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Modularity of (super)characters of small N=4 modules at c=−9 and c=−3

Determine the modular transformation properties of the characters and supercharacters of the irreducible Neveu–Schwarz and Ramond modules for the small N=4 superconformal algebra at central charges −9 and −3 constructed via inverse hamiltonian reduction from symplectic fermions; in particular, derive their behaviour under SL(2, Z) and assess modular invariance.

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Background

Using that Upr collapses to symplectic fermions at k=±1/2 and applying inverse hamiltonian reduction, the authors recover and extend classifications of irreducible modules for the small N=4 algebra at c=−9 and c=−3 and compute their (super)characters.

However, the modular analysis of these (super)characters is left open. Resolving their modularity would have implications for the associated topological quantum field theories and for understanding modular data, fusion, and categorical structures.

References

The (super)characters of the latter are easily computed, though we leave the potentially subtle analysis of their modularity to future work.

— The principal W-algebra of $\mathfrak{psl}_{2|2}$ (2509.04795 - Fehily et al., 5 Sep 2025) in Introduction, Subsection “Outline and results”, concluding paragraph on inverse reduction (around discussion of Section k=±1/2)