THREE DIMENSIONAL PERIODIC U(1) GAUGE THEORY AND STRINGS*
3차원 기간 U(1) 게이지 이론은 다중 미스너 효과로 인해 구둣질에 수렴합니다. 이 현상이 같은 물리학을 나타내는가에 대한 의문을 던지게 됩니다. semiclassicaal 계산을 통해 3차원 Sine-Gordon 모델의 동적과 관련성이 있으며, 일반적으로 이는 Z를 포함하는 단일 구성 스칼라장과 글로벌 시뮬레트리군이 있는 모든 로컬 모델의 상관성 클래스로 간주됩니다.
그러나 이 상관성 클래스에는 특정한 고정점이 없습니다. 이론을 계속 진행해 보면, 3차원 U(1) 게이지 이론은 string-like 유사 안정 상태를 보입니다. 이러한 유사 안정 상태의 파동 함수는 에너지 밀도가 1차원 곡선 위에 집중되어 있습니다.
이 연구자는, 세미클라스カル 계산을 통해 3차원 U(1) 게이지 이론과 Sine-Gordon 모델 사이의 상관성을 확인하였으며, 이를 string theory framework로 일반화할 수 있다는 의견을 제시합니다.
다만, 이러한 추론은 현존하는 field theory framework에서 universal feature를 추출하는 방법이 모호하다는 점에 주목해야 합니다.
한편, 3차원 이징 모델의 broken phase는 string theory framework로 설명할 수 있으나, 이는 ordinary field theory framework에서 이미 설명할 수 있는 상관성을 간주합니다.
결론적으로, 3차원 U(1) 게이지 이론은 string-like 유사 안정 상태를 보이며, 이를 string theory framework로 일반화하는 것이 가능함을 제시합니다.
한글 요약 끝.
THREE DIMENSIONAL PERIODIC U(1) GAUGE THEORY AND STRINGS*
arXiv:hep-th/9111055v2 27 Nov 1991RU–91–50Novemeber 26, 1991THREE DIMENSIONAL PERIODIC U(1) GAUGE THEORY AND STRINGS*byHerbert NeubergerDepartment of Physics and AstronomyRutgers UniversityPiscataway, NJ 08855-0849, U.S.A.AbstractIt will be argued that among the known systems in three dimensions that have stringlike excitations periodic U(1) pure gauge theories are the most likely candidates to lead toa string representation of their universal properties. Some recent work with F. David willalso be reviewed.
* Talk delivered at the International Symposium on Lattice Field Theory, Latt91; Na-tional Laboratory for High Energy Physics, Tsukuba, Japan; Nov. 5-10, 1991.1
Three dimensional periodic U(1) gauge theories confine due to the dual Meissnereffect [1,2]. Lattice formulations as well as continuum formulations lead to very similarphysics.
But this similarity has never been made very precise. On the basis of semiclassicalcalculations the underlying dynamics is seen to be connected to the three dimensionalSine-Gordon model.
Normally, one would view such a model as a representative of theuniversality class in three dimensions of all local models that have a single componentscalar field and a global symmetry group Z. However, no special fixed point is known inthis class – it seems that in the infrared the global symmetry either gets elevated to R(with some fine tuning) or, generically, disappears completely.If we accept that indeed no fixed point with a symmetry strictly Z exists in 3d, weconclude that the above similarity of confinement mechanisms cannot be made precise inany ordinary field theoretical continuum limit by the usual mechanism of RenormalizationGroup universality.
Of course, one can simply say that there is nothing strictly universalabout the 3d dual Meissner confining phase, period. In this talk I shall explore the oppo-site point of view, namely that there indeed is something universal, and only the correctframework for abstracting these universal features hasn’t been found yet.Any confining 3d pU(1) (three dimensional periodic U(1)) has string-like excitations.By this I mean quasi-stable states whose wave-functional shows a concentration of energydensity along a relatively smooth, one dimensional curve.The suggestion I would like to make is that maybe, the missing abstraction of theuniversal features of the various versions of the Meissner confinement mechanisms in theusual field theoretical framework can be found in a string theory.
Such a theory wouldhave, as fundamental objects, “bare” excitations associated with mathematical smoothclosed curves, which interact during their evolution by spanning two dimensional surfacesof higher genus embedded in 3d Euclidean or Minkowski flat space.There is little doubt that such surfaces cannot “go through each other” without aneffect on the Feynman probability amplitude. In ordinary field theory the paths describedby the point like excitations are transparent when they cross each other and this is crucialfor having a local field theory associated in a precise way to these paths.But, there is a possibility that the essential features of the effects of surfaces goingthrough each other, can be incorporated by adding more, intrinsic, degrees of freedomto the string.
What looks as interaction at surface crossings when the intrinsic degreesof freedom have been averaged out or frozen may appear as simple statistics when theintrinsic degrees of freedom are kept. For example, as emphasized recently by Polyakov,this is how the 2d Ising model reappears as a free field theory and most of this can begeneralized to the 3d Ising model.
To do this in a precise way is the crux of the matterwhen one attempts to associate a string theory with the strong coupling expansion surfacesappearing in any gauge model [3]. However, it may be that this cannot be done preciselyat all on the lattice, but, nevertheless, the few “glitches” that one gets are irrelevant inthe continuum limit and the correspondence to a string theory indeed does hold in thecontinuum.
