All-orders and closed-form expressions for the modified character

Develop methods to compute, to all orders in q, the modified character I_R(q)=I_M(qe^{πi},e^{πi}) introduced in the paper and derive closed-form expressions for broad classes of vertex operator algebras associated with unitary 4d N=2 superconformal field theories.

Background

The paper introduces a modified character I_R(q), defined via a special non‑Schur limit of the Macdonald index and computable from a graded‑unitarity‑motivated inner product on the VOA. The current computations are carried out case by case, and no general all‑orders or closed‑form expressions are provided.

Obtaining systematic, closed‑form control over I_R(q) would parallel the role of vacuum characters for Schur indices and enable broader applications and classifications.

References

Several important questions remain open. A first direction is to develop methods for computing this quantity to all orders and, ideally, to obtain closed-form expressions for broad classes of VOAs.

— Macdonald Index from VOA and Graded Unitarity  (2603.29829 - Jiang, 31 Mar 2026) in Conclusion, Section 5

When \omega = e{\frac{2 \pi i}{4}}= i, which corresponds to the duality defect, we conjecture the following closed form of the twisted character for arbitrary \zeta:

— Non-invertible symmetry and vertex operator algebra outer-automorphism  (2608.18926 - Maruyoshi et al., 19 Aug 2026) in Section 2.2, subsection “\(\mathcal{V}(A_1)\) case”