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

Abstract

We review the categorical approach to the BPS sector of a 4d N = 2 QFT, clarifying many tricky issues and presenting a few novel results. To a given 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 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 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.

Higher Symmetries of 5d Orbifold SCFTs

“…The states corresponding to the charges of the generators are viewed as a collection of elementary constituents, out of the bound states of which the whole spectrum of the theory can be reconstructed. Since all the states in the spectrum are formed by these bound states, their charges are integer multiples of the charges of the elementary constituents, and we can completely determine the 't Hooft screening in terms of the latter [77]. The charges of the line defects are in turn valued in a dual lattice of charges, where the duality is determined by the Dirac pairing [88,106].…”

“…By construction the integers n can be recovered by the Smith normal form of the matrix B [77]. The fact that B is antisymmetric entails that one gets the two identical electric and magnetic factors in equation (2.9).…”

“…This information is not enough to fully determine the Heisenberg algebra of non-commuting fluxes. The latter can be reconstructed exploiting the prescription on the Weyl pairing discussed in [77], to which we refer our readers.…”

“…The positive integers n can be completely determined by via a standard 't Hooft screening argument [76] -see e.g. [77]. Moreover, the quiver also captures the Weyl pairing [77] (or linking pairing [16]) from which the resulting Heisenberg algebra of non-commuting fluxes [78,79] that governs the global structure of the theories [80,81] can be reconstructed [82].…”

“…[77]. Moreover, the quiver also captures the Weyl pairing [77] (or linking pairing [16]) from which the resulting Heisenberg algebra of non-commuting fluxes [78,79] that governs the global structure of the theories [80,81] can be reconstructed [82]. Knowing the 1-form defect group of the 4d KK theory, it is easy to recover the corresponding factors of the defect group of the associated 5d SCFT:…”