arXiv:hep-th/9812162v2 6 Jan 1999
Viqar Husain and Sebastian Jaimungal1
Department of Physics and Astronomy
University of British Columbia
6224 Agricultural Road
Vancouver, British Columbia V6T 1Z1, Canada.
We study a topological field theory in four dimensions on a manifold with boundary.
A bulk-boundary interaction is introduced through a novel variational principle rather
than explicitly. Through this scheme we find that the boundary values of the bulk fields
act as external sources for the boundary theory. Furthermore, the full quantum states
of the theory factorize into a single bulk state and an infinite number of boundary
states labeled by loops on the spatial boundary. In this sense the theory is purely
holographic. We show that this theory is dual to Chern-Simons theory with an external
source. We also point out that the holographic hypothesis must be supplemented by
additional assumptions in order to take into account bulk topological degrees freedom,
since these are apriori invisible to local boundary fields.
1emails: firstname.lastname@example.org, email@example.com
There has been much recent interest in the interplay between bulk and boundary dynam-
ics. The two main directions being explored presently are (i) the Maldacena conjecture,
which postulates a relationship between a bulk string/M-theory and a boundary conformal
field theory (also known as the AdS/CFT correspondence), and (ii) the holographic hypothe-
sis [2, 3, 4], which states that all information about a theory in the bulk of a bounded region
is available, in some sense, on the boundary of the region. In particular, the AdS/CFT
correspondence has been viewed as an example of the holographic hypothesis .
The first of these directions is based in part on the observation that the symmetry group
of d + 1-dimensional anti-deSitter space-time SO(2,d) is the same as the conformal group
of Minkowski space-time in d dimensions. More specifically, a statement of the conjecture2
is [6, 7]
The left hand side of this equation is the evaluation of the Euclidean supergravity action
on the classical solutions for which the background is the (d+1)-dimensional anti-deSitter
metric. φirepresent the bulk dynamical fields in the solution. The surface integral in the
supergravity action, which is a functional of the boundary value φB
here. The right hand side is the quantum expectation value of the primary field Oiof some
conformal field theory on the boundary of AdS, where the boundary value of the bulk field
conformal field theory correlation functions from classical supergravity. It therefore provides
a classical-quantum duality for a sector of the solution space of supergravity, (– the sector for
which the metric is anti-deSitter). A key feature of this prescription is that a classical bulk
field provides, via its boundary value, an external source for a boundary quantum theory.
This feature appears in the model we discuss below.
The second direction in this bulk-boundary interplay is (at least partly) motivated by
arguments concerning black holes: The fact that the entropy of a black hole is proportional
to its area suggests the possibility that the theory describing microstates of a black hole is
either (i) a surface theory, or, (ii) a bulk theory whose states are “visible” on the bounding
surface in such a way that the entropy becomes proportional to the surface area. This is
closely connected to and motivated by the Beckenstein bound argument [8, 9].
There are in fact (at least) four possible definitions of what holography may mean:
iof φi, is a crucial input
iacts as an external source. Thus, this conjectured equality provides a way of computing
(i) For a theory defined in a bounded spatial region, all bulk degrees of freedom are
2There is a more general, and fully quantum mechanical statement of this conjecture, where the left hand
side includes functional integrals over bulk fields φiwhich have boundary values φB
anti-deSitter metrics. The statement of the correspondence given above is effectively the tree level evaluation
of the left hand side, and represents all its tests to date!
i, and over asymptotically
 C. Rovelli and L. Smolin, Nucl. Phys.331 (1991)80.
 A. Ashtekar and C. Isham, Class. Quant. Grav.9(1992)1433.
 M. Banados, K. Bautier, O. Coussaert, M. Henneaux, and M. Ortiz, Phys. Rev.
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 S. Hyun, W. T. Kim, J. Lee, “Statistical entrpy and AdS/CFT correspondence
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