3D Quantum Gravity from Holomorphic Blocks

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Etera R. Livine1, ∗and Qiaoyin Pan2, 3, 4, †

1Universit´e de Lyon, ENS de Lyon, CNRS, Laboratoire de Physique LPENSL, 69007 Lyon, France

2Department of Physics, Florida Atlantic University, 777 Glades Road, Boca Raton, FL 33431, USA

3Perimeter Institute, 31 Caroline St North, Waterloo N2L 2Y5, Ontario, Canada

4Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada

(Dated: October 11, 2022)

Three-dimensional gravity is a topological ﬁeld theory, which can be quantized as the Ponzano-

Regge state-sum model built from the {3nj}-symbols of the recoupling of the SU(2) representations,

in which spins are interpreted as quantized edge lengths in Planck units. It describes the ﬂat

spacetime as gluing of three-dimensional cells with a ﬁxed boundary metric encoding length scale.

In this paper, we revisit the Ponzano-Regge model formulated in terms of spinors and rewrite

the quantum geometry of 3D cells with holomorphic recoupling symbols. These symbols, known as

Schwinger’s generating function for the {6j}-symbols, are simply the squared inverse of the partition

function of the 2D Ising model living on the boundary of the 3D cells. They can furthermore be

interpreted, in their critical regime, as scale-invariant basic elements of geometry. We show how to

glue them together into a discrete topological quantum ﬁeld theory. This reformulation of the path

integral for 3D quantum gravity, with a rich pole structure of the elementary building blocks, opens

a new door toward the study of phase transitions and continuum limits in 3D quantum gravity, and

oﬀers a new twist on the construction of a duality between 3D quantum gravity and a 2d conformal

theory.

Contents

Introduction 2

I. The Ponzano-Regge spinfoam for 3D quantum gravity 6

A. 3D gravity with boundary 6

B. 3D quantum gravity as a discretized gauge ﬁeld path integral 7

C. Fourier transform and spin network evaluations 9

D. Topological invariance and local holography 11

1. Topological invariance from Pachner moves 11

2. Locally-holographic amplitude 13

II. A new coherent holomorphic state-integral 15

A. Generating function for spin networks 15

B. The holomorphic {12z×2}symbol and 2D Ising on a tetrahedron 17

C. A new Ponzano-Regge state-integral formula 21

D. Topological invariance of the holomorphic blocks 23

III. Geometric interpretation of the state-integral 30

A. Poles of the {12z×2}symbol 31

B. Propagator and geometric gluing 38

C. Ponzano-Regge state-integral versus state-sum models 40

Outlook 41

Acknowledgement 42

A. Spinorial phase space for loop gravity 42

∗Electronic address: etera.livine@ens-lyon.fr

†Electronic address: qpan@fau.edu

arXiv:2210.04855v1 [hep-th] 10 Oct 2022

B. Wheeler-de Witt equation of the SGF 45

C. The Ponzano-Regge model in terms of coherent blocks 46

References 48

Introduction

Spinfoam models (see e.g. [1–3] for reviews) provide a rigorous, background-independent and non-perturbative

path integral quantization of gravitational theories based on discrete topological quantum ﬁeld theories and state-

sum models. They deﬁne probability amplitudes for histories of quantum states of geometry deﬁned as entangled

collections of discrete excitations. They can be considered as a quantum version of Regge calculus for discretized

general relativity [4]. They provide transition amplitudes for spin network states in loop quantum gravity [1, 5, 6].

They also provide the triangulation weights in sum-over-random-geometries approaches to quantum gravity, such as

matrix models, tensor models and group ﬁeld theories [7, 8]. Finally, they are a natural mathematical framework for

deﬁning topological invariants e.g. [9], and have been shown to be related to non-commutative geometry, e.g. [10], to

lattice ﬁeld theories with quantum group gauge symmetries, e.g. [11, 12], and to higher gauge theories [13–16].

