Calabi-Yau algebras

By V. Ginzburg. 2006

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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].

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b9077e4ae188bfbb78ebd2f24e225140247d1fbb

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Citations in the corpus

Skew Calabi-Yau triangulated categories and Frobenius Ext-algebras (2014, background): “smooth projective variety with trivial dualizing sheaf is an important example. Other interesting examples include the bounded derived categories of ﬁnite-dimensional modules over Calabi-Yau algebras [Gi]. We are especially interested in derived categories of graded modules for those graded algebras”
Twisted Calabi-Yau property of Ore extensions (2012, background): “tion. In this case, νis called the Nakayama automorphism of A. The Nakayama automorphism is unique up to an inner automorphism. A ν- twisted Calabi-Yau algebra Ais Calabi-Yau in the sense of Ginzburg [Gin07] if and only if νis an inner automorphism of A. Graded twisted Calabi-Yau algebrasare deﬁned ”
The Hochschild cohomology ring of a Frobenius algebra with semisimple Nakayama automorphism is a Batalin-Vilkovisky algebra (2014, background): “∪ ∆(β)), for homogeneous elements α,β ∈ H∗. The BV-operator ∆ : H∗ → H∗−1 is called a generator of the Gerstenhaber bracket [ , ]. BATALIN-VILKOVISKY 5 For a Calabi-Yau algebra, V. Ginzburg showed in [17] that the Hochschild cohomology of a CalabiYau algebra has a Batalin-Vilkovisky structure. More ”
Batalin-Vilkovisky algebra structures on Hochschild Cohomology (2007, background): “n algebra of polynomial diﬀerential forms APL(M) [9], -and of the fundamental class [M] ∈H(APL(M)). Acknowledgment: We wish to thank Jean-Claude Thomas for a discussion concerning Ginzburg’s preprint [16]. We would like also to thank Yves F´elix for explaining us the wonderful isomorphism (2). 2. Ho”
Superpotentials and higher order derivations (2008, background): “up of SL(2,C) we recover the preprojective algebra, for a ﬁnite small subg roup of GL(2,C) we recover the mesh relations, and for a ﬁnite subgroup of SL(3,C) we recover the superpotential in Ginzburg [8]. Furthermore if the group is abelian in GL(n,C) we can also recover the toric result. SUPERPOTEN”
Mutation of cluster-tilting objects and potentials (2008, background): “GGGG G 0”
Singular Derived Categories of Q-factorial terminalizations and Maximal Modification Algebras (2011, background): “imilar way. Remark 4.10. We remark that since we are working in the non-local CM setting, canonical modules are not unique. Although crepancy is deﬁned with respect to the geometric canonical ωR, and [G06] deﬁnes Serre functors with respect to ωR, there are beneﬁts of allowing the ﬂexibility of diﬀe”
On the noncommutative Donaldson-Thomas invariants arising from brane tilings (2008, methodology): “ i −→j M b:k→i Ae k ⊗e iA→ M a:i→j Ae j ⊗e iA→ M i Ae i ⊗e iA→ A→ 0. The exactness should be actually shown just in the ﬁrst and the second terms. Let us show that jis injective. The map jis given by [6] x⊗y→ X b:k→i yb⊗x−y⊗bx. A general element in L i e iA⊗Ae i can be written in the form X i∈Q0 X u”
Quivers with potentials and their representations I: Mutations (2007, background): “ith the completed path algebra, so our potentials are possibly inﬁnite linear combinations of cyclic paths. The Jacobian algebras also play an important role in the recent work on Calabi-Yau algebras [4, 25, 26, 27]. In this paper we introduce and study mutations for QPs and their (decorated) repres”