Modular invariants in the fractional quantum Hall effect

We investigate the modular properties of the characters which appear in the partition functions of non-abelian fractional quantum Hall states. We first give the annulus partition function for nonabelian FQH states formed by spinon and holon (spinon-holon state). The degrees of freedom of spin are described by the affine SU(2) Kac-Moody algebra at level k. The partition function and the Hilbert space of the edge excitations decomposed differently according to whether k is even or odd. We then investigate the full modular properties of the extended characters for non-abelian fractional quantum Hall states. We explicitly verify the modular invariance of the annulus grand partition functions for spinon-holon states, the Pfaffian state and the 331 states. This enables one to extend the relation between the modular behavior and the topological order to non-abelian cases. For the Haldane-Rezayi state, we find that the extended characters do not form a representation of the modular group, thus the modular invariance is broken. We also find a new relation between the Haldane-Rezayi state and the 331 state and suggest its implications for 'The upsilon = 5/2 Enigma'. (C) 1998 Elsevier Science B.V.