Feynman diagrams and minimal models for operadic algebras

Abstract

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.

Abstract Hodge Decomposition and Minimal Models for Cyclic Algebras

“…Following the suggestion of Dan Grayson we will call a Hodge decomposition harmonious if dt = 0; it was called 'canonical' in [4] but we feel that the latter terminology may be misleading.…”

“…Moreover, any two choices of a Hodge decomposition give homotopy equivalent minimal models by Theorem 4.14 of [4]. Let us recall the notion of homotopy equivalence of operadic algebras.…”

“…It follows that the bvO-algebra structure obtained by restriction is also cyclic. The homotopy between different minimal models described in [4] (Theorem 4.14) is easily checked to be a cyclic map: the image of s and t are compatible with the extension of the inner product to the larger coefficient dg agebra…”

Modular operads and the quantum open-closed homotopy algebra

“…Moreover, this opens up the way for applications of operad homotopy theory to study of these algebras. For example, minimal models and transfer theorems have some physical relevance for gauge fixing [2,15]. The key point of the applications of the homotopy theory is that the Feynman transform is a modular operad which is always cofibrant in a suitable model structure.…”

“…1 There are now standard approaches for minimal models and transfer theorems for algebras over cofibrant operads, which hopefully carry over to our setting (e.g. [2], where the homotopy transfer is discussed).…”