Anomaly cancellation in effective supergravity from the heterotic string with an anomalous U(1)

We show that a choice of Pauli-Villars regulators allows the cancellation of all conformal and chiral anomalies in an effective field theory from Z_3 compactification of the heterotic string with two Wilson lines and an anomalous U(1).


Introduction
Starting with the determination of the full anomaly structure of Pauli-Villars (PV) regularized supergravity [1], we recently showed [2] that an appropriate choice of PV regulator fields allows for cancellation of all the T-duality (hereafter referred to as "modular") anomalies by the fourdimensional version of the Green-Schwarz term in Z 3 and Z 7 compactifications of the heterotic string without Wilson lines. 1 We further matched our results to a string calculation [3] of the chiral anomaly in those theories. Here we extend our results to a specific Z 3 compactification [4] (hereafter referred to as FIQS) with two Wilson lines and therefore an anomalous U (1), hereafter referred to as U (1) X . In the following section we briefly describe the orbifold model we are studying. In Section 3 we outline the four-dimensional Green-Schwarz mechanism and the structure of the anomaly when an anomalous U (1) is present. In Section 4 we discuss some aspects of the cancellation of ultraviolet (UV) divergences and anomaly matching that are specific to the case with an anomalous U (1), as well as some simplifications with respect to the Z 7 case studied in [2]. We summarize our results in Section 5. The full set of conditions for cancellation of UV divergences and anomaly matching are given in Appendix A, a sample solution to these constraints is presented in Appendix B, and the full spectrum for the FIQS model is displayed in Appendix C. The determination of the correct Pauli-Villars (PV) masses can have implications for soft supersymmetry breaking terms [5].

