Operads, configuration spaces and quantization

We review several well-known operads of compactified configuration spaces and construct several new such operads, C, in the category of smooth manifolds with corners whose complexes of fundamental chains give us (i) the 2-coloured operad of A-infinity algebras and their homotopy morphisms, (ii) the 2-coloured operad of L-infinity algebras and their homotopy morphisms, and (iii) the 4-coloured operad of open-closed homotopy algebras and their homotopy morphisms. Two gadgets - a (coloured) operad of Feynman graphs and a de Rham field theory on C - are introduced and used to construct quantized representations of the (fundamental) chain operad of C which are given by Feynman type sums over graphs and depend on choices of propagators.Comment: 58 page

The extended moduli space of special Lagrangian submanifolds

It is well known that the moduli space of all deformations of a compact special Lagrangian submanifold $X$ in a Calabi-Yau manifold $Y$ within the class of special Lagrangian submanifolds is isomorphic to the first de Rham cohomology group of $X$. Reinterpreting the embedding data $X\subset Y$ within the mathematical framework of the Batalin-Vilkovisky quantization, we find a natural deformation problem which extends the above moduli space to the full de Rham cohomology group of $X$.Comment: 15pages. A new citation and a correctio