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THREE MANIFOLDS AND GRAPH INVARIANTS

arXiv:hep-th/9112071v1 24 Dec 1991TIFR/TH/91–59December 1991THREE MANIFOLDS AND GRAPH INVARIANTSS. KALYANA RAMA⋆AND SIDDHARTHA SEN† ‡Tata Institute of Fundamental Research,Homi Bhabha Road, Bombay 400 005, INDIAABSTRACTWe show how the Turaev–Viro invariant can be understood within the frame-work of Chern–Simons theory with gauge group SU(2).

We also describe a newinvariant for certain class of graphs by interpreting the triangulation of a manifoldas a graph consisiting of crossings and vertices with three lines. We further show,for S3 and RP 3, that the Turaev-Viro invariant is the square of the absolute valueof their respective partition functions in SU(2) Chern–Simons theory and give amethod of evaluating the later in a closed form for lens spaces Lp,1.⋆e-mail : kalyan@tifrvax.bitnet† e-mail : sen@maths.tcd.ie‡ Permanent Address : School of Mathematics, University of Dublin, Trinity College, Dublin2, IRELAND

PrefitemFollowing the discovery of the Jones invariants in knot theory there has beenrenewed interest in trying to find new invariants associated with knots, graphsand 3–manifolds. A significant step was taken by Witten when he was able tointerpret the Jones invariants by using ideas from quantum field theory [1] .

Re-cently Turaev and Viro introduced [2] a new invariant of M3 (Md denotes a d–manifold). This invariant is combinatorial in nature and is defined as a state sumcomputed on a triangulation of the manifold and is based on the quantum 6–jsymbols associated with the quantised universal enveloping algebra, Uq(SL(2, C)).Here q = exp(i 2πk+2) , k > 0 is a complex root of unity.

The Turaev–Viro (TV)invariant is also of interest for the following reason. If one regards ji’s of the clas-sical 6–j symbol as the lengths of the sides (colors) of a tetrahedron, then in thelarge j limit the positive frequency part of the 6–j symbol becomes eiSR wherethe Regge action SR is the discretised version of the Euclidean Einstein–HilbertactionRd3x√gR for 3–d gravity [3] .

However, the sum over the coloring of thetetrahedra is divergent in the classical case but the quantum 6–j symbols providea natural cutoffj ≤k2 and regulate the divergent behaviour [4] [5] . Thus the TVinvariant is seen to be closely related to the partition function of the 3-d gravityaction.A few remarks on the relation between the TV invariant IT V and other knowninvariants of M3 are also in order.

For orientable M3 of the type M2 × [0, 1] whereM2 is closed, the authors of [4] mention that they have examined the partitionfunctions of the TV model for lower genus M2 and found them to be equal tothe absolute–value–square of the partition function of the SU(2) Chern–Simonstheory. They also mention that the equality of these two partition functions hasbeen proved by Turaev in a rather different approach.In this paper we will show how the TV invariant can be understood withinthe framework of Chern–Simons (CS) theory in which the gauge group is SU(2).We evaluate the TV invariant for the manifolds S3 and RP 3 and show that for2

these manifolds IT V (M) = I2W(M), where IW(M) is the invariant for the manifoldM obtained by calculating the absolute value of the partition function for a CSgauge theory with gauge group SU(2). We will also attempt to understand theTV invariant in terms of graphs and describe how one obtains a new invariant forcertain class of graphs using these ideas.

In our discussions we will always take M3to be a compact orientable 3–manifold with no boundary.We start by assuming that a given M3 has been triangulated and we considerthe graph G associated with the triangulation.For example if two tetrahedraT1, T2 are glued together along the face F as shown in figure 1 then we regardT1, T2 as graphs rather than as 3–dimensional objects. The important observationwe make regarding G is that G is not an arbitrary graph; it is obtained by gluingtogether a collection of tetrahedra {Ti}, represented symbolically asG =[{gij}Ti(1)where {gij} encodes gluing information.

Because of this if we want to evaluate theinvariant associated with G by evaluating G with respect to a CS measure we haveto specify what exactly the gluing process means within this framework. First ob-serve that a given tetrahedron can be regarded as a collection of Wilson lines, joinedtogether at each vertex by an appropriate invariant coupling factors as discussedby Witten [6] .

