Homotopy Theory of the Master Equation Package Applied to Algebra and Geometry: a Sketch of Two Interlocking Programs

We interpret mathematically the pair (master equation, solution of master equation) up to equivalence, as the pair (a presentation of a free triangular dga T over a combination operad O, dga map of T into C, a dga over O) up to homotopy equivalence of dgOa maps, see Definition 1 below. We sketch two general applications: I to the theory of the definition and homotopy theory of infinity versions of general algebraic structures including noncompact frobenius algebras and Lie bialgebras. Here the target C would be the total Hom complex between various tensor products of another chain complex B, C = HomB, O describes combinations of operations like composition and tensor product sufficient to describe the algebraic structure and one says that B has the algebraic structure in question. II to geometric systems of moduli spaces up to deformation like the moduli of J holomorphic curves. Here C is some geometric chain complex containing the fundamental classes of the moduli spaces of the geometric problem. We also discuss analogues of homotopy groups and Postnikov systems for maps and impediments to using them related to linear terms in the master equation called anomalies. Introduction and sketch Certain combination operads arise in the study of algebraic structures and moduli spaces. For any operad O one may define differential graded algebras over O. Let us call them dgOa’s. Fixing O they form an obvious category where the maps are dgOa maps. We will make use of a derived homotopy category based on free resolutions of dgOas and a notion of homotopy between dgOa maps. Resolutions give a procedure to replace any dgOa by a nilpotent analogue of a free dgOa. There are two similar classes of examples relevant here where the combination operad O describes compositions or tensor products of multilinear operations in application I and where the combination operad O describes gluing or union of geometric chains in application II. 1. The setup of application I Consider collections of j to k operations for various j and k positive that define examples of algebraic structures like (noncompact) frobenius algebra or Lie bialgebra. Such examples of algebraic structures, because multiple outputs appear, cannot themselves be described as algebras over operads but rather as algebras over dioperads, properads or props. Dioperads correspond to inserting only one of the multiple outputs into the inputs. Properads correspond to To appear in Algebraic Topology: Old and New, M. M. Postnikov Memorial Conference 1