Categorical Webs and $S$-duality in 4d $\mathcal{N}=2$ QFT

TL;DR

The work develops a categorical framework for 4d N=2 QFTs in which BPS objects are organized into triangle categories tied together by exact functors, capturing IR and UV viewpoints in a unified structure. Central to the approach is the exact sequence 0→D^bΓ→PerΓ→C(Γ)→0, with D^bΓ describing BPS particles, PerΓ dressing UV line operators, and C(Γ) encoding UV line operators; S-duality emerges as an auto-equivalence of the full web, enabling a computational route to identify dualities. The paper further connects these categories to class S theories via geometric realizations on Gaiotto curves, and shows how cluster characters provide concrete vevs for UV line operators, tying together wall-crossing, dualities, and line-operator physics. The framework yields new insights into UV flavor enhancements, monodromies, and the precise algebraic structure of dualities, while offering practical algorithms to compute S-duality groups and linking the mathematics of cluster algebras to the physics of BPS spectra. Overall, this categorical perspective sharpens the understanding of UV/IR relations in N=2 theories and provides computational tools for exploring dualities and line-operator data across broad classes of models, including class S theories and Argyres–Douglas systems.

Abstract

We review the categorical approach to the BPS sector of a 4d $\mathcal{N}=2$ QFT, clarifying many tricky issues and presenting a few novel results. To a given $\mathcal{N}=2$ QFT one associates several triangle categories: they describe various kinds of BPS objects from different physical viewpoints (e.g. IR versus UV). These diverse categories are related by a web of exact functors expressing physical relations between the various objects/pictures. A basic theme of this review is the emphasis on the full web of categories, rather than on what we can learn from a single description. A second general theme is viewing the cluster category as a sort of `categorification' of 't Hooft's theory of quantum phases for a 4d non-Abelian gauge theory. The $S$-duality group is best described as the auto-equivalences of the full web of categories. This viewpoint leads to a combinatorial algorithm to search for $S$-dualities of the given $\mathcal{N}=2$ theory. If the ranks of the gauge and flavor groups are not too big, the algorithm may be effectively run on a laptop. This viewpoint also leads to a clearer view of $3d$ mirror symmetry. For class $\mathcal{S}$ theories, all the relevant triangle categories may also be constructed in terms of geometric objects on the Gaiotto curve, and we present the dictionary between triangle categories and the WKB approach of GMN. We also review how the VEV's of UV line operators are related to cluster characters.

Figures (19)

• Figure 1: This figure represent the CEG of the $A_3$ cluster algebra. The dotted arrows are identifications up to permutations of the variables. The plain arrows represent mutations.
• Figure 2: Left: an electric flux tube line created by an adjoint Wilson line. Right: the adjoint flux line is broken by the creation of a gluon-antigluon pair out of the vacuum.
• Figure 3: The modular triangulation of the upper half plane and its dual graph $\widetilde{CEG}$. The picture is reproduced from fock2009cluster.
• Figure 4: The translation quiver $\mathbb Z\hat{A}_{1,1}$ ($\equiv$ the AR quiver of the transjective component of the cluster category for pure $SU(2)$). Dotted arrows stands for the action of the AR translation $\tau$. Clearly $\tau$ is the translation to the left by 2 nodes. The auto-equivalence $\xi$ is translation to the left by 1 node: $\xi^2=\tau$.
• Figure 5: The CEG of the $A_2$ Argyres-Douglas theory.

Theorems & Definitions (98)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Definition 2.8
• Definition 2.9
• Definition 2.10