Homotopy Transfer Theorem Research Articles

Differential Graded Manifolds of Finite Positive Amplitude

Abstract We prove that dg manifolds of finite positive amplitude, that is, bundles of positively graded curved $L_{\infty }[1]$-algebras, form a category of fibrant objects. As a main step in the proof, we obtain a factorization theorem using path spaces. First we construct an infinite-dimensional factorization of a diagonal morphism using actual path spaces motivated by the AKSZ construction. Then we cut down to finite dimensions using the Fiorenza-Manetti method. The main ingredient in our method is the homotopy transfer theorem for curved $L_{\infty }[1]$-algebras. As an application, we study the derived intersections of manifolds.

The inverse function theorem for curved $L$-infinity spaces

In this paper, we prove an inverse function theorem in derived differential geometry. More concretely, we show that a morphism of curved L_\infty spaces which is a quasi-isomorphism at a point has a local homotopy inverse. This theorem simultaneously generalizes the inverse function theorem for smooth manifolds and the Whitehead theorem for L_\infty algebras. The main ingredients are the obstruction theory for L_\infty homomorphisms (in the curved setting) and the homotopy transfer theorem for curved L_\infty algebras. Both techniques work in the A_\infty case as well.

Perspectives on the pure spinor superfield formalism

In this note, we study, formalize, and generalize the pure spinor superfield formalism from a rather nontraditional perspective. To set the stage, we review the notion of a multiplet for a general super Lie algebra, working in the context of the BV and BRST formalisms. Building on this, we explain how the pure spinor superfield formalism can be viewed as constructing a supermultiplet out of the input datum of an equivariant graded module over the ring of functions on the nilpotence variety. We use the homotopy transfer theorem and other computational techniques from homological algebra to relate these multiplets to more standard component-field formulations. Physical properties of the resulting multiplets can then be understood in terms of algebrogeometric properties of the nilpotence variety. We illustrate our discussion with many examples in various dimensions.

Filtered 𝐴-infinity structures in complex geometry

We prove a filtered version of the Homotopy Transfer Theorem which gives anAA-infinity algebra structure on any page of the spectral sequence associated to a filtered dg-algebra. We then develop various applications to the study of the geometry and topology of complex manifolds, using the Hodge filtration, as well as to complex algebraic varieties, using mixed Hodge theory.

English

We give a proof of the Homotopy Transfer Theorem following Kadeishvili's original strategy. Although Kadeishvili originally restricted himself to transferring a dg algebra structure to an $A_\infty$-structure on homology, we will see that a small modification of his argument proves the general case of transferring any kind of $\infty$-algebra structure along a quasi-isomorphism, under weaker hypotheses than existing proofs of this result.

Properadic Homotopical Calculus

Abstract In this paper, we initiate the generalization of the operadic calculus that governs the properties of homotopy algebras to a properadic calculus that governs the properties of homotopy gebras over a properad. In this first article of a series, we generalize the seminal notion of ${\infty }$-morphisms and the ubiquitous homotopy transfer theorem. As an application, we recover the homotopy properties of involutive Lie bialgebras developed by Cieliebak–Fukaya–Latschev and we produce new explicit formulas.

Shifted Derived Poisson Manifolds Associated with Lie Pairs

We study the shifted analogue of the "Lie--Poisson" construction for $L_\infty$ algebroids and we prove that any $L_\infty$ 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 $\operatorname{tot}\Omega^{\bullet}_A(\Lambda^\bullet(L/A))$ admits a degree $(+1)$ derived Poisson algebra structure with the wedge product as associative multiplication and the Chevalley--Eilenberg differential $d_A^{\operatorname{Bott}}:\Omega^{\bullet}_A(\Lambda^\bullet(L/A))\to \Omega^{\bullet +1}_A(\Lambda^\bullet(L/A))$ as unary $L_\infty$ bracket. This degree $(+1)$ derived Poisson algebra structure on $\operatorname{tot}\Omega^{\bullet}_A(\Lambda^\bullet(L/A))$ is unique up to an isomorphism having the identity map as first Taylor coefficient. Consequently, the Chevalley--Eilenberg hypercohomology $\mathbb{H}(\Omega^{\bullet}_A(\Lambda^\bullet(L/A)),d_A^{\operatorname{Bott}})$ admits a canonical Gerstenhaber algebra structure.

