3D Holography: from Discretum to Continuum

3D Holography: from discretum to continuum

Bianca Dittrich (Perimeter Institute) V. Bonzom, B. Dittrich, to appear V. Bonzom, B.Dittrich, S. Mizera, A. Riello, w.i.p.

ILQGS Sep 2015

1 Overview

1.Is quantum gravity fundamentally holographic?

2.Continuum: dualities for 3D gravity.

3.Regge calculus.

4.Computing the 3D partition function in Regge calculus. One loop correction. Boundary ﬂuctuations.

2 1 1 S = d3xpgR d2xphK 16⇡G 8⇡G Z Z@ 1 M M = Ar(torus) r=1 = no ↵ dependence (0.217) sol 8⇡G | 4G 1 1 S = d3xpgR d2xphK 16⇡G 8⇡G dt Z Z@ ln det( m2)= 1 d3xK(t,1 x, x)M (0.218)M 0 t = Ar(torus) r=1 = no ↵ dependence (0.217) Z Z sol 8⇡G | 4G 1 K = r dt Is quantum gravity fundamentallyln det( m2)= holographic?1 d3xK(t, x, x) (0.218) i↵ t q = e Z0 (0.219)Z • In the last yearsK =spin1 foams have made heavily use of the Generalized Boundaryr Formulation [Oeckl 00’s +, Rovelli, et al …]

Region( bdry) i↵ (0.220) • Encodes dynamicsA into amplitudes associated to (generalized)q = e space time regions. (0.219)

(0.221) bdry ( ) (0.220) ARegion bdry boundary -should satisfy (the dualized) Wheeler de Witt equation Hilbert space -gives no boundary (Hartle-Hawking) wave function

• This looks ‘holographic’. However in the standard formulation one has the gluing axiom:

‘Gluing axiom’: essential to get more complicated from simpler amplitudes

Amplitude for bigger region = glued from amplitudes for smaller regions This could be / is also called ‘locality axiom’.

35 Gluing / Locality axiom: problems in the discrete

Assumes ‘gluing axiom’: needed to get more complicated from simpler amplitudes

Amplitude for bigger region = glued from amplitudes for smaller regions

Amplitude for more complicated boundary state = Integrating over data glued from amplitudes for less associated to discretization complicated boundary state misses out on most of the continuum data. Even on the However this gluing axiom does not hold in necessary discrete data the discrete: (depending on boundary discretization). Restricting to discretization we never obtain a full resolution of identity for the continuum Hilbert space. (This leads eventually to non-local amplitudes if one wants to represent continuum physics.)

Gluing axiom / Locality axiom does not hold in the discrete. [e.g. Bahr, BD, He NJP 11 ] 4 The consistent boundary formalism [BD12, BD14, BD to appear]

We do not assume “gluing axiom” (as fundamental).

However we need a principle to connect amplitudes for less complicated and more complicated boundary data.

Main Idea: A “discretization” just determines the wave function for a small subset of the degrees of freedom. All other degrees of freedom are put into the simplest state = vacuum state.

Generalization of inductive Hilbert space techniques used in LQG. [Ashtekar, Isham, Lewandowski, …]

Main Challenge: Be consistent - observables should not depend on choice of discrete structure, which is used to compute it.

5 j j

j1 j2 j3 hcurve1 hcurve2 hcurve3 0 (0.195) | i hj1 hj2 hj3 0 (0.195) curve1 curve2 curve3 | i j exp(↵ iE )exp(↵ iE )exp(↵ iE ) 0 (0.196) 3 surf3 2 surf2 1 surf1 | i exp(↵3iEsurf3 )exp(↵2iEsurf2 )exp(↵1iEsurf1 ) 0 (0.196) j1 j2 j3 | i j1 j2 j3 hcurveVol1 hcurve2 hcurve3 0 (0.195) node | i j1 j2 j3 ↵1 ↵2 ↵3 Volnode ↵1 ↵2 ↵3 exp(↵3iEsurf3 )exp(↵2iEsurf2 )exp(↵1iEsurf1 ) 0 (0.196) ( hcurve )=1 | i { } j j j 1 2 3 ( hcurve )=1 Volnode { } ( hloops )= (hloops) ↵1 ↵2 ↵3 { } loops Q ( hloops )= (hloops) { } loopsi out in = out(@outconf 1) exp( S(conf 1)) in(@inconf 1) + ( hcurve )=1 h |P| i Q~ { } i i out(@outconf 2)out exp( inS(conf= 2)) outin((@@outinconfconf 2) 1) + exp(... S(conf(0.197) 1)) in(@inconf 1) + h |P|~ i ~ i ( hloops )= loops (hloops) { } in out out(@outconf 2) exp( S(conf 2)) in(@inconf 2) + ... (0.197) ~ Q i in out out in = out(@out conf= 1) exp( S(conf 1)) in(@inconf 1) + h |P| i P ~ i out(@outPconf 2) exp( S(conf 2)) in(@inconf 2) + ... (0.197) HFock ~ = P in out gµ⌫(t, x), (t, x) P ? ? Fock H H = gµ⌫(t, x), (t, x) P ? ? = (0.198) P H P P P HFock How to express the continuum dynamics [BD NJP 12, BD 14] g (t, x), (t, x) = (0.198) µ⌫ ( )= P P P (0.199) ? ? Avac out h out|P|;i H Boundary Hilbert space Boundary Hilbert space with low complexity with high complexity wave functions wave functions = low com( )(0.198) vac( out)= out (0.200) (0.199) P P P Avac low com A h |P|…;i

embedding of embedding of boundary highboundary com low com vac( out)= out ( high com(0.199)) vac ( low com) (0.201) (0.200) A h Hilbert|P| ;ispaces AvacHilbert spaces A

restricts to low com med com high com … ( low com) vac ( med com)(0.200) ( )(0.202) (0.201) Avac A Avac high com (cylindrical) consistency condition

32 med com( ) (0.202) Avac med com A (complete) family of consistent amplitudes deﬁnes a theory* of quantum gravity.

32 * Corresponds to a complete renormalization trajectory, with scale given by complexity parameter.

32 6 [BD NJP 12, BD 14] The consistent boundary formalism

• We now only refer to boundary: fundamentally holographic formulation?

• Question: What can we say about boundary ‘dynamics’?

• Even in the case of a topological ﬁeld theory, where we may have a chance to compute the boundary amplitudes in general?

• Relation to AdS/CFT , Flat/CFT or dS/CFT?

7 (Further) Motivation

Aim: Provide a concrete model for holography, that connects non-perturbative and / or discrete approaches to the continuum.

Strategy: Work with an explicit realization of 3D (quantum) gravity.

Puzzle: [Carlip] 3D gravity is topological. Degrees of freedom can be captured by discretization, in a discretization independent way. How does on obtain a (conformal) ﬁeld theory from this theory?

Further motivation (and an invitation): [Freidel 08] Using dualities in non-perturbative approaches? Need to develop a strategy: Questions: Can we do ﬁnite boundary? Not AdS? Work with (auxiliary) discretization?

8 Continuum theory: Thermal ﬂat 3D space

9 3D gravity - CFT correspondence

• [86 Brown-Henneaux] central extension of asymptotic charge algebra (AdS) • [Maldacena 97, …] AdS-CFT • in 3D in particular application to BTZ black holes [Strominger 97, Carlip,…]

• [Giombi-Maloney-Yin 08] one-loop partition function reproduces character of Virasoro algebra

• One-loop partition function is exact.

Extension to ﬂat space?

• [06 Barnich-Compere, … ,Barnich et al] central extension of asymptotic charge algebra for ﬂat case (BMS group at null inﬁnity )

• [15 Barnich-Gonzalez-Maloney-Oblak] one-loop partition function reproduces character of BMS group [15 Oblak]

review of continuum computation of one-loop partition function

10 1 1 S = d3xpgR d2xphK 16⇡G 8⇡G Z Z@

1 M M

. 2

3 = Ar(torus) r=1 = no ↵ dependence (0.217) T

. sol 8⇡G | 4G

AdS

)

T @

Thermal AdS ✓ (

⇡ dt

is a 1 1 2 3 Thermal2 2spinning ﬂat space

I The easiest case to start with is the one in which @ = T . ln det( m )= d xK(t, x, x) (0.218) =

M M 0 t @ The simplest manifold with such boundary is Thermal⌧ AdS3. Z Z

Take Euclidean AdS3 ) 1 =0 ✓ K = 2 2 2 2 2 2 r 2 2 2 2 2 2 2 2 ds = cosh ⇢dt + d⇢ +sinh+ ⇢d ds = dx + dx + dx = dt + dr + r d

2 t 1 2 3

with the identiﬁcation (t, ) (t + , + ✓) One-loop partition functions of 3D gravity d ⇠

⇢ periodic identiﬁcation:

As a quotient, this is just

with modulus The boundary

+ r = const = R

: Inverse temperature, 2 ✓ : Angular potential i↵

I I ( q = e (0.219)

(t, r, ) (t + ,r,+ ↵)

⇠

Angular potential

) :

+sinh =0 ⇠

✓ ,

I The resulting space2 is a solid torus with a metric of constant

⇢

, Simone Giombi

( inverse temperature and angular potential: d negative curvature t

3 moduli parameter for the (boundary) torus + r = const = R 2 2

AdS I The boundary @ is a T Region( bdry) (0.220) dt

⇢ M 1 A with modulus ⌧ = (✓ + if ).

