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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

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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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ensure that the evolved E3iand F(a) also satisfy the constraint (9). That this is indeed the

case is ensured by our variational principle, and is easiest to see directly in the covariant

picture.

Observables: The Hamiltonian of the bulk theory is a linear combination of first class

constraints. Therefore the gauge invariant observables are phase space functionals that

have weakly vanishing Poisson brackets with the constraints. For Λ ?= 0 we have not been

able to find any observables. While we do not have a proof of this,3the fact that there

is a unique solution of the bulk Dirac quantization constraint (described below) suggests

there are no bulk observables. On the other hand, for Λ = 0 there are two types of bulk

observables. One type is parametrized by loops (and are traces of holonomies), while the

other is parametrized by both loops and surfaces [11]. In this paper we will be interested in

the case of non-vanishing Λ.

Contrasted with the bulk case, there are an infinite set of boundary observables for non-

zero values of Λ. Since the boundary constraint generates Yang-Mills gauge transformations

on a, the boundary observables are traces of the holonomy of a for all loops lying on the

spatial boundary Σ2. Denoting these observables by

Tγ[a] ≡ Tr P exp

?

γa

(10)

for loops γ, their Poisson algebra is

{Tα[a],Tβ[a]} = ∆(α,β)(Tα◦β[a] − Tα◦β−1[a]), (11)

where

∆(α,β) =

?

ds

?

dt ǫab ˙ αa(s)˙βb(t) δ2(α(s) − β(t)) (12)

measures a weighted intersection number of the loops α and β, and β−1denotes traversal

of the loop in the opposite sense ( ˙ αa(s) is the tangent vector to the loop at the parameter

value s).

Consequently, the 4-dimensional theory we have outlined has an infinite number of bound-

ary observables parameterized by loops lying in the 2-boundary Σ2of Σ3. The observables

form a closed infinite dimensional Poisson algebra.

On the constraint surface, the bulk contains no local degrees of freedom: for gauge

group SU(N) there are 3(N2− 1) configuration variables Ai

aand 4(N2− 1) first class bulk

3A proof may be devised along the following lines: Write down all the basic local and non-local Gauss law

invariant variables. The local ones are combinations of the electric and magnetic fields with internal indices

contracted; there are four such variables. The non-local ones are the traces of electric field insertions between

holonomy segments such as Tr[Ea(x0)Uγ(x0,x1)Eb(x1)Uγ(x2,x3)Ec(x3)···]; these are a countably infinite

set. (In the limit of the loop γ shrinking to a point, these become functions of the local variables). Consider

the general Gauss law invariant function to be an arbitrary function of these variables, and calculate its

Poisson bracket with Ja, and see if the result can be made to vanish.

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constraints. This means that there can be at most a finite number of bulk topological

degrees of freedom. However, since there appear to be no bulk observables to evaluate on the

constraint surface, it is likely that there are no bulk degrees of freedom in this theory. This

means that the large gauge freedom may be used to set all bulk fields to zero. Nevertheless

there may still be an infinite number of surface observables on the reduced phase space.

To see this take E3ito be zero everywhere in the bulk but keep it arbitrary and non-zero

on the boundary. This gives the curvature of a via the surface constraint (9). Conversely,

given any boundary field a, the boundary field E3iis determined. This field may then be

arbitrarily extended into the interior. A particularly simple case is where E3ivanishes on

the boundary. Then the reduced phase space of our four-dimensional theory is the (finite

dimensional) moduli space of flat connections on the 2-boundary Σ2, which may be a surface

of arbitrary genus.

The boundary observable algebra given here is reminiscent of, but fundamentally different

from, the construction that gives the Kac-Moody boundary observable algebra associated

with 3-dimensional gravity. However, the Brown-Henneaux[12] construction of observables

for the latter theory is intrinsically dependent on the fall-off conditions of the bulk fields. In

particular, almost all the Brown-Henneaux observables vanish identically on solutions of 3d

gravity such as the BTZ black hole [13]. In our construction this is manifestly not the case –

the fields in the bulk may be obtained from any connection a (on Σ2), whose curvature gives

the boundary value of E3ivia the boundary constraint (9). Conversely, given bulk fields,

the boundary value of E3ifixes the curvature of the boundary connection a. As such, all

the observables are non-zero on generic solutions, unlike the case of 3-dimensional gravity.

Quantization: We will carry out a quantization in the Hamiltonian formulation described

above. There are two ways to approach this: (i) convert the classical constraints into operator

equations in a suitable representation and attempt to solve them for the quantum states, or

(ii) find a representation of the algebra of classical gauge invariant observables.

In a model such as the one described here, it is possible to carry out a “hybrid” quantiza-

tion using both of these approaches simultaneously. This is because the bulk and boundary

states have a natural separation. Specifically, the bulk constraint can be imposed as a Dirac

quantization condition (since we do not have any bulk observables to find a representation

of), while the boundary sector can be quantized by finding a representation of the algebra of

the Tα(a) observables. We will follow this procedure, and define the bulk quantum constraint

as acting by the identity on boundary states.

