Homotopy Algebras Research Articles

Weighted limits in simplicial homotopy theory

By combining ideas of homotopical algebra and of enriched category theory, we explain how two classical formulas for homotopy colimits, one arising from the work of Quillen and one arising from the work of Bousfield and Kan, are instances of general formulas for the derived functor of the weighted colimit functor.

β–γ systems and the deformations of the BRST operator

We describe the relation between simple logarithmic CFTs associated with closed and open strings, and their ‘infinite metric’ limits, corresponding to the β–γ systems. This relation is studied on the level of the BRST complex: we show that the consideration of metric as a perturbation leads to a certain deformation of the algebraic operations of the Lian–Zuckerman type on the vertex algebra, associated with the β–γ systems. The Maurer–Cartan equations corresponding to this deformed structure in the quasi-classical approximation lead to the nonlinear field equations. As an explicit example, we demonstrate that using this construction, Yang–Mills equations can be derived. This gives rise to a nontrivial relation between the Courant–Dorfman algebroid and homotopy algebras emerging from the gauge theory. We also discuss a possible algebraic approach to the study of beta-functions in sigma-models.

Cohomology theories for homotopy algebras and noncommutative geometry

This paper builds a general framework in which to study cohomology theories of strongly homotopy algebras, namely $A_\infty, C_\infty$ and $L_\infty$-algebras. This framework is based on noncommutative geometry as expounded by Connes and Kontsevich. The developed machinery is then used to establish a general form of Hodge decomposition of Hochschild and cyclic cohomology of $C_\infty$-algebras. This generalizes and puts in a conceptual framework previous work by Loday and Gerstenhaber-Schack.

A Cartan–Eilenberg approach to homotopical algebra

In this paper we propose an approach to homotopical algebra where the basic ingredient is a category with two classes of distinguished morphisms: strong and weak equivalences. These data determine the cofibrant objects by an extension property analogous to the classical lifting property of projective modules. We define a Cartan–Eilenberg category as a category with strong and weak equivalences such that there is an equivalence of categories between its localisation with respect to weak equivalences and the relative localisation of the subcategory of cofibrant objects with respect to strong equivalences. This equivalence of categories allows us to extend the classical theory of derived additive functors to this non additive setting. The main examples include Quillen model categories and categories of functors defined on a category endowed with a cotriple (comonad) and taking values on a category of complexes of an abelian category. In the latter case there are examples in which the class of strong equivalences is not determined by a homotopy relation. Among other applications of our theory, we establish a very general acyclic models theorem.

A homotopical algebra of graphs related to zeta series

The purpose of this paper is to develop a homotopical algebra for graphs, relevant to the zeta series and the spectra of finite graphs. More precisely, we define a Quillen model structure in a category of graphs (directed and possibly infinite, with loops and multiple arcs allowed). The weak equivalences for this model structure are the Acyclics (graph morphisms which preserve cycles). The cofibrations and fibrations for the model are determined from the class of Whiskerings (graph morphisms produced by grafting trees). Our model structure seems to fit well with the importance of acyclic directed graphs in many applications.

A Small Open-Closed Homotopy Algebra (Ocha)

We consider a particular finite dimensional example of an L∞ algebra in which a 2-dimensional Lie algebra acts on a 1-dimensional vector space in a non-trivial non-Lie manner. In order to understand the nature of this action, we show that this algebra is in fact an example of an open-closed homotopy algebra.

Equivalences of higher derived brackets

This note elaborates on Th. Voronov’s construction [Th. Voronov, Higher derived brackets and homotopy algebras, J. Pure Appl. Algebra 202 (1–3) (2005) 133–153; Th. Voronov, Higher derived brackets for arbitrary derivations, Travaux Math. XVI (2005) 163–186] of L ∞ -structures via higher derived brackets with a Maurer–Cartan element. It is shown that gauge equivalent Maurer–Cartan elements induce L ∞ -isomorphic structures. Applications in symplectic, Poisson and Dirac geometry are discussed.

