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5d Higgs branch and instanton magnetization

Amihay Hanany, Alessandro Tomasiello, Elias Van den Driessche

Abstract

Higgs branches of 5d $Sp(k)$ theories with $N_f$ flavours, whether at weak or strong coupling, are described by a pair of instantons transforming as pure spinors of $SO(2N_f)$. The Poisson structure is constrained by symmetry arguments and implies that these Higgs branches are algebraic integrable systems; the degeneration of the symplectic form occurs when the spinor annihilators overlap. We argue that the stratification of the Higgs branch at infinite coupling corresponds to the alignment of the instantons weights, leading to a non vanishing magnetization, and their acquisition of a mass.

5d Higgs branch and instanton magnetization

Abstract

Higgs branches of 5d theories with flavours, whether at weak or strong coupling, are described by a pair of instantons transforming as pure spinors of . The Poisson structure is constrained by symmetry arguments and implies that these Higgs branches are algebraic integrable systems; the degeneration of the symplectic form occurs when the spinor annihilators overlap. We argue that the stratification of the Higgs branch at infinite coupling corresponds to the alignment of the instantons weights, leading to a non vanishing magnetization, and their acquisition of a mass.

Paper Structure

This paper contains 33 sections, 1 theorem, 66 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. 5d Higgs branches
  3. Massless BPS spectrum.
  4. Stratification.
  5. Pure spinors
  6. Definitions
  7. Orbit.
  8. Pure spinor pairs
  9. Bispinors and forms.
  10. Type.
  11. Space of pairs.
  12. Hodge duality.
  13. Chiral ring origin
  14. Zero coupling.
  15. Infinite coupling: relations.
  16. ...and 18 more sections

Key Result

Theorem 4.1

Let $M$ be an $m$-dimensional Poisson manifold with Poisson structure $\psi$ of constant rank $2g$. Suppose that is a proper submersive Hamiltonian map of maximal rank, i.e., Then: $\blacktriangleleft$$\blacktriangleleft$

Figures (6)

  • Figure 1: General form of a linear Hasse diagram.
  • Figure 2: Weak-coupling Hasse diagram for $N_f\leq 2k$ and $N_f$ even (left) or odd (right), respectively. The order of the slice changes by two units as we go up the diagram: from $d_{N_f}$ to $d_{N_f-2}$ and so forth. Furthermore, $d_2$ can be considered as the product $a_1\times a_1$, hence showing that the Higgs branch is the union of two cones.
  • Figure 3: Weak-coupling Hasse diagram for $2k+1\leq N_f\leq 2k+3$.
  • Figure 4: Infinite-coupling Hasse diagram for $N_f\leq 2k-1$, for $N_f$ even (left) and odd (right). The choice of starting with $[-^{N_f}]$ or $[+^{N_f}]$ is inconsequential. All leaves, except the top one, have $S=0=I=\tilde{I}$. For $N_f=2k$, the two slices $a_1$ and $A_{k-N_f/2+1}$ degenerate to a single $a_2$, as described in hanany2025higgs.
  • Figure 5: Infinite-coupling Hasse diagram for $2k\leq N_f\leq 2k+3$. The choice of starting from $[+^{N_f}]$ instead of $[-^{N_f}]$ is inconsequential. All leaves, except the top one, have $S=0=I=\tilde{I}$.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Theorem 4.1: Liouville--Arnold
  • Definition 4.1: Algebraically Completely Integrable Hamiltonian System