Matching the anomalies

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

t₅^CYt₁₁^CY=t₅^Orbt₁₁^Orbe^{−b4,5+b1,5+b4,5} =t₅^Orbt₁₁^Orbe^b1,5, (5.19a) t₁₂^CYt₁₁^CY=t₁₂^Orbt^Orb₁₁ e^{−b1,5+b1,5+b4,5} =t₁₂^Orbt^Orb₁₁ e^b4,5. (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.

5.2 Example: Matching the Z7 orbifold to its blowup model 91 deviate from our standard notation in which the compactication manifold is called X to avoid confusion with the anomaly forms X),

X_2,^uni_CY:=

Z

M

X6,2, X_4,^uni_CY:=X0,4, ErX_2,^r,_CY^non := 1

6tr(iFiF), X_4,^r,_CY^non :=

Z

M

X4,4Er

They are computable from the intersection numbers on the compactication manifold M, the Bianchi identities, and the internal gauge ux F. Using the descent equations, we nd for the GreenSchwarz counterterm

a^uni_OrbX_4,^uni_Orb+X

i

τiX_i,^4,_Orb^red =a^uni_CYX_4,^uni_CY+X

r

βrX_4,^r,non_CY (5.21) Let us now discuss the four contributions to the anomalies on both sides in the following.

After that, we show how the left hand side and the right hand side combine to match the complete anomaly across both theories.

Universal orbifold anomaly I_Orb^uni

On the orbifold, we can choose a basis of U(1) charges such that there is single anoma-lous U(1)_A symmetry and the other seven U(1)'s are perpendicular to it. With this anomalous U(1)_A generator, the anomaly polynomial on the orbifold is

I_Orb^uni = 6F1 tr(iFSU(2))²+tr(iFSU(3))²+tr(iFSO(10))²−trR²+κ^IJX

I,J

FIFJ

! , (5.22) where the indicesI, J label the eight U(1) factors on the orbifold. The numerical factors κ^IJ are not given explicitly because they are not relevant in further discussions. The factor of six could be absorbed by changing the normalization of the anomalous U(1) generator T_A. However, we prefer not to do so, as otherwise we nd this factor of six in all eld redenitions in the next section.

Anomaly from eld redenition I_Orb^red

This part of the orbifold anomaly polynomial takes into account that there is a eld redenition between the states on the orbifold and in blowup which induces a change of the U(1) charges and accounts for the decoupling of Higgsed orbifold states. We calculate this change by splitting up I_Orb^red into contributions from the three types of anomalies, I_Orb^red =I_G^red+I_grav^red +I_pure^red , which we will now compute.

U(1)× G² anomaly redenition In order to compute the redenition of the U(1)×G² anomaly polynomial we need to consider the change of trQI when going from the orbifold to blowup, where the trace is taken over the elds charged under the non-Abelian group. Let us denote the U(1) charges on the orbifold by Q^γ_I, where

γ runs over all orbifold states and I labels the the U(1) generators. Likewise, we denote the U(1) charges in blowup by Q^0γ_I. Furthermore, it is convenient to introduce

∆^γ_I = Q^γ_I −Q^0γ_I, which labels the dierence between the charges in the two theories.

According to (5.8), the chargesQ^γ_I and Q^0γ_I dier byf^γ(V_I^r)whereV_I^r is the line bundle vector, or equivalently, the shifted momentum of the blowup mode at xed pointr, and the function f is given in terms of the redenitions (5.15).

The sum of the charges in blowup tr(QI)_BU = P

αQ^0α_I runs over the states α that remain massless after giving VEVs to the blowup modes. Hence, in order to recover the trace on the orbifold prior to having assigned VEVs, we also have to include a sum over the states that gain a mass in blowup, which we label by β. We thus obtain

tr(Q⁰_I) =X

α

Q^α_I −X

α

∆^α_I =X

α

Q^α_I −X

α

∆^α_I +X

β

Q^β_I −X

β

∆^β_I −X

β

Q^0β_I

=tr(QI)_Orb− X

γ=α,β

∆^γ_I −X

β

Q^0β_I , (5.23)

where we added a zero in the rst step and rearranged the terms in the second step.

Note that the last sumP

βQ^0β_I which sums over all elds that became massive in blowup vanishes identically: all massive states are vector-like with respect to their charges, so the sum always contains pairs of opposite charges. Leaving out this last term, the contribution to the 4d anomaly polynomial and the redenition part read

I_G =F^Itr(iF_G)²X

α

Q^0α_I , I_G^red ∼X

G,I

−X

γ

∆^γ_I

!

F^Itr(iF_G)² ∼X

G,I

c^G_IF^Itr(iF_G)².

(5.24)

In the sums G runs over SU(2), SU(3) and SO(10). When evaluating the sum and comparing with the orbifold result, we obtain a perfect match of all U(1)×G² anomalies of both theories, where the anomaly coecients c^G_I of (5.24) have been calculated in a specic choice of U(1) basis [61].

U(1) ×grav² anomaly redenition For the U(1)×grav² anomaly one has to include all the massless elds in the trace. This means that, in contrast to the U(1)×G² anomalies, one also has to add the contribution coming from the Abelian blowup mode charges V_I^r. The contribution to the 4D anomaly polynomial and the redenition part is then given by

I_grav∼F^ItrR²tr(Q⁰_I) =F^ItrR²X

α

Q^0α_I

=F^ItrR² X

α

Q^α_I −X

α

∆^α_I +X

β

Q^β_I −X

β

∆^β_I −X

β

Q⁰_I^β

! ,

I_grav^red ∼ − X

γ=α,β

∆^γ_I−X

r

V_I^r

!