Towards Unifying Structures in Higher Spin Gauge Symmetry

This article is expository in nature, outlining some of the many still incompletely understood features of higher spin field theory. We are mainly considering higher spin gauge fields in their own right as free-standing theoretical constructs and not circumstances where they occur as part of another system. Considering the problem of introducing interactions among higher spin gauge fields, there has historically been two broad avenues of approach. One approach entails gauging a non-Abelian global symmetry algebra, in the process making it local. The other approach entails deforming an already local but Abelian gauge algebra, in the process making it non-Abelian. In cases where both avenues have been explored, such as for spin 1 and 2 gauge fields, the results agree (barring conceptual and technical issues) with Yang-Mills theory and Einstein gravity. In the case of an infinite tower of higher spin gauge fields, the first approach has been thoroughly developed and explored by M. Vasiliev, whereas the second approach, after having lain dormant for a long time, has received new attention by several authors lately. In the present paper we briefly review some aspects of the history of higher spin gauge fields as a backdrop to an attempt at comparing the gauging vs. deforming approaches. A common unifying structure of strongly homotopy Lie algebras underlying both approaches will be discussed. The modern deformation approach, using BRST-BV methods, will be described as far as it is developed at the present time. The first steps of a formulation in the categorical language of operads will be outlined. A few aspects of the subject that seems not to have been thoroughly investigated are pointed out.

Structure of higher spin gauge interactions

In a previous paper, higher spin gauge field theory was formulated in an abstract way, essentially only keeping enough machinery to discuss gauge invariance of an action. The approach could be thought of as providing an interface (or syntax) toward an implementation (or semantics) yet to be constructed. The structure then revealed turned out to be that of a strongly homotopy Lie algebra. In the present paper, the framework will be connected to more conventional field theoretic concepts. The Fock complex vertex operator implementation of the interactions in the BRST-BV formulation of the theory will be elaborated. The relation between the vertex order expansion and homological perturbation theory will be clarified. A formal nonobstruction argument is reviewed. The syntactically derived sh-Lie algebra structure is semantically mapped to the Fock complex implementation, and it is shown that the equations governing the higher order vertices are reproduced. Global symmetries and subsidiary conditions are discussed and as a result the tracelessness constraints are discarded. Thus, all equations needed to compute the vertices to any order are collected. The framework is general enough to encompass all possible interaction terms. Finally, the abstract framework itself will be strengthened by showing that it can be naturally phrased in terms of category theory.

From Zwiebach Invariants to Getzler Relation

We introduce the notion of Zwiebach invariants that generalize Gromov- Witten invariants and homotopical algebra structures. We outline the induction procedure that induces the structure of Zwiebach invariants on the sub-bicomplex, that gives the structure of Gromov-Witten invariants on sub-bicomplex with zero differentials. We propose to treat Hodge dGBV with 1/12 axiom as the simplest set of Zwiebach invariants, and explicitly prove that it induces WDVV and Getzler equations in genera 0 and 1 respectively.

Rigidification of algebras over multi-sorted theories

We define the notion of a multi-sorted algebraic theory, which is a generalization of an algebraic theory in which the objects are of different "sorts." We prove a rigidification result for simplicial algebras over these theories, showing that there is a Quillen equivalence between a model category structure on the category of strict algebras over a multi-sorted theory and an appropriate model category structure on the category of functors from a multi-sorted theory to the category of simplicial sets. In the latter model structure, the fibrant objects are homotopy algebras over that theory. Our two main examples of strict algebras are operads in the category of simplicial sets and simplicial categories with a given set of objects.

Arbeitsgemeinschaft mit aktuellem Thema: Modern Foundations for Stable Homotopy Theory

In recent years, spectral algebra or stable homotopical algebra over structured ring spectra has become an important new direction in stable homotopy theory. This workshop provided an introduction to structured ring spectra and applications of spectral algebra, both within homotopy theory and in other areas of mathematics.

A class of strongly homotopy Lie algebras with simplified sh-Lie structures

Given a complex that is a differential graded vector space, it is known that a single mapping defined on a space of it where the homology is non-trivial extends to a strongly homotopy Lie algebra (on the graded space) when that mapping satisfies two conditions. This strongly homotopy Lie algebra is non-trivial (it is not a Lie algebra); however we show that one can obtain an sh-Lie algebra where the only non-zero mappings defining it are the lower order mappings. This structure applies to a significant class of examples. Moreover in this case the graded space can be replaced by another graded space, with only three non-zero terms, on which the same sh-Lie structure exists.

