Monopoles, Strings, and Necklaces in $SO(10)$ and $E_6$

We employ a variety of symmetry breaking patterns in $SO(10)$ and $E_6$ Grand Unified Theories to demonstrate the appearance of topological defects including magnetic monopoles, strings, and necklaces. We show that independent of the symmetry breaking pattern, a topologically stable superheavy monopole carrying a single unit of Dirac charge as well as color magnetic charge is always present. Lighter intermediate mass topologically stable monopoles carrying two or three quanta of Dirac charge can appear in $SO(10)$ and $E_6$ models respectively. These lighter monopoles as well as topologically stable intermediate scale strings can survive an inflationary epoch. We also show the appearance of a novel necklace configuration in $SO(10)$ broken to the Standard Model via $SU(4)_c\times SU(2)_L\times SU(2)_R$. It consists of $SU(4)_c$ and $SU(2)_R$ monopoles connected by flux tubes. Necklaces consisting of monopoles and antimonopoles joined together by flux tubes are also identified. Even in the absence of topologically stable strings, a monopole-string system can temporarily appear. This system decays by emitting gravity waves and we provide an example in which the spectrum of these waves is strongly peaked around $10^{-4}~{\rm Hz}$ with $\Omega_{\rm gw}h^2\simeq 10^{-12}$. This spectrum should be within the detection capability of LISA.


Introduction
Grand Unified Theories (GUTs) such as SU (4) c × SU (2) L × SU (2) R (422, for short) [1], SU (5) [2], SO(10) [3], and E 6 [4] predict the existence of topologically stable magnetic monopoles [5]. The mass and the magnetic charge carried by the monopoles depends on the underlying GUT and its symmetry breaking pattern. For instance, in breaking SU (5) to the Standard Model (SM) gauge group, the lightest monopole carries one unit of Dirac magnetic charge (and also color magnetic charge) [6], and it weighs about ten times the GUT scale M GUT . In contrast, a 422 breaking to the SM yields a stable monopole with two units of Dirac charge [7], and its mass depends on the scale of the 422 breaking which can be lower, even significantly so, than the standard GUT scale M GUT ∼ 10 16 GeV. Another interesting example of GUT scale and lighter monopoles comes from E 6 breaking via the trinification group SU (3) c × SU (3) L × SU (3) R (333, for short). This breaking produces a GUT scale Z 3 monopole that carries one unit of Dirac magnetic charge [8], as we shall verify later. The subsequent breaking of 333 to the SM gauge group yields a stable intermediate mass monopole which carries three quanta of Dirac magnetic charge [8].
The presence of topologically stable strings in these models depends on the Higgs fields that are employed to implement the symmetry breaking. A prime example is the appearance of Z 2 strings if SO(10) is broken to the SM using only tensor representations [9]. The gauge Z 2 symmetry in this case happens to be subgroup of the Z 4 center of SO (10). In supersymmetric SO(10) this Z 2 is precisely equivalent to matter parity which, among other things, provides a stable cold dark matter candidate, namely the lightest sparticle.
Composite topological defects can also appear in many GUTs and some well-known examples include monopole-antimonopole pairs connected by a string (dumbbells) [10], walls bounded by strings [11], and necklaces with monopoles acting as beads kept together on a string [12]. Consider, for instance, the breaking of SO(10) to 422 with a 54-plet of Higgs. This leaves unbroken a discrete symmetry, called C-parity, which interchanges the left and right components of any representation, accompanied by charge conjugation [11]. Under C the electric charge operator Q → −Q [11,13]. This breaking of SO (10) to 422 yields Z 2 strings. However, the subsequent breaking of the 422 symmetry to the SM group necessarily breaks this C-parity, and the strings form boundaries of domain walls [11]. Such walls can be tolerated in realistic scenarios provided they are unstable and disappear before their energy density becomes the dominant component in the universe. Another well known option, if available, is to inflate away the domain walls. It is interesting to note that observation of walls bounded by strings in 3 He has been reported recently in Ref. [14]. An example of a necklace made up of monopoles and antimonopoles connected by a Z 2 string is provided by the symmetry breaking SO(10) → SU (5) × U (1) → SU (5) × Z 2 where the last step is achieved by a Higgs 126-plet of SO (10). We will demonstrate the appearance of a new type of necklace if SO(10) breaking occurs via 422.
Of great interest, of course, is the question as to whether any of these primordial topological defects exist in nature, having either survived inflation or making an appearance after the inflationary epoch. It has been recognized [15,16] for some time that monopoles associated with an intermediate scale M I that is comparable to H, the Hubble scale during inflation, may be present in our galaxy at an observable level. This can come about if the number of e-foldings experienced during the intermediate scale phase transition is around 25-30, rather than the 50-60 e-foldings experienced by the GUT scale phase transition. Intermediate scale cosmic strings, on the other hand, can appear either in the same way as the monopoles, or even after the end of inflation. The current bound from Cosmic Microwave Background Radiation measurements on the dimensionless string tension is given by Gµ s 3.2 × 10 −7 [17], where G denotes Newton's constant and µ s is the mass per unit length of the string. Somewhat more stringent constraints on Gµ s based on pulsar timing observations have been reported in Ref. [18].
