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We explore the constraints on the spectrum of primary fields implied bymodularity of the elliptic genus of N=(2,2) 2D CFT's. We show that suchconstraints have nontrivial implications for the existence of "extremal"N=(2,2) conformal field theories. Applications to AdS3 supergravity and fluxcompactifications are addressed.

Extremal $\CN=(2,2)$ 2D conformal field theories and constraints of modularity