For ordinary four-dimensional gauge theories the prospects for this to actuallywork in a direct way are pretty dim because the field theory has unmistakable point-particle-like behavior in the ultra-violet and this seems to be impossible to reproduce in atheory that is “stringy” all the way down to zero distance. If there exists a correspondence2
to a string theory it must be of a less direct kind [4]. A more direct correspondence mighthold for N=4 supersymmetric Yang Mills: its UV finiteness may permit tuning a couplingto a stringy “double scaling limit” for its Feynman diagrams organized in powers of 1/Ncwith Nc given by the gauge group.Nevertheless, Polyakov and others went ahead and suggested that the broken phaseside of the second order transition in the 3d Ising ferromagnet is describable by a stringtheory and some of the nontrivial critical exponents of the 3d Ising model can be calculatedwithin this string theory [5].This would be a very beautiful theoretical development;however, unlike in the case of pU(1), the universal features of the transition are in thiscase describable by an ordinary (albeit strongly interacting) field theory, and therefore thestring is not necessarily needed to abstract the generic features of the transition.3d pU(1) and 3d Ising are not that different; the main dissimilitude is that whileIsing strings are unoriented pU(1) strings do carry an orientation.
One may guess thatpU(1) strings are represented by some kind of a complexified version of the fermionicrepresentation of Ising strings. I am unaware of any specific attempt to derive such arepresentation (within some specific lattice model, for example).If either 3d Ising or 3d pU(1) gauge theories have some limit where a complete setof observables can be extracted and represented by a string theory one has to ask whatthe coupling constant of this string theory would be.
Without knowing exactly what theelementary excitations along the string are we cannot realistically hope to answer this ques-tion. Indeed, if we try to identify the surfaces appearing in the strong coupling expansionsof the gauge versions of either model with world histories of strings (an identification thathas a correct analogue in 2d), we very soon are faced with the occurrence of self-crossingsand singular lines which render ambiguous the genus to be associated with the so inflictedsurfaces [3].Let us adopt the following strategy then (in our search to identify a “bare” stringcoupling): As a first step find a model where all singular lines disappear; this can be doneat the price of absolute repulsion, i.e.
self-avoidance. It is an open and important questionwhether this self-avoidance can be replaced by some interaction of purely statistical origin.If this can be done, it is likely that the genus of the surfaces in the new theory will be,with probability one, equal to the sum over genera of the connected components of theself-avoiding surface.Having identified the genus of any strong coupling diagram it ispossible then to weigh the diagrams by a fixed factor raised to the power of the genus.This factor plays the role of a “bare” (i.e.
defined at the cutoffscale) string couplingconstant. Simple arguments show that, in the Ising case this amounts to the addition oflocal multispin interactions to the basic Ising action [6].
By field theory universality theseterms, if sufficiently weak, have no effect in the continuum limit. Hence, if there exists astring representation of the broken phase of continuum φ4 in 3d it has no freely adjustablestring coupling constant, and in that case is very unlikely to be a weakly coupled string,unless, for some special reason, the string coupling must actually vanish in which case weend up with a free string theory.It was pointed out by David [7] that formulating the Ising model on the f.c.c.
latticeproduces exactly the absolute self-avoidance effect that we required above; moreover, Davidshowed that the bare string coupling, if equal to1N with integer N, could be exactly3
incorporated into a O(N) lattice gauge theory with a single “plaquette” action of a specialform defined on the dual to the f.c.c. lattice, a lattice whose elementary cell is a rhombicdodecahedron (also the shape of the first Brillouin zone of a body centered cubic lattice).This “rhombic dodecahedral lattice” is not a Bravais lattice.In view of the above we would like to investigate whether an analogous formulationcan be found for the pU(1) case [8].
Again self-avoidance can be ensured by going to aparticular gauge theory defined on a rhombic dodecahedral lattice with U(N), N ≥4,gauge group.1N plays the role of a “bare” string coupling constant as before. The theoryis dual to a Z spin model on the f.c.c.
lattice with nearest neighbor interactions and witha restriction limiting the possible differences between neighboring spins to 0, ±1. It ishoped, but not at all established convincingly, that this model is in the same “pseudo–universality” class as the 3d Z-ferromagnet, or S.0.S., model.
Again 1N appears in the dualmodel via the coupling in front of a multi-spin local interaction. It is again plausible (butin the absence of field theoretical universality less compellingly true) that, if not too large,N has no measurable effect on the relevant long distance physics and therefore, again, thephysical coupling in the alleged string theory would be fixed.
Once more we end up withthe choice between a free string theory (due to some yet undiscovered symmetry principle)or a strongly interacting one.On the technical level the U(N) model is amusing because it admits Eguchi-Kawaireduction to an eight matrix model and needs no quenching. Thus, at N = ∞, severalthings can be exactly calculated.
In particular the planar contribution to the free energycan be shown to vanish.The question whether the model before reduction has phasetransitions when N increases has not been investigated to date.Both the O(N) and the U(N) gauge theories can be shown to be insensitive to theaddition to the action of terms that break local gauge invariance down to the centers ofthe respective gauge groups. So here confinement is explicitly solely center dependent.Moreover, what seems to matter about the center is only whether it is Z2 or Zk with anarbitrary k ≥4.
This is again in agreement with standard lore.I believe that the “RSOS” model David and I introduced [8] warrants a more detailedexamination and may provide us with some clues on whether the similarities betweenthe confinement mechanisms in various realizations of pU(1) gauge theories in 3d can beabstracted into something universal within the framework of string theory. Our presentknowledge, in particular the rigorous and beautiful work of G¨opfert and Mack [9] indicatethat this cannot be done by taking an ordinary, field theoretical, continuum limit.As a first step towards testing the above speculations one might try to use the semi-classical methods pioneered by Polyakov and investigate higher spin excitations in theinstanton gas.
If there is something exactly universal about pure pU(1) in 3d it most likelyinvolves higher spin states.Acknowledgements. This work has been supported in part by the U.S. Department ofEnergy through Contract No.
DE-FG05-90ER40559. I am grateful to Fran¸cois David fora fruitful collaboration.4
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