F^ItrR² ∼c^grav_I F^ItrR²,

(5.25)

5.2 Example: Matching the Z7 orbifold to its blowup model 93 where we again added the contributions from the massive elds and used P

βQ^0β_I = 0. The index γ contains both α for massless and β for massive elds. We nd again a perfect match between the blowup polynomial and the redened one, supporting the eld redenition ansatz of (5.15).

Pure U(1) anomaly redenition A similar procedure can be applied to the pure U(1) anomalies and in this case the eld redenitions change the polynomial via

I_pure∼ 1 3!

X

I,J,K

F^IF^JF^KX

α

Q^0α_I Q^0α_JQ^0α_K

= 1 3!

X

I,J,K

F^IF^JF^K X

α

Q^α_IQ^α_JQ^α_K+X

a

q_I^aq_J^aq_K^a +X

β

Q^β_IQ^β_JQ^β_K

!

+I_pure^red

= 1 3!

X

I,J,K

F^IF^JF^Ktr(QIQJQK)_Orb+I_pure^red ,

I_pure^red ∼ 1 3!

X

I,J,K

F^IF^JF^K X

γ=α,β

(−3∆^γ_IQ^γ_JQ^γ_K+ 3∆^γ_I∆^γ_JQ^γ_K− X

γ=α,β

∆^γ_I∆^γ_J∆^γ_K

−X

a

q_I^aq_J^aq_K^a −X

β

Q^0β_IQ^0β_JQ^0β_K

!

=c^pure_{IJ K}F^IF^JF^K.

(5.26)

The factor1/3!takes care of the permutation symmetries of the sum indices as in (4.13).

The anomalies match again perfectly assuming the mass term structure explained above.

Universal blowup anomaly I_CY^uni

The universal anomaly in blowup is given by I_CY^uni=

Z

M

X₂^uniX₄^uni=− 1 12

Z

M

trR²−tr(iF²) tr(iF⁰iF⁰)tr(iF⁰)²− 1

2tr(iF⁰)²tr(iF⁰⁰iF⁰⁰)− 1

4tr(iF⁰iF⁰)trR²+⁰ ↔⁰⁰ .

(5.27)

Using the intersection numbers and the expansion of the internal ux F, we obtain I_CY^uni= 1

2 trR²−tr(iF)²

· gIF^I

, (5.28)

with anomaly coecients gI.

Non-universal local anomalies I_CY^non

Lastly, we have the non-universal axions βr to cancel the other U(1) anomalies. Their contributions are given by

I_CY^non = Z

M

X₂^rX₄^r. (5.29)

This expression is evaluated by using the Bianchi identities to express trR² in terms of tr(iF)² as

Z

Er

trR² = Z

Er

tr(iF)² = Z

M

V_r^I₁V_r^I₂Er1Er2Er. (5.30) The integration in (5.29) is performed by using the intersection numbers. We obtain

I_CY^non =1

2h^G_IF^I tr(iF)²_SO(10)−tr(iF)²_SU(2)−tr(iF)²_SU(3)

+h^pure_{IJ K}F^IF^JF^K + 1

12(h^grav_I FI)trR²,

(5.31)

where we have denoted the coecients corresponding to the mixed U(1)I×G² anomalies with h^G_I, those corresponding to the pure U(1)_I×U(1)_J×U(1)_K anomalies withh^pure_{IJ K}, and those corresponding to the mixed U(1)_I×grav²anomalies withh^grav_I . The numerical values for all the coecients evaluated in some choice of U(1) basis can be found in [61].

This concludes the calculation of the four contributions to the anomalies in (5.9).

Relation among the axions

From the above results for I_Orb^uni, I_Orb^red, I_CY^uni, and I_CY^non, we can now establish the rela-tion between the single orbifold axion a^Orb (which is the dual to the 4D KalbRamond two-form b2), the axions in blowup, and the blowup modes using the descent equa-tions (5.21). We need to make an ansatz to factorize I_Orb^red which is compatible with this interpretation. A given factorization I_Orb^red = P

rQ^r_IFIX_4,^r,red_Orb is canceled via the counterterm P

iτiX_4,^i,red_Orb. The indices i and r run over the same set (k, σ), so we use only r. Considering X_4,^uni_Orb = −6X_4,^uni_CY we make the following ansatz for relating the various axions

βr=drτr, a^uni_CY=−6a^uni_Orb+X

r

crτr. (5.32)

Here, the cr and dr are coecients in the linear combinations and the factor of −6 arises due to the normalization choice of the U(1) generators. Substituting this ansatz into (5.21), the four-form involved in the factorization is expressed as

X₄^r,_,Orb^red =crX_4,^uni_CY+drX_4,^r,non_CY . (5.33) Substituting this last expression into I_Orb^red in (5.20) yields

I_Orb^red =X

r

Q^r_IFI crX_4,^uni_CY+drX_4,^r,non_CY

. (5.34)

Looking at the whole anomaly polynomial (5.20), we impose equality of each factor on the left hand side and on the right hand side. As there are 8 anomalous U(1)s, we obtain 152 equations in total, where 8 equations arise from the 8 U(1)×grav² anomalies,

Im Dokument Exploring the Web of Heterotic String Theories using Anomalies (Seite 100-105)