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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 (37 sections, 32 theorems, 157 equations)

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