Entropy Problem in the AdS$_3$/CFT correspondence

We resolve the entropy problem in the AdS$_3$/CFT correspondence by introducing both the normalizable and non-normalizable bulk modes. On the boundary, the normalizable Liouville states gives us $c=1$ conformal field theory(CFT), whereas the non-normalizable Liouville states provide $c = {3 \ell \over 2 G}$ CFT. Such (boundary) non-normalizable modes come from non-normalizable bulk modes which serve as classical, non-fluctuating bulk background and encode the choice of local operator insertion on the boundary. Since the non-normalizable bulk modes can transfer information from bulk to boundary, it suggests that counting of non-normalizable states on the boundary at infinity leads to the entropy($S={2 \pi r_+ \over 4 G}$) of (2+1)-dimensional gravity with $\Lambda= -1/\ell^2$.