LOOP QUANTUM GRAVITY WITH COSMOLOGICAL CONSTANT
by Zichang Huang
A Dissertation Submitted to the Faculty of The Charles E. Schmidt College of Science in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
Florida Atlantic University Boca Raton, FL May 2019 Copyright 2019 by Zichang Huang
ii Loop Quantum Gravity with Cosmological Constant
by
Zichang Huang
This dissertation was prepared under the direction of the candidate's dissertation advisor, Dr. Muxin Han, Department of Physics, and has been approved by all members of the supervisory committee. It was submitted to the faculty of the Charles E. Schmidt College of Science and was accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
Muxin Han, Ph. . Dissertatio Ad isor
uc T. Wille, Ph.D. Chair, Department of Physics J~h.D:
~::;...=_~~;;,...oz:;___.__ ~___ -"_~ ____ Erik~~ Lundberg, .D. At · . ~edini, Ph.D. ean, Charles E. Schmtdt College Of Science
Khaled Sobhan, Ph.D. Interim Dean, Graduate College
111 ACKNOWLEDGEMENTS
The dissertation here has been supported by many people from different continents all over the world. I would use out of my word to thank all of them. But at first, I would like to show my graceful thanks to my advisor Dr. Muxin Han and my collaborator Dr. Antonia Zipfel. I have learned a lot from them. They have shown me a attractive world about the quantum gravity, and widen my horizon. Then I would like to thank the faculties in Florida Atlantic University physics de- partment who gave there fully support to my dissertation. Specially thanks to Dr. Jonathan Engle, Dr. Chirs Beetle, Dr. Warner Miller, Dr.Wolfgang Tichy, Dr. Luc T. Wille and Dr. Armin Fuchs, etc. Also thanks to Dr. Eric Lundberg from Math department of Florida Atlantic University for being my committee. During my stay in Florida Atlantic University, the Carlo Rovelli’s group in CPT- Marseille has also provided a great help for my research. Here is my heartfelt thanks to Prof. Carlo Rovelli, Prof. Simone Speziale, Prof. Alejandro Perez and my friend Hong- guang Liu. I would also like to thank the Prof. Canbin Liang and Prof. Yongge Ma from Beijing Normal University. Their patient guide lead me to the realm of gravity physics. Finally, I would like to thank my father Weiruo Huang and my mother Ni Dong. Their love and encouragement has always been an important support of my live.
iv ABSTRACT
Author: Zichang Huang Title: Loop Quantum Gravity with Cosmological Constant Institution: Florida Atlantic University Dissertation Advisor: Dr. Muxin Han Degree: Doctor of Philosophy Year: 2019
The spin-foam is a covariant path-integral style approaching to the quantization of the gravity. There exist several spin-foam models of which the most successful one is the Engle-Pereira-Rovelli-Levine/Freidel-Krasnov (EPRL-FK) model. Using the EPRL- FK model people are able to calculate the transition amplitude and the n-point functions of 4D geometry(both Euclidean and Lorentzian) surrounding by a given triangulated 3D ge- ometry. The semi-classical limit of the EPRL-FK amplitude reproduces discrete classical gravity under certain assumptions, which shows that the EPRLFK model can be under- stood as UV completion of general relativity. On the other hand, it is very hard to dene a continuum limit and couple a cosmological constant to the EPRL-FK model. In this dissertation, we addressed the problems about continuum limit and coupling a cosmological constant to the EPRL-FK model. Followed by chapter one as a brief in- troduction of the loop quantum gravity and EPRL-FK model, chapter two introduces our work about demonstrating (for the first time) that smooth curved spacetime geometries satisfying Einstein equation can emerge from discrete spin-foam models under an appro- priate low energy limit, which corresponds to a semi-classical continuum limit of spin-foam models. In chapter three, we bring in the cosmological constant into the spin-foam model
v by coupling the SL(2, C) Chern-Simons action with the EPRL action, and find that the quantum simplicity constraint is realized as the 2d surface defect in SL(2, C)Chern-Simons theory in the construction of spin-foam amplitudes. In chapter four, we present a way to describe the twisted geometry with cosmological constant whose corresponding quantum states can forms the Hilbert space of the loop quantum gravity with cosmological constant. In chapter five, we introduced a new definition of the graviton propagator, and calculate its semi-classical limit in the contents of spin-foam model with the cosmological constant. Finally the chapter six will be a outlook for my future work.
vi To my parents, and to my friends LOOP QUANTUM GRAVITY WITH COSMOLOGICAL CONSTANT
List of Figures ...... xi
1 Introduction ...... 1
2 Semi-Classical Continuum Limit and Emergent Gravity from Spin-Foam model ...... 6 2.1 Introduction...... 6 2.2 Spin foam models:...... 7 2.3 Large spin analysis:...... 8 2.4 Semiclassical continuum limit:...... 13 2.5 Emergent Linearized gravity:...... 15 2.6 Conclusion...... 16
3 Loop-Quantum-Gravity Simplicity Constraint as Surface Defect in Com- plex Chern-Simons Theory ...... 17 3.1 Introduction...... 17 3.2 Simplicity Constraint and Curved Tetrahedron...... 21 3.3 Quantization of Flat Connection and Simplicity Constraint...... 24
3 3.4 SL(2, C) Chern-Simons Theory on S \ Γ5 ...... 26
3.5 SL(2, C) Chern-Simons theory on M3 with Surface Defect...... 30 3.6 A Field-Theoretic Description of the Surface Defect...... 37 3.7 Surface Degree of Freedom...... 41 3.8 Conclusion...... 43
viii 4 SU(2) Flat Connection on Riemann Surface and 3D Twisted Geometry with Cosmological Constant ...... 45 4.1 Introduction...... 45 4.2 Flat Connection on Riemann Surface and Twisted Geometry...... 48 4.2.1 From Graph to Riemann Surface...... 48 4.2.2 Flat Connection on Riemann Surface and Curved Tetrahedron... 51 4.2.3 Gluing 4-holed Spheres...... 53 4.2.4 Relation with Twisted Geometry...... 54 4.3 Geometric interpretation of Twisted Geometry...... 56 4.3.1 Exponentiated Flux...... 58 4.3.2 Twist Angle and Extrinsic Curvature...... 59 4.3.3 More General Choices of base points...... 62 4.4 Symplectic Structure...... 68 4.5 Quantization...... 69
5 Spin foam propagator: A new perspective to include the cosmological constant 71 5.1 Introduction...... 71 5.2 Different proposals for a graviton propagator...... 74 5.2.1 Standard proposal and conflicts with a cosmological constant... 74 5.2.2 Perturbative truncated metric...... 76 5.3 Spin foam model with cosmological constant...... 79 5.3.1 2-point function...... 82 5.4 Asymptotic Limit of the 2-point function...... 84 5.4.1 Calculation scheme...... 85 5.4.2 First non-vanishing order of the graviton propagator...... 86 5.5 Summary...... 88
6 Outlooks ...... 91 6.1 Twistor Theory and Twisted Geometry with Cosmological Constant.... 91
ix 6.2 Emergent Graviton Propagator form Spin Foam...... 92 6.3 More Applications of the Semi-Classical Continuum Limit of the Spin- Foam model...... 92 6.4 Numerical Spinfoam and Tensor Network...... 93
Appendices ...... 95 A Appendices For Chapter Two...... 96 A.1 Spin Foam Models (SFMs) and Tensor Networks...... 96 A.2 Large Spin Analysis...... 99 A.3 Expansion of the linearized theory...... 106 A.4 Semiclassical Continuum Limit (SCL)...... 108 A.5 Convergence to Smooth Geometry...... 111 A.6 Some Topological Properties of the Triangulation...... 115 B Appendices for Chapter Four...... 119 B.1 Complex Fenchel-Nielsen (FN) Coordinate...... 119 C Appendices for Chapter Five...... 123 C.1 The area of the triangle in the geometry with a cosmological constant123 C.2 Calculation of the saddle point and the Hessian...... 124 C.3 Hessians...... 127 C.4 Chern-Simons Propagator for Non-Compact Gauge Group..... 128
Bibliography ...... 131
x LIST OF FIGURES
1.1 Cube of theoretical physics...... 1
2.1 (a) The 5-valent vertex in a 4-simplex illustrates a rank-5 tensor |Aσi. Glu- ing 4-simplices σ in K gives a tensor network TN(K, J~), where each link associates to a maximally entangled state of a pair of iτs. (b) A triangu- lation of the hypercube. The 4d hypercubic lattice with the triangulated hypercube makes K. (c) An illustration of the neighborhood N (the re- gion bounded by blue dashed lines) in the space of J~. The red curve il- ~ ~ lustrates MRegge, including J(`) as the perturbation of J(`˚). The black and blue arrows are basis vectorse ˆi(`) and ∂J~(`)/∂`, transverse and tangent to MRegge...... 9
3 3.1 Γ5-graph embedded in S ...... 21
3 3.2 M3 is obtained by gluing a number of S \ Γ5, each of which corresponds 3 to a 4-simplex in M4. The gluing of S \ Γ5’s is deduced from the gluing of 3 4-simplices in M4. In drawing the 3-manifold S \ Γ5 and M3, we imagine 3 3 to view S \ Γ5 from 4d and suppress 1 dimension. The 3-manifold S \ Γ5 has five geodesic boundary components as 4-holed spheres, coming from removing the neighborhood of five vertices of Γ5. It has ten cusp boundary components as ten annuli, coming from removing the neighborhood of ten edges of Γ5. The red curves are the annuli connecting 4-holed spheres. Two 3 S \ Γ5 can be glued through a pair of 4-holed spheres, via a certain identi- fication of holes. Each 4-holed sphere as the gluing interface corresponds to a tetrahedron shared by two 4-simplices in M4. Each hole of the 4-holed sphere (or each tunnel traveling thought the 4-holed sphere) corresponds to a triangle in the shared tetrahedron...... 31
3 3.3 This picture shows a result that we glue three S \Γ5. The yellow shell out- side indicate the ambient 3-manifold X3. Four-holed spheres B1 and R3 are 3 3 shared boundary between blue S \Γ5 and red S \Γ5. Similarly, (B5, G3) and (R4, G2) are blue-green shared boundary and red-green shared bound- ary respectively. At the center of the picture, there is a non-contractible cycle, which made π1(X3) nontrivial. There is a closed tunnel with three different colors at the center corresponds to an internal triangle shared by three 4-simplices in M4...... 34
xi 4.1 Two 4-holed spheres S a and S b glued together...... 48 4.2 An example of graph. Two 4-valence vertices A and B are connected by a link...... 49 4.3 The Riemann surface made by connecting two 4-holed spheres...... 49 4.4 An alternative graph relating to Figure 4.3 by ρ ...... 49 4.5 A Riemann surface decorated by two base points and a meridian...... 50 4.6 A Riemann surface decorated by a base point...... 50 4.7 A graph recovered from Riemann surface Figure 4.5...... 50 4.8 Graph recovered from Riemann surface Figure 4.6...... 51 4.9 Path on tetrahedron, where point 4 is a base point and edge 2-4 is a special edge...... 51 4.10 Gluing topologically two tetrahedra through the interface (1, 2, 3). The topological gluing doesn’t necessarily identify the geometry of the com- mon face...... 57 4.11 The dash circle stands for a small open sphere around point 1. The link is at the vicinity of the base point of tetrahedra...... 60 4.12 Two tetrahedra with their vertices labeled by numbers through 1 to 5..... 62
4.13 The path 4 → 2 → 1 → 5 which is labelled as p0 ...... 63
4.14 The pathp ˆ0 which is the path corresponds to the path p0 on a pair of tetra- hedra...... 63
4.15 The path p1 constructed by adding p0 with one left-handed winding..... 64
4.16 The pathp ˆ1 made by addingp ˆ0 with a left-handed winding...... 64
4.17 The path 4 → 2 → 3 → 1 → 5 which is labeled as p−1 ...... 64
4.18 The pathp ˆ−1 made by addingp ˆ0 with a right-handed winding...... 65
4.19 The path p−2 made by p0 combining with two right-handed windings.... 65
4.20 The pathp ˆ−2 made by adding two right-handed windings top ˆ0 ...... 65
3 5.1 The dual graph Γ5 of a 4-simplex lives on the (spatial) boundary, i.e. in S . It’s vertices are dual to the five tetrahedra of the 4-simplex and its edges are dual to the triangles. The figure above depicts a projection of Γ5 into the plane...... 79
xii A.1 (a) A 4-simplex σ as the building block of 4d triangulation K. (b) The 5-valent vertex illustrates a rank-5 tensor |Aσ(J~)i. (c) Gluing 4-simplices σ in K gives a tensor network TN(K, J~)...... 97
A.2 The deficit angle ε in a 2d discrete surface hinged by a point. ε , 0 demon- strates that summing the angles at the hinge fails to give 2π. One obtains a discrete curved surface when the two edges bounding ε are glued. In higher dimensions, ε is always hinged by a co-dimension-2 simplex, e.g. in 4d, ε f is hinged by a triangle f ...... 104 A.3 A visualization of a triangulated hypercube cell.The vertices of the hyper- cube are labeled by number from 0 to 15. The binary number of the vertex label is the same as the components of the vector from the origin point to the vertex...... 115
B.1 A Riemann surface S is stretched as two 4-holed spheres connected by a cylinder. γx and γy are the meridian and longitude curves of the cylinder, 0 which is useful in defining complex FN coordinate. s0,1, s0,1 and s denote the framing flags associate to the boundaries and cylinder...... 119
xiii CHAPTER 1 INTRODUCTION
Figure 1.1: Cube of theoretical physics
Modern physics has been highly successful in explaining many phenomena. For most of the modern physics theories, such as quantum mechanics, quantum field, general relativity, the type of physics constant showing up in the theory always indicates in which realm of the phenomena the theory is describing. Among those constants, the speed of light c, Newton constant G and Plank constant ~ are three important characters describing the speed range, space-time curvature effect and the quantum effect of the theory. For example, the lack of c, G and ~ in the Newton’s laws indicates that the New- ton’s laws only work for the low speed, low quantum effect, and low space-time curva- 1 ture physics. The ~ in the quantum mechanics theory reveals that the uncertainty principle plays an important role in the quantum mechanics. As a theory describing the space-time dynamic, the general relativity highly depends on c and G. If we imagine a 3-dimensional diagram, where the three axis are 1/c, G and ~, the classical mechanic, quantum mechanic, quantum field theory, special relativity and general relativity will possess the different cor- ners of a cube (Figure 1.1), considering different realms of physics they are describing. In the sense of ”scientific completeness”, we expect that there will be a theory that will occupy the corner where c, G and ~ are all non-negligible. Those physics happening at p 3 the Planck length scale lp = ~G/c , which remains obscure for the modern physics, can be applied for this theory. Since in this realm both uncertainty principle and gravitational effects are non-negligible, we main call the theory exists in this corner of the cube quantum gravity. Exploring questions in this realm, in which gravity is non-negligible but General Relativity breaks down, will lead to a deeper understanding of black hole singularities, the Big Bang and many other interesting problems. The key to those questions is a theory of quantum gravity (QG) which would revolutionize our todays understanding and largely expand the horizon of physics. Loop Quantum Gravity (LQG) is a non-perturbative approach towards quantum gravity and goes back to the famous paper [6] by Abhay Ashtekar published in 1986. Since then the field has vastly developed through the seminal contributions of Abhay Ashtekar, Carlo Rov- elli, Lee Smolin, Jorge Pullin, Jerzy Lewandowski, Thomas Thiemann and many others. Major breakthroughs were the quantization of the Hamiltonian constraint by Thomas Thie- mann [142, 147, 146, 46, 49, 48, 50, 47, 144] and the development of a path-integral formu- lation, dubbed spin foam model, going back to Reisenberger and Rovelli[123, 125, 124]. Both, covariant and canonical LQG, predict that spacetime is inherently discrete at the Planck scale (10−35 meters) and described by a quantum geometry. So far, the ideas of LQG have been successfully employed in the discussions about black hole entropy, Hawk- ing radiation and the Big Bang singularity (Big Bounce).
2 In the classical Plebanski formulation, gravity in 4 dimensions is formulated by the topological BF theory and implementing the simplicity constraint. The simplicity con- straint restricts the bivector B-field to be simple and relate to tetrad by BIJ = ∗(eI ∧ eJ), which reduces BF action to the Palatini action of gravity. In the spin-foam formulation of covariant LQG, the simplicity constraint is quantized and imposed to partition function of quantum BF theory. In Engle-Pereira-Rovelli-Livine/ Freidel-Krasnov (EPRL/FK) spin-foam model [55, 60], a linear version of simplicity con- straint is imposed in the spin-foam amplitude. Given a simplicial complex, the linear sim- plicity constraint states that for each tetrahedron t, the bivectors B-field smeared on its faces
IJ I B f share the same time-like normal vector N . It is convenient to fix the time gauge that locally in each tetrahedron, the reference frame is chosen such that NI = (1, 0, 0, 0). The time gauge breaks the local Lorentz symmetry to 3d rotation symmetry. The simplicity
IJ constraint then implies that all bivectors B f are spatial for each tetrahedron, and relate to the spatial normal of tetrahedron faces. EPRL/FK spin-foam model is obtained by quantizing the above linear simplicity con- straint and imposing weakly to BF partition function [42, 51, 52, 140]. The reason of imposing constraint weakly is that at the quantum level the components of linear simplicity constraint are not commutative. Strongly imposing the constraint results in that the solu- tion space doesn’t have enough degrees of freedom. Similar phenomena also happens in the Gupta-Bleuler formalism of quantizing electromagnetic field, and the covariant quanti- zation of strings. The quantum simplicity constraint of EPRL/FK model guarantees that (1) the boundary degrees of freedom of spin-foam amplitude match precisely with the quantum 3d geometry emerging from canonical LQG. Namely, the boundary data of EPRL/FK amplitude are SU(2) spin-network data. (2) The semiclassical large spin asymptotics of the spin-foam amplitude reproduces correctly the discrete Einstein-Regge action (without cosmological constant term) evaluated at simplicial geometries with flat 4-simplices [34, 20, 94, 81, 95].
3 However, under the EPRL/FK framework, many intriguing questions still remain open, such as the implementation of a cosmological constant, the continuum limit of LQG, or the relation between LQG and quantum entanglement. In chapter two we propose a semi-classical continuum limit of the infrared limit of the spin-foam model, and we will also show that, under the semi-classical limit of the Euclidean EPRL model, we are able to recover a linearized Einstein theory over a flat space-time. Since the spin-foam can also be considered as a network of the entangled intertwiners, the work in chapter two is also an living example of the conjecture that the gravity phenomena can naturally emerge out from the quantum entanglement. There has been several developments since 2014 on including cosmological constant in Loop Quantum Gravity (LQG) [67, 72, 69, 83, 88, 74]1. A new covariant formulation of LQG has been developed, and presented a nice relation between the covariant LQG in 4 dimensions and Chern-Simons theory on 3-manifold. In this new formalism, the spin-foam vertex amplitude is constructed by using the SL(2, C) Chern-Simons theory on 3-sphere with a Wilson graph. This new formulation using Chern-Simons theory evolves from the earlier formulation using quantum groups [78, 58, 112]. The chapter three focuses on the spin-foam amplitude constructed from the new for- malism. In particular, this work studies the quantum implementation of simplicity con- straint to the spin-foam amplitude in the presence of cosmological constant. It turns out that the simplicity constraint is realized as the 2d surface defect in SL(2, C) Chern-Simons theory used in constructing spin-foam amplitudes. By this realization of simplicity con- straint in Chern-Simons theory, we are able to construct nonperturbatively the new spin- foam amplitude with cosmological constant for arbitrary simplicial complex (with many 4-simplices). The semiclassical asymptotics of the amplitude is shown to reproduce cor- rectly the 4-dimensional Einstein-Regge action with cosmological constant term. The chapter four contains our work about defining the phase space of the loop quantum
1See e.g.[148, 86,8, 127, 118] for reviews of Loop Quantum Gravity, including both the canonical and covariant formalisms.
4 gravity with cosmological constant. The main results of this chapter are summarized as follows: The truncated LQG phase space or twisted geometry is defined on a graph dual to 3D triangulation. We define in Section 4.2.1 an 1-to-1 correspondence between a graph and a Riemann surface S with certain decoration. The new LQG phase space is defined to
2 be the moduli space of SU(2) flat connection on S, denoted by M f lat(S, SU(2)) . Aside with the phase space, calculating the first order non-vanishing order of the gravi- ton propagator or n-point function in the semi-classical limit is also important for being a standard test for the credibility of proposed spin foam model. The reliability of the EPRL- FK model has been proved through those test[2,3, 27, 25, 33, 135]. The existing results of spin foam graviton propagator rely heavily on the so-called double scaling limit where spins j are large and the Barbero-Immirzi parameter γ is small such that the area A ∝ jγ is approximately constant. However, this double scaling limit breaks down in the model including a cosmological constant3. In chapter five, we suggest to replace the metric in the propagator by an operator that only depends on the spins. The operator is only defined locally in the parameter space of the boundary data. Here in the spin-foam amplitude, the boundary state plays the role of a vacuum state for a perturbation theory over the geometry defined by the boundary state. It can be shown that the limit γ → 0 becomes superfluous for so-constructed operators. This solves the problem and enables to implement a cosmological constant.
2 M f lat(S, SU(2)) is the space of SU(2) connections A on S with vanishing curvature FA = 0, quo- tient by SU(2) gauge transformations. M f lat(S, SU(2)) is in general finite dimensional. Flat connections in M f lat(S, SU(2)) can be parametrized by holonomies along loops in S. Fixing the base point of a loop, the continuous deformation of the loop doesn’t affect the loop holonomy, since the connection is flat. So the loop holonomy only depends on the homotopy class of loops. Therefore M f lat(S, SU(2)) is isomorphic to the space of loop holonomies over the fundamental group π1(S), quotient by gauge transformations. Namely M f lat(S, SU(2)) ' Hom(π1(S), SU(2))/gauge, where Hom(π1(S), SU(2)) is the set of group homomorphisms from π1(S) to SU(2) given by the loop holonomies. 3In the spin-foam model with cosmological constant, one has to take an additional limit such that Λ → 0in the same rate as j becomes large in order to recover the correct semi-classical behavior. If one takes → P 2 additionally γ 0, in the 4-simplex Regge action f γ j f Θ f + ΛV4 , the two terms scale differently. 5 CHAPTER 2 SEMI-CLASSICAL CONTINUUM LIMIT AND EMERGENT GRAVITY FROM SPIN-FOAM MODEL
2.1 INTRODUCTION
Spin foam models are inherently discrete[32] which is why the semi-classical limit generi- cally reproduces discrete classical spacetime geometry[21, 34, 94]. However, to prove that spin foam models are indeed a UV completion of general relativity it is absolutely crucial to construct a continuum limit. One main obstacle for defining a continuum limit is the so-called flatness problem [120, 31, 98]. In order to circumvent this problem, M. Han in- troduced a modification of spin foam models in [85]. I and my collaborators used this idea in order to show that smooth gravitational wave solutions can emerge perturbatively from spin foam models. More precisely, we were able to define a sequence of spin foam models on a sequence of refined hypercubic lattices whose semi-classical limit yields a sequence of non-flat linearized Regge geometries. The classification of linearized Regge solutions and their convergence has been studied in [16]. It is shown that the solutions of linearized Regge equation converge to smooth gravitational wave geometries1 in 4d Euclidean space- time if the lattice spacing goes to zero. This implies that the sequence of Regge geometries, which we obtained from the semi-classical limit of the spin foam amplitudes, converge to smooth gravitational wave. This is a very important contribution to the field since it shows for the first time that smooth curved solutions to Einstein’s equation can emerge from spin foams. Moreover, our work confirms the idea of X. Wen[152], that gravity is emergent from fundamental quantum entanglement, since spin foams can be considered as network
1More precisely, these are solutions of Laplace equation which can be analytically continued to wave solutions. 6 of entangled quantum states (Tensor network)[90].
2.2 SPIN FOAM MODELS:
SFMs are defined over 4-dimensional (4d) simplicial triangulations K, which are obtained by gluing 4-simplices σ along their common tetrahedra τ quite similar to the gluing of tetra- hedra in 3d triangulation or triangles in 2d triangulation. Thus a triangulation K consists of simplices σ, tetrahedra τ (boundaries of σs), triangles f (boundaries of τs), edges (bound- aries of f s) and vertices. Our analysis focuses on K adapted to a hypercubic lattice in R4 in such a way that each hypercube is triangulated identically by 24 4-simplices (see Figure 2.1(b)). Consequently, K is periodic under 4d discrete translations. The same triangulation has been employed in e.g. [16, 126] to study perturbations on a flat background. A SFM is obtained by associating a state sum,
X Y Y Z(K) = A f (J f ) Aσ(J f , iτ), (2.1) J~,~i f σ to K and can be interpreted as the path integral of a triangulated manifold (here R4). In the above state sum, each triangle f is colored by an SU(2) representation J f ∈ Z+/2 and each tetrahedron τ is colored by an SU(2) intertwiner (invariant tensor) iτ. They are quantum numbers labelling histories of LQG quantum geometry states, which are the intermediate states of the path integral. J f , iτ can be related to the area of f and the shape of τ in the semiclassical interpretation [131,7, 26]. The dynamics of the model is captured in the
4-simplex amplitudes Aσ(J f , iτ) ∈ C associated to each σ. In particular, Aσ(J f , iτ) ∈ C
describes the local transition between the quantum geometry states labelled by {J f , iτ} for
f, τ on the boundary of σ. The face amplitudes A f (J f ) is the weights of the spin sum, and is often set to be 2J f + 1 for SFMs.
The amplitudes Aσ(J f , iτ) depend linearly on the intertwiners iτ and thus are rank-5 tensors on intertwiner spaces. The 4-simplices in K are glued by identifying a pair of τs in
0 P σ and σ . This implies that ~i Aσ is equivalent to the inner products between the tensors 7 | i | i P | i ⊗ | i Aσ at all σs and the maximally entangled states τ = iτ iτ iτ , where iτ are shared by pairs of σs. This yields a spacetime tensor network (TN) (Figure 2.1(a))
~ TN(K, J):= ⊗τhτ| ⊗σ |Aσ(J f )i. (2.2)
Note that the entangled intertwiners (the qudits) are the fundamental DOFs of the TN. Moreover the state sum Z(K) can now be expressed in terms of these TN, that is, Z(K) = P ~ Q J~ TN(K, J) f A f (J f ). The following demonstrates that smooth Einstein solutions can emerge from the fun- damentally entangled intertwiners. Thus it realizes the idea of emergent gravity from en- tangled qubits. In order to show this, we employ the integral representation of TN(K, J~) [19, 34, 93] (see SM II for details): Z X Y P f J f F f [X] Z(K) = A f (J f ) [dX] e , (2.3) J~ f
where F f is a function that only depends on a set of specific variables X. The details of A f
and F f depend on the specific SFM. More details on SFM, TN, and integral representation are given in Appendix A.1. Here, we focus on the Euclidean Engle-Pereira-Rovelli-Livine/Freidel-Krasnov (EPRL/FK) model [55, 60] but our results can be generalized to other SFMs, e.g., [54, 102, 67].
2.3 LARGE SPIN ANALYSIS:
LQG predicts that the geometrical areas are fundamentally discrete at the Planck scale. The
p 2 area spectrum [131,7] relates to the spins via A f = γ J f (J f + 1)`P, where γ ∈ R is the
2 2 Barbero-Immirzi parameter and `P ≡ 8πGN~. Since the semiclassical area A f `P implies
J f 1, the semiclassical analysis of SFMs is build on uniformly large (but finite) spins
J f = λ j f where λ 1 is the typical value of the spins. For the following argument, it is important to note that small perturbations J˚ + δJ of a given background spins J˚ ∼ λ 1 will still be inside this large-J regime. Moreover, the
8 Gluing many 4-simplices: spacetime foam
Spin Foam Amplitude in LQG: spins
14 15 12 13
10 11 i !35 sum over all weight product over all 4-simplicese
8 9 6 7 4 5 2 3
0 1
Figure 2.1: (a) The 5-valent vertex in a 4-simplex illustrates a rank-5 tensor |Aσi. Gluing 4-simplices σ in K gives a tensor network TN(K, J~), where each link associates to a maximally entangled state of a pair of iτs. (b) A triangulation of the hypercube. The 4d hypercubic lattice with the triangulated hypercube makes K. (c) An illustration of the neighborhood N (the region bounded by blue dashed lines) in the space of J~. The red ~ ~ curve illustrates MRegge, including J(`) as the perturbation of J(`˚). The black and blue arrows are basis vectorse ˆi(`) and ∂J~(`)/∂`, transverse and tangent to MRegge.
9 P sum J~ can be replaced through an integral by Poisson resummation formula. Thus, Z X Y P N f f J f (F f [X]+4πik f ) Z(K) = 2 [dJdX] A f (J f ) e , (2.4) ~k∈ZN f f ~ where N f denotes the number of internal f s in K. Since all k , 0 integrals in Z(K) can be
negligible (see Appendix A.2), we will set all k f = 0 hereafter.
From previous results e.g. [19, 34, 93] follows that there is a subspace of large J~ ∈ RN f that determine classical triangle areas. These spin conigurations are called Regge-like and satisfy the triangle area-length relation 1 q γJ (`) = 2(`2 `2 + `2 `2 + `2 `2 ) − `4 − `4 − `4 . (2.5) f 4 i j jk ik jk i j ik i j ik jk
2 The right-hand side determines the area a f (`)/`P of the triangle f in terms of `i j, `ik, ` jk
being the lengths (in `P unit) of 3 edges of a triangle. Since there are less edges than triangles in the bulk of K, Regge-like spins form a proper subset and Eq.(2.5) defines an
N` N embedding map R ,→ R f . N` is the number of internal edges in K. Here, we want to consider perturbations on a flat hypercubic lattice with constant spac-
1/2 2 ing (γλ) (in `P unit), which fixes all edge lengths `˚ in K . These edge-lengths in turn ˚ determine the Regge-like spins J~ = J~(`˚) by Eq.(2.5). For the study of perturbations around ˚ ˚ J~, it is sufficient to consider a neighborhood N ⊂ RN f of J~. N is constructed as follows: Firstly, smooth perturbations ` = `˚ + δ` and the embedding Eq.(2.5) define a submanifold
N f i MRegge ⊂ R of dimension N`. We choose arbitrarilye ˆ (i = 1, ··· , N f − N`) basis vectors ~ ~ P i ~ transverse to MRegge. All J = J(`) + i=1 tieˆ defines N , with J(`) ∈ MRegge and ti ∈ R. (`, ti) form a local coordinate system in N (see Figure 2.1(c)). The integral over J~ can now be split into transverse and Regge-like part as well. That R R is, dJ~ = [d`dt] J(`), where the Jacobian J(`) = |∂J~(`)/∂`, eˆi|. To avoid divergence, R ∞ we regularize the transverse integral −∞ ti by inserting a Gaussian factor parametrized by 0 < δ 1: Z Z Z Z P − δ t2 dJ~ = [d`dt] J(`) → [d`] J(`) [dt] e 4 i i . (2.6) √ √ 2(γλ)−1/2`˚ = 1, 2, 3, 2 for the cube edges, face diagonals, body diagonals, and hyperbody diagonals. 10 This regularization will be turned off by δ → 0 in the end together with the continuum limit. Eventually all physical quantities should be computed in the continuum limit to remove the triangulation dependence, so their final result should not depend on δ. Another importance of δ will be seen in a moment.