Each line may carry a different representation of SU(2). Gluingtwo tetrahedra along face F we will take to mean that the faces are connected by“walls” all carrying the trivial representation of SU(2) and the representations onthe common glued face are summed over.

By repeatedly using the factorisationtechnique of Witten [1] [6] , and observing that if a given tetrahedron is enclosedin a ball with surface S2 which intersects these trivial representation–walls thenthe Hilbert space of S2 is one dimensional, it is easy to see thatI(G) =XrepYiTi(2)where I(G) is the graph invariant and Ti is an object associated with the tetrahe-3

dron Ti defined by its six sides each of which carries a representation of SU(2). Atthis point we would like to point out the factorised nature of the invariant I(G)over the constituent tetrahedra Ti.

We will see shortly that there is a solution forTi, an object associated with the tetrahedron Ti which respects the factorisableproperty. However this solution will not have the required tetrahedral symmetryand will lead to a modified prescription for the graph invariant.

Since G does nothave any boundary each “face” of a Ti in G is glued to some other face of a Tj in G.Hence all the representations appearing in Qi Ti in equation (2) are to be summedover. Note that the graph G is essentially the spine of a triangulated manifold [2]and hence is inherently 3–dimensional.

In particular the lines in G never cross andan arbitrary number (≥3) of them can meet at a vertex.We will now attempt to understand the TV invariant in terms of planar graphswith crossings and vertices with fixed number of lines, and their invariants. Asbefore we consider the triangulation of M3 as a set of tetrahedra and “gluinginstructions”.

The set of tetrahedra with a given coloring can be viewed as a set ofgraphs and the gluing instructions a way of joining the graphs. First, a tetrahedronwith the colors (a, b, c, d, e, f) labelling its sides can be represented as a graph inany of the four ways shown in figure 2 where the crossings have a relative phasefactor [7] .

In this graph each line and region is assigned a color. The colors of aline and its neighbouring two regions as also the colors of the three lines joining ata vertex, which we call a 3–joint, form an unordered triplet.

Thus in figure 2 thetriplets are (abc), (aef), (bdf) and (cef). A triplet (abc) is said to be admissibleif |a −b| ≤c ≤(a + b).

In what follows we will consider only those colorings withadmissible triplets.We implement the gluing instructions in this case as follows. When two tetra-hedra with colors (a, b, c, d, e, f) and (a, b, c, l, m, n) are glued along their commonface with colors (a, b, c), the graphs representing them are joined together suchthat the common area and the lines will have the same colors as shown in figure3.

In this way the gluing instructions will lead to planar graphs with crossingsand 3–joints only. Since the TV invariant does not have any phase factors to be4

associated with crossings we will implement the gluing instructions in such a waythat in the resulting graph these phase factors will cancel out. This will restrictus to a certain class of graphs only as described below.

It can be easily seen thatrepresenting the tetrahedra by crossings and joints and gluing them may result inmore than one graph. It is also clear that the graph corresponding to a manifoldwithout boundary will be closed consisting only of closed lines.

The characterisa-tion of the graphs obtained by the triangulations of a given 3–manifold as describedabove is a difficult subject and is under further study.For future use we now define several quantities in a given graph. First wedefine a quantity we call character for each given crossing.

It is a sum total of thecolors of the four regions around the crossing each with an appropriate sign whichis determined to be + or −according to whether the region with that color comesto the right or left, respectively, as one moves towards the crossing along the over–line. Thus the characters of the crossings in figures 2B and 2C are (b+e−c−f) and(c + f −b−e) respectively.

Note that switching one crossing into another (allowedby tetrahedral symmetry) reverses the sign of the character. We call the sum of thecharacters of all the crossings in a graph as the character of the graph, assigning3–joints a zero character.

The charcter of the graph is closely related to the sumtotal of the phase factors associated with each crossings. In particular these phasefactors will cancel out whenever the character of the graph vanishes.

Since theTV invariant does not have any phase factors associated with the crossings it isnecessary for the graph to have a vanishing character. Henceforth we will consideronly such graphs.We also define C3 and C4 to be the total number of 3–joints and crossingsrespectively; L the total number of lines; li the total number of pieces one obtainsby cutting a given line i at all crossings and 3–joints; λ the sum of li’s and R thetotal number of regions.

