G2 manifolds

G2 manifolds
Name	G2 manifolds
Holonomy	G2
Related	Spin(7), Calabi–Yau, exceptional Lie groups

G2 manifolds are seven-dimensional Riemannian manifolds whose holonomy group is contained in the compact exceptional Lie group G2, and they play central roles in differential geometry, algebraic topology, and theoretical physics. Originating from the classification of Berger, their study connects the work of Élie Cartan, Marcel Berger, and Simon Donaldson with constructions by Dominic Joyce, Nigel Hitchin, and Edward Witten. Research on these manifolds interacts with topics addressed at institutions such as the Clay Mathematics Institute, the Isaac Newton Institute, and conferences organized by the American Mathematical Society.

Introduction

G2 manifolds were first brought to prominence in the context of holonomy classification by Marcel Berger and later became explicit objects of construction in the work of Dominic Joyce and others. Their geometry is intertwined with the representation theory of the Lie group G2 and with the exceptional series studied by Élie Cartan and Cartan's students; foundational analysis uses techniques developed by Élie Cartan, Hermann Weyl, and Cartan-inspired exterior differential systems. Modern developments involve collaborations and seminars at Harvard University, Princeton University, Stanford University, Cambridge University, and research groups led by figures such as Simon Donaldson, Richard Thomas, Nigel Hitchin, and Robert Bryant.

Definition and characterization

A Riemannian seven-manifold admits a G2 structure when its frame bundle reduces to a G2-principal subbundle; torsion-free G2 structures arise when the associated 3-form is parallel with respect to the Levi-Civita connection, equivalently producing holonomy contained in G2. Characterizations use the stabilizer of a generic 3-form in seven dimensions discovered in the work of Élie Cartan and explicated by Bryant and Salamon; analytic criteria refer to theorems of Karigiannis, results by Joyce, and PDE methods developed in research groups at Imperial College London and ETH Zurich. The existence problem invokes nonlinear elliptic systems studied by analysts such as Karen Uhlenbeck and geometric flows studied by Robert Bryant and Richard S. Hamilton.

Examples and constructions

Explicit complete examples include the asymptotically locally Euclidean constructions of Bryant–Salamon and the compact constructions introduced by Dominic Joyce via resolutions of orbifolds modelled on quotient actions studied in works associated with John Milnor and Michael Atiyah. Further methods use twisted connected sum constructions developed by Alexei Kovalev and refined by teams including Mark Haskins, Hans-Joachim Hein, and Sebastien Salamon, while conifold transitions relate to techniques explored by Philip Candelas and Per Berglund in string compactification contexts. Constructions also draw on complex geometry inputs from Calabi–Yau manifolds studied by Shing-Tung Yau and on special holonomy analogies with Spin(7) examples investigated by Robert Bryant and Simon Salamon.

Topology and invariants

Topological invariants for G2 manifolds involve Betti numbers, torsion linking forms, and characteristic classes such as the first Pontryagin class, studied in the tradition of Raoul Bott and Isadore Singer. Relations between Betti numbers and geometric structures invoke techniques from the work of William Thurston, John Milnor, and Frederick Almgren, while index-theoretic inputs use the Atiyah–Singer index theorem developed by Michael Atiyah and Isadore Singer. The study of fundamental groups and covering spaces connects with results by Hatcher and William Thurston, and calculations of invariants in explicit constructions were advanced in papers by Dominic Joyce and collaborators including Spencer Bloch and Tomasiello.

Metrics with G2 holonomy (torsion-free and torsionful)

Torsion-free G2 metrics satisfy a system of nonlinear PDEs equivalent to closure and coclosure of the defining 3-form; existence proofs utilize analytic gluing techniques and Nash–Moser-type implicit function theorems as encountered in work by Kohn, Nash, and Serge Lang-influenced analysis. Torsionful G2 structures, often termed "G2-structures with intrinsic torsion", appear in studies of flux compactifications by Edward Witten, Andy Strominger, and Joseph Polchinski in string theory contexts, with mathematical analysis contributed by Nigel Hitchin and Gil Cavalcanti. The study of Laplacian flow for G2 structures was pioneered by Robert Bryant and continued in analytic programs associated with Jeffrey Lotay and Jason Lotay.

Moduli spaces and deformation theory

The moduli space of torsion-free G2 structures is a smooth manifold under unobstructedness conditions, with dimension given by the third Betti number; deformation theory parallels Kodaira–Spencer theory for complex structures explored by Kunihiko Kodaira and Donald Spencer, and Hodge-theoretic techniques parallel those used by Pierre Deligne and Jean-Pierre Serre. Global properties of moduli spaces involve boundary phenomena studied in relation to wall-crossing influenced by research of Maxim Kontsevich and Alexei Kitaev and compactification questions tied to the work of Gerd Faltings and James McKernan. Analytic foundations reference Kuranishi models developed by Masatake Kuranishi and applications to obstruction theory connect to papers by Dominic Joyce and collaborators.

Applications in mathematics and physics

In mathematics, G2 manifolds contribute to special holonomy classification problems and to enumerative invariants analogous to Donaldson–Thomas theory associated with Simon Donaldson and Richard Thomas. In theoretical physics, compactifications of M-theory and eleven-dimensional supergravity on G2 manifolds produce models studied by Edward Witten, Cumrun Vafa, and Andrew Strominger, with implications for supersymmetry breaking and phenomenological model building examined by Michael Green, John Schwarz, and researchers in string phenomenology at CERN and SLAC. Interdisciplinary collaborations connect to geometric analysis programs at Princeton University, Caltech, and Perimeter Institute where mathematicians and physicists pursue dualities and mirror-symmetry-inspired phenomena involving G2 backgrounds.

Category:Differential geometry Category:Special holonomy Category:Seven-dimensional manifolds