Direct localization computation of the non-invertible-symmetry-twisted index

Compute the non-invertible-symmetry-twisted Schur index of four-dimensional \(\mathcal{N}=4\) \(SU(N)\) super Yang–Mills theory directly by gauge-theory localization, rather than through the proposed outer-automorphism-twisted vertex-operator-algebra character.

Background

The paper identifies the Schur index with a non-invertible duality-symmetry defect insertion with an outer-automorphism-twisted vacuum character of the associated vertex operator algebra. Because the duality defect exists only at a strongly coupled self-dual point, the authors state that the usual free-field computation does not justify tracking the S-duality action on elementary fields.

A direct localization calculation would provide an independent gauge-theory derivation of the twisted index and would test the proposed SCFT/VOA correspondence without relying on the VOA description. The authors explicitly defer this computation to future work.