[Paper Review] $L_\infty$ and $A_\infty$ structures: then and now

Jim Stasheff|arXiv (Cornell University)|Sep 4, 2018

Homotopy and Cohomology in Algebraic Topology71 references4 citations

TL;DR

This paper offers a personal, retrospective overview of the development of $A_\infty$ and $L_\infty$-structures over 55 years, tracing their origins in homotopy theory, algebraic topology, and mathematical physics. It highlights foundational contributions—especially from Sugawara, Moore, and Tamari—while emphasizing the evolution of higher homotopy algebras, their role in deformation theory, string field theory, and the emergence of cyclic and derived structures, with a focus on conceptual insights and overlooked results.

ABSTRACT

Looking back over 55 years of higher homotopy structures, I reminisce as I recall the early days and ponder how they developed and how I now see them. From the history of $A_\infty$-structures and later of $L_\infty$ -structures and their progeny, I hope to highlight some old results which seem not to have garnered the attention they deserve as well as some tantalizing new connections.

Motivation & Objective

• To provide a historical and conceptual retrospective on the development of $A_\infty$ and $L_\infty$-structures over five decades.
• To highlight foundational but underappreciated results in the theory of higher homotopy algebras.
• To explore the interplay between higher algebraic structures and their applications in geometry, topology, and mathematical physics.
• To reflect on the evolution of these structures from topological origins to modern formulations in deformation theory and string field theory.
• To emphasize the role of operads, derived brackets, and cyclic structures in unifying seemingly disparate areas of mathematics and physics.

Proposed method

• Traces the historical development of $A_\infty$-spaces and algebras from early work by Sugawara (1957) on fibrations and homotopy associativity.
• Describes the role of associahedra (Stasheff polytopes) as geometric realizations of higher associativity constraints, originally constructed by Tamari.
• Applies the framework of operads and colored operads (e.g., the Swiss cheese operad) to model open-closed homotopy algebras (OCHAs).
• Uses coderivations on differential graded coalgebras to characterize $A_\infty$-, $L_\infty$-, and OCHA-structures in a unified algebraic setting.
• Explores the use of inner products and cyclicity in $A_\infty$- and $L_\infty$-algebras, linking them to action functionals in string field theory.
• Analyzes the appearance of polytopes such as the amplituhedron and positive Grassmannian in scattering amplitude computations, connecting to associahedra.

Experimental results

Research questions

• RQ1 How did the concept of $A_\infty$-spaces evolve from early work on loop spaces and fibrations?
• RQ2 What is the significance of Tamari’s associahedra in the formalization of higher associativity?
• RQ3 How do $L_\infty$-algebras arise in deformation theory and rational homotopy theory?
• RQ4 In what way do cyclic $A_\infty$- and $L_\infty$-algebras emerge from string field theory and BV formalism?
• RQ5 What is the role of the Swiss cheese operad in unifying open and closed string structures in homotopy algebras?

Key findings

• Sugawara’s 1957 work provided the first infinite sequence of homotopy conditions for a space to be a loop space, predating the formalization of $A_\infty$-structures.
• The associahedron $K_n$, originally constructed by Tamari in 1951, became the geometric backbone of $A_\infty$-structures, though Stasheff later reinterpreted it via parameterized maps.
• The existence of $XP(3)$ in a fibration sequence is equivalent to homotopy associativity, generalizing classical projective geometry.
• Cyclic $A_\infty$-algebras were formalized by Kontsevich and Penkava, with origins in string field theory and the need for cyclic symmetry in action functionals.
• The Swiss cheese operad provides a colored operadic framework for Open-Closed Homotopy Algebras (OCHAs), modeling the interaction of open and closed strings.
• Scattering amplitudes in gauge theories are linked to the amplituhedron, a generalization of the associahedron in kinematic space, suggesting deep geometric structures in quantum field theory.