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Global Quantization in Gauge Orbit Space with

arXiv:hep-ph/9206265v2 18 Aug 1992June 26, 1992LBL-32531Global Quantization in Gauge Orbit Space withMagnetic Monopoles As a Solution to Strong CPProblem and the Relevance to UA(1) Problem ∗Huazhong ZhangTheoretical Physics GroupLawrence Berkeley LaboratoryMS 50A-3115, 1 Cyclotron RoadBerkeley, California 94720∗This work was supported by the Director, Office of Energy Research, Office of High Energyand Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy underContract DE-AC03-76SF00098.

Disclaimerii

AbstractWe generalize our discussions and give more general physical applica-tions of a new solution to the strong CP problem with magnetic monopolesas originally proposed by the author1. Especially, we will discuss aboutthe global topological structure in the relevant gauge orbit spaces to beclarified.

As it is shown that in non-abelian gauge theories with a θ term,the induced gauge orbit space with gauge potentials and gauge functionsrestricted on the space boundary S2 has a magnetic monopole structureand the gauge orbit space has a vortex structure if there is a magneticmonopole in the ordinary space. The Dirac’s quantization conditions inthe quantum theories ensure that the vacuum angle θ in the gauge theoriesmust be quantized.

The quantization rule is given by θ = 2π/n (n ̸= 0)with n being the topological charge of the magnetic monopole. There-fore, the strong CP problem is automatically solved in the presence of amagnetic monopole of charge ±1 with θ = ±2π, or magnetic monopolesof very large total topological charge (|n| ≥1092π) if it is consistent withthe abundance of magnetic monopoles.

Where in the first case with amagnetic monopole of topological charge 1 or -1, we mean the strong CP-violation can be only very small by the measurements implemented sofar. Since θ = ±2π correspond to different monopole sectors, the CP cannot be conserved exactly in strong interactions in this case.

In the secondcase, the strong CP cannot be conserved either for large but finite n. Thefact that the strong CP-violation measured so far can be only so small orvanishing may be a signal for the existence of magnetic monopoles. Wealso conjecture that the parity violation and CP violation in weak interac-tion fundamentally may intimately connected to the magnetic monopoles.The relevance to the UA(1) problem is also discussed.

The existence ofcolored magnetic monopoles may also solve the UA(1) problem. In thepresence of U(1) or monopoles as color singlets, the ’t Hooft’s solution to1

the UA(1) problem is expected. The quantization formula for the vortexstructure is also derived.

In the presence of a magnetic monopole of topo-logical charge n ̸= 0 in non-abelian gauge theories, the relevant integralfor the vortex along a closed loop in the gauge orbit space is quantized as4Nπ/n with integer N being the Pontryagin topological number for therelevant gauge functions.2

1IntroductionSince the discovery of Yang-Mills theories2, particle physics has gained greatdevelopment in the frame work of non-abelian gauge theories. One of the mostinteresting features of particle physics is the non-perturbative effects in gaugetheories such as instanton3 effects and magnetic monopoles1,4−5.

One of theother most interesting features in non-abelian gauge theories is the strong CPproblem in QCD. It is known that, in non-abelian gauge theories a Pontryaginor θ term,Lθ =θ32π2ǫµνλσF aµνF aλσ,(1)can be added to the Lagrangian density of the system due to instanton effects.With an arbitrary value of θ, it can induce CP violations.

However, the in-teresting fact is that the θ angle in QCD can be only very small (θ ≤10−9) orvanishing6. Where in our discussions of QCD, θ denotes θ+arg(detM) effectivelywith M being the quark mass matrix, with the effects of electroweak interac-tions are included.

One of the most interesting approaches to solve the strongCP problem has been the assumption of an additional Peccei-Quinn U(1)P Qsymmetry7. In this approach, the vacuum angle is ensured vanishing due to theaxions8−9 introduced.

But there has not been observational support6 to the ax-ions which are needed in this approach. Therefore, it is of fundamental interestto consider other possible solutions to the strong CP problem.One of the main purposes of this paper is to generalize the discussions ofa non-perturbative approach proposed5 by the author to solve the strong CPproblem and some relevant applications.

This is due to its physical importanceas well as the other physical relevance. The section 3 has some overlapping withour brief note Ref.

5, this is essential for the completeness due to its intimateconnection to the other aspects of the gauge theories in our discussions. Our3

approach is to show that the existence of magnetic monopoles can ensure thequantization of the θ angle and thus can provide the solution to the strong CPproblem. We will extend the formalism of Wu and Zee10 for discussing the effectsof the Pontryagin term in pure Yang-Mills theories in the gauge orbit spaces inthe Schrodinger formulation.