The ti-integral in Z(K) is a Gaussian integral and yields Z D E P λ ~j(`),F~[X] M 1 heˆi,F~[X]i2 0 [d`dX] e Dδ(`, X), Dδ = e i=1 δ J (2.7)
J 0 4π MJ Q 2 P i h i ~ i ~ { } where = ( δ ) (`) f A f (J f (`) + δ i eˆ f eˆ , F[X] ). Here, F = F f f is treated as a
complex N f -dimensional vector, and h·, ·i denotes the Euclidean inner product. Further- more, we have ignored the boundary terms in the exponent because they are unimportant in the main discussion. Since the exponent in Eq.(2.7) scales linearly in λ, we can apply the stationary phase method to Eq.(2.7). As long as the exponent in Dδ is subleading, we can directly take over the result in [19, 93, 34]. This requires to consider a regime:
λ δ−1 1. (2.8)
In this regime, the dominant contributions of Eq.(2.7) come from the critical points (`c, Xc), ~ ~ i.e. the solutions of the critical equations ReS = δXS = δ`S = 0, of the action S = h j, Fi.
Among the critical equations, ReS = δXS = 0 implies that a class of critical points can be interpreted geometrically [93, 34], i.e. we have the following equivalence for the critical points involved in the following discussion:
SFM critical point ↔ 4d simplicial geometry on K. (2.9)
Simplicial geometries, labelled by edge lengths `, are discretizations of smooth geometries on triangulation K. The flat hypercubical background geometry corresponds to a critical point (`,˚ X˚). Per- turbations thereof cover critical points (`c, Xc) that correspond to nearly-flat simplicial ge-
11 3 ometries on K . This implies F f [Xc] = iγε f (`) where ε f ’s are deficit angles, which measure discrete Riemannian curvature.
On the other hand, δ`S = 0 and Eq.(2.5) yields the equation of motion (EOM) * + ∂J~(`) X ∂a f (`) , γ~ε(`) = 0 or ε (`) = 0, (2.10) ∂` ∂` f f
and coincides with the Regge equation. Regge equation is a discretization of Einstein equation in 4d [121]. The leading asymptotic behavior of Eq.(2.7) is determined by the integrand evaluated P − M 1 heˆi,γ~εi2 at the critical point, thus is proportional to Dδ(`c, Xc) ∝ e i=1 δ . Due to δ 1, this
D i E 1/2 suppresses the contribution of the critical point (`c, Xc) exponentially unless eˆ , γ~ε . δ for all i. Since {∂J~(`)/∂`, eˆi} forms a complete basis in RN f , it follows from Eq.(2.10) that
1/2 |γε f (`)| . δ 1, (2.11)
Eqs. (2.10) and (2.11) determine the critical points (`c, Xc) that contribute essentially to Z(K), and thus are the key equations constraining the simplicial geometries emerging in the large spin limit of the model.
Eq.(2.10) can be reduced to a set of linear equations of the deficit angles ε f [16], be- cause the considered geometries are nearly-flat. That is,
M~ε = 0. (2.12)
where M is a constant N f × N f matrix. Note that this is a consequence of the nearly-flat geometries, but not a consequence of Eq.(2.11). By itself, Eq.(2.11) is compatible with the non-linear Regge equation, and excludes no nonsingular curved geometry. On a sufficiently
3The critical points considered in this work are geometrical and characterized by uniform 4-simplex ori- entations. The (oriented) 4-simplex volumes are all positive Vσ > 0 from these critical points [93]. SFM has other non-geometrical critical points, e.g. the BF type and the vector geometries with all Vσ = 0, and geometrical critical points with non-uniformly 4-simplex orientations, i.e. some Vσ < 0. The existence of critical points with non-uniform orientations is the origin of the “cosine problem” [93, 19]. However when we choose the background (`c, Xc) to be geometrical with uniform 4-simplex orientations i.e. all Vσ > 0, all critical points touched by small perturbation are still geometrical with all Vσ > 0.
12 refined triangulation, any simplicial geometry approximating a smooth geometry with typi-
2 2 cal curvature radius ρ satisfies |ε f | ' a /ρ 1, which is consistent with Eq.(2.11). Here a is the typical lattice spacing. The simplicial geometries that fail to satisfy Eq.(2.11) cannot have smooth approximation. If the regularization in Eq.(2.6) wasn’t imposed, i.e. if δ = 0 as in standard SFMs, then Eq.(2.11) would imply strict flatness ε f = 0. This strict flatness has been proven to be one of the main obstacles for recovering classical gravity from SFMs [98, 120, 31, 85].
But if δ , 0 as above, then small excitations of ε f are allowed, and therefore arbitrary smooth curved geometries may emerge from refined triangulations, while non-geometric configurations remain excluded.
2.4 SEMICLASSICAL CONTINUUM LIMIT:
The above discussion is based on a fixed triangulation K adapted to a hypercubic lattice. From this, we may construct a refined triangulation K 0 by subdividing each hypercube into 16 identical hypercubes, triangulated by simplices in the same manner as above. By refining the hypercubic lattice, we define a sequence of triangulations Kµ. Kµ0 is finer than
0 Kµ if µ < µ. The continuum limit is µ → 0 in which the vertices in the triangulation become dense in R4. The parameter µ will play the role of a mass scale in the theory.
We can now associate a SFM Z(Kµ) to each Kµ, with µ → 0 as the continuum limit of SFM. The above large spin analysis can be applied to all Z(Kµ). This gives a sequence of EOMs (2.10) (or its linearization Eq.(2.12)) and (2.11). All quantities in the equations, e.g. the spins J f , the regulator δ, the simplicial geometries, etc, depend on µ, and flow with µ → 0, which defines the semiclassical continuum limit (SCL). In particular, we will show below that the solutions to the EOM (2.10) flow to solutions of smooth Einstein equation as µ → 0. This can be derived from the fact that the solutions of linearized Regge equation converge to solutions of linearized Einstein equation as the lattice spacing a → 0 (see [16, 15, 43]). The EOMs (2.10) are already Regge equation and it only remains to relate
13 the Regge limit a → 0 and the SFM continuum limit µ → 0. In fact relating the limits is nontrivial and specifies the SCL. The regulator δ(µ) should flow to zero with µ → 0 in order to guarantee that the con- tinuum result does not depend on δ. Yet, (2.8) must still be satisfied at every step µ for
the above asymptotic analysis of Z(Kµ) to remain valid. Thus, λ(µ) has to grow faster than δ(µ)−1 .
2 The growing of λ seems to contradict the Regge limit a → 0 as the area a f = γλ j f `P and length ` obtained from Z(Kµ) grow with λ. This contradiction can be resolved by observing that length and area are dimensionful. Their numerical values depend on the choice of unit.
1 At the Planck scale, the background lattice spacing is given by `˚(µ) = (γλ(µ)) 2 `P. But in the infra-red (IR), we should zoom out to a coarser length scale. This length scale can be directly related to the parameter µ. If µ has the dimension of a mass, then µ−1 has the dimension of a length and we may set
1 −1 `˚(µ) = (γλ(µ)) 2 `P = a(µ)µ . (2.13)
So the lattice spacing a(µ) measured in terms of the µ-scale may still shrink with µ → 0. In this case da(µ)/dµ > 0, which together with (2.13) implies:
2 1 dλ − < < 0. (2.14) µ λ dµ
This inequality is not the only restriction. Recall that solutions of Regge equation arise in the leading order stationary phase approximation of Z(Kµ) as λ(µ) 1. The solutions have the (quantum) corrections of O(1/λ). The correction is bounded by C(µ)/λ(µ) with C(µ) > 0, where C(µ) grows as µ → 0 (see Appendix A.3). As a result, λ(µ) is required to grow in a faster rate, in order to keep C(µ)/λ(µ) small to suppress the 1/λ correction to Regge solutions as µ → 0. It implies
1 dλ 1 dC < . (2.15) λ dµ C dµ
2 2 In addition to the constraints (2.14) and (2.15), it follows from (2.11) and ε f ' a /ρ 14 that there should exist a bound L < ∞ s.t.
δ(µ)1/2 < L. (2.16) a(µ)2
−2 2 Otherwise, the curvature of the emergent geometry ( i.e. ρ = lim ε f (µ)/a(µ) ) would diverge. Before we continue, let us summarize the above discussion: A SCL is described by the flow of the 3 parameters λ(µ), a(µ) and δ(µ) that satisfy (2.14), (2.15), and (2.16). a(µ) and δ(µ) tend to zero in the limit µ → 0, while λ(µ) → ∞ grows faster than δ(µ)−1. This limit is well-defined because the flows satisfying the requirements always exist (shown in Appendix A.4). The SFM continuum limit µ → 0 is also an infra-red (IR) limit of SFM, since µ is a mass scale.
2.5 EMERGENT LINEARIZED GRAVITY:
The above SCL fills the gap between the continuum limits in SFM and Regge calculus. Thus, the sequence of critical points satisfying Eq.(2.10) under the SCL is the same as the sequence of Regge solutions under a → 0. The classification of Linearized Regge solutions and their convergence has been studied in [16, 15]. It is shown that the solutions of linearized Regge equation converge to smooth solution of 4d (Riemannian) Einstein equation in the limit a → 0. All the nontrivial ge- ometries obtained from the limit have curvatures as linear combinations of
Rabcd(x) = Re Wabcd exp (−k · x) , (2.17) which are Euclidian analogs of plane waves. Here k · x is the 4d Euclidean inner product
4 and k ∈ C satisfy k · k = 0. Wabcd is a traceless constant tensor that spans a 2-dimensional solution space, whose dimensions correspond to the helicity ±2 gravitons.
Recall that the main contributions to Z(Kµ) in the SCL come from critical points that satisfy linearized Regge equation, all other contributions are suppressed. Moreover, the
15 SCL maps the SFM IR limit µ → 0 to Regge calculus limit a → 0. Therefore, the above convergence result of Regge solutions can be applied to SFM as µ → 0, which shows that on a 4d flat background, the low energy excitations of SFM give all smooth solutions of linearized Einstein equation (gravitons).
2.6 CONCLUSION
In the above discussion, we have shown that from the SCL, the low energy excitations of SFM on a flat background give all smooth (linearized) Einstein solutions. It indicates that linearized classical gravity is the effective theory emerging from SFMs at low energy. Our result indicates that the SFM, being a discrete model of fundamentally entangled qudits, is a working example for the idea in emergent gravity program. Here we showed for the first time that smooth curved spacetimes can emerge from SFMs in a suited continuum limit. It suggests that SFMs have a proper semiclassical limit not only at the discrete level but also in the continuum. Our result, therefore, strengthens the confidence that covariant LQG is a consistent theory of quantum gravity. Our analysis certainly can be generalized to the nonlinear regime, and even to the case of strong gravitational field. Indeed the large spin analysis doesn’t rely on the linearization, and the EOM (2.10) is nonlinear. The emergence of black hole or cosmological solutions from SFMs can be derived by applying the Regge calculus convergence results in e.g. [65], similarly as above. These solutions will enable us to study singularities as the high energy excitations from SFMs. Finally we remark that the flows of SFM parameters λ(µ), a(µ), δ(µ) in the SCL likely relate to a renormalization group flow4. Further investigation of the relation may shed light on the renormalization of perturbative gravity.
4It may relate to the recent development of the renormalization group flow in SFM [10]. 16 CHAPTER 3 LOOP-QUANTUM-GRAVITY SIMPLICITY CONSTRAINT AS SURFACE DEFECT IN COMPLEX CHERN-SIMONS THEORY
3.1 INTRODUCTION
As an important piece of the covariant framework for coupling the cosmological constant with LQG, the quantum implementation of the simplicity constraint has been discussed by us. This work carries out the analysis of simplicity constraint for the spin-foam model with cosmological constant. The 4-dimensional spin-foam amplitude with cosmological constant is constructed by using SL(2, C) Chern-Simons theory on 3-manifold [67, 72, 69]1. In this formalism, the local Lorentz symmetry of 4d spin-foam model is translated
IJ to the SL(2, C) gauge symmetry of Chern-Simons theory. The bivector B f are naturally exponentiated and given by the holonomy of flat connection traveling transversely around the Wilson line. The tetrahedron in 4d spin-foam model corresponds to the neighborhood of the vertex where 4 Wilson lines join (see FIG.3.1 for the Wilson graph used for constructing 4-simplex amplitude). It is explained in Section 3.2 that the simplicity constraint and time gauge correspond to requiring that on the 4-holed sphere enclosing a 4-valent vertex, the gauge group of Chern-Simons is broken from SL(2, C) to SU(2). In the classical limit, the flat connections on 4-holed sphere are restricted to be SU(2). It is known that SU(2) flat connections on
1Following [67], SL(2, C) Chern-Simons theory can be viewed to be equivalent to 4d BF theory with a cosmological constant term, when the 3d space where Chern-Simons lives is the boundary of the 4d space R IJ where BF theory lives. Schematically, the BF action with cosmological constant reads S BF = B ∧∗FIJ + M4 Λ IJ 1 R IJ i R 2 B ∧∗BIJ. Integrating out B-field leads to S ∼ F ∧∗FIJ ∼ tr(A∧dA+ A∧A∧A)+c.c, where 6 Λ M4 Λ ∂M4 3 i I, J = 0, ··· , 3 are Lorentz vector indices. A = A σi is the sl2C-valued complex Chern-Simons connection. See [67] for details in the presence of Barbero-Immirzi parameter. 17 4-holed sphere is in 1-to-1 correspondence to tetrahedron geometries with constant curva- ture [74]. Thus imposing the simplicity constraint ensures the geometricity of tetrahedron at the classical level, similar to the situation of EPRL/FK model (with flat tetrahedron ge- ometries). In Section 3.3, we perform a quantization of the simplicity constraint, and define the constraint operators on the Hilbert space of SL(2,C) Chern-Simons wave functions. Similar to the situation in EPRL/FK model, we find the constraint operators are noncommutative, which motivates us to rather impose a weaker version of constraints. We propose to use the master constraint technique [143, 145, 92]. The master constraint effectively reduces the Hilbert space to a subspace, whose wave functions are equivalent to SU(2) Chern- Simons wave functions. We might view the master constraint is a Hamiltonian, for which the SU(2) Chern-Simons wave functions on the 4-holed sphere are ground states, other SL(2, C) Chern-Simons states are created as excitations similar to harmonic oscillator. In addition, we find that the weak simplicity constraint is not unique. Indeed, the so- lution of the master constraint is a coherent state peaked at the phase space point which solves the classical simplicity constraint. We know that the coherent state which saturates the Heisenberg uncertainty is not unique, e.g. the squeezed coherent states. It turns out that different ways to define coherent states peaked at classical solutions of simplicity constraint correspond to different ways of weakly imposing simplicity constraint at the quantum level.
3 In Section 3.4, we consider the graph complement 3-manifold S \ Γ5 similar to [69, 72]. We impose the quantum simplicity constraint and project the SL(2, C) Chern-Simons wave function to the space of solutions. The resulting wave function Z is proposed as a spin-foam 4-simplex amplitude with cosmological constant. We show that thanks to the simplicity constraint, the amplitude satisfies both (1) the boundary degrees of freedom match precisely with discrete 3d geometry data on the boundary of 4-simplex. The 3d geometry data is an analog of spin-network data (or semiclassically twisted geometry data) [88]. (2) The semiclassical asymptotics of the amplitude reproduce correctly the Einstein-
18 Regge action with cosmological constant term on a constant curvature 4-simplex. The situation is an generalization of EPRL/FK model to include the cosmological constant. In Section 3.5, we generalize the analysis to arbitrary simplicial complex with many 4-simplices. The spin-foam amplitude on 4d simplicial complex is an SL(2, C) Chern-
3 Simons theory on 3-manifold M3 made by gluing copies of S \ Γ5. We find interestingly, the implementation of simplicity constraint corresponds to inserting 2d surface defects to SL(2, C) Chern-Simons theory on 3-manifold. The surface defects are inserted at the gluing
3 interface (4-holed spheres) between pairs of S \ Γ5, i.e. they divide the entire 3-manifolds
3 into copies of S \ Γ5. Each surface defect restricts the Chern-Simons states, which travel
3 from one S \ Γ5 to another, to be solutions of simplicity constraint, i.e. to be equivalent to SU(2) Chern-Simons states. Because of understanding the simplicity constraint at the quantum level, we are able to formulate the spin-foam amplitude nonperturbatively on arbitrary simplicial complex, which improves the result in [83]. Because surface defects impose the quantum simplicity constraint, the two key proper- ties of 4-simplex amplitude are generalized to the general spin-foam amplitude on simpli- cial complex. The boundary data are always 3d geometry data, so the amplitude describes the quantum history of 3d geometries. The semiclassical asymptotics reproduce correctly the Einstein-Regge action with cosmological constant term on the simplicial complex. 4d simplicial geometries emerge from critical points of spin-foam amplitude, where locally each 4-simplex is of constant curvature. Interestingly, the 3-manifold M3 carrying Chern-Simons theory has a number of nontrivial cycles, each of which associates a torus cusp defect. The longitude holonomy along the B-cycle of torus cusp is noncontractible, since it associates to a noncontractible cycle of 3-manifold. It turns out that each noncon- tractible cycle corresponds to a triangle in 4d simplicial complex, and the noncontractible B-cycle holonomy corresponds to the nontrivial deficit angle hinged by the triangle. The
4d curvature is effectively created by the nontrivial cycles of 3-manifold M3. In Section 3.6, we consider the field-theoretic description of the surface defect. We can
19 define an operator insertion in the Chern-Simons path integral in terms of the continuous field theory variable. The 2d “surface operator” inserted in the path integral effectively im- plements the quantum simplicity constraint. In general, the defect of topological quantum field theory has certain dependence on the background metric, since it breaks the topolog- ical invariance to certain extend. A typical example is the framing dependence of Wilson line operators. Here the surface defect implementing the simplicity constraint also depends on a choice of metric on the 2-surface. Different choices of metrics in the field-theoretic context may be viewed as analogs of choosing different squeezed coherent states mentioned above. Thus different surface metrics for the surface defect correspond to different ways to implement weakly the quantum simplicity constraint. The semiclassical behavior is checked for the spin-foam amplitude in this field-theoretic formulation. The asymptotics again reproduce the Einstein-Regge action with cosmologi- cal constant on the entire simplicial complex. Although line defects have been widely studied in Chern-Simons theory, the results about surface defects (or domain-walls) are insufficient in the literature (some results are e.g. [103,5,4]). The surface defect appearing here has not been studied before. In Sec- tion 3.7, we investigate the surface defect by studying the propagating physical degrees of freedom on the defect 2-surface. As it is mentioned above, the surface defect reduces SL(2, C) Chern-Simons states to SU(2) in order to implement the simplicity constraint. On the defect where the gauge symmetry is broken, the previous gauge degrees of freedom become the physically propagating degrees of freedom. In other words, some additional propagating degrees of freedom have to be implemented in order to recover the original gauge symmetry on the defect. The standard example is the boundary of Chern-Simons theory, on which Wess-Zumino-Witten (WZW) model describes the propagating degree of freedom. We analyze the additional propagating degrees of freedom on the surface de- fect, which re-install the SL(2, C) gauge invariance to the model. We show that at least at the linearized level, the propagating field behaves as a 2d sigma model gauged by the
20 Chern-Simons connection.
3.2 SIMPLICITY CONSTRAINT AND CURVED TETRAHEDRON
In the spin-foam formulation without cosmological constant, the classical linear simplicity
IJ constraint requires that, given a flat tetrahedron t, each of the 4 face bivectors B f should be orthogonal to the time-like normal NI of the tetrahedron2
IJ B f NI = 0, ∀ f ⊂ ∂t. (3.1)
The time gauge may be chosen such that NI = (1, 0, 0, 0), understood as a frame choice inside the tetrahedron. The frame can be located at any point inside the tetrahedron since the tetrahedron is flat. The choice of time gauge breaks the local Lorentz symmetry down to spatial rotation
IJ i j symmetry. We have for each bivector B f = B f where i, j are 3d vector indices, and
1 a nˆi = εi jk(B ) , (3.2) f f 2 f jk wheren ˆ is a unit space-like vector. Moreover because of the closure constraint
X4 X4 IJ 0 = B f = a f nˆ f , (3.3) f =1 f =1
IJ we know that the data B f satisfying simplicity constraint endow the geometry to the tetra-
hedron t, in which a f is the face area andn ˆ f is the unit face normal vector.
1 2 3 4 5
3 Figure 3.1: Γ5-graph embedded in S .
2I, J = 0, ··· , 3 are vector indices of Lorentz group.
21 In the recent spin-foam models with cosmological constant, the 4d spin-foam 4-simplex
2 amplitude is formulated as an SL(2,C) Chern-Simons theory on S with Γ5 Wilson graph defect (Figure 3.1)[67, 69, 72]. In this formulation, each tetrahedron of the 4-simplex relates to a 4-holed sphere S enclosing a vertex of Γ5 graph. By the Chern-Simons equation of motion (in the semiclassical limit), the SL(2,C) flat Chern-Simons connection on each 4-holed sphere gives a holonomy-version of closure constraint
H4H3H2H1 = 1. (3.4)
Fixing a base point on S, H f is the holonomy of flat connection circling the f -th hole (each hole is dual to a tetrahedron face). The above formula can be viewed as a closure constraint
P IJ generalizing f B f = 0 because each SL(2,C) holonomy can be written as an exponential Λ IJJ J H f = exp( 3 B f IJ) where IJ are Lorentz generators. When the cosmological constant → P IJ Λ 0, Eq.(3.4) implies the usual closure f B f = 0 by linearization. IJ When we apply the simplicity constraint Eq.(3.1) and time gauge in this context, B f is again restricted to be spatial, thus ! ! Λ Λ i H = exp BIJJ = exp a nˆ · ~τ ∈ SU(2), ~τ = σ~ (3.5) f 3 f IJ 3 f f 2
where σ~ are Pauli matrices. Therefore the simplicity constraint and time gauge effec- tively reduce the structure group of Chern-Simons from SL(2, C) to SU(2) on each 4-holed sphere. SL(2, C) flat connections are reduced to SU(2). Eq.(3.4) becomes a product of SU(2) matrices. It has been shown in [67, 74] that the SU(2) flat connections on a 4-holed sphere S are
in 1-to-1 correspondence to the geometries of constant curvature tetrahedron, in which a f
in Eq.(3.5) is the face area,n ˆ f is the unit face normal. However since the tetrahedron is curved, a base point of tetrahedron has to be chosen in order to make sense of the frame
choice for time gauge. Thenn ˆ f is the unit face normal vector located at the tetrahedron base point. The closure constraint Eq.(3.4) and the relation Eq.(3.5) suggest that in the present of 22 cosmological constant, the flux variable used in LQG are naturally exponentiated. The exponentiated flux variable has been recently studied in e.g.[88, ?, ?] The moduli space of flat SU(2) connection is of real dimension-6, which parametrizes all degrees of freedom for constant curvature tetrahedron geometries. The eigenvalues of
SU(2) holonomies H f around the 4 holes relates to the 4 triangle areas of tetrahedron. It is shown in [74] that the shapes of tetrahedron with fixed areas are parametrized by the flat connection coordinates x, y ∈ U(1). x ∈ U(1) relates to the diagonal length of a spherical 4-sided polygon, while y ∈ U(1) relates to the “bending angle”. In the moduli space of SL(2,C) flat connections on S, the coordinates x, y are known as Fenchel-Nielsen (FN) coordinates [74, 39, 88]3, while now x, y ∈ C \{0} since they parametrize SL(2, C) flat connections. The symplectic structure of the moduli space in- duces that x, y are canonically conjugate4:
dy2 dx Ω = ∧ . (3.6) y2 x
Recall that the simplicity constraint reduces the flat connection on S from SL(2, C) to SU(2). In terms of the coordinates, the simplicity constraint implies
Re(ln x) = 0, Re(ln y) = 0. (3.7)
Namely, under the constraint, x, y become U(1) numbers parametrizing the shape of tetra- hedron.
For the completeness, the simplicity constraint also restricts the eigenvalues of H f to be U(1) numbers as well, since they relate to face areas. But it turns out that these restrictions can be easily imposed at the quantum level. The only nontrivial task is to quantize the
3The FN coordinates is defined by cutting the 4-holed sphere S into two 3-holed spheres. The flat connec- tion on S gives an SL(2, C) holonomy hx along the cut, whose eigenvalue is the FN complex length variable x. The FN twist variable y is more technical to define. In a non-technical language, It comes from a holonomy hy of flat connection traveling from one 3-holed sphere to the other, which intersects transversely hx. The diagonalization of hy gives the twist variable y. We refer to e.g. [39, 88, 69] for a mathematically precise definition. 4The square on y is conventional.
23 constraint Eq.(3.7), which we focus on in the following. For convenience, we often denote X = ln x and P = ln y2 in the following discussion.
3.3 QUANTIZATION OF FLAT CONNECTION AND SIMPLICITY CONSTRAINT
We denote by PS the phase space of SL(2, C) flat connections on S with fixed holonomy
eigenvalue around each hole. PS is of 2 complex dimension. The coordinate on PS can be chosen to be (x, y2). The symplectic structure of SL(2, C) Chern-Simons theory reads 1 ω , = tΩ + t¯Ω¯ t = k + is, t¯ = k − is k s 4π k s = (d ReP ∧ d ReX − d ImP ∧ d ImX) − (d ReP ∧ d ImX + d ImP ∧ d Re(3.8)X) . 2π 2π
The quantization of phase space PS can be carried out in a similar way as in [38]. x = exp X and y2 = exp P imply that ImX, ImP are periodic with period 2π. Weil’s criterion
of pre-quantization then requires k ∈ Z. s ∈ R leads to ωk,s being real, so that the Chern- Simons theory is unitary with respect to a standard Hermitian struction 5. As a convenient way to parametrize the complex Chern-Simons level, we write 1 − b2 is = k ∈ iR (3.9) 1 + b2 with |b| = 1. We can parametrize x, y2 and their complex conjugates by 2πi 2πi x = exp (−ibµ − m) , x¯ = exp −ib−1µ + m k k 2πi 2πi y2 = exp (−ibν − n) , y¯2 = exp −ib−1ν + n (3.10) k k where m, n ∈ R are periodic m ∼ m + k, n ∼ n + k, µ, ν are also real parameters. The
Chern-Simons symplectic form ωk,s can be re-written in terms of new variables 2π ω , = (dν ∧ dµ − dn ∧ dm) (3.11) k s k
The quantization of PS promotes the parameters µ, ν, m, n to operators µµ,ννν, m, n, whose non-vanishing commutation relation is k k µµ,νν = , [m, n] = − . (3.12) 2πi 2πi 5There is another unitary branch s ∈ iR via a nonstandard Hermitian structure [153] 24 Or in terms of x, y2
4πi 4πi xy2 = qy2x, x¯y¯2 = q˜y¯2x¯, q = exp , q˜ = exp . (3.13) t t¯
The operator algebra is represented on the space of wave functions f (µ, m) of two vari- ables. Here µ ∈ R is continuous but m ∈ Z/kZ is discrete. m only takes integer value because both of the canonical conjugate variable m and n are periodic. Operators µµ,ννν, m, n are represented by
k µ f (µ, m) = µ f (µ, m), ν f (µ, m) = − ∂µ f (µ, m) 2πi 2πi m 2πi m 2πi n e k f (µ, m) = e k f (µ, m), e k f (µ, m) = f (µ, m + 1). (3.14)
The simplicity constraint Eq.(3.7) leads to the condition µ = ν = 0 in the new parametriza- tion. To quantize the constraint, one might naively impose the operator equations µµψ = ννψ = 0 to the wave functions. However the naive operator equations trivialize the wave µ ν k function since µµ,νν = 2πi . Therefore to realize the simplicity constraint at the quantum level, we have to impose a weaker version of the constraint. This fact makes it nontrivial for the quantum implementation of simplicity constraint. Here we choose to impose the operator equation ! πµ2 (µ − iν)ψ = 0 ⇒ ψ (µ, m) = exp − f (m). (3.15) sol k
where f (m) is an arbitrary function on Z/kZ. Here the solution space is simply a k- dimensional vector space Ck, being the Hilbert space of SU(2) Chern-Simons theory of level k. The simplicity constraint at quantum level reduces SL(2, C) Chern-Simons wave function to SU(2). As an equivalent way to impose the constraint, one may also consider to impose the “master constraint” (µ2 + ν2)ψ = 0 up to “zero-point” energy 6. The solution (the depen- dence on µ) is simply the ground state of harmonic oscillator, the same as above. In this
6See [143, 145, 92] for the idea of master constraint in canonical LQG. See [55, 140] for the use of master constraint in spin-foam model to solve the simplicity constraint. 25 sense the states Eq.(3.15) may be viewed as the ground states, while the full spectrum of SL(2, C) Chern-Simons states are created by the action of “creation operator” (µ + iν). As we have seen, the constraint µ = ν = 0 at the quantum level can only be satisfied weakly. The solution of the quantum constraint is a coherent state with peakedness at µ = ν = 0. So µ = ν = 0 is satisfied only in the semiclassical limit. It is known that the coherent state peaks at a phase space point is not unique. We may choose other squeezed coherent states, which still minimize the Heisenberg uncertainty. We introduce a squeezing parameter w ∈ R, and impose (µ − iw2ν)ψ = 0 instead of Eq.(3.15), whose solution is ! πµ2 ψ(w)(µ, m) = exp − f (m). (3.16) sol w2k
We may introduce a “metric” and define a “squeezed” master constraint (w−2µ2+w2ν2). The −2µ2 2ν2 (w) above squeezed coherent state satisfies (w µ + w ν )ψsol = 0 up to the same zero-point energy as above (the zero-point energy is independent of w). The squeezing parameter introduces an ambiguity to the solution of simplicity con- straint at each 4-holed sphere S. The essential reason of the ambiguity is the noncommu- tativity of the simplicity constraints µ = 0,νν = 0. In the following Sections 3.4 and 3.5,
we keep w , 0 as a free parameter, and focus on the construction of Chern-Simons theory
with defects, as well as the geometrical reconstruction on M4. We come back to the issue of ambiguity in Section 3.6.