Retrospectively, the ﬁrst spinfoam model was constructed by Ponzano and Regge [17] and deﬁnes a discrete

topologically-invariant path integral for three space-time dimensional gravity in Euclidean signature with vanishing

cosmological constant [9, 18, 19]. Let us underline that this is not a Wick-rotated path integral but it is truly the quan-

tum theory of a positive signature metric with probability amplitudes in exp[iSgrav ]. A posteriori, the Ponzano-Regge

model has been understood as the discretized path integral for 3D gravity written in terms of veirbein-connection

variables as a topological BF theory with gauge group SU(2) [19]. There exists a Lorentzian version of this model

based on the SU(1,1) gauge group [20–23]. One can also take into account a non-vanishing cosmological constant

through a q-deformation of the gauge group [11, 12, 24]. This yields the Turaev-Viro topological invariant when the

quantum deformation parameter qis a root of unity [25]. Through this relation, the Ponzano-Regge state-sum has

been understood as a special case of the Reshetikhin-Turaev invariants [26] and thereby related to the quantization

of 3D gravity as a Chern-Simons theory as advocated by Witten [27].

The Ponzano-Regge model is constructed as a path integral over discrete 3D geometries. Considering a 3D piecewise

linear cellular complex ∆, which can be thought of as a cellular decomposition of a Riemannian 3D manifold, one has

a hierarchy of cells with dimensions between 0 and 3, which one dresses with algebraic data following the logic from

algebraic topology. Then one builds a probability amplitude for the 3D geometry from the algebraic data, such that

it is topologically invariant in the sense that it does not depend on the details of the 3D cellular complex but only on

its topology (and on boundary data). So the hierarchy of the Ponzano-Regge state-sum is:

•0-cells (points):

This level is actually put aside by spinfoam models, which focus on geometrical structures with co-dimensions

less than or equal to 2. This allows for conical singularities e.g. [28], which can be appropriately controlled in

sums over random discrete geometries e.g. [29].

•1-cells (edges):

Edges have the topology of a basic segment, devoid of any winding information. One dresses edges with

irreducible representations of the Lie group SU(2). These are labelled by half-integers j∈N

2, usually referred

to as spins. The Hilbert space Vjcarrying the representation of spin jis of dimension dj= (2j+ 1) and its

standard basis is given the spin basis labelled by the spin jand the magnetic moment mrunning from −jto

+jby integer steps.

The spin jecarried by an edge egives its quantized length in Planck unit, `e=je`P lanck . A state in Vjis

then interpreted as a quantum 3-vector of length j. This is the key to the geometrical interpretation of the

Ponzano-Regge model.

•2-cells (faces):

Faces are assumed to have the topology of a two-dimensional disk with a S1boundary. They are dressed with

intertwiners, that is SU(2)-invariant states living in the tensor product of the spins living on the edges forming

the face’s boundary:

If= InvSU(2)O

e∈∂f Vje.(1)

In the case of a triangle, consisting of three edges carrying spins j1, j2, j3, this intertwiner state is one-dimensional

if the spins satisfy the triangular inequalities and 0-dimensional in all other cases. When it is non-trivial, the

unique intertwiner state is given by the Clebsh-Gordan coeﬃcients, or equivalently the Wigner {3j}-symbols,

encoding the recouplings of the three spins. In general, intertwiner states are interpreted as the quantum version

of polygons.

•3-cells (elementary 3D regions or bubbles):

Elementary 3-cells σhave the topology of a 3-ball. Their boundary has the topology of a 2-sphere made of faces

glued together along edges. We focus on the boundary of each 3-cell and think of them as bubbles. For each

bubble, we usually introduce the dual graph Γ = (∂σ)∗

1, deﬁned formally as the 1-skeleton of the dual of the

boundary 2-complex: each face is represented as a (dual) node and each edge is represented as a (dual) link

linking two nodes. Each edge or link comes with its spin. Each face or node comes with its intertwiner state.

Such graph with representations on its links and intertwiners at its nodes, Γ{jl,ιn}, is called a spin network, as

illustrated on ﬁg.1. We deﬁne the probability amplitude for the geometry of the 3-cell as the evaluation of its

j1j2

j11

j12

j13

j14

j4j9

j10

•

ι1•

ι2

•ι3

•ι4

FIG. 1: (A portion of) the cellular decomposition of a 2-sphere made of faces glued along edges (in black and

thick) and its dual graph Γ(in red and thin) made of links and nodes. Each link lis dressed with a spin jl

and each node nis dressed with an intertwiner ιn, which together represent a spin network Γ{jl,ιn}.

boundary spin network:

Aσ[{je, ιf}e,f∈∂σ] = Tr{je}e∈∂σ O

f∈∂σ

ιf= Tr{jn}n∈ΓO

n∈Γ

ιn=EΓ[{jl, ιn}l,n∈Γ].(2)

The trace Tr here is a slightly abusive notation. It means gluing the intertwiner states using the inner product

on each edge, in the tensor product Hilbert space Ne∈∂σ Vje. In the case of a 3-simplex, or tetrahedron, the

boundary graph consists of four nodes connected to each other through six links, as illustrated in ﬁg.2. The

links carry six spins j1,..,6while the 3-valent nodes carry the corresponding unique intertwiner state numerically

given by the Clebsh-Gordan coeﬃcients. The resulting spin network evaluation is the celebrated Wigner’s {6j}-

symbol.

