Discrete gravity as a topological field theory with lightlike curvature defects
Received: November
Discrete gravity as a topological field theory with lightlike curvature defects
0 Open Access , c The Authors
1 31 Caroline Street North , Waterloo, ON N2L 2Y5 , Canada
2 Perimeter Institute for Theoretical Physics
I present a model of discrete gravity as a topological field theory with defects. The theory has no local degrees of freedom and the gravitational field is trivial everywhere except at a number of intersecting null surfaces. At these null surfaces, the gravitational field can be singular, representing a curvature defect propagating at the speed of light. The underlying action is local and it is studied in both its Lagrangian and Hamiltonian formulation. The canonically conjugate variables on the null surfaces are a spinor and a spinorvalued twosurface density, which are coupled to a topological field theory for the Lorentz connection in the bulk. I discuss the relevance of the model for nonperturbative approaches to quantum gravity, such as loop quantum gravity, where similar variables have recently appeared as well. ArXiv ePrint: 1611.02784
with; lightlike; curvature; defects; Lattice Models of Gravity; Models of Quantum Gravity; Topological Field

Boundary spinors in GR
Selfdual area twoform on a null surface
Boundary term on a null surface
Discretised gravity with impulsive gravitational waves
Glueing flat fourvolumes along null surfaces
Definition of the action
Equations of motion
Special solutions: planefronted gravitational waves
Hamiltonian formulation, gauge symmetries
Spacetime decomposition of the boundary action
Phase space, symplectic structure, constraints
Constraint algebra and gauge symmetries
Relevance for quantum gravity
Summary, outlook and conclusion
A Spinors and world tensors
1 Introduction 2 3 2.1
We will consider gravity as gauge theory for the Lorentz group. The fundamental
configuration variables in the bulk are the tetrad and the selfdual connection [1]. For a manifold
with boundaries, the action acquires boundary terms, which reconcile the variational
problem with the boundary conditions. If the boundary is null, we will see that the most natural
such boundary term is given by the threedimensional integral
∂M
rior derivative and ℓA is a spinor, whose square returns the null generators of the
threedimensional null boundary ∂M.
We will then use this boundary term to discretise gravity by truncating the bulk
geometries to field configurations that are locally flat. The bulk action vanishes and only the
threedimensional internal null boundaries and twodimensional corners contribute
nontrivially. The resulting action defines a theory of distributional fourgeometries in terms of
a topological field theory with defects that propagate at the speed of light.
Whether the continuum limit exists and brings us back to general relativity is a
difficult question. This paper does not provide an answer. We can only give some
indications in favour of our proposal: first of all, we will show that the solutions of the theory
represent fourdimensional Lorentzian geometries, which are built of a network of
threedimensional null surfaces (equipped with a signature (0++) metric) glued among bounding
twosurfaces. We will then find that every such null surface represents a potential curvature
defect. We will also show that there exist solutions of the equations of motion derived from
the discretised action that represent distributional solutions of Einstein’s equations with
nonvanishing Weyl curvature in the neighbourhood of a defect. In other words, the
Einstein equations are satisfied locally. Whether this holds on all scales, with possible higher
order curvature corrections and running coupling constants, is a more difficult question.
Sophisticated coarse graining and averaging techniques, such as those developed for Regge
calculus [2–4] and related approaches cf. [5, 6] may provide useful tools for the future.
The main motivation concerns possible applications for nonperturbative approaches to
quantum gravity, such as loop quantum gravity [7–11]. In loop quantum gravity, geometry
is described in terms of the inverse and densitised triad
Eia =
which is the gravity analogue of the YangMills electric field. In quantum gravity, it becomes
an operator, whose flux across a surface has a discrete spectrum [12, 13]. The resulting
semiclassical geometry — in the naive limit, where ~ → 0, and all quantum numbers are sent
to infinity, while keeping fixed the eigenvalues of geometric operators — is distributional.
The semiclassical electric field
Eia(p) =
has support only along the links l1, . . . , lL dual to a cellular decomposition of the spatial
manifold. There is no geometry, no notion of volume, distance and area, outside of this
onedimensional fabric of space. What is then the dynamics for these distributional geometries
at the quantum level? There are several proposals, such as those given by the covariant
spinfoam approach [11, 14–16] or Thiemann’s canonical program [9, 17, 18], but the precise
transition amplitudes are unknown. We know, however, that the threedimensional
quantum geometries represent distributional excitations of the gravitational field, and we can
expect, therefore, that the semiclassical ~ → 0 limit of the amplitudes will define a
classical theory of discrete, or rather distributional, spacetime geometries, whose Hamiltonian
dynamics is formulated in terms of gauge connection variables. This paper presents a
proposal for such a theory, and may, therefore, open up a new road towards nonperturbative
Boundary spinors in general relativity
Selfdual area twoform on a null surface
On a null surface N, the pullback of the selfdual component of the Plebański twoform
and it is important for the further development of this paper. Let us explain it in more
Consider thus an oriented threedimensional null surface N in a fourdimensional
Lorentzian spacetime manifold1 (M, gab). We assume M to be parallizable, hence there
We define the Plebański twoform
and split it into its selfdual and antiselfdual components
vectors. In terms of the Dirac gamma matrices, which may be more familiar to the reader,
we could also write
See the appendix A for further details on the notation.
which squares to
Clearly, there is an additional gauge symmetry: given the null vector ℓa, the spinor ℓA can
M be the canonical embedding of the null surface N into M, and
a, b, c, . . . are abstract tensor indices labelling the sections of the tensor bundle over either all of spacetime
we use an index notation as well: its sections carry indices A, B, C, . . . referring to the fundamental spin
( 21 , 0) representation of SL(2, C). Primed indices A¯, B¯, C¯, . . . belong to the complex conjugate spin (0, 21 )
representation. See the appendix A for further details on the notation.
This can be seen as follows: first of all we choose a second linearly independent spinor kA
ϕ∗ΣAB = µ ℓAℓB + νkAkB + iεk(AℓB).
Next, we also know that ℓa is the null generator of N. The pullback of ℓa to N thus
vanishes, which implies, in turn
This is the same as to say
We contract the free indices with all possible combinations of the spinors ℓA, kA and their
which proves the desired equation (2.5).
dimensional spatial submanifold C of N. Indeed
± Area[C] =
Whether this integral coincides with the metrical surface area
Area[C] =
dx dy g(∂x, ∂x)g(∂y, ∂y) − g(∂x, ∂y)2
depends on whether ℓa, which is future pointing, is an outgoing or incoming null generator
with respect to the induced orientation2 on C.
2That C inherits an orientation from N is immediate. We can say, in fact, that a pair of tangent vectors
choose an arbitrary future oriented timelike vector ta, which is based on C, and say that ℓa is outgoing
and this definition will not depend on the choice of ta. A straightforward calculation shows then that the
The last section gave a parametrisation (2.5) of the selfdual area twoform on a null surface
boundary action with spinors as the fundamental boundary variables.
Working in a firstorder formalism, we write the gravitational action as a functional of
is then independent of the triad, which means that the torsionless condition does not hold
equations of motion, which are the Einstein equations plus the torsionless condition. With
SM[A, e] =
We then write the bulk action in terms of selfdual variables, which are the selfdual
area twoform (2.2a) and the SL(2, C) spin connection AAB, whose curvature is the selfdual
SM[A, e] =
− 6
where cc. denotes the complex conjugate of all preceding terms.
