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Subdivision Invariant Models in Lattice

arXiv:hep-th/9302135v1 26 Feb 1993Subdivision Invariant Models in LatticeGauge TheoryDanny Birmingham 1Universiteit van Amsterdam, Instituut voor Theoretische Fysica,1018 XE Amsterdam, The NetherlandsMark Rakowski2Yale University, Center for Theoretical Physics,New Haven, CT 06511, USAAbstractA class of lattice gauge theories is presented which exhibits noveltopological properties. The construction is in terms of compact Wil-son variables defined on a simplicial complex which models a fourdimensional manifold with boundary.

The case of Z2 and Z3 gaugegroups is considered in detail, and we prove that at certain discretevalues of the coupling parameter, the partition function in these mod-els remains invariant under subdivision of the underlying simplicialcomplex. A variety of extensions is also presented.ITFA-93-03 / YCTP-P4-93February 19931Email: Dannyb@phys.uva.nl2Email: Rakowski@yalph2.bitnet

1IntroductionIn this paper, we undertake a study of certain lattice gauge theories whichhave special properties with respect to subdivision of the underlying lattice.The motivation for such a search has its roots in topological field theory(see [1] for a review), where quantum field theories have been constructedwhose observables are topological or smooth invariants of the underlyingspacetime manifold.In the pure Chern-Simons theory [2, 3], one has, inparticular, a partition function which is a topological invariant of a framingof the spacetime 3-manifold. Our interest originates primarily from the desireto see these types of structures emerge from a traditional lattice approach.Calculations in lattice gauge theory, and statistical mechanics generally,are concerned with the behaviour of systems in a continuum limit, where theunderlying lattice is subdivided into smaller and smaller units.

At any givenstage of subdivision, one has only a crude approximation to the continuumtheory. Topological field theories are, on the other hand, quite different.

Thetopology of any manifold can be captured in terms of a lattice (simplicialcomplex), and further subdivisions of that lattice in no way enhance onestopological picture of the space. It is of interest to construct lattice modelswhich also reflect this property; models in which the observables are invariantunder lattice subdivision.There is then no need to be concerned with acontinuum limit, as the model would already compute - exactly - the relevantquantities.

In other words, one would already be at the continuum limit.Here, we construct models which have the property of subdivision in-variance at certain descrete values of the coupling parameter.While ourmotivation for these particular examples stems primarily from the Chern-Simons theory, we will not establish any firm link with that theory here.Our approach is entirely self-contained and we will have no need to refer toresults in any continuum model, or to invoke general folklore in quantum fieldtheory. Although nothing in our construction forces us to consider discretegauge groups, our analysis here will focus on these simpler examples.

Wefind that the partition function of our model, which is defined on a simplicalcomplex which models a 4-manifold with boundary, is invariant under thetype 4 Alexander subdivision [4]. This is essentially a local property whichwe can prove by looking at the Boltzmann weight on a single 4-simplex.

This1

special property is restricted to discrete values of the coupling parameter. Wealso consider other types of lattice subdivision, and show that the partitionfunction of our theory on a disk is invariant under all of the Alexander moves.We begin in the next section with an overview of lattice gauge theory interms of compact Wilson variables, and provide some background on sim-plicial complexes.

Our model is then defined and we move on to considerspecific cases in succeeding sections. A simple two dimensional version isconsidered first, and then we treat the Z3 and Z2 gauge groups in four di-mensions.While all our detailed calculations are for discrete groups, wediscuss some obvious extensions both to continuous groups, and to higherdimensional analogs of the models presented here.

We close then with someconcluding remarks.2General PropertiesWe first recall the essential definitions needed in a Wilson formulation oflattice gauge theory on a simplicial complex. For a complete account of thelatter, see [5].Let [v0, · · ·, vn] denote the oriented n-simplex spanned by the geometri-cally independent set of points {vi}, called its vertices.

One can picture thesesimplices as points, line segments, triangles and tetrahedrons for n equal tozero through three. A simplex which is spanned by any subset of the verticesis called a face of the original simplex.A simplicial complex K is a collection of simplices which are glued to-gether under two restrictions.

