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A U(N) Gauge Theory in Three Dimensions as an Ensemble of Surfaces

arXiv:hep-th/9108015v1 23 Aug 1991RU–91–34July 29, 1991A U(N) Gauge Theory in Three Dimensions as an Ensemble of SurfacesbyFran¸cois David*Service de Physique Th´eorique** de SaclayF-91191 Gif-sur-Yvette CedexandHerbert NeubergerDepartment of Physics and AstronomyRutgers UniversityPiscataway, NJ 08855–0849AbstractA particular U(N) gauge theory defined on the three dimensional dodecahedral lat-tice is shown to correspond to a model of oriented self-avoiding surfaces. Using large Nreduction it is argued that the model is partially soluble in the planar limit.

* Physique Th´eorique CNRS. ** Laboratoire de l’Institut de Recherche Fondamentale du Commissariat `a l’EnergieAtomique.1

The example of three dimensional abelian gauge theory provides the best understoodmechanism for confinement at weak coupling [1-5] and, therefore, a natural place to lookfor an equivalent description in terms of a theory of real strings. As in any gauge theory, apossible place to start from when looking for string-like excitations is the strong couplingexpansion of the model regularized by discretizing space to a regular lattice.

Roughly, theclosed surfaces that generically appear in the expansion can be thought of as space-timehistories of closed strings. Depending on the nature of the gauge group these strings maybe oriented or not.The particular case of U(1)3 pure gauge theory with a single plaquette action of theVillain form [4] is exactly dual to a three dimensional spin ferromagnet; the spin degreesof freedom are integers and for this reason the model is sometimes referred to as the Z-ferromagnet [5].

The strong coupling expansion in the gauge formulation is, by duality,related term by term to to the weak coupling expansion of the Z-ferromagnet. The mostcommonly studied case is defined on a cubic lattice whose sites we denote by x, x′ andwhose bonds we represent by their end-points, < x, x′ >.

The partition function is givenby:Z =X{n(x)}+∞−∞exp{−12βX[n(x) −n(x′)]2}(1)The surfaces are made out of square plaquettes that live on the cubic lattice dual to theoriginal one and can be associated with individual spin configurations {n(x)} by providingwalls that separate the original lattice into connected, non-empty, clusters on which n hasa constant value. Any bond < x, x′ > for which n(x) ̸= n(x′) is cut by a dual plaquette.It is well known that in most gauge theories the surface interpretation of these wallconglomerates can become involved, necessitating some ad hoc definitions, and becomingquite cumbersome [6,7]; to maintain faith in the existence of a continuum string theorydescription of these surfaces one must assume that many of the above complications areirrelevant.

It would be nice to find special forms of the gauge models that avoid some ofthe complexities already at the regularized level.Even in the simple case of U(1)3 on a cubic lattice the surfaces suffer from complica-tions: plaquettes can be multiply excited in the sense that n jumps by an amount largerthan unity across them and singular lines are possible where three or more plaquettes joinat a common link. We wish to get rid of these cases and obtain a much cleaner geometricaldescription of the surfaces that appear.

We first deal with the multiply excited plaquettesby replacing the action by*Z =X{n(x)}+∞−∞exp{−12βX[n(x) −n(x′)]2k}(2)and taking the limit k →∞; this has the effect of permitting only jumps of ±1 acrossa surface and the two cases can be geometrically interpreted as being associated with* This is a generalization of an action written down by V. J. Emery and R. Swendsenfor an SOS model [8].2

the overall orientation of a closed connected surface enclosing a given spin cluster (thesurfaces are orientable). This restriction also eliminates cases where three plaquettes sharea common link.

However, singular lines where four plaquettes meet are still possible. Toavoid these configurations we place the Z-ferromagnet on a f.c.c lattice rather than on acubic one.