When a given set of three lines all begin and end in 3–joints we call them the “Baryon Orbits” (following [6] ) and we define B to be thetotal number of baryon orbits. The significance of these quantities is the following.Let N0, N1, N2 and N3 denote the number of vertices, lines, faces and tetrahedra5

respectively in the triangulation of the manifold that gave rise to the given graph.Then N3 = C3+C4, N2 = C3+λ−B, N1 = L+R and N0 = χ+R+C4+L+B−λ,where χ is the Euler characteristic of the 3–manifold. Note that χ vanishes for M3,a compact orientable 3–manifold with no boundary that we are considering here.Now given such a graph we would like to associate an object T for each 3–joint and crossing (along with a phase factor [7] ) and define a quantity as in (2)which will be invariant under Reidemeister (R–)moves.

An invariant for whichthese properties are true is obtained by identifying for each tetrahedronT ≡TW = {}q,(3)where {}q, denotes the quantum 6–j symbol. However this object has a geo-metrical shortcoming.

It does not have the symmetries of a tetrahedron. Thereis an object related to {}q with the symmetries of a tetrahedron which is noneother than the quantum Racah–Wigner coefficient upto some symmetry–preservingphase factors and is given byT ≡TT V = (−1)−(a+b+c+d+e+f)(abcdef)RW(4)for each tetrahedron colored (a, b, c, d, e, f) as in figure 2A.

But I(G) in (2) cal-culated using this object is not invariant under R–moves. The problem which weconsider is this : can we modify the RHS of (2) so that the tetrahedral symme-tries are present and it represents an invariant of the graph G i.e.

the R–movesleave I(G) invariant? This problem can be fixed, while preserving the tetrahedralsymmetry, by associating a factor (−1)2j S0jS00 with each line or region colored j andby requiring the graph to have vanishing character.

We then take the sum of thisquantity over all possible coloring with admissible triplets only. Thus the quantityI′G =XcoloringYlines,regions(−1)2j S0jS00Ycrossings,3−jointsT(5)satisfies our requirements of tetrahedral symmetry and is invariant under Reide-meister moves.Notice, however, that in this process the original factorisation6

property is lost.In the above analysis we required tetrahedral symmetry for each crossing and3–joint because they represent actual tetrahedra in the triangulation of the mani-fold which was the origin of our graph. Invariance under Reidemeister moves arenecessary because any graph is defined only upto these moves.

However since thegraphs we consider are obtained from (or can be interpreted as) the triangula-tions of a 3–manifolds it follows that I′G must be the same for any set of graphscorresponding to different triangulations of the same 3–manifold. For the case of3–manifolds it is known that any two triangulations describe the same manifoldif these two are connected by a finite number of (k, l) moves and their inverseswhere k + l = 5 [8] .

This set of moves is equivalent to Alexander moves and toMatveev moves and bubble moves [2] [8] . The (2, 3) move corresponds to splittingtwo tetrahedra OABC and XABC into three tetrahedra OXAB, OXBC andOXCA by joining the two vertices O and X.

In terms of graphs this move corre-sponds to the Reidemeister–3 move. The (1, 4) move corresponds to obtaining fourtetrahedra from one by adding a new vertex inside a tetrahedron and connectingit to its four vertices.

In terms of graphs this corresponds to the “bubble move”shown in figure 4. (The inverses of the above moves are obvious).Thus we require the quantity I′G to be invariant not only under the Reidemeistermoves but also under the bubble moves.

However invariance under the bubble moverequires that we add a vertex dependent factor and hence, the right candidate isfound to beIG = S2N000 I′G(6)where N0 = R+C4+L+B−λ is the total number of vertices as defined before. Thuswe obtain IG, an invariant of a given graph with vanishing character.

We considerin the following the examples of S3 and RP 3 in which we check explicitly that twodifferent triangulations lead to the same result. That IG is indeed independent oftriangulation for any manifold M3 has been proved by Turaev and Viro [2] , whichcan also be seen easily in our approach using the (k, l) moves.7

Thus we see that interpreting the triangulation of a manifold as a planar graphwith crossings and vertices with fixed number of lines one obtains a new invariantwhich is defined for all such graphs with vanishing character. This quantity remainsinvariant under Reidemeister moves and bubble moves for the graphs and gives TVinvariant when the graph is viewed as a triangulation of a manifold.

