$A_4$ modular invariance and the strong CP problem

We present simple effective theory of quark masses, mixing and CP violation with level $N=3$ ($A_4$) modular symmetry, which provides solution to the strong CP problem without the need for an axion. The vanishing of the strong CP-violating phase $\bar \theta$ is ensured by assuming CP to be a fundamental symmetry of the Lagrangian of the theory. The CP symmetry is broken spontaneously by the vacuum expectation value (VEV) of the modulus $\tau$. This provides the requisite large value of the CKM CP-violating phase while the strong CP phase $\bar \theta$ remains zero or is tiny. Within the considered framework we discuss phenomenologically viable quark mass matrices with three types of texture zeros, which are realized by assigning both the left-handed and right-handed quark fields to $A_4$ singlets ${\bf 1}$, ${\bf 1'}$ and ${\bf 1''}$ with appropriate weights. The VEV of $\tau$ is restricted to reproduce the observed CKM parameters. We discuss cases in which the modulus VEV is close to the fixed points $i$, $\omega$ and $i\infty$. In particular, we focus on the VEV of $\tau$, which gives the absolute minima of the supergravity-motivated modular- and CP-invariant potentials for the modulus $\tau$, so called, modulus stabilisation. We present a successful model, which is consistent with the modulus stabilisation close to $\tau=\omega$.