3 3.4 SL(2, C) CHERN-SIMONS THEORY ON S \ Γ5
3 The partition function of SL(2, C) Chern-Simons theory on S \Γ5 can be viewed as a wave
3 ¯ P 3 function ZS \Γ5 (λ`, λ`, xS, x¯S)[72, 69]. The phase space of flat connections S \Γ5 associated
3 3 to the boundary of S \ Γ5 is of complex dimension 30. The boundary ∂(S \ Γ5) is a closed 2-surface made of five 4-holed spheres S connected by ten annuli `. A convenient system
of coordinates is chosen to be the complex Fenchel-Nielsen (FN) coordinates λ`, τ` for each
2 annulus, and the xS, yS (or µS, νS, mS, nS) coordinates for each 4-holed sphere S. Here λ`
26 is the complex FN length, being the eigenvalue of meridian holonomy around the annulus `. The implementation of simplicity constraint project the Chern-Simons wave function
| 3 i → to the above solution space. The projection is done by the inner product: ZS \Γ5
| i h | 3 i ψsol ψsol ZS \Γ5 . The resulting wave function reads Z 2 Y Y πµS 3 ¯ 3 ¯ − ZS \Γ5 λ`, λ`, mS = dµS ZS \Γ5 λ`, λ`, µS, mS exp . (3.17) 5 2 R S S w k
Here λ` is nothing but the eigenvalue of H f in Eq.(3.4). The simplicity constraint about the eigenvalue of H f can be easily implemented by restricting λ` ∈ U(1) in the wave function. Now the wave function Z only depends on the data of SU(2) flat connections on Rie-
3 mann surface Σ6 = ∂S \ Γ5. It has been shown that the SU(2) flat connections on Riemann surface parametrizes the twisted geometry on 3d discrete space [88]. Thus Z is indeed qualified to be a quantum amplitude describing the evolution of 3d geometry. For the re- lation with spin-network data, it is explaned in a moment that λ` relates to the spins j`. mS ∈ Z/kZ quantizes SU(2) flat connections on 4-holed sphere, thus essentially is a label of the conformal blocks (or equivalently, 4-valent intertwiners of quantum group SU(2)q with q root of unity) [53].
3 → ∞ We consider the semiclassical limit of the resulting ZS \Γ5 (λ`, mS) as k, s . Compar- ing the semiclassical limit to the commutators Eq.(3.12) motivates us to rescale µS, νS, mS, nS by k k k k µS 7→ µS, νS 7→ νS, mS 7→ mS, nS 7→ nS (3.18) 2π 2π 2π 2π
After rescaling, mS, nS become continuous periodic variables as k → ∞.
3 ¯ ··· The semiclassical behavior of ZS \Γ5 (λ`, λ`, µS, mS) is known as [37, 154]( stands for the quantum corrections) 3 ¯ ZS \Γ5 λ`, λ`, µS, mS X Z (λ`,λ¯ `,µS,mS) X λ0 X λ¯ 0 X t d ` t¯ d ` k 0 0 = exp i ln τ + lnτ ¯ + νSdµ + nSdm + ··· . π ` λ0 π ` ¯ 0 π S S α c⊂Lα 4 ` ` 4 ` λ` 2 S (3.19) 27 3 3 The moduli space of flat connections on S \ Γ5, M f lat(S \ Γ5, SL(2, C)) 'L is understood
as the Lagrangian submanifold of the phase space. Lα is the branch of L associated to
3 the flat connection α on S \ Γ5. L can be represented as a set of polynomial equations in symplectic coordinates, whose expressions have been derived in [83]. The quantity on the
exponential is an integral of the Liouville 1-form associated to ωk,s along a contour c in Lα. 3 ¯ Insert the asymptotic express of ZS \Γ5 λ`, λ`, µS, mS in the integral Eq.(3.17),
X Z h i 3 ¯ ¯ ··· ZS \Γ5 λ`, λ`, mS = dµS exp S α λ`, λ`, µS, mS + . (3.20) 5 α R where S α λ`, λ,¯ µS, mS reads
0 0 Z (λ`,λ¯ `,µS,mS) X λ X λ¯ X X kµ2 t d ` t¯ d ` k 0 0 S S α = i ln τ` + lnτ ¯` + νSdµ + nSdm − . π λ0 π λ¯ 0 π S S π 2 c⊂Lα 4 ` ` 4 ` ` 2 S S 4 w (3.21)
In the semiclassical limit, the µS-integral Eq.(3.20) localizes asymptotically at the crit- ical points, i.e. the solutions of critical equations ∂S α/∂µS = Re(S α) = 0. The critical equations are easy to derive:
2 iw νS − µS = µS = 0 ⇒ µS = νS = 0, (3.22)
where we see that the critical equation ∂S α/∂µS = 0 is a classical version of the quantum simplicity constraint Eq.(3.15). The critical equations imply the simplicity constraint, thus require the flat connections on all S to be SU(2).
The condition µS = νS = 0 simultaneously may not be satisfied for generic branches
Lα of the Lagrangian submanifold. However it has been shown in [69] that there exists
exactly 2 branches Lα4d and Lα˜ 4d , where νS(µS = 0) = 0 can be satisfied. The SL(2, C)
3 flat connection α4d on S \ Γ5 equivalently describes the geometry of a nondegenerate 4-
simplex with constant curvature. The other flat connectionα ˜ 4d is referred to as the “parity
partner”, which corresponds to the same 4-simplex geometry as α4d, but with opposite 4d orientation.
28 Those α whose Lα doesn’t consistent with µS = νS = 0 only give exponentially sup- 3 ¯ pressed contributions to ZS \Γ5 λ`, λ`, mS in Eq.(3.20). Therefore S α λ`,λ¯ ` +··· S α¯ λ`,λ¯ ` +··· 3 ¯ 4d ( ) 4d ( ) ZS \Γ5 λ`, λ`, mS = e + e (3.23)
7 where S α4d reads Z (λ`,λ¯ `,mS) X dλ0 ¯ X dλ¯ 0 X t ` t ` k 0 S α4d = i ln τ` 0 + lnτ ¯` 0 + nSdmS , (3.24) ⊂L 4π λ 4π λ¯ 2π c α4d ` ` ` ` S
The integral in S α4d has been reduced to be of the same form as the one treated in [69, 72].
To compute S α4d , we use the geometrical interpretation of flat connections and the FN coordinates in terms of constant curvature 4-simplex geometries. This geometrical inter- pretation has been studied expensively in [67, 69]. The 10 annuli ` are of 1-to-1 corre- spondence to the 10 triangles of the 4-simplex. By the correspondence between 4-simplex
3 geometry and flat connection on S \ Γ5, the complex FN length λ` relates to the area of the triangle a(f`). The dihedral angle Θ(f`) hinged by the triangle f` corresponding to ` relates to the complex FN twist τ`. Explicitly, " # iΛ λ` = exp − a(f`) + πis` , τ` = exp −sgn(V ) Θ(f`) (3.25) 6 4
where s` ∈ {0, 1} parametrizes the lifts from PSL(2, C) to SL(2, C). sgn(V4) is the 4d
orientation of the 4-simplex, which takes different values at α4d andα ¯ 4d. Insert Eq.(3.25) in the integral Eq.(3.24), the integrand becomes proportional to
P10 `=1 Θ(f`)da(f`) except the last term in Eq.(3.24). Because all data Θ(f`), a(f`) associates to a geometrical 4-simplex, and satisfy the Schlafli¨ identity [141, 97] X10 a(f`)dΘ(f`) = Λd|V4| (3.26) `=1 P10 where V4 is the volume of the constant curvature 4-simplex, `=1 Θ(f`)da(f`) is a total derivative: X10 X Θ(f`)da(f`) = dS Regge,Λ, S Regge,Λ = a(f`) Θ(f`) − Λ|V4| (3.27) `=1 ` 7We have choose the integration contour such that the flat connections on the contour all correspond to the 4d geometries. Therefore the Schlafli¨ identity can be used in the derivation (see [69] for details). The contour is in the plane with µS = 0. 29 S Regge,Λ is the Regge action on a single 4-simplex with cosmological constant term.
The last term in Eq.(3.24) contribute the same between α4d andα ¯ 4d [69]. To remove this overall term in the asymptotics, we may consider a coherent state peaked at the phase space pointm ˚ S, n˚S, which behaves as the following when k → ∞,
P P (k) − k (m −m˚ )2− ik n˚ m ∼ 4π S S S 2π S S S φm˚ ,n˚ (mS) e . (3.28)
A candidate of φ(k) can be chosen as a product of Jacobi Theta functions (see Eq.(4.19) of
[24]) to respect the periodicity of mS. As k → ∞, the quantity
X (k) 3 λ , λ¯ , φ ZS \Γ5 ` ` mS m˚ ,n˚ (mS) (3.29) mS gives the critical equation of mS:
mS = m˚ S, nS = n˚S. (3.30)
At the critical point, the last term in Eq.(3.24) cancels the second term on the exponential in φ(k). As a result, Eq.(3.29) behaves asymptotically as
i i S Regge,Λ+··· − S Regge,Λ+··· `2 `2 e P + e P (3.31)
2 12π where `P = sΛ . The above asymptotics reproduce the result in [67, 69, 72]. The previous asymptotic results have been obtained either by pick up semiclassically the branches α4d, α¯ 4d, or by a certain ansatz of Wilson graph operator. However here we obtain the result by a systematic study of the simplicity constraint at the quantum level, and project the partition function onto the space of quantum solutions. The method used here is especially useful when generalizing the amplitude to many 4-simplices.
3.5 SL(2, C) CHERN-SIMONS THEORY ON M3 WITH SURFACE DEFECT
The correspondence between SL(2, C) flat connection on 3-manifold and 4d geometry can be generalized to arbitrary 4d simplicial manifold M4. The corresponding 3-manifold M3 30 3 corresponding to M4 can be constructed by gluing copies of S \ Γ5 (see Figure 3.2). The
3 number of glued S \ Γ5 coincides with the number of 4-simplices in M4. The gluing
3 interface between a pair of S \ Γ5 is always a 4-holed sphere S.
3 Figure 3.2: M3 is obtained by gluing a number of S \ Γ5, each of which 3 corresponds to a 4-simplex in M4. The gluing of S \ Γ5’s is deduced 3 from the gluing of 4-simplices in M4. In drawing the 3-manifold S \ Γ5 3 and M3, we imagine to view S \ Γ5 from 4d and suppress 1 dimension. 3 The 3-manifold S \ Γ5 has five geodesic boundary components as 4- holed spheres, coming from removing the neighborhood of five vertices of Γ5. It has ten cusp boundary components as ten annuli, coming from removing the neighborhood of ten edges of Γ5. The red curves are the 3 annuli connecting 4-holed spheres. Two S \ Γ5 can be glued through a pair of 4-holed spheres, via a certain identification of holes. Each 4- holed sphere as the gluing interface corresponds to a tetrahedron shared by two 4-simplices in M4. Each hole of the 4-holed sphere (or each tunnel traveling thought the 4-holed sphere) corresponds to a triangle in the shared tetrahedron.
To construct the partition function ZM3 on M3, we simply product the resulting par- 3 ¯ tition function ZS \Γ5 λ`, λ`, mS (reduced by the simplicity constraint), then identify and sum over the data mS associated to the gluing interfaces S. So we obtain a state-sum model.
X Y ¯ 3 ¯ ZM3 λ`, λ` = ZS \Γ5 λ`, λ`, mS . (3.32) 3 mS∈Z/kZ S \Γ5
In this formula, mS ∈ Z/kZ is the one before the rescaling Eq.(3.18) in the semiclassical
analysis. In general the resulting ZM3 may also depend on some leftover mS’s, in case that
M3 after gluing still has geodesic boundary components S. 31 The simplicity constraint has been implemented at the gluing interfaces S. The con- straint project the quantum states on S of SL(2, C) Chern-Simons theory to the space of solutions Eq.(3.15), which is essentially the state space of SU(2) Chern-Simons theory. Therefore the simplicity constraint introduces the defects to SL(2, C) Chern-Simons the-
3 ory. The defects is localized at the interfaces S where a pair of S \ Γ5 are glued. The defects are supported on 2-surfaces S embedded in M3. The effect of the defect is that SL(2, C) Chern-Simons theory reduces to SU(2) at the 2-surface. Schematically, the surface defect may be understood via the insertions of certain “sur- ¯ face operators” in SL(2, C) Chern-Simons theory, i.e. we write ZM3 λ`, λ` as a functional integral
Z R R it tr(AdA+ 2 A3)+ it¯ tr(A¯dA¯+ 2 A¯3) Y h i ¯ ¯ 8π M3 3 8π M3 3 ¯ ZM3 λ`, λ` = DADA e OS A, A (3.33) S
The insertions OS, located at the gluing interfaces S, play the role of the projections h i |ψsoli hψsol|. The further discussion of this operator OS A, A¯ is given in Section 3.6.
We consider the semiclassical behavior of the state-sum ZM3 as k, s → ∞. We again perform the rescaling for mS by Eq.(3.18). Then we see that as k → ∞ the sum over mS in Eq.(3.32) approximates an integral over S 1. The semiclassical asymptotics can be again
3 studied by stationary phase approximation, similar to the analysis of ZS \Γ5 . In addition to the critical equations Eq.(3.22), we have one more critical equation at each gluing interface
S, because of the integration of mS.
0 nS + nS = 0 (3.34)
3 0 where nS comes from the flat connection on the S \ Γ5 on the left of S, and nS comes from
3 the S \ Γ5 on the right of S.
0 Semiclassically nS = −nS identifies the SU(2) flat connections on the interface S from
3 the left and right S \ Γ5’s (mS has been identified). The minus sign reflects the opposite
3 orientations on S in the gluing. Thus the SL(2, C) flat connections on the copies of S \ Γ5
are glued to become a flat connection on the entire M3. 32 3 Let’s first consider M3 is obtained by gluing 2 copies of S \ Γ5 through a pair of 4-
0 3 holed spheres S, S , the fundamental group π1(M3) is given by two copies of π1(S \ Γ5)
0 0 modulo the identification of generators on S and S (π1(S) ' π1(S ) with the isomorphism denoted by I). π1(M3) is isomorphic to the fundamental group of a 1-skeleton π1(sk(M4)) from the 4d polyhedron M4 obtained by gluing a pair of 4-simplices. However here the
1-skeleton sk(M4) includes the edges of the tetrahedron shared by the pair of 4-simplices (Figure 3.2).
0 3 Given two flat connections A, A as representations π1(S \ Γ5) → SL(2, C) modulo conjugation, they are glued and give a flat connection A on M3 when they induce the
0 0 same representation to π1(S) and π1(S ) (i.e. A = A ◦ I). We reduce the flat connection on S, S0 to be SU(2), and consider A, A0 corresponds to 2 constant curvature 4-simplices
0 0 S, S . When A, A glue to A on M3, they induce the same SU(2) representation (modulo
0 conjugation) to π1(S) and π1(S ). The SU(2) flat connection reconstructs a unique geomet- rical tetrahedron of constant curvature. The constant curvature tetrahedron belongs to both S, S0, and implies S, S0 are of the same constant curvature. Therefore the flat connection
0 A on M3 effectively glues a pair of constant curvature 4-simplices S, S , and determines a 4-dimensional simplicial geometry on M4. The procedure can be continued to arbitrary ∪N 3 \ M3 = i=1(S Γ5). For each simplicial 4-manifold M4, the corresponding M3 can be constructed as in Figure 3.2. A class of flat connections A on M3 can be obtained by
3 gluing flat connections on S \ Γ5. Each A determines a 4d simplicial geometry (M4, g) obtained by gluing N 4-simplices with the same constant curvature. When the simplicial complex M4 is sufficiently refined, arbitrary smooth geometries can be approximated by the simplicial geometries. The gluing of flat connections gives extra constraint on A, A0 as well as the boundary data λ`. It is possible that a set of λ` doesn’t lead to any flat connection on M3 cor- responding to 4d simplicial geometry. In that case, we say the areas relating to λ` are non-Regge-like, otherwise we say the areas are Regge-like.
33 3 Figure 3.3: This picture shows a result that we glue three S \Γ5. The yel- low shell outside indicate the ambient 3-manifold X3. Four-holed spheres 3 3 B1 and R3 are shared boundary between blue S \Γ5 and red S \Γ5. Sim- ilarly, (B5, G3) and (R4, G2) are blue-green shared boundary and red- green shared boundary respectively. At the center of the picture, there is a non-contractible cycle, which made π1(X3) nontrivial. There is a closed tunnel with three different colors at the center corresponds to an internal triangle shared by three 4-simplices in M4.
In general, M3 can be viewed as the complement of (the open neighborhood of) certain
3 graph Γ in an ambient closed 3-manifold X3. Generically X3 is not S . It is manifest as an
3 example in Figure 3.3, when we glue three S \ Γ5. X3 has a non-contractible cycle which are generated by the gluing procedure. In another word, the fundamental group π1(X3) is non-trivial. In the case of Figure 3.3, the non-contractible cycle of X3 associates a closed
3 tunnel from connecting a number of annuli in S \ Γ5. In general each closed tunnel always
2 goes along a non-contractible cycle in X3. The tunnel gives a torus boundary T of M3.
Following the correspondence between M3 and M4, it is not hard to see that each torus boundary T 2 corresponds to an internal triangle shared by a number of 4-simplices.
The flat connection A gives the commutative meridian (A-cycle) and longitude (B- cycle) holonomies on each T 2. The commutativity implies the two holonomies can be simultaneously diagonalized. The eigenvalue λT 2 of the meridian holonomy equals to the
34 2 annulus meridian holonomy eigenvalue λ` for all λ` building T . From the correspondence
between A and simplicial 4-geometry, it is not hard to see that λT 2 relates to the area a(fT 2 )
of the internal triangle fT 2 " # 2 iΛ λ = exp − a(f 2 ) . (3.35) T 2 3 T
The eigenvalue τT 2 of the longitude holonomy can be obtained by a product of FN twists
2 τ` for all λ` building T . It has been computed in [83]. When we connect a pair of annuli
`1, `2 in X3, the FN twist of the connected annuli `1 ∪ `2 is a product of the FN twists of `1
2 and `2, and the same relation also holds for y` :
2 2 2 τ ∪ = τ τ , = . `1 `2 `1 `2 y`1∪`2 y`1 y`2 (3.36)
2 2 When a number of annuli are connected to form a T , τT = y`1∪···∪`n is the eigenvalue of , ··· , τ2 = τ the longitude holonomy, which is a product of y`1 y`n . We also have T 2 `1∪···∪`n .
Because of the relation between y` and 4-simplex hyper-dihedral angle in Eq.(3.25), τT 2
relates to the deficit angle ε(fT 2 ) hinged by the internal triangle fT 2
1 i X − sgn(V4) ε(f 2 )− πη(f 2 ) τT 2 = e 2 T 2 T , where ε(fT 2 ) = ΘS(fT 2 ). (3.37) S f ⊂S , T2
when sgn(V4) is a constant for all 4-simplices sharing fT 2 . It is shown in [83] that η(fT 2 ) is
an index taking values in {0, 1}. η(fT 2 ) = 0 everywhere on M4 means that the 4d spacetime is globally time-oriented.
3 The gluing of 4-simplices via gluing S \ Γ5 does not put any constraint on the orienta-
N tion sgn(V4) of each 4-simplex. There are 2 flat connections on M4 corresponding to the
same geometry on M4 but of different local orientations. Among them there are a pair of
flat connections give the globally oriented geometry on M4.
We again label by α4d the flat connection on M3 which corresponds to 4d simplicial
geometry, which is globally oriented (sgn(V4) is constant) and globally time-oriented (η(fT 2 )
vanishes constantly). The contribution from each α4d to ZM3 asymptotically behaves as
35 ZM3 ∼ exp S α4d , when k, s → ∞, where Z (λ,λ,¯ m) " X 02 X 0 t dλ T 2 dλ` S = i ln τ 2 + ln τ α4d π T 02 ` λ0 c⊂Lα 4 2 λ 2 ` 4d T ⊂∂M3 T `⊂∂M3 02 0 # ¯ X λ¯ 2 X λ¯ X t d T d ` k 0 + lnτ ¯T 2 + lnτ ¯` + nSdmS , (3.38) π 02 ¯0 π 4 2 λ¯ 2 λ ` 2 T ⊂∂M3 T `⊂∂M3 S⊂∂M3 The type of integral has been computed in [83]. The method of computation is similar to Eq.(3.24). Key steps are again using the geometrical interpretations Eqs.(3.37) and (3.35), as well as the Schlafli¨ identity for each 4-simplex. The result gives the Einstein-Regge
action on the simplicial complex M4, up to some additional boundary terms which corre- spond to the overall phase of the wave function (see Section 6 in [83] for details): X X isΛsgn(V4) S α = − a(f) ε(f) − Λ |V4(S)| 4d π 12 f S (3.39) Z (λ,λ,¯ m) ikΛ X ik X 0 + N(f) a(f) + nSdmS 3 2π Lα f 4d S⊂∂M3 where we have neglected the integration constant. To make the formula short, ε(f) here denotes the deficit angle for internal f or the dihedral angle for boundary f. We read the coefficient in front of Regge action to be the (inverse) Planck scale in 4d
12π `2 = . (3.40) P sΛ
N(f) indicates the leading order of S α4d is a multi-valued function since it comes from integrating the logarithmic function (see [24] for an interpretation). However if there is a quantization of area Λ X N(f) a(f) ∈ 2πZ (3.41) 3 f
The asymptotics of exp S α4d doesn’t depend on the choice of branches N(f). This area- quantization condition has been treated in [69]. It is fulfilled when the boundary condition
λ` comes from the SL(2, C) Wilson lines in X3 labelled by unitary irreps (2 j`, 2γ j`) where
j` ∈ N/2 and γ = s/k is a universal constant. The area relates the representation label by a(f) = γ j` with the correspondence between f and `. 36 The last term in Eq.(3.39) only relates to the boundary of M3 or M4. If we fix the boundary datam ˚ S, n˚S which parametrize the shapes of boundary tetrahedra, and consider
(α) the branches α on which the boundary data can be achieved, i.e. nS (m ˚ ) = n˚S, the last term in Eq.(3.39) on these branches α takes the same value, thus corresponds to an overall phase
in ZM3 [83, 69]. This overall phase can again be removed in the asymptotics by project the partition function on coherent states in mS as in Eqs.(3.28) and (3.29). The result Eq.(3.39) reproduces the earlier asymptotics result in [83], which is obtained by semiclassically picking up by hand the branches α4d. Picking up α4d semiclassically results in that the amplitude is only defined perturbatively via a semiclassical expansion. However here the result is achieved by a systematic quantization of simplicity constraint and imposing the constraint quantum mechanically to the amplitude. The resulting am- plitude on simplicial complex is a non-perturbative definition, which hasn’t been achieved in the earlier work. The above analysis shows that the branch α4d stands out from the semiclassical approximation of the non-perturbative amplitude, which gives the correct semiclassical behavior.
3.6 A FIELD-THEORETIC DESCRIPTION OF THE SURFACE DEFECT
Recall that the insertion OS in Eq.(3.33) projects Chern-Simons states on S to the ground
2 2 states ψsol of the “Hamiltonian” H = µ + ν . It has been mentioned that we can also (w) −2µ2 2ν2 w (w) introduce a squeezed version H = w µ + w ν . The ground state ψsol of H is a w=1 squeezed coherent state such that ψsol = ψsol. The squeezing parameter w introduces an ambiguity to the model at each 4-holed sphere S (in the following we equivalently understand S as a sphere with 4 marked points, which are the intersections with the Wilson- lines, see Eq.(3.44)).
Here we would like to find a field-theoretic understanding of the surface defect OS, as well as the associated ambiguity. The continuum counterparts of the conjugate variables
37 i i i i S 1,2 µ, ν are φ1 = Im(A1) and φ2 = Im(A2) (the coordinates on is chosen to be x ). h i ik φi (x), φ j (x0) = δi jδ(3)(x, x0). (3.42) 1 2 4π
2 2 R i i i i The Hamiltonian H = µ + ν has the continuous conterpart S(φ1φ1 + φ2φ2). However in order to make it coordinate-independent, we have to introduce a surface metric hab (a, b = √ R 2 ab i i 1, 2), and write S d x h h φaφb. As a result the following operator plays the role as the projector |ψsolihψsol|: " Z √ # ¯ k 2 ab i i OS[A, A; hab] = exp − d x h h φaφb , a, b = 1, 2. (3.43) 4π S
The coupling constant has to be the same as the Chern-Simons level k. If we have chosen an
independent coupling constant and scale it large, OS would have been the same as inserting
i i i i delta functions δ(φ1)δ(φ2) in the path integral. However at the quantum level φ1, φ2 cannot be constrained to zero simultaneously by the uncertainty principle, since they are canonical conjugate variables. It relates to the zero-point energy of H. Letting the coupling constant the same as k gives the sharpest projection.
Here we find that the surface metric hab is an analog or generalization of the above squeezing parameter w. Inserting O into the Chern-Simons theory breaks the topological invariance near the surface S, and makes the path integral explicitly depend on the metric hab of each S. This metric dependence is the ambiguity of imposing simplicity constraint in the field-theoretic description. It is standard that the defect in Chern-Simons theory has certain metric-dependence, by breaking the topological invariance of Chern-Simons theory. An standard example is the Wilson line defect, whose metric dependence is reflected as the framing dependence. The defect might not depend on all the metric degrees of freedom, similar to the situ- ation of Wilson lines. At the classical level, OS is both conformal and reparametrization invariant on S. The metric dependence of OS is essentially on the conformal equivalence classes of hab. Two metrics from different classes are not related by conformal transfor- mation and reparametrization. On a sphere with 4 marked points, one can always use 38 conformal transformation move 3 marked points to 0,1 and ∞. The position of the last marked point on S 2, denoted by τ, labels the conformal equivalence classes of the metric.
So the metric dependence of OS is essentially on a single complex parameter in classical theory. It is interesting to understand whether this type of metric dependence is preserved at the quantum level, or how this property receives quantum corrections. The study of this point is postponed to the future research.
Explicitly we write the spin-foam amplitude on M4 as a TQFT on 3-manifold X3 with both surface and line defects Z ¯ Y Y ¯ −i CS [X3|A,A] ¯ ¯ ZM3 = DADA e OS[A, A; hµν] W(2 jl,2γ jl)[A, A] (3.44) S l where the SL(2, C) Chern-Simons action reads Z ! h i t 2 CS X3 A, A¯ = tr A ∧ dA + A ∧ A ∧ A 8π X 3 Z3 ! t¯ 2 + tr A¯ ∧ dA¯ + A¯ ∧ A¯ ∧ A¯ (3.45) 8π X3 3
Here instead of defining the theory on M3, we write the theory on the ambient space X3 and ¯ introduce a Wilson loop operator W( jl,γ jl)[A, A] for each torus cusps. The Wilson loops are traces of holonomies in the unitary representation (2 jl, 2γ jl) of SL(2, C), where γ = s/k is the Barbero-Immirzi parameter. In case M4 has a boundary, Wilson line operators adjoint at vertices has to be introduces in X3 corresponding to the annuli cusps adjoint at 4-holed spheres as the boundary of M3. For the simplicity of the following discussion, we focus on ¯ the case that M4 has no boundary, so that W( jl,γ jl)[A, A] are all Wilson loops.
Indeed, inserting Wilson loops in TQFT on X3 is equivalent to TQFT on the complement
M3 = X3 \{l}. It is standard that the Wilson loop operator has a path integral expression [40] Z Y P R − − i ν +κ Y 1 d+A Y+ ν −κ Y¯ 1 d+A¯ Y¯ ¯ ¯ l 2 ` tr[( l l) ( ) ( l l) ( ) ] W(2 jl,2γ jl)[A, A] = DY DY e , (3.46) `
where ν, κ relate to the representation labels νl = −γ jlσ3, κl = i jlσ3 (σ3 is the 3rd Pauli matrix). Y : ×ll → SL(2, C) is a group-valued field. In the tubular neighborhood N(l) of 39 each Wilson loop, the Chern-Simons action can be written as Z Z t t tr (A⊥ ∧ dA⊥) + tr (F⊥ ∧ At) + c.c. (3.47) 8π N(l) 4π N(`) where At, A⊥ are the components of A along and perpendicular to l. F⊥ = dA⊥ + A⊥ ∧ A⊥ is the curvature. The above Chern-Simons action on N(l) is coupled with the path integral of ¯ Wilson loop. The coupled action is linear to At, At, while other ingredients in ZM3 doesn’t depend on At, A¯t. At, A¯t can be integrated to get 2 delta functions constraining F⊥ and F¯⊥ [40]:
t 1 X FT = Y (ν + κ ) Y−1δ(2)(x)dx ∧ dx 4π ⊥ 2 l l l 1 2 l t¯ 1 X F¯T = Y¯ (ν − κ ) Y¯ −1δ(2)(x)dx ∧ dx . (3.48) 4π ⊥ 2 l l l 1 2 l
We have chosen a local coordinate (x1, x2) on D so that the Wilson line goes through the origin. The constraints imply the eigenvalue of median holonomy on each T 2 to be " # 2πi λ = exp j (3.49) l k l
which is the boundary condition imposed to the theory on the complement M3 = X3 \{l}.
2 λl is the same as λT in the last section. Equivalently ZM3 can be written as a TQFT on M3 with surface defects and the above boundary condition Z ¯ Y ¯ −i CS [M3|A,A] ¯ ZM3 = DADA e OS[A, A; hµν]. (3.50) ¯ λl,λl S The above is the field-theoretic version of the wave function Eq.(3.32) defined in the pre- vious sections.
As k, s → ∞ and keeping λ` fixed, the leading contribution of ZM3 comes from the R solutions of critical equations δS = ReS = 0 when the path integral is written as eS .
ReS = 0 implies φa = 0 on each interface S, i.e. the connection reduces to SU(2) on S. At the solution of ReS = 0, the equation of motion δS = 0 is simply the same as the
Chern-Simons theory without surface defect, i.e. the connection is flat on M3
F = F¯ = 0 on M3, (3.51) 40 and satisfies the boundary condition. It has been shown in the last section that all the flat
connections satisfying the critical equations correspond to the simplicial geometries on M4, although some flat connections may not give a uniform orientation sgn(V4) on M4. As the semiclassical limit k, s → ∞, the leading contribution of each critical point is
given by evaluating the action at the critical point. In Eq.(3.44), OS = 1 at each critical point. The Chern-Simons action and the Wilson-loop action evaluated at a flat connection gives [154, 104] Z Z t dλ t¯ dλ¯ − ln τ l − lnτ ¯ l (3.52) l l ¯ 2π c⊂Lα λl 2π c⊂Lα λl
where the integration is along a contour c in the Lagrangian submanifold L'M f lat(M3, SL(2, C)).
