Categorical Tinkertoys for N=2 Gauge Theories

TL;DR

This paper classifies 4d N=2 supersymmetric gauge theories using quivers with superpotentials, linking their representation categories to geometric structures like Gaiotto surfaces, and introduces a universal subcategory capturing gauge sector universality.

Contribution

It introduces the concept of the Ringel property for quiver categories and relates light subcategories to geometric features of Gaiotto surfaces, extending classification beyond known Gaiotto theories.

Findings

Identification of a universal 'generic' subcategory for gauge theories.

Connection between light subcategories and geometric features of Gaiotto surfaces.

Extension of category gluing rules to include exceptional N=2 theories.

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 $\mathscr{L}_\lambda\subset rep(Q,W)$, indexed by points $\lambda\in N$, 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, $\mathscr{L}_{\lambda\in N}$, for each maximal degeneration of the corresponding surface $\Sigma$, 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.