The geometry of this lattice is such that its dual has exactly three plaquettesmeeting at each link; thus the bad cases we were left with disappear. The single case weneed to make a slightly ad hoc decision for is when surfaces touch at a vertex: for reasonsthat will become clear later on we decide not to regard the touching of two otherwiseseparated pieces of surface as something that connects them; in other words, when twosurfaces touch at a vertex we view the vertex as split in two, one vertex for each surfaceand a very small space open between them.We ended up with a model of random self-avoiding orientable surfaces that is verysimilar to the system shown to be equivalent to the Ising model on the f.c.c.

lattice inprevious work [9]. The difference is that in our case we have to sum over independentorientations for each connected component of the set of domain walls.This differenceis significant because it enhances the entropy of configurations made out of many smalldisconnected bubbles.

Due to self-avoidance a gas of such bubbles will exercise a pressureon a surface spanning a Wilson loop, keeping it flat, and pushing the deconfinement tran-sition present in the Z2 gauge theory dual to the Ising model to much lower temperatures,possibly all the way down to zero temperature.We now proceed to write down the U(1)3 gauge theory dual to our model:Zdual =Z{θl}Yp[1 + 2g cos(Xl∈pθl)](3)Here the θl are angles on the links l and the p’s are rhombic plaquettes on the dodecahedrallattice dual to the f.c.c. lattice.

The coupling g is given by g = exp(−12β). This action isthe simplest generalization of the Z2 action written down in ref.

[9].It was shown in the Ising case that Z2 could be replaced by O(N) and, by appropriatelyscaling g with N, a double expansion in g and N could be viewed as a sum over surfacesweighted by their total area and by the sum of the Euler characteristics of their connectedcomponents if surface touching at a site is treated as defined above.In our case thegeneralization will be to the gauge group U(N) withZU(N) =Z{Ul}Yp[1 + 2gNRe(Tr(Yl∈pUl))](4)In terms of surfaces we have:ZU(N) =X{S}gPS(A(S)) NPS χ(S) 2PS 1(5)In the above equation S denotes a connected component of the set of self-avoiding surfaces{S} and A, χ are functions of S giving the area in plaquettes and the Euler characteristic.3

The new model can be subjected to another duality transformation. The resultingexpression is somewhat complicated by the additional curvature terms.

We shall not writedown the explicit expression (it can be found by generalizing references [9] and [10]); allwe wish to stress here is that the new terms have a coupling log(N) and are local in thespin variables. Hence, their effect shouldn’t be dramatic for N small enough.As for practically any gauge theory one can extend the standard arguments [11] toshow that large N factorization will hold to any order in the coupling g. Since the groupis now U(N) one can embed in the link variables the group of lattice translations andachieve Eguchi-Kawai reduction [12].Because of the non-exponential structure of theaction, quenching should not be needed for any value of g. Quenching does appear to benecessary in the usual case when some of the new U(1) symmetries of the reduced modelget spontaneously broken by the attraction between the eigenvalues of the link matricesovercoming their kinematical repulsion [13].

Here this cannot happen because the actionwill have too weak an effect.The additional U(1)’s have to be preserved in order toensure that closed reduced loops have vanishing expectations when they correspond toopen original loops. The reduced model will consist of a finite number of matrices, withan action resembling the action of the original model.

Thus the partition function of thereduced model will be a polynomial in N and g. As a result, the purely planar contributionto the free energy per unit volume,1N2 log(Zreduced), will vanish.Let us now describe the reduction of the model in some more detail.The mainnew point to realize when generalizing from (hyper)cubic lattices is that reduction caneliminate only the degrees of freedom that are copies of each other by pure translations;one has therefore to identify the fundamental set of lattice points that generate the wholecrystal by translations only.We visualize the f.c.c. lattice as a cubic structure (with no sites yet) to which weadd vertices at the centers of all links and all cubes [14].

Each of the original cubes canbe cut into eight smaller cubes, each of which has four of its corners occupied and theother four free. Any two adjacent small cubes are mirror images of each other.

