Surface Defects and Chiral Algebras

TL;DR

<3-5 sentence high-level summary>This work develops a unified framework for understanding 4d $\mathcal{N}=2$ superconformal field theories with $(2,2)$ surface defects through the defect Schur index, showing it equals a chiral algebra module character. It connects 4d defect data to chiral-algebra operations via spectral flow, Drinfeld–Sokolov reduction, and Higgsing, providing two complementary computational routes: infrared 2d-$4d$ BPS state counting and Higgs-branch residue techniques. The authors compute defect indices for the free hypermultiplet and for Argyres–Douglas theories, and demonstrate exact matches to predicted characters of $\beta\gamma$ systems, Virasoro minimal models, and $W^{(2)}$-algebras, including twisted sectors. These results illuminate the deep correspondence between 4d SCFT defects, 2d chiral algebras, and RG flows, with practical implications for classifying and computing protected defect data in strongly coupled theories.

Abstract

We investigate superconformal surface defects in four-dimensional N=2 superconformal theories. Each such defect gives rise to a module of the associated chiral algebra and the surface defect Schur index is the character of this module. Various natural chiral algebra operations such as Drinfeld-Sokolov reduction and spectral flow can be interpreted as constructions involving four-dimensional surface defects. We compute the index of these defects in the free hypermultiplet theory and Argyres-Douglas theories, using both infrared techniques involving BPS states, as well as renormalization group flows onto Higgs branches. In each case we find perfect agreement with the predicted characters.