Consider then the variation of the action. The Einstein equations follow from the
variation of the tetrad, the variation of the connection yields the torsionless condition
∂M
at the boundary. This boundary integral must cancel against the variation of the boundary
action, otherwise the entire action is not functionally differentiable. We have assumed that
form of equation (2.5). What is then the right boundary action? The remainder (2.15)
is linear in the connection, and we expect, therefore, that the boundary action is linear
gauge covariant and linear in AABa, which suggests that the boundary term is built from
the gauge covariant derivative of some boundary fields. The only available fields at the
boundary, which are functionally independent of the connection, are the boundary spinors
complexvalued threeforms, whose boundary integrals define the most obvious candidates
to the latter by a total derivative). We are left to determine the coupling constants in front,
∂M
may seem rather dull, but it will be important for us later, when we will learn how to glue
causal regions across a bounding null surface.
reality conditions (2.9), which now turn into
If (2.21) is satisfied, the area is real and we can define the oriented area twoform on a null
surface N simply by
which can be read off the remainder (2.15) of the connection variation at the boundary.
The resulting boundary term is
∂M
where Da denotes the gauge covariant derivative
DaℓA = ∂aℓA + AABaℓB.
Notice that there is now an additional U(1)C gauge symmetry appearing: the spinors
where the gauge element z : N
→ C generates both boost and rotations preserving the null
normal iℓAℓ¯A¯. The boundary action (2.16) is not invariant under this symmetry, but we
The entire action for the gravitational degrees of freedom in a region M bounded by a null
surface N is therefore given by the expression
A are not varied in the action, they are
kept fixed in the variational principle, because they determine the boundary value (2.5) of
connection in the bulk and boundary.
In the literature, other boundary terms have been used on null surfaces as well. For
a recent survey in the metric formalism, we refer to [19] and references in there. The
boundary term (2.19) is a generalisation — it is formulated in terms of spinors, and does
not assume that the connection is torsionless, which explains the implicit appearance of
the BarberoImmirzi parameter, which enters the action (2.19) through the definition of
the momentum spinor (2.20). Notice also that additional corner terms may be necessary
as well. We will introduce them below.
Discretised gravity with impulsive gravitational waves
Glueing flat fourvolumes along null surfaces
In Regge calculus [20], the Einstein equations are discretised by cutting the spacetime
manifold M into foursimplices, and truncating the metric to field configurations that are
locally flat. The gravitational action for the entire manifold is then a sum over all such
foursimplices, each one of which contributes a bulk and boundary term. The bulk contribution
reorganised into a sum over triangles, with every triangle contributing its area times the
surrounding deficit angle.
The task is then to generalise Regge calculus and find a theory of discretised gravity
in the connection formalism, whose action is still simple enough to admit a Hamiltonian
quantisation. We will propose such a theory by dropping the assumption that the
elementary building blocks are flat foursimplices. We work instead with fourdimensional regions,
which are flat or constantly curved inside (depending on the value of the cosmological
constant), and whose boundary is null.
The theory is then specified by the matching conditions that determine the discontinuity
of the gravitational field in the vicinity of the interjacent null surface. In Regge calculus,
it is the intrinsic threedimensional geometry at the interface that is matched between the
two sides. We require this condition as well, and thus impose that
qab = qab,
where qab denotes the intrinsic threemetric from below3 the interjacent null surface N,
while qab determines the geometry from the other side.
It has signature (0++), and the null vector ℓa defines the single degenerate direction ℓa :
3The null surface N is oriented, and the null vectors ℓa are future pointing. In a neighbourhood of
N we can thus distinguish points sitting below the null surface from those lying above. The quantities
describe the boundary from above.
and ǫˆabc is the metricindependent LeviCivita density on N, which is defined for any
Areamatching condition.
We now have to convince ourselves that the matching
conditions for the spinors (3.3) are indeed equivalent to the conditions (3.1) and (3.2) for ℓa
and qab. We start with the areamatching condition (3.3a). Going back to equation (2.10)
either side, and write
In our formalism, the fundamental configuration variables are the boundary spinors ℓA
would certainly be sufficient, but they are too strong, for they also match unpheysical gauege
degrees of freedom, which are absent in (3.1). We should thus only match SL(2, C) gauge
We contract both (3e.5a) aend (3.5b) with ℓA and ℓA, going back to e(3.3a) we then obtain
the areamatching condition
of the null generators ℓa (resp. ℓa), and the matching condition e(3.3a) implies that they
both point into the same directioen, hence ℓa ∼ ℓa as desired.
Having shown that the areamatching conedition (3.3a) implies the matching (3.2) of
the null vectors, we are now left to show that the shapematching conditions (3.3b) are
on the two sides. This requires some preparation: first of all, we have to understand how
4We can always extend ℓA (resp. ℓA) with a second linearly independent spinor kA (resp. kA) into a
local spin basis such that kAℓA = 1 =ekAℓA.
To reconstruct the threemetric from the
spinors, it is more intuitive to work with densitised vectors rather than threeforms on N.
then also have tehe component twoform µ
tangent vector µ a in the complexified tangent space (T N)C through
∈ Ω2(N : C). Its densitised dual defines a
m¯aµ a = ±1.
= ±1.
ℓaka = ∓1, µ aka = 0,
µ a? The aneswer is simple: it defines a dual dyad {ma, m¯a}, which diagonalises the intrinsic
The normalisation N will be determined in a moment. The covector ma may vanish, but
this is a singular case. It implies µ a
∝ ℓa, which is the same as to say that there exists a
normalised spin dyad {kA, ℓA} such that the pullback of the selfdual twoform to N, i.e.
the nullsurface N becomes effectively twodimensional — it has no affine extension along its
degenerate as well, for it implies that the triple {ℓa, µ a, µ¯a} is linearly dependent. It then
notes the pullback to N. This is incompatible with the existence of a nondegenerate tetrad
tetrad, we proceed as follows: first of all, we fix the normalisation N of ma by demanding
The sign depends on the orientation of {ℓa, µ a, µ¯a} with respect to the fiducial volume form,
a dual basis of (T ∗N)C, such that
with all signs in (3.10), (3.11) and (3.12) matching according to the indicated pattern.
This allows us to write the pullback of the tetrad to N as
ϕ∗eAA¯
= ∓iℓAℓ¯A¯ka ± iℓAk¯A¯ m¯a ± ikAℓ¯A¯ma.
qab = 2m(a m¯b),
of the selfdual twoform to N, and get
A short calculation reveals that this parametrisation is indeed compatible with the boundary
ϕ∗eAC¯
(ϕ∗eBC¯ )b] =
= +2ℓAℓBk[a m¯b] + 2ℓ(AkB) m¯[amb],
By duality, equation (3.16b) is the same as to say that the area element is the wedge product
metric qab = (ϕ∗eAA¯)a(ϕ∗eAA¯)b. A short calculation gives
of the twodimensional codyad {ma, m¯a}. We can then, finally, also compute the induced
which concludes the reconstruction of the induced geometry of the null surface N from the
Shapematching conditions.
We are now left to show that the shapematching
conditions (3.3b) imply that the intrinsic threemetrics qab and qab match between the two sides.
