Homotopically trivializing the circle in the framed little disks

Drummond-Cole, Gabriel C.

doi:10.1112/jtopol/jtt040

Cited by 15 publications

(67 citation statements)

References 13 publications

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“…The hypercommutative algebra structure is the most important ingredient of a formal Frobenius manifold structure. A typical application of a relation between BV-algebras and hypercommutative algebras is that under some conditions like Hodge property or some trivialization of the BV-operator ∆ we obtain a Frobenius manifold structure on the cohomology of a BV-algebra; we refer to [3,2,22,17,4,8,5,7] for different aspects and different examples of this kind of correspondence and relations between them. The topological origin of all these statements looks as follows.…”

Section: Introductionmentioning

confidence: 99%

“…The homotopy quotient of the framed little discs operad by rotations is weakly equivalent to the operad of moduli spaces of stable genus 0 curves. This statement was mentioned in [23] and written in details in [7]. We are focused on the algebraic counterpart of this statement equipped with precise formulas.…”

Section: Introductionmentioning

confidence: 99%

“…This result summarizes the relations between hypercommutative algebras and Batalin-Vilkovisky algebras mentioned above. The equivalence of the homotopy categories of Hycomm-algebras and homotopy quotient of BV-algebras was given in [7] on the level of chains.…”

Section: Introductionmentioning

confidence: 99%

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Hypercommutative Operad as a Homotopy Quotient of BV

Khoroshkin

Markarian

Shadrin

2013

Commun. Math. Phys.

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We give an explicit formula for a quasi-isomorphism between the operads Hycomm (the homology of the moduli space of stable genus 0 curves) and BV/∆ (the homotopy quotient of Batalin-Vilkovisky operad by the BV-operator). In other words we derive an equivalence of Hycomm-algebras and BV-algebras enhanced with a homotopy that trivializes the BV-operator.These formulas are given in terms of the Givental graphs, and are proved in two different ways. One proof uses the Givental group action, and the other proof goes through a chain of explicit formulas on resolutions of Hycomm and BV. The second approach gives, in particular, a homological explanation of the Givental group action on Hycomm-algebras.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hypercommutative Operad as a Homotopy Quotient of BV

Khoroshkin

Markarian

Shadrin

2013

Commun. Math. Phys.

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“…Second, he announced that the arising action of changes of trivialisations of a homotopically trivial circle action on representations of the operad HyperCom coincides with the action of the Givental group. The first statement is established in [DV13,Dru11,KMS13], and the identification of the Givental action with the action of trivialisations of the circle action is obtained in [KMS13].…”

Section: Introductionmentioning

confidence: 96%

Givental action and trivialisation of circle action

Dotsenko¹,

Shadrin²,

Vallette³

2015

Journal De L’École Polytechnique — Mathématiques

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ABSTRACT. In this paper, we show that the Givental group action on genus zero cohomological field theories, also known as formal Frobenius manifolds or hypercommutative algebras, naturally arises in the deformation theory of Batalin-Vilkovisky algebras. We prove that the Givental action is equal to an action of the trivialisations of the trivial circle action. This result relies on the equality of two Lie algebra actions coming from two apparently remote domains: geometry and homotopical algebra. CONTENTS

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“…First, one can obtain the operad

f {scriptD}_{2}

as the extension of

D_{2}

by the

S^{1}

‐action:

f {scriptD}_{2} ≅ {scriptD}_{2} ⋊ S^{1}

. Then, the homotopy quotient of the latter operad by the circle action is represented by the Deligne–Mumford operad

{{\bar{M}}_{0, n + 1}}

. Finally, the quotient of

D_{2}

by the circle action is equivalent to the collection of open moduli spaces

{{scriptM}_{0, n + 1}}

, which is the open piece of the Deligne–Mumford operad that has a normal crossing divisor as its complement.…”

Section: Introductionmentioning

confidence: 99%

Toric varieties of Loday's associahedra and noncommutative cohomological field theories

Dotsenko

Shadrin

Vallette

2019

Journal of Topology

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We introduce and study several new topological operads that should be regarded as nonsymmetric analogues of the operads of little 2-discs, framed little 2-discs, and Deligne-Mumford compactifications of moduli spaces of genus zero curves with marked points. These operads exhibit all the remarkable algebraic and geometric features that their classical analogues possess; in particular, it is possible to define a noncommutative analogue of the notion of cohomological field theory with similar Givental-type symmetries. This relies on rich geometry of the analogues of the Deligne-Mumford spaces, coming from the fact that they admit several equivalent interpretations: as the toric varieties of Loday's realisations of the associahedra, as the brick manifolds recently defined by Escobar, and as the De Concini-Procesi wonderful models for certain subspace arrangements. Contents

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Homotopically trivializing the circle in the framed little disks

Cited by 15 publications

References 13 publications

Hypercommutative Operad as a Homotopy Quotient of BV

Hypercommutative Operad as a Homotopy Quotient of BV

Givental action and trivialisation of circle action

Toric varieties of Loday's associahedra and noncommutative cohomological field theories

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