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Imposing modular invariance on the partition function of a 2D conformal field theory can be used to constrain the spectrum of the theory. By restricting to a certain class of CFTs, such as sigma models with Calabi-Yau, G2, or Spin(7) target spaces, we can hope to learn something about these classes of of theories, the associated low-energy effective theories, and the abstract geometries on which they depend.

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