To show this, we first extend ℓA (resp. ℓA) with a seconde linearly independent spinor kA
ηAaηAb = ηˆ2 (−iℓAµ¯a + ikAℓa) −iℓAµ¯b + ikAℓb
a fixed fiducial volume element, we can remove the density weights and the areamatching
∈ Ω2(Ne: C2) admits the decomposition
We can now replace kA by
ηAab = µ ab − iεeabζ¯ ℓA + iεabkA.
without actually changing the canonical enormalisation kAℓA = 1 of the spin dyad {kA, ℓA}.
e
If the glueing conditions (3.3) are satisfied, we have tehues shown that there alwayes exeists
normalised spin dyads {kA, ℓA}, {kA, ℓA} on either side of the interface, such that
In other words, there are always spin dyads {kA, ℓA} and {kA, ℓA} such that the component
two sides. But now wee ealso know that the intrinsic threegeometrye qab is already uniquely
the resulting threemetrics qab and qab must agree as well. Thies ceoncludes the argument,
for it implies that the intrinsic threeemetric is the same whether we compute it from the
spinors on one side or the other. In other words
qab = qab,
which is the desired constraint (3.1) as derived from both the areamatching and
shapeis a Lorentz transformation, which is given explicitly by
hAB = ℓAkB − kAℓB : N
ℓA = hABℓB,
Before we go on to the next section, let me briefly summarise: in this section, we have
studied the conditions to glue two adjacent regions along a null surface N. In terms of
metric variables, equations (3.2) and (3.1) match the null generator and the intrinsic
threestraints: the areamatching condition (3.3a) and the shapematching condition (3.3b). The
terminology should be clear: equation (3.3a) matches the twodimensional area elements
angles dreawn on N are the same whether we compute them from the boundary spinors on
one side or the other. Notice also, that the number of constraints is the same for both
variables: the induced threemetric qab has signature (0++), hence there are five independent
matching constraints in the metric formalism. In terms of spinors we have five constraints
to three real constraints. The shapematching conditions (3.3b), on the other hand, add
only one additional complex constraint. This is not obvious from equation (3.3b), but it is
immediate when we look at (3.19). In both formalisms, we are thus dealing with the same
five number of constraints.
Definition of the action
(as introduced in e.g. equation (2.19) above), we first introduce a cellular
decomposition and cut the fourdimensional oriented manifold M into a finite family5 of closed cells
{M1, M2, . . . MN }, which are flat or constantly curved inside, i.e.
∀p ∈ Mi : FAB(p) − 3
We require, in addition, that the intersection of any two such regions Mi and Mj is at most
threedimensional. If it is threedimensional, we give it a name and call M
i ∩ M
j =: Nij an
interface, whose orientation is chosen so as to match the induced orientation from Mi. In
other words N−1 = Nji. If, on the other hand, M
ij
j intersect in a twodimensional
surface, we call it a corner C, and we shall also assume, for further consistency, that all
corners in the interior of M are adjacent to four definite such regions — four and not
three or five, simply because we require that the internal boundaries are null, in which case
all such corners arise from the intersection of two such null surfaces. See figure 1 for an
The requirement that the boundary ∂M
ij of all fourdimensional building
such internal boundary N
ij there exists a spinor6 ℓiAj : N
ij → C2 and a spinorvalued
twoin (2.2a)) admits the decomposition
5The orientation of every Mi matches the orientation of M, and every Mi is homeomorphic to a closed
fourball in R4.
The vertical position of the (ij)indices has no geometrical significance. Simplifying our notation, we
which contain no local degrees of freedom inside.
Nontrivial curvature is confined to
threeas to say that N
M is the canonical embedding of N
ij into M. That this is the same
ij is null, has been shown in section 2.1 above following the discussion of
The discontinuity of the metric across the null surface will be encoded in a discontinuity
of the spinors and the connection. Along a given null surface Nij , we will have two kinds of
[Aij ]ABa denotes the pullback of the SL(2, C) connection from the bulk Mi to the boundary
component N
mon interface N
ij ⊂ ∂Mi, and [Aji]ABa is the pullback from M
j to Nji ⊂ ∂Mj . The
comij between Mi and M
j carries then two independent SL(2, C) connections
[Aij ]ABa and [Aji]ABa. What is the relation between the two? Consider first the SL(2, C)
transformation [hij ]AB,
ℓiAj = [hji]ABℓjBi,
which brings us from one frame to the other. Such an SL(2, C) gauge transformation exists
provided the matching conditions (3.3) are satisfied, which has been shown in (3.26) above.
The spinors are gauge equivalent, but the connections may not: there is, in general, a
nonvanishing difference tensor [Cij ]ABa between the two SL(2, C) connections, and we define
it as follows:
Aij = hjidhij + hjiAjihij + hjiCjihij .
Construction of the action. The action will consists of a contribution from every
fourdimensional cell Mi, a boundary term from every interjacent null surface, and a corner
term from any two such null surfaces intersecting in a twodimensional face. The bulk
which is an sl(2, C)valued twoform with dimensions of ~. The bulk contribution to the
action is therefore nothing but the integral
the integrability condition
DΠAB = dΠAB − 2AC (A ∧ ΠB)C = 0.
The next term to add is the threedimensional boundary term from the interface
between two adjacent regions. This boundary term has a twofold job: it cancels the connection
variation from the bulk, and it imposes that the intrinsic boundary geometry is null. It
consists of the covariant symplectic7 potential
A ∧ DjiℓjAii + cc.,
plus additional constraints: the glueing conditions (3.3), which match the spinors from the
∧ πiAj ℓiAj − πjAiℓjAi + Ψiajb πiAj aπiAj a − πjAiaπjAia
arising from the ℓAvariation on N.
We sum the bulk action (3.32) with the boundary action for the spinors (3.34) and the
constraints (3.35) and also add the right corner term, which we will discuss below. The
resulting action is then given by the expression
= 2 X Z
− 2
− 2
α ℓiAm ℓiAn − ℓjAm ℓjAn + ℓAmj ℓAmi − ℓnAj ℓnAi + cc.
is the pullback of the selfdual connection from Mi to Nij ⊂ ∂Mi.
M denoting the canonical embedding. Next, there is the reality condition:
arising from the ℓAvariation of the coupled boundary plus corner terms. The resulting
boundary conditions are
Let us summarise and briefly explain the role of each term in the action (3.36). The
tion. Next, there are the integrals over the internal boundaries Nij . The variation of the
connection in the bulk yields a remainder at the boundary. The variation of the boundary
connection couples then the boundary with the bulk, yielding the constraint
The first sum P
i goes over all fourdimensional bulk regions M
second sum goes over all ordered pairs (Mi, Mj ), which share an interface Nij . This sum
crucially contains both possible orientations, i.e. P
Nij · · · = R
The last integral is the corner term, which is a sum over all quadruples (Mi, Mj , Mm, Mn)
that share a twodimensional corner such that (Mi, Mm), (Mi, Mn) and (Mj , Mm) and
and (Mm, Mn) only meet in the corner itself: M
(Mj , Mn) each share a threedimensional interface (e.g. Mi∩
for an illustration. We write P
Mm = Nim), whereas (Mi, Mj )
i ∩ M
j = Mm ∩ M
n = Cimj n. See figure 1
[ijmn] to say that any such corner appears with only one
possible orientation in the sum. This orientation is chosen arbitrarily and can be absorbed
are obtained from the twodimensional remainder
for the area twoform (2.22) as in equation (2.21) above. Finally, there are the glueing
Notice also that we have used a condensed notation in (3.36), we dropped all wedge
products, and suppressed all (ij)indices in the boundary variables. The second line in (3.36)
has to be understood, therefore, in the following sense
A ∧ Dij
pendent LeviCivita density ǫˆabc.