The FIQS model
Here we will give a brief review of the orbifold model we will consider for the rest of the paper. The FIQS model [4] is a Z 3 orbifold compactification of the 10d E 8 ⊗ E 8 heterotic string compactified to T 6 with two Wilson lines and a nonstandard embedding for the shift vector. The embeddings of the shift vector and Wilson lines are given by Where the prime indicates that the last 8 elements of the above vectors correspond to the second factor of E 8 . With these specifications, the massless spectrum of the FIQS model can be worked 1 Corrections to this paper are given in Appendix D.
out following the standard recipes [6]. The 4D gauge group is SU (3) ⊗ SU (2) ⊗ SO(10) ⊗ U (1) 8  The constants q I a are determined by requiring that q a · q b = 0 and q a · α bj = 0, where the α bj are the sixteen dimensional simple root vectors of the nonabelian gauge group factors. Thus the index b corresponds to SU (3), SU (2), or SO(10) and j runs over the rank of each group. One choice of q a 's is [7]: q 1 = 6(1, 1, 1, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0) (2.5) q 2 = 6(0, 0, 0, 1, −1, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0) (2.6) q 3 = 6(0, 0, 0, 0, 0, 1, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0) (2.7) q 4 = 6(0, 0, 0, 0, 0, 0, 1, 0)(0, 0, 0, 0, 0, 0, 0, 0) (2.8) q 5 = 6(0, 0, 0, 0, 0, 0, 0, 1)(0, 0, 0, 0, 0, 0, 0, 0) (2.9) q 6 = 6(0, 0, 0, 0, 0, 0, 0, 0)(1, 0, 0, 0, 0, 0, 0, 0) (2.10) q 7 = 6(0, 0, 0, 0, 0, 0, 0, 0)(0, 1, 0, 0, 0, 0, 0, 0) (2.11) q 8 = 6(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 1, 0, 0, 0, 0, 0) (2.12) To get the charges of the matter fields, one normalizes the U (1) a generators as where the √ 2 is inserted to adhere to the standard phenomenological normalization. For this choice, one finds that the traces Q 6 , Q 7 , and Q 8 are all nonzero. One can perform a re-definition of the generators so that only one factor of U(1) has a nonzero trace. In [4], the following re-definition was made: q (F IQS) 6 = q 6 + q 7 (2.14) Q X = 0 which is rather undesirable. Therefore, we will use a different choice such that the above mixed anomaly does not appear. In particular, we define (2. 18) In what follows, we will simply drop the superscript N and use these as the definition of the U (1) 6 and U (1) 7 generators. As a final note, the charges defined above are generally not orthogonal to one another, i.e. Tr [Q a Q b ] = 0 for some a = b. It is possible to define a new set of charges that are mostly orthogonal to one another, but we will not need to do so for our purposes. We close this section with some relations among the gauge charges q p a and modular weights q p n of the chiral superfields Φ p of the model. These will be useful in the analysis that follows. These include the universality conditions Here C a is the quadratic Casimir in the adjoint representation of the gauge group factor G a and C p a is the Casimir for the representation of the chiral supermultiplet Φ p , T a is a generator of G a , and N, N G are the number of chiral and gauge supermultiplets respectively, with, in the FIQS model, (2.20) In addition we will use the sum rules 3 Anomalies and anomaly cancellation with an anomalous U (1) The effective supergravity theory from generic orbifold compactifications with Wilson lines is anomalous under both U (1) X and T-duality: where Φ a is any chiral supermultiplet other than a diagonal Kähler modulus T i , and q a i are its modular weights. We are working in the covariant superspace formalism of ref. [8] in which the chiral multiplets Z p = T i , S, Φ a , with S the dilaton superfield, are covariantly chiral: with D A , A = a, α a fully covariant superspace derivative. In particular, under a U (1) gauge transformation where g is a hermetian superfield, and A A is the gauge potential in superspace. Gauge invariance assures that holomorphy of the superfield is maintained under (3.3). If gauge invariance is unbroken, the gauge potential A A does not appear explicitly in the superspace Lagrangian. Instead the usual Yang-Mills superfield strength W α is obtained as a component of the two-form superfield strength F AB . One can still introduce [8] a superfield superpotential V a such that but V a never appears in the Lagrangian and the chiral superfield Λ a is independent of g in (3.3). However in the presence of an anomalous U (1), gauge invariance is broken. It is easy to see that the UV divergences cannot be regulated by PV fields that all have U (1) X invariant masses. There is a quadratically divergent term proportional to D X TrT X , where D X is the auxiliary field of the U (1) X supermultiplet, which must be cancelled by the analogous term from the PV sector. Invariant masses require the coupling of PV fields with equal and opposite charges that do not contribute to (TrT X ) P V . Noninvariant masses arise from the superpotential for PV fields Φ C : with µ C constant (in the absence of threshold corrections, as for the cases considered here). If Q C X + Q C X = 0, holomorphy of (3.5) is not respected under (3.3) for a = X. For this reason we do not include the U (1) X connection in the covariant derivative (3.2). Instead of (3.3) we require under a U (1) X transformation, and the Kähler potential depends on U (1) X -charged fields through the invariant operatorsΦe Q X V X Φ. It was shown in [1] that modular noninvariant masses can be restricted to a subset of PV chiral supermultiplets Φ C with diagonal Kähler metric: and superpotential (3.5).
