  Modular invariance is known to constrain the spectrum of 2d conformal field
theories. We investigate this constraint 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. For Calabi-Yau threefolds with no enhanced
symmetry we find that there must always be non-BPS primary states of weight 0.6
or less.