In this paper, we discuss topological defects in GUTs, with emphasis on SO(10) and E 6 (see also Ref. [19,20] for a recent discussion on related topics). In Sec. 2 we show the presence of a GUT monopole carrying one unit of Dirac magnetic charge in SO(10) models, which is independent of the symmetry breaking pattern. Analogous to the SU (5) case, this monopole carries some color magnetic charge. We break the SU (4) c × SU (2) L × SU (2) R symmetry to the SM in two steps and show how an intermediate mass monopole carrying two units of the Dirac charge (Schwinger monopole) emerges from a coalescence of SU (4) c and SU (2) R monopoles bound together by flux tubes in a dumbbell configuration. This symmetry breaking pattern of 422 also yields a new type of necklace configuration consisting of alternating SU (4) c and SU (2) R monopoles connected by suitable flux tubes. A variety of other configurations is also possible including a necklace made of monopoles and antimonopoles connected by a Z 2 string. In Sec. 3 we show the presence of the GUT Dirac monopole also in E 6 models and discuss the E 6 breaking via 333, which leads to intermediate scale monopoles with three units of Dirac magnetic charge and possibly to non-superconducting stable strings. In Sec. 4 we analyze the E 6 breaking via SO(10) × U (1) ψ and show how unstable strings as well as stable strings or necklaces can appear. Sec. 5 presents a quantitative discussion of how intermediate scale monopoles, strings, and necklaces in realistic models can survive primordial inflation. In addition we discuss how gravity waves emitted by some defects may be accessible with the space based observatory LISA. Our conclusions are summarized in Sec. 6.
We will first study the breaking of SO(10) via 422 [21]. The 210 representation of SO(10) is contained in 16 × 16 = 1 + 45 + 210, and so the 422 singlet in 210 comes from (4, 1, 2)× its conjugate and (4, 2, 1)× its conjugate. One combination of these singlets gives the SO(10) singlet, and the other the 422 singlet in 210. The latter is the antisymmetric combination of these singlets and thus breaks the discrete C-parity which interchanges SU (2) L and SU (2) R and conjugates SU (4) c (C-parity, first found in Ref. [11], was later called D-parity in Ref. [22]). This is clear since the SO(10) singlet cannot break C, which belongs to SO (10), and thus it is bound to be the symmetric combination. On the other hand, the 422 singlet in the 54-plet of SO (10) comes from the product 10 × 10 = 1 s + 45 a + 54 s . Thus it originates from (1, 2, 2) × (1, 2, 2) or (6, 1, 1) × (6, 1, 1), which are both symmetric under C. One combination of them is the SO(10) singlet and the orthogonal combination is contained in 54, and so the 54-plet does not break the discrete symmetry C [11].
We choose here to employ a Higgs 210-plet for the SO(10) breaking to 422 so that no strings or subsequent walls bounded by strings are generated as in Ref. [11]. It is known [7] that the (-1,-1,-1) element of 422 coincides with the identity, and therefore three lines, one in each of the three groups between 1 and -1 constitute a closed loop, which corresponds to a magnetic monopole. We will now show that this monopole evolves to the Dirac monopole after the electroweak symmetry breaking. It carries one unit of magnetic charge as well as some color magnetic charge. (This conclusion appears to be in disagreement with Table III in Ref. [20] where it is stated that the monopole is unstable.) To make the analysis more transparent, we take the curve in SU (4) c along its X ≡ (B − L) + 2T 8 c /3 generator, where T 8 c = diag(1, 1, −2) in SU (3) c and B and L are the baryon and lepton number operators respectively. This choice is certainly equivalent to taking the curve along the generator B − L since color is unbroken. It is easy to see that X = diag(1, 1, −1, −1) in SU (4) c and the curve between 1 and -1 corresponds to a rotation by π along this generator. In SU (2) L and SU (2) R , we take rotations by π along T 3 L = diag(1, −1) and T 3 R = diag(1, −1) respectively, and the overall loop therefore corresponds to a rotation by 2π along . It is clear that this rotation brings us back to the identity element. Indeed, the group element exp(i2πT 8 c /3) = exp(2iπ/3) lies in the center of SU (3) c and exp(2iπQ) = exp(4iπ/3) acting on up-type quarks or exp(−2iπ/3) acting on down-type quarks and so the combined rotation leads to the identity element. The magnetic monopole corresponding to a rotation by 2π along the generator Q + T 8 c /3 is exactly the Dirac magnetic monopole as previously shown in Ref. [6]. Note that, in this paper, the SM was embedded in SU (5), but the argument holds for any compact group containing SU (5). The Dirac magnetic monopole, along with the ordinary magnetic field, also carries color magnetic field.