Any face of a simplex in K is required to be asimplex in K, and the intersection of any two simplices in K must be a faceof each of them.The basic fields which enter a formulation of lattice gauge theory aregroup valued maps on the 1-simplices (denoted [a, b]) with the rule thatUba = U−1ab .In order to define a theory, one requires that the group becompact with an invariant measure, and one takes the action on the “link”variables - the gauge transformations - to be given by:Uab →ga Uab g−1b. (1)2

Here, ga is a group element associated with the vertex a.We take the action of our theory to be a gauge invariant function S ofthe above link variables. The partition function is thenZ =YαZdUα exp[β S(U)] ,(2)where the index α indicates the set of independent 1-simplices.

In the caseof a discrete gauge group, the group integration (whose volume we normalizeto unity) is a discrete sum,ZdU →1|G|XU,(3)where |G| denotes the order of the group. One can further define correlationfunctions of the link variables,< Uγ1 · · · Uγp >=YαZdUα Uγ1 · · · Uγp exp[β S(U)] .

(4)In the above, a group trace is understood, whenever necessary. The key pointis that, in general, all of these quantities depend not only on the couplingparameter β, but also on the simplicial complex K.The above was an abstract description of lattice gauge theory; however,the motivation for it arises as follows.

Normally in a gauge theory the basicdynamical variable is a connection on some principal G bundle over space-time. One can construct the Wilson link variablesWCab = P exp[ZCabA] ,(5)where Cab is some path in spacetime between the endpoints a and b, and A isa connection which takes values in the Lie algebra of the group G. As usual,the symbol P denotes the path ordering.

It is straightforward to establishthat the the link variables are solutions to the differential equation:D WCab = 0 ,(6)where D is the covariant derivative.3

Let Uabc = Uab Ubc Uca be the holonomy, based at the first vertex a, aroundthe triangle determined by a, b and c, and traversed in the order from left toright . In terms of the Wilson link variables, this is represented by choosinga closed contour C in (5), and taking a group trace if necessary.

A propertyof holonomy is that in the limit when the loop becomes infinitesimally small,UC approachesexp[ZF] ,(7)where the integral is over a surface which has C as its boundary.The usual Wilson action is given byS = 12XU(U −1) + (U−1 −1) ,(8)where the sum is over all the basic holonomies on the lattice. Notice thateach term in this sum depends on a single holonomy.

It is a standard exercise[6] to show that the continuum limit of the above action is the Yang-Millstheory:S →12ZF 2 . (9)In the present discussion, we are motivated to consider a discrete versionof the Chern form, F ∧F.

The first observation is to note that one canproduce F by combinations such as U−U−1 and U−1, in the continuum limit.We will base our action on a combination of two independent holonomies,which are tied together at a point, and have no edges in common. Let usfirst, however, digress to review a product which will serve to form the analogof the wedge product of differential forms.Denote by P, one of the (r + s + 1)!

permutations of the set of vertices{v0, · · · , vr+s}, which span some (r + s)-simplex, and by Pvi the value ofthat permutation on vi.Let cr and cs be group valued maps on r- ands-simplices, respectively. The ⋆-product cr ⋆cs yields a group valued mapdefined on (r + s)-simplices, and is defined by:4

< cr ⋆cs, [v0, · · ·, vr+s] > =(10)1(r + s + 1)!XP(−1)|P | < cr, [Pv0, · · · , Pvr] > · < cs, [Pvr, · · ·, Pvr+s] > ,when the order v0 · · ·vr+s is in the equivalence class of the orientation of thesimplex [v0, · · ·, vr+s] (this determines the overall sign of the product), andwhere the sum is over all permutations of the vertices. The notation ⟨· , ·⟩is used to indicate the evaluation of the map on the accompanying simplex,and the product on the right hand side of (10) refers to multiplication in therelevant group or ring.