Equations of motion
In this section, we study the equations of motion as derived from the action (3.36). This is a
preparation for the next section, where we will find a family of explicit solutions representing
plane fronted gravitational waves, which are exact solutions of both the discretised theory
and general relativity as well.
Some of the equations of motion derived from the action (3.36) have already been
constraint (3.27), i.e.
∀p ∈ Mi : FAB(p) =
Finally, a word on the Lagrange multipliers: the continuity conditions (3.41) and (3.42)
of the momentum density
which is a spinorvalued vector density of weight one, with ǫˆabc denoting the metric
indeWe then also have the variation of the selfdual connection AAB, which gives the
integrathe internal null boundaries { ij }, where we find a number of additional constraints. The
they are obtained from the stationary points of the action (3.36) with respect to variations
added to the action in order to impose the glueing conditions across the interface, namely:
the areamatching condition (3.3a) and the shapematching condition (3.3b). Finally, we
have the relation between the bulk and the boundary, which is provided by the glueing
to the boundary. As we have seen previously in section 2.1 above, this is the same as to
say that the boundary is null.
The action (3.36) contains the boundary spinors as additional configuration variables,
and the action is stationary with respect to them provided additional equations of
motion are satisfied along the system of interfaces. For simplicity, consider only a single such
from, say, below and above the interefacee: if ϕ : M ֒
N−1) is the
Goingebacek to the definition of the action (3.36), we then see that the variation of the
boundary spinors yields the equations of motion
DaℓA =
Da ℓA =
ωa − β + i λa ℓA + ΨabπAb,
where Da (and Da) is the covariant derivative with respect to the SL(2, C) connection
in M (and M) to the inteerface N ebetween M and M.
Three immediate observations: integrability, geodesity and the expansion of the
Before we proceed, we need to develop some better intuition for this system
of equations (3.47). First of all, we can see that they are consistent with the torsionelss
geometrical interpretation. It measures the expansion ϑ(ℓ) of the null surface. This can be
seen as follows. The expansion ϑ(ℓ) can be defined by
of ϑ(ℓ) is gauge dependent — it depends on a representative ℓa of the equivalence class of
intrinsic to N. In terms of the boundary spinors, the twodimensional area element is
Going back to the equations of motion (3.47), we find
Finally, let us turn to the null generator ℓa itself. It is geodesic, and this can be seen
as follows. First of all, we write this vector field (modulo an overall normalisation) in terms
of the boundary spinors, obtaining
which follows from the areatwo from as written in terms of the boundary spinors, and
iℓAℓ¯A¯, see for instance equation (3.14) above. Now, the covariant derivative of ℓA along the
null generators is proportional to ℓA itself, which follows from
∝ ℓ
⇔ ℓAℓaDaℓA = 0,
and equation (3.47a) by noting that
∝ iℓAℓ¯A¯,
which means that the integral curves of ℓa
to the SL(2, C) connection. The SL(2, C) connection is torsionless (3.33), hence the null
≡ iℓAℓ¯A¯ are autoparallel curves with respect
generators of N are geodesics.
Difference tensor.
The null surface N bounds two bulk regions M (from above) and
M (from below). There are then two SL(2, C) connections on N, one (namely AABa) from
bfelow the interface, the other (namely AABa) from above. Their relative strength is given
by a difference tensor CABa, whose algeebraic form is determined as follows. First of all,
gauge equivalent, which means that they are related by an SL(2, C) gauge transeforme ation.
We can thus write
We then have the difference tensor (3.30) on N, which is defined as
≈ ℓA,
AABa ≈ AABa + CABa.
Subtracting the covariant differential of ℓA from the differential of ℓA, i.e. subtracting (3.47c)
This implies that CABa admits the decomposition
CABa = − β + i ℓ(AkB)λa − β + i ℓAℓBΓa,
algebraic constraint
which is a consequence of (3.57b). Finally, the difference tensor is subject to one additional
constraint: the field strengths as induced from the two sides are gauge equivalent, which is
sourced by the boundary spinors (as in (3.46)). The boundary spinors are gauge equivalent
DCAB +eCAC ∧ CC B = 0.
In the next section, we will demonstrate that explicit solutions to these equations exist:
plane gravitational waves solve the system of equations (3.47), (3.58), (3.60), (3.59) for
certain boundary spinors and a definite difference tensor CAB at the interface.
Special solutions: planefronted gravitational waves
This section is dedicated to finding explicit solutions to the equations of motion in the
neighbourhood of an interface. Rather than exploring the entire solution space, we study
only a single family of solutions, thus giving a constructive proof of existence: there are
nontrivial8 solutions to the equations of motion derived from the action (3.36), and the
particular solutions thus constructed are distributional solutions of Einstein’s equations
Consider thus a single interface N, bounding the fourdimensional regions M and M
from above and below. We set the cosmological constant to zero, hence M and M are flaft,
where {xµ } are inertial coordinates in M. The glueing conditions (3.1) imply that the
intrinsic geometry of N is the same from the two sides. The discontinuity in the metric
can, therefore, only be in the transversal direction, which motivates the following ansatz
for the tetrad across the interface
8We will find solutions with a nonvanishing distributional Weyl tensor at the interface.
Killing vector. Having solved the equations of motion for v < 0, and matched the geometries
across the interface, we are now left to solve the equations of motion for v > 0. The only
missing condition is to impose that the curvature vanishes for v > 0. A straightforward
calculation reveals that this is possible if and only if
∂uf = 0.
∂z¯f − ∂zf¯ = 0.
It is also instructive to have a look at the boundary spinors and compute them explicitly.
Going back to our initial ansatz (3.61) for the tetrad and taking also into account the
parametrisation (3.64) of the null tetrad, we immediately see that we are in a gauge for
straightforward exercise returns the momentum spinors
The twodimensional volume element on N, as given by equation (2.11), is then simply
In order to satisfy the equations of motion, this difference tensor must be equal to CABa as
derived from the equations of motion for the action (3.36). Going back to the decomposition
of CABa into ℓA and kA, as in (3.58) above, we thus find the conditions
complex. Unless the BarberoImmirzi parameter goes to zero, which is a singular limit in
the original selfdual action (2.14), the equation (3.66a) can be only imposed, therefore, for
Going back to the equations of motion for the spinors, i.e. equations (3.47) above, we can
Finally, we can now also compute the distributional curvature tensor across the
interface. Following Penrose’s conventions [23], we compute the irreducible components of the
field strength of the SL(2, C) connection, namely
is the traceless part of the Ricci tensor, which is the irreducible spin (1, 1) component of the
The resulting geometry is a solution to Einstein’s equation with a distributional source
field of a massless point particle [22].
Hamiltonian formulation, gauge symmetries
Spacetime decomposition of the boundary action
The main purpose of this paper is to open up a new road towards nonperturbative quantum
gravity. We have defined the action (3.36) and demonstrated that explicit solutions exist,
which have a nonvanishing (yet distributional) Weyl curvature (3.72) in the neighbourhood
of a null surface. The next logical step is to study the Hamiltonian formulation of the theory.
