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Instanton counting with a surface operator and the chain-saw quiver

Hiroaki Kanno, Yuji Tachikawa

TL;DR

This work advances the understanding of instanton counting in N=2 SU(N) gauge theories by incorporating general surface operators. It shows that the moduli space with a surface operator can be described by the chain-saw quiver and that the instanton partition function is assembled from fixed-point contributions labeled by Young diagrams, with the counting depending on the composition (n_I) of N. The authors connect these results to the W-algebra, demonstrating that the corresponding Verma module structure and Whittaker vector norms reproduce the partition function, and they provide explicit checks for partitions of the form [2^μ1^ν]. The framework unifies higher-dimensional gauge theory, quiver representations, and algebraic structures, offering a concrete computational tool and paving the way for extensions to broader surface operators and matter content.

Abstract

We describe the moduli space of SU(N) instantons in the presence of a general surface operator of type N=n_1+ ... +n_M in terms of the representations of the so-called chain-saw quiver, which allows us to write down the instanton partition function as a summation over the fixed point contributions labeled by Young diagrams. We find that the instanton partition function depends on the ordering of n_I which fixes a choice of the parabolic structure. This is in accord with the fact that the Verma module of the W-algebra also depends on the ordering of n_I. By explicit calculations, we check that the partition function agrees with the norm of a coherent state in the corresponding Verma module.

Instanton counting with a surface operator and the chain-saw quiver

TL;DR

This work advances the understanding of instanton counting in N=2 SU(N) gauge theories by incorporating general surface operators. It shows that the moduli space with a surface operator can be described by the chain-saw quiver and that the instanton partition function is assembled from fixed-point contributions labeled by Young diagrams, with the counting depending on the composition (n_I) of N. The authors connect these results to the W-algebra, demonstrating that the corresponding Verma module structure and Whittaker vector norms reproduce the partition function, and they provide explicit checks for partitions of the form [2^μ1^ν]. The framework unifies higher-dimensional gauge theory, quiver representations, and algebraic structures, offering a concrete computational tool and paving the way for extensions to broader surface operators and matter content.

Abstract

We describe the moduli space of SU(N) instantons in the presence of a general surface operator of type N=n_1+ ... +n_M in terms of the representations of the so-called chain-saw quiver, which allows us to write down the instanton partition function as a summation over the fixed point contributions labeled by Young diagrams. We find that the instanton partition function depends on the ordering of n_I which fixes a choice of the parabolic structure. This is in accord with the fact that the Verma module of the W-algebra also depends on the ordering of n_I. By explicit calculations, we check that the partition function agrees with the norm of a coherent state in the corresponding Verma module.

Paper Structure

This paper contains 14 sections, 87 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Instanton moduli space with a surface operator
  3. Defining data of a surface operator
  4. Instantons with a surface operator as orbifolded instantons
  5. Orbifolded instantons and the chain-saw quiver
  6. Instanton counting with a surface operator
  7. Before the orbifolding
  8. After the orbifolding
  9. Comparison with the W-algebra
  10. Structure of the W-algebra
  11. Instantons and states in the Verma module
  12. Partition function and the Whittaker vector
  13. Conclusions
  14. Details of the Drinfeld-Sokolov reduction

Figures (2)

  • Figure 1: A part of $\mathbb{Z}_M$ chain-saw quiver
  • Figure 2: A representation of the chain saw quiver for $(n_I)=(1,1)$, $\lambda^1 = (4,3,2), \lambda^2 = (2,2)$ with $d_1 = 8$ and $d_2=5$. ${\bf e}_i$ forms the basis of $V_1$, ${\bf f}_i$ forms the basis of $V_2$, and ${\bf w}_{1,2}$ is the basis of $W_{1,2}$.