Are extra dimensions timelike or spacelike?

Question

In special relativity there is a clear difference between spatial and temporal dimensions of spacetime due to the Minkowski metric diag(-1,1,1,1). In higher dimensional theories (10- and 26-dimensional string theories) does this asymmetry continue with additional dimensions being specifically time- or space-like or is there no clear difference?

Answer 1

From Polchinski's String Theory, Chapter 1:

We want to study the classical and quantum dynamics of a one-dimensional object, a string. The string moves in $D$ flat spacetime dimensions, with metric $\eta_{\mu \nu} = \mathrm{diag}(-,+,+,\cdots,+)$.

So all additional dimensions are spacelike.

Strictly speaking Polchinski is only talking about bosonic string theory at this point, but I believe the same applies to superstring theories as well. (It's been a long time since I thought about this in any detail.)