As in [2], we define a superfield Then the anomalous part of the one-loop corrected supergravity Lagrangian takes the form [1] L anom = L 0 + L 1 + L r = d 4 θE (L 0 + L 1 + L r ) ≡ d 4 θEΩ, (3.9) where E is the superdeterminant of the supervielbein, and with η = ±1 the PV signature. The operators in (3.10) are given explicitly in [1,2], except that now where Ω 0 contains the Gauss-Bonnet Chern-Simons superfield and operators composed of auxiliary superfields of the gravity supermultiplet, and is the Yang-Mills Chern-Simons superfield without the U (1) X term, and and Ω a YM is defined by its chiral projection: Ω r is composed of terms linear and higher order in ln M, and Ω D represents a "D-term" anomaly [1,2] that, together with a contribution to the Gauss-Bonnet term Ω GB , arises from uncanceled total derivatives with logarithmically divergent coefficients, requiring the introduction of a fielddependent cut-off: (3.14) L 1 is defined by its variation: TrηHΩ L + h.c., (3.15) where under (3.1) and (3.6) ln M 2 transforms as ∆ ln M 2 = H +H, (3.16) with H holomorphic. Defining we have In the presence of an anomalous U (1) X the form of f C is taken to be where k is the dilaton kähler potential, and g is defined in (3.31) below. The traces in ∆L anom can be evaluated using only PV fields with noninvariant masses or using the full set of PV fields, since those with invariant masses, H C = 0, drop out. The contribution ∆L 0 to the anomaly is linear in the parameters α C , β C , q C n , Q C X , and the trace of the coefficient of Ω 0 is completely determined by the sum rules [9] that are required to assure the cancellation of quadratic and logarithmic divergences. In (3.20) the index C denotes any chiral PV field, the index γ runs over the Abelian gauge PV superfields that are needed to cancel some gravitational and dilaton-gauge couplings, and the sum over p includes all the light chiral multiplet modular weights with q S n = 0, q T i n = 2δ i n . All PV fields with noninvariant masses have δ = 0, and most 2 with δ = 0 have α = β = q n = 0 = Q C X . For the purposes of the present analysis we can largely ignore the latter. Similarly, the cancellation of linear divergences that give rise to the chiral anomaly proportional to fixes the coefficient of Ω 0 Y M . Here G µν −iT a F a µν is the field strength associated with the fermion connection, t i = T i , λ = Λ| are the lowest components of the chiral supermultiplets T i , Λ, and a left-handed fermion f transforms as f → e φ f (3.22) under modular and U (1) X transformations; φ = − i 2 ImF for gauginos, and for chiral fermions χ p . The compensating PV contribution There is a set of chiral multiplets in the adjoint representation of the gauge group that has f = K − k; these get modular invariant masses though their coupling in the superpotential to a second set with f = k. These cancel renormalizable gauge interactions and gauge-gravity interactions, respectively. Together with a third set, that has f = 0 and contributes to the anomaly, they cancel the Yang-Mills contribution to the beta-function.
that cancels (3.21) determines the anomaly coefficient of Ω 0 Y M , since for each pair Φ C , Φ C the sum of fermion phases φ C + φ C = H C is just the holomorphic part of the variation (3.16), (3.19) of the PV mass term ∆ ln M 2 C . In the chiral formulation for the dilaton, the anomaly is cancelled by the variation of the superspace Lagrangian where Ω is the real superfield introduced in (3.9). The quantum Lagrangian varies according to so the full Lagrangian is invariant provided However the classical Kähler potential for the dilaton is no longer invariant and must be modified: where V GS is a real function of V X and of the chiral supermultiplets; it transforms under (3.1) and (3.4), (3.6) as A simple solution consistent with string calculation results [10,11] is is the Kähler potential for the moduli. The modification (3.28) is the 4d Green-Schwarz (GS) term in the chiral formulation. As discussed in [2], the 4d GS mechanism is more simply formulated in the linear multiplet formalism [8] for the dilaton. In this case the linear dilaton superfield L remains invariant, its Kähler potential is unchanged, and instead one adds a term to the Lagrangian: Only terms in the anomaly that are linear in the combinationH, wherẽ can be canceled by the Green-Schwarz term. The values of b and δ X are fixed by the conditions (3.20), (3.24) for the cancellation of divergences, together with the universality conditions (2.19), that hold for all Z 3 and Z 7 orbifold compactifications.
In contrast to L 0 , the contributions to the anomaly from L 1 and L r are nonlinear in the parameters α, β, q n , Q X , and depend on the details of the PV sector. In particular L r has no terms linear in ln M and must vanish. To insure that the anomaly coefficient depends on the T-moduli only through F (T ) we impose [2]q