In general, one could glue the intertwiner states by inserting SU(2) group elements gl(or even SL(2,C) group

elements as in [30]) along each link. This yields the spin network wave-function ψΓ

{jl,ιn}∈ C∞(SU(2)×Eσ) where

Eσcounts the number of edges on the bubble boundary (or equivalently the number of links in the boundary

graph Γ). The spin network evaluation Aσ[Γ,{jl, ιn}] then truly is the evaluation of the spin network wave-

function ψΓ

{jl,In}on trivial group elements gl=Iin SU(2), reﬂecting the fact that physical states in pure 3D

(quantum) gravity with vanishing cosmological constant have a ﬂat curvature. The Ponzano-Regge model can

indeed be understood as deﬁning the projector onto the moduli space of ﬂat SU(2) connections e.g. [22, 31].

At the end of the day, the probability amplitude of a 3D cellular complex ∆ dressed with the algebraic data of

spins on its 1-cells and intertwiners on its 2-cells is obtained by putting all those elementary building blocks together

and straightforwardly computing the product of the probability amplitudes of each 3-cells together with appropriate

weights for the edges and faces:

A[∆,{je, ιf}] = Y

(−1)2jedjeY

(−1)Pe∈∂f jeY

σAσ[{je, ιf}e,f∈∂σ ].(3)

The edge weight only depends on the spin carried by the edge and is simply the dimension of the corresponding

representation up to a sign. The face weight is a mere parity factor. The bubble weight carries the non-trivial

dynamical information of the model. Such a structure with a cellular complex dressed with representations and

intertwiner states and a probability amplitude deﬁned as the product of local amplitude for each cell depending solely

on the algebraic data it carries is called the local spinfoam ansatz for a path integral over discrete quantum geometries.

It has been shown in [7, 32] that they are Feynman diagram amplitudes of non-local non-commutative ﬁeld theories,

referred to as group ﬁeld theories or tensorial group ﬁeld theories (see [33] on recent studies of those Feynman diagrams

and the renormalization of such ﬁeld theories).

A ﬁrst important remark is that the Ponzano-Regge ansatz is topologically invariant. Indeed, let us consider the

amplitude for a 3D cellular complex ∆ obtained by summing over all possible algebraic data. More precisely, we allow

∆ to have a 2D boundary, we keep the algebraic data on ∂∆ ﬁxed while we sum over bulk spins and bulk intertwiners:

A[∆,{je, ιf}e,f∈∂∆] = X

{je,ιf}e,f ∈∆oA[∆,{je, ιf}e,f∈∂∆,{je, I =ιf}e,f∈∆o],(4)

where we have written ∆o= ∆ \∂∆ for the interior or bulk of ∆. Under appropriate gauge-ﬁxing1, this amplitude

can be shown to depend only on the boundary data and on the topology of ∆ and to never depend on the details

of the bulk cellular complex ∆o[19, 22, 34, 35]. More precisely, the amplitude for a 3D cellular complex with the

topology of a 3-ball remains the evaluation of its boundary spin network. For 3D cellular complexes with a non-trivial

topology, one needs to evaluate the spin network wave-function on the values of the non-trivial holonomies along the

non-contractible cycles of the bulk cellular complex or integrate over possible values [22, 34, 35].

The second important remark is that the Ponzano-Regge ansatz is locally holographic, in the sense that the

probability amplitudes associated to bounded regions of 3D space entirely depend on their boundary data (and their

topology). The amplitude for each elementary 3-cell σdepends by deﬁnition solely on its boundary data: the spins

jeand intertwiners ιfcarried by the edges and faces on its boundary e, f ∈∂σ. It is important to stress that

there is no new algebraic data associated to the 3-cells (no maps between intertwiner states as one could imagine).