This formalism is useful to the understanding oftopological effects in gauge theories, it has also been used with different meth-ods to derive the mass parameter quantization in three-dimensional Yang-Millstheory with Chern-Simons term10−11. Wu and Zee showed10 that the Pontrya-gin term induces an abelian background field in the gauge configuration spaceof the Yang-Mills theory.In our discussions, we will consider the case withthe existence of a magnetic monopole.

Especially, we will show that magneticmonopoles in the space will induce an abelian gauge field with non-vanishingfield strength in gauge configuration space, and there can be non-vanishing mag-netic flux through a two-dimensional sphere in the gauge orbit space. Then, theDirac quantization conditions4−5 in the corresponding quantum theories ensurethat the relevant vacuum angle θ must be quantized.

The quantization rule isderived as θ = 2π/n with n being the topological charge of the monopole tobe given. Therefore, the strong CP problem is automatically solved with theexistence of magnetic monopoles of charge ±1, or monopoles with very largetotal magnetic charges (n ≥1092π).

As we will see that an interesting featurein our derivation is that the Dirac quantization condition both in the ordinaryspace and the relevant induced gauge orbit space will be used. The relevance tothe UA(1) problem will also be discussed.

We will also discuss about the vortexstructure in the gauge configuration space in this case. As we will show thatthe vortex in the gauge orbit space must be quantized also intimately connectedto the quantization rule for the vacuum angle θ.

In the presence of a magneticmonopole of topological charge n ̸= 0, the relevant integrals for the vortex along4

a closed loops in the gauge orbit space are quantized as 4Nπ/n with integers Nbeing the Pontryagin topological numbers for the relevant gauge functions.This paper will be organized as follows.Next, we will first give a briefdescription of the Schrodinger formulation for our purpose. Then in section 2,we will clarify the topological results relevant to our discussions.

In section 3,we will show the existence of the monopole structure in the relevant gauge orbitspace and realize the relevant topological results explicitly. In section 4, we willdiscuss about the monopole structure as a solution to the strong CP problemand its relevance to the UA(1) problem.

The section 5 will be mainly discussionsof the vortex structure in the gauge orbit space in the presence of a magneticmonopole. Our conclusions will be summarized in section 6.We will now consider the Yang-Mills theory with the existence of a magneticmonopole at the origin.

The Lagrangian of the system with the θ term is givenbyL =Zd4x{−14F aµνF aµν +θ32π2ǫµνλσF aµνF aλσ}. (2)We will choose Weyl gauge A0 = 0.

This is convenient since effectively A0 is notrelevant to abelian gauge structure in the gauge configuration space with the θterm included. The conjugate momentum corresponding to Aai is thenπai = δLδ ˙Aai= ˙Aai +θ8π2ǫijkF ajk.

(3)In the Schrodinger formulation, the system is similar to the quantum systemof a particle with the coordinate qi moving in a gauge field Ai(q) with thecorrespondence10−11qi(t) →Aai (x, t),(4)Ai(q) →Aai (A(x)),(5)5

whereAai (A(x)) =θ8π2ǫijkF ajk. (6)Thus there is a gauge structure with gauge potential A in this formalism withina gauge theory with the θ term included.

According to this, the system canbe described by a Hamiltonian equation10 or in the path integral formalism11.We will not discuss about this here, since we only need the Dirac quantizationcondition for our purpose. For details, see Ref.

10 and 11.2Relevant Topological ResultsIn our discussions, We will use the convention in Ref. 10 and differential forms12where A = Aai Ladxi, F = 12F ajkLadxjdxk with F = dA + A2, and tr(LaLb) =−12δab in a basis {La | a = 1, 2, ..., rank(G)} for the Lie algebra of the gaugegroup G. In quantum theory, the Schrodinger formulation is described in thegauge orbit space with the constraint of Gauss’ law.

Let U denote the gaugeconfiguration space consisting of all the well-defined gauge potentials A thattransform as Ag = g−1Ag + g−1dg under a gauge transformation with gaugefunction g. Denoting by G the space of all the continuous gauge transformations,the gauge orbit space U/G is the quotient space of the gauge configurationspace with gauge potentials connected by continuous gauge transformations asequivalent. In the presence of a magnetic monopole, generally a singularity-freegauge potential may need to be defined in each local coordinate region.

Theseparate gauge potentials in an overlapping region can only differ by a continuousgauge transformation5. In fact, the single-valuedness of the gauge function inthe overlapping regions corresponds to the Dirac quantization condition5.

