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Giant Gravitons, Fermionic Forms and Vertex Algebras

Nick Dorey, Paul Luis Roehl

Abstract

We investigate the mathematical and physical content of the giant graviton expansion of three-dimensional $\mathcal{N}=4$ superconformal field theories in a simplifying limit. We uncover an interesting relation between the coefficients in this expansion, the Hilbert series of certain quiver varieties and the representation theory of vertex algebras. In particular, for the worldvolume theory of $N$ M2-branes at the tip of a toric hyper-Kähler four-fold cone: $X_{4}=\mathbb{C}^2 /{\mathbb{Z}_L} \times \mathbb{C}^2/{\mathbb{Z}_K}$, we derive an explicit expression for the coefficients in terms of affine fermionic forms and show that they coincide with characters of a direct sum of parafermionic W-algebras.

Giant Gravitons, Fermionic Forms and Vertex Algebras

Abstract

We investigate the mathematical and physical content of the giant graviton expansion of three-dimensional superconformal field theories in a simplifying limit. We uncover an interesting relation between the coefficients in this expansion, the Hilbert series of certain quiver varieties and the representation theory of vertex algebras. In particular, for the worldvolume theory of M2-branes at the tip of a toric hyper-Kähler four-fold cone: , we derive an explicit expression for the coefficients in terms of affine fermionic forms and show that they coincide with characters of a direct sum of parafermionic W-algebras.

Paper Structure

This paper contains 30 sections, 268 equations, 3 figures.

Table of Contents

  1. Introduction
  2. ADHM quiver
  3. Higgs and Coulomb branch limit
  4. Giant graviton expansion
  5. Relation to magnetic background charge
  6. Coefficients $Z_{m_1, m_2}$ with two non-zero wrapping numbers
  7. Twisted limit of 6d index
  8. U(N)^ L gauge theory with $K$ flavours
  9. Higgs and Coulomb branch limit
  10. Giant graviton expansion
  11. $K = 1$, $L$ arbitrary:
  12. $K$ arbitrary, $L = 1$:
  13. $K$ and $L$ arbitrary:
  14. Determination of giant graviton coefficients
  15. Fermionic Forms and Vertex Algebras
  16. ...and 15 more sections

Figures (3)

  • Figure 1: The quiver diagram for the gauge theory $\mathcal{T}_N[K,L]$. Circular nodes correspond to $U(N)$ gauge groups. The square node is a $U(K)$ flavour group. Lines between nodes correspond to bi-fundamental hypermultiplets.
  • Figure 2: The quiver diagram for the class of gauge theories generalising $\mathcal{T}_N[K,L]$. There are $K$ total flavours, distributed onto the $L$ nodes as $\sum_{A = 0}^{L-1} K_A = K$.
  • Figure 3: The quiver diagram of the mirror theory to $\mathcal{T} = \mathcal{T}_N[K,L]$, which is the theory $\mathcal{T}' = \mathcal{T}_N[L,K]$ with $K$ nodes and $L$ total flavours. The flavours are distributed over the nodes, satisfying $\sum_{\alpha = 0}^{K-1} L_{\alpha} = L$. The precise relation between the $L_{\alpha}$ and $K_A$ was mentioned in the introduction and can be found in deBoer:1996mp.