2015/08/31 by Carlsson, Erik
#14D21 #81R10 #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1509.00075
The conjectures of Alday, Gaiotto and Tachikawa and its generalizations have been mathematically formulated as the existence of an action of a W-algebra on the cohomology or K-theory of the instanton moduli space, together with a Whitakker vector. However, the original conjectures also predict intertwining properties with the natural higher rank version of the "Ext1 operator" which was previously studied by Okounkov and the author in [CO], a result which is now sometimes referred to as AGT in rank one [Alb,PSS]. Physically, this corresponds to incorporating matter in the Nekrasov partition functions, an obviously important feature in the physical theory. It is therefore of interest to study how the Ext1 operator relates to the aforementioned structures on cohomology in higher rank, and if possible to find a formulation from which the AGT conjectures follow as a corollary. In this paper, we carry out something analogous using a modified Segal-Sugawara construction for the \mathfraksl2ℂ structure that appears in Okounkov and Nekrasov's proof of Nekrasov's conjecture [NO] for rank two. This immediately implies the AGT identities when the central charge is one, a case which is of particular interest for string theorists, and because of the natural appearance of the Seiberg-Witten curve in this setup, see for instance Dijkgraaf and Vafa [DV], as well as [IKV].