For amonopole at the origin, one can actually divide the space outside the monopoleinto two overlapping regions. At a given r, the regions are two extended semi-6

spheres around the monopole, with θ ∈[π/2 −δ, π/2 + δ](0 < δ < π/2) inthe overlapping region, where the θ denotes the θ angle in the spherical polarcoordinates.As we will see that our equation for our quantization rule for the θ isdetermined by the integration on the space boundary which is topologically a2-sphere S2 for non-singular monopoles. Thus for the quantization of θ, therelevant case is that gauge potentials and gauge functions are restricted on thespace boundary S2.

We will call the induced spaces of U, G and U/G with Aand g restricted on the space boundary as restricted gauge configuration space,restricted space of gauge transformations and the restricted gauge orbit spacerespectively. Collectively, they will be called as the restricted spaces, and theunrestricted ones will be called as usual spaces.

We will use the same notationU, G and U/G for both of them for convenience, there should not be confusing.Our discussions for the monopole and vortex structures will be on the restrictedand usual spaces respectively.The topological discussions and the results we will now give are true bothfor the usual spaces and the restricted spaces. Since U is topologically trivialboth for the usual and restricted gauge configuration spaces as we will see.To establish the topological results we need, we note first that U is topo-logically trivial, thus ΠN(U) = 0 for any N. This is due to the fact that theinterpolation between any two gauge potentials A1 and A2At = tA1 + (1 −t)A2(7)for any real t is also a gauge potential, thus At ∈U (Theorem 7 in Ref.9, andRef.6).

since At is transformed as a gauge potential in each local coordinateregion, and in an overlapping region, both A1 and A2 are gauge potentials maybe defined up to a gauge transformation, then At is a gauge potential which maybe defined up to a gauge transformation in the overlapping regions, or At ∈U.7

The space U can be considered as a bundle over the base space U/G withfiber G. More generally for a bundle β = {B, P, X, Y, ¯G} with bundle space B,base space X, fiber Y, group ¯G, and projection P, let Y0 be the fiber over x0 ∈X,and let i : Y0 →B and j : B →(B, Y0) be the inclusion maps. Then we havethe homotopy sequence13 of (B, Y0, y0) given byΠN(Y0)i∗−→ΠN(B)j∗−→ΠN(B, Y0)∂∗−→ΠN−1(Y0)i∗−→ΠN−1(B) (N ≥1), (8)where ∂is the natural boundary operator, i∗, j∗and ∂∗are maps induced by i, jand ∂respectively.

Let P0 denote the restriction of P as a map (B, Y0, y0)→(X, x0, x0).Then P0j is the projection p : (B, y0) →(X, x0, B). We have the isomorphismrelationp∗: ΠN(B, Y0) ∼= ΠN(X, x0).

(9)Defining ∆∗= ∂(P0∗)−1 : ΠN(X, x0) −→ΠN−1(Y0, y0), the exact homotopysequence can be written asΠN(Y0, y0)i∗−→ΠN(B)p∗−→ΠN(X, x0)∆∗−→ΠN−1(Y0, y0)i∗−→ΠN−1(B) (N ≥1). (10)Now for our purpose with B = U, X = U/G, Y = G, and ¯G = G for thegauge group G. The choice of the base points x0 and y0 are irrelevant in ourdiscussions, since all the relevant homotopy groups based on different pointsare isomorphic.

Note that homotopy theory has also been used to study theglobal gauge anomalies 14−22, especially by using extensively the exact homotopysequences of fiber bundles and in terms of James numbers of Stiefel manifolds.More explicitly, we can now consider the following exact homotopy sequence13:ΠN(U)P∗−→ΠN(U/G)∆∗−→ΠN−1(G)i∗−→ΠN−1(U) (N ≥1). (11)Since as we have seen that ΠN(U) = 0 for any N, we have0P∗−→ΠN(U/G)∆∗−→ΠN−1(G)i∗−→0 (N ≥1).

(12)8

This implies thatΠN(U/G) ∼= ΠN−1(G) (N ≥1). (13)As shown by Wu and Zee for the usual spaces in pure Yang-Mills theory in fourdimensions,Π1(U/G) ∼= Π0(G)(14)is non-trivial, and thus θ term induces a vortex structure in gauge orbit space.This isomorphism will also be used in our discussions of the vortex structurein the presence of a magnetic monopole, but as we will see that it’s explicitrealization is more non-trivial.

It was also showed in Ref. 10 that the fieldstrength F associated with the gauge potential A is vanishing, and thus thereis no flux corresponding to F in the pure Yang-Mills theory.However, as we will show in the next section that in the presence of a mag-netic monopole, the relevant topological properties of the system are drasticallydifferent.