We can further break these two U (1)'s by the VEV of the ν c -type SM singlet component in a Higgs 16-plet, which leaves X + T 3 R unbroken (ν c represents right-handed neutrinos). To find the broken generator which is perpendicular to X + T 3 R , we must define the GUT normalized generators Figure 1: Emergence of (Schwinger) magnetic monopole with two units of Dirac charge from the symmetry breaking SU This monopole also carries color magnetic charge. An SU (4) c (red) and an SU (2) R (blue) monopole are connected by a flux tube which pulls them together to form a Schwinger monopole. The magnetic flux along the tube and the Coulomb magnetic fluxes of the monopoles are indicated. Intermediate mass monopoles such as this one may survive inflation.
Then the normalized unbroken and broken generators U and B are, respectively, The smallest broken generator with integral charges so that its periodicity is 2π is X − 2T 3 R . A rotation along this generator by 2π/3 is left unbroken by the VEV of the ν c -type Higgs. Therefore, the generated string contains magnetic flux corresponding to a rotation by 2π/3 along (X − 2T 3 R ). The magnetic fluxes of an SU (4) c and an SU (2) R monopole have to be rearranged in tubes with flux (X − 2T 3 R )/3 and Coulomb fluxes along the unbroken generator X + T 3 R . To this end, an SU (4) c monopole, which carries a full flux along X, sends 1/3 of it to an SU (2) R monopole which carries a full T 3 R flux. This latter monopole, in turn, sends 2/3 of its flux to the other one and thus a tube is generated between them which pulls them together. The rest of the fluxes are added together to give the Coulomb flux of a doubly charged (Schwinger) monopole -see Fig. 1. Note that the 1/3 of the X flux sent from the SU (4) c monopole towards the SU (2) R monopole to contribute to the tube in between cannot terminate on it but emerges as Coulomb flux from it. The same is true for the 2/3 of T 3 R flux sent from the SU (2) R monopole to the SU (4) c monopole. Finally, we have four fluxes (two corresponding to rotations by 4π/3 and 2π/3 along X, and two corresponding to rotations by 2π/3 and 4π/3 along T 3 R ), combined together to emerge as Coulomb flux from the combined monopole. This monopole corresponds to a full (2π) rotation along X + T 3 R and becomes a Schwinger monopole after the electroweak breaking. Needless to say the SU (4) c or SU (2) R monopoles can be connected by a string to their respective antimonopoles and annihilate.
Note that and thus this unbroken element belongs to the unbroken continuous subgroup, i.e. the SM group. Consequently, no unbroken discrete symmetry is left, which means that no topologically stable strings are produced since the first homotopy (fundamental) group of the vacuum manifold We only have dumbbells [10] which can transform into Schwinger monopoles.
If we inflate away the SU (4) c and SU (2) R monopoles, we can have a network of topologically non-stable strings. After the electroweak breaking, the Higgs doublets h u , h d (h u couples to the up-type quarks and h d to the down-type ones) with X = 0 and T 3 R = 1, −1, T 3 L = −1, 1 respectively develop VEVs. As we circle a string they get a phase −4π/3, 4π/3 respectively. If we add to the string 1/3 of flux along T 3 L so that the string corresponds to a rotation by 2π/3 along X − 2T 3 R + T 3 L , the phases of h u and h d change by −2π and +2π respectively around the string. Of course, this addition does not affect the ν c -type VEV of the Higgs 16-plet and also adds the minimal necessary magnetic energy on the string. For definiteness, we will assume throughout that the magnetic energy dominates over the Higgs contribution to the string energy and so these strings are superconducting [23]. We obtain left-moving and right-moving fermionic zero modes along the string via h u , h d which are the only Higgs fields coupling to quarks and charged leptons. Note that the ν c -type Higgs field couples only to right-handed neutrinos and thus does not contribute to superconductivity. Now suppose that we use the ν c ν c -type component of 126 to do the breaking of X − 2T 3 R . In this case, a rotation by 2π/6 along X − 2T 3 R leads to an unbroken element. This yields a string which contains magnetic flux corresponding to a rotation by 2π/6 along X minus flux where the last step is achieved by a 126-plet of SO (10). Notation as in Fig. 1. We display explicitly only the Coulomb magnetic flux of two of the monopoles and the magnetic flux along one of the tubes. This necklace may survive inflation.
corresponding to a rotation by 2π/3 along T 3 R . An SU (4) c and an SU (2) R monopole are then connected by two such strings with the remaining Coulomb flux in them being (X + T 3 R )/3 and 2(X + T 3 R )/3. Now if one imagines opening up one of the two strings, one finds the two monopoles connected by one string and two "loose" strings emerging from the two monopole. One can then connect these latter strings to other similar monopole-string structures in series and form necklaces [12] -see Fig. 2. Note that pairs of SU (4) c and SU (2) R antimonopoles connected by a string can also participate in the necklace with the SU (4) c antimonopole connected either to an SU (4) c monopole or an SU (2) R antimonopole, and the SU (2) R antimonopole connected either to an SU (2) R monopole or SU (4) c antimonopole. Also both tubes emerging from an SU (4) c monopole (SU (2) R antimonopole) can terminate on SU (4) c antimonopoles (SU (2) R monopoles). We thus see that a variety of necklaces can appear with different arrangements of SU (4) c and SU (2) R monopoles and antimonopoles.