Homotopy Algebras Inspired by Classical Open-Closed String Field Theory

We define a homotopy algebra associated to classical open-closed strings. We call it an open-closed homotopy algebra (OCHA). It is inspired by Zwiebach's open-closed string field theory and also is related to the situation of Kontsevich's deformation quantization. We show that it is actually a homotopy invariant notion; for instance, the minimal model theorem holds. Also, we show that our open-closed homotopy algebra gives us a general scheme for deformation of open string structures (A ∞-algebras) by closed strings (L ∞-algebras).

Open-closed homotopy algebra in mathematical physics

In this paper we discuss various aspects of open-closed homotopy algebras (OCHAs) presented in our previous paper, inspired by Zwiebach’s open-closed string field theory, but that first paper concentrated on the mathematical aspects. Here we show how an OCHA is obtained by extracting the tree part of Zwiebach’s quantum open-closed string field theory. We clarify the explicit relation of an OCHA with Kontsevich’s deformation quantization and with the B-models of homological mirror symmetry. An explicit form of the minimal model for an OCHA is given as well as its relation to the perturbative expansion of open-closed string field theory. We show that our open-closed homotopy algebra gives us a general scheme for deformation of open string structures (A∞ algebras) by closed strings (L∞ algebras).

Topological Hochschild homology of connective complex K -theory

Let ku be the connective complex K -theory spectrum, completed at an odd prime p . We present a computation of the mod ( p, υ 1 ) homotopy algebra of the topological Hochschild homology spectrum of ku .

Homotopy algebras and the inverse of the normalization functor

In this paper, we investigate multiplicative properties of the classical Dold–Kan correspondence. The inverse of the normalization functor maps commutative differential graded algebras to E ∞ -algebras. We prove that it in fact sends algebras over arbitrary differential graded E ∞ -operads to E ∞ -algebras in simplicial modules and is part of a Quillen adjunction. More generally, this inverse maps homotopy algebras to weak homotopy algebras. We prove the corresponding dual results for algebras under the conormalization, and for coalgebra structures under the normalization resp. the inverse of the conormalization.

An abstract interface to higher spin gauge field theory

A comprehensive approach to the theory of higher spin gauge fields is proposed. By explicitly separating out details of implementation from general principles, it becomes possible to focus on the bare minimum of requirements that such a theory must satisfy. The abstraction is based on a survey of the progress that has been achieved since relativistic wave equations for higher spin fields were first considered in the 1930s. As a byproduct, a formalism is obtained that is abstract enough to describe a wide class of classical field theories. The formalism, viewed as syntax, can then be semantically mapped to a category of strongly homotopy Lie algebras, thus showing that the theory in some sense exists, at least as an abstract mathematical structure. Still, a concrete physicslike, implementation remains to be constructed. Lacking deep physical insight into the problem, an implementation in terms of generalized vertex operators is set up within which a brute force iterative determination of the first few orders in the interaction can be attempted.

VERSAL DEFORMATIONS OF THREE-DIMENSIONAL LIE ALGEBRAS AS L∞ALGEBRAS

We consider versal deformations of 0|3-dimensional L∞algebras, also called strongly homotopy Lie algebras, which correspond precisely to ordinary (non-graded) three-dimensional Lie algebras. The classification of such algebras is well-known, although we shall give a derivation of this classification using an approach of treating them as L∞algebras. Because the symmetric algebra of a three-dimensional odd vector space contains terms only of exterior degree less than or equal to three, the construction of versal deformations can be carried out completely. We give a characterization of the moduli space of Lie algebras using deformation theory as a guide to understanding the picture.

Higher derived brackets and homotopy algebras

We give a construction of homotopy algebras based on “higher derived brackets”. More precisely, the data include a Lie superalgebra with a projector on an Abelian subalgebra satisfying a certain axiom, and an odd element Δ . Given this, we introduce an infinite sequence of higher brackets on the image of the projector, and explicitly calculate their Jacobiators in terms of Δ 2 . This allows to control higher Jacobi identities in terms of the “order” of Δ 2 . Examples include Stasheff's strongly homotopy Lie algebras and variants of homotopy Batalin–Vilkovisky algebras. There is a generalization with Δ replaced by an arbitrary odd derivation. We discuss applications and links with other constructions.