The anomaly and cancellation of UV divergences in the FIQS model
The full set of conditions for cancellation of the divergences and for obtaining an anomaly linear inH, Eq. (3.33), that matches the string result [3] is given in the Appendix A. In this section we outline some features of the case of Z 3 with an anomalous U (1) X . We will be primarily concerned with the contribution of ∆L 1 , Eq. (3.18), to the anomaly. This expression is nonlinear in the parameters q C n , Q C X of the PV fields, and therefore model dependent, as noted above. This was illustrated in [2] where it was shown that cancellation of the modular anomaly requires (3.34). However, the contribution cubic in Q C X is model independent. It is given by where the sum is over all PV fields, and we used the definition (3.6), (3.19) ofQ X and the fact that for any powers p, p . Cancellation of the term in TrφG · G that is cubic in from (2.19), so the anomaly (4.1) is consistent with the requirement for anomaly cancellation.
In contrast, anomaly terms quadratic in Q 2 X are model dependent. For example, in [1] it was assumed thatf C = f C for all PV fields with noninvariant masses, giving a contribution from (3.21) and (3.24) with a = X, and (2.19). Here we instead assume, in addition to (3.34), that Q C X = 0 if 1 −γ = 0, that is PV masses can be noninvariant under either T-duality or U (1) X , but not both. In this case the last term in (4.4) drops out and we recover a factor three, in agreement with the requirement for anomaly cancellation. The full set of PV fields sufficient to regulate light field couplings is described in Section 3 of [1]. These include a setŻ P =Ż I ,Ż A , with negative signature, ηŻ = −1, that regulates most of the couplings, including all renormalizable couplings, of the light chiral supermultiplets Z p = T i , Φ a . TheŻ get invariant masses through a superpotential coupling to PV fieldsẎ P with the same signature, opposite gauge charges and the inverse Kähler metric: It remains to cancel the divergences introduced by the fieldsẎ . To this end we take the following set: In the solution to the constraints given in Appendix B, the ψ C and T C are further subdivided, together with additional fields, into sets S a , a = 1, . . . , 12, some of which are charged under the nonanomalous gauge group. The φ C regulate certain gravity supermultiplet loops and nonrenormalizable coupling of chiral multiplets. These must be included together with the other PV fields introduced above in implementing the sum rules (3.20). Their contributions will be included in all the finiteness and anomaly conditions that involve only the parameters α in (4.7); otherwise they play no role in the analysis below. In the expressions given in the remainder of this section, we drop terms that contain only X α or X µν since their contributions are included in the sums (3.20) and the additional sum rule [9] In [2] we also introduced pairs Φ P , Φ P with modular invariant masses that did not contribute to the anomaly, but played an important role in canceling certain divergences. However, because the Z 3 sum rules (2.21) are much simpler than the analogous sum rules for the Z 7 case studied in [2], here we need only the set in (4.7). The quadratic and logarithmic divergences we are concerned with here involve the superfield and associated with the Yang-Mills, reparameterization and Kähler connections, Cancellation of quadratic divergences requires TrηΓ α = TrηT X = 0, (4.12) and cancellation of logarithmic divergences requires where η = +1 for light fields. Cancellation of all contributions linear and quadratic in X α is assured by the conditions in (3.20) and (4.8). The Yang-Mills contribution to the term quadratic in W α is canceled by chiral fields in the adjoint (see footnote on page 7) that we need not consider here. Finally, cancellation of linear divergences requires cancellation of the imaginary part of where G µν is the field strength associated with the fermion connection; 4 for left-handed fermions: where is the field strength associated with the Kähler connection (4.11). For a generic PV superfield Φ C with diagonal metric, its fermion component χ C transforms under (3.1) and (3.6) as In evaluating (4.14) we will use the fact that the expression 5 vanishes identically, and the expressions are total derivatives, where A a µ is an Abelian gauge field, and (4.20) The full Kähler potential forẎ , with no anomalous U (1) X , is given in [1,2]; here it takes the form q a n g n ,Ġ =αK +βg,α +β = 1, (4.21) 4 Here we neglect the spin connection whose contribution was discussed in [2]. 5 It was noted in [2] that the expression (4.18), which is in fact the T -dependent part of the chiral anomaly found in [3], vanishes. The authors of [3] attribute [12] this to their approximation that neglects higher order corrections. However if these corrections take the form whereẎ N =1,2,3 (and their counterpartsŻ N ) are gauge singlet PV fields needed [9] to make the Kähler potential and superpotential terms forŻ,Ẏ fully invariant, and the ellipsis represents terms that make no contribution to the expressions given below. Using the sum rules in (2.21) and (3.20) we obtain: Using (4.19) and (2.21), the part of XẎ that is independent of gauge charges takes the form: The modular weights for the ψ satisfy m,n g n q Pm n = gq P ψ , P η P ψ q P k l q P k n q P k n = 0, l,m,n g m g n q P l m q P l n = (q P ψ ) 2 n g n g n . Like XẎ χ , X ψ χ depends only on F, g µν and X µν , and (4.22) and (4.23) can be cancelled by some combination of the fields in (4.7), with the condition The pure T-moduli anomaly is given by (4.26) Consistency with string results [13] requires Finally, we require (4.28) Using (4.24), the condition (4.28) requires All other other contributions to ∆L 1 are required to vanish.
We conclude this section by noting that cancellation of divergences linear in the U (1) a field strengths is much simpler than for the Z 7 case considered in [2], as outlined below. The gauge charges for the FIQS ( [4]) model are listed 6 in Appendix C. The universality of the anomaly term quadratic in Yang-Mills fields strengths is guaranteed by the universality condition (2.19), as discussed in Section 3. Since gauge transformations commute with modular transformations, a set of chiral multiplets Φ b that transform according to a nontrivial irreducible representation R of a nonabelian gauge group factor G a have the same modular weights q R n such that Therefore terms linear in Yang-Mills field strengths occur only for Abelian gauge group factors. We need to cancel theẎ -loop contribution to logarithmic divergences Trη n q n g n α T a and, dropping terms proportional to the last expression in (4.19), the relevantẎ contributions to linear divergences: where we used (2.21). The last term in (4.32) is cancelled by The remaining terms in (4.32), as well as (4.31) can be cancelled by a combination of the fields in (4.7). For a = X there are additional terms proportional to (TrηT X ) P V = −TrT X .