The group element which we obtain by circling one of these strings does not belong to the SM group since its action where the last step is achieved by a 126-plet of SO (10). We assume that the monopoles from the first step of symmetry breaking are inflated away. We display explicitly only the Coulomb magnetic flux of one monopole and one antimonopole and the magnetic flux along one of the tubes. This necklace may survive inflation.
on the SM singlet ν c yields exp(iπ) = −1. Moreover, since (1, 1, −1) = (−1, −1, 1) and SU (2) L is unbroken at this stage, this element is equivalent to the generator of the Z 2 subgroup of U (1) B−L [9]. Its square is then obviously equivalent to the identity, and an extra Z 2 symmetry remains unbroken. Stable Z 2 strings without monopoles on them are also present. These strings, exactly like the ones in the necklaces above, correspond to a rotation by 2π/6 along X − 2T 3 R and are not oriented. The necklaces are themselves Z 2 strings too.
Next let us see what happens after the electroweak symmetry breaking. Recall that the Higgs doublets h u , h d have X = 0 and T 3 R = 1, −1, T 3 L = −1, 1 respectively. As we go around the string they acquire a phase −2π/3, +2π/3 respectively. If we add to the string -1/3 of flux along T 3 L such that the string corresponds to a rotation by 2π/6 along X − 2T 3 R − 2T 3 L , h u , h d remain constant around the string. Of course, this addition does not affect the ν c ν c -type VEV of 126 and also adds the minimal necessary magnetic energy along the string. Thus, only the ν c ν c -type component of 126 changes phase around the string. But this couples only to right-handed neutrinos and so these strings are not superconducting.
For an example of a monopole-antimonopole necklace formed with a Z 2 string, consider the following SO(10) breaking pattern: The first breaking, achieved by the VEVs of a 210-plet and a 45-plet along their (1,1,1) and (1,1,3) components respectively, produces a GUT monopole with one unit of Dirac magnetic charge, which presumably is inflated away. Of course, multiply charged monopoles may also be produced and inflated away. In particular, the doubly charged monopole coincides with the SU (2) R monopole we mentioned above since the corresponding loops in SU (4) c and SU (2) L are homotopically trivial. The second breaking, achieved by the VEV of the (15,1,1) component of a Higgs 45-plet, yields an intermediate scale SU (4) c monopole which carries both SU (3) c and U (1) B−L magnetic fluxes. The last breaking is done by the ν c ν c -type component of a Higgs 126-plet and the SU (4) c monopoles can form, together with the antimonopoles, a necklace tied together by a Z 2 string. Namely, an SU (4) c monopole, which carries a full magnetic flux along X, rearranges its magnetic field to form two tubes with flux (X − 2T 3 R )/6 and a Coulomb field around it with flux 2(X + T 3 R )/3. Since the SU (2) R monopoles are inflated away in this case, these tubes can only terminate on SU (4) c antimonopoles -see Fig. 3.
The latter violates C. Both these singlets can acquire VEVs and thus C will be broken, and we expect that no strings or walls bounded by strings [11] associated with C are generated.
The fundamental representation of E 6 is where with the rows being3's of SU (3) L and the columns 3's of SU (3) R , and which are an SU (3) L triplet and an SU (3) R antitriplet respectively.
One can very easily verify that the element c = (exp(i2π/3), exp(−i2π/3), exp(−i2π/3)) of the unbroken trinification subgroup H coincides with the identity element as it acts like the identity on the 27-plet and, consequently, on all the representations of E 6 . The generator of the second homotopy group π 2 (E 6 /H) = π 1 (H) = Z 3 of the vacuum manifold E 6 /H is then a loop that connects (1,1,1) with c, i.e. three curves in the three SU (3)'s from 1 to exp(i2π/3), or 1 to exp(−i2π/3), or 1 to exp(−i2π/3) respectively. Obviously, the third power of this loop is homotopically trivial, and the breaking E 6 → 333 therefore generates Z 3 magnetic monopoles.
