AGT and the Segal-Sugawara construction

The conjectures of Alday et al. [Lett. Math. Phys. 91, 167–197 (2010)] and their 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 Whitaker vector [A. Braverman et al., e-print arXiv:1406.2381 (2014); D. Maulik and A. Okounkov, e-print arXiv:1211.1287 (2012), pp. 1–276; O. Schiffmann and E. Vasserot, Publ. Math. Inst. Hautes Etud. Sci. 118, 213–342 (2013)]. 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 Carlsson and Okounkov [Duke Math. J. 161, 1797–1815 (2012)], a result which is now sometimes referred to as AGT in rank one [A. Alba et al., Lett. Math. Phys. 98, 33–64 (2011); M. Pedrini et al., J. Geom. Phys. 103, 43–89 (2016)]. 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 $s l ˆ 2 C$ structure that appears in Nekrasov and Okounkov [Prog. Math. 244, 525–596 (2006)] for general rank. 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 [e-print arXiv:0909.2453 (2009).] as well as Iqbal et al. [J. High Energy Phys. 2009, 69].