Solving the strong CP problem without axions

We formulate general conditions under which the strong CP problem is solved by spontaneous CP violation. Quark-mass matrix elements are polynomials in the CP-breaking order parameters, engineered such that their determinant is a real constant. This scheme permits only a limited number of textures. These conditions can be realized in supersymmetric theories with CP as an anomaly-free local flavour symmetry, suggesting a unified solution to the strong CP problem and the flavour puzzle. Our solution can be implemented using either modular invariance or a local U (1) symmetry. We present modular-invariant realizations where matter fields are assigned small modular weights 2 (1), utilising higher levels N=2 (N=3). Heavy quarks are in general not required, but their presence allows for models where colored particles fill non-singlet representations of the flavour group.