Tensor Constructions of Open String Theories I: Foundations

TL;DR

The paper develops a systematic tensor construction for open string theories using $A_\infty$ algebras and BV quantization. It shows that attaching an internal $A_\infty$ algebra ${\bf a}$ to the open string field theory ${\cal A}$ yields a consistent theory ${\cal A}{\otimes}{\bf a}$, whose physical content is captured by the cohomology $H({\bf a})$ and is equivalent to the Chan-Paton construction when positivity conditions hold. Positivity constraints force $H({\bf a})$ to be concentrated in degree zero and semisimple, so $H({\bf a})\cong I\oplus_{i}{\cal M}_{n_i}(\mathbb{C})$, leading to a gauge group that is a direct product of $U(n_i)$ factors; twist symmetries and projections allow $SO(n)$ and $USp(2n)$ realizations as well. The results bridge off-shell $A_\infty$-theoretic structure with on-shell physical content, underpinning a broad classification of open-string tensor theories. The work clarifies when two tensor theories are strictly or physically equivalent and sets a foundation for exploring more exotic symmetries and fermionic extensions in GZ2.

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.