Homotopy theory of the master equation package applied to algebra and geometry: a sketch of two interlocking programs

TL;DR

This paper develops a homotopy-theoretic framework for the master equation, linking algebraic structures and geometric moduli spaces, and introduces new tools for studying their equivalences and obstructions.

Contribution

It presents a novel interpretation of the master equation in homotopy theory, connecting algebraic and geometric applications, and discusses new invariants and obstructions.

Findings

Homotopy classification of algebraic structures including Frobenius and Lie bialgebras.

Application to moduli spaces of geometric objects like J holomorphic curves.

Identification of anomalies as obstructions in the homotopy framework.

Abstract

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. 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.