March 1, 2021
DFTT 18/2001
FNTT 16/2001
SISSA 56/2001/EP
hepth/0107112
Implementing holographic projections in Ponzano–Regge gravity
Giovanni Arcioni,^{1}^{1}1Address after October 2001: Spinoza Institute, Utrecht, The Netherlands ^{2}^{2}2 Email: Mauro Carfora,^{3}^{3}3Email: Annalisa Marzuoli ^{4}^{4}4Email: and Martin O’Loughlin^{5}^{5}5 Email:
Dipartimento di Fisica Teorica, Universit di Torino,
INFN, Sezione di Torino,
Via P. Giuria 1, I–10125 Torino, Italy Dipartimento di Fisica Nucleare e Teorica, Universit degli Studi di Pavia,
INFN, Sezione di Pavia,
Via Bassi, 6, I–27100 Pavia, Italy S.I.S.S.A. Scuola Internazionale Superiore di Studi Avanzati,
Via Beirut 4, I–34014 Trieste, Italy
Abstract
We consider the pathsum of PonzanoRegge with additional boundary contributions in the context of the holographic principle of Quantum Gravity. We calculate an holographic projection in which the bulk partition function goes to a semiclassical limit while the boundary state functional remains quantummechanical. The properties of the resulting boundary theory are discussed.
PACS 04.60 Nc; 11.10 Kk
Keywords: discretized holography, Ponzano–Regge model, lattice gauge theories.
1 Introduction
According to the holographic principle [1, 2]
Quantum Gravity with some prescribed asymptotic behavior can
be described by a theory defined on the boundary at infinity.
Two recent examples which display holographic behavior are Matrix Theory
[3] and the AdS/CFT correspondence [4]. In particular
regimes and by
considering suitable decoupling limits it has been possible to carry out
many checks of such a behavior and they all supported the holographic
nature of theories
containing gravity (for an exhaustive review see [5]).
This represents a new perspective which seems to
indicate that spacetime physics emerges from an underlying theory living
in lower dimensions.
The holographic principle has been originally proposed by ’t Hooft
[1] in discussing spacetimes containing black holes. In
particular, the fact that the black hole
entropy is proportional to an area supports the holographic principle and the
black hole horizon is then interpreted as a screen encoding bulk information.
This lead ’t Hooft to fomulate an Smatrix Ansatz program so as to
properly incorporate black holes when dealing with quantized gravity.
Since black holes carry the strongest possible gravitational field in any
given volume
they form a very natural
upper bound of the energy spectrum.
A very interesting feature of the ’t Hooft picture is that the theory
living on the horizon–screen of the black hole is supposed
to be some sort of
discretized model [6] which comes about when
including the transverse gravitational force in the construction of the
Smatrix. According to preliminary attempts to implement such a picture,
the operators
on the discretized horizon satisfy an angular momentum algebra and
the horizon is divided into various domains representing the various
representations
of the algebra. In the 2+1 dimensional case [7]
the horizon is described as being built up from string bits with unit lengths
and a covariant algebra of observables associated to ingoing and outgoing
radiation can be indeed constructed.
In such a framework it seems particularly interesting to examine
discretized models in
the presence of a boundary as a possible test of the holographic description of
spacetime. One of the simplest and most natural scenarios is to consider the
Ponzano–Regge model in three dimensions in the presence of a boundary. For
a general introduction to discretized gravity and to the PonzanoRegge model
in particular see the review [8].
We will work in the Euclidean sector and as a preliminary investigation
we examine
the possibility of implementing a holographic description in the presence
of a boundary. More precisely, we will try to implement some sort of
holographic
projection of bulk data precisely in the spirit of ’t Hooft ideas.
