World-sheets, manifolds, and coordinate systems

  • Thread starter Mike2
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I'm trying to understand the manifold properties of world-sheets in string theory. I'm told that world sheets are manifolds and that manifolds are locally Euclidean. So I would like to know the characteristics between the space-time coordinates of the world-sheet given as xμ verses the 2D surface parameterized by (σ,τ). Are xμ locally Euclidean? Are the coordinates (σ,τ) locally Euclidean? Remember xμ are functions of the parameters (σ,τ) or xμ=xμ(σ,τ) which defines a surface in space-time. How does this all relate to manifold theory?

Thanks.
 
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It doesn't make sense to say whether coordinates (such as the embedding spacetime coordinates xμ or the worldsheet coordinates (σ,τ)) are "Euclidean". Manifolds are Euclidean, not coordinates.

Spacetime (the target space) is Lorentzian (locally Minkowski), and so is a string worldsheet embedded within it. (Lorentzian means that its metric has the spacetime signature (-,+,+,+), as opposed to the Euclidean (+,+,+,+).) However, mathematical tricks are often performed performed to treat the worldsheet metric as Riemannian (locally Euclidean).
 
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Re: Re: World-sheets, manifolds, and coordinate systems

Originally posted by Ambitwistor
It doesn't make sense to say whether coordinates (such as the embedding spacetime coordinates xμ or the worldsheet coordinates (σ,τ)) are "Euclidean". Manifolds are Euclidean, not coordinates.
This probably explains why there is a whole chapter in Quantum Field Theory of Point Particles and Strings, by Brian Hatfield about manifold theory, but he never seems to make the connection with world-sheets. I could be wrong. I only skimmed it. I don't have it memorized.

Is it more accurate to say that manifolds CAN BE expressed locally with a euclidean metric?
 
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Re: Re: Re: World-sheets, manifolds, and coordinate systems

Originally posted by Mike2
Is it more accurate to say that manifolds CAN BE expressed locally with a euclidean metric?
I would say: "A Riemannian manifold has a locally Euclidean metric. A Lorentzian manifold has a locally Minkowskian metric. Any manifold is locally Euclidean, topologically speaking."

(There's a difference between having the topology of a Euclidean space, and having the geometry of a Euclidean space.)
 

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