Finite-\(N\) equality of the duality and \(U(1)_Y\)-type actions on VOA multiplets

Prove that the phase relating the non-invertible duality-defect action to the \(U(1)_Y\)-type action is trivial on every additional \(\mathbb{W}_p\) multiplet of the finite-rank VOA \(\mathcal{V}(A_{N-1})\), i.e. establish \(c_p=1\) for all relevant \(p\) and finite \(N\).

Background

The authors compare the duality defect Dd\mathcal{D}_d with an auxiliary U(1)YU(1)_Y-type action Y\mathcal{Y}. Their difference acts by a phase cc on each irreducible superconformal multiplet. For the stress-tensor multiplet this phase is fixed to one, but for the additional finite-rank multiplets Wp\mathbb{W}_p, the corresponding phases cpc_p are not determined by the argument.

The large-NN gravity analysis gives cp=1c_p=1 in the untruncated algebra. Extending this equality to finite NN is needed to justify identifying the twisted VOA character with the non-invertible-symmetry-twisted index for all finite-rank theories.

References

For SU(N) with N \geq 3, the multiplet \mathbb{W}p in \mathcal{V}(A{N-1}) comes from 4d other multiplet, say w_p, where c_p (A w_p = c_p w_p) is not fixed for all p. The large N consideration almost answers this. For large N, the algebra is \mathcal{W}_\infty{s,s} and all \mathbb{W}_p multiplets are not truncated. From the gravity dual which we will discuss in Section \ref{sec:largeN} in detail, we know the action S or SL(2,\mathbb{Z}) is lifted to SL(2, \mathbb{R}), whose Cartan is U(1)_Y. Thus, c_p=1 for all p. We conjecture this is preserved for finite N.

— Non-invertible symmetry and vertex operator algebra outer-automorphism  (2608.18926 - Maruyoshi et al., 19 Aug 2026) in Section 2.2, subsection “VOA automorphism”