Algebraic orbifold conformal field theories

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…

Conformal Net Realizability of Tambara-Yamagami Categories and Generalized Metaplectic Modular Categories

We show that all isomorphism classes of even rank Tambara-Yamagami categories arise as $\mathbb{Z}_2$-twisted representations of conformal nets. As a consequence, we show that their Drinfel'd centers are realized by (generalized) orbifolds of conformal nets associated with (self-dual) lattices. The quantum double subfactors of even rank Tambara-Yamagami categories are Bisch-Haagerup subfactors and we describe their (dual) principal graphs. For every abelian group of odd order the Drinfel'd centers of the associated Tambara-Yamagami categories give a fusion ring generalizing the Verlinde ring $…

More About All Current-Algebraic Orbifolds

Recently a construction was given for the stress tensors of all sectors of the general current-algebraic orbifold A(H)/H, where A(H) is any current-algebraic conformal field theory with a finite symmetry group H. Here we extend and further analyze this construction to obtain the mode formulation of each sector of each orbifold A(H)/H, including the twisted current algebra, the Virasoro generators, the orbifold adjoint operation and the commutator of the Virasoro generators with the modes of the twisted currents. As applications, general expressions are obtained for the twisted current-current …

Trigonometric Degeneration and Orbifold Wess-Zumino-Witten Model. II

The sheaves of conformal blocks and conformal coinvariants of the twisted WZW model have a factorisation property and are locally free even at the boundary of the moduli space, where the elliptic KZ equations and the Baxter-Belavin elliptic r matrix degenerate to the trigonometric KZ equations and the trigonometric r matrix,o respectively. Etingof's construction of the elliptic KZ equations is geometrically interpreted.

通过有限子群扩展融合规则：Z_N扩展共形场论的一般构造及其 orbifoldings

Extending fusion rules with finite subgroups: A general construction of $Z_{N}$ extended conformal field theories and their orbifoldings

本文提出了一种构造Z_N扩展共形场论及其orbifoldings的方法，通过非异常子群Z_n(⊂Z_N)构建了 bulk 和 chiral 理论的融合环及模分拆函数，并探讨了其在拓扑场论和边界共形场论中的应用。

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…

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…

Orbifold Construction for Topological Field Theories

An equivariant topological field theory is defined on a cobordism category of manifolds with principal fiber bundles for a fixed (finite) structure group. We provide a geometric construction which for any given morphism $G \to H$ of finite groups assigns in a functorial way to a $G$-equivariant topological field theory an $H$-equivariant topological field theory, the pushforward theory. When $H$ is the trivial group, this yields an orbifold construction for $G$-equivariant topological field theories which unifies and generalizes several known algebraic notions of orbifoldization.

Non-commutative Poisson Structures on quantum torus orbifolds

We study the Hochschild cohomology and the Gerstenhaber algebra structure on the algebraic non-commutative torus/quantum torus orbifolds resulting by the action of finite subgroups of $SL_2(\mathbb Z)$. We also examine the Poisson structures and compute the Poisson cohomology.

Orbifolds and topological defects

We study orbifolds of two-dimensional topological field theories using defects. If the TFT arises as the twist of a superconformal field theory, we recover results on the Neveu-Schwarz and Ramond sectors of the orbifold theory as well as bulk-boundary correlators from a novel, universal perspective. This entails a structure somewhat weaker than ordinary TFT, which however still describes a sector of the underlying conformal theory. The case of B-twisted Landau-Ginzburg models is discussed in detail, where we compute charge vectors and superpotential terms for B-type branes. Our construction al…

Vanishing of categorical obstructions for permutation orbifolds

The orbifold construction $A\mapsto A^G$ for a finite group $G$ is fundamental in rational conformal field theory. The construction of $Rep(A^G)$ from $Rep(A)$ on the categorical level, often called gauging, is also prominent in the study of topological phases of matter. Given a non-degenerate braided fusion category $\mathcal{C}$ with a $G$-action, the key step in this construction is to find a braided $G$-crossed extension compatible with the action. The extension theory of Etingof-Nikshych-Ostrik gives two obstructions for this problem, $o_3\in H^3(G)$ and $o_4\in H^4(G)$ for certain coeffi…

具有大间隙的CFT：来自Barnes-Wall晶格orbifold

CFTs with Large Gap from Barnes-Wall Lattice Orbifolds

本文研究通过Barnes-Wall晶格的额外特殊2群自同构构造具有大间隙的CFT，基于3-cocycle消失的猜想，构建了中心荷为128、间隙为4的holomorphic CFT。

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…