This will give interesting consequences in the quantum theory. One ofthe main topological result we will use for the restricted spaces in the presenceof a magnetic monopole isΠ2(U/G) ∼= Π1(G).

(15)Now Π2(U/G) ̸= 0 corresponds to the condition for the existence of a mag-netic monopole in the restricted gauge orbit space. In the next section, we willrealize the above topological results.

We will first show that in this case F ̸= 0,and then demonstrate explicitly that the magnetic fluxRS2 ˆF ̸= 0 can be non-vanishing in the restricted gauge orbit space, where ˆF denotes the projection ofF into the restricted gauge orbit space.9

3Monopole Structure in the Restricted Gauge OrbitSpace in the Presence of Magnetic MonopolesIn our discussions, we denote the differentiation with respect to space variablex by d, and the differentiation with respect to parameters {ti | i = 1, 2...}which A(x) may depend on in the gauge configuration space by δ, and assumedδ + δd=0. Then, similar to A = Aµdxµ with µ replaced by a, i, x, the gaugepotential in the gauge configuration space can be written as a 1-form given byA =Zd3xAai (A(x))δAai (x).

(16)Using Eq. (6), this givesA =θ8π2Zd3xǫijkF ajk(x)δAai (x) = −θ2π2ZM tr(δAF),(17)with M being the space manifold.

Since A is an abelian, then the field strengthis given byF = δA. (18)With δF = −DA(δA) = −{d(δA) + AδA −δAA}, we haveF =θ2π2ZM tr[δADA(δA)] =θ4π2ZM dtr(δAδA) =θ4π2Z∂M tr(δAδA),(19)up to a local term with vanishing projection to the relevant gauge orbit space.Usually, one may assume A →0 faster than 1/r as x →0 , then this would giveF = 0 as in the case of pure Yang-Mills theory10.

However, it is more subtle inthe presence of a magnetic monopole. Asymptotically as r →0 with a monopoleat the origin, the monopole may generally give a field strength of the form4−5,22Fij =14πr2ǫijk(ˆr)kG0(ˆr),(20)with ˆr being the unit vector for r, and this gives A →O(1/r) as x →0.Thus, one can see easily that a magnetic monopole can give a nonvanishing fieldstrength F in the gauge configuration space.10

To evaluate F, one needs to specify the space boundary ∂M in the presenceof a magnetic monopole. We now consider the case that the magnetic monopoledoes not generate a singularity in the space.

Then the effects in the case thatmonopoles are singular will be discussed. In fact, non-singular monopoles may bemore relevant in the unification theory since there can be monopoles as a smoothsolution of a spontaneously broken gauge theory similar to ’t Hooft Polyakovmonopole4.

For example, it is known that23 there are monopole solutions in theminimal SU(5) model. When the monopole is non-singular, the space boundarythen may be regarded as a large 2-sphere S2 at the spatial infinity.

For ourpurpose, we actually only need to evaluate the projection of F into the gaugeorbit space. But the evaluation of F can give more explicit understanding ofthe topological properties of the system.

The F is similar to a constant F in theordinary space, it does not give any flux through a closed surface in the space U.However, the quantum theory is based on the gauge orbit space in Schrodingerformulation , the relevant magnetic flux needs to considered in the gauge orbitspace. In fact, as we will see that the corresponding magnetic flux in the gaugeorbit space can be non-vanishing.

A gauge potential in the gauge orbit spacecan be written in the form ofA = g−1ag + g−1dg,(21)for an element a ∈U/G and a gauge function g ∈G. Then the projection ofa form into the gauge orbit space contains only terms proportional to (δa)n forintegers n. We can now writeδA = g−1[δa −Da(δgg−1)]g.(22)Then we obtainA = −θ2π2ZM tr(fδa) +θ2π2ZM tr[fDa(δgg−1)],(23)11

where f = da + a2. With some calculations, this can be simplified asA = ˆA +θ2π2ZS2 tr[fδgg−1],(24)whereˆA = −θ2π2ZM tr(fδa),(25)is the projection of A into the gauge orbit space.

Similarly, we haveF =θ4π2ZS2 tr{[δa −Da(δgg−1)][δa −Da(δgg−1)]}(26)orF = ˆF −θ4π2ZS2 tr{δaDa(δgg−1)+Da(δgg−1)δa−Da(δgg−1)Da(δgg−1)}, (27)whereˆF =θ4π2ZS2 tr(δaδa),(28)is the projection of the F to the gauge orbit space or the restricted gauge orbitspace based on the space boundary S2.Now all our discussions will be based on the restricted spaces. To see thatthe flux of ˆF through a closed surface in the gauge orbit space U/G can benonzero, we will construct a 2-sphere in it.

