Categorical tinkertoys for N=2 gauge theories

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 (Q,W) of (finite-dimensional) representations of the Jacobian algebra Q/(δ W}) should enjoy what we call the Ringel property of type G; in particular, (Q,W) should contain a universal generic" subcategory, which depends only on the gauge group G, capturing the universality of the gauge sector. More precisely, there is a family of "light" subcategories Lλ∩ (Q,W), indexed by points λ N, where N is a projective variety whose irreducible components are copies of prime1 in one-to-one correspondence with the simple factors of G. If λ is the generic point of the ith irreducible component, Lλ is the universal subcategory corresponding to the ith simple factor of G. Matter, on the contrary, is encoded in the subcategories Lλa where {λa} is a finite set of closed points in N. In particular, for a Gaiotto theory there is one such family of subcategories, Lλ εN, for each maximal degeneration of the corresponding surface Σ, 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 than the geometric gluing of surfaces, allowing for a few additional exceptional N= 2 theories which are not of the Gaiotto class. We include several examples and some amusing consequence, as the characterization in terms of quiver combinatorics of asymptotically free theories. © 2013 World Scientific Publishing Company.