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For example, in our universe, gravity and electromagnetism obey an inverse-square law due to the dimensionality of space; in higher dimensions, they would drop off faster (with the consequence that there would be no non-circular stable orbits). The nuclear force, on the other hand, is effectively one-dimensional, being restricted to flux tubes between quarks, despite living in 3D space. But, it's also a strictly short-range force.

So, is there any sensible model or set of models for how a hypothetical force could behave which is both long-range, and constrained to a lower dimension than the space it lives in? E.g., a 1D version of gravity in 3-space, or a 3D analog for electromagnetism in 4-space that would permit something like atoms to exist in a 4D universe, etc.?

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Yes, one could e.g. imagine extra compact dimensions, or a spacetime confined to a brane. Both proposals have a microscopic spacetime dimension different from an effective spacetime dimension. They have been considered in the literatur in various contexts.

E.g. in a spacetime with 3+1 large dimensions, atoms could still be stable if the extra compact dimensions are much smaller than the characteristic atomic size, cf. this Phys.SE post.

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