Consider an given element a ∈U/G,and a loop in G. The set of all the gauge potentials obtained by all the gaugetransformations on a with gauge functions on the loop then forms a loop C1 inthe gauge configuration space U. Obviously, the a is the projection of the loopC1 into U/G.

Now since Π1(U) = 0 is trivial, the loop C1 can be continuouslyextended to a two-dimensional disc D2 in the U with the boundary ∂D2 = C1.Obviously, the projection of the D2 into the gauge orbit space with the boundaryC1 identified as a single point is topologically a 2-sphere S2 ⊂U/G. With theStokes’ theorem in the gauge configuration space, We now haveZD2 F =ZD2 δA =ZC1 A.

(29)12

Using Eqs. (24) and (29) with δa = 0 on C1, this givesZD2 F =ZC1 A =θ2π2trZS2ZC1[fδgg−1].

(30)Thus, the projection of the Eq(30) to the gauge orbit space givesZS2 ˆF =θ2π2trZS2{fZC1 δgg−1},(31)where note that in the two S2 are in the restricted gauge orbit space and theordinary space respectively. This can also be obtained byZD2 trZS2 tr{δaDa(δgg−1) + Da(δgg−1)δa −Da(δgg−1)Da(δgg−1)} = 0,(32)or the projection ofRD2 F givesRS2 ˆF.

We have verified this explicitly or thetopological result that the projection ofRD2 F givesRS2 ˆF. For this one needsto use Stokes theorem in the ordinary space and the gauge configuration spacewith dδ + δd = 0, a ∈U/G or a is a constant on C1, andRD2 ˆF =RS2 ˆF in thegauge orbit space since ˆF is the projection of the F into the gauge orbit space.In quantum theory, Eq.

(31) corresponds to the topological result Π2(U/G) ∼=Π1(G) for the restricted spaces. This feature in the gauge orbit space has somesimilarity to that given in Refs.10 and 11 for the discussions of three-dimensionalYang-Mills theories with a Chern-Simons term.

We only need the Dirac quan-tization condition here for our purpose. In the restricted gauge orbit space, theDirac quantization condition givesZS2 ˆF = 2πk,(33)with k being integers.

We will now determine the quantization rule for the θ.Now let f be the field strength 2-form for the magnetic monopole. There maybe many ways to obtain non-vanishing results for the right-hand side.

For ourpurpose, one way is to restrict g to a U(1) subgroup of the gauge group, and13

obtain a non-zero topological number. Then the quantization rule for the θ willbe obtained.Let {Hi | i = 1, 2, ..., r = rank(G)} denote a basis of the Cartan subalgebrafor the gauge group G. The corresponding simple roots and fundamental weightsare denoted by {αi | i = 1, 2, ..., r} and {λi | i = 1, 2, ..., r} respectively.

Thenwe have242 < λi, αj >< αj, αj > = δij,(34)where the < λi, αj > denotes the inner product in the root vector space. Bythe theorem which states that for any compact and connected Lie group G ,any element in the Lie algebra is conjugate to at least one element in its Cartansubalgebra by a group element in G, the quantization condition for the magneticmonopole is given by23exp{ZS2 f} = exp{G0} = exp{4πrXi=1βiHi} ∈Z.

(35)WhereG0 =ZS2 f = 4πrXi=1βiHi(36)is the magnetic charge up to a conjugate transformation by a group element.Now let g(t) t ∈[0, 1] be in the following U(1) subgroup on the C1g(t) = exp{4πmtrXi,j=1(αi)jHj< αi, αi >},(37)with m being integers. In fact, m should be identical to k according to ourtopological result Π2(U/G) ∼= Π1(G) for the restricted spaces.

In this case, therelevant homotopy groups obtained are isomorphic to Z which may be only asubgroup of the homotopy groups generally for a non-abelian gauge group G. Infact for this case, the k and m should be identical since they correspond to thetopological numbers on each side. Using tr(HiHj) = −12δij andZC1 δgg−1 = 4πmrXi,j=1(αi)jHj< αi, αi >,(38)14

we obtainθ = 2πn (n ̸= 0). (39)Where we define generally the topological charge of the magnetic monopole asn = −2 < δ, β >= −2rXi=1< λi, β >,(40)which must be an integer by the quantization condition23 for the magneticmonopoles.

Where the δ is given byδ =rXi=12αi< αi, αi > =rXi=1λi. (41)The minus sign is due to our normalization convention for the Lie algebra gen-erators.

Actually, the fundamental weights {λi | i = 1, 2, ..., r} and {2αi<αi,αi> |i = 1, 2, ..., r} form the Dynkin basis and its dual basis in the root vector spacerespectively.In our definition, the topological charge of the magnetic monopole can beunderstood as follows. Up to a conjugate transformation, the magnetic chargeof the monopole is contained in a Cartan subalgebra of the gauge group.

