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Yukawa Institute for Theoretical Physics

arXiv:hep-th/9110057v1 21 Oct 1991YITP/U- 91-43October, 19913-dimensional Gravity from the Turaev-Viro InvariantShun’ya Mizoguchi†andTsukasa Tada‡Uji Research CenterYukawa Institute for Theoretical PhysicsKyoto University, Uji 611 JapanABSTRACTWe study the q-deformed su(2) spin network as a 3-dimensional quantum gravity model.We show that in the semiclassical continuum limit the Turaev-Viro invariant obtainedrecently defines naturally regularized path-integral `a la Ponzano-Regge, In which a con-tribution from the cosmological term is effectively included. The regularization depen-dent cosmological constant is found to be 4π2k2 + O(k−4), where q2k = 1.

We also discussthe relation to the Euclidean Chern-Simons-Witten gravity in 3-dimension.†JSPS fellow. e-mail: mizo@jpnrifp.yukawa.kyoto-u.ac.jp‡Soryuushi shogakukai fellow.

e-mail: tada@yisun1.yukawa.kyoto-u.ac.jp

It is well-known that 3-dimensional gravity is perturbatively trivial in the sense thatthere are no local degrees of freedom. In 3-dimension Ricci-flat space-time means trulyflat space-time, so that there are no gravitational wave modes but is only topologicalexcitation.

Topological nature of 3-dimensional gravity has already been encountered inthe old work by Ponzano and Regge on the semi-classical limit of the Racah coefficientsof su(2)[1]. In their seminal paper (and also ref.

[2]) a triangulation-independent quantitywas defined by utilizing a certain relation for 6j symbols, and was shown to be viewedas a path-integral for 3-dimensional quantum gravity in the semi-classical continuumlimit, though their expression diverges and needs some regularization. Recently, Turaevand Viro has constructed a new topological invariant from the q-deformed su(2) spinnetwork when q is a root of unity.

Their construction strongly resembles that of ref. [1],and moreover, the Turaev-Viro (TV) invariant is naturally regularized and finite dueto the restriction for the spin variables.

Therefore the TV invariant is expected to beconsidered as a regularized path-integral for 3-dimensional quantum gravity.In this letter we estimate the asymptotic behavior of the q-6j symbol by the WKBapproximation along the method of ref. [3], and see how the path-integral defined by thesu(2) spin network receives ‘quantum’ corrections from q-deformation.

We show that acontribution from the cosmological term is effectively included in the path-integral fromthe TV invariant.In ref. [1] a sum of products of four (classical) su(2) Racah-Wigner 6j symbols wasconsidered:Xx, l1, l2, l3 :allowed value(2x + 1)(2l1 + 1)(2l2 + 1)(2l3 + 1) (−1)χ×(j1j2j3xl1l2) (j6j5j1l1l2l3) (j4j2j6l2l3x) (j3j5j4l3xl1),(1)where χ = x + P3i=1 li + P6i=1 ji.By repeated application of the Biedenharn-Elliottidentity and the orthogonal relation, (1) is reduced to a single 6j symbol:∞Xx=0(2x + 1)2(j1j2j3j4j5j6).

(2)To give a meaning to the divergent expression (2) a large angular-momentum cut-offRis introduced in the summation, so that we take the equality between (1) and (2) as the1

following renormalized identity:(j1j2j3j4j5j6)=limR→∞3a4R3Xx,l1,l2,l3

Ob-viously, with fixed boundaries, IP R takes the same value in each class of triangulationconnected through the operation (3).2

In the semi-classical continuum limit xp, e (and so f) go to infinity. The 6j symbolbehaves as [1, 3](j1j2j3j4j5j6)≈(112πV )12 cos(fXi=1θiJi + π4 )(5)in the domain where Ji are uniformly large (see Figure Caption).

Here V is the volumeof the tetrahedron. Thus one may write IP R asIP R ∼ZdµYtetrahedracosfXi=1xi(π −θi) −π4(6)withdµ = ( 3a4R3)efYi=1dxi(2xi + 1).

(7)Taking only the positive frequency part of cosine, together with Pfi=1 xi(π−θi) ∼12R √gR[4], one findsIP R (positive frequency part) ∼Zdµ expi12Z √gR. (8)The above consideration is an interesting possibility to relate IP R with 3d quantumgravity, though there are some difficulties, i.e.

the appearance of i in the exponent andthe treatment of the other interference terms.Turaev and Viro defined the quantity ITV (up to the factors from the boundary) as[5]ITV≡w−2eXallowed φdYi=1|T φi |fYi=1w2i ,(9)where |T φi |, wi and w are weight assigned to tetrahedra, edges and vertices, respectively.φ stands for a configuration of edge-lengths, called coloring. φ is an allowed (‘admissible’in ref.

