Tinkertoys for Gaiotto duality

Abstract

We describe a procedure for classifying N = 2 superconformal theories of the type introduced by Davide Gaiotto. Any curve, C, on which the 6D A N −1 SCFT is compactified, can be decomposed into 3-punctured spheres, connected by cylinders. We classify the spheres, and the cylinders that connect them. The classification is carried out explicitly, up through N = 5, and for several families of SCFTs for arbitrary N . These lead to a wealth of new S-dualities between Lagrangian and non-Lagrangian N = 2 SCFTs.

Rigid holography and six-dimensional N = 2 0 $$ \mathcal{N}=\left(2,0\right) $$ theories on AdS5 × S $$ \mathbb{S} $$ 1

“…It is now known that many different types of punctures are possible in compactified six-dimensional theories, which differ by the amount of global symmetry that they give rise to. From the consideration of the global symmetries of the four-dimensional N = 2 SCFTs described above, it is clear that in our case each puncture is associated with a U(1) global symmetry, so that the punctures are minimal regular punctures in the standard terminology [31,33]. Beware that this six-dimensional N = (2, 0) A N −1 theory should not be confused with the six-dimensional N = (2, 0) A K−1 theory that we have in the bulk dual description.…”

“…So, what do we get at this singular point? To reconcile the bulk intuition with the exact field theory obstruction, we can study what happens in this limit in the four-dimensional N = 2 SCFT by using methods that were recently developed by Gaiotto and others for the study of such theories that can be described as compactified M5-branes [29][30][31][32]. A large class of four-dimensional N = 2 SCFTs, called class S, can be described using six-dimensional N = (2, 0) SCFTs compactified on Riemann surfaces with punctures (called 'UV curves'), and such a description is useful for understanding strong-weak coupling S-dualities and other symmetry properties (see [33,34] for reviews).…”

“…Similarly, when K punctures of this type come together, it has been argued [29][30][31][32] that, for large enough values of N , one obtains a weakly coupled SU(K) gauge theory. In terms of the UV curve, the torus with K coalescing punctures develops a long 'throat' (see figure 5), corresponding to a weakly coupled SU(K) gauge group.…”

“…31 The analysis below is generalizable to any discrete subgroups Γ ⊂ SU(2) ⊂ SO(4). In those cases, the number of six-dimensional tensor multiplets equals the number of non-trivial conjugacy classes of Γ.…”

Compactifications of 5d SCFTs with a twist

“…These can be evaluated using the methods of [35], and as these play a prominent rule in the proceeding discussion we have summarized them in figure 17. For the central charges we use the Higgs branch dimension and effective number of vector multiplets.…”