# Feynman diagrams and minimal models for operadic algebras

@article{Chuang2008FeynmanDA,
title={Feynman diagrams and minimal models for operadic algebras},
author={Joseph Chuang and Andrey Lazarev},
journal={Journal of The London Mathematical Society-second Series},
year={2008},
volume={81},
pages={317-337}
}
• Published 24 February 2008
• Mathematics, Computer Science
• Journal of The London Mathematical Society-second Series
We construct an explicit minimal model for an algebra over the cobar-construction of a differential graded operad. The structure maps of this minimal model are expressed in terms of sums over decorated trees. We introduce the appropriate notion of a homotopy equivalence of operadic algebras and show that our minimal model is homotopy equivalent to the original algebra. All this generalizes and gives a conceptual explanation of well-known results for A∞-algebras. Furthermore, we show that these…

## Figures and Topics from this paper

Sullivan minimal models of operad algebras
• Mathematics
Publicacions Matemàtiques
• 2019
We prove the existence of Sullivan minimal models of operad algebras, for a quite wide family of operads in the category of complexes of vector spaces over a field of characteristic zero. Our
Curved infinity-algebras and their characteristic classes
• Mathematics, Physics
• 2012
In this paper, we study a natural extension of Kontsevich's characteristic class construction for A∞- and L∞-algebras to the case of curved algebras. These define homology classes on a variant of his
Koszul duality and homotopy theory of curved Lie algebras
This paper introduces the category of marked curved Lie algebras with curved morphisms, equipping it with a closed model category structure. This model structure is---when working over an
Minimal models of quantum homotopy Lie algebras via the BV-formalism
• Mathematics, Physics
Journal of Mathematical Physics
• 2018
Using the BV-formalism of mathematical physics an explicit construction for the minimal model of a quantum L-infinity-algebra is given as a formal super integral. The approach taken herein to these
Props in model categories and homotopy invariance of structures
Abstract We prove that any category of props in a symmetric monoidal model category inherits a model structure. We devote an appendix, about half the size of the paper, to the proof of the model
Maurer–Cartan moduli and models for function spaces☆
We set up a formalism of Maurer–Cartan moduli sets for L∞ algebras and associated twistings based on the closed model category structure on formal differential graded algebras (a.k.a. differential
Gauge equivalence for complete $L_\infty$-algebras
We introduce a notion of left homotopy for Maurer--Cartan elements in $L_{\infty}$-algebras and $A_{\infty}$-algebras, and show that it corresponds to gauge equivalence in the differential graded
Unbased rational homotopy theory:a Lie algebra approach
In this paper an algebraic model for unbased rational homotopy theory from the perspective of curved Lie algebras is constructed. As part of this construction a model structure for the category of
Formality Theorem for Hochschild Cochains via Transfer Belatedly to Simon Lyakhovich on the occasion of his 50th birthday
We construct a 2-colored operad Ger + which, on the one hand, extends the operad Ger∞ governing homotopy Gerstenhaber algebras and, on the other hand, extends the 2-colored operad governing
Unimodular homotopy algebras and Chern–Simons theory
• Mathematics, Physics
• 2015
Quantum Chern–Simons invariants of differentiable manifolds are analyzed from the point of view of homological algebra. Given a manifold M and a Lie (or, more generally, an L∞) algebra g, the vector

## References

SHOWING 1-10 OF 44 REFERENCES
Dual Feynman transform for modular operads
• Mathematics, Physics
• 2007
We introduce and study the notion of a dual Feynman transform of a modular operad. This generalizes and gives a conceptual explanation of Kontsevich's dual construction producing graph cohomology
Modular Operads
We develop a \higher genus" analogue of operads, which we call modular operads, in which graphs replace trees in the deenition. We study a functor F on the category of modular operads, the Feynman
Cohomology theories for homotopy algebras and noncommutative geometry
• Mathematics
• 2009
This paper builds a general framework in which to study cohomology theories of strongly homotopy algebras, namely A∞–, C∞– and L∞–algebras. This framework is based on noncommutative geometry as
ASSOCIAHEDRA, CELLULAR W -CONSTRUCTION AND PRODUCTS OF A∞-ALGEBRAS
• 2006
The aim of this paper is to construct a functorial tensor product of A∞-algebras or, equivalently, an explicit diagonal for the operad of cellular chains, over the integers, of the Stasheff
Strongly homotopy algebras of a K\"ahler manifold
It is shown that any compact K\"ahler manifold $M$ gives canonically rise to two strongly homotopy algebras, the first one being associated with the Hodge theory of the de Rham complex and the second
Symplectic C∞-algebras
• Mathematics
• 2008
In this paper we show that a strongly homotopy commutative (or C∞-) algebra with an invariant inner product on its cohomology can be uniquely extended to a symplectic C∞-algebra (an ∞-generalisation
Associahedra, cellular W-construction and products of $A_\infty$-algebras
• Mathematics
• 2003
The aim of this paper is to construct a functorial tensor product of A ∞ -algebras or, equivalently, an explicit diagonal for the operad of cellular chains, over the integers, of the Stasheff
Loop Homotopy Algebras in Closed String Field Theory
Abstract: Barton Zwiebach constructed [20] “string products” on the Hilbert space of a combined conformal field theory of matter and ghosts, satisfying the “main identity”. It has been well known
Koszul duality for Operads
• Mathematics
• 1994
(0.1) The purpose of this paper is to relate two seemingly disparate developments. One is the theory of graph cohomology of Kontsevich [Kon 2 3] which arose out of earlier works of Penner [Pe] and