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Symmetry Mitosis and Hasse Diagram Diamonds: A Note on Brane Configurations with $\mathrm{ON}^{0}$ Planes

Sam Bennett, Amihay Hanany, Guhesh Kumaran, Lorenzo Mansi

TL;DR

Symmetry mitosis reveals a doubling mechanism for Coulomb-branch global symmetries in 3d $\mathcal{N}=4$ unitary-orthosymplectic quivers arising from brane systems with $\mathrm{ON}^0$ planes. The authors develop a framework of mitotic magnetic quivers and demonstrate it across 6d SQFTs and little string theories, producing explicit examples with $\mathrm{O}(8)\times\mathrm{O}(8)$, $\mathrm{O}(9)\times\mathrm{O}(9)$, and $E_6\times E_6$ mitoses, including full Higgs-branch Hasse diagrams for minimal $(E_6,E_6)$ conformal matter. They also generalize to LSTs and heterotic Spin$(32)$ instanton systems, showing that doubled global-symmetry blocks arise from common subquiver structures and persist under various dualities. The work connects magnetic-quiver subtraction, F-theory curve configurations, and Hilbert-series computations to uncover discrete symmetry enhancements and refined global forms on moduli spaces, with concrete brane realizations in Type IIA/IIB and Type I$'$ setups.

Abstract

This letter considers 3d $\mathcal{N}=4$ (unitary-)orthosymplectic quiver gauge theories originating from Type IIA and Type IIB brane systems with $\mathrm{ON}^0$ planes. Such theories lie outside the scope of present combinatorial techniques for Coulomb branch symmetry and symplectic stratification. It turns out that the correct prescription involves `symmetry mitosis': a common subset of nodes in two linear balanced chains source \emph{two} factors of a Coulomb branch global symmetry instead of one; the correct Coulomb branch Hasse diagram is obtained by a `doubling' procedure on that computed by naive quiver subtraction. Input from 6d SQFTs and little string theories allows for the construction of various `mitotic' magnetic quivers. The full Higgs branch Hasse diagram of minimal $(E_6,E_6)$ conformal matter is given. Additionally, a new Type I$'$ brane system using eight full D8 branes, negatively charged D6 branes, and $\mathrm{ON}^0$ planes is found corresponding to a product of $\mathrm{Spin}(32)$ instantons on $\mathbb C^2$. The corresponding 6d theory uses $\mathrm{Sp}(-1)$ gauge nodes which have the interpretation of bi-spinor matter of $\mathrm{O}(a)$ and $\mathrm{O}(12-a)$ for $a=0,1,\cdots,12$.

Symmetry Mitosis and Hasse Diagram Diamonds: A Note on Brane Configurations with $\mathrm{ON}^{0}$ Planes

TL;DR

Symmetry mitosis reveals a doubling mechanism for Coulomb-branch global symmetries in 3d unitary-orthosymplectic quivers arising from brane systems with planes. The authors develop a framework of mitotic magnetic quivers and demonstrate it across 6d SQFTs and little string theories, producing explicit examples with , , and mitoses, including full Higgs-branch Hasse diagrams for minimal conformal matter. They also generalize to LSTs and heterotic Spin instanton systems, showing that doubled global-symmetry blocks arise from common subquiver structures and persist under various dualities. The work connects magnetic-quiver subtraction, F-theory curve configurations, and Hilbert-series computations to uncover discrete symmetry enhancements and refined global forms on moduli spaces, with concrete brane realizations in Type IIA/IIB and Type I setups.

Abstract

This letter considers 3d (unitary-)orthosymplectic quiver gauge theories originating from Type IIA and Type IIB brane systems with planes. Such theories lie outside the scope of present combinatorial techniques for Coulomb branch symmetry and symplectic stratification. It turns out that the correct prescription involves `symmetry mitosis': a common subset of nodes in two linear balanced chains source \emph{two} factors of a Coulomb branch global symmetry instead of one; the correct Coulomb branch Hasse diagram is obtained by a `doubling' procedure on that computed by naive quiver subtraction. Input from 6d SQFTs and little string theories allows for the construction of various `mitotic' magnetic quivers. The full Higgs branch Hasse diagram of minimal conformal matter is given. Additionally, a new Type I brane system using eight full D8 branes, negatively charged D6 branes, and planes is found corresponding to a product of instantons on . The corresponding 6d theory uses gauge nodes which have the interpretation of bi-spinor matter of and for .
Paper Structure (10 sections, 45 equations, 2 figures)

This paper contains 10 sections, 45 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Symmetry Mitosis in Unitary Theories
  3. Symmetry Mitosis and 6d SQFTs
  4. $\mathrm{OI}_{N,k}$: An $\mathrm{O}(8)\times \mathrm{O}(8)$ Mitosis
  5. $\mathrm{OI}_{\mathrm{O}(9)}$: An $\mathrm{O}(9)\times \mathrm{O}(9)$ Mitosis
  6. $\mathrm{OI}_{E_6}$: An $E_6\times E_6$ Mitosis
  7. Symmetry Mitosis From 6d Little String Theories
  8. $\mathrm{LST}_{N,k}$: A Double $\mathrm{O}(8)\times \mathrm{O}(8)$ Mitosis
  9. Two Heterotic $\mathrm{Spin}(32)$ Instantons
  10. Higgs branch Hasse Diagrams and Mitosis

Figures (2)

  • Figure 1: A portion of the Hasse Diagram for the magnetic quiver $\mathcal{OI}^{1, 0}_{\ref{['quiver:magneticOINK']}}$ based on the Decay and Fission algorithm. Next to each magnetic quiver is its corresponding electric 6d theory, which has been written in the F-theory curve notation. Notice that Figure \ref{['fig:E6E6HasseClean']} should also occur in Figure \ref{['fig:HasseOIso8']} as a sub-diagram if the rest of the Higgsing directions were filled in.
  • Figure 2: