Identify the BRST cohomology with the doublet algebra

Prove that the BRST cohomology of the diagonal affine \(\mathfrak{sl}_2\) reduction of \(L_{k_1}(\mathfrak{sl}_2)\otimes W_{k_{so}}(\mathfrak{so}_{2n},f_{[2n-3,1^3]})\otimes bc_{\mathfrak{sl}_2}\) is isomorphic to the logarithmic doublet algebra \(A(4n-2)\) for \(n>2\), including the identification of its strong generators and operator products.

Background

The paper proves an all-orders equality between the Schur index of the (A2,D3n−2)(A_2,D_{3n-2}) Argyres–Douglas theory and the vacuum supercharacter of A(4n−2)A(4n-2), and it establishes only a graded-vector-space equivalence. It does not construct an operator-product-preserving map. Conformal gauging nevertheless determines a candidate diagonal sl2\mathfrak{sl}_2 BRST complex whose cohomology is proposed to realize the VOA.

For n>2n>2, the unresolved issue is to determine the BRST cohomology itself, establish whether it is concentrated in the expected degree, and identify its strong generators and normal-ordered products with those of A(4n−2)A(4n-2).

References

For n>2, equation eq:BRST-Ap is a proposed identification rather than a proven isomorphism. The conformal gauging construction canonically determines the BRST complex, and the level and central charge checks above show that it has the required consistency properties. The all orders supercharacter identity eq:all-orders-index-character supplies a further necessary check. It does not, however, prove that the cohomology is concentrated in the expected degree or identify the resulting operator products. A proof of eq:BRST-Ap would still require, for example, an identification of strong generators and their OPEs, or a direct computation of the BRST cohomology.

— SCFT/VOA correspondence and R-twisted reductions of $(A_2,D_{3n-2})$ Argyres--Douglas theories  (2609.04099 - Yoshida, 3 Sep 2026) in Section 2.5, subsection “\(A(4n-2)\) as a BRST cohomology”; Section 7, Summary and discussion