5.2 Example: Matching the Z 7 orbifold to its blowup model
5.2.3 Matching the spectra
For matching the massless states, we use that the states on the orbifold and in blowup are connected via eld redenitions (5.8). There are various kinds of eld redenitions that can be employed encoded in the f(tr) in (5.8). Since on the orbifold we only have
xed points that become arbitrarily widely separated in the SUGRA blowup limit (the intersection between exceptional divisors at dierent xed points vanishes identically in the Kähler cone of the blowup), we require that orbifold elds that live at a certain xed point σ are only redened with the charges of the blowup modes that reside at the same xed point. However, since in blowup we lose the information which twisted sector an orbifold state corresponds to, we allow for the inclusion of blowup modes coming from dierent twisted sectors θk. From this we nd that the following charge redenitions are realized:
QOrbk,σ 7→QCYk,σ =QOrbk,σ −Vk,σ, (5.15a) QOrbk,σ 7→QCYk,σ =QOrbk,σ +Vl,σ+Vm,σ, k 6=l 6=m6=k , (5.15b) QOrb1,σ 7→QCY1,σ =QOrb1,σ +V1,σ−V2,σ,
QOrb2,σ 7→QCY2,σ =QOrb2,σ +V2,σ−V4,σ, QOrb4,σ 7→QCY4,σ =QOrb4,σ −V1,σ+V4,σ,
(5.15c)
QOrb1,σ 7→QCY1,σ =QOrb1,σ +V1,σ+V2,σ−V4,σ, QOrb2,σ 7→QCY2,σ =QOrb2,σ −V1,σ+V2,σ+V4,σ, QOrb4,σ 7→QCY4,σ =QOrb4,σ +V1,σ−V2,σ+V4,σ.
(5.15d)
Massless states In order to illustrate the methods for matching the spectra ex-plained above, let us look at examples for the redenition. We included some orbifold states together with their redenition and the corresponding E8×E8 root in blowup in table 5.2. The full spectrum where the match is performed for all orbifold states can be found in [61]. In the match, we will use again that the resolution is an entirely local process. This means that the HRR index theorem can be split up and applied to each of the seven local resolutions of C3/Z7 individually. In this way, we can compare the spectra easily and even extract information on vector-like pairs in blowup which could not be seen by the HRR index theorem applied to the whole compact T6/Z7, as we discuss now.
First we explain how to read table 5.2. In the rst column, we assign a unique label (e.g.
Q1) to each orbifold state; for the blowup counterpart we give the irrep under which the states transform. The second column identies the twisted sectorθkfrom which the state emerges. The local multiplicity columns give the multiplicity of the states at the xed points 1-7. In blowup, they are evaluated via the local HRR theorem and on the orbifold by solving the mass conditions. The total multiplicity is the sum of the local multiplicities as obtained from the global HRR operator. Since this sees only chiral states, the total blowup multiplicity is at the same time the sum of the multiplicities of the orbifold states that are redened to the same blowup E8×E8 vector. The next column gives the shifted momenta for the orbifold states and the E8×E8 root for the blowup states. Finally, the last column gives the eld redenition that was used to match the orbifold state to its blowup counterpart.