The Gerstenhaber–Schack complex for prestacks

The aim of this work is to construct a complex which through its higher structure directly controlls deformations of general prestacks, building on the work of Gerstenhaber and Schack for presheaves of algebras. In defining a Gerstenhaber–Schack complex CGS•(A) for an arbitrary prestack A, we have to introduce a differential with an infinite sequence of components instead of just two as in the presheaf case. If A˜ denotes the Grothendieck construction of A, which is a U-graded category, we explicitly construct inverse quasi-isomorphisms F and G between CGS•(A) and the Hochschild complex CU(A˜), as well as a concrete homotopy T:FG⟶1, which had not been obtained even in the presheaf case. As a consequence, by applying the Homotopy Transfer Theorem, one can transfer the dg Lie structure present on the Hochschild complex in order to obtain an L∞-structure on CGS•(A), which controlls the higher deformation theory of the prestack A. This answers the open problem about the higher structure on the Gerstenhaber–Schack complex at once in the general prestack case.

Deformation Theory with Homotopy Algebra Structures on Tensor Products

In order to solve two problems in deformation theory, we establish natural structures of homotopy Lie algebras and of homotopy associative algebras on tensor products of algebras of different types and on mapping spaces between coalgebras and algebras. When considering tensor products, such algebraic structures extend the Lie algebra or associative algebra structures that can be obtained by means of the Manin products of operads. These new homotopy algebra structures are proven to be compatible with the concepts of homotopy theory: \infty -morphisms and the Homotopy Transfer Theorem. We give a conceptual interpretation of their Maurer-Cartan elements. In the end, this allows us to construct the deformation complex for morphisms of algebras over an operad and to represent the deformation \infty -groupoid for differential graded Lie algebras.

Pre-Lie Deformation Theory

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 underlying homology if and only if it is gauge trivial; this is the ultimate generalization of the d-dbar lemma.

A Tale of Three Homotopies

For a Koszul operad \(\mathcal {P}\), there are several existing approaches to the notion of a homotopy between homotopy morphisms of homotopy \(\mathcal {P}\)-algebras. Some of those approaches are known to give rise to the same notions. We exhibit the missing links between those notions, thus putting them all into the same framework. The main nontrivial ingredient in establishing this relationship is the homotopy transfer theorem for homotopy cooperads due to Drummond-Cole and Vallette.

What do homotopy algebras form?

In paper [4], we constructed a symmetric monoidal category SLie∞MC whose objects are shifted (and filtered) L∞-algebras. Here, we fix a cooperad C and show that algebras over the operad Cobar(C) naturally form a category enriched over SLie∞MC. Following [4], we “integrate” this SLie∞MC-enriched category to a simplicial category HoAlgCΔ whose mapping spaces are Kan complexes. The simplicial category HoAlgCΔ gives us a particularly nice model of an (∞,1)-category of Cobar(C)-algebras. We show that the homotopy category of HoAlgCΔ is the localization of the category of Cobar(C)-algebras and ∞-morphisms with respect to ∞-quasi-isomorphisms. Finally, we show that the Homotopy Transfer Theorem is a simple consequence of the Goldman–Millson theorem.

Homotopy Transfer Theorem for linearly compatible di-algebras

This paper studies the operad of linearly compatible di-algebras, denoted by \(As^{2}\), which is a nonsymmetric operad encoding the algebras with two binary operations that satisfy individual and sum associativity conditions. We also prove that the operad \(As^{2}\) is exactly the Koszul dual operad of the operad \(^{2}As\) encoding totally compatible di-algebras. We show that the operads \(As^{2}\) and \(^{2}As\) are Koszul by rewriting method. We make explicit the Homotopy Transfer Theorem for \(As^{2}\)-algebras.

Homotopy transfer and self-dual Schur modules

We consider the free 2-nilpotent graded Lie algebra $$\mathfrak{g}$$ generated in degree one by a finite dimensional vector space V. We recall the beautiful result that the cohomology $$H^ \cdot \left( {\mathfrak{g},\mathbb{K}} \right)$$ of $$\mathfrak{g}$$ with trivial coefficients carries a GL(V)-representation having only the Schur modules V with self-dual Young diagrams {λ: λ = λ′} in its decomposition into GL(V)-irreducibles (each with multiplicity one). The homotopy transfer theorem due to Tornike Kadeishvili allows to equip the cohomology of the Lie algebra g with a structure of homotopy commutative algebra.

The minimal model for the Batalin–Vilkovisky operad

The purpose of this paper is to explain and to generalize, in a homotopical way, the result of Barannikov–Kontsevich and Manin, which states that the underlying homology groups of some Batalin–Vilkovisky algebras carry a Frobenius manifold structure. To this extent, we first make the minimal model for the operad encoding BV-algebras explicit. Then, we prove a homotopy transfer theorem for the associated notion of homotopy BV-algebra. The final result provides an extension of the action of the homology of the Deligne–Mumford–Knudsen moduli space of genus 0 curves on the homology of some BV-algebras to an action via higher homotopical operations organized by the cohomology of the open moduli space of genus zero curves. Applications in Poisson geometry and Lie algebra cohomology and to the Mirror Symmetry conjecture are given.