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How can compactifying IIA (non-Chiral) and IIB (Chiral) Superstring on $T^2$ (2-torus) gives rise to ($2$ dual descriptions of) the same $\mathcal N = 2$ supergravity in $8$ dimensions? I don't see it. Could you please explain it to me or recommend me some literature to read about it?

Thank you!

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First, I have to correct you: compactifications of string theory clearly don't give you just "supergravity" – which is an inconsistent theory at short distances. They give you the full string theory, a compactified one.

Type IIA and type IIB are T-dual to each other. It means that the compactification of one on a circle $S^1$ of radius $R$ is equivalent to the compactification of the other on a circle of radius $\alpha' / R$ where $\alpha'$ is the squared string length (the Regge slope) or $1/R$ in some natural "string units" of length.

T-duality is covered in every modern textbook of string theory and to reliably explain how it works, one would have to reproduce several chapters of an introduction to string theory here which is counterproductive. It's ultimately a symmetry because it's a parity i.e. a sign flip of $\partial_z X^9$ that is only applied to the left-movers but not to the right-movers $\partial_{\bar z} X^9$. That's what interchanges $\int d\sigma \partial_\tau X^9$, a total momentum component, with $\int d_\sigma \partial_\sigma X^9$, the total winding. The left-movers and right-movers may be treated rather separately in CFTs because of the conformal symmetry.

You talk about the compactifications on $T^2$ which may be achieved by compactifying one more dimension. Because the two $S^1$ compactifications – of type IIA and type IIB – were already equivalent to each other, they remain equivalent to each other if you compactify one more dimension, of course. The moduli spaces obtained by compactifying M-theory or type IIA or type IIB on tori are all identical – and they still have a U-duality group identifying different choices of the radii and other parameters. The U-duality group for M-theory on $T^k$ is formally $E_{k(k)}(Z)$ which is only truly exceptional if $k$ is greater than five.

There is no conflict with chirality because the theory with 8+1=9 large dimensions that you get by compactifying type IIA or type IIB on a circle is already non-chiral – after all, in an odd spacetime dimension, almost all theories are non-chiral. For example, there are no Weyl (chiral) spinors in an odd space or spacetime dimension.

The only chiral vacuum among the compactifications of M-theory, type IIA, or type IIB theories on tori is the uncompactified type IIB string theory in 10 dimensions. Everything else is non-chiral. You lose the chirality once you compactify a theory on a circle. The left-right symmetry of the lower-dimensional theory is guaranteed because it may be obtained by flipping the sign of one large (remaining) dimension as well as one compactified (circular) dimension and this flip of two dimensions is just a 180-degree rotation which is always a symmetry. So parity symmetry can't be violated by the circular compactifications.

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