Note on Discrete Gauge Anomalies
논문은 고전적(global) 방정식을 대칭으로 하는 구별(discrete) 대칭을 가중(gauge)하는 문제를 다루고 있습니다. 구별 대칭이란 연속적인 변환을 반복하여 원점으로 회귀할 수 있는 변환을 말합니다. 구별 대칭은 이론물리학에서 다양한 상황에서 고려되어왔으며, 특히 스트링 이론( dây 이론)에서 유용하게 쓰입니다.
논문에서 Banks와 Dine는 구별 대칭이 가중되었을 때 발생하는 불일치(anomaly)를 연구합니다. 구별 대칭이 가중되면 불일치가 발생할 수 있으며, 이는 중력적 효과에 의하여 깨질 수 있습니다. 논문은 구별 대칭이 가중되어 발생하는 불일치를 두 가지 방법으로 다룹니다: 첫 번째는 instanton(점퍼온) 이론을 이용한 방법입니다. instanton 이론을 사용하여 구별 대칭의 불일치가 발생하지 않는다는 것을 보여주며, 이는 중력적 효과에 의해서 깨질 수 있음을 시사합니다.
두 번째 방법은 't Hooft 가 제안한 접근법으로, instanton이 가중된 구별 대칭을 위반할 경우를 고려합니다. 논문에서는 이 두 가지 방법의 결과를 비교하고, 구별 대칭이 가중되었을 때 발생하는 불일치에 관한 새로운 시각을 제시합니다.
결론적으로 논문은 구별 대칭이 가중되어 발생하는 불일치를 연구하였으며, 중력적 효과가 이러한 불일치를 깨는 것과 관련이 있음을 보여주었습니다. 이 연구는 스트링 이론 및 고전적 물리학에서 구별 대칭의 역할에 대한 새로운 이해를 제공합니다.
(8~10줄 요약)
구별 대칭은 이론물리학에서 중요한 역할을 합니다. 논문에서는 구별 대칭이 가중되어 발생하는 불일치를 연구하였습니다. 중력적 효과가 이러한 불일치를 깨는 것과 관련이 있음을 보여주었습니다. 두 가지 방법으로 구별 대칭의 불일치를 다루었으며, instanton 이론을 이용한 방법과 't Hooft 가 제안한 접근법을 비교하였습니다. 논문은 구별 대칭이 가중되었을 때 발생하는 불일치에 관한 새로운 시각을 제시하였습니다.
English summary:
The paper studies the anomaly of discrete symmetries when they are gauged in a theory. Discrete symmetries are transformations that can be repeated to return to the origin, and have been studied in various contexts in theoretical physics. In string theory, discrete symmetries often arise as relics of higher-dimensional general coordinate invariance or spontaneously broken gauge symmetries.
The authors consider two approaches to study the anomaly: one using instantons, and another based on 't Hooft's idea that instantons can break the symmetry. They compare the results from these two methods and provide a new perspective on the anomaly of discrete symmetries when they are gauged.
In conclusion, the paper studies the anomaly of discrete symmetries when they are gauged in a theory and shows that gravitational effects can break this anomaly. The study provides a new understanding of the role of discrete symmetries in string theory and classical physics.
(8~10줄 요약)
This paper investigates the anomaly of discrete symmetries when they are gauged in a theory. Two methods were used to analyze this anomaly: instanton-based approach and 't Hooft's method. The results from these two approaches were compared, providing new insights into the anomaly of discrete symmetries when they are gauged. Gravitational effects can break this anomaly, highlighting the importance of considering these effects in theoretical physics.
Note:
The original paper has some minor errors in grammar and spelling that have been corrected in the English summary.
Note on Discrete Gauge Anomalies
arXiv:hep-th/9109045v2 2 Oct 1991RU-91-42/SCIPP 91-30Note on Discrete Gauge AnomaliesTom BanksDepartment of Physics and AstronomyRutgers University, Piscataway, NJ 08855-0849, USAMichael DineSanta Cruz Institute for Particle PhysicsUniversity of California, Santa Cruz, CA 95064 USAWe consider the probem of gauging discrete symmetries. All valid constraints on suchsymmetries can be understood in the low energy theory in terms of instantons.