In order to understand the structure of these Z 3 monopoles, we define the generators T 8 Note that we use integer elements in these definitions so that a full rotation by 2π along these generators closes a circle. We see that (1/6) It is easy to check that the electric charge operator, by applying it on the various states in 27 (Y is the weak hypercharge). Finally, we see that the generator of π 1 (H) is a rotation by 2π along T 8 c /3 + Q, exactly as in the SO(10) case. As a consequence, the Z 3 monopole in E 6 , similarly to the Z 2 monopole in SO(10), carries one (Dirac) unit of ordinary magnetic flux or charge as well as color magnetic flux corresponding to the generator of the center of SU (3) c . As shown in Ref. [6] this is the ordinary Dirac monopole also carrying color magnetic charge.
We can further break 333 to SU (3) c × SU (2) L × SU (2) R × U (1) B−L (3221 B−L , for short) by giving a VEV to the N -type component of (1,3, 3) in a Higgs 27-plet. The generator in Eq. (11) remains in the unbroken subalgebra since The orthogonal broken generator is but a rotation by 2π/4 along this generator leaves N invariant and thus remains unbroken. Adding to this a rotation by 2π/4 along the unbroken generator T 8 L + T 8 R , we get an equivalent rotation by 2π/2 along T 8 L . This rotation corresponds to the group element exp(i2πT 8 L /2) = diag(−1, −1, 1) in SU (3) L , which belongs to the continuous part of the unbroken subgroup 3221 B−L . This means that no additional discrete symmetries are left unbroken. In other words, the unbroken subgroup is precisely 3221 B−L .
The second homotopy group of the vacuum manifold π 2 (333/3221 B−L ) = π 1 (3221 B−L ) 333 , which means that it consists of the homotopically non-trivial loops in 3221 B−L which are trivial in 333. The minimal loop is a 6π rotation along the generator T 8 c /3 + Q, and so the loop in SU (3) c becomes homotopically trivial and can be removed. Only the rotation along Q by 6π remains, which corresponds to a monopole with triple the ordinary magnetic charge and no color magnetic flux at all.
The subsequent breaking of SU (2) R ×U (1) B−L to U (1) Y does not generate any new topological objects provided that it is done by an SU (2) R Higgs doublet, analogous to the electroweak breaking -for a detailed explanation of this fact, see Ref. [24]. This breaking can be achieved by the VEV of a Higgs 27 along the ν c -type component of it. This belongs to an SU (2) R doublet with B − L = 1 and generates no topological defects.
The U (1) ψ intersects with SO(10) in its Z 4 center. This center is generated by −iΓ 10 , where is the chirality operator in ten Euclidean dimensions. Here we use the notation of Ref. [25], which follows the notation of Ref. [26]. The SO(10) 16-plet (1 +5 + 10) is of negative chirality and so the SU (5) singlet 1 corresponds to all σ's being -1, the5 to only one of them being -1 and all others +1, and the 10 to three of them being -1 and the rest +1. So under −iΓ 10 , 16 → i16 and, consequently, 10 → −10 and 1 → 1. Now It is easy to see that the sum of σ's coincides with the χ charge since it gives -5 for the SU (5) singlet 1, -1 for the 10, and 3 for5. So the generator −iΓ 10 of the center of SO(10) lies in U (1) χ and corresponds to a rotation by −2π/4 along it.
Also, a rotation by 2π/4 along ψ acts on the SO(10) representations as follows: 16 → i16, 10 → −10, 1 → 1 and thus coincides with −iΓ 10 . A rotation by 2π/4 along ψ together with a rotation by 2π/4 along χ is a closed loop in SO(10) × U (1) ψ . This corresponds to the smallest charge magnetic monopole generated by the breaking E 6 → SO(10) × U (1) ψ . It has 1/4 of magnetic flux along ψ and also an SO(10) flux corresponding to the inverse generator of its center iΓ 10 . A fourfold monopole, i.e. a monopole with magnetic flux equal to four times the flux of the minimal charged monopole, corresponds to a full (2π) rotation along ψ, since a full rotation along χ is homotopically trivial in SO(10).
Instead of using rotations along ψ and χ, it is more transparent to use rotation along ψ and ψ ′ . Note that ψ ′ = (χ + ψ)/4 + ψ. A rotation by 2π along (χ + ψ)/4 corresponds to the identity as we have just seen, and a rotation by 2π along ψ again is the identity as one can see from the ψ charges. The ψ direction has no common elements with the center of SU (5) since the ψ charges of the SU (5) singlets are 4 and 1. However, the ψ ′ direction has elements which coincide with the center of SU (5). Namely, exp(i2π/5) in U (1) ψ ′ coincides with the element exp(−i2πȲ /5) of the center of SU (5) withȲ = diag(2, 2, 2, −3 − 3) in SU (5). It is known [27] that χ and ψ correspond to the following GUT normalized generators Then the normalized generator for ψ ′ is and the orthogonal generator is with χ ′ = (3χ − ψ)/4. The χ ′ charges are such that the full rotation along χ ′ is a 2π rotation.