2 2⇡

I As a quotient, this is just H3/Z. Want to study partition function:

AdS

= cosh

Inverse temperature

negative curvature

The resulting space is a solid torus with a metric of constant

The simplest manifold with such boundary is Thermal

The easiest case to start with is the one in which 2

: 1 (0.221) I

I Simone Giombi One-loop partition functions of 3D gravity bdry

with the identiﬁcation ds Take Euclidean Z(, ↵)= g exp( SE) [Pictures: Giombi] D Z ~

Thermal This is expected to reproduce: Boundary Hamiltonian and angular momentum H i↵J ZVira =Tre (0.222) 11

35 ⌧I () lattice constant a any length

( b)= Xb ( b ,b ) ( ) (0.211) A D i A 1 i A b2,bi Z

2⇡ k s xk,n = e N · xs,n (0.212) s X

(2) 1 S = S(k, n, n0) rescaled 2N k,n,nX0 S(k, n, n0)=rk,nr k,n (k) (n, n0)+dk,nd k,n (k) (n, n0)+ 0 0 (rk,nd k,n + dk,nr k,n )( (k) + 2a) (n, n0) 0 0 ik↵ rk,nd k,n 2a (n, n0 + 1) (1 + n,0(e 1)) 0 ik↵ dk,nr k,n 2a (n, n0 1) (1 + n ,0(e 1)) 0 0 N (tnr0,n0 + r0,ntn0 )2a (n, n0) (0.213)

(k) =2 2 cos( 2⇡ k) , 2a =2 2 cos( 2⇡ ) (0.214) N N

2a Sdd(k, n, n0)=dk,nd k,n 2a 1 2(n, n0) (n, n0 + 1) (n, n0 1) + 0 (k) ✓ ◆✓ ik↵ ik↵ (n0, 1)(n, 0)(e 1) (n, 1)(n0, 0)(e 1) ◆

2 10 e ik↵ ··· 12 1 0 det 0 ··· 1 =2 2 cos(k↵) (0.215) ··· ··· ··· ··· ··· B eik↵ 0 12C B ··· C ↵ =0 @ A

1 1 1 1 2a 1 = 1 ik↵ ik↵ det(Mred) N 2a (k) (1 e )(1 e ) 2 k (N 1)/2 ✓ ◆ 1 3 1 2  Y S = d xpgR d xphK 16⇡G 8⇡G @ p 1 Thermal AdS1 1 ZM Z M = p4 2a 1 Classical limit: Action evaluatedik↵ ikon↵ solution 2 = Ar(torus) r=1 = no ↵ dependence (0.217) N I The easiest2a (1 case toe start)(1 withe is the) one in which @ = T . sol 8⇡G | 4G 2 k (N 1)/2 M  YThe simplest manifold with such boundary is Thermal AdS . 2 1 1 3 = (1 + (a)) Take Euclidean AdS (0.216) N O 2a (13 eik↵)(1 e ik↵) 2 1 dt 3 2 k 2(N 1)/22 2 2 2 2 ln det( m )= d xK(t, x, x) (0.218) ds = cosh ⇢dt + d⇢ +sinh ⇢d 0 t  Y t Z Z with the identiﬁcation (t, ) (t + , + ✓) K = 1 1 1 ⇠ r S = d3xpgR : Inverse temperatured2x,p✓ hK: Angular potential 16⇡G 8⇡G @ Z Z i↵ 1 M M q = e (0.219) = Ar(torus) r=1I=The resulting spaceno ↵ is adependence solid torus with a metric(0.217) of constant sol 8⇡G | 4G negative curvature

2 I The boundary @ is a T 34 M 1 with modulus ⌧ = 2⇡ (✓ + i).

I As a quotient, this is just H3/Z. There is a factor of 1/2 difference to the result one ‘wishes’ to obtain. This is due to a different boundary term one (can) adopt for a boundary at inﬁnity. [14 Detournay et al] Simone Giombi One-loop partition functions of 3D gravity As for now we do choose a ﬁnite boundary we stick with the standard boundary term. Later comments on the alternative boundary term.

35 4 G. Barnich, H. A. González, A. Maloney, B. Oblak values of the BMS charges. Only those diffeomorphisms which vanish sufficiently quickly at asymptotic infinity – in the sense that they do not change the values of these charges – are regarded as gauge transformations. One can then obtain the classical phase space by applying to the metric (2) the diffeomorphisms which do not vanish sufficiently quickly at infinity. Since there are no local degrees of freedom, the entire phase space of solutions which are continuously connected to(2) 6 is constructed byG. this Barnich, group of non-trivial H. A. González, diffeomorphisms A..Moreprecisely,since Maloney, B. Oblak the transformation law of asymptotically flat metrics under BMS3 is given by the coadjoint representation, the perturbative contributions Spiq are obtained by quan- µν1 µρ n ν1 µν1 Defining g g ρ γ x and using g µ ν1 σ 1 2cos nθ , tizing“ theB coadjointp q orbit of flat spaceB underB the“´ asymptotic´ symmetryp q group. The resulting Hilbert space will8 naturally be a representation of BMS3,andthepartition p p1q 2 dt p 3 µν1 n lnfunction det ∆ a characterm of BMS3. d x?g g Kµν1 t, σ x, γ x ´ ´ “ ż0 t ż p p qq We` therefore expect˘ that thenÿPZ perturbative partition function should be given 3 8 p ńβm by the vacuum BMS3 characterm described3 in [8].2e This1 result2cos is onenθ loop exact: p1q piq Vol R Z r ` p qs. S 0,butS 0 2forπ all i 2.Theone-loopexactnessoftheperturbativen 1 einθ 2 ‰ ““ pě { q`n“1 gravity partition function was already observedÿ in AdS3 |gravity´ [2].| µµ1 Theνν1 above argument, while appealing, has the structure of theBMSgroupbuilt Using now g g µ ν1 σ ν µ1 σ 1 2cos 2nθ , in from theB B beginning.B B It“ would` be preferablep q to verify this expectation from a direct gravitational computation,8 as this would give a check of the BMS structure of flat pp2q p 2 dt 3 µµ1 νν1 n ln det ∆space gravitym at the quantumd level.x?g Weg willg nowK describeµν,µ1ν1 thet, σ directx, γ computationx of ´ ´ “ ż0 t ż p p qq ` Sp1q,leavinghigherloopcorrectionsforfuturework.˘ nÿPZ 5 3 8p 2 pńβm 1 2cos 2cos 2 m R3 Z e nθ [AdS: 08n θGiombi-Maloney-Yin ] OneVol loop correctionr ` p q`2 p qs. “ 6π p { q` n 1 einθ [15 Barnich-Gonzalez-Maloney-Oblak] 3 Evaluation of the determinantsnÿ“1 | ´ | • although topological theory, one loop contribution might be non-vanishing, as graviton loops We follow closelymight [3], to not which be completely we refer for canceled more details. by ghost The loopsone-loop contribution Putting everythingto the partition together function according is given to by (6), the one-loop correction to the classical

