Editing Teleparallelism (section)

==Non-gravitational contexts==
There exists a close analogy of [[Riemannian geometry|geometry]] of spacetime with the structure of defects in crystal.<ref>{{cite book| title = Gauge Fields in Condensed Matter Vol II|first = Hagen| last = Kleinert| author-link = Hagen Kleinert| pages = 743–1440| date = 1989| url = http://users.physik.fu-berlin.de/~kleinert/kleiner_reb1/contents2.html}}</ref><ref>{{cite book| title = Multivalued Fields in Condensed Matter, Electromagnetism, and Gravitation| first = Hagen| last = Kleinert| author-link = Hagen Kleinert| pages = 1–496| date = 2008| bibcode = 2008mfcm.book.....K| url = http://users.physik.fu-berlin.de/~kleinert/b11/psfiles/mvf.pdf}}</ref> [[Dislocations]] are represented by torsion, [[disclination]]s by curvature. These defects are not independent of each other. A dislocation is equivalent to a disclination-antidisclination pair, a disclination is equivalent to a string of dislocations. This is the basic reason why Einstein's theory based purely on curvature can be rewritten as a teleparallel theory based only on torsion. There exists, moreover, infinitely many ways of rewriting Einstein's theory, depending on how much of the curvature one wants to reexpress in terms of torsion, the teleparallel theory being merely one specific version of these.<ref>{{cite journal| title = New Gauge Symmetry in Gravity and the Evanescent Role of Torsion| first = Hagen| last = Kleinert| author-link = Hagen Kleinert| volume = 24| pages = 287–298| date = 2010| journal = Electron. J. Theor. Phys.| doi = 10.1142/9789814335614_0016 | arxiv = 1005.1460| bibcode = 2011pchm.conf..174K| isbn = 978-981-4335-60-7| s2cid = 17972657| url = http://users.physik.fu-berlin.de/~kleinert/385/385.pdf}}</ref>

A further application of teleparallelism occurs in quantum field theory, namely, two-dimensional [[non-linear sigma model]]s with target space on simple geometric manifolds, whose renormalization behavior is controlled by a [[Ricci flow]], which includes [[Torsion tensor|torsion]]. This torsion modifies the Ricci tensor and hence leads to an [[infrared fixed point]] for the coupling, on account of teleparallelism ("geometrostasis").<ref>{{Cite journal | last1 = Braaten | first1 = E. | last2 = Curtright | first2 = T. L. | last3 = Zachos | first3 = C. K. | doi = 10.1016/0550-3213(85)90053-7 | title = Torsion and geometrostasis in nonlinear sigma models | journal = Nuclear Physics B | volume = 260 | issue = 3–4 | pages = 630 | year = 1985 |bibcode = 1985NuPhB.260..630B }}</ref>