Note that the χ ′ direction has no common elements with SU (5) since the charges of the SU (5) singlets in 27 are -1 and -4. However, the Z 4 subgroups from ψ ′ and χ ′ coincide. Namely, a rotation by 2π/4 along ψ ′ together with a rotation by 2π/4 along χ ′ lead to the identity element as one can see from the ψ ′ , χ ′ charges. It is, as we will see, more convenient to use the orthogonal generators ψ ′ , χ ′ rather than ψ, χ.
How about the previous monopole with magnetic flux (χ + ψ)/4 = (ψ ′ + χ ′ )/4? As U (1) χ ′ breaks to its Z 4 subgroup (which belongs to U (1) ψ ′ too), the χ ′ /4 flux of the monopole is confined to a tube and connects it to an antimonopole. We therefore obtain unstable dumbbells [10] which disappear. The Coulomb ψ ′ /4 and −ψ ′ /4 fluxes of the monopole and antimonopole, of course, cancel each other. However, new monopoles appear which carry U (1) ψ ′ flux. Indeed, since the Z 5 subgroup of U (1) ψ ′ belongs to SU (5), as we showed above, these monopoles correspond to a rotation by 2π/5 along ψ ′ and also carry SU (5) flux corresponding to the element exp(i2πȲ /5) of the center of SU (5). We are left at this stage only with these ψ ′ monopoles. Next we can break U (1) ψ ′ by the SO(10) singlet in a Higgs 27-plet which, of course, leaves unbroken its Z 5 subgroup contained in the center of SU (5). Then strings are formed with flux corresponding to a rotation by 2π/5 along ψ ′ which connect the ψ ′ monopoles to antimonopoles leading them to annihilate. Thus, no topological defects survive at the end. From this point on, the story proceeds as usual with the breaking of SU (5).
It is interesting to note that we could inflate away the monopoles with magnetic flux (ψ ′ + χ ′ )/4 and obtain a network of cosmic strings with magnetic flux χ ′ /4, which are not topologically stable. At the breaking of U (1) ψ ′ by the N -type component of the Higgs 27, a ψ ′ /20 magnetic flux is added along these strings in order for the phase of N to remain constant around them. This addition certainly corresponds to the minimal necessary increase of the magnetic energy of the string. At the electroweak breaking, the phases of the Higgs fields h u and h d , which have χ ′ = 2, −1 and ψ ′ = −2, −3 respectively, change around the string by 4π/5 and −4π/5 respectively. If we minimally add to the string -2/5 of flux along 2Y , the VEVs of h u , h d remain constant around it. Only the VEV of the ν c -type Higgs field changes its phase by −2π around the string. However, this string is not superconducting. Indeed, the up-type quark masses originate from the VEV of h u which remains constant around the string, and thus no transverse zero modes are generated along the string. The down-type quarks and charged lepton, although h d also remains constant, could generate zero modes since the ν c -type VEV contributes to their masses.
Recall that d c -type quarks (e-type charged leptons) exist not only in the5 in the SO (10) 16-plet, but also in the5 in the SO(10) 10-plet, which we call D c (E). Also, d-type quarks (e c -type charged leptons) exist not only in the 10 in the SO(10) 16-plet but also in the 5 in the SO(10) 10-plet, which we call D (E c ). Then the masses of the down-type quarks can be schematically written as where the mass matrix is given in terms of four 3 × 3 blocks. Three of them are of the order of the VEVs of N , ν c , and h d as indicated with constant unsuppressed coefficients. The fourth is proportional to the VEV of h d but multiplied by coefficients α ij (i, j = 1, 2, 3) which are suppressed by powers of m P , the Planck mass. This is due to fact that a direct trilinear Yukawa coupling is forbidden, in this case, by U (1) χ ′ and U (1) ψ ′ . The coefficients α ij must then necessarily contain U (1) χ ′ and U (1) ψ ′ violating SM singlet VEVs, i.e. N , ν c .
We can now apply a theorem given in Ref. [28] which says that, if a particular mass matrix element remains constant around the string, we can remove from the mass matrix the row and the column that contain it when calculating the number of transverse zero modes. In our case N and h d remain unaltered around the string, so all rows and columns can be removed and no zero modes appear. We see that the fact that ν c changes phase around the string does not generate zero modes in this case. A very similar analysis can be done for the charged leptons by replacing D c , d c , D, d in Eq. (25) by E, e, E c , e c respectively. We conclude that these strings are not superconducting.