We notethat string perturbation theory often exhibits global discrete symmetries, which are brokennon-perturbatively.Submitted to Physical Review D, Brief ReportsSep. 1991
1. IntroductionGlobal discrete symmetries have been considered in particle physics in many contexts.While no theoretical argument convincingly rules out the existance of such symmetries,like all global symmetries they are viewed with a certain skepticism.Apart from thequestion of how such symmetries might arise, it is not at all clear that symmetries ofthis type would survive gravitational effects such as wormholes.
[1] Thus, as for continuoussymmetries, it is natural to consider the gauging of these symmetries.Discrete gaugesymmetries were introduced into physics by Wegner in the context of lattice theories. [2]They appear quite frequently in compactifications of string theory, where they are oftenrelics of higher dimensional general coordinate invariance or spontaneously broken gaugesymmetries.
[3] Discrete world sheet gauge symmetries play a role in the construction oforbifolds. In this context it is particularly clear that discrete gauge symmetry coincideswith the ancient mathematical procedure of constructing new spaces by modding out amanifold by the action of a discrete group.
Krauss and Wilczek [4] pioneered the study ofdiscrete gauge symmetries in four dimensional physics. They showed that such symmetrieswould give hair to black holes and would be immune to violation by quantum gravitationaleffects like wormholes.More recently Ibanez and Ross [5] have derived constraints on low energy theoriesby requiring that all discrete symmetries be gauged.
These constraints arise because ofthe possibility that the discrete symmetries may be anomalous. Their argument involvedembedding the low energy discrete symmetry in a continuous group which is spontaneouslybroken at some high energy scale.
The anomaly constraints on this continuous symme-try, combined with the constraints on discrete charges of fermions which gain mass uponspontaneous symmetry breakdown, give the Ibanez Ross (IR) constraints. IR found thatapplied to low energy supersymmetric models, these constraints are quite restrictive.Somewhat later, Preskill, Trivedi and Wise[6] pointed out that discrete gauge symme-tries are constrained by the requirement that the ’t Hooft interaction induced by instantonsof any continuous gauge symmetry in the theory be invariant under the discrete symmetrytransformation.11 The possibility that discrete symmetries can be broken by instantons has been appreciatedfor some time; it was mentioned to one of the present authors by E. Witten, c. 1981.
Anomalies indiscrete symmetries also been discussed by Weinberg and others in the framework of technicolor. [7]1
The question immediately arises whether the Ibanez Ross constraints are related tothose of PTWW. The IR constraints are stronger, but they were derived by postulatingan embedding in a particular high energy theory.
If the constraints depend on the methodof embedding, they are not useful constraints on a low energy effective theory. If not,one would expect to be able to derive them without any reference to the high energy em-bedding theory.
Since the only low energy constraints presently known are those whichfollow from requiring that instantons of the low energy group not break the symmetry, theIR constraints might suggest some new low energy phenomena. In the present note wewill study this question.
We find that those Ibanez Ross constraints which are nonlinearin the discrete charges2 can be violated in many embedding theories. Therefore they arenot required for consistency of the low energy theory.
Failure of these constraints at lowenergy implies only that a subgroup of the full unbroken discrete gauge symmetry of themodel leaves all the low energy fields invariant. Correspondingly, it predicts constraintson the spectrum of certain massive “fractionally charged” states.
The linear Ibanez Rossconstraints are not affected by this ambiguity. They follow simply from the PTWW cri-terion that instantons of the low energy theory not violate the symmetry.
They are thusrequired for consistency of the low energy discrete gauge theory.32. FRACTIONAL CHARGES AND NONLINEAR DISCRETE ANOMALIESThe IR derivation of the cubic discrete anomaly constraints is easy to recapitulate.Suppose for simplicity that we have a ZN discrete symmetry in a low energy theory.
Weimagine that the theory arose from the spontaneous breakdown of a U(1) gauge symmetryby a Higgs field of charge N. Assume that the ratio of any two U(1) charges in the theoryis rational. Then there is a charge q (not necessarily carried by one of the fields in thetheory) such that every charge is an integer multiple of q. Normalize the U(1) generatorso that q = 1.
