The QCD axion and moduli stabilisation

We investigate the conditions for a QCD axion to coexist with stabilisedmoduli in string compactifications. We show how the simplest approaches tomoduli stabilisation give unacceptably large masses to the axions. We observethat solving the F-term equations is insufficient for realistic modulistabilisation and give a no-go theorem on supersymmetric moduli stabilisationwith unfixed axions applicable to all string compactifications and relevant tomuch current work. We demonstrate how nonsupersymmetric moduli stabilisationwith unfixed axions can be realised. We finally outline how to stabilise themoduli such that f_a is within the allowed window 10^9 GeV < f_a < 10^{12} GeV,with f_a ~ \sqrt{M_{SUSY} M_P}.Comment: 36 pages; v2: extended discussion of cosmological bound on f_a, references added, version accepted by journal; v3. factor of 2 correcte