5.2 Example: Matching the Z7 orbifold to its blowup model 87
State Sector Local multiplicity tot E8×E8root /Psh Redef
1 2 3 4 5 6 7
(3,2,1) 17 17 17 17 17 17 17 1 1,0,0,0,-1,0,0,0
0,0,0,0,0,0,0,0
Q1 untw. 17 17 17 17 17 17 17 1 1,0,0,0,-1,0,0,0
0,0,0,0,0,0,0,0
none (3,2,1) 1 -17 -17 17 -17 17 17 1 12,-12,12,12,-12,12,-12,-12
0,0,0,0,0,0,0,0 Q2 2 1 0 0 0 0 0 0 1 141 7,-7,3,3,-11,-1,-1,3
-4,-4,0,0,0,0,0,0
(5.15a) (3,2,1) 1 -17 -17 17 -17 17 17 1 12,-12,12,12,-12,-12,12,-12
0,0,0,0,0,0,0,0 Q3 1 1 0 0 0 0 0 0 1 141 7,-7,5,5,-9,3,3,5
-2,-2,0,0,0,0,0,0
(5.15a) (3,1,1) -17 17 -17 17 17 1 -17 1 0,0,0,0,-1,0,1,0
0,0,0,0,0,0,0,0
t6 4 0 0 0 0 0 1 0 1 141 -5,-5,1,1,-13,-3,1,3
-2,0,3,0,0,0,0,0
(5.15a) (3,1,1) 17 17 -17 17 -17 -17 -1 -1 12,12,-12,-12,12,12,12,12
0,0,0,0,0,0,0,0 t7 4 0 0 0 0 0 0 -1 -1 141 1,1,-7,-7,7,1,3,1
2,10,-2,0,0,0,0,0
(5.15a) (3,1,1) 17 -17 17 17 1 -87 -17 0 0,0,0,0,-1,0,0,-1
0,0,0,0,0,0,0,0
t5 4 0 0 0 0 1 0 0 1 17 -2,-2,1,1,-6,0,3,-1
-3,2,1,0,0,0,0,0
(5.15a) t12 1 0 0 0 0 1 0 0 1 17 3,3,2,2,-5,0,-1,-2
1,-3,2,0,0,0,0,0
(5.15a)
t11 2 0 0 0 0 -1 0 0 -1 17 -1,-1,4,-3,-3,0,-2,-4
2,1,-3,0,0,0,0,0
(5.15b) t18 1 0 0 0 0 0 -1 0 -1 141 -3,-3,9,-5,-5,1,-5,-1
10,0,-2,0,0,0,0,0
(5.15c) (1,1,1) 137 -17 -137 137 17 17 1 3 0,0,0,0,0,1,0,-1
0,0,0,0,0,0,0,0
s25 4 0 0 0 1 0 0 0 1 141 -3,-3,3,3,3,3,-3,-7
4,-6,-2,0,0,0,0,0
(5.15a) s26 4 0 0 0 1 0 0 0 1 141 -3,-3,3,3,3,3,-3,-7
4,-6,-2,0,0,0,0,0
(5.15a) . . .
(1,1,1) 137 -17 -137 137 17 17 1 3 0,0,0,0,0,1,0,-1
0,0,0,0,0,0,0,0
s70 2 0 0 0 1 0 0 0 1 17 1,1,-1,-1,-1,6,1,0
1,2,3,0,0,0,0,0
(5.15a) . . .
(1,1,1) 67 -17 -1 157 17 137 17 4 0,0,0,0,0,0,1,-1
0,0,0,0,0,0,0,0
s111 1 0 0 0 1 0 0 0 1 141 1,1,-1,-1,-1,-1,1,-7
-6,2,-4,0,0,0,0,0
(5.15a) s113 1 0 0 0 1 0 0 0 1 141 1,1,-1,-1,-1,-1,1,-7
-6,2,-4,0,0,0,0,0
(5.15a) . . .
(1,1,1) 137 -17 -137 137 17 17 1 3 0,0,0,0,0,1,0,-1
0,0,0,0,0,0,0,0
s112 1 0 0 0 -1 0 0 0 -1 141 1,1,-1,-1,-1,-1,1,-7
-6,2,-4,0,0,0,0,0
(5.15b) . . .
Table 5.2: Excerpt of the match of orbifold states with their blowup counterparts [61].
Let us begin with the 3 quark doublets(3,2,1). The rst eld Q1 lives in the untwisted sector. Hence it does not need to be redened. Applying the local HRR theorem to the (non-compact) resolution of C3/Z7, we get a multiplicity of1/7at each xed point.
