User:Geometry guy/Geometry by dimension

This page was taken from Mathematical landscape. Some numerology, neologisms and unjustified speculation have been removed or edited. The presence of this material here does not indicate I agree with its content. It is preserved here in case some of it can be used in future articles.

This article concerns the special features of different dimensions from the point of view of geometry, physics, and symmetry. Some believe there is an as yet undiscovered principle that connects these special features with the symmetry under which the laws of nature behave. A particular example of such a connection is the Monstrous moonshine conjecture, which linked the Monster group with modular functions and string theory. Another example is the appearance of the largest exceptional lie group, E8, as the gauge group of heterotic superstring theory.

Here we describe some of the objects in N (real) dimensions that are closely related and are believed to be linked in some unified relation between mathematics and physics.

196883 dimensions
The Monster group is the largest sporadic simple group and this is the smallest number of dimensions it acts in. It is the largest finite sporadic group. The monster group is linked with continuous objects like the J-invariant and modular forms by the Monstrous moonshine conjecture. It is conjectured to be the symmetry group of the constraint polynomial in invariance mechanics.

256 dimensions
The number of dimensions (excluding space and time) to represent all the degrees of freedom from supergravity (128 bosons + 128 fermions), which is the same as the lowest order of superstring theory.

248 dimensions
The largest exceptional Lie group E8 acts in 248 dimensions.

26 dimensions
Bosonic string theory is the original string theory consisting only of boson particles. It exists only in 26 dimensions.

The Monster group is the symmetry of a special string theory in 26 dimensions.

24 dimensions
The Leech lattice is a special lattice in 24 dimensions that is the closest arrangement of spheres in 24 dimensions. The symmetry is connected with many sporadic groups including the Monster group.

16 dimensions
The difference between the bosonic string theory and superstring theory is the number of dimensions for the gauge group lattice in heterotic string theory. The gauge group is E8xE8. E8 is the largest continous exceptional Lie group.

12 dimensions
F-Theory is a possible string theory in 12 dimensions.

11 dimensions
M-Theory is a type of string theory involving membranes postulated by Edward Witten.

Supergravity is a quantum gravity theory believed to be a special limit of M-Theory.

10 dimensions
Superstring Theory exists in 10 dimensions.

8 dimensions
The number of dimensions in Penrose's Twistor theory. (Actually 4 complex dimensions).

The number of dimensions needed to be compactified from F-Theory to get 4 dimensions.

Octonions are 8-dimensional objects.

7 dimensions
The exceptional lie group G2 acts in 7 dimensions. It is the automorphism group of the octonions.

The number of dimensions needed to be compactified from M-Theory to get 4 dimensions. They are compactified on G2 manifolds.

6 dimensions
The number of dimensions needed to be compactified from superstring theory to get 4 dimensions. They are compactified on 6-dimensional Calabi-Yau spaces.

5 dimensions
Kaluza-Klein's model for the unification of electromagnetism and gravity is a 5-dimensional theory with one dimensional compact.

4 dimensions
The number of observed space-time dimensions of the Universe with metric. The geometry is that of Minkowski.

It has symmetry group O(3,1)≈SL(2,C) and double-cover SU(2)xSU(2).

Quaternions are 4-dimensional objects.

3 dimensions
Loop quantum gravity was initially formulated as a 3-dimensional theory without time.

Euclidean geometry is formulated in 3 dimensions.

The symmetry group with with triple identity O(3)≈SU(2)≈Sp(1) acts in 3 dimensions.

2 dimensions
The number of dimensions on the world surface of a superstring.

It can be paramaterised with a complex number.

Speculation
Some believe their is a deep relation between special features in the mathematical landscape and the same objects that describe the laws of physics. and that different theories such as string theory, loop quantum gravity, invariance mechanics, twistor theory and so on will turn out to be different aspects of the same theory: the links between these theories will then correspond to relationships between objects of the mathematical landscape.

An example of this idea is the solution of the Monstrous moonshine conjecture linking the Monster group to the j-function via string theory. This has fueled further speculation that the symmetries and representations of the Monster group will be the key to uncovering the proposed connections between mathematics and physics.