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The twistor space of Penrose's twistor theory is a projective space of three complex dimensions. This can be understood as six orthogonal dimensions, three with real metric and three with imaginary metric.

The Calabi-Yau spaces which occur in many versions of string theory are also of six orthogonal dimensions - three real and three imaginary. But these spaces are compactified and finite, so they close up like the cross-section of a hosepipe.

According to Wikipedia, Ed Witten developed a twistor string theory in which "string theory may be introduced naturally into twistor space."

Is the dimensional equivalence of twistor and Calabi-Yau spaces coincidental, or does it play a significant role in Witten's model?

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  • $\begingroup$ Not sure why people are downvoting this question without commenting why. If a fairly abstruse "no" is a valid answer, why should the question not be worth asking? $\endgroup$ Apr 24, 2021 at 19:18

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It's a coincidence since a higher dimensional space time would give a higher dimensional twistor space.

However, the dimension of spacetime in string theory is fixed to ten in order to cancel the conformal anomaly. This gives a fixed internal dimension of six over a four dimensional spacetime.

In the Connes-Chamseddine model of the classical standard model, there is a non-commutative fibre that has a classical dimension of zero and a non-classical dimension of six. According to NLab, this is no coincidence.

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It's very important to recall that Witten's article is concerns (B-model) topological string theory with supertwistor space as target space; therefore, we are not talking about critical superstring theories (whose target spaces are ten dimensional).

Now, is it a 'coincidence' that the critical dimension of topological string theory is exactly the dimension that a space should have to compactify an ordinary string theory to achieve a four-dimensional vacua, namely six? Yes, it's a coincidence, or at least we don't have any good argument from topological strings that seems to support a connection between them and the observed dimension of our universe.

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