We could also inflate away the monopoles with ψ ′ /5 and SU (5) flux to get a network of strings with magnetic flux ψ ′ /5. Recall that the phase of N changes by 2π around such a string, while ν c remains constant. The phases of the electroweak doublets h u , h d change by (−2/5)2π, (−3/5)2π respectively. Adding minimally on the string 2/5 of flux along 2Y , we then see that h u remains constant around the string, while the phase of h d changes by −2π. Again, we have no zero modes from the up-quark sector. For the down-quark and charged lepton sectors, we can write mass matrices similar to the one in Eq. (25). Then, we can remove the rows and columns which contain elements proportional to ν c which leaves the 3 × 3 matrix which is proportional to α ij h d . This matrix also does not change phase around the string as one can see from the various charges of the product D c d. Thus, no transverse zero modes exist and these strings are also not superconducting. Now if the breaking to SU (5) × U (1) ψ ′ is achieved by the ν c ν c -type component of 351 ′ , the Z 8 subgroup of U (1) χ ′ remains unbroken. But the Z 4 subgroup of it is in U (1) ψ ′ , so actually the unbroken subgroup is SU (5) × U (1) ψ ′ × Z 2 . As a consequence, Z 2 strings are formed with flux χ ′ /8. Note that these are Z 2 strings, i.e. the string and antistring coincide (they are not oriented). In this case, the χ ′ /4 flux of the monopole with total flux (χ ′ + ψ ′ )/4 splits into two tubes, each with flux χ ′ /8. The monopoles can then be connected to form necklaces which are Z 2 strings themselves. We can also have simple Z 2 strings with flux χ ′ /8 without monopoles on them since Needless to say the ψ ′ monopoles will also appear at this stage as before.
We can further use the SO(10) singlet in a Higgs 27-plet to break U (1) ψ ′ . Again, we obtain flux tubes carrying ψ ′ /5 magnetic flux which connect the ψ ′ monopoles and antimonopoles and lead them to annihilation. If we inflate these ψ ′ monopoles we obtain a network of nonsuperconducting strings with flux ψ ′ /5 as we have seen above. The VEV of the N -type component does not break the extra Z 2 in Eq. (26), but merely rotates it. Actually, if we add 1/40 of the flux along ψ ′ on the string with flux χ ′ /8, the phase of the N -type component remains unchanged around it. The electroweak doublets h u , h d have χ ′ = 2, −1 and ψ ′ = −2, −3 respectively and thus, around the string, their phases change by −2π/5, +2π/5 respectively. If we minimally add to the string 1/5 of flux along 2Y , the VEVs of h u , h d remain constant around it. In summary, these strings and necklaces survive even after the electroweak breaking but they are not superconducting.

Primordial Monopoles, Strings, and Gravity Waves
As previously mentioned primordial monopoles and strings can survive inflation in realistic models. Consider, for instance, the breaking of SO(10) to the SM via the 422 subgroup, such that the Z 2 subgroup of the center of SO(10) remains unbroken. Assume that inflation is driven by an SO(10) singlet scalar field with a Higgs or Coleman-Weinberg potential and with minimal coupling to gravity [15,29]. This model predicts that the tensor-to-scalar ratio r 0.02 [30]. In other words, the Hubble parameter H during observable inflation is estimated to be of order 10 13 − 10 14 GeV, which has important implications for primordial monopoles and strings. The GUT monopoles produced during the breaking of SO(10) to 422 are inflated away, but the intermediate mass monopoles from 422 breaking at M I may survive inflation if M I ∼ H. In practice, one needs about 23−25 e-foldings for adequate suppression and still leave an observable number density of these intermediate mass monopoles [15,16] -see below. By the same token the intermediate scale Z 2 cosmic strings which are produced during the breaking of 422 to the SM can also survive the inflationary epoch. In the case the Z 2 center of SO(10) is broken, we do not have topologically stable strings. However, the monopoles produced during the breaking of 422 are connected, in the next stage of symmetry breaking, by topologically non-stable strings. The monopole-string system eventually decays by emitting gravity waves which may be detectable by future experiments (see below).
Regarding E 6 , as we have shown, the symmetry breaking E 6 → 333 yields a superheavy GUT monopole which is inflated away, at least in the inflationary scenarios we have mentioned here. However, analogous to the SO(10) case, the triply charged intermediate mass (∼ 10 14 GeV) monopoles from the breaking of 333 to the SM may be present at an observable level in our galaxy.
Other realistic examples of intermediate scale monopoles, strings and composite objects that survive an inflationary scenario can be readily constructed.
We will now give some details concerning the production and evolution of monopoles and cosmic strings as well as the gravity waves generated by topologically stable or unstable strings. The mean distance between topological defects (monopoles or strings) at production is estimated to be ∼ H −1 . During inflation it acquires an extra factor e η with η being the number of e-foldings following the generation of the defects. During the subsequent inflaton oscillations this distance is multiplied by a factor (t r /τ ) 2/3 with t r being the reheat time and τ the rollover time, and from reheating until the present time by another factor T r /T 0 (T 0 is the present temperature). So all together the mean distance between defects becomes For T r ≃ 10 9 GeV and for the SM spectrum, t r ≃ 1 GeV −1 . For the defects to enter into the horizon until today, their present mean distance in Eq. (27) should not be larger than the present time t 0 .
