Editing M-theory (section)

===Compactification on {{math|''G''<sub>2</sub>}} manifolds===

In one approach to M-theory phenomenology, theorists assume that the seven extra dimensions of M-theory are shaped like a [[G2 manifold|{{math|''G''<sub>2</sub>}} manifold]]. This is a special kind of seven-dimensional shape constructed by mathematician [[Dominic Joyce]] of the [[University of Oxford]].<ref>Yau and Nadis 2010, p. 149</ref> These {{math|''G''<sub>2</sub>}} manifolds are still poorly understood mathematically, and this fact has made it difficult for physicists to fully develop this approach to phenomenology.<ref name="Yau and Nadis 2010, p. 150">Yau and Nadis 2010, p. 150</ref>

For example, physicists and mathematicians often assume that space has a mathematical property called [[smooth manifold|smoothness]], but this property cannot be assumed in the case of a {{math|''G''<sub>2</sub>}} manifold if one wishes to recover the physics of our four-dimensional world. Another problem is that {{math|''G''<sub>2</sub>}} manifolds are not [[complex manifold]]s, so theorists are unable to use tools from the branch of mathematics known as [[complex analysis]]. Finally, there are many open questions about the existence, uniqueness, and other mathematical properties of {{math|''G''<sub>2</sub>}} manifolds, and mathematicians lack a systematic way of searching for these manifolds.<ref name="Yau and Nadis 2010, p. 150"/>