For a more complete definition of the ⋆-product, werefer to [7]; we simply note here that it is a variation of the standard cupproduct which achieves graded commutativity at the expense of associativity(the usual cup product is graded commutative only on cohomology classes).The actual number of independent terms in the above sum is given bythe number of ways one can partition the set of vertices into two parts whichcontain one vertex in common, and an easy counting yields(r + s + 1)!r! s!.

(11)Now we are equipped to return to our action. We take this to be a sumover holonomy pairsS =X(U −U−1) ⋆(U −U−1) ;(12)a trace is understood when required, and the sum here is over all elementary4-simplices in the simplicial complex.

The ⋆-product ensures that the twofactors are independent holonomies with one point in common. Of course,one could equally well consider an action of the formS =X(U −1) ⋆(U −1) ,(13)or several other possibilities, each of which yield the same continuum limit.Note, however, that the quantity (U −U−1)abc has the additional feature thatit is antisymmetric in its last two indices; this follows simply from Uacb = U−1abc.As we have seen, the holonomy U is a group valued map on 2-simplices.

Theactions defined above, in terms of two independent holonomies, are thereforenaturally defined on a 4-simplex.5

We should emphasize that achieving a continuum limit akin to the formF ∧F is purely motivational, and we shall not draw on any properties ofcontinuum theories here. It is not clear, a priori, whether any of the familiarcontinuum properties will translate onto a finite lattice.

We note that anaction similar to the above was studied for the case of a torus, in a differentcontext [8]. Our aim is simply to take the above lattice definition, and provethat it has certain topological features.When we evaluate the star product on a given 4-simplex, the antisym-metrization produces generically 5!

terms. With the action (12), the symme-tries present reduce that to 15 distinct terms.

We will take as the Boltzmannweight of this theory, evaluated on the simplex [v0, v1, v2, v3, v4], the normal-ization given by:W[0, 1, 2, 3, 4]=B[0, 1, 2, 3, 4] B[0, 1, 3, 4, 2] B[0, 1, 4, 2, 3]B[1, 0, 2, 4, 3] B[1, 0, 3, 2, 4] B[1, 0, 4, 3, 2]B[2, 0, 1, 3, 4] B[2, 0, 3, 4, 1] B[2, 0, 4, 1, 3]B[3, 0, 1, 4, 2] B[3, 0, 2, 1, 4] B[3, 0, 4, 2, 1]B[4, 0, 1, 2, 3] B[4, 0, 2, 3, 1] B[4, 0, 3, 1, 2] ,(14)where,B[0, 1, 2, 3, 4] = exp[β (U −U−1)v0v1v2 (U −U−1)v0v3v4] . (15)Our analysis deals with the general case of complex coupling β.

We will,however, still use the phrase “Boltzmann weight” in this more general con-text.Using the Boltzmann weight defined above, we can compute the partitionfunction for the theory defined on a simplicial complex K. The central issueof interest here is to examine how this function behaves upon subdivisionof the complex. For ease of illustration, let us consider a single 4-simplex[v0, v1, v2, v3, v4].A convenient basis of subdivision operations, known asAlexander moves [4], are available, and these allow a direct analysis of thisquestion.

The Alexander moves can be described in turn by:Type 1 Alexander subdivision:[v0, v1, v2, v3, v4] →[x, v1, v2, v3, v4] + [v0, x, v2, v3, v4] ,(16)6

Type 2 Alexander subdivision:[v0, v1, v2, v3, v4] →[x, v1, v2, v3, v4] + [v0, x, v2, v3, v4] + [v0, v1, x, v3, v4] , (17)Type 3 Alexander subdivision:[v0, v1, v2, v3, v4]→[x, v1, v2, v3, v4] + [v0, x, v2, v3, v4] + [v0, v1, x, v3, v4]+[v0, v1, v2, x, v4] ,(18)Type 4 Alexander subdivision:[v0, v1, v2, v3, v4]→[x, v1, v2, v3, v4] + [v0, x, v2, v3, v4] + [v0, v1, x, v3, v4]+[v0, v1, v2, x, v4] + [v0, v1, v2, v3, x] . (19)One can picture the move of type 1 as the introduction of an additionalvertex x, which is placed at the center of the 1-simplex [v0, v1], and is thenjoined to all the remaining vertices of the 4-simplex.Moves 2 to 4 in-volve a similar construction, where x is placed at the center of the simplices[v0, v1, v2], [v0, v1, v2, v3], and finally [v0, v1, v2, v3, v4].

