The charge conjugation orbifold $V_{Zα}^{+}$ is rational when $<α,α>/2$ is prime

Rational Generalised Moonshine from Abelian Orbifoldings of the Moonshine Module

We consider orbifoldings of the Moonshine Module with respect to the abelian group generated by a pair of commuting Monster group elements with one of prime order $p=2,3,5,7$ and the other of order $pk$ for $k=1$ or $k$ prime. We show that constraints arising from meromorphic orbifold conformal field theory allow us to demonstrate that each orbifold partition function with rational coefficients is either constant or is a hauptmodul for an explicitly found modular fixing group of genus zero. We thus confirm in the cases considered the Generalised Moonshine conjectures for all rational modular f…

置换轨形中扭曲模的融合积：II

Fusion Products of Twisted Modules in Permutation Orbifolds: II

本文针对 CFT 型简单有理 $C_2$-余有限顶点算子代数的张量积，确定了任意置换生成循环群作用下扭曲模的融合积。

The relationship between skew group algebras and orbifold theory

Let V be a simple vertex operator algebra and let G be a finite automorphism group of V. In [DY], it was shown that any irreducible V-module is a completely reducible V^G-module where V^G is the G-invariant sub-vertex operator algebra of V. In this paper, we give an alternative proof of this fact using the theory of skew group algebras. We also extend this result to any irreducible g-twisted V-module when g is in the center of G and V is a g-rational vertex operator algebra.

Congruence Property in Orbifold Theory

Let $V$ be a rational, selfdual, $C_2$-cofinite vertex operator algebra of CFT type, and $G$ a finite automorphism group of $V.$ It is proved that the kernel of the representation of the modular group on twisted conformal blocks associated to $V$ and $G$ is a congruence subgroup. In particular, the $q$-character of each irreducible twisted module is a modular function on the same congruence subgroup. In the case $V$ is the Frenkel-Lepowsky-Meurman's moonshine vertex operator algebra and $G$ is the monster simple group, the generalized McKay-Thompson series associated to any commuting pair in t…

Algebraic orbifold conformal field theories

We formulate the unitary rational orbifold conformal field theories in the algebraic quantum field theory framework. Under general conditions, we show that the orbifold of a given unitary rational conformal field theories generates a unitary modular category. Many new unitary modular categories are obtained. We also show that the irreducible representations of orbifolds of rank one lattice vertex operator algebras give rise to unitary modular categories and determine the corresponding modular matrices, which has been conjectured for some time.

Reconstruction and Local Extensions for Twisted Group Doubles, and Permutation Orbifolds

We prove the first nontrivial reconstruction theorem for modular tensor categories: the category associated to any twisted Drinfeld double of any finite group, can be realised as the representation category of a completely rational conformal net. We also show that any twisted double of a solvable group is the category of modules of a completely rational vertex operator algebra. In the process of doing this, we identify the 3-cocycle twist for permutation orbifolds of holomorphic conformal nets: unexpectedly, it can be nontrivial, and depends on the value of the central charge modulo 24. In add…

Modular invariance of trace functions in orbifold theory

The goal of the present paper is to provide a mathematically rigorous foundation to certain aspects of rational orbifold conformal field theory, in other words the theory of rational vertex operator algebras and their automorphisms. Under a certain finiteness condition on a rational vertex operator algebra V which holds in all known examples, we determine the precise numbers of g-twisted sectors for any automorphism g of V of finite order. We prove that the trace functions and correlations functions associated with such twisted sectors are holomorphic functions in the upper half-plane and, und…

Modularity in orbifold theory for vertex operator superalgebras

This paper is about the orbifold theory for vertex operator superalgebras. Given a vertex operator superalgebra V and a finite automorphism group G of V, we show that the trace functions associated to the twisted sectors are holomorphic in the upper half plane for any commuting pairs in G under the C_2-cofinite condition. We also establish that these functions afford a representation of the full modular group if V is C_2-cofinite and g-rational for any g in G.

Flatness and Semi-Rigidity of Vertex Operator Algebras

In vertex operator algebra theories, most of the general theorems are proved under the assumptions of rationality and C_2-cofiniteness. In this paper, we obtain several general theorems without the assumption of rationality so that we can use them for proving rationality of given C_2-cofinite vertex operator algebras. For example, we apply them to orbifold models and show that if g is a finite automorphism of a rational C_2-cofinite vertex operator algebra T of CFT-type with T'\cong T and a fixed point subVOA T^g is C_2-cofinite, then T^g is also rational.

