Pre-Calabi-Yau structures and moduli of representations

Yeung, Wai-Kit

doi:10.48550/arxiv.1802.05398

Cited by 8 publications

(11 citation statements)

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“…Moreover, P. Seidel also informed us that in [Sei12] he gave the same definition (Definition 3.5), calling them boundary algebras. In the infinite dimensional case, some relations between pre-CY structures and symplectic/Poisson geometry appear in [Yeu18;IK20].…”

Section: Introductionmentioning

confidence: 99%

Pre-Calabi-Yau algebras and topological quantum field theories

Kontsevich¹,

Takeda²,

Vlassopoulos³

2021

Preprint

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We introduce a notion generalizing Calabi-Yau structures on Ainfinity algebras and categories, which we call pre-Calabi-Yau structures. This notion does not need either one of the finiteness conditions (smoothness or compactness) which are required for Calabi-Yau structures to exist. In terms of noncommutative geometry, a pre-CY structure is as a polyvector field satisfying an integrability condition with respect to a noncommutative analogue of the Schouten-Nijenhuis bracket. We show that a pre-CY structure defines an action of a certain PROP of chains on decorated Riemann surfaces. In the language of the cobordism perspective on TQFTs, this gives a partially defined extended 2-dimensional TQFT, whose 2-dimensional cobordisms are generated only by handles of index one. We present some examples of pre-CY structures appearing naturally in geometric and topological contexts.

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Section: Introductionmentioning

confidence: 99%

Pre-Calabi-Yau algebras and topological quantum field theories

Kontsevich¹,

Takeda²,

Vlassopoulos³

2021

Preprint

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“…As an application, we construct in [11] shifted symplectic and Poisson structures on derived moduli stacks of representations using an explicit trace map.…”

Section: Map Of Dg Modules Overmentioning

confidence: 99%

Shifted symplectic and Poisson structures on global quotients

Yeung¹

2021

Preprint

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For a derived stack obtained as a quotient of a derived affine scheme by a reductive group, we show that shifted symplectic structures can be characterized by the Cartan-de Rham complex. For nonreductive groups, we also show the analogous statement for Getzler's extension of the Cartan-de Rham complex. Dually, we construct a Cartan model for polyvector fields on global quotients by reductive groups, and show that shifted Poisson structures can be characterized by it.

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“…In [21], 5.3, Toën sketches an argument for the particular case of the main theorem when C is both smooth and proper, and describes a version of the second corollary. In [24], Theorem 4.67, Yeung proves a version of the main theorem for a certain substack of M C . In [20], Shende and Takeda develop a local-to-global principle for constructing absolute and relative Calabi-Yau structures on dg categories of interest in symplectic topology and representation theory.…”

Section: Introductionmentioning

confidence: 95%