Modular Data and Verlinde Formulae for Fractional Level WZW Models I

TL;DR

The paper resolves long-standing questions about the modular properties and Verlinde formula for fractional-level $\\uhat{sl}(2)$ theories, especially at $k=-\tfrac{1}{2}$ and $k=-\tfrac{4}{3}$, by treating characters as distributions and embracing an extended representation category that includes indecomposable modules. It derives a continuous-spectrum Verlinde framework with a projective SL$(2,\mathbb{Z})$ action on typical and atypical characters, and demonstrates perfect agreement between the resulting fusion rules (Grothendieck fusion) and explicit modular data. The work constructs infinite families of modular invariant partition functions via simple-current extensions and extended algebras $\mathbb{W}^{(b)}$, linking the fractional-level theories to rational/logarithmic CFTs and coset relations with triplet algebras. It also refines and, in some cases, corrects previously conjectured fusion rules (notably at $k=-\tfrac{4}{3}$), providing a robust framework for understanding logarithmic behavior in fractional-level WZW models and their potential geometric realizations. Overall, the results position fractional-level theories as tractable, building-block models for bulk and boundary logarithmic CFTs with well-controlled modular structure and a practical Verlinde calculus.

Abstract

The modular properties of fractional level affine sl(2)-theories and, in particular, the application of the Verlinde formula, have a long and checkered history in conformal field theory. Recent advances in logarithmic conformal field theory have led to the realisation that problems with fractional level models stem from trying to build the theory with an insufficiently rich category of representations. In particular, the appearance of negative fusion coefficients for admissible highest weight representations is now completely understood. Here, the modular story for certain fractional level theories is completed. Modular transformations are derived for the complete set of admissible irreducible representations when the level is k=-1/2 or k=-4/3. The S-matrix data and Verlinde formula are then checked against the known fusion rules with complete agreement. Finally, an infinite set of modular invariant partition functions is constructed in each case.

Figures (2)

• Figure 1: Depictions of the admissible irreducible $\widehat{\mathfrak{sl}} \left( 2 \right)_{-1/2}$-modules. Each labelled state declares its $\mathfrak{sl} \left( 2 \right)$-weight and conformal dimension (in that order). Conformal dimensions increase from top to bottom and $\mathfrak{sl} \left( 2 \right)$-weights increase from right to left.
• Figure 2: Depictions of the admissible irreducible $\widehat{\mathfrak{sl}} \left( 2 \right)_{-4/3}$-modules. Each labelled state declares its $\mathfrak{sl} \left( 2 \right)$-weight and conformal dimension (in that order). Conformal dimensions increase from top to bottom and $\mathfrak{sl} \left( 2 \right)$-weights increase from right to left. We have shifted the middle row to emphasise the conjugation symmetry. Moreover, the modules of this row do behave in many respects as if they should be assigned a half-integer spectral flow index.