This suggeststhat any graph with vanishing character can be interpreted as a triangulation of3–manifold and has IG as its invariant. (However, as mentioned earlier, the fullclassification of graphs which correspond to the triangulation of a 3–manifold isa difficult subject and is under further study).As can be seen from such aninterpretation IG is invariant not only under Reidemeister moves and bubble movesbut also under ‘cutting offthe crossings and 3–joints and interconverting them andgluing them back preserving the “gluing instructions” and the character of thegraph’ – such a process will in general lead to a different graph.

This processof reducing by symmetry transformations the original graph to a set of standardsimple objects whose invariants can be evaluated easily is quite akin to the skeinrelations in calculating the usual graph invariants. To evaluate IG completely weneed one further identintyXa,bbacS0aS00S0bS00=1S200S0cS00(7)where a, b and c label the line and its neighbouring regions respectively.We illustrate the above procedures for a simple case of two tetrahedra gluedtogether as in figure 5a.

(Actually, this figure represents one way of triangulatingS3).This graph has a vanishing character.Replacing the two tetrahedra (3–joints) by two different crossings we obtain the graph of figure 5b which has a nonvanishing character and hence is not admissible. Switching one of the crossingslead to an admissible graph with vanishing character (figure 5c) which, under aReidemeister move (same as summing over b1), leads to the graph in figure 5d.Note that one could have obtained the graph in figure 5d from that in figure 5a ina single step by using the Reidemeister move for the 3–joint.

Thus this example,8

as well as all the others that we have tried, indicates that irrespective of how oneinitially represents the given set of tetrahedra as a graph preserving the gluinginstructions one always gets the same result for IG after a sufficient number ofsymmetry operations – as one must if IG were a genuine invariant of a graph withvanishing character (or equivalently of a triangulated manifold).We now evaluate the TV invariant for S3 and RP 3. The 3–sphere S3 can beviewed as the boundary of a 4–simplex with the vertices denoted by 0, 1, 2, 3 and 4.Thus S3 consists of five tetrahedra whose vertices are (0123), (0124), (0134), (0234)and (1234).

The graph corresponding to this triangulation is given in figure 6. Thegraph invariant IS3 (which is the same as TV invariant) for S3 can be evaluatedeasily using the Reidemeister moves and the result isIS3 = S200.

(8)The 3–projective space RP 3 can be triangulated as shown in figure 7 wherethe lines labelled the same, and the corresponding faces, are to be identified. TheTV invariant for RP 3 can be calculated easily and is given byIRP 3 =2k + 2(1 + (−1)k)sin2π2(k + 2)(9)where k is related to q, the root of unity, as defined earlier.One can obtain another triangulation of S3 and RP 3 by first triangulatinga lens space Lp,q since the manifolds S3 and RP 3 are special cases of Lp,q with(p, q) = (1, 0) and (2, 1) respectively.

The lens space Lp,q is obtained as follows [9].Consider a region of 3–space bounded by two spherical caps meeting in anequatorial circle. Rotate the lower cap onto itself through an angle of 2πqpradiansand then reflect it about the equatorial plane onto the upper cap.

The resultingmanifold thus obtained is the lens space Lp,q.One possible triangulation Lp,q,which suffices for our purposes, is shown in figure 8 with the lines labelled thesame and the corresponding faces identified [10] . In figure 8, i ∈Z/p where i is9

the subscript of the label βi. Using this triangulation the expression for the TVinvariant ILp,q can be written in terms of T for each tetrahedron.

For (p, q) = (1, 0)and (2, 1), this expression can be evaluated and the results agree with equations(8) and (9) , as they should. However, this agreement would not be there if thevertex factor in equation (6) were absent.Though we find it hard to evaluatethe expression for ILp,q in general, we are able to evaluate it for another case,(p, q) = (3, 1).

But in this particular instance, with the triangulation for L3,1 givenas in figure 8, the proof for the invariance of ILp,q is not valid for reasons givenin [2] and hence IL3,1 evaluated as above is not the right answer.In a different context Danielsson [11] has evaluated IW(M) for M = S3 andLp,1, where IW(M) is the invariant for the manifold M obtained by calculating theabsolute value of the partition function for a CS gauge theory with gauge groupSU(2). They can be written asIW(S3) =r2k + 2sinπk + 2IW (Lp,1) =2k + 2k+1Xn=0sin2 πnk + 2exp(iπpn22(k + 2)).