Euclidean action is 1 1 1 2 1 11 0 Sp qS = ln det ∆p q dln3x detpgR∆p q ln det d∆2pxqp. hK (6) “´2 16⇡G ` ´8⇡2G @ 8 ZM 8 2Z M 2 ń2βm 1 0 inθ ´ inθ p1q Here2e∆p q, ∆pcosq, ∆p2qnareθ thecos kineticn1 θ operatorsm“0 which1 arisee in the linearizede expansion S r p q´= Laplaciansp Ar(torus)qs for spin r2,=1 spin= 1 and spin 0no ↵ dependence . (0.217) n 1 einθsol2 8⇡G | n 14G einθ 1 eínθ “ nof1 general relativity around the flat backgroundÝÑ n 1 metricˆ (2), including` ghosts.˙ They ÿ“ | ´ | ÿ“ 1 ´ ´ 1 are the Laplacian operators for a massless,S traceless= symmetricd tensor,3xpgR a vector, and d2xphK The two divergentascalar,respectively(seealsoe.g.[11]). sums can be made convergent2 by replacing161⇡dtG θ3 by θ iϵin8 the⇡G first@ ln det( m )= ZMd xK(t, x, x) Z M (0.218) 1 i θ itϵ ` sum, and by θ Theiϵ determinantsin the second are evaluated one. Defining using theq heate kernel0p1` approaq givesch, which is straight- = Z Ar(torus)Z r=1 = no ↵ dependence (0.217) ´ sol “8⇡G | 4G forward in flat space. The heat kernels K, Kµµ1 and Kµν,µ1ν1 are defined to be • evaluate8 determinants2n by heat kernel2n approach (regulate with mass term) solutions to the di1fferential equations1 q q¯ S•puseq method of images to get heat kernel, on quotient space from heat kernels on R^3 p0qn 12 qn 1 1 q¯n (summation“ n∆1 overˆm imagest K òft; thex, x action˙0, of the group, gives the sumdt over k below) pÿ“ ´ ´B´ q p ´q“ 2 1 3 p1q 2 ln1 det( m )= d xK(t, x, x) (0.218) ∆ m t Kµµ1 t;x, x 0, t p ´ ´Bq p q“ 0 or equivalently, p2q 2 1 Z Z Summing∆ the threem contributionst Kµν,µ1ν1 t ;together,x, x 0and, taking mass to zero: p ´ ´Bq p q“ 8 p1q 1 eS . with q = ei↵ Depends only on (0.219) 1 qk 2 angular potential. “ k“2 ź | ´ | 13 From (3) we can then read off the 1-loop partition function around flat space:

8 1 ´ Sp0q`Sp1q`Op!q β 1 Z β, θ e ! e 8G! 1 O ! . r s“ “ 1 qk 2 p ` p qq kź“2 | ´ |

This expression matches the vacuum BMS3 character, as computed in [8], for a Euclidean time translation by β and central charge c2 3 G.Itcanalsobeobtained “ { as the ﬂat limit of the one-loop partition function in AdS3 [3].

1This analytic continuation is quite natural considering thefactthatangularpotentialswhich are real in Lorentzian signature become imaginary in Euclidean signature. The partition function Zrβ, θs is most naturally viewed as a function of complex angular potential.

35 Remarks

• the calculation does not make use of the topological nature of the theory

• Can we change that? • Understand better contributing degrees of freedom and structure of the result? (Why k >1?)

• ﬁnite boundaries?

• address more directly the puzzle: how to get ﬁeld theory (on boundary) from topological theory?

14 Regge calculus

(at least in 3D) semi-classical limit of spin foams

Linearized Regge calculus:

[Roceck, Williams 81] [BD, Freidel, Speziale 07 : in the context of GBF and graviton propagator]

15 Regge calculusSet up: Regge calculus[61 Regge] classical version of the Ponzano Regge model [68 Ponzano, Regge] (Regge in semi-class. limit of spinfoam models, [Conrady, Hellmann, Gomes Talks]) •approximate space time by piecewise flat triangulation length variables on edges fix geometry •• discretization of general relativity discrete action defines dynamics •• use an (arbitrary) triangulation • assign length variables to edges of the triangulation: defines (piecewise flat) geometry S = dDx⇥g 1 R cont 2 ⇤ ⇥

• dynamics defined by the (here 3D) Regge action: S = F V discr h h hinges h simplices deficit angle tetrahedron SRegge = ledge✏edge ledge!edge ✏ =2⇡ ✓ 4d:triangles edge edge 3d: edges tetrahedra bulkXvolume edge of deficit angle volumebdryX of edge X triangle/edge 4-simplex/ dihedral angle tetrahedron

6 boundary 1 S = d3xpgR d2xphK Eucl 2 Z Z tetrahedron !edge = ⇡ ✓edge 16 tetrahedraX Regge calculus: Hamilton-Jacobi functions

S = l ✏ l ! Regge edge edge edge edge bulk edge bdry edge X X continuum analogue:

Schlaeﬂi identity for tetrahedron contraction of the ledge✓ =0 tetrahedron: edge variation of Ricci tensor e X gives total divergence

Variation of action: ! EOM: SRegge = ✏edge ledge =0 3D gravity is ﬂat.

SRegge sol = ledge!edge | bdryX edge Hamilton-Jacobi function (of continuum GR) for piecewise ﬂat boundary data. Independent of choice of bulk triangulation. (Together with local form of action shows topological nature of the theory.) 17 Using this result for the 3 2move(3.24),weidentify: − (3) ∂ω01 W0 = H(01),(01) = (3.28) − ∂l01 (2) ∂ωb ∂ω01 ∂l01 (U0)b,b′ = + (3.29) − ∂lb′ ∂lb ∂lb′

(2) Hb,b′

! ∂ω"# 01$ (V )(01),b = . (3.30) − ∂lb Furthermore requiring form invariance of (3.24) implies

1 (2) P3 2 exp H(ij)(km) λij λkm (3.31) − ∝ ⎧− 2 ⎫ [reviews by Hamber, Williams] (ij)=(01),(km)=(01) Quantum⎨ ̸ (Regge̸ calculus ⎬ such that in order to show that (3.24)⎩ is invariant (on the lev1 el of the action),⎭ one has to show Z = µ(l)exp( S (l)) that U = H(2),whichimplies Regge D Regge Z ~ • ‘heated’ debate on the correct1 measure [Lund-Regge, Hamber-Williams, Menotti et al, …] ∂ω01 ∂l01 ∂ω01 − ∂ω01 ∂ω01 In the initial configuration with four tetrahedra, there are four bulk edges, and hence four equations of motion. These, again require that the (four) bulk deficit angles have to vanish, i.e. = that. the complex has to be flat. We know that we can easily const(3.32)ruct such solutions by placing − ∂lb ∂lb′ ∂l01 ∂lb ∂lb′ avertexintothe(flat)tetrahedron(1234)anddeterminingthe lengths of the four bulk edges. , - There is, of course, a three–dimensional parameter space of where to place the inner vertex • [11 BD, Steinhaus ]: Measure should be chosen suchexactly, so hence that the solutions (at areleast not uniquely the determined. linearized) This is the welltheory known gauge is freedom in Regge calculus on flat solutions [43, 44, 45, 40, 10], a discrete remnant of the diffeomorphism Note that we have already provenindependent that of (3.32) choice holds of (bulk) due triangulation. to the identitysymmetry in the (3.12). continuum. From This this it follows shows that of the four form equations of motions only one is independent and that we have to expect three null modes in the Hessian matrix of the system, signifying three gauge degrees of freedom. Further discussions and extensions to the case with invariance of the action. cosmological constant can be found in [10, 15, 19]. For the invariance of the measure µ(l)in(3.24)weexaminethecontributionfromtheGaus-4 4

0 sian integral: 3 3 1 √ √ √ √ 1 2π 2π 2π 12π V2¯V3¯V4¯ 2 = = = 2 (3.33) • [11 BD, Steinhaus ]: Key point: Figure 2: 4 1move.Thetetrahedronissplitintofourbyplacingoneadditional vertex inside (3) ∂ω01 l01 V− ¯V¯ det(W0) H the0 tetrahedron0 and1 connecting it to the remaining vertices in the boundary giving four internal The (3D) Hessian(01),(01) for Pachner− ∂ movel01 configurationsedges (dashed). (local changes of the triangulation) . factorizes/ over tetrahedra/ and edges. Hence, choosing the measure factor as 2.3 Pachner moves in 4D This allows to construct a triangulation invariant2.3.1 measure!4 2 move − This Pachner move is very similar to the 3 2movein3D,seeFig.3.Theinitialconfiguration − le is one with four• 4–simplices,function (01234), of background (01235), (01245) and (flat)(01345), solution sharing one edge (01). All e the other edges are boundary edges. Removing this edge and introducing a tetrahedron (2345), √12π we obtain a configuration with two 4-simplices (02345) and (12345) sharing this tetrahedron. µ(l)= As there is• onlymatches one bulk edge exactly we again haveasymptotics only one(3.34) equation of ofmotionfortheinitialPonzano Regge √V configuration. A (flat) solution can always be constructed in the following way: The boundary 1τ τ triangulation is the same as for two 4–simplices sharing one tetrahedron. Such a configuration can always (i.e. foramplitude all boundary edgefor lengths one satisfyingtetrahedron theappropriateinequalities)beem- bedded into flat 4D space. We can hence straightforwardly determine the distance between the 1 vertices 0 and 1 in the induced metric, which defines the lengthoftheedge(01). we obtain a partition function, invariant under 3 2Pachnermoves.Here18 In some exceptional casese theredenotes might be also solutions the with edgescurvature [15], however this − seems to be rather a discretization artifact. For the perturbative solutions around flat space and τ the tetrahedra in the triangulation. we are interested in, we can note that the linearized equations of motion have a unique (flat) solution for all (linearized) boundary perturbations.

