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-infinity algebras. Further, 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- February 2008
- DOI:
- 10.48550/arXiv.0802.3507
- arXiv:
- arXiv:0802.3507
- Bibcode:
- 2008arXiv0802.3507C
- Keywords:
-
- Mathematics - Algebraic Topology;
- Mathematics - Quantum Algebra;
- 18D50;
- 57T30;
- 81T18;
- 16E45
- E-Print:
- 18 pages