Strong homotopy Lie algebras, homotopy Poisson manifolds and Courant algebroids

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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“…For recent developments, see [58], [25] and [57]. References [29,46,4] are also related to the present paper.…”

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confidence: 88%

Shifted Derived Poisson Manifolds Associated with Lie Pairs

Bandiera

Chen

Stiénon

et al. 2019

Commun. Math. Phys.

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“…One could address this issue in the context of twisted QP manifolds, introduced in [22]. Furthermore, twisted Poisson structures may be also understood in the context of L ∞ algebras and homotopy Poisson structures, as in [35] and [36]. It would be interesting to adopt this perspective also for the twisted R-Poisson structures studied here.…”

Section: Jhep09(2021)045mentioning

confidence: 99%

Topological field theories induced by twisted R-Poisson structure in any dimension

Chatzistavrakidis

2021

J. High Energ. Phys.

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We construct a class of topological field theories with Wess-Zumino term in spacetime dimensions ≥ 2 whose target space has a geometrical structure that suitably generalizes Poisson or twisted Poisson manifolds. Assuming a field content comprising a set of scalar fields accompanied by gauge fields of degree (1, p − 1, p) we determine a generic Wess-Zumino topological field theory in p + 1 dimensions with background data consisting of a Poisson 2-vector, a (p + 1)-vector R and a (p + 2)-form H satisfying a specific geometrical condition that defines a H-twisted R-Poisson structure of order p + 1. For this class of theories we demonstrate how a target space covariant formulation can be found by means of an auxiliary connection without torsion. Furthermore, we study admissible deformations of the generic class in special spacetime dimensions and find that they exist in dimensions 2, 3 and 4. The two-dimensional deformed field theory includes the twisted Poisson sigma model, whereas in three dimensions we find a more general structure that we call bi-twisted R-Poisson. This extends the twisted R-Poisson structure of order 3 by a non-closed 3-form and gives rise to a topological field theory whose covariant formulation requires a connection with torsion and includes a twisted Poisson sigma model in three dimensions as a special case. The relation of the corresponding structures to differential graded Q-manifolds based on the degree shifted cotangent bundle T*[p]T*[1]M is discussed, as well as the obstruction to them being QP-manifolds due to the Wess-Zumino term.

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“…This bracket •, • fails to be a Lie bracket, but it can be completed to a Lie 2-algebra. This Lie 2-algebra was found in [LSX] by the authors in the study of homotopy Poisson structures; see [LSX,Example 4.10] for more details.…”

Section: So That We Havementioning

confidence: 69%