2.3.2 5 1 move − The 5 1isagainanalogoustothe4 1movein3D.Intheinitialconfigurationfive4–simplices − − 3.5 4 1 move share one vertex (0) which is adjacent to five bulk edges. Removing this vertex and the adjacent − edges we are left with just one simplex (12345), see also Fig. 4

For 3D gravity, in addition to the 3 2move,wehavealsotoconsiderthe4 1move.This8 − − move amounts to the subdivision of one tetrahedron, denoted by (1234), into four by adding one additional vertex (0), placing it inside the original tetrahedron and connecting it with all of the remaining vertices, see section 2.2.2. In contrast to the 3 2move,theedgelengthsofthenewedges,i.e.thepositionofthe new − vertex inside the original tetrahedron, is not uniquely ﬁxed. In fact the action is invariant under translations of the vertex (0) inside the tetrahedron (1234), such that one expects the Hessian matrix to have three null eigenvectors. In order to compute this matrix, terms of the form ∂ωe ∂le′ have to be evaluated just as in the 3 2move.Followingasimilarderivationasintheprevious − section, one arrives at the following terms:

16 Gauge Symmetries in Regge •vertex translations act on ﬂat solutions ➡solutions are non-unique ➡discrete representation of diffeo. symmetry [Rocek and Williams 81,Diffeomorphism BD, Freidel, Speziale 07] symmetry in Regge calculus ➡Hessian of action evaluated at ﬂat solutions has null eigenvalues

3D vertexRegge EOM translation impose flatness. Given a flat solution with bulk vertices,acting the position on flat of any of these vertices can be changed as long as the geometry remains flat. Thus one expects three gauge modessolution per vertex. In the linearized theory this results in three null modes per vertex. [Roceck-Williams 81]

This ‘vertex translation’ symmetry is the discrete remnant of the continuum diffeomorphism symmetry and is responsible Does this also hold forfor curvedthe topological solutions? nature of the theory.

[Bahr, BD 09]: This diffeomorphism symmetry is broken for 4D Regge solutions with curvature. This has severe repercussions (e.g. 8 triangulation dependence) and necessitates a continuum limit.

19 The solid torus partition function a la Regge

20 Questions to answer

• Can we take a boundary at ﬁnite radius?

• Do we need to take the continuum limit on the boundary? In angular and in time direction?

21 Set-up

• Choose a triangulation of the solid torus. Allow a very ﬁne triangulation on the boundary but choose bulk triangulation as simple as you can think of.

V1 = V2 = V3 =: V

To get ﬂatness (around time-like axis):

2 L ! 2⇡ splits into three a := =1 cos tetrahedra 2R2 N

number of prisms in a time slice

• Evaluation of classical Regge action on solution is straightforward and works with simplest triangulation (even just using three tetrahedra). Reproduces continuum result. No continuum limit (on boundary) necessary.

22 Hessian of the Regge action I

• The one-loop correction is determined by (determinant of) the Hessian of the Regge action @S @✓tetra H = Regge = e ee0 @l @l @l e e0 tetra e0 X

L2 lele0 a = H = M dimension free, only depends on 2 ee0 12V ee0 2R factors (partially) cancelled by measure term

[similar scaling behaviour found in BD, Freidel, Speziale 07 for cuboids]

• Only ratio L/R matters, this is ﬁxed by the number N of prism in one time slice. • Two null vectors corresponding to overall scaling symmetry and scaling in time direction only.

23 Hessian of the Regge action II

"1 + a 1 0 0 1 " a "1 a 0 0 0 0 0 1 "1 + a 0 0 "1 1 " a a 0 0 0 0 0 0 0 "a 0 a 0 0 1 " a 0 0 1 "1 0 0 0 "a 0 a 0 "1 "a "1 0 1 1 " a "1 a 0 "1 1 0 0 "1 0 "1 1 "1 1 " a 0 a 1 "1 0 0 1 1 0 "1 a a 0 0 0 0 0 0 0 "1 0 0 ! 1 $$ In[53]:= 2 0 0 1 " a "1 0 0 0 0 0 0 0 MatrixForm 2 " 0 0 0 "a "1 1 0 0 "1 1 0 1 2 2 " " 0 0 0 "1 0 lel1e0 "1 0 1 1 0 1 L H = M 2 2 2 ee0 ee0 " a = 0 0 1 0 "1 0 0 1 0 0 1 1 2 12V 2 2 2 2R 0 0 "1 1 1 "1 0 0 1 1 1 "1 2 2

M =

2 ("1 + a) 2 0 0 2 (1 " a) "2 2 a 0 0 0 0 0 r1 2 2 ("1 + a) 0 0 "2 2 (1 " a) 2 a 0 0 0 0 0 r2 0 0 "2 a 0 2 a 0 0 2 (1 " a) 0 0 2 "2 r3 0 0 0 "2 a 0 2 a 0 "2 "2 a "2 0 2 r4 prism 2 (1 " a) "2 2 a 0 "2 2 0 0 "2 0 "2 2 d1 M = "2 2 (1 " a) 0 2 a 2 "2 0 0 2 2 0 "2 d2 2 a 2 a 0 0 0 0 0 0 0 "2 0 0 t 0 0 2 (1 " a) "2 0 0 0 0 0 0 1 0 τ1 0 0 0 "2 a "2 2 0 0 "2 "1 0 2 τ2 0 0 0 "2 0 2 "2 0 "1 "1 0 1 λ1 0 0 2 0 "2 0 0 1 0 0 "1 1 λ2 0 0 "2 2 2 "2 0 0 2 1 1 "2 η

24 1 1 S = d3xpgR d2xphK 16⇡G 1 8⇡G 1 Z 3 Z@ 2 p S = 1 M d xpgR M d x hK 16⇡G 8⇡G @ = Ar(torus)Z r=1 = Zno ↵ dependence (0.217) sol 8⇡G 1 M| 4G M = Ar(torus) r=1 = no ↵ dependence (0.217) sol 8⇡G | 4G dt ln det( m2)= 1 d3xK(t, x, x) (0.218) 0 t Z Z1 dt ln det( m2)= d3xK(t, x, x) (0.218) 1 K = 0 t r Z Z 1 K = r q = ei↵ (0.219)

q = ei↵ (0.219)

( ) (0.220) ARegion bdry ( ) (0.220) ARegion bdry bdry (0.221)

bdry (0.221) H i↵J ZVira =Tre (0.222)

H i↵J Hessian of theZVira Regge=Tre action III (0.222) e ple tetra Ztetra exp( le(⇡ ✓ )) (0.223) ⇠ pV e Hessian for the full triangulationQ tetra e • re-sort (bulk) variables by associatingX them to vertices (s=angular step, n=time step) on the bdry e ple tetra • variable transformationZtetra to absorb1 scalingexp( factorsle(⇡ ✓e )) 2 (0.223)1 µ⌫ 2D ⇠ 2p1Vtetra µ⌫i 2⇡ k s 2D S = d xpg( g @µ@⌫ + R) (0.243) x(Sk,= n)=d Qxpg( ge @Nµ@·e x⌫(s,+ n) R) (0.224)(0.243)2 p 2 X Z N s Z • Fourier trafo X 1 2⇡ 2 i N k 2s 2 2 x(k, n)= K2⇡ + KV↵ e+ V 2· ⇡ x(s, n) K(0.224)+ KV + V 1 i N (⌫ 2⇡ k) n i k s x(k, ⌫)= Se↵ epNT · e N · x(s, n) S(0.225)e↵ (0.244) (0.244) p ˆ2 s2 ˆ2 2 ⇠ Tˆ2K2 + Aˆ2V 2 NNT n ⇠ T KX+ A V X