The probability amplitude of the 3-cell is the evaluation of its spin network Aσ[{je, ιf}e,f∈∂σ ] = EΓ[{jl, ιn}l,n∈Γ].

Moreover, this property is true also for non-elementary 3-cells, that is for every bounded 3D region. Indeed, as long

as one considers a bounded 3D region Rwith the topology of a 3-ball, the topological invariance property of the

Ponzano-Regge amplitude, detailed above, implies that the amplitude A[R,{je, ιf}e,f ∈∂R] is simply the evaluation of

the boundary spin network on ∂R. This means that any bounded 3D region with the topology of a 3-ball behaves

exactly as an elementary 3-cell and its probability amplitude always only depends on its boundary data and never on

the details of its bulk cellular decomposition. In this setting, the local holography principle is deeply interlaced with

the topological invariance of the theory2.

In the present work, we propose to revisit the Ponzano-Regge model and write it in terms of coherent boundary

states for each 3-cell instead of pure spin networks sharply peaked on lengths. Those coherent states will be peaked

on both intrinsic geometries - the edge lengths - and extrinsic geometry - the dihedral angles between faces, which

deﬁne a discrete measure of extrinsic curvature. This reformulation has two important features:

•Using a coherent superposition of boundary spin networks, deﬁned as an inﬁnite series over the spins controlled

by couplings dual to the spins, actually amounts to considering a generating function for the spin network

evaluations. This is the same logic as for a simple quantum harmonic oscillator, in which matrix elements of an

operator in the coherent state basis. That is

hz|ˆ

O|˜zi=X

n,m∈N

¯zn˜zm

√n!m!hn|ˆ

O|mi(5)

1The sum over bulk spins and bulk intertwiners is usually divergent, just as Feynman diagrams in quantum ﬁeld theory. It is possible

to render those amplitudes ﬁnite by q-deforming SU(2) at root of unity, with q= exp(2iπ/N + 2) for an integer N∈N. This gives the

Turaev-Viro topological invariant [25] and is interpreted as switching on a non-vanishing positive cosmological constant Λ >0. Even

without quantum deforming the gauge group, one can still identify the translational gauge symmetry responsible for the divergences and

gauge-ﬁx them - typically by ﬁxing the value of the spins on the edges belonging to a maximal tree in ∆o, in which case the gauge-ﬁxed

amplitudes never depend neither on the choice of the gauge-ﬁxing tree nor on the bulk cellular complex ∆o[9, 19, 36, 37].

2A non-topological invariant model could still be locally holographic if one introduces the possibility of a non-trivial renormalization ﬂow

under coarse-graining, meaning that the probability amplitude of a bounded region would still be an evaluation of the boundary spin

network wave function but that evaluation would now depend on extra parameters reﬂecting the size of the region (and perhaps other

basic coarse-grained observables of the bulk geometry). The key would be that there would be only a ﬁnite number of extra parameters

and that this number would be the same for all regions.

can be understood as a generating function for the matrix elements hn|ˆ

O|micontrolled by the complex couplings

zand ˜z. Here, we focus on 3-valent boundary graphs, for which we don’t need intertwiner labels, so that spin

network evaluations EΓ[{jl, ιn}l,n∈Γ] simply depend on the spins on the boundary graph links. We will simply

write EΓ[{jl}l∈Γ]. Then we deﬁne coherent spin network evaluations similarly as for the harmonic oscillator as:

EΓ[{Yl}l∈Γ] = X

{jl∈N

Y2jl

lW[{jl}l∈Γ]EΓ[{jl}l∈Γ],(6)

with the couplings Yl∈Cand weights W[{jl}l∈Γ] possibly involving factorials of the spins [30, 38, 39]. Gen-

erating functions is a powerful mathematical tool. For instance, they typically map the asymptotic behaviour,

here at large spins i.e. the semi-classical regime for length scales very large compared to the Planck length, onto

poles of the generating function.