Re-stricting to each U(1) subgroup generated by a generator Hi (i=1, 2,...,r) in thebasis of the Cartan subalgebra, the monopole has a topological number ni cor-responding to the Dirac quantization condition. Then generally the topologicalnumber n in our definition is given byn =rXi=1ni.

(42)Obviously, we expect that this is the natural generalization of the topologicalcharge to the non-abelian magnetic monopole. To the knowledge of the author,such an explicit general definition Eq.

(40) in terms of the fundamental weightsof the Lie algebra for the topological charge of non-abelian magnetic monopolesis first obtained by the author.15

As a remark, our derivation has been topological. Our quantization rulecan also be obtained by using constraints of Gauss’ law.

This more physicalapproach and the discussion of its physical relevance will be given elsewhere.4Magnetic Monopoles as A Solution to the Strong CPProblem and the Relevance to the UA(1) ProblemAs we have seen that in the presence of magnetic monopoles, the vacuum angleθ must be quantized. The quantization rule is given by Eq.(39).

Therefore, weconclude that the existence of magnetic monopoles can provide a solution to thestrong CP problem. In the presence of magnetic monopoles with topologicalcharge ±1, the vacuum angle of non-abelian gauge theories must be ±2π, theexistence of such magnetic monopoles gives a solution to the strong CP problem.The existence of many monopoles can ensure θ →0, and the strong CP problemmay also be solved.

In this possible solution to the strong CP problem withθ ≤10−9, the total magnetic charges present are |n| ≥2π109. This may possiblybe within the abundance allowed by the ratio of monopoles to the entropy26,but with the possible existence of both monopoles and anti-monopoles, the totalnumber of magnetic monopoles may be larger than the total magnetic charges.Generally, one needs to ensure that the total number is consistent with theexperimental results on the abundance of monopoles.In the above discussions, we consider the case that magnetic monopolegenerates no singularity in the space, for example, with monopole as a smoothsolution in a spontaneously broken gauge theory.

If we consider the magneticmonopoles as a singularity similar to the Wu-Yang monopole27 which is the firstnon-abelian monopole solution found, then the space boundary can be regardedas consisting of an infinitesimal inward 2-sphere around each magnetic monopole16

and a large 2-sphere at the spatial infinity. In the space outside the monopole, each infinitesimal inward sphere effectively gives a contribution equivalent toa monopole of opposite topological charge.

Then the total contribution of theinfinitesimal spheres and the contribution from the large sphere at the spatialboundary are all cancelled in the relevant integrations.Therefore, only theexistence of non-singular magnetic monopoles may provide solution to the strongCP problem.Moreover, note that our conclusions are also true if we add an additional θterm in QED with the θ angle the same as the effective θ in QCD if there existDirac monopoles as color singlets, or a non-abelian monopoles with magneticcharges both in the color SU(3) and electromagnetic U(1). Then the explana-tion of such a QED θ term is needed.

The effect of the term proportional toǫµνλσFµνFλσ in the presence of magnetic charges was first considered28,29 relevantto chiral symmetry. Then, the effect of a similar U(1) θ term was discussed forthe purpose of considering the induced electric charges30 as quantum excitationsof dyons associated with the ’t Hooft Polyakov monopole and generalized mag-netic monopoles23,31.

Especially, the generalized magnetic monopoles are usedto consider the possibility of quarks as dyons in a spontaneously broken gaugetheory31. An interesting feature is that31 if quarks are dyons in a spontaneouslybroken gauge theory, then their electric charges will not be exactly fractionallyquantized, instead they will carry extra charges proportional to θ. Moreover,two meson octets, one baryon octet and one baryon decuplet free of magneticcharges were constructed31 from quarks as dyons.

For our purpose, we expectthat if a QED θ term is included, it may be possibly an indication of unificationfor the color gauge symmetry and electromagnetic U(1) symmetry. A θ termneeds to be included in the unification gauge theory since Π3(G) = Z for theunification group G, monopoles with magnetic charges involving the QED U(1)17

symmetry are generated through spontaneous gauge symmetry breaking. Gen-erally, such an arbitrary θ term in QED may not be discarded since it is nota total divergence globally in the presence of magnetic charges and as we haveseen that the θ term physically can have non-perturbative effects.We would like to emphasize that in our approach with magnetic monopolesas a solution to the strong CP problem, the UA(1) problem can also be solved.The UA(1) problem was originally solved by t’ Hooft3 with the fact that theUA(1) is not a symmetry in the quantum theory due to the axial anomaly32, theconserved UA(1) symmetry is not gauge invariant and its spontaneous breakingdoes not generate a physical light meson.In our solution with the presenceof colored magnetic monopoles, the UA(1) symmetry is explicitly broken28.

