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Quote Clifford Johnson D-brane page 108

This group includes the T-dualities on all of the $d$ circles, linear redefinitions of the axes, and discrete shifts ofthe $B$-field. The full space of torus compactifications is often denoted: $${\cal M} = O(d, d, Z)\backslash O(d, d)/[O(d) × O(d)].$$

The $/[O(d) × O(d)]$ meant module the group by equivalence class of $[O(d) × O(d)]$ but what was $\backslash O(d, d)$? From the various context it seemed to be an extension. But why not just write $\otimes$ or $\oplus$? What does $\backslash O(d, d)$ mean?

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  • $\begingroup$ @KP99 I didn't quite understand that part. The reading seemed to suggest that the element was $G^Z$ as in isomorphism of $g_{i}^{1}\sim g_i^{2}\in O(d,d,Z)$ etc. How could substruct the $g_i\in O(d,d)$ then? Even it could how would the group still be closed? $\endgroup$
    – ShoutOutAndCalculate
    Commented Nov 14, 2022 at 1:19
  • $\begingroup$ @Ghoster I googled because of the red ares and the symbol was supposed to be "setminus". It immediately made a bit sense to me, then it did not... so I'm hesitating to call it "setminus" without understanding what's going on. $\endgroup$
    – ShoutOutAndCalculate
    Commented Nov 14, 2022 at 1:29
  • $\begingroup$ @Ghoster Got it. "backslash". ^_^! $\endgroup$
    – ShoutOutAndCalculate
    Commented Nov 14, 2022 at 1:33

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The notation in the title stands for left and right cosets of a group $G$ wrt. a subgroup $H$.

$H_1\backslash G/H_2$ denotes a double coset of a group $G$ wrt. two subgroups $H_1$ and $H_2$; one from left and one from right.

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  • $\begingroup$ But wasn't $O(d,d,\mathbb{Z})\backslash O(d,d)\cong \mathbb{Z}$? How could it be modded by $O(d)\times O(d)$ then? $\endgroup$
    – ShoutOutAndCalculate
    Commented Nov 14, 2022 at 2:29

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