(10)Comparing our results for IS3 and IRP 3 with equation (10) after setting p = 2 (seebelow for evaluation of IW (Lp,1)) we find thatIM = I2W(M)(11)for M = S3 and RP 3.The sum in equation (10) , denoted by 14σp,r, r = k + 2 can be evaluated asfollows. (This procedure is due to R. Balasubramanian [12] .) First, it can be seenthatσp,r = 12(G(p, 0, 4r) −G(p, 4, 4r))(12)where G(a, b, l) = Pl−1n=0 exp(i2πl (an2 +bn)).

Now we state the following properties10

of G(a, b, l) [12] . In the following all the variables are integer valued and (x, y)denotes the greatest common divisor of two integers x and y.

(1) If (a, l) does not divide (b, l) then G(a, b, l) = 0. (2) G(ca, cb, cl) = cG(a, b, l).Using the properties (1) and (2) we will restrict ourselves to the case where(a, l) = 1.

(3) Let ω be an integer such that 2aω + b is a multiple of l. Such an ω isguaranteed to exist for any a, b and l provided (2a, l) divides (b, l). Since (a, l) = 1,this implies that ω exists if either l is odd or if both b and l are even.ThenG(a, b, l) = exp(−i2πl aω2)G(a, 0, l).

(4) If l is even and b is odd then G(a, b, l) = exp(−i2πl aµ2)G(a, 0, l) where µ isan integer such that 2aµ + (b −1) is a multiple of l.(5) For l odd |G(a, 0, l)| =√l. (6) Let l = 2λ.

Then |G(a, 0, l)|2 + |G(a, 1, l)|2 = 2l and |G(a, 0, l)| = 0 (or√2l) if aλ is odd (or even).Using the above properties (1) – (6) of G(a, b, l), IW(Lp,1) in (10) can beevaluated for any p and r. For example, one can obtain all the values of IW(Lp,1)listed in [11] for some specific values of p and k.Moreover, assuming that equation (11) is true for any manifold ( see [4] ), theTV invariant for the lens space Lp,1 can also be obtained by the above method.ACKNOWLEDGEMENTSIt is a pleasure to thank Sumit Das for many discussions and for bringing thereferences [2] [4] to our attention. We would also like to thank R. Balasubramanianfor giving us his result on the summation in equation (10) .11

REFERENCESN1 E. Witten, Comm. Math.

Phys. 121 (1989) 351.

N2 V. G. Turaev and O.Y. Viro, State sum Invariants of 3–Manifolds and Quantum 6–j Symbols, LOMIPreprint (1990).

N3 T. Regge, Nuovo Cemento 19 (1961) 558; G. Ponzano and T.Regge, in Spectroscopic and Group Theoretic Methods in Physics, Ed. F. Bloch(North–Holland, Amsterdam,1968).

N4 H. Ooguri and N. Sasakura, PreprintKUNS 1088, HE(TH) RIMS-778 (August 1991). N5 S. Mizoguchi and T. Tada,Preprint YITP/U-91-43 (October, 1991).

N6 E. Witten, Nucl. Phys.

B322 (1989)629; ibid B330 (1990) 285. N7 A. N. Kirillov and N. Yu.

Reshetikhin, Adv. Seriesin Math.

Phys. Vol.

7, Ed. V. G. Kac, World Scientific (1988) 285–339.

N8 M.Gross and S. Varsted, Preprint NBI-HE-91-33 (August 1991). N9 H. Seifert andW.

Threlfall, Seifert and Threlfall : A Text Book of Topology, Academic Press,New York, NY, U.S.A, 1980. N10 J. R. Munkres, Elements of Algebraic Topology,The Benjamin/Cummings Publishing Company Inc., Menlo Park, CA, U.S.A,1984.

N11 U. Danielsson, Phys. Lett.

B220 (1989) 137. N12 R. Balasubramanian,private communication.12

FIGURE CAPTIONSFigure 1 : Gluing procedure.Figure 2 : Representation of a tetrahedron by graphs.Figure 3 : Gluing procedure for graphs.Figure 4 : Bubble move.Figure 5 : One example of graph manipulations.Figure 6 : Graph of S3.Figure 7 : Triangulation of RP 3.Figure 8 : Triangulation of Lp,q (subscripts for β ∈Z/p).13

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