Ifthe strong CP problem is solved by the pure electromagnetic U(1) monopolewith a θ term included with θ being the same as the effective θQCD, then theUA(1) symmetry is not explicitly broken and the t’ Hooft’s solution to the UA(1)problem can be applied. Thus UA(1) problem can be solved in our approach tosolve the strong CP problem with the existence of magnetic monopoles.5Vortex Structure in the Gauge Orbit SpaceIn this section, we will discuss about the vortex structure in the gauge or-bit space.It is known that33 there can be vortex structures in some three-dimensional field theories with the boundary of the space being topologicallya circle.

The discussions in Ref.10 in the gauge orbit space are for the pureYang-Mills theory with a θ term. We will consider the case in the presence ofmagnetic monopoles.In order to discuss about the vortex structure in the gauge orbit space inthe gauge theories we are interested in, we need to consider the integration of ˆA18

along a closed loop ˆC in the gauge orbit space U/G. As in Ref.10, such a loopˆC can be constructed by projection.

Let C denote an open path in the gaugeconfiguration space U with gauge potentials A and Ag as the two end points,where g ∈G is a gauge function. Obviously, the projection of C into the U/Ggives a closed loop ˆC with the two end points of C identified as a single point.Thus we have10 topologicallyZˆCˆA ∼=ZC A.

(43)In pure Yang-Mills theory, one can verify that10 the A can be written as the dif-ferentiation of the Chern-Simons secondary topological invariant34 in the gaugeconfiguration space. Thus in the case of pure Yang-Mills theory, the Chern-Simons secondary invariant can be regarded as a gauge function in the gaugeconfiguration space.

Now in the presence of a magnetic monopole, we obtainδ{θW[A]} = −θ2π2ZM tr(δAF) +θ4π2ZM dtr(AδA),(44)orA = δ{θW[A]} −θ4π2ZM dtr(AδA),(45)whereW[A] = −14π2ZM tr(AdA + 23A3)(46)is the Chern-Simons secondary topological invariant. This givesZC A =ZC δ{θW[A]} −θ4π2ZCZ∂M dtr(AδA),(47)orZC A = θ{W[Ag] −W[A]} −θ4π2ZCZS2 tr(AδA).

(48)NowW[Ag]−W[A] =112π2ZM tr(g−1dg)3+ZM dα2[A, g] = 2N[g]+ZS2 α2[A, g], (49)19

whereN[g] =124π2ZM tr(g−1dg)3,(50)andα2[A, g] = −14π2tr(Ag−1dg). (51)ThusZC A = 2θN[g] −θ4π2ZS2 tr(Ag−1dg) −θ4π2ZCZS2 tr{AδA}.

(52)Now since C is an open path in U from A to Ag, the integralRC A generallycontains two parts. The first part is topologically invariant as we will see, thesecond term depends only on the end points A and Ag or the gauge function g.The third therm or the second and third term together is generally path depen-dent, but it does not contain any non-vanishing topological invariant, namely itis a path-dependent local term.

This can be seen as follows. Since the space Uis topologically trivial, the open path C in U with the two end points fixed canbe continuously deformed into the straight intervalAt = tAg + (1 −t)At ∈[0, 1].

(53)We only need to verify this by evaluating the integral with the C being thestraight interval. Then topologicallyZC tr{AδA}∼= −Z 10 At(Ag −A)dt = −tr{(Ag −A)Z 10 [(tAg + (1 −t)A)dt}= −tr{12(Ag −A)(Ag + A)} = −tr[(Ag)2 −2AAg −A2] = 2tr(AAg).

(54)With Ag = g−1Ag + g−1dg, this can be written asZC tr{AδA} ∼= 2tr{Ag−1Ag + Ag−1dg}. (55)Thus topologicallyZC A ∼= 2θN[g] + I2,(56)20

whereI2 = −θ4π2ZS2 tr{(2Ag−1Ag + 3Ag−1dg}. (57)One can now easily see that the second part I2 of the integral contains no non-vanishing topological invariant.

Since A and g are independent each other, theg as a mapping g : S2 →G can be continuously deformed into a constantmapping. The I2 is topologically equivalent to an integral with the integrandproportional to trA2 = 0.

Thus, up to topologically trivial terms, we obtainZˆCˆA ∼= 2θN[g]. (58)It is known that the integral N[g] is topologically invariant when M is compact-ified as a 3-sphere S3.

