Categorical Tinkertoys for N=2 Gauge Theories

TL;DR

This work develops a categorical framework to classify 4d ${\cal N}=2$ quiver gauge theories with gauge group $G$ by demanding a Ringel-type property: the abelian category ${\mathsf rep}(Q,\mathcal{W})$ contains a universal, perturbative subcategory capturing the gauge sector and a family of light subcategories $\mathscr{L}_{\lambda}$ indexed by a parameter space $N$. It connects the physics of light (mutually local vector multiplets and half-hypermultiplets) and heavy (dyon) states to representation-theoretic structures, notably Ringel's canonical algebras for $SU(2)$ and their generalizations to $G$-canonical algebras. The paper builds a two-pronged theory: (i) a gauge-functor-based canonical construction yielding $G$-tubes and their matter sectors, and (ii) a robust 2d/4d correspondence linking the Coxeter element and Stokes data to the BPS spectrum, ensuring consistency with known dualities and UV behavior. It also treats non-gentle and non-quiver theories, non-Lagrangian fixtures, and tropical geometric interpretations (Gaiotto degenerations) as part of a broad program toward categorical tinkertoys for ${\cal N}=2$ theories. The results unify the perturbative universality of the gauge sector with a controlled, modular approach to matter, spectra, and dualities, enabling explicit analyses of BPS states in wide classes of theories, and revealing deep connections to preprojective and gentle algebras, as well as to the 2d/4d correspondence.

Abstract

In view of classification of the quiver 4d N=2 supersymmetric gauge theories, we discuss the characterization of the quivers with superpotential (Q,W) associated to a N=2 QFT which, in some corner of its parameter space, looks like a gauge theory with gauge group G. The basic idea is that the Abelian category rep(Q,W) of (finite-dimensional) representations of the Jacobian algebra $\mathbb{C} Q/(\partial W)$ should enjoy what we call the Ringel property of type G; in particular, rep(Q,W) should contain a universal `generic' subcategory, which depends only on the gauge group G, capturing the universality of the gauge sector. There is a family of 'light' subcategories $\mathscr{L}_λ\subset rep(Q,W)$, indexed by points $λ\in N$, where $N$ is a projective variety whose irreducible components are copies of $\mathbb{P}^1$ in one--to--one correspondence with the simple factors of G. In particular, for a Gaiotto theory there is one such family of subcategories, $\mathscr{L}_{λ\in N}$, for each maximal degeneration of the corresponding surface $Σ$, and the index variety $N$ may be identified with the degenerate Gaiotto surface itself: generic light subcategories correspond to cylinders, while closed-point subcategories to `fixtures' (spheres with three punctures of various kinds) and higher-order generalizations. The rules for `gluing' categories are more general that the geometric gluing of surfaces, allowing for a few additional exceptional N=2 theories which are not of the Gaiotto class.

Figures (10)

• Figure 1: $\mathbb{P}^1$ components in Kodaira's singular fiber of type IV.
• Figure 2: The ideal triangulation of the sphere with three boundary components corresponding to the 'prism' quiver \ref{['prisms']}. The arcs of the triangulation are the purple curves. The WKB geodesics corresponding to the three light stable $SU(2)$$W$--bosons are draw in red. The corresponding bands $C$ are given by the cyclically ordered list of arcs crossed by each WKB geodesic.
• Figure 3: The strings of two bricks with dimension $\mathfrak{d}$. All others are obtained from these two by acting with the ${\mathbb Z}_3$ automorphism of the prism quiver $Q$.
• Figure 4: Two inequivalent degeneration of the $g=3$, $b=2$ surface: the surface is decomposed into $3$--punctured spheres (blue), punctured disks (green) and infinitely thin cylinders (red). Each such maximal degeneration defines a weakly coupled regime.
• Figure 5: The quiver $\mathbb{A}_4$.