The Derivation algebra and automorphism group of the generalized Ramond N=2 superconformal algebra

The Adapted Ordering Method in Representation Theory

In 1998 the Adapted Ordering Method was developed for the representation theory of the superconformal algebras. This method, which proves to be very powerful, can be applied to most algebras and superalgebras, however. It allows: to determine maximal dimensions for a given type of singular vector space, to identify all singular vectors by only a few coefficients, to spot subsingular vectors and to set the basis for constructing embedding diagrams. In this article we present the Adapted Ordering Method for general algebras and superalgebras which admit a triangulation and review briefly the res…

Determinant Formula for the Topological N=2 Superconformal Algebra

The Kac determinant for the Topological N=2 superconformal algebra is presented as well as a detailed analysis of the singular vectors detected by the roots of the determinants. In addition we identify the standard Verma modules containing `no-label' singular vectors (which are not detected directly by the roots of the determinants). We show that in standard Verma modules there are (at least) four different types of submodules, regarding size and shape. We also review the chiral determinant formula, for chiral Verma modules, adding new insights. Finally we transfer the results obtained to the …

A family of simple non-weight modules over the twisted $N=2$ superconformal algebra

We construct a class of non-weight modules over the twisted $N=2$ superconformal algebra $\T$. Let $\mathfrak{h}=\C L_0\oplus\C G_0$ be the Cartan subalgebra of $\T$, and let $\mathfrak{t}=\C L_0$ be the Cartan subalgebra of even part $\T_{\bar 0}$. These modules over $\T$ when restricted to the $\mathfrak{h}$ are free of rank $1$ or when restricted to the $\mathfrak{t}$ are free of rank $2$. We provide the sufficient and necessary conditions for those modules being simple, as well as giving the sufficient and necessary conditions for two $\T$-modules being isomorphic. We also compute the acti…

Singular dimensions of the N=2 superconformal algebras. I

Verma modules of superconfomal algebras can have singular vector spaces with dimensions greater than 1. Following a method developed for the Virasoro algebra by Kent, we introduce the concept of adapted orderings on superconformal algebras. We prove several general results on the ordering kernels associated to the adapted orderings and show that the size of an ordering kernel implies an upper limit for the dimension of a singular vector space. We apply this method to the topological N=2 algebra and obtain the maximal dimensions of the singular vector spaces in the topological Verma modules: 0,…

Resolutions and Characters of Irreducible Representations of the N=2 Superconformal Algebra

We evaluate characters of irreducible representations of the N=2 supersymmetric extension of the Virasoro algebra. We do so by deriving the BGG-resolution of the admissible N=2 representations and also a new 3,5,7...-resolution in terms of twisted massive Verma modules. We analyse how the characters behave under the automorphisms of the algebra, whose most significant part is the spectral flow transformations. The possibility to express the characters in terms of theta functions is determined by their behaviour under the spectral flow. We also derive the identity expressing every $\hat{sl}(2)$…

Equivariant K3 Invariants

In this note, we describe a connection between the enumerative geometry of curves in K3 surfaces and the chiral ring of an auxiliary superconformal field theory. We consider the invariants calculated by Yau--Zaslow (capturing the Euler characters of the moduli spaces of D2-branes on curves of given genus), together with their refinements to carry additional quantum numbers by Katz--Klemm--Vafa (KKV), and Katz--Klemm--Pandharipande (KKP). We show that these invariants can be reproduced by studying the Ramond ground states of an auxiliary chiral superconformal field theory which has recently bee…

广义q-Weyl代数上正迹的一种不同方法

A different approach to positive traces on generalized q-Weyl algebras

该研究拓展了q-Weyl代数正扭曲迹的研究范围，考虑更一般的反线性自同构选择，证明对应四维规范理论标准Schur指标的自同构下正迹具有唯一性，可为超共形场论参数计算提供支撑。

关于奇数Leech格CFT的对称性

On the Symmetry of Odd Leech Lattice CFT

本文证明奇数Leech格顶点算子代数自同构群中的Mathieu群子群是非分裂扩张，并研究了不变流，受月光现象启发。