# 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}, year={2008}, volume={81} }

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…

## 33 Citations

Sullivan minimal models of operad algebras

- MathematicsPublicacions 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
- 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

- Mathematics
- 2015

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…

Props in model categories and homotopy invariance of structures

- Mathematics
- 2008

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…

Minimal models of quantum homotopy Lie algebras via the BV-formalism

- MathematicsJournal 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…

Gauge equivalence for complete $L_\infty$-algebras

- Mathematics
- 2018

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…

Maurer-Cartan Moduli and Theorems of Riemann-Hilbert Type

- MathematicsAppl. Categorical Struct.
- 2021

We study Maurer–Cartan moduli spaces of dg algebras and associated dg categories and show that, while not quasi-isomorphism invariants, they are invariants of strong homotopy type, a natural notion…

Unbased rational homotopy theory:a Lie algebra approach

- Mathematics
- 2015

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…

## References

SHOWING 1-10 OF 50 REFERENCES

Dual Feynman transform for modular operads

- Mathematics
- 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

- Mathematics
- 1994

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

- Mathematics
- 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

- Mathematics
- 1998

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

- Mathematics
- 2001

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…