Perturbation theory in differential homological algebra II

A. We study the shifted analogue of the "Lie-Poisson" construction for L∞ algebroids and we prove that any L∞ algebroid naturally gives rise to shifted derived Poisson manifolds. We also investigate derived Poisson structures from a purely algebraic perspective and, in particular, we establish a homotopy transfer theorem for derived Poisson algebras.As an application, we prove that, given a Lie pair (L, A), the space tot Ω • A (Λ • (L/A)) admits a degree (+1) derived Poisson algebra structure with the wedge product as associative multiplication and the Chevalley-Eilenberg differential d Bott A arXiv:1712.00665v2 [math.QA]

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“…Proof. This was proved in [41], see also [17,20] for some related results. For completeness, we sketch a proof of this fact.…”

Section: Homotopy Transfer For Derived Poisson Algebrasmentioning

confidence: 67%

“…We start by recalling the standard Perturbation Lemma. To the best of our knowledge, this result first appeared implicitly in the paper [47], and explicitly in [7,16]: for the treatment given here, see also [17,20,41].…”

Section: Homotopy Transfer For Derived Poisson Algebrasmentioning

confidence: 90%

Shifted Derived Poisson Manifolds Associated with Lie Pairs

Bandiera

Chen

Stiénon

et al. 2019

Commun. Math. Phys.

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“…18 In Sec. In this case an alternative route is to look for perturbative solutions of the equation of motion.…”

Section: Introductionmentioning

confidence: 99%

Invariants from classical field theory

Díaz

Leal

2008

Journal of Mathematical Physics

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We introduce a method that generates invariant functions from perturbative classical field theories depending on external parameters. Applying our methods to several field theories such as abelian BF, Chern-Simons and 2-dimensional Yang-Mills theory, we obtain, respectively, the linking number for embedded submanifolds in compact varieties, the Gauss' and the second Milnor's invariant for links in S^3, and invariants under area-preserving diffeomorphisms for configurations of immersed planar curves.Comment: 20 pages, 1 figure, to appear in J. Math. Phy

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“…This is reminiscent of the homotopy perturbation lemma with the infinitesimal cocycle as "initiator" [3,12]. As an application to star-products, the Moyal's formula was obtained in this way.…”

Section: Discussionmentioning

confidence: 99%

On deformation of associative algebras and graph homology

2007

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Deformation theory of associative algebras and in particular of Poisson algebras is reviewed. The role of an "almost contraction" leading to a canonical solution of the corresponding Maurer-Cartan equation is noted. This role is reminiscent of the Homotopical Perturbation Lemma, with the infinitesimal deformation cocycle as "initiator."Applied to star-products, we show how Moyal's formula can be obtained using such an almost contraction and conjecture that the "merger operation" provides a canonical solution at least in the case of linear Poisson structures.

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Perturbation theory in differential homological algebra II

Cited by 131 publications

References 1 publication

Shifted Derived Poisson Manifolds Associated with Lie Pairs

Shifted Derived Poisson Manifolds Associated with Lie Pairs

Invariants from classical field theory

On deformation of associative algebras and graph homology

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