α labels the branch of L where the flat connection locates. λl, τl is the eigenvalues of merid- ian and longitude holonomies on the T 2 boundary. As a result, the contribution of a critical point gives the same result as Eqs.(3.38) and (3.39) up to an overall constant (removing the boundary terms in Eqs.(3.38) and (3.39)). The leading contribution of the flat connection
α4d gives the Regge action with cosmological constant on M4 i X X ZM ∼ exp a(f) ε(f) − Λ |V4(S)| . (3.53) 4 2 `P f S
3.7 SURFACE DEGREE OF FREEDOM
The surface defect introduced in Eq.(3.43) explicitly breaks the SL(2, C) gauge invariance into SU(2) on the surface S. Then from the field theory point of view, the gauge degree of freedom becomes the physically propagating degree of freedom on S, similar to the case of 2d Wess-Zumino-Witten model as the boundary field theory of Chern-Simons theory in 3d bulk. In other words, introducing additional degree of freedom on S recovers the SL(2, C) gauge invariance on S. We consider the infinitesimal gauge transformation of SL(2, C) connection, which turns out to be sufficient for the present purpose
i i i i jk j k i i i i jk j k δξAµ = Dµξ = ∂µξ + ε Aµ ξ , δξA¯µ = D¯ µε¯ = ∂µξ¯ + ε A¯µ ξ¯ (3.54) 41 We consider the background field (A, A¯) being a critical point of the path integral, which i i ¯ satisfies φa = 0 on S. So we have Aa is an SU(2) connection, and Da = Da is an SU(2) covariant derivative on S with respect to the background field. Therefore the gauge trans-
i formation of φa is
i 1 i i i δξφ = D ξ − ξ¯ ≡ D ϕ (3.55) a 2i a a i 1 i − ¯i where ϕ = 2i ξ ξ is a scalar in adjoint representation of SU(2).
The infinitesimal gauge transformation of OS in Eq.(3.43) gives Z √ ! Z √ k 2 ab i i k 2 ab i i 3 δϕ − d x h h φaφb = − d x h h Daϕ Dbϕ + o(ϕ ) (3.56) 4π S 4π S
i at the critical background field with φa = 0 on S. If we add it to the exponent of Eq.(3.43)
and redefine OS by h i Z " Z √ Z √ !# ¯ i k 2 ab i i k 2 ab i i OS A, A, hab := Dϕ exp − d x h h φaφb − δϕ d x h h φaφb (3.57). 4π S 4π S h i Now OS A, A¯, hab is invariant under SL(2, C) gauge transformation (any gauge trans- formation can be compensated by a shift of gauge parameter ϕ). Expanding the term √ R 2 ab i i δϕ S d x h h φaφb at the critical background field gives Eq.(3.56) as the leading or- der for small ϕ. The additional term in ϕ looks like a (gauged) linear sigma-model on S. h i When we insert the above complete operator OS A, A¯, hab into Eq.(3.44), the additional degree of freedom ϕ on S doesn’t modify our previous semiclassical analysis. The Chern- Simons connections A that we are interested in are nontrivial on all 4-holed spheres S.
Turning on a nontrivial background field Aa , 0 on S makes ϕ massive, whose mass term is given by
i jk ilm ab j l k m ab jl km jm kl j l k m ε ε h AaAbϕ ϕ = h δ δ − δ δ AaAbϕ ϕ
h ab km l l m ki k m = h δ AaAb − Aa Ab ϕ ϕ (3.58)
One can diagonalize the mass matrix in the square bracket by orthogonal transformation N, i.e. 42 −1 h T T i T T N tr(A hA)1 − A hA N = tr(A hA)1 − diag(x1, x2, x3), where x1,2,3 ≥ 0. and tr(A hA) =
8 x1 + x2 + x3 > 0 . So the eigenvalues of the mass matrix are all positive. ϕ being massive motivates us to integrate out ϕ, which at the semiclassical level projects to the ground state ϕ = 0.
The surface defect OS modifies the equations of motion by adding a singular term
i i F(A) = δ(t) dt ∧ J hab, φa, ϕ (3.59)
where t is the coordinate transverse to S, and the location of S corresponds to t = 0. The
i i i critical equation φa = 0 and ϕ = 0 on S implies J hab, φa, ϕ = 0 on the right-hand side of the equation of motion. Thus the equation of motion reduces to the flatness Eq.(3.51). So we conclude that all the critical flat connections on M3 studied in the last section are still critical, even when we take into account the additional degree of freedom ϕ on the surface defect.
3.8 CONCLUSION
In this chapter we study the quantization and implementation of LQG simplicity constraint in spin-foam model in the presence of cosmological constaint. Spinfoam amplitudes with cosmological constant are formulated as complex Chern-Simons theories on certain class of 3-manifolds. Implementation of quantum simplicity constraint results in surface de- fects in the Chern-Simons theory. These surface defects guarantee the amplitude have the correct semiclassical limit, which reproduces the Einstein-Regge action with cosmological constant on 4d simplicial complex. This work relates LQG simplicity constraint to surface defects in Chern-Simons theory. Although line defects have been widely studied in Chern-Simons theory, surface defects (or domain-walls) are however not sufficiently studied in the literature. The surface defect
8 T i A hA is a positive semi-definite matrix when A , 0. xi may vanishes since Aa may not be a nondegen- T erate matrix. But if all x1,2,3 = 0, it would lead to (AN) h(AN) = 0, which implies AN = 0 and A = 0, since N is invertible.
43 appearing here has not been studied before. We have done some preliminary investigations of the surface defect by studying the propagating physical degrees of freedom on the defect surface. We show that at the linearized level, the propagating field behaves as a 2d sigma model gauged by Chern-Simons connection. This formalism makes it possible to define rigorously the spin-foam amplitude with cosmological constant. The present definition of the amplitude either uses the infinite di- mensional path integral [67] or uses a semiclassical expansion [72]. However it is known
3 that the Chern-Simons partition function ZS \Γ5 can be expressed as a finite dimensional
3 integral [83]. Now the spin-foam amplitude is constructed by projecting ZS \Γ5 onto the so- lution of simplicity constraint by Eq.(3.17), which is also a well defined operation. So the entire spin-foam amplitude can be written as a finite dimensional integral, whose finiteness is ready to be studied. The research on this aspect is currently undergoing.
44 CHAPTER 4 SU(2) FLAT CONNECTION ON RIEMANN SURFACE AND 3D TWISTED GEOMETRY WITH COSMOLOGICAL CONSTANT
4.1 INTRODUCTION
From the work on implementing the cosmological constant in loop quantum gravity, a new covariant formulation of LQG has been developed, and presented a nice relation between LQG in 4D and Chern-Simons theory on 3-manifolds. In this new formulation, the LQG amplitude can semi-classically reproduce the amplitude given by Einstein-Hilbert action with a cosmological constant. One of the interesting topic is to understand the relation between this new covariant for- mulation and the canonical LQG. As the first step of the whole project, we introduce a new phase space of LQG defining on 3D spatial slice. This phase space also relates to the phase space of Chern-Simons theory which is the moduli space of at connections on 2D Riemann surface. We also find that SU(2) flat connections on (decorated) 2D Riemann surface are shown to be equivalent to the generalized twisted geometries in 3D space with cosmological constant. Various at connection quantities on Riemann surface are mapped to the geometri- cal quantities in discrete 3D space. For the LQG without cosmological constant, the twisted geometry is a useful geometrical parametrization[62, 106, 132, 77, 61, 105, 140]. So, we propose that the moduli space of SU(2) at connections on Riemann surface generalizes the phase space of twisted geometry including the cosmological constant. The Riemann surface S considered here is generally made by gluing a number of 4- holed spheres. On each 4-holed sphere, the flat connection is dual to a constant curvature tetrahedron geometry ([75], reviewed in Section 4.2.2). The (curved) tetrahedron closure
45 condition is imposed by the flatness of connections. So the new LQG phase space and twisted geometry in this paper are already at the level of gauge invariant twisted geometries.
When the 4-holed spheres are glued to form S, we define a longitude holonomy Gab of flat connection, which travels from one 4-holed sphere to another (see Figure 4.1). It is
shown in Section 4.2.3 that Gab can be written in the following form
ξabτ3 −1 Gab = Mabe Mba , (4.1)
In its geometrical interpretation, Mab, Mba ∈ SU(2) rotatez ˆ = (0, 0, 1) to the unit normals of a pair of glued faces from two tetrahedra, by the relation between flat connection on 4-
holed spheres and constant curvature tetrahedra. The above form of Gab closely resembles the geometrical interpretation of the link holonomy in the usual twisted geometry. Here
ξab plays the role of the twisted angle. The above formula suggests the close relation between SU(2) flat connection on Riemann surface and the generalized twisted geometry with cosmological constant, provided that the tetrahedra are of constant curvature in the present context. Indeed the relation is made precise in Section 4.2.4, where we also identify the triangle area from the flat connection, and point out that a generalized area-matching condition is satisfied here. In Section 4.3, we derive the relation between flat connection variables on Riemann sur- face and 3D geometrical variables of interest in LQG, in particular, fluxes and holonomies of Ashtekar-Barbero connection. Firstly, We show in Section 4.3.1 that in presence of cosmological constant, the usual LQG flux is naturally replaced by an exponentiated flux, which is the proper variable for a curved triangle face. The exponentiated flux is identified
to the meridian holonomy Hab of the flat connection on the Riemann surface.
Secondly, we shown in Section 4.3.2 that the longitude holonomy Gab on Riemann surface is identified with the LQG holonomy GAB of Ashtekar-Barbero connection along a link traveling from the interior of one tetrahedron to another. The Ashtekar-Barbero
i i i i connection Aα = Γα + γkα contains the extrinsic curvature kα. The 4D hyper-dihedral angle 46 i ΘAB between a pair of neighboring tetrahedra, as the discrete version of kα, turns out to
be proportional to the twist angle ξab in Eq.(4.1), i.e. ξab = γΘAB where γ is the Barbero- Immirzi parameter. As an consequence from the curvature of tetrahedroa and tetrahedron faces, the area- normal description of each constant curvature tetrahedron involves a choice of base point among the tetrahedron vertices, where the face normals are located. Different choices of
tetrahedra’s base points lead that the path of GAB is non-unique from one tetrahedron to another, in contrast to the situation of flat tetrahedra. This subtlety is discussed in Section
4.3.3. It turns out that different choices of paths of the LQG holonomy GAB are in 1-to-1
correspondence to the paths of Gab of the flat connection on the Riemann surface.
In Section 4.4, we discuss the symplectic structure of the new LQG phase space M f lat(S, SU(2)) from SU(2) flat connection on Riemann surface. We show that the natural symplectic struc- ture Ω on M f lat(S, SU(2)) (derived from Chern-Simons theory) results in that the triangle
1 area and twist angle ξab are the canonical conjugate variables , resembling the symplectic structure of twisted geometry. Combining the canonical variables of flat connection on 4-holed sphere dual to the curved tetrahedron in [75], it shows the symplectic structure
Ω on M f lat(S, SU(2)) generalizes from the usual LQG to the situation with cosmological constant Λ. As the limit of Λ → 0, twisted geometry variables with flat tetrahedra and the symplectic structure are recovered from the symplectic coordinates of M f lat(S, SU(2)) and their geometrical interpretations.
Finally, we carry out the quantization of M f lat(S, SU(2)) in Section 4.5. The discussion here mainly focuses on the quantization of the symplectic coordinates which correspond to twisted geometry variables2. Quantizing the twisted geometry variables is shown to be the same as a quantum torus. Both the triangle area and twisted angle are quantized and have discrete spectra. The cosmological constant gives a cut-off to the area spectrum. Given the
1They relate to the famous Fenchel-Nielsen coordinates of flat connections, see Appendix B.1 for expla- nation. 2 More complete discussions on quantizing M f lat(S, SU(2)) are given in e.g. [1, 53].
47 Hab Hba
Gab
5C 5D
Figure 4.1: Two 4-holed spheres S a and S b glued together.
relation between twist angle ξab and extrinsic curvature, the discreteness of ξab leads to the discreteness of the hyper-dihedral angle ΘAB at the quantum level. It might relate to the discreteness of time in LQG, as proposed in [134].
4.2 FLAT CONNECTION ON RIEMANN SURFACE AND TWISTED GEOME- TRY
4.2.1 From Graph to Riemann Surface
As the first step to build a bridge between twisted geometries on graphs and generalized twisted geometries on Riemann surfaces, we define a bijection between graphs and Rie- mann surfaces. A graph contains a collation of links and vertices, while vertices are the end-points of links. Naively one can define a map ρ from the set of graphs to the set of Riemann surfaces as follows: Firstly we relate each n-valent vertex to a 2D-sphere with n holes. Then we relate each link to a cylinder connecting a pair of holes on different n-holed spheres. By doing so we promote a graph to a surface, which defines ρ. However, the map ρ is surjective but not injective. The Riemann surface made by two n-holed spheres connected by a cylinder is topolog- ically equivalent to a (2n-2)-holed sphere which means, its pre-image in ρ may be either a graph with a single (2n-2)-valence vertex or a graph with two n-valence vertices connected
48 AB
Figure 4.2: An example of graph. Two 4-valence vertices A and B are connected by a link.
Figure 4.3: The Riemann surface made by connecting two 4-holed spheres.
by one link. So ρ−1 is not single-valued. For example, assuming we have a graph made by a pair of 4-valence vertices connected by a link as it is showed in Figure 4.2, ρ maps this graph to a Riemann surface as Figure 4.3. However, since Figure 4.3 is topologically equivalent to a 6-holed sphere, the graph contains a single 6-valence vertex like Figure 4.4 also maps to Figure 4.3 by ρ. So the pre-image of Figure 4.3 is nonunique. ρ is not a bijection. But we can construct a mapρ ˆ between graphs and Riemann surfaces as a bijection by decorating Riemann surfaces. Namelyρ ˆ is a map between the set of graphs and the set of decorated Riemann surfaces. More concretely, the idea of constructingρ ˆ is nearly the same as ρ, except thatρ ˆ relates a link to a cylinder decorated with a meridian, and relates a n-valence vertex to a n-holed sphere decorated with a base point located on it. The identical
B
A
Figure 4.4: An alternative graph relating to Figure 4.3 by ρ
49 Figure 4.5: A Riemann surface decorated by two base points and a meridian.
Figure 4.6: A Riemann surface decorated by a base point.
Riemann surfaces with different decorations (meridians and base points) are understood as different decorated Riemann surfaces. As for the mapρ ˆ−1, the graph is uniquely recovered by connecting all base points on decorated Riemann surface with links, under the condition that each decorated meridian should only intersect one link. Soρ ˆ maps Figure 4.2 to a decorated Riemann surface Figure 4.5 and maps Figure 4.4 to a different decorated surface Figure 4.6. It eliminates the ambiguity we mentioned before by distinguishing different decorations. The inverse of the map brings Figure 4.5 and Figure 4.6 to Figure 4.7 and Figure 4.8 respectively without any ambiguity.
Figure 4.7: A graph recovered from Riemann surface Figure 4.5.
Although the naive map ρ is not a bijection between graphs and Riemann surfaces,ρ ˆ is a bijection between graphs and decorated Riemann surfaces. Twisted geometries are defined on a cellular decomposition of 3D space. The cellular decomposition is dual to 50 Figure 4.8: Graph recovered from Riemann surface Figure 4.6.
Figure 4.9: Path on tetrahedron, where point 4 is a base point and edge 2-4 is a special edge. a graph. Because of the bijection between graphs and decorated Riemann surfaces. The cellular decomposition, where twisted geometries live, is equivalently understood as dual to a decorated Riemann surface. Therefore we are able to use decorated Riemann surface to study twisted geometries on discrete 3D space. Because all Riemann surfaces are decorated in the following discussion, in the rest part of the paper, we use the term “Riemann surface” to refer to the decorated Riemann surface.
4.2.2 Flat Connection on Riemann Surface and Curved Tetrahedron
Given a 4-holed 2D-sphere denoted by S a, according to [75, 68, 70, 73], there is a bijection between a flat SU(2) connection defined on S a and a convex constant curvature tetrahedra geometry as far as the non-degenerate geometry is concerned. The 4-holed sphere is considered as a decorated Riemann surface which has a base point. Denote by Hi the SU(2) holonomy along the loop which starts from the base point of the sphere, goes around the i-th hole3 on the sphere, and returns to the base point, we have Y Hi = e. (4.2) i
3i, which is a number from 1 to 4, labels 4 holes on the sphere
51 On the other hand, if we have a convex tetrahedra, by choosing a base point and a special edge we can define the closed paths along the boundary of each face. More specifically, if the base point is contained in the boundary of the face4, the path will start from the base point, go around the face and end at the base point. If the base point is not contained in the boundary of the face5 the path will start from the base point, go along the special edge, move around the face, return back to base point through the special edge again. See Figure 4.9. The spin-connection holonomies of those paths will obey the relation
Y U∂ fi = e, (4.3) i which is identical to Eq.(4.2) of the SU(2) flat connection holonomies. It suggests an
identification between the flat connection holonomies Hi on S a and the spin-connection
holonomies U∂ fi on tetrahedron. Furthermore, [70] also shows that for the constant curvature tetrahedron whose faces
are of vanishing extrinsic curvature, U∂ fi relates to the area Ai and normal Ni of the face fi by ! Λ i H = U∂ = exp A N τ ∈ SU(2), (4.4) i fi 3 i i − i where Ni is the unit surface normal located at the base point, and τi = 2 σi (σi is the i-th Pauli matrix). It turns out that (4.2) or (4.3) is the closure condition for the constant curvature tetrahe- dron. Namely given the data Ai, Ni satisfying (4.2) or (4.3), a unique convex tetrahedron of constant curvature Λ can be reconstructed [70, 75]. As Λ tends to be small, (4.2) reduces to the usual closure condition of the flat tetrahedron
X i AiN = 0. (4.5) i 4Like the face (1, 2, 4), (2, 3, 4) and (1, 2, 4) in FIG4.9. 5Like face (1, 2, 3) in Figure 4.9.
52 4.2.3 Gluing 4-holed Spheres
We expect that if we have a Riemann surface made by connecting two 4-holed spheres
S a, S b by a 2D-cylinder, the flat connection on this surface have a geometry interpretation as gluing two curved tetrahedra. Figure 4.1 showed an example. The middle dash line implies the decoration-meridian.
Hab stands for the holonomy around a hole on S a. Hba goes around the corresponding hole
on S b which is glued to the hole on S a. Gab is the holonomy connecting two base points. We consider the flat SU(2) connection on the Riemann surface. The flatness implies
−1 Hab = GabHbaGab . (4.6)
all Hab, Gab, Hba belong to SU(2).
Hab can be digonalized as xab 0 = −1 Hab Mab Mab (4.7) −1 0 xab
Similarly for H ba xba 0 = −1 Hba Mba Mba (4.8) −1 0 xba
Both of the matrices Mab, Mba belong to SU(2).
As a consequence of Eq.(4.6), the eigenvalues of Hab and Hba are identical
xab = xba. (4.9)
−1 Inserting Eqs.(4.7) and (4.8) into (4.6) shows that the combination Mab Gab Mba commutes −1 −1 with diag(xab, xab ). It implies Mab Gab Mba is diagonal − e iξab/2 0 −1 Mab Gab Mba = , (4.10) ξ / 0 ei ab 2 which means
ξabτ3 −1 Gab = Mabe Mba . (4.11) 53 Here we note that the parameter ξab is not uniquely determined by Gab. In Eqs.(4.7)
0 α τ3 and (4.8), xab is invariant under the “gauge transformation” Mab → Mabe and Mba →
00 α τ3 0 00 Mbae . As a result, there is a gauge parameter α = α − α appearing in equation (4.10) so that ξab → ξab − α. Eq.(4.11) becomes
(ξab−α)τ3 −1 Gab = Mabe Mba . (4.12)
It turns out that ln xab and ξab are a pair of symplectic coordinates of SU(2) flat connec- tions on Riemann surface. The freedom of α corresponds to the freedom in choosing the
ξab = 0 in the coordinate system. The details are given in Appendix B.1.
4.2.4 Relation with Twisted Geometry
The equation (4.11), coming from the SU(2) flat connection on Riemann surface, suggests a relation with the twisted geometry in 3-dimensions. As it is introduced in [63], the twisted geometry defined on a graph Γ has the phase
∗ space S 2 ⊗ S 2 ⊗ T S 1 on each link. The phase space can be parametrized by the collection
−1 i −1 i of variables (N, N˜ , j, ξ). N = nτ3n = N τi and N˜ = n˜τ3n˜ = N˜ τi indicate the unit normals Ni, N˜ i of the 2-face dual to the link. Ni, N˜ i associates to the two ends of the link. n, n˜ are the rotations transformingz ˆ = (0, 0, 1) to the vectors Ni, N˜ i. j is the area of the 2-face. The twist angle ξ relates to the link holonomy g by
g = neξτ3 n˜−1. (4.13) which shares the similarity with Eq.(4.11)
The SU(2) flat connection on each of the pair of 4-holed spheres S a, S b relates to the geometry of a constant curvature tetrahedron. Gluing S a, S b and obtaining the Riemann surface Figure 4.1 suggest the topological gluing of two tetrahedra through a common face as Figure 4.10. Eq.(4.4) relates Hab to the geometry of a face of the tetrahedron associated to S a. Then Eq.(4.7) implies that ! iΛ x = exp − A , Ni τ = M τ M−1 (4.14) ab 6 ab ab i ab 3 ab 54 i where Aab is the area of the face. Nab is the unit normal of the face located at the tetrahedron ~ base point. The SU(2) matrix Mab is thus the rotation transformingz ˆ = (0, 0, 1) to Nab, playing precisely the same role as n in twisted geometry. A similar interpretation is valid
0 00 for Hba. Mba plays the same role asn ˜. The angles α , α are the freedom of rotations in ~ ~ the plane perpendicular to Nab, Nba As a result, the relation Eq.(4.11) of Gab on Riemann surface resembles the twisted geometry equation (4.13). Gab relates the two unit normals ~ ~ ~ ~ Nab, Nba of the gluing interface, where Nab (Nba) is located at the base point of the left ~ ~ (right) tetrahedron. Gab again presents a twist angle ξab between Nab, Nba, in the same way as the twisted geometry. The resemblance between twisted geometry and SU(2) flat connection on Riemann surface may be summarized by the following:
n ↔ Mab, n˜ ↔ Mba, (4.15) j ↔ ln xab, ξ ↔ ξab.
It suggests that the twisted geometry can be generalized to the situation with constant cur- vature tetrahedron. The generalized twisted geometry relates naturally to the SU(2) flat connection on Riemann surface. The twist angle ξ in the usual twisted geometry has been interpreted as the extrinsic curvature of the spatial slice, when g is the holonomy of the Ashtekar-Barbero connection along the link [133, 106]. A similar interpretation can be
obtain for ξab from Gab on Riemann surface, which is discussed in the next section.
The identification xab = xba is a generalization of the area-matching condition in twisted
geometry. There is a key subtlety in comparing xab = xba and the usual area-matching. xab ∈ 12π relates Aab via an exponential. The periodicity restricts Aab [0, |Λ| ]. However there is no restriction to guarantee that the constant curvatures Λ are the same from the pair
of tetrahedra. It may happen that the first tetrahedron corresponding to S a is spherical
(Λ > 0), while the second tetrahedron corresponding to S b is hyperbolic (Λ < 0). Due to
this subtlety, xab = xba doesn’t restrictively implies the area matching Aab = Aba, but rather
55 contains an ambiguity. More precisely, it implies that
|Λ| |Λ| A = A or A = 2π − A . (4.16) ab ba 3 ab 3 ba
The source of this ambiguity is the proper interpretation of tetrahedron face area from the closure condition Eq.(4.3). The details can be found in [70, 75] 6.
At the quantum level, in the spin-foam model with cosmological constant [68], xab ∈ N 2πi relates to the spins jab /2 by xab = exp( k jab), where the integer k is the Chern-Simons
level. So xab = xba implies jab = jba which is the same identity as the spin-network. The
above ambiguity of area-matching comes from the ambiguity in interpreting the spins jab as areas at the level of constant curvature tetrahedron. However the ambiguity Eq.(4.16) is resolved dynamically in the semiclassical limit of spin-foam amplitude, which exhibits more constraints than the twisted geometry, and implies the gluing of tetrahedra with both area and shape matchings of their faces [68]. In the present paper, we work at the level of twisted geometry, and admit that there is an ambiguity Eq.(4.16) of area-matching when the twisted geometry is generalized to curved
tetrahedra. The “generalized area-matching condition” is xab = xba in the present context. As another important remark, the SU(2) flat connection on Riemann surface automati- cally implies the closure condition Eq.(4.3). Therefore the twisted geometry obtained from SU(2) flat connection on Riemann surface is at the level of the gauge invariant twisted geometry (a` la [63]), in which the tetrahedron closure condition has been implemented.
4.3 GEOMETRIC INTERPRETATION OF TWISTED GEOMETRY
In [133] and [106], a relation between twist angle and extrinsic curvature has been estab- lished for the usual twisted geometry. The twist angle ξ can be interpreted as γΘ, where Θ is the hyper-dihedral (boost) angle between two tetrahedra, and γ is the Barbero-Immirzi parameter. The hyper-dihedral angle Θ is a discrete version of extrinsic curvature as kαβ.
6 −1 In the case of small area Aab, Aba |Λ| , the second possibility cannot hold, which resolves the ambi- guity. 56 B4 B1 B3 A1
A3
B2
A4
A2
Figure 4.10: Gluing topologically two tetrahedra through the interface (1, 2, 3). The topological gluing doesn’t necessarily identify the geome- try of the common face.
The flux variable X in LQG is interpreted as jNi where j is the area and Ni is the normal of the face. In this section, we show that the geometric interpretation of twisted geometry on graph remains valid in the generalized twisted geometry from flat SU(2) connection on Riemann surface. One of the key difference between a geometry made piecewise by curved tetrahedra and a traditional Regge geometry is that, instead of the flatness inside each tetrahedra for the usual Regge case, it is curved inside each of the tetrahedra in our case. Just as the difference between special relativity and general relativity, we can no longer directly compare or inner product vectors at different points in space. Comparing vectors at different points involves a parallel transportation. Consider in 4D spacetime, a 3D spatial slice triangulated by constant curvature tetra- hedra. For each tetrahedron, instead of defining a general space-like normal vector for its surfaces, we have to specify the space-like normal vector of each surfaces at a certain base point of the surface. Similarly, the 4D normal of the tetrahedron should be defined at the base point of the tetrahedron.
57 4.3.1 Exponentiated Flux
R ∧ The flux X of a face f is used to be defined as f e e in the usual context of LQG or twisted geometry. However it has been suggested in [75, 44], that in the presence of cosmological constant or constant curvature tetrahedron, X should be replaced by a suitable version of exponentiated flux.
i i Indeed we consider the 3D spin connection Γα determined by the triad eα, and define
i 7 the holonomy of Γα along the boundary of each tetrahedron face . By non-abelian stokes theorem, we have I ! U∂ f =P exp Γ ∂ f Z ! (4.17) =P exp U(x)RU−1(x) , f where P stands for path ordering in the first line and surface ordering in second line. R is
i the curvature obtained from Γα. A path system has been chosen on f such that for each point x ∈ f , there is a path px connecting x to the base point of U∂ f . U(x) is the parallel transportation by Γ along the path px. Within each tetrahedron, the geometry is of constant curvature, which implies R = Λ ∧ 3 e e. Therefore Z ! Λ −1 U∂ f = P exp U(x)(e ∧ e)U (x) (4.18) 3 f Y " # Λ αβ −1 = P exp U(x) eαeβδA(x) U (x) . x 3 We have discretized the integral in the above. Each point x is contained in a plaquette
αβ whose area is δA(x). Here U(x) parallel transport eαeβ to the base point of U∂ f along px. Moreover f , as a face of constant curvature tetrahedron, is a flatly embedded surface (vanishing extrinsic curvature) in 3d constant curvature space. f being a flatly embed-
Λ αβ −1 ded surface implies that the quantity 3 U(x) eαeβ U (x) (located at the base point) is independent of x [68].
7If the base point is not contained in the face, then the holonomy will use the special edge to connect the base point and the face as it is showed in the last picture in Figure 4.9. 58 At the base point, the wedge product of two orthonormal frame vectors along the inter-
~ i face gives the surface normal N which in SU(2) representation is N τi. So equation (4.18) reduces to ! Λ i U∂ = exp AN τ (4.19) f 3 i where A is the area of the surface. In the usual twisted geometry the flux variable is given ~ ~ ~ by X = AN where N is the normal of a flat tetrahedron face. The new variable U∂ f , which is natural in the present context of curved tetrahedron, is manifestly an exponentiated flux variable, with N~ being the face normal at the base point. This actually proves the equation (4.4). By the correspondence between constant cur-
vature tetrahedron and SU(2) flat connection on 4-holed sphere [75], U∂ f is identified with
the holonomy Hab of flat connection. For a pair of tetrahedra topologically glued as in Figure 4.10, their interface f = (1, 2, 3) has two exponentiated fluxes U∂ f and U˜ ∂ f associated to two different tetrahedra. They relate ~ ~ respectively to two different normals Nab and Nba located at the base points of tetrahedra A and B. In the reconstruction of constant curvature tetrahedron geometry from flat SU(2) con- nection on S a or S b, all the resulting face normals of tetrahedron A and tetrahedron B are located at their base points respectively.
4.3.2 Twist Angle and Extrinsic Curvature
Take Figure.4.10 as an example of gluing two tetrahedra, tetrahedron A is gluing with tetrahedron B through the interface labeled by (1, 2, 3), which means the point A1 identifies with B1 after the gluing, so do the points A2, A3 and B2, B3. We firstly consider a simple case: We set tetrahedron A and tetrahedron B share the same base point at A1 as well as B1. In the following, this point is often mentioned as point
1 for abbreviation, and the interface (1, 2, 3) will be denoted as fAB.
