Metric of n-sheeted AdS_3: Constructing BTZ

[craigthone]

suppose the AdS_3 metric is given by
$$ds^2 =d\rho^2+cosh^2\rho d\psi^2 +sinh^2 \rho d\phi^2$$
what is the n-sheeted space of it? Can the n-sheeted BTZ be constructed from it by identifications as n=1 case?

Thanks in advance.

 

[haushofer]

Maybe you should give the definition of an n-sheeted space here to get more responses.

 

[craigthone]

About n-sheeted space, I do not know the precise defination. I will give some introductions.
consider a 3-dimensional space B_1 whose boundary is M_1. Then the manifold M_n is defined as n-folded cover of M_1: taking n copies of M_1, cutting each of them apart at a region A, and gluing them in cyclic order. Then the bulk solution B_n whose boundary is M_n is the n-sheeted space of B_1.

For example:
BTZ solution can be wriiten as :
$$ ds^2=r^2 d\tau^2 +(r^2+1)^{-1} dr^2 + (r^2+1) d\phi^2$$
n-sheeted BTZ is given by:
$$ ds^2= r^2 d\tau^2 +(n^2r^2+1)^{-1} n^{2}dr^2 + (n^2r^2+1) n^{-2} d\phi^2$$