Permutation orbifolds of Virasoro vertex algebras and $W$-algebras

We study permutation orbifolds of the $2$-fold and $3$-fold tensor product for the Virasoro vertex algebra $\mathcal{V}_c$ of central charge $c$. In particular, we show that for all but finitely many central charges $\left(\mathcal{V}_c^{\otimes 3}\right)^{\mathbb{Z}_3}$ is a $W$-algebra of type $(2, 4, 5, 6^3 , 7, 8^3 , 9^3 , 10^2 )$. We also study orbifolds of their simple quotients and obtain new realizations of certain rational affine $W$-algebras associated to a principal nilpotent element. Further analysis of permutation orbifolds of the celebrated $(2,5)$-minimal vertex algebra $\mathca…

Modularity of trace functions in orbifold theory for Z-graded vertex operator superalgebras

We study the trace functions in orbiford theory for Z-graded vertex operator superalgebras and obtain a modular invariance result. More precisely, let V be a C_2-cofinite Z-graded vertex operator superalgebra and G a finite automorphism group of V. Then for any commuting pairs (g,h) in G, the hσ-trace functions associated to the simple g-twisted V-modules are holomorphic in the upper half plane where σis the canonical involution on V coming from the superspace structure of V. If V is further g-rational for every g n G, the trace unctions afford a representation for the full modular group SL(2,…

Quantization, orbifold cohomology, and Cherednik algebras

We compute the Hochschild homology of the crossed product $\Bbb C[S_n]\ltimes A^{\otimes n}$ in terms of the Hochschild homology of the associative algebra $A$ (over $\Bbb C$). It allows us to compute the Hochschild (co)homology of $\Bbb C[W]\ltimes A^{\otimes n}$ where $A$ is the $q$-Weyl algebra or any its degeneration and $W$ is the Weyl group of type $A_{n-1}$ or $B_n$. For a deformation quantization $A_+$ of an affine symplectic variety $X$ we show that the Hochschild homology of $S^n A$, $A=A_+[\hbar^{-1}]$ is additively isomorphic to the Chen-Ruan orbifold cohomology of $S^nX$ with coef…

对尼梅依尔晶格顶点算子代数的系统轨道构造

Systematic Orbifold Constructions of Schellekens' Vertex Operator Algebras from Niemeier Lattices

本文系统构造了所有70个中心荷为24且具有非零权重一空间的强理性整体顶点算子代数，通过尼梅依尔晶格顶点算子代数及其短自守变换的轨道构造，并证明其由权重一空间的李代数结构唯一确定。

Conformal Orbifold Theories and Braided Crossed G-Categories

We show that a quantum field theory A living on the line and having a group G of inner symmetries gives rise to a category GLoc A of twisted representations. This category is a braided crossed G-category in the sense of Turaev. Its degree zero subcategory is braided and equivalent to the usual representation category Rep A. Combining this with [29], where Rep A was proven to be modular for a nice class of rational conformal models, and with the construction of invariants of G-manifolds in [60], we obtain an equivariant version of the following chain of constructions: Rational CFT -> modular ca…

Representations of the vertex operator algebra V_{L_{2}}^{A_{4}}

The rationality and C_2-cofiniteness of the orbifold vertex operator algebra V_{L_{2}}^{A_{4}} are established and all the irreducible modules are constructed and classified. This is part of classification of rational vertex operator algebras with c=1.

On the tensor structure of modules for compact orbifold vertex operator algebras

Suppose $V^G$ is the fixed-point vertex operator subalgebra of a compact group $G$ acting on a simple abelian intertwining algebra $V$. We show that if all irreducible $V^G$-modules contained in $V$ live in some braided tensor category of $V^G$-modules, then they generate a tensor subcategory equivalent to the category $\mathrm{Rep}\,G$ of finite-dimensional representations of $G$, with associativity and braiding isomorphisms modified by the abelian $3$-cocycle defining the abelian intertwining algebra structure on $V$. Additionally, we show that if the fusion rules for the irreducible $V^G$-m…

Orbifolds and minimal modular extensions

Let $V$ be a simple, rational, $C_2$-cofinite vertex operator algebra and $G$ a finite group acting faithfully on $V$ as automorphisms, which is simply called a rational vertex operator algebra with a $G$-action. It is shown that the category ${\cal E}_{V^G}$ generated by the $V^G$-submodules of $V$ is a symmetric fusion category braided equivalent to the $G$-module category ${\cal E}={\rm Rep}(G)$. If $V$ is holomorphic, then the $V^G$-module category ${\cal C}_{V^G}$ is a minimal modular extension of ${\cal E},$ and is equivalent to the Drinfeld center ${\cal Z}({\rm Vec}_G^α)$ as modular te…

顶点算子代数上的等价关系，II：Witt等价与Orbifold

Equivalence Relations on Vertex Operator Algebras, II: Witt Equivalence and Orbifolds

本文定义强有理VOA与RCFT的Witt等价，论证其为拓扑操作关联的必要条件并猜想充分性，还用于对称性、量子Galois理论及p-邻域研究。

A new Rational Conformal Field Theory extension of the fully degenerate W_{1+\infty}^{(m)}

We found new identities among the Dedekind eta-function, the characters of the W_{m} algebra and those of the level 1 affine Lie algebra su(m)_{1}. They allow to characterize the Z_{m}-orbifold of the m-component free bosons u(1)_{K_{m,p}} (our theory TM) as an extension of the fully degenerate representations of W_{1+infty}^{(m)}. In particular, TM is proven to be a Gamma _{theta}-RCFT extension of the chiral fully degenerate W_{1+infty}^{(m)}.