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), {mathcal{T}}_{3,frac{3}{2}} , emerging in this duality splits into a free piece and an interacting piece, {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 {mathcal{T}}_X by bootstrapping its chiral algebra, {}_{mathcal{X}}left({mathcal{T}}_Xright) , 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 {mathcal{T}}_X and, by studying this quantity in the limit of small S1, we make contact with a proposed S1 reduction. Along the way, we discuss various properties of {mathcal{T}}_X : as an mathcal{N} = 1 theory, it has flavor symmetry SU(3) × SU(2) × U(1), the central charge of {}_{mathcal{X}}left({mathcal{T}}_Xright) 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 {mathcal{T}}_X theories (giving us a surprisingly close AD relative of Gaiotto’s TN theories), but it does lead to some open questions in the context of the chiral algebra/4D mathcal{N} =2SCFT correspondence.
Highlights
As one takes the gauge coupling to infinity, Argyres and Seiberg found that, instead of getting a weakly coupled S-dual description in terms of another su(3) gauge theory with fundamental matter, one instead finds a dual consisting of an su(2) theory coupled to a doublet of hypermultiplets and an su(2) ⊂ e6 factor of the global symmetry of the Minahan-Nemeschansky E6 superconformal field theory (SCFT) [4]
), we see that there must be at least one operator, OaI, transforming in the 2 × 8 representation of the global symmetry
One strong piece of evidence in favor of our conjecture in the previous section is that there exists a set of operator product expansions (OPEs) among the generators described there that is consistent with Jacobi identities of the type described in (2.8)
Summary
In order to get more detailed information about the TX theory, we compute its Schur index using the S-duality described in figure 1 and figure 2. Where the measure of integration is the SU(3) Haar measure, Iflavors is the index of the three fundamental flavors, I(A1,D4) is the index of the (A1, D4) theory, and Ivect is the vector multiplet index (see appendix B for detailed expressions). We use this expansion to conjecture the generators of the associated chiral algebra, χ(TX ). We bootstrap this chiral algebra and show that it is consistent (in the sense that it obeys Jacobi identities of the form reviewed in (2.8)). We will argue that it is the unique such chiral algebra with the generators we conjecture and the anomalies required from the discussion in the introduction and section 2.27
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