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An admissible level $\widehat{\mathfrak{osp}} \left( 1 \middle\vert 2 \right)$-model: modular transformations and the Verlinde formula

David Ridout, John Snadden, Simon Wood

Abstract

The modular properties of the simple vertex operator superalgebra associated to the affine Kac-Moody superalgebra $\widehat{\mathfrak{osp}} \left( 1 \middle\vert 2 \right)$ at level $-\frac{5}{4}$ are investigated. After classifying the relaxed highest-weight modules over this vertex operator superalgebra, the characters and supercharacters of the simple weight modules are computed and their modular transforms are determined. This leads to a complete list of the Grothendieck fusion rules by way of a continuous superalgebraic analogue of the Verlinde formula. All Grothendieck fusion coefficients are observed to be non-negative integers. These results indicate that the extension to general admissible levels will follow using the same methodology once the classification of relaxed highest-weight modules is completed.

An admissible level $\widehat{\mathfrak{osp}} \left( 1 \middle\vert 2 \right)$-model: modular transformations and the Verlinde formula

Abstract

The modular properties of the simple vertex operator superalgebra associated to the affine Kac-Moody superalgebra at level are investigated. After classifying the relaxed highest-weight modules over this vertex operator superalgebra, the characters and supercharacters of the simple weight modules are computed and their modular transforms are determined. This leads to a complete list of the Grothendieck fusion rules by way of a continuous superalgebraic analogue of the Verlinde formula. All Grothendieck fusion coefficients are observed to be non-negative integers. These results indicate that the extension to general admissible levels will follow using the same methodology once the classification of relaxed highest-weight modules is completed.

Paper Structure

This paper contains 24 sections, 25 theorems, 167 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The basic Lie superalgebra $\mathfrak{osp} \super{1}{2}$
  3. A brief review of $\mathfrak{sl} \lparen2\rparen$
  4. A brief review of $\mathfrak{osp} \super{1}{2}$
  5. The Weyl group
  6. The affine Kac-Moody superalgebra $\widehat{\mathfrak{osp}} \super{1}{2}$
  7. The affine algebra
  8. Generalised Verma modules and vertex operator superalgebras
  9. Simple weight modules
  10. Automorphisms
  11. The affine minimal model $\mathsf{B}_{0\vert 1} \lparen2 , 4\rparen$: Modules
  12. Zhu's algebra
  13. Classifying $\mathsf{B}_{0\vert 1} \lparen2 , 4\rparen$-modules
  14. The affine minimal model $\mathsf{B}_{0\vert 1} \lparen2 , 4\rparen$: Characters
  15. Highest-weight characters
  16. ...and 9 more sections

Key Result

Theorem 1

Every simple $\mathfrak{sl} \lparen2\rparen$ weight module is isomorphic to one of the following mutually non-isomorphic modules:

Figures (1)

  • Figure 1: Submodule inclusions in the $k=-\frac{5}{4}$ Verma modules $\mathcal{V}^{\mathrm{NS}}_{-5/4,0}$ (left) and $\mathcal{V}^{\mathrm{NS}}_{-5/4,-1/2}$ (right). Each vertex indicates a singular vector generating a Verma submodule. The pairs $(\lambda,\Delta)$ attached to each vertex give the $\mathfrak{osp} \super{1}{2}$-weight $\lambda$ and conformal weight $\Delta$ of the corresponding singular vector.

Theorems & Definitions (52)

  • Theorem 1: Classification of simple $\mathfrak{sl} \lparen2\rparen$ weight modules
  • Theorem 2: Classification of simple $\mathfrak{osp} \super{1}{2}$ weight modules
  • proof
  • Proposition 3: The Sugawara Construction
  • Theorem 4: Kac-Kazhdan Determinant Formula KacStr79BowRep97
  • Proposition 5
  • Proposition 6
  • proof
  • Proposition 7
  • proof
  • ...and 42 more