Macdonald index and chiral algebra

Jaewon Song

doi:10.1007/jhep08(2017)044

Abstract

For any 4d mathcal{N} = 2 SCFT, there is a subsector described by a 2d chiral algebra. The vacuum character of the chiral algebra reproduces the Schur index of the corresponding 4d theory. The Macdonald index counts the same set of operators as the Schur index, but the former has one more fugacity than the latter. We conjecture a prescription to obtain the Macdonald index from the chiral algebra. The vacuum module admits a filtration, from which we construct an associated graded vector space. From this grading, we conjecture a notion of refined character for the vacuum module of a chiral algebra, which reproduces the Macdonald index. We test this prescription for the Argyres-Douglas theories of type (A1, A2n) and (A1, D2n+1) where the chiral algebras are given by Virasoro and widehat{mathfrak{su}}(2) affine Kac-Moody algebra. When the chiral algebra has more than one family of generators, our prescription requires a knowledge of the generators from the 4d.

Highlights

JHEP08(2017)044 associated to the 4d N = 2 SCFT, one can compute the Schur index for non-Lagrangian
We conjecture a notion of refined character for the vacuum module of a chiral algebra, which reproduces the Macdonald index
We test this prescription for the Argyres-Douglas theories of type (A1, A2n) and (A1, D2n+1) where the chiral algebras are given by Virasoro and su(2) affine Kac-Moody algebra

Summary

Lagrangian theories

We discuss the superconformal theories constructed from gauge theory. Which is exactly the same as the Macdonald index of a vector multiplet Up to this point the weight w(X) for the generator X seems to coincide with its scaling dimension. The chiral algebra of the Lagrangian gauge theory at the zero coupling is given by the tensor product of the chiral algebras for the free vectors and hypermultiplets under the Gauss’ law constraint. The refined character can be obtained by taking the tensor product of that of the free theories and integrating over the gauge group with Haar measure. This is exactly the same procedure to obtain the superconformal index for a Lagrangian theory.

Conclusions and open problems

A Macdonald index of the Argyres-Douglas theory