On the Integrability of Four Dimensional N = 2 Gauge Theories in the Omega Background

We continue to investigate the relationship between the infrared physics of N = 2 supersymmetric gauge theories in four dimensions and var-ious integrable models such as Gaudin, Calogero-Moser and quantum spin chains. We prove interesting dualities among some of these inte-grable systems by performing different, albeit equivalent, quantizations of the Seiberg-Witten curve of the four dimensional theory. We also discuss conformal field theories related to N = 2 4d gauge theories by the Alday-Gaiotto-Tachikawa (AGT) duality and the role of conformal blocks of those CFTs in the integrable systems. As a consequence, the equivalence of conformal blocks of rank two Toda and Novikov-Wess-Zumino-Witten (WZNW) theories on the torus with punctures is found. ar X i