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Irreducible modules over N=2 superconformal algebras from algebraic D-modules

Haibo Chen, Xiansheng Dai, Dong Liu, Yufeng Pei

Abstract

In this paper, we introduce a family of functors denoted $\mathscr{F}_b$ that act on algebraic D-modules and generate modules over N=2 superconformal algebras. We prove these functors preserve irreducibility for all values of $b$, with a few clear exceptions described. We also establish necessary and sufficient conditions to determine when two such functors are naturally isomorphic. Applying $\mathscr{F}_b$ to N=1 super-Virasoro algebras recovers the functors previously introduced in \cite{CDLP}. Our new functors also facilitate the recovery of specific irreducible modules over N=2 superconformal algebras, including intermediate series and $U(\mathfrak{h})$-free modules. Additionally, our constructed functors produce several new irreducible modules for N=2 superconformal algebras.

Irreducible modules over N=2 superconformal algebras from algebraic D-modules

Abstract

In this paper, we introduce a family of functors denoted that act on algebraic D-modules and generate modules over N=2 superconformal algebras. We prove these functors preserve irreducibility for all values of , with a few clear exceptions described. We also establish necessary and sufficient conditions to determine when two such functors are naturally isomorphic. Applying to N=1 super-Virasoro algebras recovers the functors previously introduced in \cite{CDLP}. Our new functors also facilitate the recovery of specific irreducible modules over N=2 superconformal algebras, including intermediate series and -free modules. Additionally, our constructed functors produce several new irreducible modules for N=2 superconformal algebras.
Paper Structure (16 sections, 19 theorems, 56 equations)

This paper contains 16 sections, 19 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Some basic definitions of superalgebras
  4. $N=2$ superconformal algebras
  5. Weyl (super)algebra
  6. Constructions of functors
  7. Properties of functors
  8. Preservation of irreducibility
  9. Isomorphisms
  10. Applications and examples
  11. Applications
  12. Examples
  13. Intermediate series modules
  14. $U(\mathbb{C}L_0)$-free modules
  15. Fraction modules
  16. ...and 1 more sections

Key Result

Lemma 2.2

The $N=2$ Neveu-Schwarz algebra $\hat{\mathcal{G}}[{\frac{1}{2}}]$ and the $N=2$ Ramond algebra $\hat{\mathcal{G}}[0]$ are isomorphic to each other by the spectral shift isomorphism: $\delta: \hat{\mathcal{G}}[{\frac{1}{2}}] \rightarrow \hat{\mathcal{G}}[{0}]$, defined by where $m\in\mathbb{Z},p\in\frac{1}{2}+\mathbb{Z}$.

Theorems & Definitions (31)

  • Definition 2.1
  • Lemma 2.2: SS
  • Definition 2.3
  • Lemma 2.4: BLZ
  • Definition 2.5: CW
  • Proposition 2.6: CDLP
  • Lemma 3.1: DVV
  • Remark 3.2
  • Lemma 3.3
  • proof
  • ...and 21 more