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Twisted geometries are discrete geometries that play a role in loop quantum gravity and spin foam models, where they appear in the semiclassical limit of spin networks.[1][2][3] A twisted geometry can be visualized as collections of polyhedra dual to the nodes of the spin network's graph.[4] Intrinsic and extrinsic curvatures are defined in a manner similar to Regge calculus, but with the generalisation of including a certain type of metric discontinuities: the face shared by two adjacent polyhedra has a unique area, but its shape can be different. This is a consequence of the quantum geometry of spin networks: ordinary Regge calculus is "too rigid" to account for all the geometric degrees of freedom described by the semiclassical limit of a spin network.

The name twisted geometry captures the relation between these additional degrees of freedom and the off-shell presence of torsion in the theory, but also the fact that this classical description can be derived from Twistor theory, by assigning a pair of twistors to each link of the graph, and suitably constraining their helicities and incidence relations.[5][6]
References

L. Freidel and S. Speziale (2010). "Twisted geometries: A geometric parametrisation of SU(2) phase space". Phys. Rev. D. 82 (8): 084040. arXiv:1001.2748. Bibcode:2010PhRvD..82h4040F. doi:10.1103/PhysRevD.82.084040.
C. Rovelli and S. Speziale (2010). "On the geometry of loop quantum gravity on a graph". Phys. Rev. D. 82 (4): 044018. arXiv:1005.2927. Bibcode:2010PhRvD..82d4018R. doi:10.1103/PhysRevD.82.044018.
E. R. Livine and J. Tambornino (2012). "Spinor Representation for Loop Quantum Gravity". J. Math. Phys. 53 (1): 012503. arXiv:1105.3385. Bibcode:2012JMP....53a2503L. doi:10.1063/1.3675465.
E. Bianchi, P. Dona and S. Speziale (2011). "Polyhedra in loop quantum gravity". Phys. Rev. D. 83 (4): 044035. arXiv:1009.3402. Bibcode:2011PhRvD..83d4035B. doi:10.1103/PhysRevD.83.044035.
L. Freidel and S. Speziale (2010). "From twistors to twisted geometries". Phys. Rev. D. 82 (8): 084041. arXiv:1006.0199. Bibcode:2010PhRvD..82h4041F. doi:10.1103/PhysRevD.82.084041.
S. Speziale and Wolfgang M. Wieland (2012). "The twistorial structure of loop-gravity transition amplitudes". Phys. Rev. D. 86 (12): 124023. arXiv:1207.6348. Bibcode:2012PhRvD..86l4023S. doi:10.1103/PhysRevD.86.124023.

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