First of all, we note that the bulk action (3.32) is topological. All physical degrees of
freedom sit, therefore, either at the system of null surfaces { ij } or at the twodimensional
Cmn . As long as we are concerned with the canonical analysis on only one
such null surface N alone, we can then also work with a simplified action, which is found
We insert the equations of
motion (3.45) for the selfdual twoform back into the bulk action, thus obtaining
− 2 M
FAB ∧ F AB =
2 ∂M
Tr A ∧ dA +
A ∧ A ∧ A
2 SCS, ∂M[A],
which is the selfdual ChernSimons action with complexvalued coupling constant
Riemann curvature tensor. Both components vanish everywhere except at the null surface
and the traceless part of the Ricci tensor is determined to be
Tab =
The appearence of the SL(2, C) ChernSimons action in a fourdimensional theory may
come as a surprise, but it has been anticipated by several authors, who have suggested that
the socalled Kodama state, which is the exponential of the ChernSimons functional for the
underlying gauge group, plays a significant role for quantum gravity in fourdimensions [24–
29]. This paper confirms these early expectations.
dimensional cells {
M1, M2, . . . }. Each one of them has has the topology of a four ball,
ij : ∂M
i = S
Nij . For definiteness, consider only one such threesurface N at the
boundary between two bulk regions, say M and M. Integrating out the selfdual twoform
action (4.1) alongfthe interjacent null surface N. Each one of these SL(2, C) ChernSimons
either side. The resulting coupled action is therefore given by
2 SCS, N[A] − 2 SCS, N[A]+
All terms in this boundary action have a straightforward geometrical interpretation:
variation with respect to the selfdual connection imposes that the field strength FAB on N
geometric, i.e. compatible with the existence of a signature (0++) null metric qab on N.
Finally, we have the glueing conditions (3.3), which are obtained by demanding that the
conditions impose11 continuity across the interface: the intrinsic geometry of N is the same
whether we compute it from the boundary spinors on either side of the interface.
Next, we write the action in a Hamiltonian form. This requires a clock — a foliation
N ≃ [0, 1] ×
vector field ta. This should strike us as a surprise: a threedimensional null surface always
has a preferred time direction: the direction of its null generators. So how can it be that
our action (4.3), which is meant to be an action for a null surface N, lacks such a preferred
structure? The answer is simple: in our theory, there is no metric formulation to begin with,
the metric is a derived or composite field. It exists only onshell — only if the reality
condi11See section 3.1 above.e
10The same is true for F AB and −π(AℓB)/γ.
taking into account also the corner terms and boundary conditions at the twodimensional intersections
(see figure 1) will be left for the future.
qab on N. Only for those configurations that satisfy the reality conditions (2.21) can such
a metric be defined, according to the construction that has been given in section 3.1 above.
We then choose a time function t, which is a mere coordinate, and a transversal vector
ponents. With a slight abuse of notation, we denote the pullback of the SL(2, C) bulk
connection AAB ∈ Ω1(M : sl(2, C)) to the t = const. slices simply by
the threeboundary N. Equation (4.4) defines the spatial components of the connection.
Its tcomponent13 defines the Lagrange multiplier
Finally, we also have the velocity
with Lt denoting the Lie derivative along the vector field ta.
AABa = ϕt∗[AAB]a,
A˙ ABa = ϕt∗[LtAAB]a,
ℓ˙A = LtℓA,
In the following, we restrict ourselves to those parts of configuration space, where the
area element (2.11) is nondegenerate on S, hence we assume
in terms of components
Finally, we define the velocities
Notice, the component functions U a and Va have different density weights: Va is a oneform
is a twoform (hence a density) on S. Both U a and Va are Lagrange multipliers, since
the boundary action (4.3) contains no derivatives of them. Yet they are not completely
arbitrary: the matching conditions (3.3) and the reality conditions (2.21) impose constraints
on them. The reality conditions (2.21), which follows from the variation of the action (4.3)
U a = U¯ a.
On the other hand, there are the matching conditions (3.3), which follow from the variation
the other side oef the interface, i.e.
Va = V a,
U a = U a,
χeAa = ϕt∗[etyπA]a = Ue aπA + V aℓA.
It is then useful to dualise the component functions U a and Va. We take the canonical
LeviCivita density ǫˆab on S, and define
N a := ǫˆabU b,
J a := ǫˆabVb,
∈ T S is a tangent vector, while J a
∈ Ω2(S : T S) is a vectorvalued density.
We insert the 2 + 1 decompositions for both the connection (i.e. (4.4), (4.5)) and for
the boundary spinors (i.e. (4.7), (4.13)) back into the boundary action, and get
γ ǫˆab AABaA˙ ABb − ΛABFABab + π
− ϕℓA
γ ǫˆab AABaA˙ ABb − ΛABF ABab + π
− ϕℓA
− ℓDaℓ
S∂N =
are the time components of the Lagrange multipliers imposing the areamatching
constraint (3.3a), and the reality conditions (2.21).
where ˇǫab is the inverse LeviCivita density on S, implicitly defined through ǫˆacˇǫbc = δb .
a
Poisson brackets (4.17) and (4.18) and their complex conjugate, e.g. {
canonical variables vanish.
Next, we have the constraints. The variation with respect to the sl(2, C) Lie algebra
Variation with respect to N yields the scalar constraint
The Lagrange multiplier N a gives rise to the vector constraint
S[N ] =
ΛAB h γ ǫˆabFABab + πAℓBi =! 0,
ΛAB h γ ǫˆabF ABab + πAℓBi =! 0.
Going back to the 2+1 split (4.15) of the action, we can immediately read off the symplectic
structure. First of all, we see, that the spinors ℓA, ℓ
A are canonical conjugate to the
A = ϕt∗[πA] and π = eϕ∗[πA]. The fundamental Poisson
which is given by
πA(p), ℓB(q) = − ǫABδ(2)(p, q),
AABa(p), ACDb(q) = +
AABa(p), ACDb(q) = − γˇǫabδC(AδDB)δ(2)(p, q),
Ha[N a] =
Finally, we have the matching constraints
M [ϕ] =
Ma[J a] =
− ℓADaℓ
A =! 0,
which are obtained from the variation of the action (4.15) with respect to the Lagrange
multipliers ϕ and J a.
Thus far concerning the constraints. The evolution equations, on the other hand,
assume a Hamiltonian form as well: for any phase space functional F , its time evolution is
governed by the Hamilton equations
F = {H, F } ,
where the corresponding Hamiltonian is the sum over all constraints of the system
H = S[N ] + Ha[N a] +
Constraint algebra and gauge symmetries
Having defined the Hamiltonian, we proceed to calculate the Poisson algebra among the
constraints and check whether the constraints are preserved under the Hamiltonian flow (4.23).
For the Gauss constraints (4.19) the situation is straightforward.
We recover two
functions on S, we find
where we have defined the sl(2, C) Lie bracket
GAB[Λ1AB], GCD[Λ2CD] = GAB [Λ1, Λ2]AB ,
GAB[Λ1AB], GCD[Λ2CD] = GAB [Λ1, Λ2]AB ,
[Λ1, Λ2]AB = [Λ1]AC [Λ2]C B − [Λ2]AC [Λ1]C B.
