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  Questions about T- and U- dualities and the non-compact U(1)

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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(10d,10d;Z). This T-duality group does not commute with the SL(2,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 Td is SO(d,d;Z). So is Witten here referring to a compactification of a 10-dimensional theory on T10d, i.e. a (10d)-torus, so that there one gets a d-dimensional non-compact manifold times a (10d)-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 E6. A proper subgroup of E6 that contains SO(5,5) would have to be SO(5,5) itself or SO(5,5)×R (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 E6.

Question 2: Is R the same as (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 E6 that contains SO(5,5) is SO(5,5) itself or SO(5,5)×R? What is the motivation for including SO(5,5)×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,R) for Type-IIB SUGRA in D=10 and it is SL(2,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!]

This post imported from StackExchange Physics at 2016-07-31 16:03 (UTC), posted by SE-user leastaction
asked Jul 31, 2016 in Theoretical Physics by leastaction (425 points) [ no revision ]
Those questions are mostly unrelated and this question hence too broad. As for question 2, R× or R are common notations for the group of units of a ring R. R× is isomorphic to R,+ through the logarithm.

This post imported from StackExchange Physics at 2016-07-31 16:03 (UTC), posted by SE-user ACuriousMind

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