Dyson Sphere: The Empire has two, although neither by that name, the concept *there* having been devised by Cirys Minaxianos. One, a Cirys bubble, is at Esilmúr (Imperial Core); a thin layer of light-sail material surrounds the star, held up by photon and solar wind pressure in much the same manner as a statite, and managed by magnetic manipulation and selective releases of solar wind. The material, in turn, is coated with energy-producing solar foil, powering massive arrays of antimatter production facilities.
The other, a three-layered incomplete Cirys swarm is at Corícal Ailék, where the Transcend keeps its brain: the innermost being golden-winged energy-collection complexes skimming the corona of the star and powering the rest of the swarm, the middle layer being fractal chandeliers of organic crystal nanocircuitry, providing the Transcend with its massive processing power, and the outermost being the hexterranes, a lattice of hexagonal habitat-plates providing living space here at the heart of the light.
Another reader question:
A thought hits me: If the Empire has the power to shepherd stars and (at least theoretically) to destroy them, does that mean that it also might have the capability to move them?
The destroying them (in theory, but it’s a good theory) isn’t so relevant in this context. It is a sad reflection of the nature of the universe that destroying things tends to be pretty easy, at least compared to creating them. That’s entropy for you.
As for moving stars. Well, theoretically, there are several possibilities. For example, you could use the Cirys bubble (a solar-sail-material-based dynamic Dyson sphere, similar to this) technology in use at the Esilmúr energy production facility along with the star-stabilizing plasmonics at use in stellar husbandry arrays to build a functioning Shkadov thruster.
Doing this would require solving several of what I believe technarchs traditionally refer to as “interesting engineering problems”, but it wouldn’t require any radically new scientific breakthroughs to make work. Just time, genius, and an Imperial assload of cash.
(In somewhat more radical ideas – a stargate moves mass around, and stars are, well, mass. Given certain constraints on energy requirements (because stars are a lot of mass) and the need to sink rather vastier amounts of kinetic energy (because stars are a lot of mass) than usual to avoid nasty intrinsic problems – and you’ll note no-one’s stargate-jumping planets around, either – this almost certainly involves solving a great many more interesting engineering problems than the former one. But again, nothing fundamental stops you from doing it, either.)
All of which is to say: moving stars isn’t a realized capability, but while it’s currently restricted to the drawing board and wild speculative fiction, it’s certainly a realizable one. Analogically speaking, should the necessity suddenly turn up (“it’s coming right at us!”), they just have to run the Manhattan Project; they don’t have to discover nuclear fission, first.