Theta-problem and the String Swampland

Understand

In the context of $\mathcal{N}=2$ supergravity without vector multiplets coupled to hypermultiplets, the coupling constant of graviphoton $\tau$ is apriori a free parameter.

• Stringy realization of this and using a mathematical conjecture leads to the statement that $j(\tau)\in \mathbb{R}$ so that the $\theta$-angle is $0$ or $\pi$.
• We conjecture that for any consistent realization of $\mathcal{N}=2$ supergravity theories coupled only to hypermultiplets this is the case and the rest belong to the swampland.
• This leads to the speculation that the $\theta$-angle for QCD or QED may also be fixed to $0$ for quantum gravitational consistency.