Calabi-Yau algebras

Abstract

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].

few comments added, misprints corrected

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Cited by in corpus (203)

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• Exchange graphs and Ext quivers
• Wall Crossing of BPS States on the Conifold from Seiberg Duality and Pyramid Partitions
• Consistency conditions for brane tilings
• Dimer models and the special McKay correspondence
• Stability conditions and quantum dilogarithm identities for Dynkin quivers
• The representation type of Jacobian algebras
• The cohomological Hall algebra of a preprojective algebra
• Relative singularity categories I: Auslander resolutions
• Quiver Yangian from Crystal Melting
• Stable categories of Cohen-Macaulay modules and cluster categories
• Skew Calabi-Yau triangulated categories and Frobenius Ext-algebras
• Deformed Calabi-Yau Completions
• Donaldson-Thomas theory for Calabi-Yau 4-folds
• The Calabi-Yau Landscape: from Geometry, to Physics, to Machine-Learning
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• Higher Nakayama algebras I: Construction
• Mirror symmetry: from categories to curve counts
• Purity for graded potentials and quantum cluster positivity
• Stability conditions and the $A_2$ quiver
• Auslander-Reiten theory revisited
• Gerasimov's theorem and N-Koszul algebras
• On the noncommutative Donaldson-Thomas invariants arising from brane tilings
• The Hochschild cohomology ring of a Frobenius algebra with semisimple Nakayama automorphism is a Batalin-Vilkovisky algebra
• Theorie homotopique des DG-categories
• Crepant resolutions and brane tilings I: Toric realization
• On Serre duality for compact homologically smooth DG algebras
• On two cohomological Hall algebras
• Cluster characters II: A multiplication formula
• D-Branes on Toric Calabi-Yau Varieties
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• Quantum cluster variables via vanishing cycles
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• A note on consistency conditions on dimer models
• Cellular resolutions of noncommutative toric algebras from superpotentials
• A Comprehensive Survey of Brane Tilings
• On Generalized Cluster Categories
• 2-CY-tilted algebras that are not Jacobian
• The motivic Donaldson-Thomas invariants of (-2) curves
• Derived equivalences from mutations of quivers with potential
• Quivers with potentials and their representations I: Mutations
• C-vectors via $\tau$-tilting theory
• The Nakayama automorphism of the almost Calabi-Yau algebras associated to SU(3) modular invariants
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• Crossed-products of Calabi-Yau algebras by finite groups
• Relative singularity categories II: DG models
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• Lectures on Noncommutative Resolutions
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• A theory of generalized Donaldson-Thomas invariants. II. Multiplicative identities for Behrend functions
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• Growth of Graded Twisted Calabi-Yau Algebras
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• A 3-Calabi-Yau algebra with G_2 symmetry constructed from the octonions
• Deformations of Koszul Artin-Schelter Gorenstein algebras
• On higher rank Donaldson-Thomas invariants
• An introduction to Noncommutative Projective Geometry
• Mutation of cluster-tilting objects and potentials
• A Calabi-Yau algebra with $E_6$ symmetry and the Clebsch-Gordan series of $sl(3)$
• On pseudo-Anosov autoequivalences
• AS-regularity of geometric algebras of plane cubic curves
• Singular Derived Categories of Q-factorial terminalizations and Maximal Modification Algebras
• Superpotentials and Higher Order Derivations
• The A-infinity Deformation Theory of a Point and the Derived Categories of Local Calabi-Yaus
• Stability Conditions and Branes at Singularities
• Curvature of the space of stability conditions
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• Relative Calabi-Yau completions
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• Skew group algebras of Jacobian algebras
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• A criterion for homogeneous potentials to be 3-Calabi-Yau
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• Twisted Calabi-Yau property of Ore extensions
• Orientation data for coherent sheaves on the local projective plane
• Classification of quadratic and cubic PBW algebras on three generators
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• Duality for Differential Operators of Lie-Rinehart Algebras
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• Asymptotic shifting numbers in triangulated categories
• Equivalence of A-Maximization and Volume Minimization
• Graded 3-Calabi-Yau algebras as Ore extensions of 2-Calabi-Yau algebras
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• Relative Singularity Categories
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• A Note on Support in Triangulated Categories
• Finite-dimensional algebras are (m>2)-Calabi-Yau-tilted
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• Hodge numbers and Hodge structures for Calabi-Yau categories of dimension three
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• Hochschild (co)homologies of dg $K$-rings and their Koszul duals
• Nakayama automorphism and applications
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• A Simple Construction of Derived Representation Schemes
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• Van Den Bergh isomorphisms in String Topology
• Systolic inequalities for K3 surfaces via stability conditions
• The Calabi-Yau property of Hopf algebras and braided Hopf algebras
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• The braid group for a quiver with superpotential
• Derived categories and Calabi-Yau algebras
• An algebraic description of the bispectrality of the biorthogonal rational functions of Hahn type
• A Geometric Approach to Hochschild Cohomology of the Exterior Algebra
• A complete classification of $3$-dimensional quadratic AS-regular algebras of Type EC
• Quivers with potentials and actions of finite abelian groups
• Non-commutative Courant algebroids and Quiver algebras
• Circle and line bundles over generalized Weyl algebras
• Calabi-Yau algebras viewed as deformations of Poisson algebras
• Quantum dilogarithms and partition q-series
• From Calabi-Yau dg Categories to Frobenius manifolds via Primitive Forms
• C-vectors and dimension vectors for cluster-finite quivers
• Noncommutative mirror symmetry for punctured surfaces
• Dualities of artinian coalgebras with applications to noetherian complete algebras
• Returning Arrows for Self-injective Algebras and Artin-Schelter Regular Algebras
• Calabi-Yau coalgebras
• Poincare-Birkhoff-Witt Deformations of Calabi-Yau Algebras
• Cocommutative Calabi-Yau Hopf algebras and deformations
• Sheaves of Twisted Cherednik Algebras as Universal Filtered Formal Deformations
• D-critical locus structure on the Hilbert schemes of some local toric Calabi-Yau threefolds
• Skew Calabi-Yau property of normal extensions
• Recollements from generalized tilting
• Cluster tilting objects in generalized higher cluster categories
• Higher APR tilting preserves $n$-representation infiniteness
• Module categories over finite pointed tensor categories
• Cluster algebras arising from cluster tubes II: the Caldero-Chapoton map
• Batalin-Vilkovisky structure on Hochschild cohomology with coefficients in the dual algebra
• Twisted bi-symplectic structure on Koszul twisted Calabi-Yau algebras
• Stably Calabi-Yau properties of derivation quotient algebras
• Hopf-Galois objects of Calabi-Yau Hopf algebras
• Groups actions, and D equivalences of categories of coherent sheaves of symplectic resolutions
• On the boundary algebras of the Jacobian algebras of the bordered marked surfaces
• On Gromov-Yomdin type theorems and a categorical interpretation of holomorphicity
• Ext algebra of Nichols algebras of type $A_2$
• D-critical loci for local toric Calabi-Yau 3-folds
• Batalin--Vilkovisky algebra structures on the Hochschild cohomology of generalized Weyl algebras
• A note on the bijectivity of antipode of a Hopf algebra and its applications
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• Noncommutative derived Poisson reduction
• Non-commutative projective Calabi-Yau schemes
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• A new family of algebras whose representation schemes are smooth
• Cleft extensions of Koszul twisted Calabi-Yau algebras
• On the homology of almost Calabi-Yau algebras associated to SU(3) modular invariants
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• Derived Picard groups of homologically smooth Koszul DG algebras
• On homological smoothness of generalized Weyl algebras over polynomial algebras in two variables