If we arrange the spin one half fermions in the theory into a collection ofleft handed doublets, then the anomaly cancellation condition may be written:Xq3L = −(Xq3i +X¯qi3 +Xq3a)(2.1)2 We use terminology appropriate to an abelian discrete group. As we will see later, the correctconstraints can be stated in a way which does not depend on the nature of the group.3 In order to demonstrate this correspondence between PTWW and IR we have had to correcta factor of two in one of the linear IR equations, which makes the constraint somewhat stronger.2
On the left hand side of this equation we sum over the U(1) charges of all the states in thetheory which are left massless after spontaneous symmetry breakdown. The heavy stateson the right hand side are divided into those which get Majorana masses qa, and thosewhich pair up with another left handed field to make a Dirac mass term.
Since the massterms must be made gauge invariant by multiplying them by a single valued function ofthe Higgs field, the charges of the heavy fields satisfy:2qa = 0 mod N(2.2)qi + ¯qi = 0 mod N(2.3)From this it follows thatXq3L = mN + nN 38(2.4)where m and n are integers. There is nothing incorrect about this equation or its derivation.However, it does not refer solely to information about the low energy theory.
The integernormalization of charges may implicitly imply things about the high energy theory in whichthe light particles are embedded. In particular, suppose that in the above normalizationall of the light particles have charges which are multiples of an integer L which dividesN.
Then the effective symmetry group of the low energy theory is Z NL . The anomalyconstraint is nonlinear in the charges and the cubic anomaly constraint for Z NL is notsatisfied.
Similar remarks apply to bilinear constraints involving two ZN charges and alow energy U(1) generator.4Models in which the effective symmetry group of the full theory is larger than thatof the low energy theory are rather common.In string theory models constructed by“modding out” a conformal field theory by the action of a discrete symmetry have sectorstwisted under the action of the discrete group. These sectors need not contain any lightparticles (as is the case, for example, when one mods out a Calabi-Yau space by the actionof a freely acting group).
The symmetry acting in this sector can be larger than that ofthe original conformal field theory. For example, if the original conformal theory had aZN × ZM symmetry, and one mods out by the action of the ZN, the twisted sectors mayexhibit a ZN×M symmetry.4 As remarked by Ibanez and Ross, these constraints are rendered uninteresting anyway bythe ambiguity in normalization of U(1) charges.3
The nonlinear IR constraints are not totally devoid of interest. If we believe that agiven low energy discrete symmetry must be gauged, then their failure implies the exis-tences of new fractionally charged states and an enlarged symmetry group at high energy.5However, we have not found a way to rewrite the constraint so that it throws much lighton the nature of these states.
In general, there will be many ways to satisfy the constraintby adding different high energy sectors to the theory. For example, we can always makea ZN symmetry consistent by embedding it in a ZN2 theory in which all the low energyfields carry ZN2 charge which is equal to 0 mod N.3.
INSTANTONS AND DISCRETE GAUGE SYMMETRIESThe linear IR constraints do not suffer from the difficulty that we encountered in theprevious section. The rescaled constraints of the ZN theory are precisely those appropriateto the low energy Z NL theory.
It is easy to see that the linear constraint involving lowenergy nonabelian gauge groups is almost identical with that of PTWW, namely that the’t Hooft effective Lagrangian[8] be invariant under the discrete group. It is perhaps worthstressing that if this condition is not satisfied, not only is the symmetry broken in theone-instanton sector, but gauging the symmetry would give an inconsistent theory.
Thisfollows from ’t Hooft’s argument[8] that the effect of a dilute instanton-anti-instanton gason low momentum fermion Green’s functions in any topological sector can be summarizedby insertion of this effective Lagrangian.The PTWW constraint is stronger by a factor of 2, than that of IR, but we canextract this extra factor from the IR method as well.Indeed, the source of the extrafactor of 12 in IR’s equation is heavy Majorana fermions. All such fields must transformas real representations of the low energy nonabelian gauge group.
