Homological perturbation theory for algebras over operads

Berglund, Alexander

doi:10.2140/agt.2014.14.2511

Cited by 56 publications

(94 citation statements)

References 28 publications

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“…As an application, we show in Sec. 4 that "∞-categorical" analogs of the existence and uniqueness statements that comprise the Homotopy Transfer Theorem [1,2,14,15] follow as a corollary of our Theorem 2. In more detail, suppose we are given a cochain complex A, a homotopy algebra B of some particular type (e.g, an A ∞ , L ∞ , or C ∞ -algebra) and a quasi-isomorphism of complexes φ : A → B.…”

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confidence: 72%

Homotopical Properties of the Simplicial Maurer–Cartan Functor

Rogers

2018

MATRIX Book Series

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We consider the category whose objects are filtered, or complete, L ∞algebras and whose morphisms are ∞-morphisms which respect the filtrations. We then discuss the homotopical properties of the Getzler-Hinich simplicial Maurer-Cartan functor which associates to each filtered L ∞ -algebra a Kan simplicial set, or ∞-groupoid. In previous work with V. Dolgushev, we showed that this functor sends weak equivalences of filtered L ∞ -algebras to weak homotopy equivalences of simplicial sets. Here we sketch a proof of the fact that this functor also sends fibrations to Kan fibrations. To the best of our knowledge, only special cases of this result have previously appeared in the literature. As an application, we show how these facts concerning the simplicial Maurer-Cartan functor provide a simple ∞-categorical formulation of the Homotopy Transfer Theorem.

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confidence: 72%

Homotopical Properties of the Simplicial Maurer–Cartan Functor

Rogers

2018

MATRIX Book Series

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“…If there was 11 any ghost number 0 state λ in H lc , it could generate the gauge transformation δΨ lc = c 0 K lc λ + m lc 2 (Ψ lc , λ) + m lc 2 (λ, Ψ lc ) + · · · . (4.12) 10 The minimal model theorem for A ∞ ensures uniqueness of the minimal model of these cyclic A ∞ algebras. In other words, (4.4) and (4.10) have the same S-matrix at the tree level.…”

Section: Reduction Of Gauge Symmetry and Covariancementioning

confidence: 99%

“…One can lift any coderivation, cohomomorphism or homotopy contraction to a thick map in a natural and trivial way. See [10] for further details. Clearly, it is compatible with differentials, compositions, and C-linear structure…”

Section: Tensor Trick and Thick Mapmentioning

confidence: 99%

“…Hence, because of the lemma, the reduced theory reproduces the same string amplitudes as Witten's string field theory. 10 Note that m lc : T (H lc ) → T (H lc ) and m lc defines the vertices of the light-cone string field theory via m lc = m lc 1 + m lc 2 + m lc 3 + · · · + m lc n + · · · . One can identify m lc acting on T (H lc ) with ι m lc π acting on the physical subspace of T (H cov ), for which we also write T (H lc ) ⊂ T (H cov ).…”

Section: Light-cone Reduction and A ∞ Structurementioning

confidence: 99%

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Light-cone reduction of Witten’s open string field theory

Matsubara

2019

J. High Energ. Phys.

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We elucidate some exact relations between light-cone and covariant string field theories on the basis of the homological perturbation lemma for A ∞ . The covariant string field splits into the light-cone string field and trivial excitations of BRST quartets: The latter generates the gauge symmetry and covariance. We first show that the reduction of gauge degrees can be performed by applying the lemma, which gives a refined version of the no-ghost theorem of covariant strings. Then, we demonstrate that after the reduction, gauge-fixed theory can be regarded as a kind of effective field theory and it provides an exact gauge-fixing procedure taking into account interactions. As a result, a novel lightcone string field theory is obtained from Witten's open string field theory.1 In this direction, there are some interesting investigations. See [4-6] for example.show that a novel light-cone string field theory appears within the covariant formulation, from which we cannot extract the old light-cone formulation itself directly.Note that gauge-fixed theory can be regarded as a kind of effective theory. For a given gauge theory S[Ψ], one can obtain a gauge-fixed action S red [ψ] by integrating out the gauge degrees ψ g of Ψ = ψ + ψ g as( 1.1) where Vol g denotes its gauge volume. It will give an exact gauge-fixing procedure taking into account interactions. The reduced action S red [ψ] reproduces the same amplitudes as the original action S[Ψ]. As we will show, the homological perturbation lemma provides us exact treatment of this formal procedure. In particular, by applying the lemma for A ∞ algebras, an A ∞ effective field theory is directly obtained from the original A ∞ field theory. In this paper, we construct an action for light-cone string field theory explicitly as the classical part of such an effective action for the Witten's theory.In section 2, we review the relation between the BRST operator and the light-cone kinetic operator. There exist similarity transformations connecting these. In section 3, we explain the homological perturbation lemma for A ∞ . We show that the reduction of gauge degrees can be described by applying the lemma, which provides a refined version of the no-ghost theorem of covariant strings. In section 4, we explain the light-cone reduction of interacting theory. A novel light-cone string field theory is constructed, which has an A ∞ action. Appendix A is devoted to explaining basic facts of the homological perturbation and its application to similarity transformations.In this paper, we write [[A, B]] for the graded commutator of A and B,where the upper index of (−) A denotes A's degree. The graded commutator will be defined for states, operators or mathematical operations appropriately.

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“…Notice that homotopical transfer of structure produces a weakly equivalent L ∞ [1] algebra, with F : (W, R) → (V, Q) an explicit weak equivalence. Using a more refined version of the above theorem [3] it is also possible to construct a…”

Section: Review Of L ∞ [1] Algebrasmentioning

confidence: 99%

Nonabelian higher derived brackets

Bandiera

2015

Journal of Pure and Applied Algebra

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Abstract. Let M be a graded Lie algebra, together with graded Lie subalgebras L and A such that as a graded space M is the direct sum of L and A, and A is abelian. Let D be a degree one derivation of M squaring to zero and sending L into itself, then Voronov's construction of higher derived brackets associates to D a L∞ structure on A[−1]. It is known, and it follows from the results of this paper, that the resulting L∞ algebra is a weak model for the homotopy fiber of the inclusion of differential graded Lie algebras i :. We prove this fact using homotopical transfer of L∞ structures, in this way we also extend Voronov's construction when the assumption A abelian is dropped: the resulting formulas involve Bernoulli numbers. In the last section we consider some example and some further application. Introductionbe the inclusion of a differential graded Lie subalgebra, recall that its homotopy fiber is the differential graded Lie algebra (dgla) In this paper we find explicit formulas under the additional assumption that A ⊂ M is a graded Lie subalgebra of M , then the L ∞ structure on A[−1] is given (after décalage) by the family of degree one symmetric brackets Φ(D) i :where S i is the symmetric group, ε(σ) = ε(σ; a 1 , . . . , a i ) is the Koszul sign and the B j are the Bernoulli numbers. ThewhereOur first main result is that with these definitions Date: September 9, 2013.

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Homological perturbation theory for algebras over operads

Cited by 56 publications

References 28 publications

Homotopical Properties of the Simplicial Maurer–Cartan Functor

Homotopical Properties of the Simplicial Maurer–Cartan Functor

Light-cone reduction of Witten’s open string field theory

Nonabelian higher derived brackets

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