Intrinsic VOA computation of the full Macdonald index

Determine whether there exists an intrinsic method, defined purely within the associated vertex operator algebra of a four-dimensional N=2 superconformal field theory, to compute the full Macdonald index I_M(q,T) (beyond the Schur limit), analogous to the identification of the Schur index with the vacuum character.

Background

The SCFT/VOA correspondence identifies the Schur index of a 4d N=2 SCFT with the vacuum character of its associated VOA, but a general VOA-based derivation of the Macdonald index has been elusive. The paper highlights this as a longstanding issue and proposes a method only for a special non‑Schur limit of the Macdonald index, leaving the general problem unresolved.

The main technical difficulty is that the Macdonald index depends on SU(2)_R quantum numbers, which are obscured in the VOA description due to the topological twist. The authors’ method recovers a special limit (I_R) but not the full I_M(q,T), motivating the open problem.

References

However, the SCFT/VOA correspondence presents at least two important puzzles and open questions. This naturally leads to the second question: is there an intrinsic way to compute the Macdonald index purely within the framework of the VOA, just as for the Schur index? These two issues have remained open for more than a decade since the proposal of the SCFT/VOA correspondence.

— Macdonald Index from VOA and Graded Unitarity  (2603.29829 - Jiang, 31 Mar 2026) in Introduction, Section 1