The Dynkin index ofany such representation is an even integer (in the normalization in which the Dynkinindex counts the number of fermion zero modes in an instanton with topological chargeone), and this gives an extra factor of two on the right hand side of the equation thatprecisely cancels the 12 coming from the discrete gauge charge of a Majorana field. Thus,5 Here we assume that there is some scale at which we can consider the discrete symmetryto be embedded in a four dimensional continuous gauge group.
Since we have shown that thenonlinear constraints depend on the nature of the high energy theory, it is not clear that theirimplications are the same when the symmetry comes from geometrical considerations as in KaluzaKlein theories.4
the corrected IR condition coincides exactly with the low energy PTWW condition, andis valid independently of the manner in which the theory is modified at high energy.This derivation of the discrete anomaly constraints makes it clear that they probeonly nonperturbative gauge dynamics, a fact which is obscured by the IR derivation.Indeed, from the low energy point of view, any discrete global symmetry can be gaugedin perturbation theory. It is the dilute instanton gas which violates anomalous discretesymmetries in weakly coupled theories.
The PTWW derivation also shows us that weshould only expect anomalies in discrete abelian groups that act by the same phase on allfermions in the same representation of the low energy nonabelian gauge group. Any othertransformation can be written as such a “flavor blind” phase times a transformation whichleaves the ‘t Hooft interaction invariant.Similar considerations apply to the linear gravitational anomaly of discrete symme-tries.
The linear IR constraint on discrete-gravitational anomalies can be derived by notingthat the minimal gravitational instanton which is a spin manifold (so that fermion fieldsare well defined), has two fermion zero modes per Weyl field. There is one weak point inthis argument for the discrete gravitational anomaly.
As for gauge instantons, the PTWWargument demonstrates the existence of a problem in a particular topological sector. Inthe gauge case we were able to promote this into an argument of inconsistency for the fulltheory by considering a dilute gas.
We do not know if a similar dilute gas argument worksin the gravitational case. [9] The mathematical classification of four dimensional gravita-tional instantons which might satisfy cluster decomposition has not yet been carried out.Even if we were to find such instantons, it is not completely clear that Euclidean consid-erations make sense in quantum gravity, where the action is unbounded from below.
Onthe other hand, we have examined many field theoretic and string theoretic models withgauged discrete symmetries and have not been able to find any which violate this condi-tion. We have not been able to find consistent high energy embeddings for low energytheories which violate the linear gravitational IR constraint, as we were able to do for thenonlinear constraints.
Thus we believe that it is probably correct as it stands.If we accept this argument, our considerations make it easy to generalize the linear IRconditions to discrete R symmetries in supergravity. Such symmetries arise, for example,in Kaluza-Klein theories and string theories, where a surviving discrete subgroup of thehigher-dimensional Lorentz symmetry will in general transform spinors non-trivially.
Todetermine the linear condition, we need only count the number of gravitino zero modes inthe background instanton field5
Gauge instantons have no gravitino zero modes, while a minimal gravitational in-stanton, with signature 16 has precisely 2. Thus the discrete-nonabelian gauge anomalycondition will remain the same for R symmetries.
The anomaly constraint for the gravi-tational anomaly of discrete R symmetries will be modified to2Xqi + 2q3/2 = 0 mod N(3.1)where the sum is over all of the fermionic fields belonging to chiral or gauge multipletsof supersymmetry, and q3/2 is the discrete charge of the gravitino. We emphasize thatfrom our point of view, the latter equation depends on an assumption about the spectrumof zero modes in allowed gravitational instanton backgrounds.
Since we do not have aclassification of clustering instantons in four dimensional supergravity, this analysis mustbe regarded as provisional. The true gravitational anomaly constraint will be that the‘t Hooft effective Lagrangian for quantum gravitational instantons be invariant under thediscrete symmetry that one is proposing to gauge.4.
Discrete Symmetries in String TheoryWe have mentioned string theory several times to illustrate the issues discussed inthis paper.String theory provides numerous examples of gauged discrete symmetries.One might try to turn the reasoning around and ask whether discrete gauge symmetriesin string theory are ever anomalous. Our interest here is in string models which are freeof perturbative anomalies, i.e.
modular invariant. We know of no general argument thatinsures that discrete gauge symmetries in such models are anomaly free.
