N $$ \mathcal{N} $$ = 2 S-duality revisited

Abstract Using the chiral algebra bootstrap, we revisit the simplest Argyres-Douglas (AD) generalization of Argyres-Seiberg S-duality. We argue that the exotic AD superconformal field theory (SCFT), T 3 , 3 2 $$ {\mathcal{T}}_{3,\frac{3}{2}} $$ , emerging in this duality splits into a free piece and an interacting piece, T X $$ {\mathcal{T}}_X $$ , even though this factorization seems invisible in the Seiberg-Witten (SW) curve derived from the corresponding M5-brane construction. Without a Lagrangian, an associated topological field theory, a BPS spectrum, or even an SW curve, we nonetheless obtain exact information about T X $$ {\mathcal{T}}_X $$ by bootstrapping its chiral algebra, X T X $$ {}_{\mathcal{X}}\left({\mathcal{T}}_X\right) $$ , and finding the corresponding vacuum character in terms of Affine Kac-Moody characters. By a standard 4D/2D correspondence, this result gives us the Schur index for T X $$ {\mathcal{T}}_X $$ and, by studying this quantity in the limit of small S 1, we make contact with a proposed S 1 reduction. Along the way, we discuss various properties of T X $$ {\mathcal{T}}_X $$ : as an N $$ \mathcal{N} $$ = 1 theory, it has flavor symmetry SU(3) × SU(2) × U(1), the central charge of X T X $$ {}_{\mathcal{X}}\left({\mathcal{T}}_X\right) $$ matches the central charge of the bc ghosts in bosonic string theory, and its global SU(2) symmetry has a Witten anomaly. This anomaly does not prevent us from building conformal manifolds out of arbitrary numbers of T X $$ {\mathcal{T}}_X $$ theories (giving us a surprisingly close AD relative of Gaiotto’s T N theories), but it does lead to some open questions in the context of the chiral algebra/4D N $$ \mathcal{N} $$ =2SCFT correspondence.