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| Special holonomy | |
|---|---|
| Name | Special holonomy |
| Caption | Parallel transport on a sphere illustrating holonomy |
| Field | Differential geometry |
| Introduced | 1955 (Berger) |
| Notable | Élie Cartan, Marcel Berger, Shing-Tung Yau, Eugenio Calabi |
Special holonomy Special holonomy refers to Riemannian manifolds whose holonomy groups are proper subgroups of the full orthogonal group, a phenomenon central to modern differential geometry, complex geometry, and mathematical physics. Rooted in the work of Élie Cartan and systematized by Marcel Berger, special holonomy connects to conjectures and theorems by Eugenio Calabi, Shing-Tung Yau, and constructions by Dominic Joyce, with implications across research associated to Institute for Advanced Study, Princeton University, and Harvard University.
The study of special holonomy grew from investigations by Élie Cartan in the early 20th century and was crystallized by Marcel Berger in 1955, influencing subsequent work by Calabi, Yau, and Simon Donaldson. It relates to parallel spinors in the context of Paul Dirac, parallel forms connected to Élie Cartan's exterior calculus, and reduction of structure groups studied by Hermann Weyl and Élie Cartan's schools. Developments intersect research at institutions like Courant Institute, Department of Mathematics, University of Cambridge, and Massachusetts Institute of Technology.
Berger's classification enumerated candidates for irreducible, non-symmetric Riemannian holonomy groups; his list includes groups studied by Klaus Friedrich, Jean-Pierre Bourguignon, Robert Bryant, and S. Salamon. The candidates—subgroups of SO(n), namely U(n), SU(n), Sp(n), Sp(n)·Sp(1), G2, and Spin(7)—were analyzed using techniques advanced at Princeton University, University of Oxford, University of Cambridge, and IHÉS. Subsequent proofs and counterexamples involved scholars such as Eugenio Calabi, Shing-Tung Yau, Dominic Joyce, Nigel Hitchin, and Christoph Bär. Berger's constraints build on representation theory developments by Hermann Weyl, classification work by Élie Cartan, and holonomy computations influenced by Kobayashi and Nomizu.
Riemannian manifolds with holonomy in U(n), SU(n), Sp(n), G2, or Spin(7) correspond to geometric structures studied by Kunihiko Kodaira, André Weil, John Milnor, and Michael Atiyah. Kähler manifolds with U(n) holonomy connect to results by A. Weil and moduli problems pursued by Pierre Deligne, David Mumford, and Maxim Kontsevich. Hyper-Kähler metrics with Sp(n) holonomy were developed by Nigel Hitchin, Alekseevsky, and Calabi. Exceptional holonomy groups G2 and Spin(7) were constructed by Dominic Joyce and further analyzed by Robert Bryant and Simon Salamon. Theory ties to index theorems by Atiyah-Singer and gauge theory developments by Simon Donaldson and Edward Witten.
Eugenio Calabi conjectured existence of Ricci-flat Kähler metrics on manifolds with vanishing first Chern class; Shing-Tung Yau proved the Calabi conjecture, a milestone recognized by Fields Medal-level work and connected to research groups at Harvard University and Princeton University. Calabi–Yau manifolds with SU(n) holonomy underpin studies by Philip Candelas, Paul Green, Xenia de la Ossa, and Cumrun Vafa in string theory, and are central to mirror symmetry developed by Maxim Kontsevich, Kontsevich and Soibelman, Strominger–Yau–Zaslow, and Bershadsky. Moduli spaces of Calabi–Yau metrics involve contributions from David Morrison, Mark Gross, Ron Donagi, and Sheldon Katz, while compactification scenarios engage Edward Witten, Juan Maldacena, and Andrew Strominger.
Exceptional holonomy groups G2 and Spin(7) were linked to differential forms studied by Élie Cartan and representation theory advances by Élie Cartan's classification. Explicit metrics with G2 holonomy were produced by Robert Bryant and compact examples constructed by Dominic Joyce; further analytic techniques were advanced by Kovalev, Alexei Kovalev, and Mark Haskins. Spin(7) examples were given by Bryant and compactified by researchers like Joyce and Jason Lotay. These manifolds are central to work by Edward Witten, Michael Atiyah, Gordon L. Kane, and Steven Weinberg in applications to string/M-theory, and to geometric analysis pursued by Richard Schoen and Klaus Uhlenbeck.
Construction methods include Yau's solution techniques credited to Shing-Tung Yau, orbifold resolution techniques by Dominic Joyce, and gluing constructions by Alexei Kovalev and Ronald Stern. Constructions draw on algebraic geometry from Kunihiko Kodaira, birational techniques by Shigefumi Mori, and deformation theory developed by Kuranishi and applied by M. Reid. Examples and explicit metrics were developed by Robert Bryant, Nathaniel Hitchin, Simon Donaldson, Yuri Manin, and Klaus Friedrich. Techniques from symplectic geometry and mirror symmetry involve inputs from Maxim Kontsevich, Denis Auroux, Paul Seidel, and Dusa McDuff.
Special holonomy permeates models in String theory, M-theory, and compactification schemes used by Edward Witten, Cumrun Vafa, Joseph Polchinski, and Juan Maldacena. Calabi–Yau compactifications underlie phenomenology explored by Philip Candelas and Gary Horowitz; G2 compactifications inform M-theory vacua considered by Michael Atiyah and Edward Witten. In geometry, links to gauge theory and invariants connect to Simon Donaldson, Edward Witten, Clifford Taubes, and Kronheimer–Mrowka work. Special holonomy informs advances in moduli spaces studied by Pierre Deligne, Maxim Kontsevich, and Curtis McMullen, and has inspired collaborations across Institute for Advanced Study, IHÉS, and Perimeter Institute.