The congruence subgroup property is established for the modular representations associated to any modular tensor category. This result is used to prove that the kernel of the representation of the modular group on the conformal blocks of any rational, C2-cofinite vertex operator algebra is a congruence subgroup. In particular, the q-character of each irreducible module is a modular function on the same congruence subgroup. The Galois symmetry of the modular representations is obtained and the order of the anomaly for those modular categories satisfying some integrality conditions is determined.

Unitary vertex operator algebras are introduced and studied. It is proved that most well-known rational vertex operator algebras are unitary. The classification of unitary vertex operator algebras with central charge c ≤ 1 is also discussed.

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The extensions of the affine vertex operator algebras L sl2 (k, 0) and the preunitary vertex operator algebras with central charges c < 1 are classified. In particular, the unitary vertex operator algebras with central charges c < 1 are classified.p i=0

In this paper, mirror extensions of rational vertex operator algebras are considered. The mirror extension conjecture is proved.