Understanding Compactified Dimensions in Superstring and M-Theory

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In summary, superstring theory includes a 10-dimensional space-time with 6 compactified spatial dimensions in Calabi-Yau manifolds and 4 expanded dimensions, with one additional dimension in M-theory. The shape of the compactified dimensions in M-theory must have G2 holonomy in order to preserve N=1 supersymmetry and cannot always be represented as a direct product of a circle and Calabi-Yau manifold. This highlights the lack of a definitive definition for the dimension of the compact space.
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Kevin_Axion
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In superstring theory there is a 10 dimensional space-time with 6 compactified spatial dimensions in Calabi-Yau manifolds and 4 expanded dimensions with three being space and one temporal. Now with m-theory there is one more dimension so does that mean there are 7 compactified dimensions (I know it appears illogical to ask this question but I always here String Theorists saying there are 6 compactified dimensions, most likely because they aren't discussing m-theory). If so what is the shape of these compactified dimensions?
 
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Well you can get usual 10D string theory by compactifying M-theory on a circle.
 
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If one wants to get down from d=11 to d=4 and preserve N=1 supersymmetry, then the 7-dimensional compact manifold needs to have G2 holonomy. This group is the analog of SU(3) for a six-dimensional Calai-Yau manifold.

Such "G2-manifolds" are generically not equivalent to a direct product of a circle times a Calabi-Yau manifold, so a priori yield a different class of theories in d=4.
However some of such compactifications are dual to "ordinary" compactifications of d=10 strings on Calabi-Yau spaces. This shows again that there is no absolute notion of what the dimension of the compact space is.
 

Related to Understanding Compactified Dimensions in Superstring and M-Theory

1. What are compactified dimensions?

Compactified dimensions are additional spatial dimensions that are hidden or "compactified" at a very small scale in certain theories, such as superstring theory and M-theory. These dimensions are thought to be curled up or "wrapped" in a way that makes them undetectable at our current level of observation.

2. Why are compactified dimensions important in superstring and M-theory?

In superstring and M-theory, the existence of compactified dimensions is crucial to resolving the theoretical inconsistencies that arise when attempting to unify the four fundamental forces of nature (gravity, electromagnetism, strong nuclear force, and weak nuclear force). By incorporating these extra dimensions, these theories are able to mathematically describe the universe in a more complete and consistent way.

3. How many compactified dimensions are there?

The number of compactified dimensions varies depending on the specific version of superstring or M-theory being studied. Most theories propose 6 or 7 compactified dimensions, in addition to the 3 dimensions of space and 1 dimension of time that we are familiar with.

4. Can compactified dimensions be observed or measured?

At our current level of technology, it is not possible to directly observe or measure compactified dimensions. However, some scientists are working on experiments and theories that may one day provide indirect evidence for their existence.

5. How does the concept of compactified dimensions impact our understanding of the universe?

The existence of compactified dimensions challenges our traditional understanding of space and time, and forces us to think beyond the three-dimensional world that we experience. It also has profound implications for our understanding of the fundamental laws of physics and the origin and structure of the universe.

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