It is straightforward to show that N[g] is topologicallyinvariant with g as a mapping g : M →G from the space manifold to the gaugegroup G in our discussion. The only change is that a small variation for thegauge function gives an additional boundary term which is vanishing due to thefact that the space boundary ∂M is topologically a 2-sphere and Π2(G) = 0.To obtain non-vanishing results for N[g], we need to restrict to the gauge func-tions with g →0.

Then for the space manifold is effectively compactified asa 3-sphere S3, and N[g] is the Pontryagin topological number corresponding tothe homotopy group Π3(G). ThusZˆCˆA ∼= 2θN,(59)with N being integers.

This corresponds to the isomorphism relationΠ1(U/G) ∼= Π0(G) = Π3(G) = Z,(60)where G is the space of all the gauge transformations g satisfying g →0. Withthe quantization rule θ = 2π/n we obtained, we can writeZˆCˆA ∼= 4Nπn(n ̸= 0),(61)21

with n being the topological charge of the monopole. Therefore, there can bevortex structure in the gauge orbit space.

In the presence of magnetic monopolesthe vortex must be topologically quantized with the quantization rule given bythe above equation. In the presence of a monopole of topological charges ±1,the vortex is quantized as ±4πN.

In the presence of many monopoles with verylarge total topological charges n, the vortex can be only very small or vanishing.Our discussions in the presence of magnetic monopoles are more non-trivial thanthe case in pure Yang-Mills theory , especially for the explicit realization of thetopological isomorphism due to the local terms involved.In the above discussions, the magnetic monopoles are regarded non-singularin the space. In fact, one can easily see that in the presence of singular monopolesthe quantization rule is given by Eq.

(59), but as we have seen that in this caseθ can be arbitrary.As a remark, note that in QED or more generally an abelian gauge theorywith N = 0 since Π3(U(1)) = 0, there is no corresponding topological vortex inthe gauge orbit space even in the presence of magnetic monopoles.6ConclusionsWe have discussed extensively about the topological structure in the relevantgauge orbit space of gauge theories with a θ term. The presence of a magneticmonopole in the ordinary space can induce monopole and vortex structures inthe restricted and usual gauge orbit spaces.

The Dirac quantization conditionsensure that the vacuum angle θ must be quantized. The quantization can pro-vide a solution to the strong CP problem with the existence of one monopole oftopological charge ±1, or many monopoles if it is consistent with the abundanceof magnetic monopoles.

The UA(1) problem may also be solved with the exis-22

tence of colored magnetic monopoles, or by t’ Hooft’s solution if the magneticmonopoles are of only U(1) charges as color singlets. Therefore, the fact thatthe strong CP-violation can be only so small or vanishing may be a signal for theexistence of magnetic monopoles.

An interesting feature is that in the presenceof one magnetic monopole of charge ±1, θ = ±2π according to the quantizationrule obtained. The cases of n = ±2 may also possibly solve strong CP problem.But when the vacuum angle is θ = ±π, other than the Strong CP problem itmay have other effects35 different from the case of θ = ±2π or vanishing, forexample, on quark masses, but these are usually discussed without the presenceof monopoles.

In the axion approach to solve the strong CP problem, the vac-uum angle should be vanishing, and there has been argument35 that the vacuumenergy is minimized at vanishing vacuum angle.We have also derived the quantization formula for the vortex by using ourquantization rule for the θ angle. Thus, as we have shown that the monopolestructure and vortex structure in the restricted gauge orbit spaces and the usualgauge orbit spaces are connected through our quantization rules.As a remark, note that usually if strong interaction conserves CP, then the2π or π may be expected equivalent to −2π or −π since they are related by aCP operation and θ may be expected to be periodic with period 2π.

However,according to our quantization rule, this is not true due to the fact that ±2πor ±π correspond to different monopole sectors.If the strong CP problemis solved by a monopole of topological charge ±1 or ±2, this means the CP-violation can be only very small in the measurements implemented so far, theCP can not be exactly conserved, since the θ = ±2π or θ = ±π correspondto two different physical systems. If the strong CP problem is solved due tothe existence of many monopoles, then the observation of strong CP violationgives an indirect measurement of the abundance of magnetic monopoles.

For23

any finite number of magnetic monopoles, the CP cannot be exactly conservedin the strong interactions. The strong CP-violation may provide informationabout the structure of the universe.As a conjecture, we expect that the parity violation and CP violation inweak interaction may intimately connected to magnetic monopoles also.The author would like to express his gratitude to Y. S. Wu and A. Zee forvaluable discussions and suggestions.

The author is also grateful to O. Alvarezfor his invitation.24

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