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Inquantum gravity,quantum geometry is the set of mathematical concepts that generalizegeometry to describe physical phenomena at distance scales comparable to thePlanck length. Each theory of quantum gravity uses the term "quantum geometry" in a slightly different fashion.
String theory uses quantum geometry to describe exotic phenomena such asT-duality and other geometric dualities,mirror symmetry,topology-changing transitions, minimal possible distance scale, and other effects that challenge intuition. Generally, string theory is initially explored on a compact six-dimensional manifold to restrict the algebraic data needed for computation. By utilizescompactifications, string theory describes geometric states, where a compactification is a spacetime that looks four-dimensional macroscopically even if its actual dimension is higher. One goal in exploring string compactifications is to findvacuum solutions where the space is maximally symmetric.[1]
When computing these vacuum solutions, preservingsupersymmetry gives a first-order system of equations which can partially give the second-order equations of motion. This supersymmetry enables the use of differential geometry methods by usingtransition functions[disambiguation needed]. Because the six-dimensional manifold cannot be covered with a single coordinate system, transition functions are grouped together into differentG-structures. To define the G-structure, infinitesimal parameters for supersymmetry calledspinors are introduced to enable stability during the transition.[1] More technically, quantum geometry refers to the shape of aspacetime manifold as experienced byD-branes, which includes quantum corrections to themetric tensor, such as the worldsheetinstantons. For example, the quantum volume of a cycle is computed from the mass of abrane wrapped on this cycle.[citation needed]
In an alternative approach to quantum gravity calledloop quantum gravity (LQG), the phrase "quantum geometry" usually refers to theformalism within LQG where the observables that capture the information about the geometry are well-defined operators on aHilbert space. In particular, certain physicalobservables, such as the area, have adiscrete spectrum. LQG isnon-commutative.[2]It is possible (but considered unlikely) that this strictly quantized understanding of geometry is consistent with the quantum picture of geometry arising from string theory.[citation needed]
Another approach, which tries to reconstruct the geometry of space-time from "first principles" isDiscrete Lorentzian quantum gravity.
