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Modular classes of Q-manifolds: a review and some applications
Differential Geometry · Submitted 9 May 2017
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We review the basic ideas and then apply this technology to various examples, including $L_{\infty}$-algebroids and higher Poisson manifolds.
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The abstract, sentence by sentence
A Q-manifold is a supermanifold equipped with an odd vector field that squares to zero. The notion of the modular class of a Q-manifold -- which is viewed as the obstruction to the existence of a Q-invariant Berezin volume -- is not well know. We review the basic ideas and then apply this technology to various examples, including $L_{\infty}$-algebroids and higher Poisson manifolds.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | A Q-manifold is a supermanifold equipped with an odd vector field that squares to zero. | background 1.00 | 0.05 | 0.03 | asserted 1.00 |
| s1 | The notion of the modular class of a Q-manifold -- which is viewed as the obstruction to the existence of a Q-invariant Berezin volume -- is not well know. | gap 0.83 | 0.06 | 0.06 | asserted 0.99 |
| s2 | We review the basic ideas and then apply this technology to various examples, including $L_{\infty}$-algebroids and higher Poisson manifolds. | contribution 0.99 | 0.88 | 0.26 | asserted 0.99 |
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The paper
Original abstract
A Q-manifold is a supermanifold equipped with an odd vector field that squares to zero. The notion of the modular class of a Q-manifold -- which is viewed as the obstruction to the existence of a Q-invariant Berezin volume -- is not well know. We review the basic ideas and then apply this technology to various examples, including $L_{\infty}$-algebroids and higher Poisson manifolds.
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How the literature uses it
2 arXiv papers cite this in 3 sentences
Papers citing it · 1 sentence without a confident label
Cites it as background 2 papers
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Much of this section is taken from [ref] and references therein.
Modular Classes of Q-Manifolds, Part II: Riemannian Structures & Odd Killing Vectors Fields · Introduction · 0.96
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For a review of modular classes of Q-manifolds and further references the reader may consult [ref] .
Homological sections of Lie algebroids · Modular classes of Q-Lie algebroids · 1.00
What it cites · 18 sentences without a confident label
Compares with 5 papers
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This example is in agreement with Damianou & Fernandes [ref] .
Integrable hierarchies and the modular class · Definition and main properties of the modular class · 0.95
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Restricting attention to Lie algebroids we obtain [equation] which is in agreement with the classical literature, e.g. [ref] .
Homology and modular classes of Lie algebroids · $L_∞$-algebroids · 0.93
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A different notion of the modular class of a Schouten (odd Poisson) manifold can be found in the work of Khudaverdian & Voronov [ref] .
On odd Laplace operators · Higher Poisson manifolds · 0.92
Cites as background 7 papers
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First let us recall the definition of a higher Poisson manifold (cf. [ref] ) A higher Poisson manifold is a pair $(M, 𝒫)$, where $𝒫 ∈C^∞(Π^*M)$ is an even (pseudo)multivector field, known as a homotopy Poisson structure, that satisfies the Poisson condition $𝒫, 𝒫 =0$, where the bracket is the canonical Schouten bracket on $Π^*M$.
Higher derived brackets and homotopy algebras · Higher Poisson manifolds · 0.94
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First let us recall the definition of a higher Poisson manifold (cf. [ref] ) A higher Poisson manifold is a pair $(M, 𝒫)$, where $𝒫 ∈C^∞(Π^*M)$ is an even (pseudo)multivector field, known as a homotopy Poisson structure, that satisfies the Poisson condition $𝒫, 𝒫 =0$, where the bracket is the canonical Schouten bracket on $Π^*M$.
Higher Poisson Brackets and Differential Forms · Higher Poisson manifolds · 0.94
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The closely related notion of the modular class of a Lie algebroid was introduced by Evens, Lu and Weinstein [ref] .
Transverse measures, the modular class, and a cohomology pairing for Lie algebroids · Introduction · 0.97
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