2⇡ ↵ 1 i (⌫ ktaking) n i 2care⇡ k s of twisted ↵ NT 2⇡ · N x(k, ⌫)= 2 1e00 2De · x(s, n) 2 1(0.225)00 2D S0 =pNNdv :=xTp⌫g( g k@0@0 +periodiﬁcation R) in timeS0 = d(0.226)xp(0.245)g( g @0@0 + R) (0.245) n22⇡ 2 Z X Z ↵ v := ⌫2 k shifted2 time frequency (0.226)2 2 2 K + KV + V 2 K + KV + V Se0↵ 2⇡ Se0↵ (0.246) (0.246) ⇠ Aˆ2V 2 ⇠ Aˆ2V 2

(2) 1 1 t (2) 1 1 t S = xeHee xe = (ˆx(k, ⌫)) M˜ (k,S ⌫) =(ˆx( k, x⌫))H x = (0.247)(ˆx(k, ⌫)) M˜ (k, ⌫) (ˆx( k, ⌫)) (0.247) 2 0 0 2 · · e ee0 e0 e,e k,⌫ 2 2 · · X0 X Xe,e0 Xk,⌫ original ﬂuctuation rescaled and Fourier Fourier transform (k + k )(⌫ + ⌫ ) (0.248) variables 35transformed0 0 variables leads to (k + k0)(⌫ + ⌫0) (0.248)

37 37 1 1 S = d3xpgR d2xphK 16⇡G 8⇡G Z Z@ 1 M M = Ar(torus) r=1 = no ↵ dependence (0.217) sol 8⇡G | 4G

dt ln det( m2)= 1 d3xK(t, x, x) (0.218) t Z0 Z 1 1 S = d3xpgR d2xphK 1 16⇡G 8⇡G K = 1 Z 3 1 Z@ 2 p r S = 1 Md xpgR Md x hK 16⇡G 8⇡G @ = Ar(torus)ZM r=1 = Z M no ↵ dependence (0.217) sol 81⇡G | 4G 1 1 = Ar(torus) r=1 = no ↵ dependence (0.217) i3↵ sol2 p8⇡G | 4G S = q1 = ed xp3gR 1 d x 2 hKp (0.219) S =16⇡G d xpgR 8⇡G @ d x hK 2 1 dt 3 16⇡ZGM 8⇡ZGM@ ln det( m )= d xK(t, x, x) (0.218) 1 ZM Z M dtt = Ar(torus)1 = no ↵lndependence det( m2)= Z01 (0.217)Z d3xK(t, x, x) (0.218) r=1 1 sol =8⇡G Ar(torus)| r=1 =4G no ↵ dependence 0 t (0.217) sol 8⇡G K| = r 4G Z Z 1 Region( bdryK) = r (0.220) A 2 1 dt 3 i↵ ln det( m )=2 1 dt d xK3 (t, x, x) q = e (0.218) (0.219) ln det( m )=0 t d xK(t, x, x) i↵ (0.218) Z 0 Zt q = e (0.219) 1 Z Z K = r 1 K = r ( ) (0.220) bdry ARegion (0.221)bdry Region( bdry) (0.220) q = ei↵ A (0.219) q = ei↵ (0.219) bdry (0.221) H i↵J bdry (0.221) ZVira =TrRegione ( bdry) (0.222)(0.220) A Region( bdry) H(0.220)i↵J A ZVira =Tre (0.222) H i↵J ZVira =Tre (0.222) (0.221) bdry pl (0.221) ple bdry e e tetra Z e exp( l (⇡ ✓tetra)) Ztetra p exp((0.223)le(⇡ ✓e )) (0.223) tetra e e ⇠ peVtetrale tetra ⇠ pVtetra Ztetra Q exp( e le(⇡ ✓e )) (0.223) e H i↵J ⇠ p X Q Z =Tre Vtetra e (0.222) ViraXZ =Tr e H i↵J Q X (0.222) Vira 1 i 2⇡ k s x(k, n)= e N · x(s, n) (0.224) p1 i 2⇡ k s x(k, n)= N s e N · x(s, n) (0.224) 1 pl i 2⇡ k s pN X x(k, n)= e eepN · x(s, n) tetra s (0.224) Ztetra e exp(le le(⇡ ✓e tetra)) 2⇡ (0.223)↵ pN 1 Xi (⌫ k) n i 2⇡ k s Ztetra⇠ psVtetra exp( le(⇡ ✓e )) NT 2⇡ (0.223)N e x(k, ⌫)= e 2⇡ ↵ · e · x(s, n) (0.225) XQ⇠ pVtetra 1 i (⌫ k) n i 2⇡ k s Q X e pNNT NT 2⇡ N 2aX= k=1 x(k, ⌫)= n e (0.229)· e · x(s, n) (0.225) pNNT X 2a = k=1 2a =1k=1 2⇡ (0.229) (0.229)n i N k2s⇡ x(k, n)=2⇡ ↵1 e 2⇡i· xk(ss, n) X ↵ (0.224) 1 x(k,i N n)=(⌫ 2⇡ k) n ei Nk s x(s, n) (0.224) x(k, ⌫)= e T pN · e N ·x(s, n) v := ⌫ (0.225)↵ k (0.226) pNs 2· ⇡ 2⇡ pNNT X! s=exp(i k) v := ⌫ k (0.230) (0.226) n 2⇡ 2⇡ Xk N 2⇡ !k =exp(X i1 k!)k =exp(ii 2⇡k)(⌫ ↵ k) n 2⇡ (0.230) (0.230) N NN 2⇡ 2⇡ ↵ i k s p x(k, ⌫)= 1 e Ti (⌫ · ke)n N i 2· ⇡ xk (ss, n) 0 2a N(0.225)k,0 0 x(k, ⌫)= e NT 2⇡ · e N · x(s, np) (0.225) ··· pNNT ˜ 2a0 Nk,0 2apNkk,0 k + 20a(1 !v) pNNnT M2⇡(k, ⌫)= 1 ······ 2⇡ X↵ n 2⇡ 2apN0k,0 k +2ak(1 !v ) k + 2a(1k !v) ! =exp(i !vv) =exp(M˜i (k,v⌫))= 0 (0.231) (0.231) ······1 !v =exp(v :=i⌫ vv) kX NT (0.231) 0 +(0.226)2a(1 ! 1) Hessian of theNT ReggeNT ↵ action 0IV ···k ···v ···k ···1 v2⇡:= ⌫ k↵ @ (0.226) ··· A v :=⌫2⇡ k ··· ···(0.226) ··· ··· (0.227) 2⇡ @ A (0.227) trdtrdtrd(0.232) (0.232) (0.232) 0 2apN 0 0 2apNk,0 k,0 0 1 p 0 2apNk,0 0 ··· k···=2 !k !k (0.228) ˜ 2apN 2a Nk,0 k + 2a(1k +!2a)(1 !v) ··· 1 ˜ M(k, ⌫)=k,0 2apNk,k0 k k1 kv+ 2a(1 !kv)···=2 !k ! (0.228) M(k, ⌫)= ˜ 0 + 2a(1 ! ) ··· k M(k, ⌫0)=0 + 2a(1 k! 1) v 1 k 1··· 0 0 k 0 vk + 2a(1 !v )k k 1 ··· 1 ··· ··· ······ ··· ··· ···@ ··· ··· ··· ··· ··· ··· 2A ···a = k=1 (0.227) (0.229) @ @ A 2a =A(0.227)k=1 (0.227) (0.229) 2⇡ 1 !k =exp(i2⇡ k) (0.230) k =2 !k !k 1 N (0.228) Lattice Laplacian in angular direction k =2 !k ! !k =exp(i k) (0.228) (0.230) k N 2⇡ !v =exp(i2⇡ v) (0.231) 2a = k=1 !v =exp(i NTv(0.229)) (0.231) 2a = k=1 NT (0.229) 35 2⇡ 35 ! =exp(i k2)⇡ 35 (0.230) k ! =exp(Ni k) (0.230) k N • Rather weak coupling between time slices.2⇡ !v =exp(i 2v⇡) (0.231) • Need time coupling to get dependence!v =exp( onN angularTi v) twist variable (0.231) NT • (so should not take limit of vanishing 2 a =here)k =1 (0.229) 35 35 2⇡ ! =exp(i k) (0.230) k N 26

2⇡ !v =exp(i v) (0.231) NT

36 trd36 (0.232)

36 Null vectors: diffeomorphism symmetry

Null vectors (for each time step) correspond to vertex displacements: • involving d(k=0) and t variables: displacement of bulk vertex along axis • involving r(k=+1), d(k=+1) variables: displacement of bulk vertex in radial direction • involving r(k=-1), d(k=-1) variables: displacement of bulk vertex in (some other) radial direction

Only modes k=0,+1,-1 affected by gauge symmetry (3 null vectors per bulk vertex)

We can integrate out the d-variables for k>0.

