Weight-two coefficient and descendant tower for the extremal two-point function

Compute the scalar \(\alpha_k\) multiplying the Virasoro vector in the weight-two term of the extremal affine \(sl(2)\) two-point operator product, and derive the complete descendant-by-descendant Laurent tower beyond that order.

Background

For even level k, the paper determines the leading Laurent coefficient of the extremal two-point function and proves that the next coefficient vanishes. It shows that the subsequent weight-two coefficient is controlled by the projection of the operator product onto the conformal vector, with an undetermined scalar αk\alpha_k. The paper outlines a finite mode and dual-basis computation but does not carry it out, either generally or at the smallest nontrivial level.

References

What remains open is only a single scalar \alpha_k.

— Modular properties of affine \(\SLA{sl}{2}\) torus \(n\)-point functions  (2609.01496 - Zuevsky, 1 Sep 2026) in Remark following Proposition 4.4, within Section 4.3

We leave both directions, and the corresponding computations, for future work.

— Modular properties of affine \(\SLA{sl}{2}\) torus \(n\)-point functions  (2609.01496 - Zuevsky, 1 Sep 2026) in Remark 4.5 (Remark \ref{rmkgeneraln-outlook})

Whether the new sub-leading data of \cref{secsl2} produces non-congruence representations, as it does already at n=1 for dimension three and above (KSW Theorem 5.5(3), Theorem 5.6), is exactly the kind of question for which the general families of KSW Theorem 5.6 (levels k=pt-2) would be the natural place to look once the sub-leading two-point coefficients of \cref{rmkgeneraln-outlook} are computed.

— Modular properties of affine \(\SLA{sl}{2}\) torus \(n\)-point functions  (2609.01496 - Zuevsky, 1 Sep 2026) in Section 5, paragraph beginning “Concerning congruence”