On the space of elliptic genera

Invariance under modular transformations and spectral flow restrict the possible spectra of superconformal field theories (SCFT). This paper presents a technique to calculate the number of constraints on the polar spectra of N=(2,2) and N=(4,0) SCFT's by analyzing the elliptic genus. The polar spectrum corresponds to the principal part of a Laurent expansion derived from the elliptic genus. From the point of view of the AdS_3/CFT_2 correspondence, these are the states which lie below the cosmic censorship bound in classical gravity. The dimension of the space of elliptic genera is obtained as the number of coefficients of the principal part minus the number of constraints. As an additional illustration of the technique, the constraints on the spectrum of N=4 topologically twisted Yang-Mills on CP^2 are discussed.