[5]) coloring if any set of triple which forms a face of a tetrahedron satisfies thetriangle inequality, and if the sum of any triple is an integer less than k −2. They provedthat ITV is independent of triangulation if|T φi | = (−1)P6l=1 j(i)l(j(i)1j(i)2j(i)3j(i)4j(i)5j(i)6)qw2i = [2xi + 1]q,w2 = −2k(q −q−1)2,(10)3

where {······} stands for a (restricted) q-6j symbol with q = eπik [6], and [n]q is a q-integer[n]q = qn−q−nq−q−1 . If q is a root of unity in general, some Clebsch-Gordan coefficients of therepresentation of Uq(sl(2)) diverge.

Therefore one must, and can successfully, restrictthe domain of the spin variables to a finite set ({0, 12, 1, . .

. , k −12} in our case).It is surprising that IP R and ITV are in the same form.

Moreover, since(j(i)1j(i)2j(i)3j(i)4j(i)5j(i)6)q=(j(i)1j(i)2j(i)3j(i)4j(i)5j(i)6)+ O(k−2)[2xi + 1]q=2xi + 1 + O(k−2)−2k(q −q−1)2=k32π2(1 + O(k−2)) ,(11)ITV approaches IP R in the k →∞(q →1) limit with a =8π23 , where k plays thesame role as R. In other word, the TV invariant ITV provides a natural regularizedpath-integral for 3-dimensional quantum gravity.Now let us study the asymptotic behavior of q-6j symbols. We start from the fol-lowing relation, which is a special case of quantum analogue of the Biedenharn-Elliotidentity for restricted q-6j symbols [6]§:(j1j2j3j4j5j6)q(j1j2j312j3 + ∆bj2 + ∆a)q=Xζ=j4± 12(−1)j1+2j2+2j3+j4+j5+j6+ζ+ 12+∆a+∆b[2ζ + 1]q×(j1j2 + ∆aj3 + ∆bζj5j6)q(12j2j2 + ∆aj6ζj4)q×(12j3 + ∆bj3j5j4ζ)q∆a,b = ±12.

(12)Combining the equations above, one obtains the following recursion relation:[2j1]qg(j1 + 1)(j1 + 1j2j3j4j5j6)q§ Nomura also discussed identities of q-6j and asymptotic behavior when a part of the arguments islarge [10].4

+2h(j1)(j1j2j3j4j5j6)q+[2j1 + 2]qg(j1)(j1 −1j2j3j4j5j6)q= 0(13)g(j) = {[j2 + j3 + 1 + j]q[j2 + j3 + 1 −j]q×[j + j2 −j3]q[j −j2 + j3]q×[j5 + j6 + 1 + j]q[j5 + j6 + 1 −j]q×[j + j5 −j6]q[j −j5 + j6]q}12(14)h(j) =n[2j + 1]q[2j + 2]q[2j]q×[j3 + j4 + j5 + 2]q[j3 + j5 −j4 + 1]q−[2j]q[j + j5 + j6 + 2]q[j + j5 −j6 + 1]q×[j + j2 + j3 + 2]q[j + j3 −j2 + 1]q−[2j + 2]q[j6 −j5 + j]q[j5 + j6 −j + 1]q×[j2 + j3 −j + 1]q[j −j2 + j3]qo/2 . (15)Since we are considering restricted q-6j symbols, the spin variables are truncated atk.

Therefore, to take large angular momentum limit we have to take k also large. Hencewe substitute j1, j2, .

. .

for λj1, λj2, . .

. and k for λk at the same time, then we sendλ →∞, keeping the ratio ji/k ≪1.

Defining the q-analog of triangle-area:Fq(a, b, c) = 14q[a + b + c]q[a −b + c]q[a + b −c]q[−a + b + c]q ,(16)we can express g(j) in a more compact form:g(j) = 16Fq(j, j2 + 12, j3 + 12)Fq(j, j5 + 12, j6 + 12). (17)Recalling the distinction between J and j (Fig.1), we obtaing(λj)=16qFq(λJ, λJ2, λJ3)Fq(λJ −1, λJ2, λJ3)Fq(λJ, λJ5, λJ6)Fq(λJ −1, λJ5, λJ6)×h1 + O(λ−2)i.

