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The colored parenthesized braid operad is a sequence of full subcategories of the fundamental groupoids of the configuration spaces of the complex plane, and a conformal block, a physical quantity of chiral two-dimensional conformal field theory, is a sheaf on the configuration space, which is mathematically formulated in terms of a vertex operator algebra and its modules. In this paper we show that the colored parenthesized braid operad acts weakly 2-categorically on a module category of a vertex operator algebra whose object is a module $M$ such that $M$ and $M^\vee$ are $C_1$-cofinite. If this action induces a 1-categorical action, then the module category inherits a balanced braided tensor category structure, which is shown for a rational $C_2$-cofinite vertex operator algebra. This gives an alternative proof of a result of Huang and Lepowsky., 117 pages, 30 figures

It is known that there are 48 Virasoro algebras acting on the monster conformal field theory. We call conformal field theories with such a property, which are not necessarily chiral, code conformal field theories. In this paper, we introduce a notion of a framed algebra, which is a finite-dimensional non-associative algebra, and showed that the category of framed algebras and the category of code conformal field theories are equivalent. We have also constructed a new family of integrable conformal field theories using this equivalence. These conformal field theories are expected to be useful for the study of moduli spaces of conformal field theories., 80 pages, 1 figure. Compared to version 1, version 2 has the different abstract and an additional Remark (Remark 5.13)

The main purpose of this paper is a mathematical construction of a non-perturbative deformation of a two-dimensional conformal field theory. We introduce a notion of a full vertex algebra which formulates a compact two-dimensional conformal field theory. Then, we construct a deformation family of a full vertex algebra which serves as a current-current deformation of conformal field theory in physics. The parameter space of the deformation is expressed as a double coset of an orthogonal group, a quotient of an orthogonal Grassmannian. As an application, we consider a deformation of chiral conformal field theories, vertex operator algebras. A current-current deformation of a "vertex operator algebra" may produce new vertex operator algebras. We give a formula for counting the number of the isomorphic classes of vertex operator algebras obtained in this way. We demonstrate it for some holomorphic vertex operator algebra of central charge $24$., 49 pages, 1 figure. In version 3, only the introduction has been slightly changed