Tensor Constructions of Open String Theories I: Foundations

TL;DR

This paper investigates the algebraic structures underlying open string theories, establishing a foundational framework using homotopy associative algebras and demonstrating their equivalence and physical implications.

Contribution

It clarifies the algebraic framework of open string field theory and connects homotopy algebra structures to physical equivalences and gauge groups.

Findings

Homotopy associative star algebra can be attached to open string theories.

Off-shell equivalence corresponds to homotopy equivalence of algebras.

String theories with certain algebraic properties are physically equivalent to familiar gauge theories.

Abstract

The possible tensor constructions of open string theories are analyzed from first principles. To this end the algebraic framework of open string field theory is clarified, including the role of the homotopy associative A_\infty algebra, the odd symplectic structure, cyclicity, star conjugation, and twist. It is also shown that two string theories are off-shell equivalent if the corresponding homotopy associative algebras are homotopy equivalent in a strict sense. It is demonstrated that a homotopy associative star algebra with a compatible even bilinear form can be attached to an open string theory. If this algebra does not have a spacetime interpretation, positivity and the existence of a conserved ghost number require that its cohomology is at degree zero, and that it has the structure of a direct sum of full matrix algebras. The resulting string theory is shown to be physically equivalent to a string theory with a familiar open string gauge group.