The Perfect Mirror: Quantum Fields on a Donut and the Geometry Within ⚡ экспресс
It is shown that for a CFT on a flat open solid torus, the two-point function in the Weyl frame is in exact correspondence with a finite geodesic entirely lying inside the bulk AdS. The exactness is kinematic and requires neither large N, nor strong coupling, nor heavy operators — semiclassical bulk dynamics is not invoked. The standard boundary-anchored relation is a singular limit of this exact pair. For a free scalar field, the mode expansion along S¹ generates an infinite tower of effective masses on H, whose complicated propagators exactly resum to the same simple higher-dimensional geodesic expression. Together with another exact pair between the disconnected entanglement entropy and the area of the minimal cross-section of the entanglement wedge, discovered on the same torus, the result points to a promising program of searching for exact pairs in AdS/CFT.
On a donut-shaped ring, the theory lives on the surface, while inside hides a curved space — like a saddle widening toward the edge. According to the holographic principle of Juan Maldacena, everything happening inside can be described by data on the boundary.
It turns out that for such a shape, this connection becomes a perfect mirror: a simple measure of one point's influence on another on the surface exactly equals the length of the shortest path through the curved inner space. This is a strict equality, not an approximation, and it works even for the simplest fields, where the most complex calculations unexpectedly simplify down to a single line.
The same holds for quantum entanglement: its measure — entanglement entropy — between two parts of the surface matches the area of a special cross-section inside. This builds an exact dictionary: information on the boundary is a reflection of the geometry in the bulk. Such an approach promises simplified calculations and helps understand quantum gravity near black holes.
🎯 The donut shape lets you hide extra dimensions—like in an old videogame where a character goes off one edge of the screen and instantly pops out on the other side.
🎬 The holographic principle echoes 'The Matrix' and the novels of Alastair Reynolds: reality could be just a projection from a surface.