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  AdS space as an example of klein geometry

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If we start with the definition of the coset $AdS_{D+2}:=\frac{O(2,D)}{O(1,D)}$ , How do we derive the constraint equation for the AdS coordinates $\mu \nu -(X^{i})^{2}=R^{2}$ ?

asked Jan 7, 2018 in Theoretical Physics by anonymous [ no revision ]

1 Answer

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Consider a vector in $\mathbb{R}^{2,D}$ with norm $-R^{2}$ , The set of all vectors with this norm are rotated into each other by the $O(2,D)$ Rotations. Use this group to make the vector in the form $X=(1,0,0...)$ It is obvious that the isotropy group that leaves this invariant is $O(1,D)$ and thus we get the equivalence because these vectors with the specified norm are in one to one correspondance with the group transformations modulo the isotropy group.

answered Jan 8, 2018 by anonymous [ no revision ]

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