In order to discuss the extrinsic curvature k and its relation with the twist angle ξab, we
59 1
ε
Figure 4.11: The dash circle stands for a small open sphere around point 1. The link is at the vicinity of the base point of tetrahedra. consider a Regge geometry on the spatial slice, which is made by tetrahedra with constant curvature Λ. We zoom into a pair of glued tetrahedra as Figure (4.10). In the present situation, both of the 4D (timelike) normals of two tetrahedra are defined at the common base point 1, so does the derivative of 4D normals. Although in the usual Regge geometry the extrinsic curvature is smeared on the entire interface [133], it is reasonable now to regulate the smeared extrinsic curvature in a neighborhood at the base point, since the extrinsic curvature is the derivative of 4D normal. It is also consistent with the semiclassical geometry emerged from spin-foam [68], in which the discrete extrinsic curvature is defined at the base point, instead of being defined at the common face. By the above argument, the discrete extrinsic curvature is given by Z 3 2 kαβ(x) = ΘAB Nα(x)Nβ(x)δ (x, f(σ))d σ, (4.20) f f is the neighborhood of point 1 on face fAB. Nα(x) is the 3D normal vector field on fAB. Nα is not a constant since the face is curved. kαβ only has the component normal to fAB because it describes the change of 4D normals across fAB. The plane where 4D normal rotates is orthogonal to fAB. ΘAB is the boost angle (hyper-dihedral angle) between the 4D-normals of the two tetrahedra [133, 106].
i i i We define a holonomy GAB of Ashtekar-Barbero connection Aα = Γα + γkα traveling within a small neighborhood at the common base point. GAB is along an infinitesimal link
60 which intersect fAB transversely. Z α i βi GAB = P exp d (Γα + γe kαβ)τi, (4.21) eαi is a triad defined in the neighborhood.
i At the vicinity of the base point, we can choose a smooth triad field eα in tetrahedron
i i A and extend smoothly to tetrahedron B. The 3D spin connection Γα determined by eα is a
8 i smooth field when crossing fAB . Thus in GAB, the contribution from the spin connection Γα is tiny since is infinitesimal. The main contribution comes from the extrinsic curvature. Combing (4.20), we have Z Z βi 3 α 2 GAB 'P exp γe ΘABNαNβδ (x, f(σ)))τi d d σ f i = exp γΘABN τi (4.22)
i βi where N = e Nβ is located at the intersection between and f. As the limit → 0, the intersection approaches to the common base point (point 1). We are free to perform a gauge transformation at one end of on the tetrahedron B side.
ατ3 −1 The gauge transformation is written as V = Mab(Mbae ) where the notions Mab, Mba, α
9 are explained in a moment. After the gauge transformation GAB → GABV is written as
−1 i − − γΘAB Mab N τi Mab ατ3 1 GAB = Mabe e Mba (4.23) (γΘAB−α)τ3 −1 ' Mabe Mba ~ As the limit → 0, Mab ∈ SU(2) has been set to be the rotation transformingz ˆ to N. ~ ~ N ≡ Nab is understood as the normal of fAB at the base point of tetrahedron A. Mba ∈ SU(2) ~ rotations the local frame in tetrahedron B, and rotationz ˆ to a new vector Nba, understood as
the normal of fAB at the base point of tetrahedron B (as the limit → 0). Namely Mab, Mba have the same geometrical meaning as the ones interpreted previously in Eq.(4.11). The angle α is again the rotation freedom in the plane perpendicular to the face normal.
8 i The 3D curvature of Γα is smooth except at each internal edge. The internal edge is the hinge of 3D deficit angle. 9 The periodicity of γΘAB on the exponential reflects the compact-ness of the space of SU(2) Ashtekar- Barbero connection. 61 1
4 5
2 3
Figure 4.12: Two tetrahedra with their vertices labeled by numbers through 1 to 5.
Comparing Eqs.(4.23) and (4.12), we find the flat connection holonomy Gab on Rie-
mann surface can be identified with the (infinitesimal) holonomy GAB of Ashtekar-Barbero connection in 3D space discretized by constant curvature tetrahedra. It relates the twist
angle ξab to the hyper-dihedral boost angle ΘAB by
ξab = γΘAB. (4.24)
4.3.3 More General Choices of base points
The path of GAB being infinitesimal is an artifact from assuming tetrahedra A and B
to share the same base point. If two tetrahedra have different base points, GAB will be not infinitesimal. Consequently, we also need to take the different shapes of the path into consideration.
In general, GAB is defined as the holonomy of Ashtekar-Barbero connection along the path connecting two base points of two tetrahedra. When two points coincide, the path of GAB is defined un-ambiguously as in previous section. However when two base points doesn’t coincide, the path connecting them may be non-unique. Consider again two tetrahedra glued together. For convenient, we label the vertices by the number through 1 to 5 as it is showed in Figure 4.12. As mentioned in Section.4.2.2, the base point and special edge have to be specified on each tetrahedra so that the tetrahedron geometry relates to a flat connection on 4-holed sphere. Let’s define point 4 and point 5 in Figure 4.12 as the base point for two tetrahedra respectively and define edge 4 − 2 and edge 5 − 1 as the respective special edges.
62 We may choose the path of GAB to sequentially pass through 4 → 2 → 1 → 5 as it
10 is showed in Figure 4.13 . Let’s denote this path as p0. However, there is another path
passing through 4 → 2 → 3 → 1 → 5 as it is showed in Figure 4.17, denoted by p−1.
Clearly path p−1 is just path p0 plus an additional closed winding 1 → 2 → 3 → 1. Similarly we may add a closed winding 1 → 3 → 2 → 1 to make the path looks like
Figure 4.15, denoted by p1, or add two more closed windings 1 → 2 → 3 → 1 to p0 to get a path like Figure 4.19, denoted by p−2. We define the winding 1 → 2 → 3 → 1 as right-handed and 1 → 3 → 2 → 1 as left-handed. These examples indicate that (1) the path connecting the base points 4 and 5 is not unique, and (2) the paths connecting 4 and 5 can be classified by the number of windings along the boundary of interface, when the paths contain the special edges of two tetrahedra. We can make any path by adding right-handed or left-handed windings to p0. We label the path between 4 and 5 by pi, where i ∈ Z is the winding number.
1
4 5
2 3
Figure 4.13: The path 4 → 2 → 1 → 5 which is labelled as p0
Gab
Figure 4.14: The pathp ˆ0 which is the path corresponds to the path p0 on a pair of tetrahedra.
10 For convenience, we set the path of GAB always contain the special edges when the base points are not on the interface. It is consistent with the choice of path for Gab on Riemann surface shown in Figure 4.9.
63 1
4 5
2 3
Figure 4.15: The path p1 constructed by adding p0 with one left-handed winding.
Gab
Figure 4.16: The pathp ˆ1 made by addingp ˆ0 with a left-handed winding.
The above paths pi are in 1-to-1 correspondence to the paths for Gab of the flat connec- tion on Riemann surface.
On a Riemann surface like Figure 4.7, the holonomy Gab is along the path connecting two base points of the 4-holed spheres. The paths is again not unique. With the pair of base points fixed, the homotopy classes of the paths are again classified by the windings along the meridian11. For instance, we may have a path connecting the base points as it is showed in Figure 4.14, denoted byp ˆ0. We may draw some different pathsp ˆi by addingp ˆ0 with right-handed windings (i < 0) or left-handed windings (i > 0) along the meridian. Figure 4.14, Figure 4.16, Figure 4.18 and Figure 4.20 are the examples of these paths.
1
4 5
2 3
Figure 4.17: The path 4 → 2 → 3 → 1 → 5 which is labeled as p−1
As the arrangement of the pictures in previous page indicates, a bijection can be defined
11 Within each homotopy class, the paths give the same Gab because the connection is flat. 64 Gab
Figure 4.18: The pathp ˆ−1 made by addingp ˆ0 with a right-handed winding.
1
4 5
2 3
Figure 4.19: The path p−2 made by p0 combining with two right-handed windings.
Gab
Figure 4.20: The pathp ˆ−2 made by adding two right-handed windings top ˆ0
65 by relating pi, being the path of GAB on tetrahedra, top ˆi which is the path of holonomy Gab on Riemann surface.
However, no matter along which path the holonomy GAB on tetrahedra is defined, the common features are that (1) the path goes from one tetrahedron to the other, and (2) the path pass through at least one vertex of the interface triangle for at least once. So we are
able to choose one vertex P of the interface fAB. The vicinity of P is understood to contain the intersection between the path and the interface, as the path travels from one tetrahedron to the other. More precisely, we perform a regularization so that the path is not precisely along the edges of tetrahedra but is rather located slightly inside the tetrahedra. It has an infinitesimal distance from the tetrahedron edges.
The (regularized) path of GAB can be divided into three segments. The 1st segment
(with holonomy GPB) connects the base point of tetrahedron B to the vicinity of the chosen interface vertex P, the 2nd segment is the infinitesimal link in the vicinity of P, and intersects transversely the interface fAB, the same as the one in Section 4.3.2. The 3rd segment (with holonomy GAP) connects the vicinity of P to the base point of tetrahedron
12 A . As it is mentioned in the last paragraph, the 1st and 3rd segments of GPB and GAP are not precisely along the edges of tetrahedra, but slightly located inside the tetrahedron B and A.
Denote the separation of GAB as
GAB = GAPGGPB (4.25)
i i i Recall GAB is the holonomy of Ashtekar-Barbero connection Aα = Γα + γkα. In the dis-
cretization, the extrinsic curvature Eq.(4.20) is only located at the interface fAB, and inside
the neighborhood f of the chosen vertex P (So the hyper-dihedral angle ΘAB is defined
i at P). So the extrinsic curvature kα only contributes G, while GAP, GPB only receive the
i contribution from the spin connection Γα, since they are slightly inside the tetrahedra A and
12 When one of the base point is on the interface, we only need to divide the path of GAB into two segments, the link and the segment connecting to the other base point.
66 B. Thus GAP or GPB are parallel transports relating the reference frames at different points −1 ~ inside tetrahedron A or B. GAP parallel transports the interface normal Nab from the base ~ point of tetrahedron A to the vicinity of P, and GPB parallel transports Nba from the base point of tetrahedron B to the vicinity of P.
Within the vicinity of P, G has been computed in Eqs (4.22) and (4.23). As a result, we obtain again
(γΘAB−α)τ3 −1 GAB = Mabe Mba , (4.26)
Here Mab, Mba is different from the ones in (4.23) up to the additional parallel transports
GAP, GPB. But the new Mab, Mba have the right geometrical meaning as the ones interpreted ~ ~ previously in Eq.(4.11). Namely Mab (Mba) rotationsz ˆ to the normal Nab (Nba). The normals ~ ~ Nab, Nba of the interface fAB are located respectively at the base points of tetrahedra A and
B. ΘAB is the hyper-dihedral boost angle at P.
Comparing to Eq.(4.12), identifying Gab to GAB relates the twist angle to the hyper- dihedral angle
ξab = γΘAB. (4.27)
There is a useful remark: The above discussion starts from the flat connection on Rie- mann surface and proposes the 3D geometrical interpretation to the flat connections. How- ever it may be helpful to consider a reverse logic: One may start from the 3D discrete geometry with constant curvature tetrahedra, and construct the holonomy GAB of Ashtekar-
Barbero connection, as well as the exponential flux U∂ f . The discussion in Sections 4.3.2 and 4.3.3 shows Eq.(4.26) for GAB, while the discussion in Section 4.3.1 shows Eq.(4.19) for U∂ f . Then Eq.(4.6) follows once we identify GAB = Gab and U∂ f = Hab. But Eq.(4.6) characterizes the flat connection on Riemann surface. Thus the 3D discrete geometry re- lates to the SU(2) flat connection on Riemann surface. The relation between 3D discrete geometry and SU(2) flat connection on Riemann sur- face suggests that in the presence of cosmological constant, the phase space of LQG, con- 67 sisting of the holonomies and fluxes, is equivalent to the moduli space M f lat(S, SU(2)) of SU(2) flat connections on Riemann surface.
4.4 SYMPLECTIC STRUCTURE
1 1 ∗ 1 The usual twisted geometry phase space P = S ⊗ S ⊗ T S quotient out Z2 and the kernel of the symplectic structure is symplectomorphic to T ∗SU(2), which is the phase space of LQG on an edge, at the non-gauge-invariant level. But we still need to pick out the subspace fulfilling the closure condition and quotient out the SU(2)V gauge equivalence which is generated by closure condition on each tetrahedron. Finally at the gauge-invariant level, LQG phase space is T ∗SU(2)E//SU(2)V . However, in this paper we relate the SU(2) flat connection on Riemann surface to the generalized twisted geometry with curved tetrahedra. The flat connection on Riemann surface automatically take the closure condition Eq.(4.2) into account. So the discussion in this paper is directly at the gauge invariant level. For a closed Riemann surface S (relating to a closed graph byρ ˆ), the moduli space of SU(2) flat connections M f lat(S, SU(2)) is a symplectic space, whose symplectic struc- R k ∧ ture is Ω = 4π S tr [δ1A δ2A] (k becomes Chern-Simons level in quantum theory). Ω can be derived from Chern-Simons theory on S × R. We propose that in the presence of cosmological constant Λ, M f lat(S, SU(2)) is a generalization of the LQG phase space T ∗SU(2)E//SU(2)V base on the graphρ ˆ(S). It is easy to check that they have the same di- mension, by Eq.(4.15) translating the flat connection variables to twist geometry variables.
What’s more, the symplectic form Ω can be parametrized by xab and ξab defined in Sec- tion 4.2.3, which have twisted geometry interpretations. The variables xab and ξab relates to the complex Fenchel-Nielsen(FN) coordinates of flat connections (see Appendix B.1).
The FN coordinates are the symplectic coordinates on M f lat(S, SU(2)). As a result, the sympectic form can be expressed as ik Ω = − dξ ∧ d ln x + ··· (4.28) 2π ab ab 68 ··· stands for the symplectic coordinates for the flat connection on individual 4-holed
spheres S a, S b. The coordinates in ··· equivalently parametrizes the shapes of constant curvature tetrahedra associated to S a, S b, which has been studied extensively in [75].
Geometrically ξab and ln xab relate to the hyper-dihedral angle Θab and area Aab of the
2πi interface fAB. By using the relation xab = exp( k jab), the above symplectic structure Ω, derived from M f lat(S, SU(2)), reproduces the right Poisson bracket for the twist geometry
{ jab, ξab} = 1 (4.29)
2π ∝ | | 2 2 Here k Λ `P and jab`P is proportional to the area Aab up to the ambiguity mentioned below Eq.(4.16).
Other canonical variables in M f lat(S, SU(2)) describe the shapes of tetrahedra. They has been studied in [75], and shown to be a proper generalization from the case of flat tetrahedra.
The above discuss suggests that M f lat(S, SU(2)) is indeed the right phase space of LQG or twisted geometry in the presence of cosmological constant.
4.5 QUANTIZATION
Given that M f lat(S, SU(2)) is the right phase space for LQG with cosmological constant, we would like to understand the quantization of the phase space and its implication to quantum 3d geometry.
The quantization of M f lat(S, SU(2)) has been well-understood in the development of
Chern-Simons theory with compact gauge group. M f lat(S, SU(2)) is also the phase space of Chern-Simons theory on S × R. See e.g. [1, 53] for the results of quantization. However instead of provide a full exposition of the quantum theory, we rather focus on quantizing the quantities which have geometrical interpretations in twisted geometry, e.g. the face area and twisted angle.
Because the phase space M f lat(S, SU(2)) is compact, the proper coordinates of the
2 phase space relating the area Aab and twisted angle ξab are the exponentials xab and yab = 69 e−iξab (see Appendix B.1 for details). The symplectic structure implies
k {ln x , ln y2 } = (4.30) ab ab 4π
2 ∈ In quantum theory, the quantization of xab, yab U(1) is the same as a quantum torus. × k 2 ∧ The prequantum line bundle over U(1) U(1) has a curvature 2π d ln yab d ln xab. Weil’s integrality criterion then implies that k ∈ Z. We choose the xab-polarization such that the wave function is written as f (ln xab), satisfying both periodicity and invariant under Weyl
13 reflection f (ln xab) = f (− ln xab) = f (ln xab + 2πi). The periodicities in both ln xab and 2 iπ 2iπ ··· ln yab implies that ln xab can only take k + 1 discrete values ln xab = 0, k , k , , iπ, i.e.
2πi j 1 k x = e k ab , j = 0, , ··· , . (4.31) ab ab 2 2
Given the relation between jab and the area Aab. the above implies the discrete area spec- k`2 P ∝ | |−1 trum with a cut-off 2 Λ . 2 In the same way, in the yab-polarization where the wave function is f (ln yab), one find
2 2 iπ 2iπ ··· 2 −iξab ln yab can only take k + 1 discrete values ln yab = 0, k , k , , iπ. Given that yab = e , we obtain a discrete spectrum of twist angle ξab
π 2π ξ = 0, , , ··· , π (4.32) ab k k
Provided the relation ξab = γΘAB, the quantization implies a discrete spectrum of hyper- dihedral angle ΘAB, which is a new phenomena in the presence of cosmological constant. It might relate to the discreteness of time in LQG, as proposed in [134].
13 → −1 The Weyl reflection xab xab is a redundancy of the coordinate xab. 70 CHAPTER 5 SPIN FOAM PROPAGATOR: A NEW PERSPECTIVE TO INCLUDE THE COSMOLOGICAL CONSTANT
5.1 INTRODUCTION
Spin foam models [32, 119] aim at a path integral description of Loop Quantum Gravity (LQG). Despite tremendous developments in recent years, most models struggle in includ- ing a cosmological constant Λ. Yet, the empirical data clearly hints at a non-vanishing, positive cosmological constant. In order to develop more realistic models, it is therefore of great importance to incorporate a cosmological constant. As shown in e.g. [139, 79, 41, 30] a cosmological constant might also serve as a natural regulator through a quantum group structure, which has been established in Euclidean spin foam models. For Lorentzian sig- nature the connection between a quantum group and a cosmological constant is so far unknown. But an alternative approach towards including a cosmological constant has been suggested in [68, 76, 71, 73]. The guiding idea of [68, 76, 71, 73] is to express the Lorentzian spin foam action with cosmological constant as a SL(2, C)-Chern-Simons the-
ory evaluated on a specific graph observable Γ5 (see Figure 5.1), which can be interpreted as the dual graph of a constantly curved 4-simplex. Due to the lack of experimental and observational data, testing the semiclassical prop- erties of a proposed model of quantum gravity is crucial to justify the assumptions made. In spin foam models there exist essentially two standard test of this kind: On the one hand, the spin foam amplitude of a semiclassical state is governed by a phase depending on the discrete Regge action in the limit where spins are large (see e.g. [21, 19, 94, 93, 34]). As shown in [68], the model [68, 76, 71, 73] reproduces the correct Regge-phase with cosmo-
71 logical constant in a semiclassical limit where spins j (i.e. areas) and the Chern-Simons coupling |h| become large w.r.t. ~. On the other hand, the first non-vanishing order of the spin-foam graviton n-point function should reproduce the one of Regge calculus in the semi-classical limit. In fact, it was this latter test (see [2,3]) that revealed the shortcom- ings of the Barrett-Crane model [17, 18] and led to the development of the Engle-Pereira- Rovelli-Levine (EPRL) model [57, 56, 115]. 1 The aim of this paper is to establish the graviton propagator for the model with cosmological constant[68, 76, 71, 73]. Although semi-classical behavior of spin-foams is expected to be achieved in the large spin limit, the existing results on graviton n-point function requires more input in taking the limit. Namely, it requires a double scaling limit to recover the semi-classical graviton n- point function. The double scaling limit consist of taking large spins j but at the same time small Barbero-Immirzi parameter γ, such that the kinematical area A ≈ γ j stays constant. While this limit works fine in models without cosmological constant (see e.g. [2,3, 27, 25, 33, 80, 107, 108]) it is bound to break in models that include a cosmological constant. This can be easily seen from the following argument: In the spin-foam model with cosmological constant[68, 76, 71, 73, 84, 87], one has to take an additional limit such that Λ → 0 in the same rate as j becomes large in order to recover the correct semiclassical behavior, i.e. the Regge action on a constantly curved 4-simplex. Then if one takes additionally γ → 0, in
P 2 the 4-simplex Regge action f γ j f Θ f + ΛV4 , the two terms scale differently. At least
2 2 when the cosmological constant is small, the 4-simplex volume V4 behaves as γ j while
the area is γ j, so both don’t scale (Θ f doesn’t scale). But Λ scales to zero, which eliminates the cosmological constant term in the first non-vanishing order of the graviton propagator. However, in the standard proposal of spin-foam propagator which employs a particular choice of the metric operator, the double-scaling limit is necessary in order to suppress non-classical contributions in the first non-vanishing order (see e.g. [2,3, 27, 25, 33] ).
1See e.g. [2,3, 27, 25, 33, 135] for the recent results on spin-foam graviton propagator and 3-point function. 2 f denotes a triangle in the 4-simplex. Θ f denotes a dihedral angle.
72 As the calculations in section 5.4 reveals, this is also the case in the model [68] if one follows the standard proposal. Yet, as argued above, sending γ to zero will eliminate also the cosmological constant term. To resolve this dilemma, it seems necessary to reconsider the definition of the graviton propagator. In the traditional approach [2,3, 27, 25, 33] the propagator is constructed out of the met- ric operator of canonical Loop Quantum Gravity [148,8, 86, 129]. While this is certainly a viable choice it posses several questions. Firstly, there exists no proof that canonical and covariant LQG are compatible in the sense that operators of the canonical theory can be directly mapped to operators in spin foam models. On the other hand, the metric in the canonical theory is defined on the kinematical level. But spin foam models supposedly solve all the constraint and therefore should yield the expectation values of physical not kinematical operators. For these reasons one might very well consider a metric operator that is directly adapted to the spin foam setting and does not make immediate use of the canonical theory. This point of view is in particular supported if spin foam models are interpreted as truncated theory theories for discrete quantum gravity, whose relation to a full theory of quantum gravity can only be recovered in a continuum limit. Understanding spin-foam graviton propagator in the semiclassical continuum limit is a research undergo- ing, based on the recent result in [85]. We suggest to replace the metric in the propagator by an operator that only depends on the spins. The operator is only defined locally in the parameter space of the boundary data. In other words, the operator is specifically tied to the boundary state and a neighbor- hood in the parameter space of the boundary data. It is natural from the perturbative QFT perspective in which the perturbative QFT operators are usually defined upon a choice of vacuum of the theory. Here in the spin-foam amplitude, the boundary state plays the role of a vacuum state for a perturbation theory over the geometry defined by the boundary state. By a simple argument it can be shown that the limit γ → 0 becomes superfluous for so- constructed operators. This solves the problems discussed above and enables to implement
73 a cosmological constant. In section 5.2 we will review the original construction of the graviton propagator and discuss alternatives directly adapted to spin foam models. These different choices for a graviton propagator are then analyzed in the context of the recently proposed model with cosmological constant [68], which we will briefly review in section 5.3. As shown in [68], the model reproduces the correct Regge-phase with cosmological constant in a semiclassi- cal limit where spins j (i.e. areas) and the Chern-Simons coupling |h| become large w.r.t. ~. It is, hence, ideally suited to demonstrate the problems of the double scaling limit in the presence of a cosmological constant. As shown in section 5.4, the expected semiclassical result can only be reproduced for the modified graviton propagator and not for the orig- inal one, giving further evidence that a different construction of the propagator might be necessary. The paper concludes with a discussion of these findings in section 5.5.
5.2 DIFFERENT PROPOSALS FOR A GRAVITON PROPAGATOR
5.2.1 Standard proposal and conflicts with a cosmological constant
Recall that a spin foam amplitude provides a map from the Hilbert space H∂R induced on
the discrete boundary ∂R of a region R into C. That is, W : Φ 7→ hW|Φi for Φ ∈ H∂R. The expectation value of an observable O, in the sense of the general boundary proposal [36], is then given by hW|O|Φi hOi = . (5.1) hW|Φi Thus the metric 2-point function or graviton propagator is of the form
Gαβγδ(x, y) = hqαβ(x)qγδ(y)i − hqαβ(x)ihqγδ(y)i . (5.2)
Note that we are here working with rescaled inverse density-two metric qαβ = det h hαβ
rather than the boundary metric hαβ on ∂R in order to allow for a direct comparison with canonical LQG. As any operator in LQG, qαβ must be regularized. Since in the first order
αβ α αβ α βi αβ formalism q is obtained by contracting the densitized co-triads Ei , i.e. q = Ei E , q 74 will be smeared over the surfaces dual to the edges of the graph γ over which the boundary
state Φ is defined. In the following, we will restrict to graphs Γ5 that are dual to a 4-simplex σ, since the 4-simplex amplitude is the most fundamental one in all spin foam models. In this setup, the discretized metric at the node n is of the form
ab a b qn := Ei (n)Ei (n)
a where Ei (n) is the co-triad smeared over the triangle ∆na in σ that is shared by the tetrahedra
τn and τa. It follows that the discrete graviton propagator can be generically written as
abcd ab cd ab cd Gmn = hqn qm i − hqn ihqm i . (5.3)
a In older approaches the co-triads Ei (n) are replaced by the flux operators of canonical LQG, i.e. they act as the right invariant vector fields on the edges of the boundary spin network (see e.g [130, 29,2, 27] for details). The first non-vanishing order in these approaches is then found by performing an asymptotic analysis for large spins, that is j → λ j for λ >> 1, and takes the generic form
3 −1 4 3 0 ˆ G(λ) ≈ (γλ) q, j q, j (HRegge) j j0 + γ λ (H + O(γ)), (5.4)
3 where HRegge is the Hessian of the Regge action as a function of j, where q, j is the deriva- ab ˆ tive of the expectation value of qn with respect to j and where H is independent of γ. In order to suppress the non-classical term proportional to Hˆ previous works now enforce the
2 p additional limit γ → 0 keeping the area A = γlp j( j + 1) approximately constant. For models with a cosmological constant we expect a similar result with the difference that the Hessian now depends on the action i X S Regge = − AabΘab − ΛV4 , (5.5) l2 p (ab)
where Aab is the area, Θab stands for the dihedral angle, and where V4 is the 4d volume. Indeed, this is exactly what we did find for the recently proposed model by Haggard, Han,
3without cosmological constant and γ = 1 75 Riello and Kaminski [68, 76, 71, 73] if one considers instead of the pure large j-limit the
Λ double scaling limit j → λ j and Λ → λ (see section 5.4 for details). But now the limit γ → 0 can no longer be considered since the Regge action is no longer linear in γ. While the area is linear in γ the 4d volume scales as area squared and, hence, depends quadratically on γ. Consequently, the area term would be much greater than the volume term in the limit γ → 0, which would suppress the cosmological constant term in the Regge action as well as in the Hessian. This is obviously not what we expect. The above considerations suggest that there is a generic problem in deriving the gravi- ton propagator when a cosmological constant is included, which is not restricted to the model [68, 76, 71, 73] analyzed in greater detail in the subsequent section. Consequently, we should revisit the construction of the graviton propagator itself. Recall that the densi-
α tized co-triad Ei is defined on the kinematic level since it originates from the 3+1 decom- position before the Hamiltonian constraint is applied. But the 2-point function should yield the expectation value for an incoming and an outgoing graviton excitation on a coherent boundary state on the dynamic level. Moreover, there is no formal proof that canonical and covariant LQG are compatible in the sense that operators carry-over from canonical to covariant LQG. So, it is not a priori clear whether the canonical flux operators are the only viable choice to define the metric 2-point function. Instead one could choose an approach in which the metric operator is based on variables that are more inherently defined in the spin foam model.
5.2.2 Perturbative truncated metric
The most promising candidate, which can solve the problems mentioned above, is a metric operator q( j) that only depends on the area-variables, i.e. spins j. Since the only non-zero
derivative with respect the system variables is in this case q( j), j, the first non-vanishing order of the asymptotic expansion (5.4) takes the form
3 −1 G(λ) ≈ λ q, j q, j0 (HRegge) j j0 , (5.6) 76 which only contains the expected term. Since areas and surface normal determine a 4- simplex uniquely up to translation and inversion, an example of a metric operator in the above scenario can be
n i j (qξ)ab = δi j(γ jnanna)(γ jnbnnb) (5.7)
i where nab is the normal to the triangle ∆ab. This choice might be too simple in the sense that it depends heavily on the normals, which are fixed by the boundary data. A less trivial proposal is to express the edge-lengths in terms of the area variables and construct the metric by those edge-area relations. Since a 4-simplex is uniquely fixed by 10 independent edge lengths, the discrete metric is also uniquely determined by those lengths. In particular, this means that we can deter-
i mine the normals nab in (5.7), which are related to the discretized co-triads, as functions of the lengths. On the other hand, there are exactly 10 areas in a 4-simplex, so that it looks tempting to express the metric in terms of the areas by solving the inverse of the Heron formula (5.8) and the corresponding 4-simplex constraints. Heron’s formula 4 is given by
1 p A = (l + l + l )(−l + l + l )(l − l + l )(l + l − l ), (5.8) i j 4 k l m k l m k l m k l m where Ai j stands for the area constructed by the edges lk, ll, lm. But, as a second order equa- tion, the inverse of Heron’s formula has more than one solution and the solution of the full system of equations is therefore ambiguous (see e.g. [22] ). In fact, expressing the metric purely in terms of areas faces the same problem as earlier attempts to define the Regge action in terms of areas (see e.g. [128, 109]), namely that the areas are subjected to hidden constraints (see e.g. [22][110][122][151]). Thus, we also need to consider variables fixed by the boundary state, e.g. the dihedral angles5 as suggested in [45]. This is possible since the ambiguity mentioned in [22] is discrete. For ten given areas there are multiple choices of the edge lengths to reconstruct a 4-simplex but there is no continues deformation be-
4Heron’s formula works for the flat tetrahedron. In the spherical case, the area can be determined by the edge-lengths through L’Huilier’s theorem. In hyperbolic case, one can also get a similar relation through the hyperbolic law of cosine. The details are discussed in the appendix.C.1. 5This is also a viable choice for curved simplices, see [9] 77 tween these choices. For a chosen set of edge-lengths, the perturbation on the geometry cannot transform itself into another 4-simplex geometry that match the same areas. More specifically, for a fixed boundary state, the solution of the inverse of the Heron’s formula must be uniquely chosen to match the edge-lengths of the boundary state. Under this cir- cumstance, we can construct the metric as a function of the area variables which are valued in the neighborhood of the exact areas given by the edges lengths of the fixed boundary state. Within the neighborhood of the given areas, the variation of the areas will not change the choice of the inverse solution of the Heron’s formula. So the exact form of the metric function will also remain. However, a so-constructed metric is only locally defined in the parameter space of boundary data since it is only valid for a specific choice of boundary data. In order to show that it is still sensible to define a propagator by using a boundary- data-dependent metric operator as described above, let us briefly revisit the generalized boundary proposal [36] underlying the construction of the graviton propagator. As pointed out in [29], there is no preferred vacuum state in a background-independent quantum grav- ity theory. Instead, the 2-point function is evaluated on a specific geometry encoded in the boundary states, which can be interpreted as the ‘vacuum state’ around which we are con- sidering quantum perturbations. This means that for each different boundary geometry, we obtain a different 2-point function as a result of a perturbative truncated field theory. For such a perturbatively defined truncated field theory, we may therefore consider a metric operator in the above sense. The metric operator is defined upon a choice of the vacuum state on which the perturbation theory is defined. More specifically, within a truncated field
theory, defined by a specific boundary state |Φ > j, the metric operatorq ˆ j defined by the area variables j lead to the following 2-point function:
abcd ab cd ab cd (G j)nm = h(q j)n (q j)m i − h(q j)n ih(q j)m i j . (5.9)
ab i j Here (q j)n is given by Eq.(5.7), while viewing nna, nnb as functions of j. Then the truncated
78 1 2 3 4 5
Figure 5.1: The dual graph Γ5 of a 4-simplex lives on the (spatial) bound- ary, i.e. in S 3. It’s vertices are dual to the five tetrahedra of the 4-simplex and its edges are dual to the triangles. The figure above depicts a projec- tion of Γ5 into the plane. expectation value is hW|O|Φi j hOi j = . (5.10) hW|Φi j From (5.6) it then follows immediately that the first non-vanishing order of the asymptotic expansion of (5.9) matches the expected Regge-like form. As shown in section 5.4, the second term in (5.4) vanishes if the metric operator only depends on the areas since Hˆ depends on the derivatives of q with respect to the variables distinct from j. Consequently, the limit γ → 0 becomes superfluous. The same argumenta- tion is also valid for constantly curved simplices in the model [68, 76, 71, 73]. Note that the boundary data in this model fixes the sign of the cosmological constant, which determines whether the 4-simplex is spherical or hyperbolic. Furthermore, each face of the 4-simplex is flatly embedded in an ambient space S3 or H 3 [76], so that it is still possible to construct an unique metric operator from the area variables for a fixed boundary. The asymptotic analysis is then very similar to the flat case and will be discussed in the rest of this article.
