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I'm studying Witten's paper, "String Theory Dynamics in Various Dimensions" (arXiv:hep-th/9503124), and have a few questions from this paper about T- and U- dualities.

On page 3, in the last paragraph, Witten says

...in $d < 10$ dimensions, the Type II theory (Type IIA and Type IIB are equivalent below ten dimensions) is known to have a $T$-duality symmetry $SO(10-d, 10-d; \textbf{Z})$. This $T$-duality group does not commute with the $SL(2, \textbf{Z})$ that is already present in ten dimensions, and together they generate the discrete subgroup of the supergravity symmetry group that has been called $U$-duality.

Question 1 (just to make sure I get this right): From Vafa's lectures (https://arxiv.org/abs/hep-th/9702201), I understand that the T-duality group corresponding to compactification of a Type-II theory on a $d$-torus $T^d$ is $SO(d, d; \textbf{Z})$. So is Witten here referring to a compactification of a $10$-dimensional theory on $T^{10-d}$, i.e. a $(10-d)$-torus, so that there one gets a $d$-dimensional non-compact manifold times a $(10-d)$-dimensional torus?


In the footnote on pages 3-4, Witten says

For instance, in five dimensions, T-duality is $SO(5, 5)$ and U-duality is $E_6$. A proper subgroup of $E_6$ that contains $SO(5, 5)$ would have to be $SO(5, 5)$ itself or $SO(5,5) \times \textbf{R}^*$ ($\textbf{R}^*$ is the non-compact form of $U(1)$), so when one tries to adjoin to $SO(5,5)$ the $SL(2)$ that was already present in ten dimensions (and contains two generators that map NS-NS states to RR states and so are not in $SO(5,5)$) one automatically generates all of $E_6$.

Question 2: Is $\textbf{R}^\star$ the same as $(\mathbb{R}, +)$ described on Stackexchange here? Is the notation standard? Where else can I find it? (Some string theory text?)

Question 3: How do I know whether the proper subgroup of $E_6$ that contains $SO(5,5)$ is $SO(5,5)$ itself or $SO(5,5) \times \textbf{R}^*$? What is the motivation for including $SO(5,5) \times \textbf{R}^*$ in the first place?

Question 4: I understand that the $SL(2)$ being referred to is the S-duality group for Type-IIB in $D = 10$ dimensions (so it is $SL(2, \mathbb{R})$ for Type-IIB SUGRA in $D = 10$ and it is $SL(2, \mathbb{Z})$ for Type-IIB string theory in $D = 10$). But I am confused by the phrase "when one tries to adjoin to $SO(5,5)$ the $SL(2)$ that was already present in ten dimensions...". My question is very silly: why are we trying to adjoin a T-duality group in 5 dimensions with an S-duality group in 10 dimensions in the first place? [Disclaimer: I am fairly certain I have misunderstood Witten's point (and the language) here, so I welcome a critical explanation!]

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    $\begingroup$ Those questions are mostly unrelated and this question hence too broad. As for question 2, $R^\times$ or $R^\ast$ are common notations for the group of units of a ring $R$. $\mathbb{R}^\times$ is isomorphic to $\mathbb{R},+$ through the logarithm. $\endgroup$ – ACuriousMind Jul 31 '16 at 10:54
  • $\begingroup$ Can this question be reopened if I edit it to cut it down to just one question? $\endgroup$ – leastaction Aug 18 '16 at 19:13