In this connection it is worthwhile stressing that the continuum counterpart
of the Ponzano–Regge model is three–dimensional
Chern–Simons theory [9]
and it is well known that for gauge group
this latter induces on the boundary a WZW model
and eventually Liouville field theory. This sort of ”intrinsic” holographic
behaviour has also led to the conjecture that a suitable higher dimensional
Chern–Simons theory with some prescribed symmetries
can be a possible candidate for a holographic description of Mtheory
[10]. Chern–Simons theory has indeed a very big symmetry group
resembling gravity diffeomorphism invariance and this should lead to a
holographic behavior when putting a suitably conditioned boundary in the
theory. In view of these remarks it seems therefore interesting to analyze
the presence of a boundary in a discretized version of Chern–Simons
theory starting precisely from the three dimensional case.
The paper is organized as follows: in Section 2 we briefly recall some
aspects of the
Ponzano–Regge model in the presence of boundary. In Section 3
we separate the pathsum into contributions from the
bulk, intermediate contributions that “communicate from bulk to boundary”
and purely boundary terms. We then take the semiclassical limit of
the bulk part of the pathsum and discuss the resulting action. In Section
4 we make some comments on the resulting split pathsum and the
significance of the various terms.
2 Ponzano–Regge gravity in the presence of boundaries
The original Ponzano–Regge model [11], associated with the partition
function of
–dimensional Euclidean gravity for a closed simplicial manifold ,
can be
extended to deal with manifolds with a non–empty boundary
(where could be the disjoint union of a finite number of
components).
The basic idea is to consider the triangulation induced on the boundary
by the bulk simplicial
decomposition of the threemanifold since the tetrahedra that have one
or more of their
triangular faces in common with the boundary induce a boundary triangulation.
The weight of the path sum associated with this boundary triangulation can
be determined in a variety of ways. Via the canonical formalism whereby
one considers a twomanifold that is evolving in Euclidean time [12].
Otherwise we can use a general topological construction
[13, 14],
in which one enforces invariance of the path sum including boundary
under the discretized version of diffeomorphisms [15]. Finally,
via the discretization of the corresponding BF (firstorder formalism for
threedimensional gravity) theory, imposing suitable
boundary conditions on the path integral [16]. In each of
these cases, the path sum includes sum over the boundary spins, and thus
the boundary geometry has identical weights. In the case of [12] and
[16], this boundary sum remains also a function of the connection
since boundary conditions were chosen such that the connection was fixed
on the boundary.
We give in the rest of this section a brief review of these results for the
path sum with fluctuating boundary.
For simplicity we will take the connection to be trivial, although
in general a global gauge choice of this type cannot be made due to
possible non–trivial holonomy along one–cycles in the boundary two–fold.
Let be a Piecewise Linear () pair of a fixed
topological type, and consider a particular triangulation
with edges,
faces and tetrahedra labelled by suitable elements of the
Racah–Wigner algebra of
(or of its deformed counterpart, a root of
unity [17]).
More precisely, we weight the contribution of any such
triangulation to the partition
function (or state sum)
by introducing the following functional [13]:
(1)  
Here denote the number of vertices, edges and
tetrahedra which
lie entirely in the interior of (), while
are the number of
vertices, edges and triangular faces in ; is the
number of tetrahedra
which have faces in .
The set of spin variables labels edges in , while the
set is associated
with edges in ; each spin variable runs over
.
The cut–off plays here the same role as in Ponzano–Regge
asymptotic formula [11],
i.e. ( an arbitrary constant), and
all spin variables in (1)
are bounded above by .
Each of the symbol in (1) has a precise group theoretical
and geometrical meaning,
namely:

[respectively ], the dimension of the –th [respectively –th] irreducible representation of , gives the contribution to the measure of each edge belonging to [respectively ].

The Racah–Wigner symbol
(2) represents, apart from the phase factor , , a tetrahedron in .
The notation stands for a tetrahedron which has some of its faces in , and thus the entries of such symbol depend on its place in the triangulation. 
The Wigner symbol
(3) apart from the phase factor , , is associated with a triangular face in , and each variable (representing the projection of the corresponding with respect to a fixed quantization axis) satisfies the usual requirement in integer steps.