27 ⌧I () lattice constant a

any length

( b)= Xb ( b ,b ) ( ) (0.211) A D i A 1 i A b2,bi Z

2⇡ k s xk,n = e N · xs,n (0.212) s ⌧I () X lattice constant a ⌧I () (2) 1 lattice constant a S = S(k, n, n0) any length rescaled 2N k,n,n0 2a =anyk=1 length (0.229) X S(k, n, n0)=rk,nr k,n (k) (n, n0)+dk,nd k,n (k) (n, n0)+ 0 0 ( )= X ( ) ( ) (0.211) b bi b1,bi b2,bi (rk,nd k,n + dk,nr k,n )( (k) + 2a) (n, n0) A D A A 0 0 ( )= Z X ( ) ( ) (0.211) ik↵ b bi b1,bi b2,bi A D A A rk,nd k,n0 2a (n, n0 + 1) (1 + n,0(e 1)) 2⇡ Z !k =exp(i k) (0.230) ik↵ dk,nr k,n 2a (n, n0 1) (1 + n ,0(e 1)) N 2⇡ k s 0 0 xk,n = e N · xs,n (0.212) 2⇡ k s N (tnr0,n + r0,ntn )2a (n, n0) (0.213) x = se N x (0.212) 0 0 k,n X · s,n s 2a = k=12⇡ X (0.229) !v =exp(i v) (0.231) NT (2) 1 S = S(k, n, n0) rescaled(2) 21N (k) =2 2 cos( 2⇡ k) , 2a =2 2 cos( 2⇡ ) (0.214) Srescaled = S(k, n, n0) N N 2⇡ 2N k,n,nX0 2a!=k =exp(i k) k,n,n0 (0.229)(0.230) After integratingk=1 N S( k,out n, n0)= d-variablesrk,nrXk,n (k) (n, n0)+dk,nd k,n (k) (n, n0)+ 0 0 S(k, n, n0)=rk,nr k,n (k) (n, n0)+dk,nd k,n (k) (n, n0)+ trd 0 (0.232) 0 (rk,nd k,n0 + dk,nr k,n0 )( (k) + 2a) (n, n0) (rk,nd k,n0 + dk,nr k,n 0 )( (k) + 2a) (n, n0) 2a 2⇡ ik↵ ! =exp(i v) rk,nd k,n0 2a (n,(0.231) n0 S+dd 1)(k,(1 n, + nn,0)=0(ike↵ dk,n1))d k,n0 2a 1 2(n, n0) (n, n0 + 1) (n, n0 1) + v 2⇡ latticerk,nd Laplaciank,n 2a in(n, time n0 + direction 1) (1 + withn,0( eangular twist1)) !k =exp(i k)NT 0 (0.230) ik↵ (k) ik↵ ✓ ◆✓ N dk,nr k,n0 2a (n, n0 1) (1 + n0,0(e 1)) dk,nr k,n0 2a (n, n0 1) (1 + n0,0(e 1)) 2a ik↵ ik↵ ˜ N (t r + r t )2a (n, n ) (0.213) (n0, 1)(n, 0)(e 1) (n, 1)(n0, 0)(e 1) (M 0(l, ⌫)) = 2a (1 ) N (t nr 0,n0+ r 0,nt n)20 a (n,(0.233) n0) 0 (0.213) rr v n 0,n0 0,n n0 2⇡ k ◆ !v =exp(trdi v) (0.231)(0.232) NT

2⇡2⇡ 2⇡ 2⇡ ik↵ • vanishes for k=+1 and k=-1 modes as ( k )) =2=2 2 2 cos( cos( k k) ) and, , 2a2=2a =22 cos(2 cos() ) 2 (0.214)10(0.214) e 2a NN N N ··· trd˜ (0.232) 12 1 0 (M 0(l, ⌫))rr = 2a v(1 ) (0.233) det 0 ··· 1 =2 cos(k↵) (0.215) k • For computation of determinant we need: ··· ··· ··· ··· ··· B eik↵ 0 12C B C 2a ··· NT˜ 1 @ A (M0(l, ⌫))rr = 2a1 v(1 ) (0.233) 2 ! ! =2 k2 cos(↵k) . vanishes(0.234) for ↵ =0 v v ⌫=0 Y independent of number of time steps

NT 1 1 2 ! ! =2 2 cos(↵k) . (0.234) v v • higher⌫=0 order in a terms: Y

N 2 A 2a 1 1 = . (0.235) k 4 2a kY=2 ✓ ◆

36 36 ⌧I () lattice constant a

any length

( b)= Xb ( b ,b ) ( ) (0.211) A D i A 1 i A b2,bi Z

2⇡ k s xk,n = e N · xs,n (0.212) s X

(k) =2 2 cos( 2⇡ k) , 2a =2 2 cos( 2⇡ ) (0.214) N N

2a Sdd(k, n, n0)=dk,nd k,n 2a 1 2(n, n0) (n, n0 + 1) (n, n0 1) + 0 (k) ✓ ◆✓ ik↵ ik↵ (n0, 1)(n, 0)(e 1) (n, 1)(n0, 0)(e 1) ◆

One loop2 correction10 e ik↵ ··· 12 1 0 det 0 ··· 1 =2 2 cos(k↵) (0.215) ··· ··· ··· ··· ··· B eik↵ 0 12C B ··· C @ A ↵ =0 from previous integration Finally: (for N odd) 1 1 1 1 2a 1 = 1 det(M ) N 2a (k) (1 eik↵)(1 e ik↵) red 2 k (N 1)/2 ✓ ◆  Y p 1 1 1 = p4 2a N 2a (1 eik↵)(1 e ik↵) 2 k (N 1)/2  Y 2 1 1 = (1 + (a)) (0.216) N O 2a (1 eik↵)(1 e ik↵) 2 k (N 1)/2  Y factors get absorbed into measure term

29 6 G. Barnich, H. A. González, A. Maloney, B. Oblak

µν1 µρ n ν1 µν1 Defining g g ρ γ x and using g µ ν1 σ 1 2cos nθ , “ B p q B B “´ ´ p q 8 p p1q 2 dt p 3 µν1 n ln det ∆ m d x?g g Kµν1 t, σ x, γ x ´ ´ “ ż0 t ż p p qq ` ˘ nÿPZ 3 8 p ńβm m 3 2e 1 2cos nθ Vol R Z r ` p qs. “ 2π p { q` n 1 einθ 2 nÿ“1 | ´ | µµ1 νν1 Using now g g µ ν1 σ ν µ1 σ 1 2cos 2nθ , B B B B “ ` p q 8 pp2q p 2 dt 3 µµ1 νν1 n ln det ∆ m d x?g g g Kµν,µ1ν1 t, σ x, γ x ´ ´ “ ż0 t ż p p qq ` ˘ nÿPZ 3 8p pńβm 5m 3 2e 1 2cos nθ 2cos 2nθ Vol R Z r ` p q` p qs. “ 6π p { q` n 1 einθ 2 nÿ“1 | ´ |

Putting everything together according to (6), the one-loop correction to the classical Euclidean action is

8 ńβm 8 2inθ ´2inθ 2e cos 2nθ cos nθ 0 1 e e Sp1q m“ . r p inq´θ 2 p qs inθ ínθ “ 1 n 1 e ÝÑ 1 n ˆ1 e ` 1 e ˙ nÿ“ | ´ | nÿ“ 1 ´ ´ 1 S = d3xpgR d2xphK The two divergent sums can be made convergent by replacing16⇡G θ by θ iϵin8 the⇡G first@ ZM Z M 1 ipθ1ìϵq ` sum, and by θ iϵ in the second one. Defining =q e Ar(torus)gives r=1 = no ↵ dependence (0.217) ´ sol “8⇡G | 4G 8 2n 2n p1q 1 q q¯ S n n , “ 1 n ˆ1 q ` 1 q¯ ˙ dt nÿ“ ´ ´ ln det( m2)= 1 d3xK(t, x, x) (0.218) 0 t or equivalently, Features Z Z 8 p1q 1 i↵ eS . with q = e (0.219) “ 1 qk 2 źk“2 | ´ | From (3) we can then read off the 1-loop partition function around flat space: • much simpler than continuum calculation: 8 1 0gauge1 fixing only for k=0,+/-1 modes, no (further) regularization necessary, at finite ´ Sp q`Sp qÒp!q β 1 Z β, θ e ! boundary e 8G! 1 O ! . r s“ “ 1 qk 2 p ` p qq kź“2 | ´ | • identifies contributing degrees of freedom and explains structure of the result This expression matches the vacuum BMS3 character, as computed in [8], for a • need to take continuum limit in angular direction: Euclidean time translation by β and central charge c2 3 G.Itcanalsobeobtained for getting correction term small“ and{ for getting the product over all modes as the flat limit of the one-loop partition function in AdS3 [3].

