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arXiv:hep-th/9812162v2 6 Jan 1999

Topological Holography

Viqar Husain and Sebastian Jaimungal1

Department of Physics and Astronomy

University of British Columbia

6224 Agricultural Road

Vancouver, British Columbia V6T 1Z1, Canada.

Abstract

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: husain@physics.ubc.ca, jaimung@physics.ubc.ca

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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[1],

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 [5].

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]

?

exp−

?

AdSd+1LSUGRA(φi(φB

i))

?

=

?

exp

?

∂AdSd+1OiφB

i

?

CFT

.(1)

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

φB

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

1

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?

?

Figure 1: Two Wilson lines piercing the boundary at the same points where path (a) is

trivial while (b) wraps around a handle.

“visible” on the boundary of the spatial region through a physical process. This may

be done via a “screen mapping” [4, 10].

(ii) For an n-dimensional theory, T1, containing bulk and boundary degrees of freedom,

there is an (n−1)-dimensional theory, T2, which captures all the degrees of freedom of

T1. Then the possibility is open that T2 is itself defined on a manifold with boundary,

having both bulk and surface degrees of freedom. This is obviously different from (i).

(iii) The same as (ii) with the extra condition that T2 is a theory defined strictly on the

boundary of the region on which T1 is defined. In this case, T2 has only bulk degrees

of freedom, since the boundary of a boundary vanishes.

(iv) All the degrees of freedom of a theory in a bounded region are associated with its

boundary. In this case holography is automatic.

The AdS/CFT correspondence appears to fall in category (iii). However, the Beckenstein

bound argument, which requires entropy to be proportional to the bounding area, appears

to be consistent with all four possibilities.

An interesting possibility which should be taken into account in a definition of holography

is the case of theories which have bulk topological degrees of freedom, associated for example

with handles. It is then possible that these are not visible on the boundary. An example is a

Wilson line observable with end points on the boundary which may or may not wrap around

a bulk handle (see Figure 1). Then only (ii) seems viable as a definition of holography, and

may exclude the AdS/CFT case (iii).

In this paper we describe a theory which is holographic in the sense of both (ii) and

(iv) above. The quantum states factorize into bulk and boundary states, with a unique

bulk state. It has the unusual property that all its quantum states are effectively associated

with loops lying on the spatial boundary. The dynamics of the loop states is trivial, so

their worldlines are cylindrical. Furthermore, the bulk states give rise to external sources

for the boundary quantum theory in a natural manner. This latter feature is similar to one

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of the key features in the AdS/CFT correspondence, albeit in a simpler context. Models of

this type may be used to probe the limitations of the holographic hypothesis and perhaps

the AdS/CFT correspondence. One such limit, as briefly mentioned above, appears to be

that bulk topological degrees of freedom are not captured by the values of local fields on

boundary.

The Model: The theory we consider is a topological field theory on a four-dimensional

manifold lR×Σ3, defined by an unusual specification of the variational principle. The action

is

S[A,B;a] =

?

lR×Σ3Tr

?

B ∧ F(A) +Λ

2

B ∧ B

?

+ k

?

lR×Σ2Tr

?

a ∧ da +2

3a ∧ a ∧ a

?

(2)

where B is a Lie-algebra valued two form, A and a are Lie-algebra valued one forms, and

F(A) denotes the curvature two-form. Σ2is the 2-boundary of the “spatial” surface Σ3.

This action contains no explicit interaction terms between the bulk and boundary fields.

However, the action alone does not determine the equations of motion or the subsequent

canonical structure, since it must be supplemented by a variational principle. Particular

choices of the variational principle can lead to a situation in which the bulk BF-theory is

coupled to the boundary Chern-Simons theory. We will invoke the particular scheme where

the field a must be varied in accord with the variations of the field A on the boundary.

The variation of this action is

δS = Tr

?

lR×Σ3[δB ∧ (F(A) + ΛB) + δA ∧ (DAB)] + Tr

?

lR×Σ2[k F(a) ∧ δa + B ∧ δA], (3)

the variational principle is well-defined if we require that

δa = δA|bd.

δAS ≡ S[A + δA,B;a + δA|bd.] − S[A,B;a],(4)

and the usual requirement that all surface terms in the variation vanish. The constraints on

the variations of the fields may be viewed as giving rise to the equations of motion for the

boundary theory. Ordinarily, a variational principle is supplemented by conditions such as

the vanishing of the variations of certain fields on the boundary. Our prescription is unusual

only in that it fixes the variation of certain fields to equal certain other variations on the

boundary.

A question concerning our approach so far is why we do not, perhaps more simply,

consider the above action with a = A at the outset. The reason is that doing this gives

a different theory: functional differentiability requires that the fields A and B satisfy the

condition F(A) = B on the boundary. In our case on the other hand, the bulk field B

provides a source for the independent boundary curvature F(a).

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The phase space variables are identified by writing

S[A,A0,B,B0;a,a0] =

?

dt

? ?

Σ3Tr

?

?

?

B ∧˙dA + B0∧ (F(A) + Λ B) + A0∧ DAB

?

+ k

Σ2Tra ∧˙da + a0∧ F(a)

?

+

?

Σ2Tr A0∧ B

?

(5)

In this expression we have written A = A0+ A, B = B0+ B and a = a0+ a, where the

fields carrying a subscript 0 contain the ‘time’ components, and the other fields contain the

spatial components of the respective forms;˙d represents the time component of the exterior

derivative; F(·) is the spatial part of the curvature two form; and DAis the spatial part of

the covariant derivative. The boundary contribution has two terms: the first is the Chern-

Simons action in Hamiltonian form, and the second is from an integration by parts in the

bulk part of the action. This latter term provides a bulk source for the boundary curvature.

The canonical structure of the theory is obvious from (5): Ai

the canonically conjugate variables in the bulk, ai

boundary; (a,b,··· are spatial indices 1,2,3 in the bulk, and 1,2 on the boundary; i,j,···

are Lie algebra indices). The time component fields (a0,A0and B0) appear as Lagrange mul-

tipliers, and varying these fields gives the phase space constraints. Since Σ3has a boundary,

there is an additional boundary constraint arising from functional differentiability of the

action (recall that a0must be varied with A0(4)). The relevant variations are

aand Eai≡ ǫabcBi

2are canonically conjugate on the

bcare

1and ai

δA0S =

?

?

dt

??

?

Σ3Tr {δA0∧ DAB} +

?

Σ2Tr {δA0∧ (B + k F(a))}

?

(6)

δB0S = dt

Σ3Tr {δB0∧ (F(A) + Λ B)}(7)

¿From the above it is clear that the Hamiltonian is a linear combination of constraints.

The bulk constraints are

Gi≡ (DaEai)

???Σ3= 0 ,Jai≡

?

ǫabcF(A)i

bc+ Λ Eai????Σ3= 0(8)

In addition to these there is a surface constraint

Hi≡

?

E3i+ k ǫabF(a)i

ab

????Σ2= 0,(9)

due to the presence of the surface integral in (6).

It is clear that both the bulk and boundary constraints are first class and provide the

complete prescription for classical Hamiltonian evolution. The bulk phase space variable

E3iprovides a source for the boundary curvature, and the boundary fields a evolve via the

first class constraint (9). This evolution is a gauge transformation on a. As a result the

bulk evolution of (A,E) is consistent with the boundary evolution of a. To see this more

explicitly, start from an initial classical configuration where E3ifixes F(a). We must now

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