5.3 SPIN FOAM MODEL WITH COSMOLOGICAL CONSTANT
Before turning to the analysis of the various graviton propagators let us briefly review the model [68, 76, 71, 73] with which we are working. The model can be derived from the
79 action 6(see [68] for details) Z " ! # " ! # 1 1 Λ 1 SΛBF := − ≺ 1 − ? B ∧ F[A] − 1 − ? B ∧ B , (5.11) 2 M γ 6 γ where B is a bivector, F[A] is the curvature of an SL(2, C)-connection A on space-time M and γ is the Barbero-Immirzi parameter. This reduces to the gravitational Holst-action once the so-called simplicity constraint B = e ∧ e is imposed, with e being a tetrad on M. Following the general strategy in modern spin foam models [57, 56, 115], one now derives the path integral of (5.11) on a given (simplicial) discretization of M. In the following, we
7 restrict our attention to a single curved 4-simplex and its dual graph Γ5. The amplitude
associated to a boundary state ΨΓ5 is given by Z
hΛBF|ψΓ5 i = DA DΠ exp(−iSΛBF) ψΓ5 ,
1 where D denotes the path integral measure and Π = 1 − γ ? B. Integrating out Π and splitting into self- and anti-self-dual part yields Z ! h h¯ hΛBF|ψΓ i = DA DA¯ exp −i W[A] − i W[A¯] ψΓ [A, A¯] (5.12) 5 2 2 5 with Z 1 2 W[A]:= tr(A ∧ dA + A ∧ A ∧ A) 4π S 3 3 12π 1 and h = Λ γ + i . The simplicity constraint is now imposed on the boundary spin ~ ~ ¯ network, restricting ψΓ5 to a state Γ5( j, i|A, A) solely labeled by SU(2)-spin-network data (~j,~i) in exactly the same manner as in the Engle, Pereira, Rovelli, Levine (EPRL) model ~ [57, 56, 115]. If we choose coherent boundary data, i.e. Levine -Speziale intertwiners iLS ~ ~ then Γ5( j, iLS |A, A¯) is given by Z Y5 Y ~ ~ Γ5( j, iLS |A, A¯) = dga Pab(ga, gb, Gab) SL(2,C) = < a 1 a b (5.13) Z Y5 Y † −1 ¯ := dga h jab, −~nab|Y ga Gab[A, A] gb Y| jab,~nbai . SL(2,C) a=1 a
8 ciated to the triangle ∆ab as seen from τa and | j,~ni represent the coherent states defined by Perelomov [116, 117]. Furthermore, Y is the EPRL-map [115], that maps the SU(2)-
j,γ j irreducible H j into the SL(2, C)-irreducible H , and G[A, A¯] denotes the holonomy of the Chern-Simons connection. Since the Chern-Simons term in (5.12) is invariant under local SL(2, C) transformations (modulo 2πZ) and (5.13) is invariant under the gauge trans- formation
−1 ga 7→ G(a)ga ; Gab 7→ G(a)GabG(b) (5.14) R Q we can fix ga = 1 and drop the infinite integral dga so that the Λ-deformed EPRL- amplitude [68] is finally given by Z ! h h¯ Y hWΛ |σ; ~j,~ni = DADA¯ exp −i W[A] − i W[A¯] P (G ) . (5.15) EPRL 2 2 ab ab a , where Pab(Gab) can be expressed as † Pab(Gab) = h jab, −~nab|Y Gab[A, A¯] Y| jab,~nbai. (5.16) Following [68, 21], Pab can be expressed as exponential by using the representation of the coherent states Y| j,~ni in terms of homogeneous functions f (z) of two complex variables T (z0, z1):= z on which g ∈ SL(2, C) acts by its transpose (i.e. g f (z) = f (g z)). More precisely, r (2 j + 1) hz|Y| j, ξi := f j(z)( j,γ j) = < z, z >iγ j−1− j< z¯, ξ >2 j, ξ π 9 T T where ξ is the unit spinor associated to ~n , and J maps (z0, z1) to (−z¯1, z¯0) . This implies Z jab jab T ( j ,γ j ) P = dz f (z )( jab,γ jab) f (G z ) ab ab (5.17) (ab) ab Jξab ab ξba ab ab CP1 8 In the piecewise linear setting ~nab is the outward pointing normal of ∆ab. In the case of a constantly curved tetrahedron the closure condition is given by a cyclic conditions on the holonomies hab starting at a base point and encircling ∆ab. Since ∆ab are flatly embedded surfaces these holonomies are completely determined by the normals ~nab at the base point and the areas of ∆ab. See [76] for details. 9 I.e. hξ|σξ~ i = ~n(ξ) where σi are the Pauli matrices 81 i − ∧ − with measure dz = 2 (z0dz1 z1dz0) (¯z0dz¯1 z¯1dz¯0). After a further change of variables, z → GT z and z → z¯, which is introduced for later convenience, we therefore obtain10 Z Z ~ S [A,A¯,~z;~j,~ξ] hWΛEPRL|σ; j,~ni = DA DA¯ dµ(z) e (5.18) (CP1)10 with −1 † h h¯ X < Jξab, (G ) zab >< zab, ξba > S [A, A¯,~z; ~j, ~ξ] = − i W[A] − i W[A¯] + 2 j ln ab ab − − 1 1 2 2 1 † 1 † 2 2 (ab) < (Gab ) zab, (Gab ) zab > < zab, zab > < z , z > + γ ab ab i jab ln −1 † −1 † < (Gab ) zab, (Gab ) zab > (5.19) and Y −(2 j + 1) dµ(z) = ab dz . πk k2k −1 † k2 ab (ab) zab (Gab ) zab 5.3.1 2-point function As discussed in section 5.2, the metric operator in the traditional approach [130, 29,2, 27] a is constructed from the canonical flux operator Eb which act as the right invariant vector fields on the links (ab). That is, for a single link in (5.13) one finds (analogously to [25]): † a i h jab, −~nab(ξ)|Y GabY(Eb) | jab,~nbai Z !2iγ jab −1 † 2 2 ! jab 2 j + 1 kz k hJξab, (G ) zabi hzab, ξbai = ab dz˜ ab ab Ai ( j , z ) π ab k −1 † k h −1 † −1 † ih i ab ab ab (Gab ) zab (Gab ) zab, (Gab ) zab zab, zab (5.20) i i i hσ zab,ξbai i σ with A ( jab, zab) = γ jab and τ = . Note that the same is true for the new ab hzab,ξbai 2i i metric operator (5.7) discussed in section 5.2 only that Aab now depends solely on jab and not on zab. Thus in both cases we find: Z h | a · b| ~ i D D ¯ S i j WΛEPRL En En σ; j,~n = A A dµ(z) e δi j Ana Anb Z (5.21) h | a · b c · d | ~ i D D ¯ S i j k l WΛEPRL En En Em Em σ; j,~n = A A dµ(z) e δi j Ana Anb δkl Amc Amd . 10See equation (5.20) in [68] 82 The 2-point function (5.3) depends of course on the choice of the boundary state. In order to test the semiclassical properties of (5.3) it is therefore important to choose a state with appropriate semiclassical properties. As advertised in [130], a good choice11, that is peaked on intrinsic and extrinsic geometry alike, is given by a superposition of the states |σ; ~j,~ni of the form X |Φ i = Φ |σ; ~j,~ni 0 ~j, ~j0 ~j X X ( j − j ) ( j − j ) X − ab 0ab cd 0cd − 0 − − | ~ i = exp γα(ab)(cd) √ √ i (γφab( jab j0ab) jabθab) σ; j,~n j0ab j0cd ~j (ab)(cd) (ab) (5.22) 0 Here, φab are the dihedral angles of the tetrahedra, which encode the extrinsic curva- 12 ture , and α(ab)(cd) is a complex 10×10 matrix with positive definite real part. Furthermore, we used the freedom of choice in the phase of the spinors ξab to add an additional phase e−i jabθab , whose purpose is to cancel the non-Regge-like phase13 in the asymptotic limit of the model (see section 14 of [68]). With this choice of a boundary state the semiclassical 2-point function is given by P R S n m P R S n P R S m ~ DA DA¯ dµ(z) e tot q q ~ DA DA¯ dµ(z) e tot q ~ DA DA¯ dµ(z) e tot q abcd j ab cd j ab j cd Gmn = P R − P R P R , D D ¯ S tot D D ¯ S tot D D ¯ S tot ~j A A dµ(z) e ~j A A dµ(z) e ~j A A dµ(z) e (5.23) where we introduced the short notation: n i j qab := δi jAnaAnb (5.24) 11These states are closely related to the so-called complexifier coherent states discussed in e.g. [11, 12]. Also see [28]. 12see section e.g. section 10.2 of [68] for the detailed construction 13This non-Regge-like phase depends only on the boundary data and plays the same role as the phase −i jabΠab e appearing in the Lorentzian EPRL model [21], where Πab is zero if both normals of the tetrahedra τa and τb are future or both past pointing and otherwise equals π. In fact, it is possible to impose a similar restriction (Regge-phase convention) on the phases of ξab as in [21], so that jabθab reduces to jabΠab. 83 and X − − X 1 jab j0ab jcd j0cd 0 S tot = − γα(ab)(cd) √ √ − i (γφ ( jab − j0ab) − jabθab) + S [A, A¯,~z; ~j, ~ξ] . 2 ab (ab)(cd) j0ab j0cd (ab) (5.25) 5.4 ASYMPTOTIC LIMIT OF THE 2-POINT FUNCTION In the following we derive the first non-vanishing contribution of the asymptotic expansion abcd ¯ ¯ of Gmn (λ) for large λ where j 7→ λ j, h 7→ λh and h 7→ λh. Under this rescaling of the n 2 n parameters, the action (5.25) scales as S tot(λ) = λS tot(1) and qab scales as λ qab in the old as well as in the new proposal. If the spins j become large one can furthermore approximate the sums over the spins in (5.25) by integrals using the Euler-Maclaurin formula, so that 1 Gabcd(λ) λ4 mn R R R R R R D D ¯ µ λS tot n m D D ¯ µ λS tot n D D ¯ µ λS tot m ~j A A d (z) e qab qcd ~j A A d (z) e qab ~j A A d (z) e qcd ∼ R R − R R R R . D D ¯ µ λS tot D D ¯ µ λS tot D D ¯ µ λS tot ~j A A d (z) e ~j A A d (z) e ~j A A d (z) e (5.26) However, the asymptotics of (5.26) is only well-defined if the critical points of the inte- grants are isolated. Therefore, we need to fix the only remaining gauge freedom14 under the local transformation A → Ag(x) = g(x)Ag(x)−1 + gdg(x) by e.g. imposing the gauge fixing condition discussed in [13] on W[A] and W[A¯] (see appendix C.4). This modifies the path-integral measure by adding a Faddeev-Popov determinant. But due to the normaliza- tion in (5.26) the dependence on the measure factor drops out, so that the Faddeev-Popov determinant can be safely ignored. Hence, the details of the gauge fixing are not of impor- tance for the following and will be suppressed for the sake of simplicity. 14 { } Γ In the action (5.19) this invarianceR is broken at the vertices vi of 5 due to the gauge fixing (5.14). But { } vi is a set of measure zero in S 3 so that a gauge fixing of the Chern-Simons connection can be imposed 3 equivalently on the punctured sphere S \{vi}. 84 5.4.1 Calculation scheme Before the first non-vanishing order (5.26) is calculated explicitly, let us briefly explain the reasoning behind the calculation. Suppose S (x, y), x ∈ Rdx, y ∈ Rdy is a complex valued function, and it is smooth in the neighborhood of the point (x ˚, y˚), which is a solution of the critical conditions Re(S ) = 0 and δS = 0. Then the asymptotic expansion of the integral R I = dxdy u(x, y)eλS (x,y), where u is a function with compact support in a neighborhood of the critical point, can be derived by expanding every integral separately. More specifically, in the first step, we only expand the integral in x and leave y as free parameters. As the action S is a complex valued function, the number of critical conditions δxS = 0 is twice as many as the number of the type-x variables. So for y close toy ˚, there is no guarantee 15 that δxS = 0 has a solution. However, the almost-analytical extension of the action has a solution for critical condition δx˜S = 0, which is denoted asx ˜(y). According to Theorem 2.3 in [111]16 the asymptotic expansion in x is given by Z Z λS (x,y) 2π dx λS (x ˜(y),y) I = dxdy u(x, y)e = ( ) 2 dy u˜(y) e (5.27) λ with iIndHxx " ! # e 1 1 00 − 1 u˜(y) = √ u(x ˜, y) + u (x ˜, y)H 1 + D + O( ) . (5.28) λ xi x j xi x j λ2 |detHxx| 2 x˜=x˜(y) Here Hxx is Hessian with respect to x, dx is the rank and Ind H is the index of the Hessian H, 00 = ∂2 /∂ ∂ 0 = 17 uxi x j u xi x j and D depend linearly on u and its first order derivative ux atx ˜ x˜(y). In the next step the integral of y is expanded around its critical point. The almost-analytical extension also grants the existence of a solution of δy˜S (x ˜(˜y), y˜) = 0. 15An almost analytic extension of a function is in general not uniquely defined. However, the asymptotic expansion of I does not depend on the details of the chosen extension (see e.g [99, 111] for a proof). 16 Also see Theorem 7.7.12 in [99] and [82] 17 Explicitly D is given by 0 000 −1 i j −1 kl 5 000 000 −1 im −1 jn −1 kl D(y, z) = u R (H ) (H ) + uR R (H ) (H ) (H ) | = , , i jkl xx xx 2 i jk mnl xx xx xx x˜ x˜(y z) − − 1 − i − j where R(k, y) = S (k, y) S (x ˜, y) 2 (Hxx)i j(k x˜) (k x˜) . See e.g. [99] for details. 85 As the almost-analytical extended parametery ˜ can be valued arbitrarily in the neigh- borhood ofy ˚, there may be more than one critical point for the system. However, according to theorem 2.3 in [111] (or theorem 3.1 in [82]), the only critical point (x ˜(y), y) that also solves Re(S ) = 0 is the critical point on the real axis, i.e. (x ˚, y˚). Thus, imposing the re- ality condition (x ˜(y), y) = Re(x ˜(y), y) in (5.29) singles out the correct critical point and the asymptotic expansion of (5.27) is given by dx+dy ! ˜ 2π 2 ei(IndHyy+IndHxx) I = eλS λ |detH˜ | · |detH | yy xx (5.29) ! 1 00 − 00 − 1 · u + (u H 1 + u˜ H˜ 1 + D˜ ) + O( ) . λ xi x j xi x j yiy j yiy j λ2 2 (x ˜(˜y),y˜) ˜ ˜ Here, H is the Hessian of the action S (x ˜(˜y), y˜),u ˜yi,y j stands for the second derivative of the function u(x ˜(˜y), y˜) and D˜ is a function that only depends on u and its first order deriva- tives. Note that all the multiplicative prefactors in (5.29) cancel in the graviton propagator (5.26). Moreover a short calculation shows that also all terms that are not proportional to n 0 m 0 (qab) (qcd) drop out so that finally R R R 1 dxdy q(x, y) p(x, y) eλS (x,y) dxdy q(x, y) eλS (x,y) dxdy p(x, y) eλS (x,y) G(λ) = R − R R λ4 dxdy eλS (x,y) dxdy eλS (x,y) dxdy eλS (x,y) 1 h i 1 = p0 q0 H−1 + p˜0 q˜0 H˜ −1 + O( ). λ xi x j xi x j yi y j yiy j λ2 (5.30) Thus, we only need to determine the critical point and calculate the inverse Hessians H−1 and H˜ −1 in order to obtain the asymptotics of G(λ). The x-variables represent z, z¯ and the fields A, A¯ and the role of the y-variables is taken by the spins j. 5.4.2 First non-vanishing order of the graviton propagator As mentioned in the previous section, we only need to compute the critical point of the action, the Hessians and the derivatives of (5.24). The critical points and Hessians are obviously independent from the concrete implementation of the metric, only the derivatives of the q’s depend on this choice. The fix points with respect to connection and z-variables 86 has been calculated in [68] (see appendix C.2). As proven in [68], the path integral is peaked on flat connections with defects associated to the edges of the graph Γ5 (see Fig. 5.1). These can be interpreted as describing the parallel transport along the edges/faces of a constantly curved 4-simplex, where the sign of the curvature is determined by the boundary data. Furthermore, the action S [A, A¯,~z; ~j, ~ξ] in (5.25) reduces to the Regge-action at the critical point. So it only remains to calculate the critical point with respect to the spins. 0 It is easy to see that the real part of (5.25) only vanishes if jab = jab. Recall that in the general scheme, discussed above, the critical conditions with respect to j are enforced after the critical conditions for z and A have been evaluated. Thus the action S [A, A¯,~z; ~j, ~ξ] in (5.25) reduces to the Regge-action and we obtain ∂ ∂S total ab S Regge = −iγφ0 + = 0. (5.31) ∂ jab crit ∂ jab The Hessians are straight forward to calculate and the explicit expressions are given in appendix C.3. In the standard proposal, where the metric operator is defined through the i i hσ zab,ξbai canonical one, A equals γ jab (see section 5.3.1) which implies ab hzab,ξbai i iφanσ ξan iφan e e ξan δ n = γ2 − i i z¯na qab j0na j0nb nan nbn |zna| |zna| δ qn = 0 zan ab (5.32) n δA(x)qab = 0 n δA¯(x)qab = 0 . Thus, only the inverse Hessians H−1 and H−1 are needed. Even though H = 0 the z¯adz¯cd jad jcd z¯adz¯cd inverse does not vanish since the Hessian is not block diagonal in z and A-variables (see equations (C.16)-(C.25)). However, Hzz¯, HzA, HzA¯, HAA etc. are all of the form λ(Hˆ +O(γ)), ˆ −1 where H is independent of γ and λ, while Hzz and Hz¯z¯ vanish. This implies that Hz¯z¯ must −1 ˆ −1 be of the form λ (Hz¯z¯ + O(γ)) as well so that −1 4 3 ˆ −1 δz¯q δz¯q Hz¯z¯ = γ λ (Hz¯z¯ + O(γ)) . 87 ˜ On the other hand, the type-y Hessian H jab jcd is given by (ab)(cd) γα δ jab δ jcd S tot = − √ √ + δ jab δ jcd S Regge, (5.33) crit j0ab j0cd crit P where S Regge = (ab) γ( jabΘab − γΛV4) and V4 is the 4-volume of the simplex for γ = 1. Consequently, the first non-vanishing order of the graviton propagator with the standard definition of the metric operator is given by λ ≈ n 0 m 0 ˜ −1 + γ4λ3 ˆ + O γ . (5.34) G( ) (˜qab) jcd (˜qcd) je f H jcd je f (H ( )) The first term in (5.34) is the expected semiclassical expression and scales as γ3λ3 except for the term proportional to the volume, which scales as γ4λ3. The additional second term in (5.34) would be suppressed if we send γ to zero in addition to the above limit18. Unfor- tunately, this would also suppress the volume term and can therefore not be used to recover the expected result. n In the alternative metric proposal, discussed in section 5.2, qab is independent ofz ¯ so that the second term in (5.34) drops out automatically, since in this case δz¯q in (5.32) vanishes, and the additional limit γ → 0 can be avoided. This also applies to the standard EPRL model without cosmological constant. To summarize, we observe that we can only recover the expected semiclassical expression of the graviton propagator in the model [68, 76, 71, 73] if the metric operator is defined purely by the areas, which strengthens our view point that a different construction of the graviton propagator might be necessary. 5.5 SUMMARY In this chapter we explored the semiclassical limit of the metric 2-point function for the model [68, 76, 71, 73], which includes a cosmological constant. If the metric operator is defined through the flux operators of canonical LQG alike previous works on the graviton propagator, then the first non-vanishing order in the limit j, |h| → ∞ contains an additional 18This is the so-called double scaling limit considered in previous calculations [2,3, 27, 25, 33] without cosmological constant. 88 term next to the expected semiclassical contribution. In contrast to models without a cos- mological constant, this additional term can not be suppressed by taking the limit γ → 0 since this limit would also suppress the cosmological constant in the semiclassical action. We therefore suggested to reconsider the definition of the graviton propagator itself. Since spin foam models are build from discretized model, it is tempting to interpret them as truncated theories whose true nature becomes transparent only in a continuum limit. Moreover, the 2-point function is a tool deeply rooted in perturbative quantum field theory. In this light, we may interpret the semi-classical boundary states as “vac- uum states” of a truncated theory around which we are considering quantum perturbations. So each boundary state gives rise to a different truncated theory. One might therefore won- der whether it wouldn’t be more appropriate to define the metric operator in a way that is directly adapted to the boundary state and the so defined truncated theory. Since we are working in a discretized setting, the first guess would be to define the metric by the edge lengths of the discretization over which the semi-classical states are peaked. However, this would not give rise to a proper operator as spin foams depend only on the area, i.e. the spins. Even though the edge lengths are not uniquely solvable in terms of the areas, it is still possible to define the metric in terms of the areas since the ambiguity is discrete and can be uniquely fixed through the boundary states. The simplest choice would be to define i the tetrad operator as jnab, where nab is the outward pointing normal of the triangle shared by tetrahedron τa and τb. This choice might be too simple in the sense that it heavily de- pends on the boundary data and, hence, might suppress interesting quantum fluctuations. Yet, area and normals to the faces heavily overdetermine the 4-simplex so a more quan- tum definition of the metric operator might be found, whose dependence on the boundary state is less restrictive. In any case if the metric operator only depends on the spins and no other variables of the path integral then the second non-classical term in the propagator is no longer present and the expected semi-classical result is recovered. This would also supersede the limit γ → 0 in models without cosmological constant. 89 To conclude, it seems to be necessary to redefine the graviton propagator by considering a more truncated scenario in order to recover the semi-classical expression in models with cosmological constant. Of course, the considerations here are only valid for a single 4- simplex and a final conclusion should be only drawn after implementing a continuum limit. 90 CHAPTER 6 OUTLOOKS In my dissertation, we have proposed the a semi-classical continuum limit of the spin-foam model, and reproduced a smooth solution of linearized Einstein Equation from the semi- classical continuum limit of the spin-foam model. We explored the simplicity constraint, phase space and propagator of the loop quantum gravity with the cosmological constant. In this chapter, several future plans will be shown. 6.1 TWISTOR THEORY AND TWISTED GEOMETRY WITH COSMOLOGI- CAL CONSTANT Twistor theory was proposed by Roger Penrose as a possible path to quantum gravity [114, 113]. Penrose even proposed that twistor space should be the basic arena from which space- time itself should emerge. It has been claimed that the twisted geometries give a geometric interpretation of twistors[61]. Through the twistorial representation, it is also possible to discuss the dynamic of the twisted geometries[106]. Inspired by those ideas I plan to develop a twistor representation for the twisted geometry with cosmological constant, whose phase space should correspond to the moduli space of SU(2) flat connection on 2D Riemann surfaces, and study its dynamic.This can provide another way to approach the canonical formulation of the LQG with cosmological constant. It will also provide an living example of Penrose’s idea that space-time emerges from the twistor space. 91 6.2 EMERGENT GRAVITON PROPAGATOR FORM SPIN FOAM One of my previous work showed that one can construct an Euclidean gravitational wave like solution from the Spin Foam models by taking the semi-classical continuum limit. A prerequisites of this work is the existence of a Regge geometry that converges to a smooth gravitational wave like geometry in its continuum limit[16]. In [126], the continuum limit of this Regge geometry corresponds to a spin-2 field theory. With the help of this result, I plan to derive the connection between spin-2 fields and the semi-classical continuum limit of the spin foam models. Thus my aim is to derive the graviton (spin-2 particle) propagator the from the spin foam model. 6.3 MORE APPLICATIONS OF THE SEMI-CLASSICAL CONTINUUM LIMIT OF THE SPIN-FOAM MODEL Loop quantum gravity was initially designed as an UV completion of Einstein’s general relativity. So it is always good to ask ourselves whether we can reconstruct smooth so- lutions of Einstein Equation. As it is introduced, the semi-classical continuum limit of the spin-foam models reproduces smooth geometries as long as one can define a sequence of Regge geometries converging to it. In my previous work, I reconstructed a specific smooth solution of the linearized Laplace equation in the Euclidean space-time from a Eu- clidean spin-foam model by using a existing continuum limit of a type of periodic Regge geometries. Generally, I suspect for any convergent Regge geometry one can find the corre- sponding semi-classical continuum limit of the spin-foam model. Apart from the periodic Euclidean Regge geometry, it would be highly interesting to see whether one can reproduce Lorentzian geometries with cosmological constant from spin foam models. This could be done by e.g. applying our semi-classical continuum limit scheme to Kasner cosmology for which converging Regge solution exists[66]. 92 6.4 NUMERICAL SPINFOAM AND TENSOR NETWORK One of the main obstacles in applying the EPRL spin foam model for solving realistic physical problems is the complexity of computations, especially for very refined triangula- tion with many 4-simplices and very high cut-offs of the spin variables. In the spin foam formulation the boundary data of a single 4-simplex is given by 10 coherent states | j,~n > which correspond to the areas and surface normals of the 4-simplex. If we neglect the face amplitude, the EPRL amplitude for one 4-simplex can be considered as a summation over all possible 15-j symbols where the 10 spins assigned to the triangles are given by the boundary data and the remaining 5 describe the intertwiner degrees of freedoms. When the triangulation contains more than one 4-simplices, one has to sum the spins for all faces that are shared by the 4-simplices. As we increase the refinement of the triangulation, the computation complexity will increase since more and more spin sums of the bulk faces and intertwiners have to be dealt with. This makes it necessary to introduce a cut-off of for this spin sums. For analyzing the semi-classical limit of the system, this cut-off must corre- spond to a large spin value, which highly complicates numerical calculations. However, the tensor network algorithm, which has been developed to improve efficiency of simulating quantum entanglement[150], may prove very useful to avoid this problem. Following the multi-scale renormalization ansatz (MERA), a multilayer network struc- ture is introduced to describe the degree of freedom for an entangled quantum many-body system. By doing so, one can assimilate symmetries such as invariance under translations or rescaling, in the computation which results in substantial gains in computational effi- ciency. Furthermore, one is also able to construct MERA state to represent SU(2) invariant tensor [138, 137, 136]. In their work the, a rank 4 invariant tensor is represented as a tensor network comprises an SU(2) invariant vector and three split tensors. Each split tensor is decomposed into a degenerate part which contains the degree of freedom independent with the SU(2) symmetry and a structural part which is a generalized Clebsch-Gordon coeffi- cient. Among this structure, I plan to optimize the degenerate part of the tensor network, 93 so that some of the intertwiner degrees of freedom, which are not significantly contribute to the spin foam amplitude, can be dropped out. Of course, several numerical trails are necessary in order to design the degenerate part. However, once I find those degenerate coefficients, I can pre-store them in the computer and accelerate the computing. 