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CalabiYau algebras
"... Abstract. We introduce some new algebraic structures arising naturally in the geometry of CY manifolds and mirror symmetry. We give a universal construction of CY algebras in terms of a noncommutative symplectic DG algebra resolution. In dimension 3, the resolution is determined by a noncommutative ..."
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Abstract. We introduce some new algebraic structures arising naturally in the geometry of CY manifolds and mirror symmetry. We give a universal construction of CY 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 CY 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 CY algebras involving quivers, 3dimensional McKay correspondence, crepant resolutions, Sklyanin algebras, hyperbolic 3manifolds and ChernSimons. Examples related to quantum Del Pezzo surfaces are discussed in [EtGi].
Hochschild and cyclic homology of YangMills algebras
, 906
"... The aim of this article is to compute the Hochschild and cyclic homology groups of the YangMills algebras YM(n) (n∈N≥2) defined by A. Connes and M. DuboisViolette in [CD1], continuing thus the study of these algebras that we have initiated in [HS]. The computation involves the use of a spectral se ..."
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The aim of this article is to compute the Hochschild and cyclic homology groups of the YangMills algebras YM(n) (n∈N≥2) defined by A. Connes and M. DuboisViolette in [CD1], continuing thus the study of these algebras that we have initiated in [HS]. The computation involves the use of a spectral sequence associated to the natural filtration on the universal enveloping algebra YM(n) provided by a Lie idealtym(n) inym(n) which is free as Lie algebra.
Solotar A., Representation theory of Yang–Mills algebras
"... The aim of this article is to describe families of representations of the YangMills algebras YM(n) (n ∈ N≥2) defined by A. Connes and M. DuboisViolette in [CD]. We first describe irreducible finite dimensional representations. Next, we provide families of infinite dimensional representations of YM ..."
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The aim of this article is to describe families of representations of the YangMills algebras YM(n) (n ∈ N≥2) defined by A. Connes and M. DuboisViolette in [CD]. We first describe irreducible finite dimensional representations. Next, we provide families of infinite dimensional representations of YM(n), big enough to separate points of the algebra. In order to prove this result, we prove and use that all Weyl algebras Ar(k) are epimorphic images of YM(n).
Barduality M.Movshev
, 2008
"... In this note we give a homological explanation of ”pure spinors ” in YM theories with minimal amount of supersymmetries. We construct A ∞ algebras A for every dimension D = 3, 4, 6,10, which for D = 10 coincides with homogeneous coordinate ring of pure spinors with coordinate λ α. These algebras are ..."
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In this note we give a homological explanation of ”pure spinors ” in YM theories with minimal amount of supersymmetries. We construct A ∞ algebras A for every dimension D = 3, 4, 6,10, which for D = 10 coincides with homogeneous coordinate ring of pure spinors with coordinate λ α. These algebras are Bardual to Lie algebras generated by supersymmetries, written in components. The algebras have a finite number of higher multiplications. It is typical for generic A ∞ algebra. The main result of the present note is that in dimension D = 3,6, 10 the algebra A ⊗Λ[θ α] ⊗Matn with a differential d is equivalent to BatalinVilkovisky algebra of minimally supersymmetric YM theory in dimension D reduced to a point. This statement can be extended to nonreduced theories. 1
Bonn
, 2008
"... This is a next paper from a sequel devoted to algebraic aspects of YangMills theory. We undertake a study of deformation theory of YangMills algebra Y Ma “universal solution ” of YangMills equation. We compute (cyclic) (co)homology of Y M. 1 Introduction. YangMills algebra Y M was introduced in ..."
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This is a next paper from a sequel devoted to algebraic aspects of YangMills theory. We undertake a study of deformation theory of YangMills algebra Y Ma “universal solution ” of YangMills equation. We compute (cyclic) (co)homology of Y M. 1 Introduction. YangMills algebra Y M was introduced in [3]. We in [7] rediscovered it supersymmetric version analyzing HoweBerkovits construction [4] [1] of YangMills theory using pure spinors. The algebra Y M is by definition a quotient of a free Lie algebra Free(V),
Barduality M.Movshev
, 2008
"... In this note we give a homological explanation of “pure spinors ” in YM theories with minimal amount of supersymmetries. We construct A ∞ algebras A for every dimension D = 3, 4, 6,10, which for D = 10 coincides with homogeneous coordinate ring of pure spinors with coordinate λ α. These algebras are ..."
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In this note we give a homological explanation of “pure spinors ” in YM theories with minimal amount of supersymmetries. We construct A ∞ algebras A for every dimension D = 3, 4, 6,10, which for D = 10 coincides with homogeneous coordinate ring of pure spinors with coordinate λ α. These algebras are Bardual to Lie algebras generated by supersymmetries, written in components. The algebras have a finite number of higher multiplications. It is typical for generic A ∞ algebra. The main result of the present note is that in dimension D = 3,6, 10 the algebra A⊗Λ[θ α]⊗Matn with a differential d is equivalent to BatalinVilkovisky algebra of minimally supersymmetric YM theory in dimension D reduced to a point. This statement can be extended to nonreduced theories. 1