Testing Macdonald index as a refined character of chiral algebra

Abstract

We test in (A n−1 , A m−1 ) Argyres-Douglas theories with gcd(n, m) = 1 the proposal of Song's in [1] that the Macdonald index gives a refined character of the dual chiral algebra. In particular, we extend the analysis to higher rank theories and Macdonald indices with surface operator, via the TQFT picture and Gaiotto-Rastelli-Razamat's Higgsing method. We establish the prescription for refined characters in higher rank minimal models from the dual (A n−1 , A m−1 ) theories in the large m limit, and then provide evidence for Song's proposal to hold (at least) in some simple modules (including the vacuum module) at finite m. We also discuss some observed mismatch in our approach.

Infinitely many 4d $\mathcal{N}=2$ SCFTs with $a=c$ and beyond

“…As we have seen already, it is possible to compute the Schur index of Γ(G) theory if it does not have any flavor symmetry. A useful way of computing the Schur index is to use 4d/2d correspondence that maps the index of class S theory to the correlation function of 2d topological field theory [74][75][76], which has been further extended to the Argyres-Douglas theories [31,33,34,109,115]. For some cases, the Macdonald index, which is slightly more refined then the Schur index, can be computed in this way.…”

Superconformal surfaces in four dimensions

“…In [42,43,109] the index was computed via a conjectural formula in terms of the 2d-4d BPS spectrum in the Coulomb branch of the theory, being thus applicable to non-Lagrangian theories as well. Finally, for vortex defects a prescrisption to compute index was given in [37] and used in a variety of different theories [42,48,[110][111][112]. Since some of the examples we will consider below are precisely vortex defects we will give a brief summary of how they are obtained.…”

“…This construction motivated the prescription of [37], whereby the Schur index of vortex defects in T is computed from that of T U V by taking a certain residue in the fugacity associated to the u(1) f , and stripping off the index of the decoupled free hypermultiplets that arise in the infrared. This prescription has been used to compute the Schur index of different vortex defects in [42,48,112], which were then matched to the characters of non-vacuum modules of χ(T ). Note that a particular theory T can be embedded in different T U V theories and the vortex defects created from different ultraviolet theories can be distinct.…”

“…Recall that this refinement involves refining the index by a Cartan that is not preserved by the chiral algebra, and thus its interpretation in chiral algebra is not clear. In [112] the conjectured prescription of [128] was used to attempt to recover the Macdonald index from the chiral algebra, but the authors found disagreements with the expressions for the superconformal indices in some examples.…”

Deformation quantizations from vertex operator algebras

“…Then the supercurrents are effectively removed, and similarly the canonical stress-energy tensor ultimately decouples. 31 The VOA's character reads…”