In differential geometry, a G₂ manifold is a seven-dimensional Riemannian manifold

Riemannian manifold

In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

 with holonomy group G₂. The group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

  is one of the five exceptional simple Lie group

Simple Lie group

In group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.A simple Lie algebra is a non-abelian Lie algebra whose only ideals are 0 and itself...

s. It can be described as the automorphism group of the octonions, or equivalently, as a proper subgroup of SO(7) that preserves a spinor

Spinor

In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...

 in the eight-dimensional spinor representation or lastly as the subgroup of GL(7) which preserves a positive, nondegenerate 3-form, . The later definition was used by R. Bryant. Non-degenerate may be taken to be one whose orbit has maximal dimension in . The stabilizer of such a non-degenerate form necessarily preserves an inner product which is either positive definite or of signature . Thus, is a subgroup of . By covariant transport, a manifold with holonomy has a Riemannian metric and a parallel (covariant constant) 3-form, , the associative form. The Hodge dual, is then a parallel 4-form, the coassociative form. These forms are calibrations in the sense of Harvey-Lawson

Calibrated geometry

In the mathematical field of differential geometry, a calibrated manifold is a Riemannian manifold of dimension n equipped with a differential p-form φ which is a calibration in the sense that...

, and thus define special classes of 3 and 4 dimensional submanifolds, respectively. The deformation theory of such submanifolds was studied by McLean.

Properties

If M is a -manifold, then M is:

• Ricci-flat
• orientable
• a spin manifold

History

Manifold with holonomy was firstly introduced by Edmond Bonan in 1966, who constructed the parallel 3-form, the parallel 4-form and showed that this manifold was Ricci-flat. The first complete, but noncompact 7-manifolds with holonomy were constructed by Robert Bryant

Robert Bryant

Robert Bryant is an American mathematician specializing in differential geometry. As of 2007, he has served as the chairman of the Mathematical Sciences Research Institute . He is known for his work in exterior differential systems, special holonomy, and Finsler geometry. He is also a senior...

 and Salamon in 1989. The first compact 7-manifolds with holonomy were constructed by Dominic Joyce

Dominic Joyce

Dominic D. Joyce is a British mathematician, currently a professor at the University of Oxford and a fellow of Lincoln College since 1995. His undergraduate and doctoral studies were at Merton College. He undertook a DPhil in geometry under the supervision of Simon Donaldson, completed in 1992...

 in 1994, and compact manifolds are sometimes known as "Joyce manifolds", especially in the physics literature.

Connections to physics

These manifolds are important in string theory

String theory

String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...

. They break the original supersymmetry

Supersymmetry

In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...

 to 1/8 of the original amount. For example, M-theory

M-theory

In theoretical physics, M-theory is an extension of string theory in which 11 dimensions are identified. Because the dimensionality exceeds that of superstring theories in 10 dimensions, proponents believe that the 11-dimensional theory unites all five string theories...

 compactified on a manifold leads to a realistic four-dimensional (11-7=4) theory with N=1 supersymmetry. The resulting low energy effective supergravity

Supergravity

In theoretical physics, supergravity is a field theory that combines the principles of supersymmetry and general relativity. Together, these imply that, in supergravity, the supersymmetry is a local symmetry...

 contains a single supergravity supermultiplet

Supermultiplet

In theoretical physics, a supermultiplet is formally a group representation of a supersymmetry algebra. It consists of a collection of particles, called superpartners, corresponding to operators in a quantum field theory which in superspace are represented by superfields.Superfields were...

, a number of chiral supermultiplets equal to the third Betti number

Betti number

In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....

 of the manifold and a number of U(1) vector supermultiplets equal to the second Betti number.

See also: Calabi-Yau manifold

Calabi-Yau manifold

A Calabi-Yau manifold is a special type of manifold that shows up in certain branches of mathematics such as algebraic geometry, as well as in theoretical physics...

, Spin(7) manifold