We consider the inflationary scenario with a Coleman-Weinberg potential of Ref. [29] with the coupling λ 3 in Eqs. (5) and (6) of this reference much smaller than λ 2 . In this case, Eq. (6) of this reference reduces to To be more specific, we will take as an example a particular viable realization of this scenario which appears in the fourth line of Table 4 in Ref. [31]. In this case, the inflationary scale is V 1/4 0 ≃ 1.75 × 10 16 GeV, which implies that H ≃ 7.25 × 10 13 GeV. Also A = 1.43 × 10 −14 corresponding to λ 2 ≃ 6.14 × 10 −7 . The VEV of the inflaton M in Ref. [29] or v in Ref. [31] is M ≃ 29.4 m P ≃ 7.17 × 10 19 GeV. From the formula of Ref. [29], we then obtain that λ 0 = 1.9 × 10 −13 , and from Eq. (12) of the same reference that The requirement that the defects eventually enter the horizon (i.e. they are not inflated away) gives η 67.7. From the formula η = 3c/λ 0 of Ref. [15], we conclude that the parameter c ∼ (M d /M ) 2 4.3 × 10 −12 , where M d is the breaking scale corresponding to the defects. This scale should then satisfy the inequality M d 1.5 × 10 14 GeV. In the case of strings (M d ≡ M s ), this gives for the dimensionless string tension Gµ s ≃ (M s /m P ) 2 3.7 × 10 −9 .
Note that the GUT scale is given by M GUT ∼ λ 1/2 2 M (see Ref. [29]). For the particular example we are discussing, M GUT ∼ 5.6×10 16 GeV. Of course, this is just an order of magnitude estimate since we do not know the precise values of the couplings a and b in the potential in Eq. (5) of Ref. [29] and the GUT gauge coupling constant.
For models predicting the existence of topologically stable Z 2 strings, we can employ Fig. 1 of Ref. [18], which holds for strings surviving until the present time. We see that strings with Gµ s 1.5 × 10 −11 , namely M s 9.45 × 10 12 GeV, are allowed by the current experimental bounds. It is important to note that topologically stable strings with Gµ s 10 −20 will be possibly measurable by LISA and BBO in the future.
The number density n m of topologically stable magnetic monopoles can be estimated as in Ref. [15]. At production it is expected to be ∼ H 3 . During inflation, the monopoles are diluted by a factor exp (−3η m ), where η m is the number of e-foldings from the time of monopole production until the end of inflation. During inflaton oscillations, n m is multiplied by another factor (t r /τ ) −2 (this was not taken into account in Ref. [15]). The final relative monopole number density is Requiring that r does not exceed 10 −30 (the Parker bound) [32] and for the numerical example discussed here, we find that η m 23.5. This implies that, for topologically stable monopoles, the parameter c m ∼ (M m /M ) 2 1.5 × 10 −12 , and the corresponding symmetry breaking scale M m 8.77 × 10 13 GeV .
Next let us consider SO(10) broken via SU (4) c × SU (2) L × SU (2) R without topologically stable Z 2 strings. In this case, during the breaking of SU (4) c × SU (2) L × SU (2) R to SU (3) c × U (1) B−L × SU (2) L × U (1) R , we have the formation of SU (4) c (red) and SU (2) R (blue) monopoles at a scale M m . These monopoles are subsequently partially diluted by inflation. At a breaking scale M s , where SU (3) c × U (1) B−L × SU (2) L × U (1) R reduces to the SM gauge group, these monopoles are connected by strings forming random walks with step about the horizon size at subsequent times. Later, the monopoles enter the horizon connected in pairs by one string segment. After this time, the monopole pairs with the string segment behave like pressureless matter. The strings eventually decay to gravity waves and the monopoles merge to form either Schwinger monopoles or simply annihilate if they are a red or blue monopole with the corresponding antimonopole.

Conclusions
Grand Unified Theories with a unified (single) gauge coupling constant such as SU (5), SO (10), and E 6 all predict the existence of a topologically stable magnetic monopole that carries a single unit of Dirac magnetic charge (quantized with respect to the electron charge). This superheavy GUT scale magnetic monopole also carries color magnetic charge, and this conclusion holds independent of the symmetry breaking pattern of the underlying GUT model. In SU (5), this magnetic monopole happens to be the lightest one with mass ∼ M GUT /α GUT ∼ 10 17 GeV, where α GUT (∼ 1/10) is the GUT fine structure constant.
In models such as SO(10) or E 6 , where the symmetry breaking proceeds via one or more intermediate steps, magnetic monopoles can appear that carry two or three units of the Dirac magnetic charge and their masses are related to the intermediate scale. Hence they are lighter than the GUT magnetic monopole with one unit of charge. We have observed that intermediate mass (∼ 10 14 GeV or so) magnetic monopoles and cosmic strings of similar mass scale may be present in our galaxy at an observable level. We have depicted scenarios which give rise to superconducting cosmic strings as well as composite objects including a novel type of necklace that can survive inflation. The gravity waves emitted by some of these topological defects may be observable with the space based observatory LISA.