Editing Holonomy (section)

===Applications===

====String Theory====
Riemannian manifolds with special holonomy play an important role in [[string theory]] [[compactification (physics)|compactifications]].
<ref>{{citation | author = Gubser, S. | title = Special holonomy in string theory and M-theory | editor=Gubser S.|display-editors=etal}}
+{{citation |  title=Strings, branes and extra dimensions, TASI 2001. Lectures presented at the 2001 TASI school, Boulder, Colorado, USA, 4–29 June 2001.  | place=River Edge, NJ | publisher=World Scientific | isbn=978-981-238-788-2 | pages=197–233 | year=2004 | arxiv=hep-th/0201114 | last1=Gubser | first1=Steven S. }}.</ref>
This is because special holonomy manifolds admit [[Covariance and contravariance of vectors|covariantly]] constant (parallel) [[spinor]]s and thus preserve some fraction of the original [[supersymmetry]]. Most important are compactifications on [[Calabi–Yau manifold]]s with SU(2) or SU(3) holonomy. Also important are compactifications on [[G2 manifold|G<sub>2</sub> manifold]]s.

====Machine Learning====
Computing the holonomy of Riemannian manifolds has been suggested as a way to learn the structure of data manifolds in [[machine learning]], in particular in the context of [[manifold learning]]. As the holonomy group contains information about the global structure of the data manifold, it can be used to identify how the data manifold might decompose into a product of submanifolds. The holonomy cannot be computed exactly due to finite sampling effects, but it is possible to construct a numerical approximation using ideas from [[spectral graph theory]] similar to Vector Diffusion Maps. The resulting algorithm, the Geometric Manifold Component Estimator ({{Smallcaps|GeoManCEr}}) gives a numerical approximation to the de Rham decomposition that can be applied to real-world data.<ref>{{citation |last1=Pfau |first1=David |last2=Higgins |first2=Irina |last3=Botev |first3=Aleksandar |last4=Racanière |first4=Sébastien |title=Disentangling by Subspace Diffusion |journal=Advances in Neural Information Processing Systems |date=2020 |arxiv=2006.12982}}</ref>