Macdonald Index from VOA and Graded Unitarity

Abstract: The SCFT/VOA correspondence provides a powerful framework for studying 4d $\mathcal N=2$ superconformal field theories (SCFTs) through the mathematical machinery of 2d vertex operator algebras (VOAs). It captures the Schur operators of the underlying SCFT, whose spectrum is encoded by the Schur index and its refinement, the Macdonald index. While the Schur index is identified with the vacuum character of the associated VOA, a general VOA-based derivation of the Macdonald index has remained elusive. In this letter, we propose a novel and intrinsic method for recovering a special non-Schur limit of the Macdonald index directly from the VOA. The construction requires no additional assumptions and applies whenever the underlying 4d theory is unitary. We test the proposal in a variety of examples, and further extend it to the case with surface defects, suggesting a notion of graded unitarity in the presence of defects. Our method also introduces a new class of series for general VOAs, analogous to but distinct from the conventional character, and potentially useful in broader contexts.

• The paper introduces a novel prescription to derive a modified Macdonald index from VOA data by leveraging graded unitarity.
• It constructs an anti-linear automorphism that encodes CPT properties and recovers SU(2)_R information lost under the topological twist.
• The framework is validated across free multiplets, minimal models, and Argyres-Douglas theories, illustrating its universal applicability.

Macdonald Index from VOA and Graded Unitarity: An Expert Summary

Introduction and Motivation

The SCFT/VOA correspondence provides a powerful structural bridge between four-dimensional $\mathcal N=2$ SCFTs and two-dimensional VOAs, mapping the spectrum of Schur operators in 4d to the states/operators in 2d. While the Schur index of the 4d theory is equivalent to the vacuum character of the associated VOA, the more refined Macdonald index—sensitive to the $SU(2)_R$ and $U(1)_r$ quantum numbers—has lacked a universal, intrinsic VOA-based description. This gap is particularly noteworthy since the $SU(2)_R$ quantum number is topologically twisted away in the standard VOA construction. Additionally, there is a long-standing puzzle associated with the mapping of unitarity: unitary 4d SCFTs can be associated with non-unitary VOAs due to a sign flip in the central charge.

This work introduces a new intrinsic prescription for recovering a special non-Schur limit of the Macdonald index—denoted $\mathcal{I}_R(q)$—directly from VOA data, leveraging the notion of graded unitarity. The construction is universal and applies to all unitary 4d $\mathcal N=2$ SCFTs and their associated VOAs, and it naturally generalizes to include surface defects (non-vacuum modules).

Formal Construction and Prescription

The proposal is based on defining an anti-linear involutive automorphism $\phi$ of the VOA, which generalizes complex conjugation and encodes the CPT properties of the parent SCFT. Specifically, $\phi$ leaves the vacuum and the stress tensor invariant, reverses the $U(1)_r$ charge, acts anti-linearly, and satisfies $\phi^2 = (-1)^{2h}$ on an operator of weight $SU(2)_R$0. This automorphism allows the definition of a Hermitian form via

$SU(2)_R$1

where $SU(2)_R$2 is the coefficient in the OPE.

Graded unitarity, recently clarified in the SCFT/VOA context, constrains the inner product of physical operators to pick up a sign $SU(2)_R$3, where $SU(2)_R$4 and $SU(2)_R$5 are the $SU(2)_R$6 and $SU(2)_R$7 quantum numbers, and $SU(2)_R$8 is the conformal weight. This leads to a construction of two indices, derived from the Gram matrix of $SU(2)_R$9 over all operators—$U(1)_r$0 (the Schur index/character) and $U(1)_r$1 (the "modified character" corresponding to the non-Schur Macdonald limit): $U(1)_r$2 where $U(1)_r$3 are the counts of positive/negative eigenvalues of the Gram matrix in each subspace of fixed conformal weight $U(1)_r$4 and fermion number $U(1)_r$5.

$U(1)_r$6 corresponds to evaluating the Macdonald index at the special point $U(1)_r$7, and has integer series coefficients closely tied to the $U(1)_r$8 charge.

Application to Explicit Examples

A diverse set of illustrative cases is evaluated:

• Free vector multiplet: The VOA is that of symplectic fermions. The construction leads directly to $U(1)_r$9 for the Schur index and $SU(2)_R$0 for the modified character, in agreement with the respective limits of the Macdonald index.
• Free hypermultiplet: Here the VOA is the symplectic boson algebra. The prescription yields $SU(2)_R$1 (Schur) and $SU(2)_R$2 (modified), matching the corresponding specializations.
• $SU(2)_R$3 Argyres-Douglas Theories: These correspond to $SU(2)_R$4 Virasoro minimal models. The computation matches known results for the Macdonald index via fermionic sum representations and demonstrates that the prescription remains valid even for minimal models outside the SCFT/VOA window.
• $SU(2)_R$5 Argyres-Douglas Theories: The relevant VOA is the $SU(2)_R$6 minimal model. The modified character computed matches known limits of the Macdonald index for a broad range of parameters, matching results from the Macdonald polynomial techniques.
• $SU(2)_R$7 Theories: The VOA is the Kac-Moody algebra $SU(2)_R$8 at fractional negative level. The prescribed index agrees with explicit Macdonald index calculations for various $SU(2)_R$9.
• Interacting theories from gauged diagonal flavor symmetries (e.g., $\mathcal{I}_R(q)$0): These cases respect the prescription, with the Gram matrix analysis matching detailed Macdonald index expansions.

In all cases, the model-independent component is the role of the anti-linear automorphism and the graded sign, which encode the $\mathcal{I}_R(q)$1 information lost under the topological twist.

Generalization to Surface Defects

The method is extended to modules corresponding to surface defect insertions, with a detailed discussion for Virasoro minimal models. In this context, the anti-linear automorphism acts on the state space, and the Hermitian structure enables computation of defect Macdonald indices, confirming graded unitarity persists in the presence of defects. Explicit calculations for surface modules demonstrate systematic agreement with defect Macdonald indices computed by combinatorial and representation-theoretic methods.

Structural and Theoretical Implications

This work establishes the existence of a new, intrinsic, "modified character" for arbitrary VOAs naturally associated with the graded Hermitian structure induced from the UV SCFT. This object is not simply a reweighting of the VOA character, but encodes refined quantum number dependence absent in the conventional character.

From the perspective of representation theory, it suggests that the spectrum of any unitary-induced VOA admits a signed decomposition governed by the $\mathcal{I}_R(q)$2 grading, which robustly persists across various types of unitary 4d parent theories and their topological-defect generalizations.

The analysis is suggestive of rich interplay between the modular, automorphic, and spectral properties of characters and their modifications under these graded prescriptions. Potential avenues include the exploration of modular invariance or modular differential equations satisfied by the modified character—an aspect that is fundamental for classification results in VOA theory [Beem–Rastelli, (Beem et al., 2017)].

Conclusion

This work introduces a universal, mathematically rigorous mechanism for extracting a special non-Schur limit of the Macdonald index directly from VOA data, grounded in a new insight into graded unitarity and Hermitian structures on VOAs. The framework is algorithmically effective, as demonstrated by explicit examples ranging from free fields to strongly-coupled Argyres-Douglas theories. Its applicability to modules encoding surface defects highlights the naturalness of the construction. Systematic study of the modified character, its modular properties, and its implications for the classification and structure of VOAs arising from 4d SCFTs constitutes a promising direction for future research.

Reference: "Macdonald Index from VOA and Graded Unitarity" (2603.29829)