There is, in addition, atype 0 move which is effected by replacing a vertex of the simplicial complexby a new vertex. This can be considered as a degenerate case, and need notconcern us in the following.According to Alexander [4], two simplicial complexes are said to be equiv-alent if and only if it is possible to transform one into the other by a sequenceof these moves.

Hence, any function of K which is invariant under thesemoves yields a combinatorical invariant of the simplical complex.3A Toy Model in Two DimensionsLet us consider a simpler two dimensional version of the type of models wewant to explore. Here, it is natural to consider an action which depends ona single holonomy, and for the gauge group Z3 = {1, exp[2πi/3], exp[4πi/3]},we will take the Boltzmann weight evaluated on the 2-simplex [v0, v1, v2] tobe given by:W[v0, v1, v2] = exp[β (U −U−1)v0v1v2] ,(20)7

where Uv0v1v2 is the holonomy combination Uv0v1Uv1v2Uv2v0.It is now a simple matter to explore the subdivision properties of thistheory. First, consider the Boltzmann weight under the type 2 Alexandermove where[v0, v1, v2] →[v3, v1, v2] + [v0, v3, v2] + [v0, v1, v3] .

(21)The picture of this subdivision operation is simply that of adding a new v3vertex to the center of the original, and then connecting that to the othervertices by the addition of three new links. This present theory enjoys thespecial property that:W[v0, v1, v2] = 127XUv0v3,Uv1v3,Uv2v3W[v3, v1, v2] W[v0, v3, v2] W[v0, v1, v3] ,(22)when the coupling takes values such that s3 = 1, where s = ei β√3 is aconvenient scale parameter.

This property is not obvious, but can be checked,by hand, in a straightforward way. Notice that at the special points, theBoltzmann weight itself is Z3-valued for every link configuration, and thiswill be the pattern in all our examples.The situation under type 1 Alexander subdivision is slightly trickier.Here, a given 2-simplex [v0, v1, v2] is broken into two pieces;[v3, v1, v2] + [v0, v3, v2] .

(23)The picture is that of splitting the [v0, v1] edge by the introduction of the v3vertex at its center, and then connecting that to v2 with a new link. Onecan ask how the Boltzmann weight or partition function behaves under thismove.

Again, one can verify by hand thatW[v0, v1, v2] = 13XUv2v3W[v3, v1, v2] W[v0, v3, v2] ,(24)when s3 = 1 and Uv0v1 = Uv0v3Uv3v1. We were fortunate in this case to havea simple relation between the boundaries of the original simplex, and that ofthe simplices after type 1 subdivision; the extra constraint on the product oftwo of the new link variables just reflects that relationship.8

Of course, the action (20) vanishes for the group Z2, and so, in line withthe analysis of the previous section, one may wish to examine the Z2 modelwith Boltzmann weight:W[v0, v1, v2] = exp[β (U −1)v0v1v2] . (25)The immediate observation, however, is that the above is simply thestandard Wilson action for this group.

Nevertheless, if one examines thismodel with a complex coupling β, then a simple analysis reveals that theBoltzmann weight is invariant under the Alexander moves of type 1 and 2.This takes places when the coupling satisfies s2 = 1, where s = e−2β, andUv0v1 = Uv0v3Uv3v1.It is well known that two-dimensional Yang-Mills theory is invariant underarea-preserving subdivisions, although such a result is obtained for β realand positive, and using a heat kernel form of the action [9, 10]. This simpleexample indicates that it might be fruitful to re-examine the usual Wilsonaction and search for special subdivision properties at complex couplings.One other point that is especially transparent in this model concerns ourchoice to include the −1 term in the defining action; one might ask whetherthat really has any significance.Suppose we had defined the BoltzmannweightW ′[v0, v1, v2] = exp[β Uv0v1v2] .