The partition function to be associated with the manifold is found by summing the contributions of the type (1) over all admissible angular momentum configurations for the simplicial decomposition , under the condition that all spin variables – together with the ’s – are bounded above by a fixed . In the spirit of the Ponzano–Regge approach, in order to remove the cut–off we formally take the limit of the state sum, and thus the expression of the “regularized” partition function can be formally written as:
(4) 
This state sum reduces to the Ponzano–Regge functional in the case
, while we would recover the expression found in
[12] if we keep fixed the triangulation on
the boundary manifold, i.e. for
.
In the asymptotical limit (all spin variables large) (4)
represents the semiclassical
partition function of Euclidean gravity in the presence of a boundary
with the action
discretized according to the Regge prescription [18],
(cfr. [19] for the extension of Regge
Calculus to manifolds with boundary). More precisely, for a
triangulation
the action contains the Einstein–Regge action of the bulk,
namely (where ’s are the deficit angles),
and a suitable contribution
due to the boundary .
To discuss the role of the boundary terms in the partition
function (4) we refer now to [20]
(see also [16, 14, 12]),
where the state sum model induced on
the closed boundary manifold was derived.
It turns out that each triangular
face in a triangulation has to be
associated with the following product of two
symbols
(5) 
Here and are two different sets of momentum
projections associated with the
same angular momentum variables , and
.
The expression of the functional associated with a particular
triangulation is
(6)  
where are the numbers of vertices, edges and triangles in , respectively. Summing over all admissible triangulations we get the partition function of the closed dimensional theory which reads
(7) 
where the regularization is carried out according to the previous prescription. This expression can be evaluated in a staightforward way, providing the expression
(8) 
where is the Euler character of the manifold
.
Thus the partition function of the dimensional closed model induced
on the boundary by
(4) contains the only topological invariant
which is significant in the present context (recall also that the Regge action
in dimension is just ).
Note that the regularization
prescription used above is ill defined in general as there is no natural
way to remove the cutoff.
The naturally regularized counterpart of the closed Ponzano–Regge model
is the
quantum invariant introduced in [17] which turns out to be
related to a double
Chern–Simons–Witten theory at level where is
the deformation parameter.
Moreover, the limit for of the Turaev–Viro
functional corresponds to a semiclassical
partition function containing an Einstein–Regge term plus a
volume term with a positive
cosmological constant related to [28] (cfr. also
[21, 22] on these issues).
In [13], [14] the deformed counterparts of
the state sums (4) and
(8) have been defined, and in particular (8)
corresponds to the quantum invariant
(9) 
where .
The state sums associated with simplicial dissections of
manifolds based
on the recoupling theory of quantum angular momenta have a rich
topological structure which
is non trivially implemented by exploiting a suitable set of
operations on simplices
(topological moves).
In general, given two compact dimensional simplicial
–manifolds (with or
without boundaries) Pachner proved that they are –homeomorphic
if and only if
their underlying triangulations are related to each other either by a
finite set of bistellar moves
(in the closed case, [23]) or by a finite set of
elementary shellings
(in the case with boundary, [15]). Since in the
–type and the
topological equivalence class of any compact manifold are in
correspondence, in the present context
we can simply speak of ”topological equivalent” or ”homeomorphic” manifolds.
The next step consists in recognizing that all the moves we are
referring to can be translated
into algebraic identities which are encoded in the structure of
the partition functions
written above.
Recall that in the closed dimensional
case (see [11], [17])
the bistellar moves can be expressed algebraically in terms of the
Biedenharn–Elliott
identity (representing the moves
[ tetrahedra joined along a common face]
[ tetrahedra joined along a common edge])
and of both the BE identity and the
orthogonality conditions for symbols, which represent the barycentic move
together with its inverse, namely [ tetrahedron] [
tetrahedra]. Thus the Ponzano–Regge functional, namely
(4) for , is formally a topological invariant (and the Turaev–Viro state sum
is in fact a well–defined topological invariant).