The fact that a fractional multiplicity appears is linked to the fact that C3/Z7 is non-compact. The real multiplicity is given by summing over the contribution from the various local patches and they are always found to be integer as it should be. In fact, the fractional multiplicities are rather nice: they tell us that the eld Q1 lives to 1/7 at each of the 7 xed points, i.e. the eld is democratically smeared out over all xed points, as one would expect for an untwisted eld. The elds Q2 and Q3 both live at the rst orbifold xed point. Both are redened to a unique root vector via (5.15a) at the rst xed point (and of the second respectively rst twisted sector). By looking at the local multiplicity operator, we see a multiplicity of one at the rst xed point.
Hence the local multiplicity operator exactly sees the orbifold state. At the other xed points, we see fractional multiplicities of ±1/7, which however sum to zero and thus the overall multiplicity is one. These fractional non-existing states can be interpreted as those untwisted states which were projected out on the orbifold. As long as they sum to zero, we will ignore them in the following. If they do not sum to zero but to one, they indicate an untwisted sector eld, as seen for the eld Q1.
For the triplets(3,1,1)there are states that transform in the fundamental 3as well as in the anti-fundamental 3. It suces to investigate the triplet weights since the anti-triplets weights correspond to their negatives. Thus, a positive multiplicity indicates
a triplet state whereas a negative multiplicity indicates an anti-triplet state. As an example for this, we look at the states t7 and t6 which transform in the (3,1,1) and (3,1,1). Their overall multiplicity is -1 and 1, and the local multiplicity operator reveals that these states live at xed points 7 and 6, respectively. This is again readily conrmed from the orbifold spectrum.
Something conceptually new happens for the orbifold states t5, t12, t11, and t18. Al-though these four states are redened to the same root the total multiplicity is zero.
This happens because the HRR index can only count the net number of states which is 2−2 = 0. However, the local HRR index gives some insight into what is happening.
The three states t5, t12, t11 all live at xed point 5 on the orbifold. As there are two left-chiral and one right-chiral state the local multiplicity is 1. For the one right-chiral statet18, there is a local multiplicity of -1 at xed point 6. Hence the overall multiplicity is zero. The multiplicities of all other states can be worked out in a similar manner.
Higgsed states Vector-like states can acquire a mass in the blowup procedure from trilinear Yukawa couplings. On the orbifold, these couplings can be calculated using CFT techniques. This leads to a set of selection rules which can be used to check which couplings are allowed. The selection rules on the Z7 orbifold arise from requiring gauge invariance, compatibility with the space group, and conservation of H momentum.
Conservation of Rcharge will be discussed below. Gauge invariance simply amounts to the requirement that the sum of the left-moving shifted momenta of the strings involved in the coupling is zero.
The space group selection rule requires that the product of the constructing space group elements of the states involved in the Yukawa coupling must be the identity element (1,0). For trilinear couplings this states that the allowed couplings are of the form
(k= 1, σ1)◦(k = 2, σ2)◦(k = 4, σ4), with σ1+ 2σ2 + 4σ4 = 0 mod 7. (5.16) If the coupling involves states which reside all at the same xed point (σ1 =σ2 =σ4), the space group selection rule is trivially fullled. However, there also exist solutions to (5.16) for states coming from three dierent xed points. Since these couplings arise from worldsheet instantons [100, 101], they are suppressed by a factor of the forme−aiα0 where ai are the moduli which govern the sizes of the CY or of the lattice underlying the orbifold. As it turns out, in our case H momentum is conserved for the trilinear couplings if the space group selection rule is fullled.
The conservation rule of the R charge dened in (2.46) reads X
ζ
Rζi = 1, (5.17)
where ζ runs over the three states involved in the Yukawa coupling. Equation (5.17) is trivially fullled for states without oscillators if the space group rules are. However, in a compact orbifold this symmetry will be broken down to a subgroup by the torus lat-tice. Therefore the formerly forbidden couplings are expected to be suppressed by the size of the lattice. If the lattice is factorizable, the remaining symmetry is the discrete
5.2 Example: Matching the Z7 orbifold to its blowup model 89 rotation of the three two-tori. In this case the selection rule needs only be satised up to multiples of the order of the orbifold group. For the non-factorizable SU(7) lattice of the Z7 orbifold, we checked that the symmetry is broken completely except for the Z7 itself, so (5.17) should not be imposed on the orbifold.
