An ordinary donut surface, but not smooth — instead, pierced by invisible layers, like puff pastry where each layer is slightly rotated relative to its neighbor. That's how a contact structure on a three-dimensional manifold works: at each point, a plane is defined, and these planes twist smoothly, touching each other according to a special law. In the new work, this construction is placed inside a three-dimensional complex space — a world where each point has three coordinates, but with imaginary numbers. Inside the surface, a knot is singled out — a closed line along which the layers become 'complex-smooth,' like glass. This knot serves as the spine of a book whose pages are disks filling the surface.
From this knot, as from a thread, a quantum system hangs. The authors construct a special bundle over the entire three-dimensional space — a 'blanket' of complex numbers attached to each point. On the knot, this bundle is cut, and from its sections (like patterns on the thread), the space of quantum states is assembled. It turns out to be finite-dimensional, like a box with a finite number of cells. The key player here is a field of directions called the Reeb field. If the surface satisfies an additional condition (becomes Sasakian), this field behaves like a geodesic flow — the shortest paths in a curved world. It is precisely this that models Einstein's gravity, where matter tells space how to curve, and space tells matter how to move.
The entire construction relies only on the choice of surface, contact structure, and volume form (like choosing units of measurement). No external fields or manual tweaking. Moreover, the authors introduce an invariant — a 'fingerprint' of the contact structure that distinguishes even those three-dimensional tori (donuts) that previously seemed identical. This same invariant serves as a quantum label that is preserved under deformations. Thus, the geometry of a single knot links the quantum counting of states with the gravitational dance of the Reeb field, hinting at a unified language for the two pillars of physics.
Contact structures arise naturally in optics: a light wavefront is a contact manifold, where rays are perpendicular to the front, and polarization defines an additional plane. So the geometry from the article resonates with how light is focused by lenses.
In Ian McDonald's novel 'River of Gods,' quantum states and gravity are interwoven in the geometry of extra dimensions, where 'knots' of reality dictate physical laws.