Unitarity of strings and non-compact Hermitian symmetric spaces

TL;DR

This paper demonstrates the unitarity of string theories on certain Hermitian symmetric non-compact spaces G/K, extending previous results from SL(2,R) to more general groups, and clarifies the structure of physical state spaces.

Contribution

It generalizes unitarity proofs for string theories on SL(2,R) to broader classes of Hermitian symmetric spaces G/K, establishing conditions for unitarity and the structure of physical states.

Findings

Unitarity established for specific discrete representations.

Generalization of unitarity results from SL(2,R) to G/K spaces.

Physical states are contained within a subspace of the G/K state space.

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 the subspace of the G/K state space.