A ug 2 00 3 Open - string vertex algebras , tensor categories and operads

We introduce notions of open-string vertex algebra, conformal open-string vertex algebra and variants of these notions. These are " open-string-theoretic, " " noncommutative " generalizations of the notions of vertex algebra and of conformal vertex algebra. Given an open-string vertex algebra, we show that there exists a vertex algebra, which we call the " meromorphic center, " inside the original algebra such that the original algebra yields a module and also an intertwining operator for the meromorphic center. This result gives us a general method for constructing open-string vertex algebras. Besides obvious examples obtained from associative algebras and vertex (super)algebras, we give a nontrivial example constructed from the minimal model of central charge c = 1 2. We establish an equivalence between the associative algebras in the braided tensor category of modules for a suitable ver-tex operator algebra and the grading-restricted conformal open-string vertex algebras containing a vertex operator algebra isomorphic to the given vertex operator algebra. We also give a geometric and operadic formulation of the notion of grading-restricted conformal open-string vertex algebra, we prove two isomorphism theorems, and in particular, we show that such an algebra gives a projective algebra over what we call the " Swiss-cheese partial operad. " 0 Introduction In the present paper, we introduce and study " open-string-theoretic, " " non-commutative " generalizations of ordinary vertex algebras and vertex operator algebras, which we call " open-string vertex algebras " and " conformal open-string vertex algebras. " This is a first step in a program to establish 1 the fundamental and highly nontrivial assumptions used by physicists in the study of boundary (or open-closed) conformal field theories as mathematical theorems and to construct such theories mathematically. See [H9] and [HK2] for definitions of open-closed conformal field theory in the spirit of the definition of closed conformal field theory first given by Segal [S1]–[S3] and Kontsevich in 1987 and further rigorized by Hu and Kriz [HK1] recently. More recently, Moore suggested in [M3] that in order to generalize a certain formula relating a nonlinear σ model and the K-theory on its target space to conformal field theories without obvious target space interpretation, one should define some kind of algebraic K-theory for " open string vertex operator algebras. " We hope that the notions and results in the present paper will provide a solid foundation for the formulation and study of such a K-theory. Vertex (operator) algebras …