94 APPENDICES 95 APPENDIX A APPENDICES FOR CHAPTER TWO A.1 SPIN FOAM MODELS (SFMS) AND TENSOR NETWORKS In 4 dimensions, the main building block of a triangulation K is a 4-simplex σ (see Figure A.1(a)), whose boundary ∂σ contains 5 tetrahedra τ and 10 triangles f . K is obtained by gluing a (large) number of σ through pairs of their boundary tetrahedra. In the following K itself should be understood as purely combinatorial or topological while the discrete geometry is encoded in the associated state sum Z(K) of the SFM. Generically, Z(K) takes the form X X Y Y Z(K) = A f (J f ) Aσ(J f , iτ), (A.1) J~ ~i f σ where the summand products over all triangles f and all 4-simplices σ in the triangulation ~ ~ K. The SFM data (J, i) assigns each triangle f an SU(2)-representation labelled by J f ∈ Z+/2, and assigns each tetrahedron τ ⊂ K an SU(2)-intertwiner (rank-4 invariant tensor) iτ, i.e. ∈ ⊗ · · · ⊗ ≡ H inv . iτ InvSU(2)[VJ1 VJ4 ] J1···J4 (A.2) Each σ associates to a 4-simplex amplitude Aσ(J f , iτ), which depends on 10 J f and 5 iτ P assigned to f, τ ⊂ ∂σ. The weight A f (J f ) of J~ is usually called the face amplitude. Both, the face amplitude A f (J f ) and the 4-simplex amplitude Aσ(J f , iτ) are model de- pendent. In the following we mainly focus on the Euclidean Engle-Pereira-Rovelli-Levine (EPRL) model [55, 60]. In this model, the 4-simplex amplitude Aσ is given by the contrac- tion of 5 Spin(4) invariant tensors Iτ, that depend on iτ (τ = 1, ··· 5). That is, Aσ(J f , iτ) = tr (I1 ⊗ I2 ⊗ I3 ⊗ I4 ⊗ I5) , (A.3) where Iτ is given by Z 4 ±··· ± Y (J+) (J−) n+n− m1 m4 + − f + f − f f n1···n4 Iτ = dh dh D + + (h )D − − (h )Cn f iτ . m f n f m f n f f =1 96 ± ± (J f ) ± The above integral integrates over 2 copies of SU(2) with Haar measures dh . D ± ± (h ) m f n f + − ± n f n f are Wigner D-matrices for the representation J f and Cn f are Clebsch-Gordan coefficients + − ··· interpolating between (J f , J f ) and J f ( f = 1, , 4) which are subject to the constraint 1 J± = |1 ± γ|J . (A.4) f 2 f ∈ ∈ ± ∈ Here, γ R is the Barbero-Immirzi parameter. If γ = p/q (p, q Z), then J f Z/2 implies J f ∈ qZ for p + q odd or J f ∈ qZ/2 for p + q even. (a) (b) (c) Figure A.1: (a) A 4-simplex σ as the building block of 4d triangulation K. (b) The 5-valent vertex illustrates a rank-5 tensor |Aσ(J~)i. (c) Gluing 4-simplices σ in K gives a tensor network TN(K, J~). Note that Aσ(J f ) with fixed J f ’s is a linear map from 5 invariant tensors iτ to C. In other words, Aσ(J f ) is a rank-5 tensor state (see Figure A.1(b)) |Aσ(J f )i ∈ H inv ⊗ H inv ⊗ H inv ⊗ H inv ⊗ H inv . J1 J2 J3 J4 J4 J5 J6 J7 J7 J3 J8 J9 J9 J6 J2 J10 J10 J8 J5 J1 (A.5) 97 Thus, the 4-simplex amplitude can be written as an inner product Aσ(J f , iτ) = hi1 ⊗ · · · ⊗ i5|Aσ(J f )i. (A.6) P The above relation allows us to write the summand of J~ in Eq.(A.1) as a tensor net- work. We observe that a pair of σ, σ0 is glued in K by identifying a pair of tetrahedra 0 0 τ = τ = σ ∩ σ . Correspondingly, a pair of invariant tensors in Aσ, Aσ0 is identified by setting iτ = iτ0 and summing over iτ in Z(K). The identification and summation may be formulated by inserting a maximal entangled state at each τ X |τi = | i ⊗ | i ∈ H inv ⊗ H inv . : iτ iτ J1 J2 J3 J4 J1 J2 J3 J4 (A.7) iτ This yields a tensor network, ~ TN(K, J):= ⊗τhτ| ⊗σ |Aσ(J f )i, (A.8) where the tensors Aσ at the vertex is contracted with |τi at the edges (see Figure A.1(c)). In other words, the EPRL pair of |iτi in |τi is associated to the two ends of the edge in Figure 0 1 A.1(c), and contracted with the pair |Aσi, |Aσ0 i (σ ∩ σ = τ) at the two ends . Inserting (A.8) into (A.1) finally gives X Y ~ Z(K) = TN(K, J) A f (J f ). (A.9) J~ f Note that both, TN(K, J~) and Z(K), are wave functions of boundary SFM data if ∂K , ∅, or numbers if ∂K = ∅. Due to the presence of the maximal entangled states |τi, the tensor network formu- lation (A.9) allows to interpret SFMs as models of entangled qubits (or more precisely qudits). Recent advances in condense matter suggest that entangled qubits and their quan- tum information might be fundamental, while gravity might be emergent phenomena (see 1To compare with the usual definition of SFM, the network in Figure A.1(c) is the 1-skeleton of the 2- complex dual to K. Note that the network Figure A.1(c) was oriented in the usual definition of SFM, where iτ associated to the target of each edge was the dual hiτ|. Here we have encoded the duality map |iτi 7→ hiτ| at the target of each oriented edge into |Aσ(J~)i, in order to formulate TN(K, J~) as a projected entangled pair states (PEPS) [?, 96]. 98 e.g. [152]). Our results demonstrate that SFMs are concrete examples, in which grav- ity emerges from fundamentally entangled qubits, and therefore relate quantum gravity to quantum information. An important step in establishing the results of this paper is to analyze the behavior of (A.9) for large spins. This is best studied in the integral representation of TN(K, J~) [19, 93]: Z P ± ~ ± f J f F f [gστ, ξτ f ] TN(K, J) = dgστdξτ f e (A.10) ± 2 where gστ ∈ SU(2) × SU(2) and ξτ f ∈ C are normalized spinors, < ·|· > is the Hermitian inner product and F f is expressed as h i X h ± − −1 − F f gστ, ξτ f = (1 − γ) ln < ξτ f (gστ) gστ0 ξτ0 f > σ, f ⊂σ i + −1 + +(1 + γ) ln < ξτ f (gστ) gστ0 ξτ0 f > . (A.11) The above integral representation is valid for γ < 1. For γ > 1 one obtains a similar expression (see [93]). A.2 LARGE SPIN ANALYSIS The LQG area spectrum is given by 2 p 2 a f = γ`P J f (J f + 1), `P = 8πGN~, (A.12) 2 Thus the limit ~ → 0, for which `P a f , corresponds to large spins J f 1. This is why the semiclassical analysis of SFMs focuses on the large-J regime of the state-sum Eq.(A.9). Obviously any perturbation J˚+δJ of a given SFM configuration J˚ 1 remains in this large J-regime. P Since the action f J f F f scales linearly under the rescaling J f = λ j f (here λ is un- derstood as the typical value of spins on K), the asymptotic behavior of Z(K) for λ 1 can be studied by performing a stationary phase analysis. Moreover, we may use Poisson 99 resummation formula to replace the sum over large spins by an integral. If there are N f triangles in K and γ = p/q (p, q ∼ O(1)), where p + q is odd (i.e. J ∈ qZ), then we obtain Z X 1 P Y f J f (F f [X]+2πik f /q) Z(K) = [dJdX] e A f (A.13) qN f ~k∈ZN f f For p+q even and J ∈ qZ/2 one simply has to replace q by q/2 in the above formula. Here, ± X is a short-hand notation for the integration variables X = (gστ, ξτ f ). There is a subspace of J~ ∈ RN f in the large J-regime that has the interpretation of classical triangle areas. These J~s are called Regge-like and satisfy the triangle area-length relation: 1 q γJ (`) = 2(`2 `2 + `2 `2 + `2 `2 ) − `4 − `4 − `4 . (A.14) f 4 i j jk ik jk i j ik i j ik jk 2 The right-hand side is the classical area a f (`)/`P of the triangle f , whose vertices are la- belled by i, j, k. `i j is the length (in Planck unit) of the edge connecting the vertex i and the vertex j of a triangle. As shown in [35, 93], only those Regge-like J~s lead to classi- cal geometries in the asymptotic expansion of Z(K). Moreover, Eq. (A.14) determines a proper subspace in RN f since generically the number of edges in K is less than the number of triangles. In particular, this applies to the triangulation used in the main text of this paper (see Section A.6 for details). We are interested in perturbations `˚+δ` of a fixed background `˚. `˚ is defined in the letter as a triangulated flat geometry. By Eq.(A.14), the perturbations generates the submanifold N f N f MRegge in the space of spins R . Any spin in a neighborhood N ⊂ R of MRegge can be i uniquely decomposed into a tangential and a transverse part to MRegge. Lete ˆ denote the basis vectors transverse to MRegge and N` the number of edges, then XM ~ ~ i J = J(`) + tieˆ , ti ∈ R (A.15) i=1 i with M = N f − N`. So (`, t ) form a local coordinate system in N where non-Regge-like ~ J have at least one ti , 0. This choice of coordinate system is very helpful in studying perturbations on flat spacetime in SFM. 100 We can use this local coordinates (`, ti) to split the integral over J~ into tangential and R R transverse part. That is, dJ~ = [d`dt] J(`), where J(`) = |∂J~(`)/∂`, eˆi(`)| is the Jaco- bian. We regulate the transverse integral of ti by inserting a Gaussian factor Z Z Z Z P − δ t2 dJ~ = [d`dt] J(`) → [d`] J(`) [dt] e 4 i i . (A.16) to avoid divergence. The regulator will be turned off (δ → 0) together with the continuum limit. In addition it should satisfy λ δ−1 1, (A.17) for reasons that are explained below. The ti-integral is simple and can be directly integrated: Z − δ PM 2 PM i Y PM 1 i i i=1 t + i=1 tiΨ i=1 δ Ψ Ψ [dt] e 4 i (k) A(J f ) = e (k) (k) F , (A.18) f where !M 4π Y 2 X F (`, X) = A J (`) + eˆi Ψi . (A.19) δ f f δ f (k) f i i h i ~ ~ i ~ { } ∈ N f Here we used the short-hand notation Ψ(k) = eˆ , F + 2πik/q with F = F f f C and Euclidean inner product h·, ·i. Inserting (A.18) into Z(K) yields X Z D E λ ~j(`),F~[X]+2πi~k/q (k) Z(K) = [d`dX] e Dδ (`, X), ~k PM 1 i i − (k) i=1 δ Ψ(k)Ψ(k) N f Dδ (`, X) = e F (`, X) J(`) q . (A.20) Here, we have ignored all boundary terms in the exponent, since they do not play an im- portant role in the main discussion. The integrals in Z(K) can be analyzed by using stationary phase approximation. Due to λ δ−1 1, the asymptotic behaviors of the integrals are determined by the critical points of D E S = ~j(`), F~[X] + 2πi~k/q . (A.21) 101 The critical points xc are solutions of the critical equations Re(S ) = 0; δ`S = δXS = 0. (A.22) The critical equations are as follows: The equation of motion δ`S = 0 gives * + ∂~j(`) , F~[X] + 2πi~k/q = 0, (A.23) ∂` and Re(S ) = δXS = 0 yields (see [93] for details) i) The closure condition X 0 = j f nˆτ f (A.24) f ⊂τ ii) The gluing condition ± ± gστ nˆτ f = gστ0 nˆτ0 f (A.25) 1 2 3 where the unit vectorn ˆτ f =< ξτ f |σ~ |ξτ f > and σ~ = (σ , σ , σ ) are the Pauli matrices. ± A class of SFM critical points ( j f , gστ, ξτ f ) ≡ ( j f , X) satisfying Eqs.(A.24) and (A.25) determine simplicial geometries on K [94]. In brief, Eq.(A.24) endows all tetrahedra τ with a geometry, in which the triangle areas are proportional to j f , andn ˆτ f is the unit normal vector of f in τ. Eq.(A.25) guarantees that the 5 tetrahedra on the boundary of each 4-simplex σ can be consistently glued. Consequently, Eq.(A.25) assigns a geometry ± to σ where gστ are the geometrical parallel transports that relate the local reference frames at the centers of σ and τ. Since the tetrahedra are shared by neighboring 4-simplices, i.e. τ = σ ∩ σ0, the geometries of σ, σ0 are glued and reconstruct a geometry on the entire K. Note that a solution of Eqs.(A.24) and (A.25) always exists because here the spins j f = j f (`) in Z(K) are Regge-like. Moreover, this solution corresponds to a simplicial geometry with edge-lengths `. In the following, we consider the same triangulation K as in the main text of this paper, 4 ~˚ which is adapted to a hypercubic lattice in R . J f = J f (`˚) is determined by a flat background 102 1/2 geometry on K, where `˚ are the edge-lengths and (γλ) `P is the constant lattice spacing of the hypercubic lattice. The background flat geometry `˚ corresponds to a critical point (˚j f , X˚); and perturbations (˚j f + δ j f , X˚ + δX) thereof, where ˚j f + δ j f is determined by the edge-length perturbation `˚ + δ` in Z(K) formulated by Eq.(A.20). The critical points considered in this work are geometrical and characterized by uni- form 4-simplex orientations. The (oriented) 4-simplex volumes are all positive Vσ > 0 from these critical points [93]. SFM has other non-geometrical critical points, e.g. the BF type and the vector geometries with all Vσ = 0, and geometrical critical points with non- uniformly 4-simplex orientations, i.e. some Vσ < 0. The existence of critical points with non-uniform orientations is the origin of the “cosine problem” [93, 19]. However when we choose the background (˚j f , X˚) to be geometrical with uniform 4-simplex orientations i.e. all Vσ > 0, all critical points touched by small perturbation are still geometrical with all Vσ > 0. The leading order asymptotics of the integral is given by the evaluation of eS at the P critical points. At the geometrical critical points with all Vσ > 0, S = f S f is obtained by [34, 93, 85], + − + − λS f = iJ f Φ f + Φ f + iγJ f Φ f − Φ f + 2πiJ f k f /q. (A.26) ± ± ∼ ± + ± − ∼ + ± − Each Φ f is an angle defined modulo 2π: Φ f Φ f + 2π. So Φ f Φ f Φ f Φ f + 4π, and a + ± − → + ± − → simultaneous transformation Φ f Φ f Φ f Φ f +2π amounts to a shift of k f k f +(q+ p) since γ = p/q. But shifting k f by an integer doesn’t change Z(K), since Z(K) sums all integer k f . In order to avoid this ambiguity, we fix the following range of angles: + − + − Φ f + Φ f ∈ [−2π, 2π], Φ f − Φ f ∈ [−π, π]. (A.27) ± P ± ± As geometrical interpretations of these angles, Φ f = σ φτστ0 mod 2π, where φτστ0 satisfy [19] + − φτστ0 − φτστ0 = π − θ f (σ) ∈ [−π, π]. (A.28) 103 0 Here, θ f (σ) is the 4d dihedral angle between the two tetrahedra τ and τ within σ. Eq. + − − − P (A.27) implies Φ f Φ f = n f π σ, f ⊂σ θ f (σ) mod 4π, where n f is the number of σ sharing f . Note that n f is always even for the triangulation K considered in this paper (see Section A.6). Then shifting by multiples of 2π and 4π gives + − X Φ f − Φ f = n f π − θ f (σ) − 4πu − 2πv σ, f ⊂σ X = 2π − θ f (σ) = ε f (A.29) σ, f ⊂σ for certain u, v ∈ Z. The deficit angle ε f hinged by f is a discrete description of Riemann curvature in simplicial geometry (Figure (A.2)). Since we consider only critical points that are perturbations on the flat geometry, |ε f | must be smaller than π. ε Figure A.2: The deficit angle ε in a 2d discrete surface hinged by a point. ε , 0 demonstrates that summing the angles at the hinge fails to give 2π. One obtains a discrete curved surface when the two edges bounding ε are glued. In higher dimensions, ε is always hinged by a co-dimension-2 simplex, e.g. in 4d, ε f is hinged by a triangle f . + − ± To determine Φ f + Φ f , we consider all gστ whose σs and τs share a single internal triangle f . We can form a loop holonomy: ± σ ≡ ± ± ± ··· ± = Φ± ˆ ± σ G f ( ) gστgτσ1 gσ1τ1 gτkσ exp i f X f ( ) (A.30) ˆ ± ± · ± −1 ± where X f (σ) = gστ(ˆnτ f σ~ )(gστ) . At a critical point, G f (σ) is given [34, 93] by Gˆ f (σ) = exp ∗Xˆ f (σ) θ f exp πm f Xˆ f (σ) (A.31) 104 in the vector representation Gˆ f (σ) ∈ SO(4), and m f ∈ {0, 1} labels two different types of critical points [91]. ˆ ∈ + − ∈ × ± 1 ± − The lifting of G f (v) SO(4) to (G f (v), G f (v)) SU(2) SU(2) yields Φ f = 2 m f π θ f ± ± ∈ { } k f π, where k f 0, 1 labels the lift ambiguities. Thus: + − + − Φ f − Φ f = θ f − (k f − k f )π, + − + − Φ f + Φ f = πm f − (k f + k f )π. (A.32) At the critical point (˚j f , X˚), which corresponds to the flat background geometry deter- ˚ ± ± mined by `, we set m f = k f = 0. This also fixes m f = k f = 0 for critical points in the neighborhood of (˚j f , X˚) that are reached by small perturbations (˚j f + δ j f , X˚ + δX). Since F f at a critical point evaluates to F f = iγε f , (A.33) the action S at these critical points reduces to λS f = iγJ f ε f + 2πiJ f k f /q. (A.34) Inserting the above result in the equation of motion Eq.(A.23) leads to * + ∂~j(`) , γ~ε + 2π~k/q = 0. (A.35) ∂` (k) The function Dδ (`, X) in Eq.(A.20) plays the role of an integration measure, and takes the following value at each critical point: − PM 1 h i ~ i2 − M − (k) i=1 δ eˆ ,γ~ε+2πk/q 1 N f Dδ = e 4πδ J q . (A.36) This implies that the essential contributions to Z(K) come from critical points that satisfy |heˆi, γ~ε + 2π~k/qi| . δ1/2. (A.37) ∂~j(`) i Recall that ∂` ∀` is a basis tangent to MRegge, ande ˆ is a basis transverse to MRegge. So ~ ∂ j(`) i N f ∂` , eˆ form a basis of R . Eqs.(A.35) and (A.37) imply 1/2 |γε f + 2πk f /q| . δ . (A.38) 105 Note that the deficit angles ε f are all small for small perturbations (˚j f + δ j f , X˚ + δX) of (k) the flat geometry. Therefore, Eq.(A.38) can only be satisfied for k f = 0. So Dδ suppresses the ~kth integral in the perturbative regime unless ~k = 0. The critical points contributing to ~k , 0 integrals are far away from the perturbative regime. As a result, the equation of motions (A.23) obtained from perturbations of the flat back- ground all correspond to EOMS with ~k = 0. These coincides with the Regge equation of discrete gravity: X ∂a f (`) ε (`) = 0, (A.39) ∂` f f ~ 2 where we set k = 0 and a f = γJ f `P in Eq.(A.23). Additionally, the solution is subject to the following constraint from SFM: 1/2 |γε f | . δ . (A.40) To summarize, the solutions of Eqs.(A.39) and (A.40) are the critical points that contribute dominantly to Z(K). A.3 EXPANSION OF THE LINEARIZED THEORY The large spin analysis uses the stationary phase approximation, which is an 1/λ asymptotic expansion of integrals in Eq.(A.20). We focus on the expansion of the integral with ~k = 0, at the level of the linearized theory. We write δX = δX(`) + δX, where δX(`) solves the critical equations δXS = Re(S ) = 0. By this change of variables, h i 1 T S = S `˚ + δ`, X˚ + δX(`) + δX HXXδX + ··· . (A.41) 2 From the discussion in the last section, we know that S [`˚ + δ`, X˚ + δX(`)] is the Regge action. At the quadratic order, 1 T S [`˚ + δ`, X˚ + δX(`)] = δ` H``δ` + ··· (A.42) 2 106 has been studied in [126], in which the Hessian matrix H`` was shown to be degenerate. The kernel of the Hessian contains (1) the space of solutions of linearized Regge equation, and (2) 4 zero modes corresponding to the diffeomorphisms in the continuum, and (3) 1 zero mode of hyperdiagonal edge-length fluctuation. We obtain the following bound of error for the large spin analysis in the last section 2 Z ! N Z ! N π 2 h i− 1 π 2 λS 2 ⊥ 2 k k k 2 C [d`dX] e Dδ(`, X) − det (HXX) det K`` [dδ` ] Dδ δ` , X(δ` ) ≤ . λ λ λ (A.43) ⊥ ⊥ Here K`` is the nondegenerate part of H``, and N = rank(K``) + rank(HXX). The integral R [dδ`k] is over solutions of linearized Regge equations and zero modes. C > 0 bounds the 1/λ correction [100]. The semiclassical approximation by Regge solutions is valid when the 1/λ corrections are negligible, i.e. when Cδ/λ is small. The bound relates to the derivatives of Dδ by [100] C c 2 = sup ∂Dδ + sup ∂ Dδ . (A.44) λ λ 2 −2 −M where c is a constant. Since ∂ Dδ ∼ δ from the exponential (δ in Eq.(A.20) is an overall constant), λ δ−2 1 (A.45) has to be satisfied to validate the expansion, which makes Eq.(A.17) more precise. Eq.(A.43) is the expansion at the level of linearized theory, whose asymptotics is an integral over critical solutions (solutions of EOM and zero modes). It indicates that the critical solutions contribute dominantly to the SFM. In this paper we mainly discuss the convergence of critical solutions under the semiclassical continuum limit. In a companion paper [89], we report the result of graviton propagator and the continuum limit, in which we apply gauge fixings to remove zero modes. 2 Eq.(A.43) assumes the non-degeneracy of the Hessian matrix HXX after removing gauge redundancies. This non-degeneracy is supported by some numerical evidences. A general proof of non-degeneracy for the Hessian in SFM is lacking in the literature. In case it happens that HXX is degenerate, there are additional zero modes coming from SFM variables X. Then the effective theory is the Regge gravity coupling to these additional zero modes, when go beyond the linearization. But in this paper, we focus on the sector of linearized Regge gravity and the continuum limit. 107 A.4 SEMICLASSICAL CONTINUUM LIMIT (SCL) The above discussion is based on a fixed triangulation K. We may construct a refined triangulation K 0 which is adapted to a refined hypercubic lattice in the same way as K is adapted to the original hypercubic lattice. The refined hypercubic lattice is simply given by subdividing each hypercube into 16 identical hypercubes. By refining the hypercubic 0 lattice we define a sequence of triangulations Kµ where Kµ0 is finer than Kµ if µ < µ. In the continuum limit µ → 0 the vertices in the triangulation become dense in R4. A sequence of SFMs is defined by associating an amplitude Z(Kµ) to each Kµ. Since the above large spin analysis is valid for all Z(Kµ), it gives a sequence of Eqs.(A.39) and (A.40) on the sequence of Kµ: X ∂a f (µ) ε (µ) = 0, |γε (µ)| . δ1/2(µ). (A.46) ∂` f f f All quantities in the equations, e.g. the spins J f , the regulator δ, and the simplicial geome- tries, etc, depend on µ, and flow toward µ → 0. We set the triangulation label µ to be a mass scale such that µ−1 is a new length unit. −2 Then a f (µ) = α f (µ)µ . The lattice spacing a(µ) is given by the background flat geometry on Kµ: 1 −1 `˚(µ) = (γλ(µ)) 2 `P = a(µ)µ . (A.47) We define the semiclassical continuum limit (SCL) as the flow of the 3 parameters λ(µ), a(µ), δ(µ), where a(µ), δ(µ) → 0 and λ(µ) → ∞ (λ(µ) δ(µ)−2) for µ → 0. In addition, these flows should satisfy 2 1 dλ − < < 0, (A.48) µ λ dµ 1 dλ 1 dC < , (A.49) λ dµ C dµ δ(µ)1/2 bounded from above. (A.50) a(µ)2 Here, C(µ) is the bound in Eq.(A.43), which now depends on µ for the expansion of Z(Kµ). 108 The constraint Eqs.(A.48)-(A.50) are necessary due to the following reasons: Firstly, the motivation for the SCL is to relate the SFM continuum limit µ → 0 to the continuum limit a → 0 in Regge calculus, so that we can apply the convergence result in Regge calculus to the solutions of Eqs.(A.39) and (A.40). Obviously, this requires that the lattice space a(µ)2 ∝ λ(µ)µ2 → 0 as µ → 0. Thus, d dλ 0 < λ(µ)µ2 = µ2 + 2µλ (A.51) dµ dµ which yields Eq.(A.48). Secondly, the 1/λ correction has to be small for all µ, in order that classical Regge solutions are the leading orders of Z(Kµ). It is important to have Regge solutions at all µ to apply the convergence result in Regge calculus. This demands Eq.(A.43) to be valid for all Z(Kµ) with C(µ)/λ(µ) being always small. C(µ) ∼ δ(µ)−2 grows when the triangulation is refined. Thus, λ(µ) is required to grow in a faster rate in order to suppress C(µ)/λ(µ) as µ → 0. This requires ! d C(µ) C dλ 1 dC 0 < = − + (A.52) dµ λ(µ) λ2 dµ λ dµ which gives 1 dλ 1 dC < . (A.53) λ dµ C dµ This condition guarantee that Eq.(A.43) is valid at all µ, with the 1/λ correction being always small, i.e. the following bound holds in the continuum limit µ → 0: C(µ) C(1) < , (A.54) λ(µ) λ(1) where µ = 1 is the starting point of the flow. Thirdly, the simplicial geometry should approximates a smooth geometry. If this is the case then the typical curvature radius ρ of the smooth geometry relates to the deficit angle −2 −2 of the simplicial geometry by ρ ' ε f a . The regulator δ and conditions (A.40) and (A.50) guarantee that the curvature ρ−2 of the emergent geometry is bounded (geometry is nonsingular) as µ → 0. 109 Eqs.(A.48)-(A.50) have nontrivial implications for the SCL: In order that a satisfactory flow λ(µ) exists, Eqs.(A.48) and (A.49) have to be consistent, i.e. 1 dC 2 > − , (A.55) C dµ µ which yields a restriction to the assignment of µ to Kµ. Since µ is assigned to a sequence of triangulations Kµ ≡ Kµn ≡ Kn (µn−1 > µn), the variable µ ≡ µn is actually discrete. In the above, we have assumed that C(µn) and λ(µn) can be continued to differentiable functions C(µ) and λ(µ). Integrating Eq.(A.55) leads to Z µ − Z µ − n 1 1 dC n 1 2 dµ > − dµ (A.56) µn C dµ µn µ which implies the following constraint on µn: " # 1 µ − C(µ ) 2 n 1 > n . (A.57) µn C(µn−1) Note that, µn satisfying this constraint always exists. Once we have a satisfactory assignment of µ to Kµ, the running behavior of λ(µ) is constrained by 2 1 dλ 1 dC − < < . (A.58) µ λ dµ C dµ In addition, Eqs.(A.50) and (A.17) requires δ(µ) to satisfy λ(µ)−1/2 δ(µ) ≤ L2λ(µ)2µ4 (A.59) −2 −2 1/2 2 where Lγ `P is the bound of δ(µ) /a(µ) . The existence of a satisfactory δ(µ) requires that λ(µ)5/2 µ−4. (A.60) which is another constraint for the flow λ(µ). A flow λ(µ) satisfying both constraints Eqs.(A.58) and (A.60) always exists. The fol- lowing provides a satisfactory example of λ(µ). Consider the ansatz: λ(µ) = λ(1) µ−2+u, (A.61) 110 where λ(1) is the initial value of λ(µ) at µ = 1. Eq.(A.58) implies 1 dC 2 − u u > 0, > − , ∀1 ≤ s ≤ m + 1. (A.62) C dµ µ The second inequality certainly can be satisfied by a suitable assignment of µ to Kµ, by 2 2−u a similar derivation showing Eq.(A.55) can be satisfied (replacing µ by µ ). It doesn’t restrict the value of u. But combining (A.60), we obtain an upper bound of u: 2 0 < u < . (A.63) 5 If u is within the above range then we obtain a satisfactory flow λ(µ) = λ(1)µ−2+u, which q u/2 2 −1/2 1−u/2 ≤ 2 2u implies a(µ) = µ γλ(1)`P, and λ(1) µ δ(µ) L µ . This example illustrates that flows λ(µ), a(µ), δ(µ), which satisfy Eqs.(A.48)-(A.50) always exist. So the SCL of SFM is well-defined. A.5 CONVERGENCE TO SMOOTH GEOMETRY The equations of motion from SFM’s contain the Regge equation X ∂α f (µ) ε (µ) = 0. (A.64) ∂` f f In the SCL the lattice spacing a(µ) goes to zero with µ → 0. Therefore the behavior of SFM critical points in the SCL is closely related to the convergences of solutions to Regge equation in the continuum limit a → 0. The latter has been studied in [16, 15] for the linearized theory on a flat background. In the following, we review the results in [16, 15] and apply them to our case. The following discussion often suppresses the label µ but uses the lattice space a to label the continuum limit. Regge’s equation can be written as a set of linear equations of ε f for small perturbations on a flat background, i.e. X ε f cot (ϑ`) = 0, (A.65) f,`⊂ f 111 where ϑ` is the internal angle of the triangle f opposite to the edge ` and evaluated on the flat background `˚. In addition to Eq.(A.65) the deficit angles ε f should satisfy the (linearized) Bianchi identity X h dei ε f U[bcma]U = 0 (A.66) f f,`⊂ f where m is the outward-normal of ` in the plane of f and Uab is an antisymmetric tensor associate to f that is given by Uab = vawb − vbwa. (A.67) The unit vectors v, w are mutually orthogonal and orthogonal to f . Note that the Bianchi identity is satisfied automatically, if one uses edge-length vari- ables to describe the system. However, here it is more convenient to use deficit angles as the system variables. In this case, the Bianchi identity is an additional constraint. This formulation of linearized Regge equation using deficit angles is equivalent to the one using edge lengths, because a set of linearized deficit angles satisfying the Bianchi identities can construct a linearized (piecewise-flat) metric, unique up to linearized diffeomorphisms (4 zero modes mentioned in Section A.3). [23]. Conversely, from the linearized metric, one can construct the linearized deficit angles. Given the periodic nature of the triangulation K, we consider the periodic configuration of ε f with the shift ωi(a) along the body principles of a hypercube. The shift relates ε f and ε f 0 for parallel f in two neighboring hypercubes by 0 ε f = ωi(a) ε f . (A.68) Here i = 1, 2, 4, 8 label the 4 body principles of the hypercube. Eq. (A.68) can be more conveniently written by introducing the short hand notation Ω(a) = ( ω1(a), ω2(a), ω4(a), ω8(a) ) . (A.69) 112 and computing the Fourier transform of Eq.