Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Equivalence between Chain Categories of Representations of Affine sl(2) and N=2 Superconformal Algebras

B L Feigin, A M Semikhatov, I Yu Tipunin

TL;DR

The paper addresses the problem of relating representation theories of the affine ${\widehat{s\ell}}(2)$ algebra and the ${N=2}$ superconformal algebra by establishing an equivalence of their highest-weight-type module categories modulo spectral flow. It develops the Kazama–Suzuki (KS) and inverse anti-KS mappings, showing how Verma, relaxed, and massive Verma modules correspond across the two algebras when organized into spectral-flow chains. The main contributions include a complete classification of singular vectors in relaxed ${\widehat{s\ell}}(2)$ modules, the explicit construction of maps between twisted Verma and topological Verma modules, and the proof of category equivalences between Verma-chain and higher-level HW-type categories for both algebras. These results enable transferring structural information and (potentially) fusion rules between the two theories, enriching the understanding of conformal-block constructions and dualities in ${\widehat{s\ell}}(2)$ and ${N=2}$ contexts.

Abstract

Highest-weight type representation theories of the affine sl(2) and N=2 superconformal algebras are shown to be equivalent modulo the respective spectral flows.

Equivalence between Chain Categories of Representations of Affine sl(2) and N=2 Superconformal Algebras

TL;DR

The paper addresses the problem of relating representation theories of the affine algebra and the superconformal algebra by establishing an equivalence of their highest-weight-type module categories modulo spectral flow. It develops the Kazama–Suzuki (KS) and inverse anti-KS mappings, showing how Verma, relaxed, and massive Verma modules correspond across the two algebras when organized into spectral-flow chains. The main contributions include a complete classification of singular vectors in relaxed modules, the explicit construction of maps between twisted Verma and topological Verma modules, and the proof of category equivalences between Verma-chain and higher-level HW-type categories for both algebras. These results enable transferring structural information and (potentially) fusion rules between the two theories, enriching the understanding of conformal-block constructions and dualities in and contexts.

Abstract

Highest-weight type representation theories of the affine sl(2) and N=2 superconformal algebras are shown to be equivalent modulo the respective spectral flows.

Paper Structure

This paper contains 37 sections, 32 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. The ${\widehat{s\ell}}(2)$ and $N\!=\!2$ algebras, their automorphisms and Verma modules
  3. ${\widehat{s\ell}}(2)$
  4. ${\widehat{s\ell}}(2)$ Verma modules, spectral flow transform, and extremal diagrams
  5. Singular vectors in ${\widehat{s\ell}}(2)$ Verma modules
  6. Relaxed Verma modules
  7. 'Charged' singular vectors
  8. Relation with the previous constructions of some ${\widehat{s\ell}}(2)$ modules
  9. Singular vectors in relaxed Verma modules
  10. Codimension-$2$ cases.
  11. Going 'past the highest-weight'.
  12. Codimension $3$.
  13. Auxiliaries
  14. Fermions (ghosts)
  15. The 'Liouville' scalar
  16. ...and 22 more sections

Key Result

Theorem II.4

I. ([KK]) A singular vector exists in the Verma module ${\cal M}_{j,k}$ over the affine $s\ell(2)$ algebra if and only if $j={\fam\ssffam j}^+(r,s,k)$ or $j={\fam\ssffam j}^-(r,s,k)$, where II. ([MFF]) All singular vectors in the module ${\cal M}_{{\fam\ssffam j}^\pm(r,s,k),k}$ are given by the explicit constructions:

Theorems & Definitions (42)

  • Definition II.1
  • Definition II.2
  • Definition II.3
  • Theorem II.4
  • Definition II.6
  • Definition II.8
  • Lemma II.9
  • Theorem II.10
  • Lemma II.11
  • Theorem II.12
  • ...and 32 more