Extremal N=(2,2) 2D Conformal Field Theories and Constraints of   Modularity
  We explore the constraints on the spectrum of primary fields implied by modularity of the elliptic genus of N=(2,2) 2D CFT's. We show that such constraints have nontrivial implications for the existence of "extremal" N=(2,2) conformal field theories. Applications to AdS3 supergravity and flux compactifications are addressed. 
Matthias R. Gaberdiel, Sergei Gukov, Christoph A. Keller, Gregory W.   Moore, Hirosi Ooguri