In [20] the identities representing the bistellar
moves in have been established
(and they will be given explicitly in the next section).
They are associated with the flip move
[ triangles] [ triangles]
(a pair of triangles joined to form a
quadrilateral are transformed into two triangles joined along the
other diagonal)
and with the barycentric move and its inverse, namely [ triangle]
[ triangles]. Then the fact that (7) is actually a
topological invariant
can be recognized on the basis of its manifest invariance under any
finite set of
such moves.
Turning now to the case of , the topological
transformations
that have to be taken into account are the elementary shellings
and their inverse moves
introduced in [15]. These operations
involve the cancellation of one tetrahedron at a time
from a given triangulation
.
In order to be
deleted, the tetrahedron must have some of its dimensional faces
lying on the boundary . Moreover, for each elementary shelling
there exists an inverse move which corresponds to the attachment of a new
tetrahedron to a suitable component in . In [13]
the identities corresponding to such moves have been found, and the
expression of the
state sum given in (4) and (1) is actually manifestly
invariant under any finite set of such moves. The conclusion is that, owing to
Pachner’s results, (4) is a topological () invariant of
the pair . Notice also that, since
reduces to the Ponzano–Regge partition function in the case , it is automatically invariant also under bistellar moves
in the bulk .
3 Projecting on the boundary and the decoupling limit
In the spirit of the holographic principle we would like
to decouple – or more properly disentangle – the theory living
on the boundary from the bulk gravity theory. We therefore look for some way
of projecting the theory described in the previous section onto the boundary.
Notice that the dimensional state sum given in (7)
is not generated in this way since it simply represents
a theory obtained from the dimensional functional (1)
by restricting it to the boundary and by summing freely over all
triangulations of
the surface , up to
regularization.
To set up a true decoupling procedure one has to examine more carefully
the structure of the functional (1) describing the state
of a single
triangulation . In particular,
one expects to recognize bulk pieces, boundary pieces and interaction
terms bulk–boundary. Bulk pieces are going to describe the
fluctuations of the
spins ’s in the interior of the manifold, boundary pieces the
fluctuations of the spins ’s (and of their corresponding ’s)
on the boundary, and the interaction terms
the ”coupling” between such bulkboundary fluctuations.
The interaction bulk–boundary is clearly given by those tetrahedra which
have some of their components on the boundary triangulation, and in
particular by terms
of the type in (1). Such symbols may have
in principle
a varying number of spins of type or . But, as discussed at the
end of the previous
section, we can take advantage of
the invariance of in (4)
under elementary shellings. In particular we can always transform a
given triangulation
in
such a way that
each tetrahedron which shares with the boundary a number of faces
has actually only one face in .
Call the resulting dissection standard triangulation to be
denoted from now on by
.
Note however that in a standard triangulation there are two distinct
types of tetrahedra
having edges in :

The tetrahedra which are involved in the definition of the standard triangulation itself, namely the ones which share with the boundary one of their faces and the corresponding three edges. Their number is equal to the number of triangular faces in , which have been denoted by in (1), namely we have
(10) 
The tetrahedra which have exactly one edge in and no corresponding face (there could be a varying number of such tetrahedra for any edge in ). Denote their total number by .
We shall call the
coupling tetrahedra.
The symbols to be associated with these types of
coupling tetrahedra in the state
functional of a standard triangulation can be always cast –
by making use of the symmetry
properties of the symbols – in two particular forms
(11) 
where the phase factor is explicitly given by , and
(12) 
with the phase factor given by .
Thus, if we denote by
(13)  
(14) 
we can rewrite the state functional (1) for a standard triangulation by setting
(15)  
where the edges of the coupling tetrahedra
have been labelled by and , respectively.
We have introduced and as there
are only independent edges of type J in
due to the obvious identification of edges in common
to more than one of these coupling tetrahedra.
Ponzano–Regge gravity is a discretized counterpart of a
second order theory in
which the variables associated with the edges play the role of the
metric tensor [18].
