# Questions about T- and U- dualities and the non-compact U(1) [closed]

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!]

## closed as too broad by ACuriousMind♦, sammy gerbil, Gert, CuriousOne, Cosmas ZachosAug 1 '16 at 14:36

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• 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. – ACuriousMind Jul 31 '16 at 10:54
• Can this question be reopened if I edit it to cut it down to just one question? – leastaction Aug 18 '16 at 19:13