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The Superconformal Index of the (2,0) Theory with Defects

Mathew Bullimore, Hee-Cheol Kim

TL;DR

This work computes the six-dimensional $\mathcal{N}=(2,0)$ theory on $S^1\times S^5$ in the presence of codimension-2 and -4 defects by localizing the corresponding 5d $\mathcal{N}=2$ SYM partition function on $S^5$, and specializes to a chiral-algebra limit where the index matches 2d chiral algebras. By organizing contributions around the three fixed circles of the squashed $S^5$, the authors derive exact forms for perturbative and instanton sectors, including Ramified (orbifold) instantons for codimension-2 defects. They demonstrate that defect configurations map precisely to irreducible modules of the $W_N$-algebra and of the affine $\widehat{\mathfrak{su}}(N)$ algebra at certain levels, with vacuum, degenerate, and semi-degenerate modules realized by specific defect insertions. The results provide a unified 6d origin for a broad class of chiral-algebra representations and illuminate the role of defects as chiral vertex operators in the associated $W_N$ and affine algebras, with potential applications to class S and 4d/3d correspondences. The analysis also outlines modular features and avenues for extending to generic punctures $\rho$ and more general defect configurations.

Abstract

We compute the supersymmetric partition function of the six-dimensional $(2,0)$ theory of type $A_{N-1}$ on $S^1 \times S^5$ in the presence of both codimension two and codimension four defects. We concentrate on a limit of the partition function depending on a single parameter. From the allowed supersymmetric configurations of defects we find a precise match with the characters of irreducible modules of $W_N$ algebras and affine Lie algebras of type $A_{N-1}$.

The Superconformal Index of the (2,0) Theory with Defects

TL;DR

This work computes the six-dimensional theory on in the presence of codimension-2 and -4 defects by localizing the corresponding 5d SYM partition function on , and specializes to a chiral-algebra limit where the index matches 2d chiral algebras. By organizing contributions around the three fixed circles of the squashed , the authors derive exact forms for perturbative and instanton sectors, including Ramified (orbifold) instantons for codimension-2 defects. They demonstrate that defect configurations map precisely to irreducible modules of the -algebra and of the affine algebra at certain levels, with vacuum, degenerate, and semi-degenerate modules realized by specific defect insertions. The results provide a unified 6d origin for a broad class of chiral-algebra representations and illuminate the role of defects as chiral vertex operators in the associated and affine algebras, with potential applications to class S and 4d/3d correspondences. The analysis also outlines modular features and avenues for extending to generic punctures and more general defect configurations.

Abstract

We compute the supersymmetric partition function of the six-dimensional theory of type on in the presence of both codimension two and codimension four defects. We concentrate on a limit of the partition function depending on a single parameter. From the allowed supersymmetric configurations of defects we find a precise match with the characters of irreducible modules of algebras and affine Lie algebras of type .

Paper Structure

This paper contains 20 sections, 131 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Computational Method
  3. The Superconformal Index
  4. $S^5$ Partition Function
  5. Computation of $S^5$ Partition Function
  6. Codimension 4 Defects
  7. Codimension 2 Defects
  8. $W_N$ - Algebra Characters
  9. Vacuum Module
  10. Degenerate Modules
  11. Semi-Degenerate Modules
  12. Affine Characters
  13. Vacuum Module
  14. Highest Weight Modules
  15. Discussion
  16. ...and 5 more sections

Figures (5)

  • Figure 1: A summary of the configurations of defects referred to in the introduction. Codimension 2 defects are shown in blue and codimension 4 defects are shown in red.
  • Figure 2: Linear quiver
  • Figure 3: Chain-saw quiver
  • Figure 4: A summary of the configurations of a codimension 2 and codimension 4 defects that in the chiral algebra limit reproduce characters of irreducible modules of the $W_N$ - algebra.
  • Figure 5: A summary of the configurations of a codimension 2 and codimension 4 defects that in the chiral algebra limit reproduce characters highest weight modules of the affine $\widehat{\mathfrak{su}}(N)$ at level $k = - N - b^{\pm2}$.