(26)Then a simple scaling of our previous result yields the relation,W ′[v0, v1, v2] = 18 sXUv0v3,Uv1v3,Uv2v3W ′[v3, v1, v2] W ′[v0, v3, v2] W ′[v0, v1, v3] ,(27)which we can interpret as subdivision invariance up to a scaling factor. Thislends some support to our original geometrical motivation which suggestedthe combination (U −1).

If one were to search more generally for othermodels with interesting subdivision properties, it would be important not todiscard models which had this additional scaling behavior.9

4The Z3 ModelWe will consider in this section the type of model we outlined in the sectionon General Properties; i.e., a gauge theory whose action is based on twoindependent holonomies which are tied together at a point and have no edgesin common. Pictorially, one might refer to this as a “bowtie” configuration.The setting here is on a simplicial complex which models a four dimensionalmanifold with boundary (which can also be empty) and we will focus on thegroup Z3, which we represent multiplicatively as the cube roots of unity, 1,exp[2πi/3] and exp[−2πi/3].

We were motivated in our choice of action bythe familiar Chern form, and we saw earlier that we had some freedom inwriting a discrete analog. Our calculations in this section will be based onthe action (12), and we will take the Boltzmann weight for a given orderingof vertices to be given by:B[0, 1, 2, 3, 4] = exp[β(U −U−1)v0v1v2 (U −U−1)v0v3v4] ,(28)and we will insert that into the expression (14) to get a quantity W[0, 1, 2, 3, 4]which takes into account all the different permutations of the ⋆-product.

Itwill be useful in the following analysis to introduce a scale parameter whichwe take in this model to be the quantity s = exp[−3 β].The behaviour of the theory under the type 4 Alexander move parallelsthat of the simple two dimensional model under the type 2 move. Let usinvestigate the Boltzmann weight of this model in the same way.

Take the4-simplex [v0, v1, v2, v3, v4] and its corresponding type 4 subdivision which isthe sum of five 4-simplices:[v5, v1, v2, v3, v4] + [v0, v5, v2, v3, v4] + [v0, v1, v5, v3, v4] +[v0, v1, v2, v5, v4] + [v0, v1, v2, v3, v5] . (29)The property of this theory is thatW[v0, v1, v2, v3, v4] = 135XUi5W[v5, v1, v2, v3, v4] W[v0, v5, v2, v3, v4]W[v0, v1, v5, v3, v4]W[v0, v1, v2, v5, v4]W[v0, v1, v2, v3, v5] ,(30)when s3 = 1 and where the sum is over the 5 links which join to the new v5vertex.

We do not know an elementary way of seeing this at the present time.10

However, it is simple enough to write a computer program to check this kindof relation and verify it for all choices of “boundary data”, and this we havedone using Mathematica [11]. The relation is quite simple owing to the factthat both the boundary of the original simplex and its type 4 subdivisionare identical.

This property guarantees that when we evaluate the partitionfunction of the theory on any simplicial complex, that the number one gets isinvariant under all type 4 subdivisions. We emphasize again that this specialproperty only holds for the values of the coupling β such that s3 = 1.While we do not yet have such a complete understanding of the subdivi-sion properties of this model under the other Alexander moves at the level ofBoltzmann weights, we can nevertheless offer some computational evidencewhy it is interesting.

Here, we will compute exactly the partition function ofthe theory on the 4-disk which provides an example with a boundary topo-logically equivalent to S3. One can model the disk as a simplicial complexwith a single 4-simplex, or through more complex subdivisions.The following results for the partition function (2) were calculated usingMathematica.

One aspect of lattice gauge theory that is important to takeadvantage of in these computer studies is the freedom to gauge fix some linkcomponents. The issue of gauge fixing in the Wilson formulation is particu-larly simple and elegant and does not introduce any murky questions whichcould undermine the rigour of our analysis.