The vector constraint, on the other hand, gives rise to twodimensional diffeomorphisms
modulo SL(2, C) gauge transformations. If N a and M a denote vectorvalued test functions
on S, we find after a straightforward calculation that
Ha[N a], Hb[M b]o = −Ha[LN M a]+
N M a = [N, M ]a is the Lie derivative.
of twodimensional diffeomorphism and local SL(2, C) gauge transformations. First of all,
{Ha[N a], πA} = Da(N aπA) = LN↑ πA,
Hb[N b], AABao =
of the Lie deerivaetive intoethe spin bundle. The third Poisson bracket (4.28b), on the other
hand, returns a diffeomorphism only onshell — only if, in fact, the Gauss constraint (4.19)
is satisfied, in which case
N bˇǫabǫˆcdF ABcd = N bF ABba = LN↑ AABa,
For the Gauss constraint, the situation is easier. There we get the fundamental
transwhich are the generators of right translations along the fibres of the SL(2, C) principal
A and AABa is completely analogous.
We then also have the matching consetraient (4.22ae), which Poisson commutes with the
SL(2, C) connections AABa and AABa, but generates the U(1)C transformations
= +ϕℓA,
{M [ϕ], Ma[J a]} = 2Ma[ϕJ a].
immediately, as feor instanece
Concerning the algebra of constraints, we only need to consider two further Poisson
{S[N ], Ma[J a]} =
{Ha[N a], Mb[J a]} =
− Da (J aℓA) N bDbℓA+
If we perform a partial integration (N a and J a all have compact support) and bring all
covariant derivatives on one side, this can be simplified to
− N aDaℓAJ bDbℓA + Da J aℓA N bDbℓAi.
{Ha[N a], Mb[J a]} = − Ma [LN J a] +
where the Lie derivative of the vectorvalued density J a on S is defined by
LN J a = 2Db N [bJ a] + N aDbJ b.
The equations (4.25), (4.27), (4.32), (4.33) and (4.35) give already all relevant Poisson
brackets. All other Poisson brackets among the constraints can be inferred trivially from
either (4.28) or (4.30). So as for e.g.
{Ha[N a], S[M ]} = −S[LN M ],
which is an immediate consequence of (4.28).
First class and second class constraints.
We have now collected all Poisson brackets
that are necessary to identify the first class and second class constraints of the system. The
vector constraint Ha[N a] (see (4.28)) generating twodimensional diffeomorphisms on S is
SL(2, C) gauge transformations (4.30) on either side of the ineterfaece. Equally, for the
matching constraint: M [ϕ] is first class and generates the U(1)C transformations (4.31) of
the boundary spinors.
We are then left with the scalar constraint S[N ] (as in (4.20)) and the matching
condition Ma[J a] (as in (4.22b)). The scalar constraint is second class, which is a consequence
of (4.35). For the matching condition, the situation is more complicated. The constraint
them is second class, all others are again first class. This can be seen as follows: in general,
we will have that ℓADaℓA does not vanish14 on phase space. We can then parametrise the
densitised vector J a as follows
= z()ǫˆabℓADbℓA + z()d2v qab(β + i)ℓ¯A¯Dbℓ¯A¯,
where qab is a fiducial signature (++) twometric on S and d2v denotes the corresponding
area element. The only relevant Poisson bracket for determining the second class component
z() and z() for which the Poisson bracket {Ma[J a(z(), z())], S[N ]} vanishes onshell 15 for
≈ 0,
vant Poisson brackets involving Ma[J a(z(), 0)] weakly vanish as well, which is an immediate
consequence of (4.28), (4.30) and (4.31). Hence the constraint
straints (4.19), (4.20), (4.21), (4.22a), (4.22b) firstclass.
15Given all constraints (4.19), (4.20), (4.21), (4.22a), (4.22b) are satisfied.
That (4.41) defines another first class constraint follows from (4.33) and
≈ 0.
All other constraints Poisson commute with (4.41). This can be inferred already from the
infinitesimal gauge transformations (4.28), (4.30) and (4.31).
We are thus left to identify the single second class component of Ma[J a]. It is given
by the expression
from (4.33) through
MahJ a 0, iy() i, S[N ]o + cc. ≈
≈ 4
6= 0,
which does not vanish unless ℓADaℓA = 0.
Gauge symmetries.
We have now identified all first class constraints of the system:
vector constraint (4.21) and the matching ceonditeion (4.22a), which generate two copies of
internal SL(2, C) gauge transformations (on either side of the interface), twodimensional
diffeomorphisms of S and U(1)C gauge transformations (4.31). We then have additional
firstclass constraints, which can be identified with those components (4.40) and (4.41) of
Ma[J a] that Poisson commute with the scalar constraint S[N ]. The geometric meaning
of local SL(2, C) gauge transformations, U(1)C transformations and twodimensional
diffeomorphisms is clear,16 but what kind of gauge transformations are generated by those
components of Ma[J a] that are first class?
The answer is hidden in an additional gauge symmetry, which appears in the bulk. The
bulk action (3.32) has in fact more symmetries than just fourdimensional diffeomorphisms
and local SL(2, C) transformations. It enjoys a further gauge symmetry, which renders the
entire theory topological. The action is invariant under the infinitesimal shifts
equation (2.5) and (2.20).
A → ez/2ℓA and ηA → e−z/2, see
of the boundary action breaks this shift symmetry, but only partially. To understand this
more explicitly, let us first define
illustrate that Ma[J a] generates a version of the shift symmetry (4.45) on the boundary.
Using the Poisson brackets (4.17) and (4.18), a short calculation gives
1 J bˇǫbaℓAℓB,
o =
o =
= J a(Daℓ(A)ℓB) +
Notice that the last line is nothing but the covariant exterior derivative of the first.
Comfor the shift symmetry in the bulk, with the gauge parameter J a at the boundary by
The matching constraint Ma[J a] therefore generates a shift transformation (4.45) with
gauge parameter (4.49). Notice, however, that this shift symmetry is broken partially by
the addition of the reality conditions (4.20) at the boundary. Only if J a is of the particular
form of (4.40) or (4.41), do we get a symmetry preserving the constraint hypersurface. For
generic values of J a, the conditions (4.40) and (4.41) will be violated, and the Hamiltonian
vector field {Ma[J a], ·} will lie transversal to the constraint hypersurface.
Dimension of the physical phase space. In summary, the system admits four types
of gauge constraints, and two second class constraints. The scalar constraint S[N ] and the
y()component (4.43) of the matching constraint Ma[J a] are second class. We then have the
first class constraints, which are the vector constraint Ha[N a], generating diffeomorphisms
side of the interface, the matching coenstraeint M [ϕ], generating U(1)C transformations and
the remaining components (4.40) and (4.41) of Ma[J a], which generate the residual shift
symmetry (4.48) at the boundary. The situation is summarised in the table 1 below.
The counting proceeds as follows: we have two second class constraints and nineteen
first class constraints; there are two independent components of the vector constraint, two
Dirac classification
two first class constraints
three Cvalued first class constraints
three Cvalued first class constraints
one Cvalued first class constraint
one is second class, three are first class
one first class constraint
2 × 2 = 4
2 × 6 = 12
2 × 6 = 12
2 × 2 = 4
1 + 2 × 3 = 7
directions. This renders the boundary theory topological. There are no local degrees of freedom.
times six independent constraints generating the SL(2, C) transformations on either side
of the interface, two independent components of the U(1)C generators M [ϕ] (the smearing
function ϕ : S → C is complex) and three additional first class constraints, namely (4.40)
and (4.41) generating the residual shift symmetry (4.48) at the boundary. The kinematical
dimensions, every first class constraint removes two degrees of freeedom,ewhiech leaves us
the scalar constraint S[N ] and the second class component (4.43) of Ma[J a], which leaves
us with no local degrees of freedom along the threedimensional interface. This renders
the boundary theory topological. All physical degrees of freedom can only appear at the
twodimensional corners.