Thus, in the spirit of an holographic
scenario, one has to take a suitable limit on some ”metric variables” living
close to the boundary
in order to be able to decouple the boundary theory from the bulk.
The collection of the coupling tetrahedra is
obviously a natural candidate to play this role.
However, we easily recognize that there is a further class of tetrahedra
living close to the boundary,
namely those tetrahedra with edges in , but which have one of
their
vertices in (the weights of these vertices are
included in the factor
of (15)). Denote this new
collection by and
call them coupling tetrahedra too.
Their number runs, say, from to , and each of them
corresponds to a
symbol of the form
(16) 
where no confusion should arise with the generic symbol introduced in (2). Then, if we denote by
(17) 
the first two groups of terms in the right–hand side of (15) can be rewritten as
(18)  
where again takes into account the additional necessary identifications of edges between the tetrahedra and the tetrahedrae . On the right–hand side of (18) we have then defined
(19) 
The role of the weights of such edges, labelled again by
to simplify the notation, will be discussed below.
In order to set up a decoupling procedure in (15)
(taking into account
(18)), we have to
keep fixed all labels together with the
variables of type , while rescaling
the variables belonging to the coupling tetrahedra
by the same factor . More precisely,
we have different asymptotic expressions for each type of coupling tetrahedra
(see [11], [24] and the original references therein), namely

For a tetrahedron we get
(20) where . Such an expression tells us that by rescaling the three variables by a large , this symbol goes into a weighted Wigner symbol with the same ’s, and with momentum projections depending on differences of ’s according to
(21) 
For a tetrahedron we get
(22) where and is the angle between the the edge labelled by and the quantization axis. Thus we get that the asymptotic behavior of such a symbol goes like a product of a Kronecker and a Wigner –function with entries given by
(23) 
Finally, the asymptotic expression for a tetrahedron is nothing but the original Ponzano–Regge formula [11], namely
(24) is the Euclidean volume of the tetrahedron spanned by the six edges , and is the angle between the outward normals to the faces which share (these angles can be obviously expressed in terms of the ’s). Note that the sum under the exponential is the Regge action for the tetrahedron .
Since we can freely choose the decoupling parameter at this level, we carry out the rescaling both on the three types of symbols according to (20), (22), (24) and on the phase factors and weights in (15) and (18). Then the functional which turns out to be associated with the resulting configuration is
(25)  
where each term containing is the approximation for
the corresponding volume factor in (24).
This expression is an almost factorized product of two groups of terms
i) A Ponzano–Regge–like
state functional for a dimensional bulk triangulation to be closed with
boundary terms depending on some of the fixed edges ,
(see below for more details). The sum of this
state functional over all possible triangulations with fixed labels
on the boundary
(up to regularization) will provide in the bulk a family of gravity
partition functions
depending on the decoupling parameter .
ii) A second functional, represented by the remaining terms in
(25), containing contributions resembling the state
functional for a dimensional closed triangulation given in (7)
(i.e. a pair of Wigner symbols for each triangular face).
Referring to i), we recognize that the topological union of the
coupling tetrahedra,
, fills in a thick shell
close to the boundary
. The linear extension of such a shell is of the
order of the decoupling
parameter , and some of the edges considered in (18)
happen to lie
on the boundary of the 3dimensional triangulation
(26) 
For each value of the decoupling parameter, denote by the fixed 2dimensional triangulation which closes up (26) giving rise to the pair
(27) 
Then any such a pair is topologically equivalent to the original pair as can be easily seen by exploiting the invariance under elementary shellings discussed at the end of Section 2. Moreover, with respect to , represents an inner boundary with
(28) 
For what concerns the state functional to be associated
with (27), note that only contributions from the edges in
– and not from its faces – have to be taken into
account. This is due to the fact that
the contribution to the state functional of an edge on a fixed boundary
amounts simply to [12], which further reduces to
in our case.