On the otherhand, we have explained above that such an anomaly would signal an inconsistency. Thusit is possible that there is an additional consistency condition for string models, which isnon-perturbative in nature.We have examined a number of models for this possibility, and have indeed foundnumerous examples where the linear IR conditions are not satisfied.
However, in all ofthese cases, it it possible to cancel the anomaly. String compactifications always containat least one axion field, usually called the “model-independent axion,” which couples to thetopological charge of the various gauge groups.
In all of the examples we have examined,it is possible to cancel the anomaly by assigning a nonhomogeneous transformation lawto the axion under the discrete symmetry. In other words, an instanton in these theoriesgives rise to an expectation value for a fermionic operator, O, which is not invariant under6
the discrete symmetry.However, because the axion couples to the topological charge,O is multiplied by a factor of the form eia, where a is the axion field (in a suitablenormalization). If we assign a transformation law to the field a, of the form a →a+2πq/N(in the case of a ZN symmetry), the full instanton amplitude is gauge invariant.
Such anon-linear transformation law means that the gauge symmetry is spontaneously broken ata high energy scale (of order the Planck scale). Perturbation theory, on the other hand,exhibits an unbroken discrete symmetry to any finite order; this symmetry (which is nota gauge symmetry) is explicitly broken by non-perturbative effects.
This in itself may bephenomenologically interesting, since it suggests that it is natural to postulate approximatediscrete symmetries.It is perhaps worthwhile to give one example of the phenomenon we are describing.For this, consider the O(32) theory compactified on a textbook[3] example of a Calabi-Yau compactification, described by a quintic polynomial in CP 4. At a special point inthe moduli space, this model has a large discrete symmetry group, including four Z5symmetries.
It is straightforward to check that these symmetries all satisfy the linear IRconditions. Now mod out this theory by a freely-acting discrete symmetry.
In particular,ref. [3] defines a Z5 symmetry called A.
Include also a Wilson line. This Wilson line can bedescribed as follows.
In the fermionic formulation of the heterotic string, there are 26 free,left-moving fermions in this compactification. Group them as 6 complex fermions and 14real ones.
If α is a fifth root of unity, the Wilson line rotates three of the complex fermionsby α, three by α3, and leaves the rest untouched. This choice is modular invariant.
Itleaves an unbroken gauge group SU(3) × SU(3) × O(14) × U(1)2. It also leaves unbrokenthe four original Z5 symmetries.
A straightforward calculation shows that, for an instantonembedded in any of the three gauge groups, the appropriate operator O transforms as α2under each of the Z5 symmetries. Since the model-independent axion couples in the sameway to each of the gauge groups, letting a →a + 6π/5 cancels the anomaly.5.
ConclusionsIt is sometimes argued that any discrete symmetries which might play a role in lowenergy physics will be gauge symmetries. Following Ibanez and Ross, we have consideredthe constraints which must be satisfied if this is to be the case.
We have seen that onlyconditions which can be derived from low energy considerations (i.e. instantons of low en-ergy gauge groups and possibly gravitational instantons) hold independent of assumptionsabout the high energy theory.7
On the other hand, we have also provided some evidence that it makes sense, as inthe work of Preskill, Trivedi, Wilczek and Wise, to postulate discrete symmetries whichare broken only by small, non-perturbative effects. Indeed, we have seen that this is acommon phenomenon in string theory.
This is analogous to the situation with Peccei-Quinnsymmetries. In field theory, in both cases, it seems somewhat unnatural to postulate theexistence of symmetries which are broken “a little bit.” In string theory, this is a commonoccurrence.Finally, we have noted that the anomaly conditions may provide a non-perturbativeconstraint on string compactifications, but we have not exhibited a modular invariantmodel which fails to satisfy these conditions.
Similarly, we are not aware of any pertur-batively consistent string vacuum whose low energy field theory suffers from a nonper-turbative SU(2) anomaly. Perhaps in string theory perturbative anomalies are the wholestory.AcknowledgementsWe thank Nathan Seiberg for many helpful conversations.
This work was supportedin part by DOE grants DE-FG05-90ER40559. and DE-AM03-76SF00010.8
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