The SUGRA theory on the blowup side is, however, only valid in the large volume limit. In particular, we expect that the R charge selection rule (5.17), which is broken by the orbifold lattice, is still a valid symmetry in the large volume limit. We therefore expect the states which are supposed to get a mass via such suppressed couplings on the orbifold, to appear as massless states in the multiplicity operator in blowup. By comparing the spectra we indeed nd that the index theorem sees massless states for which the orbifold theory predicts non-local mass terms or mass terms which do not satisfy (5.17). To illustrate the absence of both types of mass terms in blowup we look at suitable examples.
As an example for mass terms not satisfying (5.17) consider the singlet states s25, s26, s70, s111, s112 and s113, see table 5.2. These states are all oscillator states which explains their degeneracy and which makes them sensitive to a possible R symmetry.
Together with the blowup modes s68 and s27, there are the following orbifold trilinear superpotential couplings when imposing only gauge and space group invariance and the H momentum rule:
(s111 s112 s113)
a11s68 a12s68 a13s27
a21s68 a22s68 a23s27
a31s68 a32s68 a33s27
s25
s26
s70
, (5.18)
where the aij are coecients which are naively of order one. Now when one gives a VEV to the blowup modes s68 and s27, these couplings give rise to a rank three mass matrix and thus one would expect all 6 singlets to become massive and disappear from the chiral spectrum in blowup. However, when we look at the roots to which these singlets can be redened, the local HRR index reveals that there are four states at the resolved xed point where the singlets in question were localized. Therefore four of these singlets must stay massless in the heterotic supergravity limit α0 → 0. This means that the above mass matrix has to have only rank one, such that just one pair of singlets is decoupled. One could explain this by assuming that all coecients aij
are equal, but this assumption is a priori not justied and would lead to mixing of the elds during redenition. It is probably more sensible to argue that the local HRR index sees states only in the large volume limit where the R symmetry (5.17) is exact.
Imposing R symmetry here would set all coecients to zero except fora21and a23 and therefore naturally explain the rank one mass matrix at this place.
The local R charge selection rule (5.17) is only relevant for oscillator states, as states satisfying the space group selection rule have P
ζqish,ζ = 1and hence (5.17) is fullled for states without oscillators. Interestingly, the states which have oscillators often allow for more than one possible redenition (5.15). Imposing (5.17) in conjunction with consistency of the local blowup spectra singles out a unique eld redenition.
Using these redenitions, we were nally able to establish a perfect match between the
anomalies on the orbifold and in blowup, which we take as a strong cross-check that the above discussion is valid.
To illustrate the presence of the instantonic non-local mass terms, we investigate the triplet states t5, t12, t11, and t18 encountered above. From the employed redenitions we nd
t5CYt11CY=t5Orbt11Orbe−b4,5+b1,5+b4,5 =t5Orbt11Orbeb1,5, (5.19a) t12CYt11CY=t12OrbtOrb11 e−b1,5+b1,5+b4,5 =t12OrbtOrb11 eb4,5. (5.19b) The coupling of t5 and t12 with t18 is non-local as the states reside at dierent xed points. Hence this coupling is not captured by the multiplicity operator. The redeni-tions clearly show that in blowup where bk,σ 1, the couplings (5.19) provide a mass term which vanishes in the blowdown limit bk,σ 1 in units of α0. This means that from the blowup perspective a linear combination of t5 and t12 pairs up with t11 and lifts the exotic state from the massless particle spectrum in blowup. This behavior is also conrmed from the orbifold perspective. The appearance ofb1,5 (5.19a) shows that t5 from the θ4 sector and t11 from the θ2 sector couple to the blowup mode from the θ sector as dictated by the space group selection rule. Likewise, for the second mass term (5.19b) we nd a coupling between t12 from the θ sector, t11 from the θ2 sector, and the blowup mode from the θ4 sector as indicated by b4,5.