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I'm following Witten's essay and he writes:

Let us try to make such a theory with one spacetime dimension instead of four. The choices for a one-manifold are quite limited:

and then gives this picture:

enter image description here

My questions are:

  1. Why those are the two options? I understand that topologically those are the only option, but as I understood we care only about diffeomorphisms (that are stronger than Homeomorphisms). Did I understood it correctly and its just give the same result in this case?

  2. What does it mean that those are 1d manifolds in the 4d spacetime? are they represent only a passage of time in a specific location? or do they live in the 4d spacetime in a way when they have components in all the 4 coordinates?

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  • $\begingroup$ This is a math question, not a physics question, so it is off topic. However, even as a math question, it argues from a faulty premise, since there are several other one-dimensional manifolds—variations on "long line." en.wikipedia.org/wiki/Long_line_(topology) $\endgroup$
    – Buzz ♦
    Commented Jun 11, 2021 at 5:13
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    $\begingroup$ You can find a proof of it here : jstor.org/stable/2322421?origin=crossref $\endgroup$
    – Slereah
    Commented Jun 11, 2021 at 7:45
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    $\begingroup$ @ziv You can find a beautiful proof of this fact in the Aprendix "Classifying one dimensional manifolds" in the book "Topology from the differentiable viewpoint" by Milnor. $\endgroup$
    – Ramiro Hum-Sah
    Commented Jun 11, 2021 at 19:21
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    $\begingroup$ @buzz : the long line is not second countable and hence not (according to the most widely used definition) a manifold. $\endgroup$
    – WillO
    Commented Jun 11, 2021 at 22:09

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  1. Yes, it is a theorem that in one dimension there are only two homeomorphism classes of manifolds, and it is a separate theorem that there are only two diffeomorphism classes of manifolds.

  2. Witten is positing a one-dimensional spacetime. It is not a priori imbedded in any higher dimensional spacetime.

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