1 • result for alpha-dependent part is independent of number of time slices! This analytic continuation is quite natural considering thefactthatangularpotentialswhich[free particle in a ‘perfect discretization’: are real in Lorentzian signature become imaginary in Euclidean signature. The partition functionBahr, BD, Steinhaus 11] Zrβ, θs is most naturally viewed as a function of complex angular potential. • Can we see the contributing degrees of freedom in the effective action for the boundary ﬂuctuations?

35 2a = k=1 (0.229)

2⇡ ! =exp(i k) (0.230) k N

2⇡ !v =exp(i v) (0.231) NT

trd (0.232)

2a (M˜ 0(l, ⌫))rr = 2a v(1 ) (0.233) k

NT 1 1 2 ! ! =2 2 cos(↵k) . (0.234) v v ⌫=0 Y

N 2 A 2a 1 1 = . (0.235) k 4 2a BoundarykY=2 Fluctuations:✓ ◆ what to expect

1 bumps= 2 p conformal Se↵ = d x hK mode (0.236) 8⇡G @ Z M

• functional of the 2D boundary geometry (non-uniqueness due to ‘inside vs outside bumps’) [related to Freidel 08] • invariant under boundary diffeomorphism: should only depend on conformal metric mode

• This is harder to evaluate than it seems (using metric variables)! [Skenderis, Solodukhin 99: for adS one should obtain the effective action by integrating out Liouville ﬁeld, heavily use of Fefferham Graham expansion] [Carlip 05: for adS: Liouville ﬁeld arises because it describes position of boundary (using Fefferham-Graham expansion) and describes breaking of 3D diffeomorphism symmetry giving rise to entropy]

[Riello (unpublished notes): for ﬂat case, can consider an auxiliary ﬁeld which describes position of boundary from the central axis]

• Suggest that our r variable is related to (Liouville?) auxiliary ﬁeld. However to show this directly is involved due to (broken) diffeo invariance in angular direction. 31

36 2a = k=1 (0.229)

2⇡ ! =exp(i k) (0.230) k N

1 =2 ! ! (0.229) k k k 2⇡ !v =exp(i v) (0.231) NT

2a = k=1 (0.229) 1 =2 ! ! (0.229) k k k 2a = k=1 (0.230) trd (0.232) 2⇡ !k =exp(i k) (0.230) 2a = k=1 (0.230)2N⇡ ! =exp(i k) (0.231) k N 2⇡ ˜ 2a 2⇡ !v =exp(i (Mv)0(l, ⌫))rr = 2a v(1 )(0.231) (0.233) ! =exp(i k) (0.231)NT k k N 2⇡ !v =exp(i v) (0.232) NT

2⇡ trdN 1 (0.232) !v =exp(i v) (0.232)T NT 1 2 ! ! =2 2 cos(↵k) . (0.234) trd v v (0.233) ⌫=0 ˜ Y 2a trd (M 0(l, ⌫))rr (0.233)= 2a v(1 ) (0.233) k

NA 2 2a (M˜ 0(l, ⌫))rr = 2a v(1 )2a 1 (0.234) 2a NT 1 1k = . (0.235) (M˜ 0(l, ⌫))rr = 2a v(1 ) 1(0.234) k 4 2a 2 !v ! =2k=2 ✓2 cos(↵k) ◆. (0.234) k v Y ⌫=0 N Y1 T N 1 1 T 2 !v !v =2 2 cos(1↵k) . (0.235) 1 NA 2 2 2 !v ! =2 2 cos(↵k) . (0.235)S = d xphK (0.236) Boundary v Fluctuations: ⌫=0 what2a e↵ 1 we get ⌫=0 Y 1 = 8⇡G. @ (0.235) Y k 4 2a Z M kY=2 ✓ ◆ NA 2 NA 2 2a 1 • the effective action2a can be1 ‘easily computed’:1 encoded= in 3x3 matrix. for the three types(0.236) 1 = . 1 (0.236)2 ˆ ˆ k p4 T 2=a T " ,A= A" (0.237) of boundary edgesk 4 2a k=2Se↵ = d x hK (0.236) k=2 ✓ ◆ ✓ 8⇡G ◆@ Y Y Z M • diffeomorphism symmetry in time direction is exactly realized, even on the discrete level 1 2 p 1 Se↵ = d x hK S =T = Tˆ" ,A(0.237)d=2xApˆ"hK (0.237)(0.237) 8⇡G @ e↵ ˆ Z M 8⇡G @ A✏ • diffeomorphism symmetry in angular direction isZ moreM!k complicated=exp i kto recover:=exp(i✏K) , (0.238) R ! T = Tˆ" ,L= Lˆ" (0.238) ˆ Aˆ✏ ˆ 2⇡Tˆ✏ • boundary continuum limit !k =expT = T "i,Lk ==exp(L" i✏K) , (0.238)(0.238) R!v! =exp i v =exp(i✏V ) , (0.239) ! ˆ Lˆ✏ 2⇡T ✏ !v =exp i v =exp(i✏V ) , a =1 cos(✏K) (0.239) (0.240) !k =exp i k =exp(i✏K) , (0.239) R ! Lˆ✏ ! !k =exp i k a =1=exp(cos(i✏K✏K) ,) (0.240)(0.239) 2⇡Tˆ✏ R ! =exp i v =exp(i✏V ) , (0.240)! v • expand in epsilon,! and truncate at lowest (zeroth)2⇡Tˆ✏ order a =1 cos(✏K) !v =exp i v(0.241)=exp(i✏V ) , (0.240) ! • the resulting matrix still breaks angular diffeomorphisms a =1 cos(✏K) (0.241) R2 K2 + KV + V 2 • matrixEigenvalue entries = are rational function in k,v: terms of(0.242) homogeneity degree 0 and 2 Aˆ2 V 2 R2 K2 + KV + V 2 • take only the degree 2 terms: theseEigenvalue dominate = as k,v = 0,1, 2, … (0.242) 36 Aˆ2 V 2 36 36 • the resulting matrix has now two null modes, corresponding to spatial and time diffeos’s 36 32 1 =2 ! ! (0.229) k k k

2a = k=1 (0.230)

