TY - JOUR

T1 - Feynman diagrams and minimal models for operadic algebras.

AU - Chuang, Joseph

AU - Lazarev, Andrey

PY - 2010/4

Y1 - 2010/4

N2 - 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 results carry over to the case of algebras over modular operads; the sums over trees get replaced by sums over general Feynman graphs. As a by-product of our work we prove gauge-independence of Kontsevich's ‘dual construction’ producing graph cohomology classes from contractible differential graded Frobenius algebras.

AB - 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 results carry over to the case of algebras over modular operads; the sums over trees get replaced by sums over general Feynman graphs. As a by-product of our work we prove gauge-independence of Kontsevich's ‘dual construction’ producing graph cohomology classes from contractible differential graded Frobenius algebras.

U2 - 10.1112/jlms/jdp073

DO - 10.1112/jlms/jdp073

M3 - Journal article

VL - 81

SP - 317

EP - 337

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

SN - 0024-6107

IS - 2

ER -