(18)Thus we rewrite the relation (13), keeping terms up to order λ−1, as follows:(Fq(λJ1 + 1, λJ2, λJ3)Fq(λJ1 + 1, λJ5, λJ6)[2(λJ1 + 1)]q) 12 (λj1 + 1λj2λj3λj4λj5λj6)q5

+(Fq(λJ1 −1, λJ2, λJ3)Fq(λJ1 −1, λJ5, λJ6)[2(λJ1 −1)]q) 12 (λj1 −1λj2λj3λj4λj5λj6)q+h(λj1)8 {[2(λJ1+1)]q[2(λJ1−1)]q}1/2{Fq(λJ1, λJ2, λJ3)Fq(λJ1, λJ5, λJ6)}−1/2{[2λJ1]q}1/2×(λj1λj2λj3λj4λj5λj6)q= 0. (19)Introducingϕ(j1)≡(Fq(λJ1, λJ2, λJ3)Fq(λJ1, λJ5, λJ6)[2λJ1]q)1/2 (λj1λj2λj3λj4λj5λj6)q(20)c(J1)≡−h(λj1)16 {[2(λJ1 + 1)]q[2(λJ1 −1)]q}1/2 Fq(λJ1, λJ2, λJ3)Fq(λJ1, λJ5, λJ6),(21)we arrive at a difference equation for j1:n∆2 + 2 −2c(J1)oϕ(j1) = 0.

(22)Let us estimate the solution of the above equation (22) in the large k limit. Whenq →1, c(J1) approaches its ‘classical’ value cos θ1 [3], so one expands c(J1) asc(J1) = cos θ1 + ̺ + O(k−4) ,(23)where the next-leading term ̺ is of order k−2.

Sincecos(θ −̺sin θ)=cos θ cos( ̺sin θ) + sin θ sin( ̺sin θ)=cos θ 1 −̺22 sin2 θ + . .

. !+ sin θ ̺sin θ −̺33!

sin3 θ + . .

. !=cos θ + ̺ + O(k−4) ,(24)(22) becomes in the large k limit as follows:∆2 + 2 −2 cos(θ1 −̺sin θ1)ϕ(x) = 0.

(25)Here we have changed the variable from J1 to x.According to ref. [3], we solve thedifference equation (25) by WKB approximation.

The result isϕ(x) ≈Cqsin(θ1 −̺sin θ1)cosZ(θ1 −̺sin θ1)dx + π4,(26)6

where C is a normalization. Thus we obtain the expression for the asymptotic formulaof q-6j symbol:(j1j2j3j4j5j6)q∼C′√V cos Zθ1(x)dx −Z̺(x)sin θ1dx + π4!.

(27)In the above expression C′ is a quantity that approaches to C =1√12π as q →1 andcould be a function of Ji at k−2 order. The second term inside the cosine correspondsto a regularization counter term, while the spin dependence of C′ may be regarded as acorrection for the measure.

The integral of θ1 in the cosine is proved to be P θiJi [3].Next we estimate ̺/ sin θ in the large k and angular momenta limit. Defining ǫq =π26k2,then[λJ]q =λJ −ǫqλJ3 h1 + O(λ−2)i,(28)so thatc(J1)=−h(λj1)16 {[2(λJ1 + 1)]q[2(λJ1 −1)]q}1/2 Fq(λJ1, λJ2, λJ3)Fq(λJ1, λJ5, λJ6)=−λ5 (hc + ǫqhq) {1 + ǫq (8J21 + 2 (J22 + J23 + J25 + J26))}2λJ1 · 16Fc(λJ1, λJ2, λJ3)Fc(λJ1, λJ5, λJ6)=cos θ1 −ǫqhq + (8J21 + 2 (J22 + J23 + J25 + J26)) hc2J1 · 16Fc(J1, J2, J3)Fc(J1, J5, J6)+ O(λ−2) ,(29)where hc and hq are the following:hc=−2J1(J41 −J21J22 −J21J23 + 2J21J24 −J21J25−J22J25 + J23J25 −J21J26 + J22J26 −J23J26)(30)hq=4J1(4J61 −3J41J22 −J21J42 −3J41J23 −J21J43 + 12J41J24 + 2J21J44 −3J41J25−8J21J22J25 −J42J25 + 2J21J23J25 + J43J25 −J21J45 −J22J45 + J23J45 −3J41J26+2J21J22J26 + J42J26 −8J21J23J26 −J43J26 −J21J46 + J22J46 −J23J46).