2⇡ ! =exp(i k) 2a = k=1 (0.231) (0.229) k N 2a = k=1 (0.229) 2⇡ 2⇡ !k =exp(i k) (0.230) !v =exp(i v) N (0.232) 2⇡ NT !k =exp(i k) (0.230) N 2⇡ !v =exp(i v) (0.231) NT 2⇡ trd (0.233) !v =exp(i v) (0.231) NT 2a = k=1 (0.229) trd (0.232) ˜ 2a 2⇡ trd(M 0(l, ⌫))rr = 2a v(1 (0.232)) !k =exp(i k) (0.234) (0.230) k N 2a = k=1 (0.229) 2a (M˜ 0(l, ⌫)) = 2a (1 ) (0.233) rr v 2⇡ !v =exp(i kv) (0.231) 2⇡ 2a NT ˜ ! =exp(i kN)T 1 (0.230) (M 0(l, ⌫))krr = 2a v(1 ) 1 (0.233) N k 2 !v !v NT=21 2 cos(↵k) . (0.235) 1 trd (0.232) 2⇡ ⌫=0 2 !v ! =2 2 cos(↵k) . (0.234) Y v !v =exp(i v) ⌫=0 (0.231) NT 1 NT Y 2a 1 ˜ 2 !v !v =2 2 cos(N ↵2k) . (0.234)(M 0(l, ⌫))rr = 2a v(1 ) (0.233) A k ⌫=0 2a NA 2 1 Y trd 1 = (0.232)2a. 1 (0.236) k 4 N 12a1 = . (0.235) k=2 ✓ ◆ T k 4 2a Y k=2 ✓ ◆ 1 N 2 Y 2 !v !v =2 2 cos(↵k) . (0.234) A 2a 1 2a (M˜ 0(l, ⌫)) = 2a (1 ) ⌫=0 (0.233) 1 rr = v . Y(0.235) k 4 2a k 1 k=2 ✓ ◆ 2 1 2 Y S = d xSpe↵hK= NA 2 d xphK (0.237) (0.236) e↵ 8⇡G 2a 1 N 1 8⇡G @ @ 1 = . (0.235) T Z M Z M 1 k 4 2a 2 !v 1 ! =2 2 cos(↵k) . (0.234)k=2 ✓ ◆ v 2 p Y ⌫=0 Se↵ = d x hK (0.236) 8⇡G @ ˆ 1ˆ Y Z M T = T " ,A= A" 2 p (0.237) T = Tˆ" ,L= Lˆ" Se↵ = d x hK (0.238) (0.236) NA 2 8⇡G @ 2a 1 Z M 1 = . (0.235) ˆ k ˆ 4 2a k=2T = T " ,A= A" (0.237)T = Tˆ" ,A= Aˆ" (0.237) BoundaryY ✓ ◆ Fluctuations: whatAˆ✏ we get ! =exp i k =exp(i✏K) , (0.238) Lˆ✏ k R 1 2 p ! Se↵ = d x hK!k =exp i k =exp(i✏K(0.236)) , ˆ (0.239) 8⇡G @ R ! ˆ A✏ Aˆ✏Z M !2k⇡T=exp✏ i k =exp(i✏K) , (0.238) !v =exp i v =exp(R i✏V ) , (0.239) !k =exp i k =exp(i✏K) , (0.238) ! R ! 2⇡Tˆ✏ ! T = Tˆ" ,A= Aˆ" (0.237) !v =exp i v =exp(i✏V ) , a =12⇡Tˆ✏cos(✏K) (0.240) (0.240) 2⇡Tˆ✏ ! !v =exp i v =exp(i✏V ) , (0.239) !v =exp i v =exp(i✏V ) , (0.239) ! ! a =1 cos(✏K) (0.241) • after this limiting procedureAˆ✏ there is just one eigenvalue for the normala =1 metriccos( deformation✏K) mode: (0.240) !k =exp i ak =1=exp(cos(i✏✏KK)) , (0.240)(0.238) R ! R2 K2 + KV + V 2 2⇡Tˆ✏ 2 2 Eigenvalue2 = (0.241) R K + KV + V 2 !v =exp i v =exp(Eigenvaluei✏V ) , = (0.239) Aˆ2 V (0.242) ! 2 2 Lˆ V 2 1 µ⌫ 2D S = d xpg( 1g @µ@⌫ + R) (0.243) a =1 cos(✏K) S =(0.240)d2xpg(2 gµ⌫@ @ + 2DR) (0.242) Z 2 µ ⌫ 36 Z R2 K2 + KV + V 2 36 2 2 • consider discretized and linearized Liouville theory and compareK Kthe2 ++ KVresultingKV ++ V 2 effective action Eigenvalue = 2 (0.241)Se↵ (0.243) Aˆ2 V Se↵ ˆ2 2 ˆ2 2 (0.244) ⇠⇠TˆT2KK2 ++ LAˆ2VV 2

2 1 36µ⌫ 2D 2 1 00 2D S = d xpg( g @ @ + R) S0 = (0.242)d xpg( g @ @ + R) (0.244) 2 µ ⌫ 2 0 0 Z 2 1 00 2D 2 Z 1 µ⌫ 2D S0 = d xpg( g @0@0 + R) (0.245) S = d xpg( g @µ@⌫ + R) (0.243) 2 2 2 2 2 2 K + KV + V Z 2 K + KV + V Z Se↵ (0.243)Se0↵ (0.245) ⇠ Tˆ2K2 + Aˆ2V 2 ⇠ Aˆ2V 2 2 2 2 2 2 K + KV + V K + KV + V Se0↵ (0.246) 2 1 00 2D ⇠ ˆ2 2 SS0 =e↵ d xpg( g @ @ + R) (0.244)(0.244) 36L V ⇠ Tˆ2K22 + Lˆ02V 20 Z 2 2 1 1 2 K + KV + V S(2) = x H xMatches= our(ˆx( k,result.⌫))t M˜ (k, ⌫) (ˆx( k, ⌫)) (0.247) 2 Se0↵ 1 00 2D e ee0 (0.245)e0 S0 = d xpg(⇠g @0@0Aˆ2+V2 R) 233 (0.245) 2 · · 2 Xe,e0 Xk,⌫ Z 36 2 2 (k + k0)(⌫ + ⌫0) (0.248) 2 K + KV + V Se0↵ (0.246) ⇠ Lˆ2V 2

1 1 S(2) = x H x = (ˆx(k, ⌫))t M˜ (k, ⌫) (ˆx( k, ⌫)) (0.247) 2 e ee0 e0 2 · · Xe,e0 Xk,⌫

(k + k0)(⌫ + ⌫0) (0.248)

37 Remarks

Include cosmological constant

• The calculation can be easily generalized to dS or AdS background. [wip Bonzom, BD, Mizera, Riello] • One has to use Regge calculus with homogeneously curved tetrahedra, only this will give triangulation independence. [Bahr, BD 09a, Bahr, BD 09b]

Full non-perturbative version?

• The full (non-linear) quantum version of Regge calculus is given by the Ponzano Regge (spin foam) model. [Ponzano, Regge 68] Need to consider BF-theory with boundary. [Bonzom, Smerlak, unpublished notes]

• Using (3D) spherical tetrahedra corresponds to the Turaev Viro model.

• Using (3D) hyperbolic tetrahedra: analytical continuation of Turaev Viro (results)? [eg Geiller, Noui 13] 34 Conclusions

• in 3D discrete models provide elegant method to evaluate partition functions

• provides an explicit model for holography (based on topological ﬁeld theory)

• provides toy model to understand continuum limit in background independent theories in consistent boundary formulation [BD 12,14]

• conﬁrm [Carlip 05 ] interpretation: auxiliary ﬁeld (radial edges) gives rise to entropy

• regaining angular diffeomorphism symmetry is involved: Use different building blocks with one side being curved? [similar to Bahr, BD 09b]

35 Many further directions to explore

• introduce subdivision in radial direction: explicit model for holographic renormalization [Skenderis] / MERA-AdS [Vidal, Swingle, …]

• (BTZ) black holes: identify degrees of freedom responsible for entropy (at the horizon) [Carlip]

• applications to dS?

• Applications to 4D: need to take full continuum limit. • By comparing to corresponding continuum computation, ﬁx the measure for 4D Regge calculus. [14 BD, Kaminski, Steinhaus: Discretization independence implies non-locality in 4D gravity]

• Hamiltonian analysis of charges in (Ponzano-) Regge realization: on ﬁnite boundaries [recent paper for AdS continuum: Andrade, Marolf]

[canonical Regge calculus: and Hamiltonian constraints: 09 Bahr, BD, 12, 13 BD, Hoehn; constraint algebra of 3D constraints in the discrete: 13 BD, Bonzom; quantum Hamiltonians and algebra of quantum Hamiltonians: 09 Bonzom, Freidel; 14 BD, Esterlis ]

36 1 1 S = d3xpgR d2xphK 16⇡G 8⇡G Z Z@ Alternative1 M boundary M term = Ar(torus) r=1 = no ↵ dependence (0.217) sol 8⇡G | 4G

1 1 1 S = d3xpgR d2xphK 16⇡G 2 8⇡G @ ZM Z M (0.218) [14 Detournay, Grumiller, Scholler, Simon] • derive an alternative variational principle for infinite boundary dt ln det( m2)= 1 d3xK(t, x, x) (0.219) • for finite boundary this does however changet the equations of motion Z0 Z 1 K = r • in particular for Regge calculus: -the triangulation we use turns out to be inconsistent (for finite R) with the alternative boundary (due to induced deficit angleq at= horizontalei↵ edges) (0.220) -moreover it gives not the background action one wishes for (due to induced deficit angle at time axis)

( ) (0.221) • can use alternative triangulation:ARegion gluebdry the current triangulation with an outer ring built from prisms and cuboids

• need furthermore insert a ring bdryof wedges (prisms and pyramides alternating)(0.222) to be consistent at ﬁnite radius R

This issue does not arise in adS where the counter boundary term is just a functional of the boundary metric. H i↵J ZVira =Tre 37 (0.223)

pl Z e e exp( l (⇡ ✓tetra)) (0.224) tetra ⇠ p e e Vtetra e Q X

1 i 2⇡ k s x(k, n)= e N · x(s, n) (0.225) p N s X

2⇡ ↵ 1 i (⌫ k) n i 2⇡ k s x(k, ⌫)= e NT 2⇡ · e N · x(s, n) (0.226) p NNT n X

↵ v := ⌫ k (0.227) 2⇡

0 2apNk,0 0 p ··· ˜ 2a Nk,0 k k + 2a(1 !v) M(k, ⌫)= 1 ··· 0 + 2a(1 ! ) 0 k v k ···1 ··· ··· ··· ··· @ A (0.228)