Hitchin systems in N = 2 field theory

This note is a short review of the way Hitchin systems appear in four-dimensional N = 2 supersymmetric field theory. The literature on the Hitchin system and its role in quantum field theory is a vast one. Restricting attention just to the role of Hitchin systems in N = 2 supersymmetric field theory (thus neglecting such fascinating topics as T-duality on the Hitchin fibration and its relation to the geometric Langlands program [36, 35, 41, 25], the use of Hitchin systems in N = 4 super Yang-Mills [2, 3, 4], the role of Higgs bundles in F-theory [14], . . . ) cuts things down somewhat but still leaves an enormous pool of papers and topics from which to choose. Selfishly, I have chosen to focus here on points which I think I understand, which in practice mainly means points on which I have worked. In particular, although this review is meant for a special volume devoted to the AGT correspondence, I will have very little to say about that. This is not because I think there is nothing to say — on the contrary, works such as [46, 51] have demonstrated that there clearly is — but because I do not know precisely what to say. In one sentence, the relation between N = 2 theories and Hitchin systems is that the Hitchin system arises as the moduli space of the N = 2 theory compactified on a circle. My aim in this note is to explain a dictionary between various aspects of the field theory (its Coulomb branch, its line defects, its surface defects, . . . ) and their manifestations in the Hitchin system (the Hitchin base, some distinguished holomorphic functions, some distinguished hyperholomorphic bundles, . . . ), along with a few ways in which this dictionary gives insight into aspects of the Hitchin system. My perspective on this subject has been heavily influenced by a long and enjoyable collaboration with Davide Gaiotto and Greg Moore. It is a pleasure to thank them for this collaboration and for the many things that they have taught me.