  We study the linear response of interacting active Brownian particles in an
external potential to simple shear flow. Using a path integral approach, we
derive the linear response of any state observable to initiating shear in terms
of correlation functions evaluated in the unperturbed system. For systems and
observables which are symmetric under exchange of the $x$ and $y$ coordinates,
the response formula can be drastically simplified to a form containing only
state variables in the corresponding correlation functions (compared to the
generic formula containing also time derivatives). In general, the shear
couples to the particles by translational as well as rotational advection, but
in the aforementioned case of $xy$ symmetry only translational advection is
relevant in the linear regime. We apply the response formulas analytically in
solvable cases and numerically in a specific setup. In particular, we
investigate the effect of a shear flow on the morphology and the stress of $N$
confined active particles in interaction, where we find that the activity as
well as additional alignment interactions generally increase the response.
