraic braid twist group actions for 3-fold crepant resolutions

    math.AG 2026-03 unverdicted novelty 7.0

    For crepant resolutions X(1,3,9) and X(1,3,13) the derived categories admit faithful braid twist group actions of types D and E induced by spherical object configurations.