The duallattice is made out of vertices that sit at the unoccupied corners of the little cubes andat their centers. The bonds on this lattice connect these new centers to the new corners.Two adjacent cubes have two new corners in common and together with the two newcenters they build up an elementary rhombic plaquette.

The smallest shape enclosed bythe rhombi is a dodecahedron and the dodecahedra fill the space exactly.It is clear now that the dodecahedral lattice has at least two kinds of vertices, onewith eight links connected to it (a little cube corner) and another with only four (a littlecube center). What is slightly less obvious is that there are really two kinds of links withcoordination number equal to four, related to each other by reflection through a plane.These two kinds cannot be mapped one into the other by a pure translation.

We thereforeend up with a reduced model consisting of three vertices and eight oriented links. Theeight links start from a central vertex, C, and are connected in two groups of four totwo additional sites, referred to as L(eft) and R(ight).

On each of the links we have aU(N) matrix or its hermitian conjugate, depending on the direction we traverse the link.Denoting the C—L(R) four link variables by Uα (Vα) the partition function of the reduced4

model becomes:Zreduced =Z4Yα=14Yβ=1{dUαdVβ[1 + 2gNRe(TrUαU†β VαV †β )]}(6)Any loop on the original lattice can, modulo translations, be identified by a sequence of linktraversals after an arbitrary starting point has been picked on the loop. A C—R(L) linktraversal must be followed by a R(L)—C one respectively, but a R(L)—C link traversal canbe followed either by a C—R or a C—L one.

The sequence of link passages can be takenover to the reduced lattice. We only need to make sure now that sets of links that wouldcorrespond to a curve with different end points on the original lattice can be distinguished,even after reduction, from the reduced image of an originally closed curve.

For this we needadditional U(1) symmetries in the reduced model under which only the reduced images ofclosed curves will give a singlet after taking the U(N) trace. To ensure that a curve indeedcloses there are three conditions corresponding to the three independent coordinates ofthe “end point” that must be identical with the “starting point”.

Hence we need threeadditional U(1)’s. The U(1) “counting” goes as follows: The reduced model has five U(1)’s,Uα →eiφα+iψuUα;Vα →e−iφα+iψvVα.

(7)The original model had two non-gauge U(1) symmetries corresponding to the multipli-cation by a phase of all the C—R link variables and by another phase of all the C—Llink variables. These two U(1)’s are obviously present in the reduced model too, leaving5 −2 = 3 new U(1)’s, the exact needed number.Armed with the knowledge that factorization holds, one can now simply replay theEguchi-Kawai [12] derivation of the equivalence of the reduced model to the original one.As we already mentioned, there is no reason to suspect that the additional U(1)’s will breakspontaneously and therefore quenching won’t be necessary.

In view of the polynomial formof the action it is plausible that the model is essentially soluble and that explicit expressionsfor the expectation values of the traces of many Wilson loop operators can be written down.We are not going to pursue these matters any further here.Instead we turn to making several observations about the structure of the model.The partition function of the original model wouldn’t change if we change the spacethe link variables take values in from U(N) to SU(N) as long as N ≥4. Moreover, nochange in Wilson loop averages will occur if the Haar integration measure for each linkvariable is altered by multiplication by exp[ρ(Ul)] where ρ is a class function also invariantunder multiplication of its argument by an element of the center of the group.

A similarremark holds for the model of ref. [9].This shows that we have real sensitivity onlyto the center of the group, in accordance with one of the more popular mechanisms forconfinement.

Note that there is a difference between the case that the center is strictly Z2and when the center contains Z4. The Z3 case seems special and indeed its dual would bea Z3 spin model which, in three dimensions, in the simple cases, will have no continuousphase transitions.There is a non-trivial issue that has to be brought up regarding the expected impor-tance of the restriction |n(x) −n(x′)| ≤1 on the magnitude of the jump between nearest5

neighbors in the Z-ferromagnet. We would like the restriction to have no dramatic effectwhen β is very large, in particular not to have a deconfinement transition at a finite β.Superficially it seems that the restriction, if anything, will only aid confinement becauseit helps the n(x) →n(x) + n0 symmetry of the dual spin system to stay broken.