(31)Fc(a, b, c) is the ‘classical’ value of Fq(a, b, c), that is, the area of a triangle whose edgesare of length a, b and c.The numerator of the second term of (29) proved to be:8J31 · 16Fc(J4, J2, J6)Fc(J4, J5, J3) cos θ4. (32)7

Using the formula∂V∂J1= −J1 · 16Fc(J4, J2, J6)Fc(J4, J5, J3) cos θ4144V,(33)we obtain̺=−ǫq × 8J31 · 16Fc(J4, J2, J6)Fc(J4, J5, J3) cos θ42J1 · 16Fc(J1, J2, J3)Fc(J1, J5, J6)=24ǫq∂V∂J1sin θ1. (34)Inserting (34) into (27), we conclude thatITV (positive frequency part) ∼Zdµ′ exp i12Z √g R −8π2k2!

!,(35)where dµ′ = dµ + O(k−2).It is a plausible result that the cosmological term appears as a regularization counterterm though our approximation for q-6j symbol is valid only in the region ji ≪k and allthe ji are uniformly large. If the sum-region with respect to edge-lengths is unbounded,any configuration with high deficit angles would be allowed.

In our case the summationis truncated at length k so that the configuration is chopped offif it is spiky enough;consequently, the virtual processes with large volume are suppressed.A crucial question is the following: “what does it mean if you gather only negativefrequency factors to obtain a ‘Lorentzian’ action (i×classical action), though you havebeen considering a Euclidean system?” In 2 + 1-dimension the Einstein action coincideswith the Chern-Simons (CS) action with gauge group G under some appropriate field-identification, where G is the isometry group of space-time, i.e. G = ISO(2, 1), SO(3, 1)and SO(2, 2) if space-time is Minkowski, de-Sitter and anti-de-Sitter, respectively [7].However, in 3-dimension with Euclidean signature the Lagrangian for the SO(4) CStheory, which is supposed to be the Euclidean version for space-time with positive cos-mological constant, is pure imaginary; hence the equivalence to the Einstein gravity issubtle [7].

On the other hand, we started from the TV invariant and interpreted it asthe path-integral in the semi-classical continuum limit. Since amplitude for any processis topologically invariant, so should be the action.

Therefore it would be better to con-sider the Regge action appeared in our model as the CS action, rather than the Einstein8

action. The strange i factor may indicate the subtlety in the correspondence betweenthe Euclidean gravity and a CS theory.Indeed, there is an evidence that the TV invariant is related to a CS theory.Ifone evaluates ITV in a 3-manifold F × [0, 1] where F is a triangulated 2-surface, thenITV naturally induces a representation of the modular group of F [5].

Comparing therepresentation with that of the Jones polynomial, Turaev and Viro conjectured that theirinvariant would be related to the ‘square’ of the Hilbert space of a topological field theoryfor the Jones polynomial, i.e. the SU(2) CS theory with Wilson lines [8].

Recently, ithas been shown in ref. [9] that the TV invariants can be constructed from the SO(4)(= SU(2) × SU(2)) CS theory with Wilson lines in the large k limit.We would like to thank M. Ninomiya for very fruitful discussions and continuousencouragement.

We also thank M. Hayashi, A. Hosoya, S. Iso, S. Sawada, J. Soda andH.

Suzuki for discussions.One of us (T. T.) also thanks A. Kirillov and H. Oogurifor discussions, and appreciate a kind hospitality of RIMS 91 Project.This work issupported in part by JSPS and Soryuushi shogakukai.xllljjj4jjj3212416531θθFig.1. The 6j symbol(j1j2j3j4j5j6)is represented by a tetrahedron whose edges are oflength Ji = ji + 12. θi is the angle between the outer normal of the two faces which havethe edge Ji in common.

The subdivision (3) is also illustrated.9

References[1] G. Ponzano and T. Regge, in:Spectroscopic and group theoretical methods in physics,ed. F. Bloch (North-Holland, Amsterdam, 1968).

[2] B. Hasslacher and M. J. Perry, Phys. Lett.

B103 (1981) 21. [3] K. Schulten and R. G. Gordon, Jour.

Math. Phys.

16 (1975) 1961; ibid 1971. [4] T. Regge, Nuovo Cimento 19 (1961) 551.

[5] V. G. Turaev and O. Y. Viro, LOMI preprint. [6] A. N. Kirillov and N. Yu.

Reshetikhin, in Adv. Ser.

in Math. Phys, ed.

G. Kac, vol.7 (1988). [7] E. Witten, Nucl.

Phys. B311 (1988) 46.

[8] E. Witten, Commun. Math.

Phys.121(1989)351. [9] H. Ooguri and N. Sasakura, KUNS-1088, August 1991.

[10] M. Nomura, J. Math.

Phys. 30 (1989) 2397.10

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