A remark on simplicity of vertex algebras and Lie conformal algebras

Simplicity of quadratic Lie conformal algebras

In this paper, simplicity of quadratic Lie conformal algebras are investigated. From the point view of the corresponding Gel'fand-Dorfman bialgebras, some sufficient conditions and necessary conditions to ensure simplicity of quadratic Lie conformal algebras are presented. By these observations, we present several classes of new infinite simple Lie conformal algebras. These results will be useful for classification purposes.

Simple current extensions beyond semi-simplicity

Let V be a simple VOA and consider a representation category of V that is a vertex tensor category in the sense of Huang-Lepowsky. In particular, this category is a braided tensor category. Let J be an object in this category that is a simple current of order two of either integer or half-integer conformal dimension. We prove that $V\oplus J$ is either a VOA or a super VOA. If the representation category of V is in addition ribbon, then the categorical dimension of J decides this parity question. Combining with Carnahan's work, we extend this result to simple currents of arbitrary order. Our n…

Twisted Hamiltonian Lie Algebras and Their Multiplicity-Free Representations

We construct a class of new Lie algebras by generalizing the one-variable Lie algebras generated by the quadratic conformal algebras (or corresponding Hamiltonian operators) associated to Poisson algebras and a quasi-derivation found by Xu. These algebras can be viewed as certain twists of Xu's generalized Hamiltonian Lie algebras. The simplicity of these algebras is completely determined. Moreover, we construct a family of multiplicity-free representations of these Lie algebras and prove their irreducibility.

A remark about the anomalies of cyclic holomorphic permutation orbifolds

Using a result of Longo and Xu, we show that the anomaly arising from a cyclic permutation orbifold of order 3 of a holomorphic conformal net $\mathcal A$ with central charge $c=8k$ depends on the "gravitational anomaly" $k\pmod 3$. In particular, the conjecture that holomorphic permutation orbifolds are non-anomalous and therefore a stronger conjecture of Müger about braided crossed $S_n$-categories arising from permutation orbifolds of completely rational conformal nets are wrong. More general, we show that cyclic permutations of order $n$ are non-anomalous if and only if $3\nmid n$ or $24|c…

研究$W$--代数中共形嵌入与坍缩水平的新方法

New approaches for studying conformal embeddings and collapsing levels for $W$--algebras

本文提出研究仿射顶点代数到仿射$W$-代数共形嵌入的新方法，证明中心荷相等是共形嵌入的充要条件，并给出判定共形水平坍缩的充分条件，进而证明Creutzig猜想并刻画钩型$W$-代数的半单分解。

非幺正 Wightman CFT 与非幺正顶点代数

Non-unitary Wightman CFTs and non-unitary vertex algebras

本文建立了带拟初级生成族的 Möbius 顶点代数与单位圆上不必幺正的 Möbius 协变 Wightman CFT 之间最一般的范畴等价，并证明非幺正 Reeh–Schlieder 定理。