Givental action and trivialisation of circle action

Dotsenko, Vladimir; Shadrin, Sergey; Vallette, Bruno

doi:10.5802/jep.23

Cited by 9 publications

(16 citation statements)

References 21 publications

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“…REMARK. Such homotopy trivializations applied to Batalin-Vilkovisky algebra structures were proved to play a key role in the description of Givental action on Cohomological Field Theories in [DSV15bis].…”

Section: Lemma 3 In Any Model Category a Cofibrant Replacement Of Amentioning

confidence: 99%

Pre-Lie Deformation Theory

Dotsenko¹,

Shadrin²,

Vallette³

2016

MMJ

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In this paper, we develop the deformation theory controlled by pre-Lie algebras; the main tool is a new integration theory for pre-Lie algebras. The main field of application lies in homotopy algebra structures over a Koszul operad; in this case, we provide a homotopical description of the associated Deligne groupoid. This permits us to give a conceptual proof, with complete formulae, of the Homotopy Transfer Theorem by means of gauge action. We provide a clear explanation of this latter ubiquitous result: there are two gauge elements whose action on the original structure restrict its inputs and respectively its output to the homotopy equivalent space. This implies that a homotopy algebra structure transfers uniformly to a trivial structure on its The classical computations hold here, so one has ln(e λ ) = λ and e ln(1+λ) = 1 + λ for any λ ∈ Hom(A ¡ , End(V )), which concludes the proof.In this case, the exponential group G exists in the bigger algebra a A,V , and is isomorphic to the gauge group Γ. By the weight grading property, its action on Maurer-Cartan elements is well defined and given by the conjugationSince the Maurer-Cartan elements correspond to homotopy A-module structures, the question is now how to give a homotopical interpretation to the gauge group and its action. Let α and β be two homotopy A-module structures on V , i.e. two Maurer-Cartan elements. One defines a more general notion of maps between two homotopy A-modules, called ∞-morphisms and denoted α β, by degree 0 maps f : A ¡ → End(V ) satisfying the equationThese can be composed by the formula g ⋆ f and ∞-isomorphisms are the ∞-morphisms such that the first component is invertible, i.e. f (0) (I) ∈ GL(V ). When this first component is the identity map, we call them ∞-isotopies and we denote their set by ∞−iso.Theorem 1. For any Koszul algebra A and for any chain complex (V, d), the group of ∞-isotopies is isomorphic to the gauge groupand the Deligne groupoid is isomorphic to the groupoid whose objects are homotopy A-modules and whose morphisms are ∞-isotopiesProof. The first assertion follows directly from Proposition 2. Two Maurer-Cartan elements α and β are gauge equivalent if and only if there is an ∞-isotopy between the two homotopy A-module structures on V . Indeed, there exists λ ∈ Hom(A ¡ , End(V )) 0 such that β = e λ .α = e λ ⋆ α ⋆ e −λ if and only if

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Section: Lemma 3 In Any Model Category a Cofibrant Replacement Of Amentioning

confidence: 99%

Pre-Lie Deformation Theory

Dotsenko¹,

Shadrin²,

Vallette³

2016

MMJ

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“…Proof The proof is analogous to [, Lemma 2]. In a nutshell, the proof amounts to combining topological recursion relations from Proposition in a somewhat imaginative way.

□

…”

Section: Givental Group Action On Prefixnchypercom‐algebrasmentioning

confidence: 98%

“…This section follows the general plan of in order to prove that the Givental action is equal to the action of gauge symmetries in a certain homotopy Lie algebra. Let us assume that

A

is a chain complex with zero differential.…”

Section: Givental Group Action On Prefixnchypercom‐algebrasmentioning

confidence: 99%

“…Proof The proof is analogous to [, Theorem 5]. Basically, there are two main steps that one has to undertake.…”

Section: Givental Group Action On Prefixnchypercom‐algebrasmentioning

confidence: 99%

“…In addition, algebra structures over the operad

prefixHyperCom

— genus zero cohomological field theories — admit an unexpectedly rich group of symmetries due to Givental coming from different trivialisations of the circle action of the homotopy quotient . The latter group of symmetries can also be used to encode Koszul braces on commutative algebras and a particular type of homotopy

prefixBV

‐algebras, so‐called commutative homotopy

prefixBV

‐algebras .…”

Section: Introductionmentioning

confidence: 99%

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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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