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4d N=2 Gauge Theories and Quivers: the Non-Simply Laced Case

Sergio Cecotti, Michele Del Zotto

TL;DR

The paper addresses constructing BPS quivers with superpotentials for 4d ${\cal N}=2$ theories with non-simply laced groups ${B_n},{C_n},{F_4},{G_2}$. It develops a geometric and categorical framework based on twisted (monodromic) fibrations and folding, introducing a specialization mechanism to incorporate matter representations and to generate new quiver theories. A central outcome is explicit quivers ${Q(\mathfrak{g})}$ with superpotentials ${\cal W(\mathfrak{g})}$ for BCFOG SYM, together with a detailed description of the light category $\mathscr{L}(\mathfrak{g})$, its monodromy, and how auxiliary fields render the construction tractable. The work also presents a versatile set of tools—monodromy, preprojective algebras for valued graphs, and specialization—that enable systematic generation of quivers for non-simply laced theories and their coupling to matter, with broader implications for geometric engineering and the BPS spectrum.

Abstract

We construct the BPS quivers with superpotential for the 4d N=2 gauge theories with non-simply laced Lie groups (B_n, C_n, F_4 and G_2). The construction is inspired by the BIKMSV geometric engineering of these gauge groups as non-split singular elliptic fibrations. From the categorical viewpoint of arXiv:1203.6743, the fibration of the light category L(g) over the (degenerate) Gaiotto curve has a monodromy given by the action of the outer automorphism of the corresponding unfolded Lie algebra. In view of the Katz--Vafa `matter from geometry' mechanism, the monodromic idea may be extended to the construction of (Q, W) for SYM coupled to higher matter representations. This is done through a construction we call specialization.

4d N=2 Gauge Theories and Quivers: the Non-Simply Laced Case

TL;DR

The paper addresses constructing BPS quivers with superpotentials for 4d theories with non-simply laced groups . It develops a geometric and categorical framework based on twisted (monodromic) fibrations and folding, introducing a specialization mechanism to incorporate matter representations and to generate new quiver theories. A central outcome is explicit quivers with superpotentials for BCFOG SYM, together with a detailed description of the light category , its monodromy, and how auxiliary fields render the construction tractable. The work also presents a versatile set of tools—monodromy, preprojective algebras for valued graphs, and specialization—that enable systematic generation of quivers for non-simply laced theories and their coupling to matter, with broader implications for geometric engineering and the BPS spectrum.

Abstract

We construct the BPS quivers with superpotential for the 4d N=2 gauge theories with non-simply laced Lie groups (B_n, C_n, F_4 and G_2). The construction is inspired by the BIKMSV geometric engineering of these gauge groups as non-split singular elliptic fibrations. From the categorical viewpoint of arXiv:1203.6743, the fibration of the light category L(g) over the (degenerate) Gaiotto curve has a monodromy given by the action of the outer automorphism of the corresponding unfolded Lie algebra. In view of the Katz--Vafa `matter from geometry' mechanism, the monodromic idea may be extended to the construction of (Q, W) for SYM coupled to higher matter representations. This is done through a construction we call specialization.

Paper Structure

This paper contains 24 sections, 156 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Geometric preliminaries
  3. $Q$--systems and Dirac integrality conditions
  4. Review of the light category $\mathscr{L}\subset \mathsf{rep}(Q,{\cal W})$
  5. The light category of non--simply laced ${\cal N}=2$ SYM
  6. Monodromic light categories
  7. Preprojective algebras for non--simply laced (i.e. valued) graphs
  8. The structure of $\mathsf{mod}\,{\cal P}(\mathfrak{g})$
  9. Example: $B_2$
  10. Example: $G_2$
  11. Example: $F_4$
  12. Monodromy of $\mathscr{M}(\mathfrak{g})$
  13. 'Auxiliary' fields and the final form of the light category $\mathscr{L}(\mathfrak{g})$
  14. The quiver superpotential ${\cal W}$ for $BCFG$ SYM
  15. Other applications of the monodronic construction
  16. ...and 9 more sections

Figures (2)

  • Figure 1: Dynkin diagrams foldings. The graphs on the left are called the 'parent graph' $G_\text{parent}$ of the graphs $G$ on the right.
  • Figure 2: The quivers of SYM with non--simply laced gauge group. The corresponding superpotentials are given in section \ref{['SEc:wwww']}.