Consider first the bulk constraints and use the connection representation; Aiare treated

as configuration variables, and their conjugate momenta Eiare treated as functional deriva-

tive operators

Ei→ −i

δ

δAi.(13)

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We assume that the quantum states may be written as the product

Ψ = ψΣ3(A) ⊗ ψΣ2(α),

where α denotes the parameterization of boundary states (to be discussed below). The bulk

constraint Jaigives the condition

??

ǫabcF(A)i

bc− iΛ

δ

δAia

??????Σ3ψΣ3

?

⊗ {I|Σ2 ψΣ2(α)} = 0 (14)

where I is the boundary identity operator. The unique solution of this constraint is,

ψΣ3(A) ≡ exp

?

−i

Λ

?

Σ3Tr

?

A ∧ dA +2

3A ∧ A ∧ A

?

−

i

2Λ

?

Σ2Tr (A1A2)

?

(15)

This state also satisfies the bulk Gauss constraint. The surface term in the exponential is

necessary to guarantee that the functional derivative gives ǫabcFi

bulk Gauss law invariance. The solution in the case where Σ3is compact without boundary

is the bulk part of this functional, and has been discussed in [14].

This Chern-Simons state is not directly related to the Chern-Simons part of the origi-

nal action (5): the state is still a solution of the bulk constraints if Σ3has no boundary.

Furthermore, although (15) is a solution of the bulk constraint, the functional does not

transform trivially under the generators of the constraint (i.e. the Poisson bracket of the

constraint with the functional does not vanish). This is unlike the Wilson loop functional,

which is simultaneously a Gauss law invariant classical observable, as well as a quantum

state satisfying the quantum Gauss constraint.

(The latter result might seem surprising at first, and in apparent violation of the intuition

derived from the Gauss law. However, it is also illustrated in a simple quantum mechanical

example4. Consider the constraint equation,

bc, and does not spoil the

(ˆ x + αˆ p)ψ = 0

where α is some dimensionful constant. The solution to this constraint in the x representation

is

ψ ∝ exp

?

−i

2αx2

?

This function is clearly not invariant under the transformation x → x + α generated by the

constraint. However, the canonical transformation ˜ p = x+αp, ˜ x = x/α reduces the constraint

to ˜ p˜ψ = 0 whose solution is˜ψ(˜ x) = constant, which does commute with constraint.)

Turning now to the boundary dynamics, we choose to quantize this sector by finding a

representation of the algebra of the boundary observables Tα[a]. This is easiest to do in

4The authors would like to thank W. Unruh for pointing out this example.

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the loop representation, and we follow here the prescription used in the approach to non-

perturbative quantum gravity [15, 16]. The holonomy observablesˆTαare defined to act on

loop states |β > by

ˆTα|β >:= i¯ h∆(α,β)

?

|α ◦ β > −|α ◦ β−1>

?

. (16)

¿From this definition it follows that

[ˆTα,ˆTβ] = ∆(α,β)

?ˆTα◦β−ˆTα◦β−1

?

. (17)

Thus, in this approach the boundary states are the loop kets |α >, and the full quantum

state of the theory is the product

|A,α >= ψ[A]|α > .(18)

The loop states |α > are not all independent: the states are traces of holonomies in the

connection representation and are subject to the Mandelstam identities induced by the trace

relations on SU(N) matrices. Furthermore, because there is a unique bulk state, the labeling

of quantum states is effectively only by loops. An inner product on this space of states may

be defined as < α|β >= δαβ. This completes the description of the quantum theory.

Discussion: The model we have described has a number of unusual features which are

useful to compare with Chern-Simons theory on a manifold with boundary, and with 2+1

gravity in particular. These latter theories have the property that they are topological in

the bulk and, with particular fall-off conditions on the fields, induce a Kac-Moody algebra

of observables (which are all constants of motion) on the boundary. These theories thus

have non-trivial bulk and boundary observables, (if the bulk has non-trivial topology). The

boundary observables may be viewed as the observables of a two-dimensional boundary

conformal field theory. Apparently for this reason, these theories have been viewed as an

example of the AdS/CFT correspondence [17], and therefore an example of holography.

However this does not correspond to holography for any of the possible definitions given

above. What would be required for a correspondence with one of these definitions is the

specification of a 2-dimensional theory that has both the bulk and boundary observables of

3-dimensional gravity. (See [18] in this regard.)

For the case of our 4-dimensional model, the 3-dimensional theory that has the same

observables algebra is Chern-Simons theory coupled to an external source Ja(which plays the

role of E3i), ie. the action is the Chern-Simons one with the additional term

Consequently, in addition to viewing our model as an example of type (iv) holography, we

can also view it as type (iii) holography.

In summary, we have discussed some aspects of holography in a 4-dimensional model in

which all degrees of freedom are associated with loops on a 2-dimensional boundary. We have:

1

Λ

?

R×Σ2 AaJa.

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(i) pointed out that topological bulk observables are missed by any boundary theory that is

directly induced by the bulk fields, (ii) suggested that this shortcoming may be side-stepped

by broadening sufficiently the definition of holography, and (iii) given a 3-dimensional theory

that has the same observable algebra as our 4-dimensional model. This provides a concrete

example of duality: theories in different spacetime dimensions having the same classical and

quantum observable algebra.

This work was supported by the Natural Science and Engineering Research Council of

Canada, and by a University of British Columbia Graduate Fellowship. We thank G. W.

Semenoff and W. G. Unruh for discussions.

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