How-ever, there might be a flaw in the argument because the model can also be viewed as arestricted Z4 spin model. * To see this, let us work for the moment in a finite volumewith free boundary conditions and fix n(x0) at some site x0 to zero.

Consider the set of“pure gauge fields” (on the f.c.c. lattice) consisting of the differences n(x) −n(x′) acrossoriented bonds and associate to each such link the angle θ(x, x′) = π2 [n(x) −n(x′)].

Onecan think about these angles as a set of “pure gauge fields” for the gauge group Z4. Settingθ(x0) = 0 one can construct a unique Z4 spin configuration that would gauge transformθ(x, x′) to zero everywhere.

The set {θ(x)}θ(x0)=0 is in one to one correspondence withthe set {n(x)}n(x0)=0 if the angle θ(x) is not let to rotate by more than ninety degreesalong any bond. The difference between the restricted and unrestricted model can be alsoseen in another way: in the restricted model averages of Wilson loops that carry a chargelarger than two vanish exactly.If one thinks about the model as a model of real surfaces representing boundary freemembranes in a fluid one may interpret the additional factor of two per connected com-ponent as arising from the averaging over a degree of freedom internal to the surface.

Forexample, one could imagine that on each surface there lives an interacting two dimensionalIsing system whose self-coupling is infinite, but whose degrees of freedom are otherwisedecoupled from the medium. The surface entropy factor arises from summing over thetwo possible states of the magnetization in each connected component of the membrane.

**From this point of view one can generalize the Z2 model of ref. [9] even further by admittingp states per surface and increasing thus the entropy factor to p per connected component.When formulated in terms of bulk spin variables this model can be viewed as consistingof spins that can take values on a homogeneous Bethe lattice of coordination p. Whenmoving across an elementary bond a spin value can at most jump to a nearest neighbor onthe Bethe lattice.

The case we described in more detail in this note corresponds to p = 2.Suppose that the class of models discussed in the present note, as well as more tra-ditional formulations, all are related to each other by admitting a continuum limit that isdescribed by a string theory. Polyakov has conjectured that the three dimensional Isingmodel is described in the critical regime by a fermionic free string theory [1,17].

These twosituations are different: While the Ising string would describe a system that is known tobe completely described by an ordinary (but strongly interacting) field theory, the U(1)gauge case probably admits no continuum field theoretical description in the limit wherethe scale is set by the string tension (the regularized form of the field theory is more or lessa three dimensional Sine-Gordon model, hence perturbatively non-renormalizable). Thereexists a decorated loop operator in the Z2 case (at least on the cubic lattice) that obeys alinear loop equation (up to self-intersections, and these are not rapidly generated); thereexists no known analogue in the U(1) case (the Schwinger-Dyson equation for the Wilsonloop will rapidly generate self-intersections).

Our present note and the previous paper* Here we generalize some observations made in ref. [15].

** This case would represent a particular limit of a model studied in ref. [16].6

on the Z2 case [9] have shown that the models admit the introduction of a parameterthat might be interpreted as a “bare” string coupling constant; the critical properties ofthe models seem insensitive to small variations in this coupling in both cases, indicatingthat if a “physical” string coupling does make its appearance eventually, it will have anintrinsically determined value that cannot be tuned at will.It would be interesting to formulate precise numerical tests for the conjectures thateither theory is represented by a self-consistent, complete string theory. Some attemptsin this direction have been made in references [18].

The simplest approach conceptuallywould be to try to see some sign of Regge behavior, for example by identifying a few lowlying resonances of moderate spin. The U(1) case seems to be under good control numer-ically, beyond bulk properties, as the basic ideas about confinement have been recentlyconvincingly tested quantitatively [19], so there is some hope.

