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F-Theory, as I understand it, is a realisation of Type IIB String Theory as a 12-dimensional theory in such a way that the $SL(2,\mathbb Z)$ symmetry becomes natural because Type IIB String Theory is F-Theory compactified on a torus.

Question 1: However, this explains only why F-Theory needs to be 12-dimensional. Why does one of these extra infinitesimal dimensions need to be temporal, though.

Question 2: I've also heard that F-Theory isn't exactly a 2-time theory. However, that doesn't make sense to me, as, since the metric signature needs to be $(-1,-1,1,1,1,1,1,1,1,1,1,1)$, i.e. mostly plus, with 2 temporals and 10 spatials. Or am I missing something obvious, here?

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You can't really assign "the only correct signature" to the two extra infinitesimal dimensions of F-theory, so the signature is 11+1. The uniform signature of both of these dimensions is also needed for the moduli space to be $SL(2,R)/SO(2,R)$. Note that $SO(2,R)$ in the denominator is only a symmetry if they have the same signature.

When treated as a compactification, we think of them as of being spacelike. On the other hand, when it comes to the reality and chirality of spinors in the would-be 12-dimensional theory, they mimic what you would expect from the signature 10+2. One couldn't have a 32-component real spinor in 11+1 dimensions (spinors are complex in that dimension).

Those things don't contradict each other because the dimensions are infinitesimal. The proper length of the dimensions is really $0=0i$, it is compatible with both signatures.

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