(A.68) on the hypercubic lattice (aZ)4. This yields, Z π a d4k P ε = i i kiniaε , ∈ Z. f (n) 4 e f (k) ni (A.70) − π (2π) a Thus each “plane wave” corresponds to Ω(a) = eik1a, eik2a, eik4a, eik8a and tends to (1, 1, 1, 1) in the limit a → 0. In the following we will assume the same limit behavior, i.e. Ω(a) → (1, 1, 1, 1), and that the derivative Ω0(0) exists at zero for general Ω(a). Note that Ω(a) is complex because the ki ’s are complex in Euclidean signature, as we see in a moment. Due to periodicity, Eqs.(A.65) and (A.66) reduces to a set of linear equations for ε f ’s within a single hypercube. Let this hypercube be denoted by cell(0), then X M [Ω(a)] f 0 f ε f (a) = 0. (A.71) f ⊂cell(0) In the following we consider complex solutions of the above equation and their conver- gence. The physical solutions are the real parts of those solutions, and converge when the complex solutions converge. By selecting a solution ε f (a) of Eq.(A.71) for each a we can generate a sequence of linearized Regge configurations. The convergence of this sequence is closely related to the convergence of the associated discrete Riemann curvature tensors. The discrete curvature is defined as a tensor-valued distribution that maps a smooth function f of compact support to the tensor Rabcd[f] given by X Z f → ε f [UabUcd] f fζ ≡ Rabcd[f]. (A.72) f f Here ζ is the area measure of f and Uab is the bi-vector of the triangle f . One can now show that the sequence of solutions to Eq.(A.71) converges for a → 0 if Rabcd converges as 2 a distribution provided that ε f /a remains bounded [16, 15]. Note that in the SCL defined above the latter condition is automatically satisfied due to the regulator δ and Eq.(A.50). It is more convenient to consider a stronger convergence for the sequence of solutions 2 ε f (a). Namely we require that ε f (a)/a converges for all f as a → 0, which clearly implies the above convergence criterion. 113 In [15] it was shown that for any family of vectors Ω(a), for which Ω(0) = (1, 1, 1, 1) 0 (0) and Ω (0) exist, and any solution ε f of Eq.(A.71) at a finite a0 there exists a sequence of (0) 2 → solutions ε f (a) of Eq.(A.71) such that ε f (a0) = ε f . Moreover, the limit ε f (a)/a as a 0 exists for all f and the discrete curvature tensor Rabcd converges to 0 Rabcd(x) → Wabcd exp −Ω (0) · x , (A.73) where Wabcd is a traceless complex constant tensor, and · is the 4d Euclidean inner product. There are 3 possible cases for different k ≡ Ω0(0) ∈ C4. Case 1: If k , 0 satisfies k·k = 0 then Wabcd spans a 2-dimensional solution space, where the dimension corresponds to the helicity ±2 of gravitons. Note that k has to be complex, otherwise k · k = 0 would imply k = 0. Let U and V denote the real and imaginary part of the tensor W, and m and l the real and imaginary part of k. The real part of Eq.(A.73) is Uabcd exp(−l · x) cos(m · x) +Vabcd exp(−l · x) sin(m · x). (A.74) The appearance of exp(−l · x) is due to the difference between Minkowskian and Euclidean signatures. Case 2: For k , 0 and k · k , 0, the solution space is 1-dimensional and Rabcd converges to zero. Case 3: For k = 0 the vector Ω(a) = (1, 1, 1, 1) is a constant and Rabcd converges to a nonzero constant. The solution space corresponds to the full 10-dimensional space of traceless tensors Wabcd. The geometries in Case 1 are smooth solutions of linearized Einstein equation, as Eu- clidean analog of plane waves. They correspond to the nontrivial low energy excitations from SFM under SCL. Case 2 with Rabcd = 0 doesn’t change the flat background geometry and, thus, correspond to purely gauge fluctuations of the triangulation in the flat geometry. 114 The solutions in Case 3 deserves some further explanation. Although those solutions appear in addition to the “plane wave” geometries Eq.(A.74), they only associate to k = 0. So the set of solutions in case 3 is of measure-zero in the space of all solutions. The space of all solutions in the continuum limit is infinite-dimensional, although the solution space with a fixed k is finite-dimensional. A generic linear combination Z 4 Rabcd(x) = dk δ (k · k) Re Wabcd(k) exp (−k · x) (A.75) C4 is insensitive to the value of Wabcd(0) (solution in Case 3). The above Rabcd(x) is a Euclidean analog of a realistic gravitational wave that is not a purely plane wave but has a distribution Wabcd(k). Among the zero modes mentioned in Section A.3, 4 diffeomorphisms have been taken care in the above analysis because of using deficit angle variables, which leads to ±2 helic- ities. The hyperdiagonal zero mode has the same behavior as in Case 2, i.e. it converges to zero curvature Rabcd = 0 [16]. A.6 SOME TOPOLOGICAL PROPERTIES OF THE TRIANGULATION 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 Figure A.3: A visualization of a triangulated hypercube cell.The vertices of the hypercube are labeled by number from 0 to 15. The binary number of the vertex label is the same as the components of the vector from the origin point to the vertex. The analysis in this paper is based on a fixed type of triangulation K. In this section we 115 collect a couple of useful properties of K. K is adapted to a 4-dimensional hypercube lattice in which each lattice cell is a triangu- lated hypercube (Figure A.3). Each vertex of the hypercube is labelled by a number from 0 to 15. Note that the vertex number written in binary form (n1, n2, n3, n4) with ni = 0, 1 yields the components of the vector from the origin to the vertex. Thus the vertex num- bers define 15 lattice vectors at the origin, which are edges and various diagonals of the hypercube and subdivide the hypercube into 24 4-simplices. The triangulation K is made from the hypercube lattice by simply translating the triangulation from one hypercube to another. In order to simplify the problem, one can consider K as a N4 lattice. Among those hypercubic cells, a hypercube whose lattice components contain 0 or N − 1 lies on the boundary of the lattice. A hypercube whose lattice components do not contain 0 or N − 1 is in the bulk. A single triangulated hypercube has 65 edges, 110 triangles and 24 4-simplices. How- ever, the numbers of the edges and the triangles per bulk cell in the lattice is smaller than those numbers for a single hypercube since triangles and edges are shared by different hy- percube cells. If there are n edges or triangles parallels to each other in a single triangulated hypercube then each of those edges or triangles will be shared by n hypercube cells in the bulk of the lattice. Thus the effective weight of those edges or faces in a cell is 1/n. For example, in a single hypercube the triangle (4, 5, 15) is the only triangle that is paral- lel to (0, 1, 11). One finds that the shift vector between (0, 1, 11) and (4, 5, 15) is (0, 1, 0, 0). In the bulk of the lattice, the triangle (4, 5, 15) of cell with lattice coordinate (t, x − 1, y, z) coincides with the triangle (0, 1, 11) of the cell (t, x, y, z). Similarly, the triangle (0, 1, 11) in the cell (t, x + 1, y, z) coincides with (4, 5, 15) in cell (t, x, y, z). Thus the bulk cell (t, x, y, z) only posses half of the triangle (0, 1, 11) and half of (4, 5, 15). Similar arguments work for all the other faces and edges in the bulk of the lattice K. So in the lattice, each bulk hypercube only posses 15 edges and 50 triangles. Furthermore, we can define a coincide number ψ of a triangle f where ψ = m + 1 if one 116 triangle f coincide with m triangles coming from other cells. The maximum value of ψ( f ) is equal to one plus the number of the triangles that are parallel to f in a single isolated hypercube3. For any triangle f in a bulk cell, ψ( f ) must be equal to its maximum value. But in a boundary cell, not all the triangles have maximum ψ( f ). Those triangles lie in the boundary triangles. In an N4 lattice, the boundary hypercubes contribute 356 + 574(N − 2) + 310(N − 2)2 + 56(N − 2)3 boundary triangles and 80 + 148(N − 2) + 84(N − 2)2 + 14(N − 2)3 boundary edges. So in the bulk, there are 50N4 − (356 + 574(N − 2) + 310(N − 2)2 + 56(N − 2)3) triangles and 15N4 − (80 + 148(N − 2) + 84(N − 2)2 + 14(N − 2)3) edges. When N tends to be large, the ratio between the number of bulk edges and the number of bulk triangles will converge to 3 : 10 . Furthermore one can show that every bulk triangle is shared by an even number of 4- simplices because any triangle within a single triangulated hypercube must be shared by 1,2,4 or 6 4-simplices. Definen ˜( f ) to be the total number of 4-simplices within a hypercube that are sharing the triangle f . We call f of type-1 ifn ˜( f ) = 1, or of type-2 ifn ˜( f ) , 1 respectively. There are 24 type-1 triangles in a single hypercube. Table A.1 lists all of those triangles and the triangles parallel to them. Table A.1: Each column of the table shows 4 triangles that parallel to each other. The triangles appears in the first two lines are type-1 and the triangles in the last two lines are type-2. type-1 (1,5,13) (1,3,7) (1,3,11) (1,9,13) (1,9,11) (2,6,7) (2,3,7) (2,3,11) (2,10,11) (4,5,7) (4,6,7) type-1 (2,6,14) (8,10,14) (4,6,14) (2,10,14) (4,12,14) (8,12,13) (8,9,13) (4,5,13) (4,12,13) (8,9,11) (8,10,11) type-2 (0,4,12) (0,2,6) (5,7,15) (3,11,15) (5,13,15) (10,14,15) (10,11,15) (0,1,9) (0,8,9) (0,1,3) (0,2,3) type-2 (3,7,15) (9,11,15) (0,2,10) (0,8,12) (0,8,10) (0,4,5) (0,1,5) (6,7,15) (6,14,15) (12,13,15) (12,14,15) Obviously some of the triangles are shared by different hypercubes. For those triangles one should add upn ˜( f ) in different hypercubes in order to count how many 4-simplices are sharing the face f . Table A.1 shows that each of the type-1 triangle must be parallel to another type-1 triangle and two type-2 triangles. From this we may conclude: 3The maximum value of ψ( f ) also equals one over the weight of the triangle f . 117 • Any triangle shown in the Table A.1 is shared by 4 hypercubes. In two of those hypercubes, the triangle is type-1 and in the other two hypercubes, it is type-2. • The triangles listed in the same column are shared by the same number of 4-simplices. P Explicitly, the triangle (x, y, z) is shared by f n˜( f ) of 4-simplices, where f stands for all the triangles that are in the column and contain triangle (x, y, z). Moreover, P f n˜( f ) must be even since it can be expressed as 1 + 1 plus two even number. • For the other type-2 triangle in the Table A.1, the number of 4-simplices shared by it should be the sum of 2, 4 or 6, which is also even. Thus in the bulk of K, every triangle is shared by an even number of 4-simplices. 118 APPENDIX B APPENDICES FOR CHAPTER FOUR B.1 COMPLEX FENCHEL-NIELSEN (FN) COORDINATE Consider a Riemann surface S shown in Figure B.1, S may be stretched into a pair 4-holed spheres connected by a cylinder. The stretching leads to the definition of a pair of complex Fenchel-Nielsen (FN) coordinates on M f lat(S, SL(2, C)) the moduli space of (framed) flat SL(2, C) connections on S [101, 39]. Figure B.1: A Riemann surface S is stretched as two 4-holed spheres connected by a cylinder. γx and γy are the meridian and longitude curves of the cylinder, which is useful in defining complex FN coordinate. 0 s0,1, s0,1 and s denote the framing flags associate to the boundaries and cylinder. A framed SL(2, C) flat connection on S is an SL(2, C) flat connection A with a choice of flat section s (called the framing flag) in an associated flag bundle (1) over every boundary components of S (the holes) and (2) over the cylinder connecting the two 4-holed spheres [39, 59]. The flat section s may be viewed as a C2 vector field, defined up a complex rescaling and satisfying the flatness equation (d − A)s = 0. Each s from either a boundary component or the cylinder may be extend to be a flat section on the entire S by the flatness equation, although s from different origin result in different flat sections on S. Let s be the framing flag on the cylinder in Figure B.1. Obviously s at a point p on the cylinder is the the eigenvector of the holonomy H(p) along the meridian curve γx based at 119 p. The eigenvalue x ∈ C of H(p) defines the complex FN length variable, i.e. x 0 p p p −1 H( ) = M( ) M( ) . (B.1) 0 x−1 Here M(p) ∈ SL(2, C) used for diagonalization depends on the base point p of H(p). The first column of M(p) is just s(p) up to a normalization. Although s is defined as the framing flag on the entire cylinder, from the view point of two 4-holed spheres, we are motivated to define two different framing flags by choosing two different initial values s(p0) ands ˜(p1) for the flatness equation. The flat sections developed from s(p0) ands ˜(p1) are denoted by s ands ˜ respectively. The two framing flags s, s˜ on the cylinder come from the two framing flags associated to the two holes of 3-holed spheres connected by the cylinder. We find that s ands ˜ are only different by a rescaling, which is denoted by y ∈ C. Indeed, we denote by G(p1, p0) the holonomy along γy traveling from p0 1 to p1. Then G(p1, p0)s(p0) = ys˜(p1). (B.3) It is clear that the parameter y depends on the choices and normalizations of s(p0) ands ˜(p1). As the variable canonical conjugate to x, the complex FN twist τ is defined in the following way: Consider a longitude curve γy traveling along the cylinder connecting two p p 0 points 0, 1 on two different 4-holed spheres (see Figure B.1). Let s0,1, s0,1 be the framing flags for 2 pairs of boundary components in ∂S, τ is defined by (see e.g. [39])2 0 0 (s0 ∧ s ) (s1 ∧ s)(s ∧ s) τ = 0 1 . (B.4) ∧ 0 ∧ ∧ 0 (s0 s)(s0 s) (s1 s1) 1 It is easy to observe that G(p1, p0)H(p0) = H(p1)G(p1, p0) by the flatness of the connection A. By Eq.(B.1), we have ! x 0 M(p )−1G(p , p )M(p ) 1 1 0 0 0 x−1 ! x 0 = M(p )−1G(p , p )M(p ). (B.2) 0 x−1 1 1 0 0 where the first columns of M(p0), M(p1) are s(p0), s˜(p1). It implies the diagonalization −1 −1 M(p1) G(p1, p0)M(p0) = diag(y, y ). Eq.(B.3) is obtained by restricting the attention to the first column of M(p)’s. 2For convenience, τ defined here is different to the one in [39] by a minus sign 120 where the SL(2, C) invariants s∧ s0 are evaluated at a common point after parallel transport- 0 ∧ 0 ∧ ing s and s . Without loss of generality, we evaluate the first ratio with factors (s0 s0), (s0 0 ∧ p ∧ 0 ∧ ∧ 0 p s), (s0 s) at 0, and evaluate the second ratio with factors (s1 s), (s1 s), (s1 s1) at 1. The evaluation involves both s(p0) and s(p1) at two ends of γy, while the parallel transporta- tion between s(p0) and s(p1) depends on a choice of contour γy connecting p0, p1. Different γy may transform s(p1) → xs(p1). One can show that x, τ are canonical conjugate variables of the holomorphic Atiyah- R ∧ Bott-Goldman symplectic form Ω = S tr [δ1A δ2A], i.e. the reduction of the symplectic from to x, τ gives [64, 70] dτ dx Ω = ∧ + ··· . (B.5) τ x We evaluate the twist variable τ by evaluating the first ratio in Eq.(B.4) at p0, while evaluating the second ratio at p1: (s (p ) ∧ s0 (p )) τ = 0 0 0 0 p ∧ p 0 p ∧ p (s0( 0) s( 0))(s0( 0) s( 0)) p ∧ p p p 0 p ∧ p p p (s1( 1) G( 1, 0)s( 0))(s1( 1) G( 1, 0)s( 0)) p ∧ 0 p (s1( 1) s1( 1)) 0 h (s0(p0) ∧ s (p0)) = y2 0 p ∧ p 0 p ∧ p (s0( 0) s( 0))(s0( 0) s( 0)) 0 (s1(p1) ∧ s˜(p1))(s (p1) ∧ s˜(p1))i 1 . (B.6) p ∧ 0 p (s1( 1) s1( 1)) The quantity in the bracket only depends on the flat connections and framing flags located in the pair of 4-holed spheres. The data of framing flags are assumed a priori. Only the holonomy G(p1, p0) traveling from one 4-holed sphere to the other has a nontrivial intersec- tion with meridian holonomy H, while H essentially has no intersection with any holonomy located in a single 4-holed sphere. Therefore the quantity in the bracket Poisson commutes with x, so we can rewrite the symplectic structure as dy2 dx Ω = ∧ + ··· . (B.7) y2 x We introduce on C2 the Hermitian inner product hs, s0i = s¯1 s01 + s¯2 s02, and normalize s(p0) ands ˜(p1) by hs(p0), s(p0)i = hs˜(p1), s˜(p1)i = 1. Under this normalization, we restrict 121 our attention to SU(2) flat connections as a subspace in M f lat(S, SL(2, C)), and we want to understand what is the restriction of the variables x, y. Firstly, it is obvious that x ∈ U(1). The matrices M(p0), M(p1) ∈ SU(2) are written as s1(p ) −s¯2(p ) 0 0 M(p0) = , 2 1 s (p0)s ¯ (p0) s˜1(p ) −s˜¯2(p ) 1 1 M(p1) = (B.8) 2 1 s˜ (p1) s˜¯ (p1) The flat connection being SU(2) implies G(p1, p0) ∈ SU(2), then implies y ∈ U(1). We have written y = e−iξ/2 in Eq.(4.10). 122 APPENDIX C APPENDICES FOR CHAPTER FIVE C.1 THE AREA OF THE TRIANGLE IN THE GEOMETRY WITH A COSMO- LOGICAL CONSTANT As it is pointed out in [76], all faces of the curved 4-simplex are flatly embedded in the ambient space S3 or H3. In the spherical case this means each of the triangle faces of the 4-simplex is a part of the great 2-sphere of S3 and it is enclosed by the edges which are the great cycles of the 2-sphere. Geometrically this kind of triangle is called a spherical triangle. The area of a spherical triangle is given by the equation D = R2(A + B + C − π) (C.1) where A, B, C are the interior angles of the triangle. The radius of the great 2-sphere can √ be given by the cosmological constant through R = 1/ Λ. By redefining the unit properly we can normalize the radius to 1. By using the spherical sine law and spherical cosine law, one can get an expression of the area in terms of the edge-lengths. In chapter.VIII of the book [149], this is given by the L’Huilier’s theorem r D 1 1 1 1 tan = tan s tan (s − a) tan (s − b) tan (s − c), (C.2) 4R2 2 2 2 2 1 where a, b, c are edge lengths and s = 2 (a + b + c). For the hyperbolic case, the area is given as D = R2(π − A − B − C) (C.3) where A, B, C are still interior angles, R2 is given by R2 = 1/Λ. Similarly after choosing a proper unit to normalize the cosmological constant, the hyperbolic sine law and hyperbolic cosine law can convert the area as p D 1 − cosh2 a − cosh2 b − cosh2 c + 2 cosh a cosh b cosh c tan( ) = , (C.4) 2R2 1 + cosh a + cosh b + cosh c 123 where a, b, c are edge lengths [14]. For a 4-simplex the sign of the determinant of the tetrahedron gram matrix can suffi- ciently show whether it is spherical geometry, hyperbolic geometry or flat geometry [76]. The elements of the gram matrix are the cosine function of the 2d dihedral angles which are given by the boundary data. So from a fixed boundary data, one can uniquely decide which equation among (5.8), (C.2) and (C.4) is needed to construct the metric in terms of the areas as the scheme we discussed in section 5.2. Small perturbation of the boundary data doesn’t change the sign of Λ in each tetrahedron. So the metric operator is essentially defined for a neighborhood of boundary data, consistent with the proposal in section 5.2. C.2 CALCULATION OF THE SADDLE POINT AND THE HESSIAN On the critical point the conditions Re(S ) = 0 and δS = 0 are satisfied. For the action (5.25), Re(S tot) = 0 is equivalent to 1 X j − j j − j − γα ab√ 0ab cd√ 0cd 2 (ab)(cd) (ab)(cd) j0ab j0cd −1 † (C.5) X < Jξab, (G ) zab >< zab, ξba > + 2 j ln ab = 0 ab − − 1 1 1 † 1 † 2 2 (ab) < (Gab ) zab, (Gab ) zab > < zab, zab > The first term in (C.5) is quadratic which has its maximum value 0 at j = j0. The second term in (C.5) is less than or equal to 0 due to the Cauchy-Schwartz inequality. Thus, all terms have to vanish separately, which forces j0ab = jab, Jξab ∝ zab and ξba ∝ zba. Since ξ is normalized this implies iφab iφba e −1 † e Jξab = (G ) zab and ξba = zab . (C.6) | −1 † | ab | | (Gab ) zab zab Combining the two equations yields |zab| ξ = −ei(φab−φba) G Jξ (C.7) ab | −1 † | ab ba (Gab ) zab −1 † −1 − where we used JGab = Gab J and J = J. The condition δS total = 0 stands for the derivatives w.r.t the four types of variables, z, A and j respectively. The derivative w.r.t z 124 is1, −1 † † −1 < Jξab, (Gab ) Jzab > < (GabGab) zab, Jzab > δ(z )S total = 2 jab − jab(1 + iγ) ab −1 † † −1 < Jξab, (G ) zab > < (G Gab) zab, zab > ab ab (C.8) † −1 < Jz , ξ > < Jzab, (G Gab) zab > +¯ 2 j ab ba − j (1 + iγ) ab , ab ab † − < zab, ξba > 1 < (GabGab) zab, zab > which splits into two independent equation for and ¯. Equation (C.6) implies < Jzab, ξba >= † −1 † −1 0, which means that < Jzab, (GabGab) zab >=< (GabGab) zab, Jzab > has to vanish as well. But this implies in turn that −1 † < Jξab, (Gab ) Jzab >= 0, (C.9) in the term proportional to . Again by using that the ξab’s are normalized and by using (C.6), we find that e−iφab e−iφba < (G−1)†z , Jξ >= < z , ξ > | −1 † | ab ab ab | | ab ba (Gab ) zab zab −1 † (C.10) |(G ) zab| −1 ab i(φab−φba) ⇔ < zab, Gab Jξab >= e < zab, ξba > . |zab| Because zab and Jzab are orthogonal, (C.10) and (C.9) combine into −1 † |(G ) zab| i(φab−φba) ab Jξab = e Gabξba. (C.11) |zab| The derivatives with respect to A and A¯ can be calculated by using the technique of [68]. We find: δS tot ih µνρ i h −1 † † i (2)µ |Re S = = F [A(x)]−(1+iγ) jcd < (G ) τiG Jξcd, Jξcd > δ (x) = 0 ∀x ∈ lcd i ( tot 0) νρ c,s0 c,s0 lcd δAµ(x) 16π (C.12) and ¯ δS tot ih µνρ i h −1 † † i (2)µ |Re S = = F¯ [A¯(x)]+(1−iγ) jcd < Jξcd, (G ) τiG Jξcd > δ (x) = 0 ∀x ∈ lcd , i ( tot 0) νρ c,s0 c,s0 lcd δA¯µ(x) 16π (C.13) 1 Since z is an element on a Riemann surface CP1, the variation of z is perpendicular to z itself, i.e. δz = Jz. 125 where the δ-function is defined by Z 1 dlµ (2)µ (3) − δ l (x) = δ (x l(s)) ds. (C.14) 0 ds Equations (C.12) and (C.13) imply that the SL(2, C)-connection is flat on S 3 except on the edges of the Γ5 graph (see Fig. 5.1). This can be related to the closure condition for curved tetrahedra [68]. The last variation with respect to j yields (ab)(cd) ∂S total X γα ( jcd − j0cd) ∂S ∂S Regge = − √ √ − i(γφab − θab) + = −iγφab + = 0. ∂ j 0 ∂ j 0 ∂ j ab crit (cd) j0cd j0ab ab crit ab (C.15) In the last step we used that the action (5.19) evaluated at the critical points reduces to the Regge-action with a cosmological constant plus a non-Regge like phase (see [68] for details). Since the non-Regge like phase only depends on the boundary data the phase θab can be chosen in such a way that it cancels the non-Regge like phase leading to the above result. 126 C.3 HESSIANS The Hessians at the critical point are given by: Hz¯cdz¯e f =0, (C.16) Hze f zcd =0, (C.17) −1 −1 † (iγ − 1) (iγ + 1)Ge f (Ge f ) H =δ(e f )(cd) j − , (C.18) zcdz¯e f e f | |2 | −1 † |2 ze f (Ge f ) ze f − iφe f (1 + iγ) je f e h −1 −1 † † † −1 −1 †i µ H i = (G (G ) τiG Jξe f ) − < Jξe f , Ges τiG Jξe f > (G Jξe f ) δ (x), ze f Aµ(x) | −1 † | e f es0 es0 0 es0 e f le f (Ge f ) ze f (C.19) − −iφe f (1 + iγ) je f e h −1 −1 −1 −1 i µ H i = (G τiG Jξe f )− < Jξe f , Ges τiG Jξe f > (G Jξe f ) δ (x), z¯e f Aµ(x) | −1 † | s0 f es0 0 es0 e f le f (Ge f ) ze f (C.20) − iφe f (1 iγ) je f e h −1 −1 † −1 † † −1 †i µ H ¯i = (G τiG Jξe f ) − < Jξe f , (G ) τiG Jξe f > (G Jξe f ) δ (x), ze f Aµ(x) | −1 † | s0 f es0 es0 es0 e f le f (Ge f ) ze f (C.21) − −iφe f (1 + iγ) je f e h −1 −1 † † −1 † † −1 i µ H ¯i = (G (G ) τiG Jξe f )− < Jξe f , (G ) τiG Jξe f > (G Jξe f ) δ (x), z¯e f Aµ(x) | −1 † | e f es0 es0 es0 es0 e f le f (Ge f ) ze f (C.22) where s0 stands for a point on the edge le f with a coordinate x. Furthermore, −1 † † −1 † † H ¯ j i = (1 + iγ) jcd < ((Ges ) τiGes )Jξe f , ((Ges0 ) τ jG 0 )Jξe f > Aν(y)Aµ(x) 0 0 0 es0 −1 † † −1 † † µ ν − < Jξe f , ((Ges ) τiGes )Jξe f >< Jξe f , ((Ges0 ) τ jG 0 )Jξe f > δ (x)δl (y), 0 0 0 es0 le f e f (C.23) 0 where s stands for a y point on the edge l . The Hessian element H i j and H i j 0 e f Aµ(x1)Aν(x2) A¯µ(x1)A¯ν(x2) are ih µλρ i H j i = (δ j F [A(x)]) Aν(y)Aµ(x) 16π Aν(y) λρ −1 † −1 † † −(1 + iγ) jab < P((Gas ) τ jG(Gs s0 ) τiG 0 )Jξab, Jξab > (C.24) 0 0 0 as0 −1 † † −1 † † µ ν − < (Gas ) τ jGas Jξab, Jξab >< (Gas0 ) τiG 0 Jξab, Jξab > δ (x)δl (y) 0 0 0 as0 lab ab 127 ¯ ih µνρ i H ¯ j ¯i = (δ ¯ j F¯ [A¯(x)]) Aν(y)Aµ(x) 16π Aν(y) νρ h −1 † −1 † † + (1 − iγ) jab < Jξab, P((G ) τi(G 0 ) τ jG )Jξab > as0 s0 s0 as0 −1 † † −1 † † µ ν − < Jξab, (Gas ) τiGas Jξab >< Jξab, ((Gas0 ) τ jG 0 )Jξab > δ (x)δl (y), 0 0 0 as0 lab ab (C.25) where P stands for the path order on the edge lab. Note that the contributions from the i i Chern-Simons term, i.e. δ j F [A(x)] and δ j F¯ [A¯(x)], are degenerated as long as the Aν(y) λρ A¯ν(y) νρ remaining gauge freedom under the local transformation A → Ag(x) = g(x)Ag(x)−1 +gdg(x) is not fixed. Therefore, we have to impose a gauge fixing e.g. by following [13]. C.4 CHERN-SIMONS PROPAGATOR FOR NON-COMPACT GAUGE GROUP In order to get a non-degenerate propagator for the SL(2, C) Chern-Simons action, all gauges need to be fixed. Fortunately, the gauge fixing procedure of Chern-Simons the- ory for a non-compact group was already derived in the1990s by Bar-Natan and Witten [13]. This gauge fixes the maximum compact subgroup of SL(2, C) in order to obtain a positive definite Hermitian inner product. Assume the connection can be expressed as A = A0 + B, where A0 is the critical value, and introduce the following gauge fixing term Z h V = dµ Tr[¯c ∗ D(0) ∗ TB], (C.26) 4π where µ is Riemann measure on the mannifold defining Chern-Simons theory, ∗ is the Hodge star and T is a projector that projects sl(2, C) onto su(2). Moreover,c ¯ is a Lagrange multiplier of the gauge fixing D(0)TB = 0. With this gauge fixing the Chern-Simons action 128 can then be written as 2 Z h h i S = − i W[A] + Tr(− B ∧ D(0)B − iφ ∗ D(0)T ∗ B + c¯ ∗ D(0)T ∗ D(0)c) CS 2 4π 2 Z h¯ h¯ i − i W[A¯] + Tr(− B¯ ∧ D¯ (0)B¯ − iφ˜ ∗ D¯ (0)T ∗ B¯ + c˜¯ ∗ D¯ (0)T ∗ D¯ (0)c˜) 2 4π 2 Z Z h 1 i = − i W[A] − Tr(− B0 ∧ D(0)B0 + 2iφ0 ∗ TD(0) ∗ TB0) + Tr(¯c0D(0) ∗ TD(0)c0) 2 2 2 Z Z h¯ 1 i − i W[A¯] − Tr(− B¯ 0 ∧ D¯ (0)B¯ 0 + 2iφ˜0 ∗ T D¯ (0) ∗ T B¯ 0) + Tr(c˜¯0D¯ (0) ∗ T D¯ (0)c˜0). 2 2 2 (C.27) √ √ √ Here, B0 = B h/4π is an sl(2, C) algebra valued 1-form, c0 = c h/4π,c ˜0 = c˜ h/4π, √ √ √ c¯0 = ∗c¯ h/4π and c˜¯0 = ∗c˜¯ h/4π are sl(2, C) algebra valued functions, and φ0 = Tφ h/4π √ and φ˜0 = Tφ˜ h/4π are sl(2, C) algebra valued 3-forms. Following [13], we can now introduce a positive definite scalar product for the P-forms u and v as Z (u, v) = − Tr(u ∧ ∗Tv). (C.28) Define H := (B0, φ0) ∈ Ω1 ⊕ Ω3 and let H˜ (B¯ 0, φ˜0) denote it’s conjugate then h i 0 0 S CS = − i W[A] + (H, Lˆ −H) − (¯c , ∆ˆ 0c ) 2 2 (C.29) h¯ i 0 0 − i W[A¯] + (H˜ , Lˆ −H˜ ) − (c˜¯ , ∆ˆ c˜ ), 2 2 0 (0) (0) (0) (0) where Lˆ − = (∗TD + D ∗ T)J, ∆ˆ 0 = ∗TD ∗ TD and where J equals −1, if it acts on a function or 3-form, and equals 1, if it acts on a 1-form or a 2-form. Effectively, this promotes H and H˜ to the new variables of our theory. The first component of H are just A or A¯ for H˜ , which means that for those components δHS = 0 or δH˜ S = 0 impose the critical conditions (C.12) and (C.13). The derivative with respect to φ or φ˜ yield the gauge fixing condition D(0) ∗ TB0 = 0 and D(0) ∗ T B¯ 0 = 0. As it is mentioned in (5.14), we have already chosen a gauge on each vertex to absorb ga into the holonomy Gab. 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