Unitarity of strings and non-compact Hermitian symmetric spaces

Abstract

If G is a simple non-compact Lie Group, with K its maximal compact subgroup, such that K contains a one-dimensional center C, then the coset space G/ K is an Hermitian symmetric non-compact space. SL(2, R)/U(1) is the simplest example of such a space. It is only when G/ K is an Hermitian symmetric space that there exists unitary discrete representations of G. We will here study string theories defined as G/ K′, K′= K/ C, WZNW models. We will establish unitarity for such string theories for certain discrete representations. This proof generalizes earlier results on SL(2, R) , which is the simplest example of this class of theories. We will also prove unitarity of G/ K conformal field theories generalizing results for SL(2, R)/U(1) . We will show that the physical space of states lie in a subspace of the G/ K state space.