A Suggestion for a Functorial Formulation of String Field Theory

We make a first suggestion for an axiomatic approach to string field theory in the form of what we will call a string field functor. We give a plausibility argument for the existence of a string field functor and show that it has to be unique up to a natural gauge freedom. We prove some elementary properties of string field functors and argue that such a functorial formulation is only possible on the full moduli space of three dimensional extended topological quantum field theories and not on the smaller so called extended moduli space of string theory. We also give a heuristic argument that the suggested axiomatics of a string field functor should give the BRSTinvariant antibracket of ghost number −1 which was singled out as the most important structural feature of full moduli space in [Wit 1992]. In addition, we show that the axiomatics of string field functors is closely related to the approach of developing algebraic and noncommutative geometry in the