In theoretical physics, quantum geometry is the set of new mathematical concepts generalizing the concepts of geometry whose understanding is necessary to describe the physical phenomena at very short distance scales (comparable to Planck length). At these distances, quantum mechanics has a profound effect on physics.
Each theory of quantum gravity uses the term “quantum geometry” in a slightly different fashion. String theory, a leading candidate for a quantum theory of gravity, uses the term quantum geometry to describe exotic phenomena such as T-duality and other geometric dualities, mirror symmetry, topology-changing transitions, minimal possible distance scale, and other effects that challenge our usual geometrical intuition. More technically, quantum geometry refers to the shape of the spacetime manifold as seen by D-branes which includes the quantum corrections to the metric tensor, such as the worldsheet instantons. For example, the quantum volume of a cycle is computed from the mass of a brane wrapped on this cycle. As another example, a distance between two quantum mechanics particles can be expressed in terms of the Lukaszyk–Karmowski metric.
In an alternative approach to quantum gravity called loop quantum gravity (LQG), the phrase “quantum geometry” usually refers to the formalism within LQG where the observables that capture the information about the geometry are now well defined operators on a Hilbert space. In particular, certain physical observables, such as the area, have a discrete spectrum. It has also been shown that the loop quantum geometry is non-commutative.
It is possible (but considered unlikely) that this strictly quantized understanding of geometry will be consistent with the quantum picture of geometry arising from string theory.
Another, quite successful, approach, which tries to reconstruct the geometry of space-time from “first principles” is Discrete Lorentzian quantum gravity.