Relevance for quantum gravity
So far, we have only been studying the classical theory. The main motivation concerns,
however, nonperturbative approaches to quantum gravity, such as, in particular, loop quantum
gravity. Let me explain and justify this expectation, without going into the mathematical
Loop quantum gravity can be based either on the phase space [1, 7] for an SL(2, C)
connection or the phase space for an SU(2) connection [30, 31]. The complex variables have
the advantage that local Lorentz invariance is manifest, though we then also need to impose
additional reality conditions, which are otherwise already solved implicitly (see e.g. [32, 33]
for a recent analysis on the issue).
On the phase space for the complex variables, the symplectic structure is determined
by the fundamental Poisson brackets for the selfdual variables
which can be derived from the topological bulk action (3.32) as well. It was then noted [34,
35] that the theory can be discretised, or rather truncated, by requiring that the connection
17The symplectic structure is determined by (4.17) and (4.18).
be flat everywhere except along the onedimensional edges {Ei} of a cellular decomposition
to the system of edges {Ei}, modulo SL(2, C) gauge invariance at the nodes p1, . . . , pN of
= ǫAB,
= −ǫAB,
One then postulates Poisson brackets
and shows that the symplectic reduction with respect to the areamatching constraint
One of them is
vector to the face fl dual to the link. This normal is often required to be timelike, if
it is, however, null rather than timelike, the condition simplifies: there is then always a
A on fl, such that nAA¯ = iℓAℓ¯A¯, which implies that either ω
Al are interchangeable, ℓA is unique
modulo U(1)C transformations, and we can, therefore, always restrict ourselves to the case
e.g. [36–38], a generalisation to null surfaces was proposed as well cf. [39].
19By convention ΠAB is assigned to the fibre over the initial point of the underlying link.
l
transport along the link going from the intersection p towards either endpoint l(0) or l(1).
This in turn suggests to identify the conjugate spinors with the twodimensional surface
implies to view the faces fl dual to the links as twodimensional cross sections of
threedimensional null surfaces Nij . For any such face fl there is then a threedimensional internal
side of a null interface N(ij)(l) shining out of fl.
shining out of the faces fl dual to the links of the graph. The Poisson brackets (4.53),
which were previously postulated, can be then derived from the Poisson brackets (4.17)
of the discrete spinors is analogous to equation (3.46). The same happens for the link
holonomy; the SL(2, C) parallel transport (4.52b) along a link is analogous to the SL(2, C)
gauge transformtion (3.25) across the interface. Equally for the constraints: the
areamatching condition and the reality conditions appear both in the discrete theory on a
graph (as in (4.54) and (4.55)) and in the threedimensional boundary theory (as in (4.22a)
and (4.20)). There is no doubt that this correspondence must be worked out in more detail.
So far, I find the analogy encouraging. It suggests that the kinematical structure of loop
quantum gravity — graphs, operators and spinnetwork functions — can be all lifted along
null surfaces obtaining a fully covariant picture of the dynamics in terms of a topological
field theory with defects.
Summary, outlook and conclusion
This paper developed a model for discrete gravity in four spacetime
dimensions where the only excitations of geometry are carried along curvature defects propagating
at the speed of light. The resulting theory has no local degrees of freedom in the bulk,
nontrivial curvature is confined to threedimensional internal boundaries, which represent a
system of colliding null surfaces.
The theory is similar to Regge calculus [20] and other discrete approaches, such as
’t Hooft’s model of locally finite gravity [40], causal dynamical triangulations and causal
sets [41, 42], but there are fundamental differences. First of all, and most crucially, we
have a field theory for the Lorentz connection rather than a lattice model for the metric.
This field theory is topological and the underlying spacetime manifold splits into a union
of fourdimensional cells {
Mi} : M = SiN=1
Mi, whose geometry is either flat or constantly
cell, there are no local degrees of freedom. Nontrivial curvature is confined to internal
boundaries N
ij = M
i ∩ Mj , which are threedimensional.
The underlying action (3.36) is local, and splits into a sum over all fourdimensional
building blocks, inner threeboundaries and twodimensional corners. The internal
boundary terms are necessary to have a wellposed variational principle. The problem of finding
the correct dynamics for the curvature defects boils then down to finding the right
boundary action, which cancels the connection variation from the bulk and consistently glues the
Mi} across their boundaries.
What is then the right boundary term? We are viewing gravity as a YangMills gauge
theory for the Lorentz group. At a boundary, a YangMills gauge connection couples
naturally to its boundary charges. Consider, for example, a configuration where the YangMills
electric field is squeezed into a Wilson line. Wherever this Wilson line ends and hits a
twodimensional boundary, a colour charge appears that cancels the gauge symmetry from
the bulk. For an SL(2, C) Lorentz connection, the relevant charge is spin, which suggests
to look for an action with spinors as the fundamental boundary variables. We proposed
such an action in section 2 for a boundary that is null. That the internal boundaries are
null rather than spacelike or timelike is well desired, it imposes a local notion of causality:
the field strength of the SL(2, C) connection is trivial everywhere except at the internal
boundaries, which are null and represent, therefore, the world sheets of curvature defects
propagating at the universal speed of light. This was further justified in section 3.4, where
we gave a constructive proof of existence: we showed that there are explicit solutions of
the equations of motion derived from the action (3.36), which represent impulsive
gravitational waves. These are exact solutions of Einstein’s equations in the neighbourhood of an
interface, and may describe e.g. the gravitational field of a massless point particle [22].
The model is specified by the action for the internal boundaries. This action assumes
a surprisingly simple form. It defines, in fact, nothing but the symplectic structure for
a spinor ℓA and its canonical momentum, which is (in three dimensions) a spinorvalued
interpretation: the bilinear iℓAℓ¯A¯ defines the null generator of the interface, the spin (1,0)
area element (2.22). The spinors at the interface are not completely independent, they
are subject to certain constraints. First of all, we have the reality conditions (2.21), that
the interface. Furthermore, there are the glueing conditions (3.3) that match the intrinsic
threegeometry across the interface. The only metric discontinuity is in the transversal
direction. Adding the glueing conditions to the action has a further effect: given a boundary
is SL(2, C) invariant, but it violates this additional U(1)C gauge symmetry. The symmetry
is restored by the areamatching condition (3.3a), which is added to the action by replacing
A more thorough analysis of the gauge symmetries was performed in section 4. First
of all, we noticed that the action in the bulk (3.32) is topological. We then integrated out
at the threedimensional interfaces. Next, we performed a 2 + 1 split of the boundary
action and identified the symplectic structure of the theory. All fundamental variables
appear twice, because every such interface N
ij bounds two bulk regions M
induce boundary variables from either side. We found that the SL(2, C) connection
becomes Poisson noncommutative at the boundary, while the configuration spinor ℓA has the
we found the canonical Hamiltonian (4e.24), wheich is a sum over the constraints of the
system, which consist of the vector constraint generating twodimensional diffeomorphisms,
a pair of SL(2, C) Gauss constraints generating SL(2, C) gauge transformations on either
side of the interface, the areamatching condition generating U(1)C transformations of the
boundary spinors, and finally the three firstclass components (4.40), (4.41) of the glueing
conditions (4.22b), which generate the residual shift symmetry (4.45), (4.48). All of these
constraints are firstclass, the reality condition (4.20), on the other hand, is second class,
reduction removes, therefore, forty dimensions from the kinematical phase space, which is
fortydimensional as well. This brought us to the conclusion that the theory has no local
degrees of freedom, neither in the bulk nor at the threedimensional internal boundaries.
