World-sheets, manifolds, and coordinate systems

[Mike2]

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 x^μ verses the 2D surface parameterized by (σ,τ). Are x^μ locally Euclidean? Are the coordinates (σ,τ) locally Euclidean? Remember x^μ are functions of the parameters (σ,τ) or x^μ=x^μ(σ,τ) which defines a surface in space-time. How does this all relate to manifold theory?

Thanks.

 

[Ambitwistor]

It doesn't make sense to say whether coordinates (such as the embedding spacetime coordinates x^μ 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).

 

[Mike2]

Originally posted by Ambitwistor
It doesn't make sense to say whether coordinates (such as the embedding spacetime coordinates x^μ 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?

 

[Ambitwistor]

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.)