TENSOR DECOMPOSITION, PARAFERMIONS, LEVEL-RANK DUALITY, AND RECIPROCITY LAW FOR VERTEX OPERATOR ALGEBRAS

For a semisimple Lie algebra sl(n), the basic representation L-(sln) over cap (1, 0) of the affine Lie algebra (sl(n)) over cap is a lattice vertex operator algebra. The first main result of the paper is to prove that the commutant vertex operator algebra of L-(sln) over cap (l, 0) in the l-fold tensor product L-(sln) over cap (1, 0)(circle times l) is isomorphic to the parafermion vertex operator algebra K(sl(l), n), which is the commutant of the Heisenberg vertex operator algebra L-(h) over cap (n, 0) in L-(sll) over cap (n, 0). The result provides a version of level-rank duality. The second main result of the paper is to prove more general version of the first result that the commutant of L-(sln) over cap(l(1) + ... + l(s), 0) in L-(sln) over cap(l(1), 0) circle times ... circle times L-(sln) over cap(l(s), 0) is isomorphic to the commutant of the vertex operator algebra generated by a Levi Lie subalgebra of sl(l1) + ... + l(s) corresponding to the composition (l(1), ... , l(s)) in the rational vertex operator algebra L(sl) over cap l1 + ... + ls (n, 0). This general version also resembles a version of reciprocity law discussed by Howe in the context of reductive Lie groups. In the course of the proof of the main results, certain Howe duality pairs also appear in the context of vertex operator algebras.