Relevance for quantum gravity. The proposal defines a topological gauge theory with
defects. Solutions of the equations of motivation represent distributional spacetime
geometries, where the gravitational field is trivial in fourdimensional causal cells, whose boundary
is null. The geometry is discontinuous across these internal boundaries, which represent
curvature defects propagating at the speed of light.
Our main motivation concerns possible applications for nonperturbative approaches
to quantum gravity, such as loop quantum gravity. We have a few indications supporting
this idea: first of all, the model has a kinematical phase space, whose canonical structure is
extremely close to recent developments in loop quantum gravity. In [36–38, 43–45], a new
representation of loop quantum gravity has been introduced with spinors as the fundamental
configuration variables. This construction was bound to the discrete phase space on a
graph. An interpretation was missing for what these spinors are in the continuum. This
paper closes this gap and provides a continuum interpretation: the loop gravity spinors are
the canonical boundary degrees of freedom of the gravitational field on a null surface.
The most interesting indication in favour of our proposal concerns its dynamical
structure. The theory is topological and this suggests that the transition amplitudes, which are
formally given by the path integral
ordinary integrals (or sums) over the moduli of the theory. This would be reminiscent of
quantum gravity in three dimensions, where the PonzanoRegge amplitudes can be written
as a product over SU(2) group integrals for each edge times SU(2) delta functions imposing
the flatness of the connection (see [46] for a recent derivation). Such moduli exist, and the
simplest example is the fourvolume20
4Vol[Mi] =
of a given fourcell Mi, or the trace of the SL(2, C) holonomy around the perimeter of a
corner. The existence of such nonlocal observables is an important hint that the formal
definition of the path integral (5.2) has a mathematical precise meaning and defines a
socalled spinfoam model, which is given by certain fundamental amplitudes assigned to the
adjacency relations of the underlying cellular decomposition of the fourmanifold M (such
as in three dimensions where the 6jsymbol defines the vertex amplitude for the
Ponzano
Perspectives. The amplitudes (5.2) are defined for a given and fixed family of
fourdimensional cells {
Mi}, which are glued among bounding interfaces { ij }. This
combinaN
torical structure is an adhoc input, which enters the classical action (3.36) as an external
background structure. How are then different discretizations {
Mi} with different
combinatorial structures supposed to be taken into account? There are two possible answers to
this question: in the first scenario, the full theory will be defined through a continuum
limit, which sends the number of fourdimensional cells to infinity. The definition of the
theory would then most likely include some sort of renormalisation group flow, which would
give a prescription for how to take this limit in a rigorous manner. The main conceptual
difficulty with such an approach is that there is no fundamental lattice scale entering the
20That the fourvolume 4Vol[Mi] defines an observables is straightforward to see: clearly, it is invariant
under local Lorentz transformations and diffeomorphisms that preserve the bulk region Mi. It is also
integral ∝ R
action (3.36). Indeed, it is the gravitational field itself that determines the size of the
individual building blocks, and this makes it difficult to identify the correct variables and the
correct notion of scale to study the renormalization group flow. Therefore, more
sophisticated tools and techniques such as those developed for Regge calculus [2–4] and socalled
spinfoam models [5, 6, 47, 48] may be required.
The second possibility, which I find more appealing, is a more radical idea. In this
scenario, the amplitudes for a given and fixed configuration of fourcells would be seen as
Feynman amplitudes for an auxiliary quantum field theory. To define the entire theory,
one would then sum over an infinite, but most likely very preferred class of combinatorical
structures, which would arise from the perturbative expansion of the auxiliary field theory.
The approach would be conceptually very similar to group field theory [49–51], where the
gravitational path integral on a given simplicial discretisation arises from the perturbative
expansion of a quantum field theory over a group manifold.
Finally, there is one obvious open question that I have avoided altogether, namely how
the two physical degrees of freedom of general relativity should come out of the model.
This question is certainly related to the previous question regarding the continuum limit,
but some hints of an answer should already appear at the level of the microscopic theory,
which is defined by the action (3.36). This action was constructed such that the solutions
of the equations of motion represent fourdimensional distributional geometries, where the
curvature is trivial in fourdimensional cells, which are glued among bounding nullsurfaces.
The geometry is described in terms of SL(2, C) gauge variables (an SL(2, C) connection in
the bulk coupled to spinors at the internal null boundaries). If we then take the quotient
by the internal SL(2, C) gauge transformations, we are left with a theory that can only
be described by a metric and a connection. The connection satisfies the torsionless
condition (3.33), hence we expect that the only relevant degrees of freedom are captured by the
metric, which is now locally flat. We then saw in section 3.2 that special solutions of the
equations of motion exist that have a nontrivial distributional curvature tensor at the
defect: the resulting Weyl tensor is of Petrov type IV, thus describing transverse gravitational
radiation. But we then also saw that there are solutions where both the traceless part of
the Ricci tensor (3.73) and the Weyl tensor (3.72) are nonvanishing, the Ricci tensor being
tributional matter (as in e.g. string theory), with a distributional energy momentum tensor
gravity with more than just two propagating degrees of freedom. A minimal example for
such a theory is given by the Starobinsky model [52] of inflation, which has three
propagating degrees of freedom (which are given by the two polarisations of gravitational radiation,
and one additional spin0 scalar mode). I find this idea very promising and exciting, and it
is, in fact, the line of reasoning that I am currently investigating. A more rigorous analysis
will be presented in an upcoming article, which is currently under preparation.
This research was supported in part by Perimeter Institute for Theoretical Physics.
Research at Perimeter Institute is supported by the Government of Canada through the
Department of Innovation, Science and Economic Development and by the Province of Ontario
through the Ministry of Research and Innovation.
Spinors and world tensors
Following Penrose’s notation, we write ℓA with A, B, C, . . . to denote a twocomponent
spinor that transforms under the fundamental representation of SL(2, C), primed indices
A¯, B¯, C¯, . . . refer to the complex conjugate representation. The indices are raised and
group action. Our conventions are
ℓA = ǫBAℓB, ℓA = ǫABℓB, ℓ¯A¯ = ǫ¯B¯A¯ℓ¯B¯ , ℓ¯A¯ = ǫ¯A¯B¯ ℓ¯¯ ,
B
with ǫAC ǫBC = δA. The relation between spinors and internal Minkowski vectors vα is
B
matrix representation
the generalised Pauli identity
α maps an internal Lorentz vector vα ∈ R4 into an antihermitian21
map. This isomorphism can be generalised to any world tensor. It maps the Lorentz
cover of the restricted Lorentz group) into a proper orthochronous Lorentz transformation
= 1,
σAC¯ ασ¯C¯Bβ = −δBAηαβ − 2ΣABαβ,
are the selfdual generators of SL(2, C). Equation (A.3) implies that the matrices
relations of the Lorentz group. Indeed, we have
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