The construction of the partitionfunction in terms of group integrations and a gauge invariant action allowsone to fix arbitrarily the links on a maximal tree [6]. Roughly speaking, thisis any collection of links which does not include a closed path and which can-not be extended by the addition of other links.

From a practical perspective,this significantly reduces the number of group integrations (which are justfinite sums in this case) that we must perform. Let us now list, in turn, theresults of our calculation for the 4-disk.For the representation of the disk in terms of a single 4-simplex, we findthe partition function:Z = 136 (221 + 120(s + s−1) + 60(s2 + s−2) + 54(s5 + s−5) + 20(s6 + s−6)) .

(31)11

Under a type 1 Alexander subdivision of that simplex, we find:Z=139 (4215 + 2256(s + s−1) + 1596(s2 + s−2) + 720(s3 + s−3)+660(s4 + s−4) + 1068(s5 + s−5) + 664(s6 + s−6) + 456(s7 + s−7)+48(s8 + s−8) + 162(s10 + s−10) + 72(s11 + s−11) + 32(s12 + s−12)) . (32)Under a type 2 Alexander subdivision, the computation yields:Z=1310 (9641 + 5544(s + s−1) + 4482(s2 + s−2) + 2610(s3 + s−3)+2178(s4 + s−4) + 3222(s5 + s−5) + 2286(s6 + s−6) + 1764(s7 + s−7)+738(s8 + s−8) + 522(s9 + s−9) + 666(s10 + s−10) + 270(s11 + s−11)+294(s12 + s−12) + 54(s13 + s−13) + 30(s15 + s−15) + 18(s16 + s−16)+18(s17 + s−17) + 8(s18 + s−18)) .

(33)For the case of the disk represented by four 4-simplices which are the type 3subdivision of the original simplex, we have:Z=1310 (10293 + 4680(s + s−1) + 3756(s2 + s−2) + 2064(s3 + s−3)+2508(s4 + s−4) + 2544(s5 + s−5) + 1840(s6 + s−6) + 1992(s7 + s−7)+1638(s8 + s−8) + 1104(s9 + s−9) + 1080(s10 + s−10) + 600(s11 + s−11)+320(s12 + s−12) + 72(s13 + s−13) + 60(s14 + s−14) + 16(s15 + s−15)+24(s16 + s−16) + 72(s18 + s−18) + 8(s21 + s−21)) ,(34)and finally for the partition function on the simplical complex resulting fromthe fourth Alexander move, we have:Z=1310 (11841 + 5460(s + s−1) + 2640(s2 + s−2) + 780(s3 + s−3)+2250(s4 + s−4) + 2034(s5 + s−5) + 600(s6 + s−6) + 1560(s7 + s−7)+2520(s8 + s−8) + 2970(s9 + s−9) + 1560(s10 + s−10) + 600(s11 + s−11)+180(s12 + s−12) + 90(s13 + s−13) + 60(s15 + s−15) + 60(s16 + s−16)+120(s17 + s−17) + 60(s18 + s−18) + 60(s19 + s−19)) . (35)12

Although these results may appear at first glance to have neither rhymenor reason, we expect that they will have special properties at the pointss3 = 1. Indeed, all five of the functions in (31) - (35) reduce to the simpleform,Z(s) = 134 ( 29 + 26 (s + s−1)) ,(36)at those particular values of s. When s = 1, or equivalently β = 0, we havea trivial subdivision invariant point, and Z = 1, but at the other two cuberoots of unity, we find the value Z = 1/27.As a second example, let us consider the 4-dimensional sphere S4.

Wemodel S4 as the boundary of a 5-simplex:∂[v0, v1, v2, v3, v4, v5]=[v1, v2, v3, v4, v5] + [v0, v3, v2, v4, v5] + [v0, v1, v3, v4, v5]+[v0, v1, v2, v5, v4] + [v0, v1, v2, v3, v5] + [v0, v1, v2, v4, v3] ,(37)where we have chosen to write it in a way such that each 4-simplex is pos-itiviely oriented according to the order given by its vertices.The precisedefinition of the boundary operator ∂can be found in [5]. The result of thiscomputation isZ = 1310(33309 + 12300(s9 + s−9) + 570(s18 + s−18)) .

