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In this paper, we construct an action of the fundamental groupoid of the Swiss-cheese operad (the parenthesized permutation and braid operad) on $C_1$-cofinite module categories of a vertex operator algebra. Based on this result, we show that when a boundary conformal field theory has locally $C_1$-cofinite chiral symmetry, all operator product expansions converge absolutely on open regions in the configuration spaces of the upper half-plane and define real analytic functions (the correlation functions), independent of orders and parentheses of OPEs., Comment: 56 pages

Full Vertex Operator Algebras (full VOA) are extensions of two commuting Vertex Operator Algebras, introduced to formulate compact two-dimensional conformal field theory. We define unitarity, polynomial energy bounds and polynomial spectral density for full VOA. Under these conditions and local $C_1$-cofiniteness of the simple full VOA, we show that the correlation functions of quasi-primary fields define tempered distributions and satisfy a conformal version of the Osterwalder-Schrader axioms, including the linear growth condition. As an example, we show that a family of full extensions of the Heisenberg VOA satisfies all these assumptions., Comment: 49 pages, 2 TikZ figures. Minor revisions about formal variables, updating funding information and bibliography

Conformal blocks, physical quantities of chiral 2d conformal field theory, are sheaves on the configuration spaces of the complex plane, which are mathematically formulated in terms of a vertex operator algebra, its modules and associated D-modules. We show that the operad of fundamental groupoids of the configuration spaces, the parenthesized braid operad, acts on the conformal blocks by the monodromy representation. More precisely, let $V$ be a vertex operator algebra with $V=\bigoplus_{n\geq 0} V_n$, $\dim V_n <\infty$, $V_0=\mathbb{C}\bf{1}$ and $V\text{-mod}_{\mathrm{f.g}}$ the category of $V$-modules $M$ such that $M$ is $C_1$-cofinite and the dual module $M^\vee$ is a finitely generated $V$-module. We show that the parenthesized braid operad weakly 2-categorically acts on $V\text{-mod}_{\mathrm{f.g}}$, and consequently $V\text{-mod}_{\mathrm{f.g}}$ has a structure of the (unital) pseudo-braided category. Moreover, if $V$ is rational and $C_2$-cofinite, then $V\text{-mod}_{\mathrm{f.g}}$ is a balanced braided tensor category, which gives an alternative proof of a result of Huang and Lepowsky., Comment: 121 pages. In v2, add Appendix B, relax the assupmtion of Main theorem, revise Introduction

In this paper, we construct various simple vertex superalgebras which are extensions of affine vertex algebras, by using abelian cocycle twists of representation categories of quantum groups. This solves the Creutzig and Gaiotto conjectures in the case of type ABC. If the twist is trivial, the resulting algebras correspond to chiral differential operators in the chiral case, and to WZW models in the non-chiral case., Comment: 45 pages

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., Comment: 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 $24$., Comment: 66 pages, 1 figure. In version 4, add several results and the order of the sections was changed. Adv. Math (2023)

In physics, it is believed that the consistency of two dimensional conformal field theory follows from the bootstrap equation. In this paper, we introduce the notion of a full vertex algebra by analyzing the bootstrap equation, which is a "real analytic" generalization of a $\mathbb{Z}$-graded vertex algebra. We also give a mathematical formulation of the consistency of four point correlation functions in two dimensional conformal field theory and prove it for a full vertex algebra with additional assumptions on the conformal symmetry. In particular, we show that the bootstrap equation together with the conformal symmetry implies the consistency of four point correlation functions. As an application, a deformable family of full vertex algebras parametrized by the Grassmanian is constructed, which appears in the toroidal compactification of string theory. This give us examples satisfying the above assumptions.

We introduce the notion of a genus and its mass for vertex algebras. For lattice vertex algebras, their genera are the same as those of lattices, which plays an important role in the classification of lattices. We derive a formula relating the mass for vertex algebras to that for lattices, and then give a new characterization of some holomorphic vertex operator algebras., Comment: 36 pages, to appear in Math. Z

Herein we study conformal vectors of a Z-graded vertex algebra of (strong) CFT type. We prove that the full vertex algebra automorphism group transitively acts on the set of the conformal vectors of strong CFT type if the vertex algebra is simple. The statement is equivalent to the uniqueness of self-dual vertex operator algebra structures of a simple vertex algebra. As an application, we show that the full vertex algebra automorphism group of a simple vertex operator algebra of strong CFT type uniquely decomposes into the product of certain two subgroups and the vertex operator algebra automorphism group. Furthermore, we prove that the full vertex algebra automorphism group of the moonshine module over the field of real numbers is the Monster., Comment: 13 pages