  Effects of the theta parameter are studied in Witten's model of holographic
4d Yang-Mills, where theta is the coefficient of the CP-breaking topological
term. First, the gravity background, including the full backreaction of the RR
form dual to the theta parameter, is revisited. Then, a number of observables
are computed holographically: the ground-state energy density, the string
tension, the 't Hooft loop, the light scalar glueball mass, the baryon mass
scale, the critical temperature for deconfinement - and thus the whole (T,
theta) phase diagram - and the entanglement entropy. A simple rule is provided
to derive the theta corrections to (at least) all the CP-neutral observables of
the model. Some of the observables we consider can and have been in fact
studied in pure 4d Yang-Mills on the lattice. In that framework the results,
obtained in the small theta regime, are given up to very few powers of theta^2.
The corresponding holographic results agree qualitatively with available
lattice data and signal an overall mass scale reduction by theta. Moreover,
being exact in theta, they provide a benchmark for higher order corrections in
Yang-Mills.