(38)When we restrict s to be a cube root of unity, we find that Z = 1.5A Novel Z2 ModelWe already considered a two dimensional Z2 based model in an earlier section;here we would like to extend that to four dimensions. Of course, actions whichdepend on the combination U −U−1 necessarily lead to a trivial theory forthis group, but there is another problem if we use the U −1 combination inconcert with the ⋆-operator.

For abelian groups generally, the holonomy Uabcis invariant under cyclic permutations of the indices, so that the base pointof the holonomy does not enter. In the case of Z2, we also have that U = U−113

for all group elements, so the holonomy combination is in fact symmetric inall indices. We tacitly avoided this in two dimensions and simply definedhow to evaluate the Boltzmann weight on a given 2-simplex.

It is interestingthat one can do something similar in four dimensions as well, and we willtake as a matter of definition the expression (14) for W[0, 1, 2, 3, 4] togetherwith a new quantity,B[0, 1, 2, 3, 4] = exp[β (U −1)v0v1v2 (U −1)v0v3v4] . (39)This has the effect of selecting essentially half the terms which would appearin the ⋆-product, and “discarding” those which would have entered withopposite sign.

The bottom line as to whether this is a fruitful line of thoughtis whether we can achieve similar success in terms of subdivision properties.As in all the models, we will find a certain scale combination convenient, andwe take s = exp[4 β] in this section.The first order of business is to analyze the subdivision properties underthe type 4 Alexander move. It turns out that this model also enjoys the simplerelation (30) for its Boltzmann weight at the points s2 = 1.

Although thenumber of link variables is much smaller than in the Z3 example, the numberis nevertheless quite large, and we also employed a computer program toverify this claim.It is also straightforward to analyze the other subdivision properties ofthis model in the explicit calculation of the partition function on a 4-disk.We find, in the same way, that for a single 4-simplex this theory yields thepartition function,Z = 126 (11 + 15s2 + 27s5 + 10s6 + s15) . (40)Under the first Alexander move, where there are two 4-simplices, we find,Z=129 (31 + 54s2 + 33s4 + 48s5 + 12s6 + 96s7 + 24s8 + 105s10+72s11 + 22s12 + 6s20 + 8s21 + s30) .

(41)The type 2 Alexander move applied to the original simplex leads to,Z=1210 (29 + 45s2 + 54s4 + 45s5 + 27s6 + 108s7 + 18s8 + 72s9 + 54s10+18s11 + 153s12 + 36s13 + 27s14 + 102s15 + 135s16 + 45s17 + 13s18+9s25 + 18s26 + 12s27 + 3s36 + s45) ,(42)14

while for the type 3 move we find,Z=1210 (19 + 12s2 + 48s4 + 24s5 + 16s6 + 48s7 + 15s8 + 72s9 + 42s10+84s12 + 48s13 + 96s14 + 24s15 + 30s16 + 96s17 + 28s18 + 75s20 + 104s21+84s22 + 4s24 + 4s30 + 24s31 + 12s32 + 8s33 + 6s42 + s60) . (43)Lastly, the type 4 Alexander move, which we already know is an invarianceof the partition function at s2 = 1 from our more general analysis, leads to,Z=1210 (16 + 30s4 + 15s5 + 30s6 + 60s9 + 45s10 + 70s12 + 90s14 + 60s15+70s18 + 120s19 + 30s20 + 90s22 + 27s25 + 75s26 + 130s27 + 20s36+30s37 + 5s39 + 10s48 + s75) .

(44)All of these results become much more transparent when we restrict themto the points where s2 = 1. It is remarkable that all of the above polynomialsreduce to the simple formula,Z(s) = 124 ( 9 + 7 s) .

