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.