•The coherent spin superpositions, or equivalently the generating functions, that we consider here allow for

exact analytical resummation of the Ponzano-Regge amplitudes as rational functions in the couplings. They are

actually the generalization of Schwinger’s generating function for the {6j}-symbols [40, 41], they were introduced

as coherent spin network states in [42–44] and showed to lead to exact closed formula for spinfoam models in

[45, 46]. At the end, the evaluations EΓ[{Yl}l∈Γ], for speciﬁc well-chosen weights W[{jl}l∈Γ], were shown to be

given by the inverse squared partition function of the 2D Ising model with inhomogeneous couplings tan−1Yl

on the boundary graph [39]. For instance, the generating function for the {6j}-symbols corresponds to the

inverse of the square of the 2D Ising model on the tetrahedron with six variables Y1,..,6living on the edges and

representing the strength of the coupling between the four triangles [47]. Then, in general for arbitrary 3-cells

and their boundary graphs, this provides formulas for the Ponzano-Regge amplitude as holomorphic functions

of couplings living on the boundary of the 3-cells. These formulas are at the heart of the proposed holographic

duality between 3D quantum gravity deﬁned by the Ponzano-Regge path integral and the 2D Ising model [39].

Building on those previous works, we show how to glue the holomorphic amplitudes associated to the 3-cells - or in

short, holomorphic blocks- deﬁned as the evaluation of the coherent spin network superpositions on their 2d boundary.

This gluing is done in a topologically invariant way, that is so that overall amplitudes of a 3D region do not depend on

the chosen bulk cellular decomposition, and ultimately reproduces the sum over spins of the original Ponzano-Regge

formulation.

This reformulation oﬀers a new twist to the story of the Ponzano-Regge path integral. We indeed formulate it as a

topological net of 2D Ising partition functions glued together: each 3-cell deﬁnes a 2D Ising model on its boundary,

then those 3-cells, and thus those 2D Ising models, are glued together in a topologically-invariant fashion. We refer

to this construction as a topological Ising net. It would be enlightening to investigate in the future how general such

topological Ising nets can be, whether they can be deﬁned in any dimension, using arbitrary powers of the Ising

partition function, if they can be generalized to other condensed matter models and whether we can depart from

topological invariance in a controlled way with a non-trivial, yet integrable, renormalization ﬂow encoding the fusion

of the 3-cell algebraic structure and amplitudes.

Moreover, one can look at this construction from the perspective of (ﬁnite distance) holographic dualities `a la

AdS/CFT correspondence. As the 2D Ising model becomes a conformal ﬁeld theory (CFT) in its critical regime, the

exact equivalence of the present formulation between the 3D quantum gravity and the 2D Ising partition function,

which holds for every value of Ising couplings, can be understood as a non-critical version of the gauge-gravity

holography. Interpreting the Ising partition function for non-critical couplings as a non-critical version of conformal

blocks, the 3D Ponzano-Regge path integral is realized as gluing such 2D non-critical blocks. This version of holography

holds for discrete quantized geometries and not only at the level of ﬁeld theories in the continuum limit (see e.g. [48, 49]

for holographic duality at the ﬁeld theory level). More recent work following this line of thought and investigating the

holographic behaviour of the Ponzano-Regge path integral can be found in [50–52]. A hope is that this reformulation

will lead to new developments in the investigation of the phase diagram of 3D quantum gravity and the implementation

of quasi-local holography in spinfoams and loop-gravity-inspired path integrals for quantum gravity.

In section I, we review the standard formulation of the Ponzano-Regge state-sum as a path integral for discretized

3D gravity in its ﬁrst-order formulation in terms of vierbein-connection variables. We show that the Ponzano-

Regge amplitude for a 3D region is the spin network evaluation on the 2D boundary of the region, and that gluing

neighbouring 3D regions is implemented by a fusion of those spin network evaluations done in a topologically-invariant

way, which leads to a locally holographic formulation of 3D quantum gravity.

In section II, we introduce the generating function for spin network evaluations and compute it as a rational

holomorphic function. The {6j}-symbol for the tetrahedron becomes a holomorphic {12z×2}-symbol, equal to the

inverse squared partition function of the 2D Ising model on the tetrahedron. We write the Ponzano-Regge model in

terms of those holomorphic blocks and show the topological invariance of this new formulation.

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摘要：

3DQuantumGravityfromHolomorphicBlocksEteraR.Livine1,andQiaoyinPan2,3,4,y1UniversitedeLyon,ENSdeLyon,CNRS,LaboratoiredePhysiqueLPENSL,69007Lyon,France2DepartmentofPhysics,FloridaAtlanticUniversity,777GladesRoad,BocaRaton,FL33431,USA3PerimeterInstitute,31CarolineStNorth,WaterlooN2L2Y5,Ontario,Canada...

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