Since the f.c.c. lattice is astack of two dimensional triangular lattices the identification of states of higher spin mightbe easier here than in the cubic case.Our main purpose in the present note was to show that three dimensional gaugetheories are an interesting place to look for new understandings of systems of fluctuatingsurfaces or of string theories.Acknowledgements.

This research was supported in part by the DOE under grant #DE-FG05-90ER40559. FD would like to thank the theoretical particle physics group atRutgers University for hospitality while part of this work was done.

HN would like tothank E. Witten for several useful discussions.7

References[1] A. M. Polyakov, Gauge Fields and Strings, Contemporary Concepts in Physics,Volume 3, Harwood Academic Publisher, (1987). [2] A. M. Polyakov, Nucl.

Phys. B120 (1977) 429.

[3] M. E. Peskin, Ann. Phys.

(NY) 113 (1978) 122; R. Savit, Phys. Rev.

Lett. 39(1977) 55; T. Banks, R. Myerson, J. Kogut, Nucl.

Phys. B129 (1977) 493.

[4] J. Villain, J. Phys.

(Paris) 36 (1975) 581. [5] M. G¨opfert, G. Mack, Comm.

Math. Phys.

82 (1982) 545. [6] J.-M. Drouffe, J.-B.

Zuber, Phys. Rep. 102 (1983) 1.

[7] I. Kostov, Nucl. Phys.

B265[FS15] (1986) 223; K. H. O’Brien, J.-B. Zuber, Nucl.Phys.

B253 (1985) 621. [8] V. J. Emery, R. Swendsen, Phys.

Rev. Lett.

39 (1977) 1414. [9] F. David, Europhys.

Lett. 9 (1989) 575.

[10] T. Hofs¨ass, H. Kleinert, J. Chem. Phys.

86 (1987) 3565. [11] See Appendix B in [6].

[12] T. Eguchi, H. Kawai, Phys. Rev.

Lett. 48 (1982) 1063.

[13] G. Bhanot, U. M. Heller, H. Neuberger, Phys. Lett.

113B (1982) 47. [14] N. W. Ashcroft, N. D. Mermin, Solid State Physics Hoft-Saunders InternationalEditions, New York, NY (1976).

[15] E. Domany, D. Mukamel, A. Schwimmer, J. Phys.A. Math.Gen.13 (1980)L311.

[16] S. Leibler, D. Andelman, J. Physique 48 (1987) 2013. [17] A. M. Polyakov, Phys.Lett.

82B (1979) 247, Phys.Lett.103B (1981) 211;Vl. S. Dotsenko, A. M. Polyakov, in Advanced Studies in Pure Mathematics16, Academic Press (1988); Vl.

S. Dotsenko, Nucl. Phys.

B285 (1987) 45; A. R.Kavalov, A. G. Sedrakayan Nucl. Phys.

B285[FS19] (1987) 264; A. G. SedrakayanLAPP-TH-314/90 Annecy preprint (1990); E. Fradkin, M. Srednicki, L. Susskind,Phys. Rev.

D21 (1980) 2885; S. Samuel J. Math.

Phys. 21 (1980) 2806; C.Itzykson, Nucl.

Phys. B210[FS6] (1982) 477; P. Orland Phys.

Rev. Lett.

59(1987) 2393. [18] M. Caselle, R. Fiore, F. Gliozzi, P. Provero, S. Vinti, DFTT 26/90 Torino preprint(1990); R. Schrader Jour.

of Stat. Phys.

40 (1985) 533; G. M¨unster, in Prob-abilistic Methods in Quantum Field Theory and Quantum Gravity Editedby P. H. Damgaard et al. , Plenum Press, New York, (1990).

[19] A. Duncan, R. Mawhinney, Phys. Rev.

D43 (1991) 554; R. J. Wensley, J. D. Stack,Phys. Rev.

Lett. 63 (1989) 1764; T. Sterling, J. Greensite, Nucl.

Phys. B220(1983) 327; T. A. DeGrand, D. Toussaint, Phys.

Rev. D22 (1980) 2478.8

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