(45)The two roots of unity, +1 and −1 yield the values 1 and 1/8 respectivelyfor the partition function of the disk.Again, we can compute the partition function on S4, and in this instancewe find:Z=1210(16 + 60 s6 + 45 s10 + 15 s12 + 180 s14 + 20 s18 + 180 s20 + 180 s24+45 s28 + 27 s30 + 180 s32 + 60 s42 + 15 s54 + s90) . (46)Observe that the polynomial for S4 contains even powers of s only.

Whens2 = 1, the partition function therefore assumes the value Z = 1.6GeneralizationsHere, we discuss some generalizations and extensions of the models consid-ered above. The framework we have outlined is obviously quite general, and15

one can immediately consider corresponding theories based on the traditionalcontinuous gauge groups. The only constraint, as we have noted, is that thegroup should have an invariant measure and finite volume, or new ideas arerequired.

On a finite lattice, the partition function is a well defined number,and the issue is whether these other gauge groups allow for special subdivi-sion invariant points. The behaviour of these theories in the continuum limitfor generic couplings is a far more difficult question, and we will not addressthat here.

From a purely topological perspective, one might adopt the pointof view that behaviour away from the special points is irrelevant.As our final example, let us consider a model in six dimensions. Clearly,one can define a model in any even dimension, either with discrete or con-tinuous gauge groups.

The construction is quite straightforward, and simplyinvolves taking the star product of three independent holonomies, viz.,S =X(U −U−1) ⋆(U −U−1) ⋆(U −U−1) . (47)Such an action will be evaluated on a 6-simplex, and the sum is over allthe elementary 6-simplices in the simplicial complex.Again, it would beof interest to examine the subdivision properties of the associated partitionfunctions.7Concluding RemarksAs we have seen, the interesting subdivision properties of the models pre-sented here are based upon a few key ingredients.

In particular, one is guidedto employ the star product operator in seeking a lattice transcription of theChern-form. As a result, one finds (in four dimensions) an action which de-pends on two holonomies.

Furthermore, a crucial ingredient is to considerthese models for a general complex coupling parameter; indeed, the interest-ing subdivision properties are present when the associated scale parameteris a root of unity. It should be emphasized that the Boltzmann weight itselfis a group valued object at these special points, and this plays an importantrole in the analysis.The novel features uncovered in these models should warrant further in-vestigation.

Of most importance, perhaps, is a detailed examination of the16

properties of the Boltzmann weight under the remaining Alexander moves.The challenge to come to a similar level of understanding for more compli-cated groups is also well defined. One would also like to see if a relationexists with the models presented in [12, 13], and to explore the subdivisionproperties of other correlation functions.References[1] D. Birmingham, M. Blau, M. Rakowski, and G. Thompson, Phys.

Rep.Vol. 209 (1991) 129.

[2] A. Schwarz, Lett. Math.

Phys. 2 (1978) 247.

[3] E. Witten, Comm. Math.

Phys. 121 (1989) 351.

[4] J.W. Alexander, Ann.

Math. 31 (1930) 292.

[5] J. Munkres, Elements of Algebraic Topology, Addison-Wesley, MenloPark, 1984. [6] M. Creutz, Quarks, Gluons and Lattices, Cambridge University Press,1983.

[7] D. Birmingham and M. Rakowski, Phys. Lett.

299B (1993) 299. [8] P. Di Vecchia, K. Fabricius, G.C.

Rossi, and G. Veneziano, Nucl. Phys.B192 (1981) 392.

[9] A. Migdal, Zh. Eksp.

Teor. Fiz.

69 (1975) 810 (Sov. Phys.

JETP. 42(1976) 413).

[10] E. Witten, Commun. Math.

Phys. 141 (1991) 153.

[11] S. Wolfram, Mathematica, A System for Doing Mathematics by Com-puter, Addison-Wesley, Redwood City, 1991. [12] R. Dijkgraaf and E. Witten, Commun.

Math. Phys.

129 (1990) 393. [13] D. Altschuler and A. Coste, Commun.

Math. Phys.

150 (1992) 83.17

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