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, the 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.
In an alternative approach to quantum gravity called loop quantum gravity (LQG), the phrase quantum geometry usually refers to the calculation within LQG that shows that certain physical observables, such as the area, have a discrete spectrum of eigenvalues. This phrase is often used as synonymous with loop quantum gravity as a whole. It is possible but considered unlikely by many that this strictly quantized understanding of geometry will be confirmed by string theory.
