Tinkertoys for Gaiotto duality

We describe a procedure for classifying 4D N=2 superconformal theories of the type introduced by Davide Gaiotto. Any punctured curve, C, on which the 6D (2,0) SCFT is compactified, may be decomposed into 3-punctured spheres, connected by cylinders. The 4D theories, which arise, can be characterized by listing the ``matter" theories corresponding to 3-punctured spheres, the simple gauge group factors, corresponding to cylinders, and the rules for connecting these ingredients together. Different pants decompositions of C correspond to different S-duality frames for the same underlying family of 4D \mathcal{N}=2 SCFTs. We developed such a classification for the A_{N-1} and the D_N series of 6D (2,0) theories. We outline the procedure for general A_{N-1} and D_N, and construct, in detail, the classification through A_4 and D_4, respectively.