A Pedagogical Introduction to the AGT Conjecture

Abstract

The AGT conjecture is a natural identification between certain information-rich objects in 4d N=2 supersymmetric gauge theories and Liouville conformal field theory. Though relatively easy to state, it rests upon a host of concepts and computational techniques developed over the span of 25 years, many of which were only recently uncovered. This thesis consists of two parts. The first half is written for the seasoned researcher who wishes to review the essential concepts, understand the statement of the conjecture, and observe how a proof of a simple but non-trivial subcase is performed. The second half is written for the curious student: it consists of a number of appendices dedicated to explaining in greater detail the topological objects appearing the history and statement of the conjecture, to proving many group- and representation-theoretic statements whose conclusions are frequently used in the literature but whose proofs appear rarely, and to introducing a rarely-studied but critically important matter representation: the trifundamental.