If we now select
(29) 
(30)  
Notice that here we are forced to include only
vertex weights,
(number of vertices of the inner boundary), namely those vertices which are
actually in the new .
As remarked before, we have now the possibility of getting at once
the partition function
of the bulk on applying the standard formal procedure in the case
of a fixed boundary, i.e.
(31) 
This family of Euclidean gravity partition functions
encodes information about
the decoupling parameter in its fixed boundary, while free fluctuations
of all the other spin variables in the bulk are obviously allowed.
In the decoupling limit to be discussed below, the state
sum (31) should be sharply peaked on (semi)classical
configurations, namely on states (30) in which all spin variables
in have been rescaled by choosing some finite ,
with .
But such conditions do not affect the ”topological character”
of (31), since Ponzano and Regge showed that also the asymptotic
symbols satisfy all the algebraic identities [11], and thus the
bistellar
moves in can work as well as before.
Coming back now to the factorization of the state functional (25),
and taking into account (30), we can formally write
(32)  
where we have set
(33) 
to denote the (fluctuating) outer boundary, in agreement
with the notation for the (frozen) inner boundary.
The expressions of the last two terms on the right–hand side of
(32) can be recognized from (25), taking into
account (29), (33) and (30). Thus

In the projection map
(34) we collect all the terms of (25) which depend explicitly from (the phase factors containing can be dropped e.g. by choosing an even ).

The remaining terms of (25), not yet considered, are
(35) where the extra–factor due to the vertices of – which compensates the modification of the weights in (30) – has been included. This expression represents the functional to be associated with a holographic state generated by disentangling the external boundary from the original dimensional theory.
As a first remark, note that the projection map includes
combinatorial quantities related
not only to the topology of the manifold
, but also to the triangulations of ,
and of the shell between them. Thus, in particular,
the numbers , ,
, , in (34)
cannot be related to each
other in a unique way unless we choose a priori the topological
type and the triangulations.
Anyway, we may say that the onset of a decoupling regime is achieved
when the value of the limit
(36) 
is small as compared with the other functionals
appearing in the factorization (32). In other words, we require that
the gravitational contribution
of the shell made up by the coupling tetrahedra is negligible when
approaching the large
–limit both with respect to the bulk contribution (30)
and also with respect to the residual functional (35)
surviving on the
external boundary. Such a kind of behavior can be translated into
suitable sets of
selection rules (or more properly, in
holographic language, a sort of ”screen map” [2])
on the triangulations involved in (34).
Indeed, given a topological type , we can in
principle select – by making use
of both bistellar moves in the interior of and shellings and/or inverse
shellings in the boundary – exactly those standard triangulations
which satisfy the
constraints, and in the decoupling limit the bulk–boundary transfer
process is efficiently performed
only by such classes of triangulations. We shall return to the physical
interpretation of
this point in the next section.
An inspection of the holographic state functional (35)
shows that it is not
independent of the triangulation .
This feature is an obvious consequence of the factorizazion
prescription (32),
which breaks
the topological invariance of the original
. Even if such an
invariance can be restored in the bulk according to the procedure
given in (31),
by comparing (35) with the bidimensional
functional (6) we
see that in the present case a number of new significant
quantities appear (the phase factors are
unimportant since we could redefine in a suitable way the Wigner symbols).
To address this issue in more details, recall from Section 2 and
from [20] that the
dimensional bistellar moves are

The flip move, which tranforms a pair of contiguous triangles into another pair of triangles by keeping their common boundary quadrilateral fixed. Denote this move by
(37) The algebraic identity representing this move is
(38) where Latin letters denote angular momentum variables, Greek letters the first set of momentum projections and primed Greek letters the second one. A label associated with the triangles involved in the move has been added.

The Alexander (or barycentric) move which amounts to adding one vertex in the interior of a triangle giving rise to three triangles , , bounded by the original one. We denote this operation and its inverse move by
(39) and the corresponding identity reads
(40)