We discuss partition functions of 2d conformal field theories. Modular invariance is known to constrain the spectrum of such theories. We investigate these constraints systematically, using the linear functional method to put new improved upper bounds on the lowest gap in the spectrum. We also consider generalized partition functions of N = (2,2) superconformal theories and discuss the application of our results to Calabi-Yau compactifications. This talk is based on 1209.4649 with H.Ooguri and 1307.6562 with D.Friedan.

For arbitrary gauge groups, I will describe how one can check at the one-instanton level that the Nekrasov partition function of pure N=2 super Yang-Mills is equal to the norm of a certain coherent state of the corresponding W-algebra. For non-simply-laced gauge groups, this coherent state is in the twisted sector of a simply-laced W-algebra. I will also talk about the case of gauge theories with hypermultiplets.