Open-string vertex algebras, tensor categories and operads

Abstract

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 vertex 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.''

53 pages

References in corpus (7)

• Dirichlet branes, homological mirror symmetry, and stability
• On the structure of open-closed topological field theory in two dimensions
• Conformal Correlation Functions, Frobenius Algebras and Triangulations
• Some Comments on Branes, G-flux, and K-theory
• K-Theory from a physical perspective
• Spinor construction of the c = 1/2 minimal model
• Differential equations and intertwining operators

Cited by in corpus (33)

• Duality and defects in rational conformal field theory
• Braided tensor categories and extensions of vertex operator algebras
• Tensor categories for vertex operator superalgebra extensions
• Cardy algebras and sewing constraints, I
• A mathematical theory of gapless edges of 2d topological orders. Part I
• Morita classes of algebras in modular tensor categories
• The logarithmic triplet theory with boundary
• Gapless edges of 2d topological orders and enriched monoidal categories
• Categorification and correlation functions in conformal field theory
• A mathematical theory of gapless edges of 2d topological orders. Part II
• A topological phase transition on the edge of the 2d ${\mathbb{Z}}_2$ topological order
• Cardy condition for open-closed field algebras
• Full field algebras, operads and tensor categories
• Meromorphic open-string vertex algebras
• Open-closed field algebras
• Closed and open conformal field theories and their anomalies
• Meromorphic open-string vertex algebras and Riemannian manifolds
• On modules for meromorphic open-string vertex algebras
• Topological and conformal field theory as Frobenius algebras
• Meromorphic open-string vertex algebras and modules over two-dimensional orientable space forms
• Uniqueness of open/closed rational CFT with given algebra of open states
• Duality Theorems for Infinite Braided Hopf Algebras
• Double Bicrosssum of Braided Lie algebras
• Modular invariants for group-theoretical modular data. I
• The boundary phase transitions of the 2+1D ${\mathbb{Z}}_N$ topological order via topological Wick rotation
• A mathematical formalism for the Kondo effect in WZW branes
• Representations of the category of modules over pointed Hopf algebras over S_3 and S_4
• Duality Theorem and Hom Functor in Braided Tensor Categories
• Representation theory of vertex operator algebras and orbifold conformal field theory
• The Factorization of Braided Hopf Algebras
• A cohomology theory of grading-restricted vertex algebras
• Mirror Extensions of Vertex Operator Algebras
• Braided commutative algebras over quantized enveloping algebras