The final anomaly in the FIQS model
In Appendix A we show that is possible to cancel all the ultraviolet divergences from theẎ fields with a choice of the set (4.7) such that the fields with noninvariant masses have the properties Then, including the results of [2], the anomaly due to the variation of (3.9) takes the form where Ω = Ω YM − Ω GB + Ω g , where Ω g is defined in (4.26), and b spin governs the contributions from PV masses, as opposed to those arising from uncancelled divergences: with 8π 2 b = 6 in the FIQS model. In the absence of an anomamous U (1), Λ = 0, the anomaly can be cancelled by the four dimensional GS mechanism as described in [2]. However with Λ = 0, the anomaly as written in (5.3) is no longer universal and cannot be cancelled by the GS term alone. However all of the "D-terms", in other words the full expression Ω , can be removed [14] by adding counterterms to the Lagrangian, giving a universal anomaly which can now be cancelled by the GS term. 7 The results for the Gauss-Bonnet and Yang-Mills terms are well-established [10] and result from the universality conditions (2.19).

Conclusions
We have shown that a suitable choice of Pauli-Villars regulator fields allows for a full cancellation of the chiral and conformal anomalies associated, respectively, with the linear and logarithmic divergences in the effective supergravity theory from a Z 3 orbifold compactification with Wilson lines and an anomalous U (1). A future work [13] will compare this result with that obtained directly from string theory.  The elimination of ΩD further obviates the need for a modification of the linear-chiral duality transformation, a possibility condsidered in Appendix B of [2] and Appendix E of [1].
A Conditions for the cancellation of ultraviolet divergences and the evaluation of the anomaly

A.1 Notation
We pair PV fields according to their mass terms. A pair of PV fields (Φ P , Φ P ) has a superpotential coupling and a Kähler potential with an identical definition holding for f P but with primes on the constants {α P , β P , q P n }. While we will not use it often, summing over the index C means summing over PV fields and then their primed partners whereas summing over P means summing over only the unprimed or primed fields, depending on the quantity being summed. For example, However, to reduce clutter, we will abbreviate the above. When summing over primed and unprimed fields, we will use "Tr". When summing over only primed or unprimed ones, we will use "Sum". Thus the above would be written as

Tr[ηα] = Sum[ηα] + Sum[ηα ]. (A.5)
We will also encounter sums over various combinations of U(1) charges, U(1) X charges, and modular weights. To abbreviate these, especially when dealing with the quantum numbers of the light fields, we will define

A.2 Conditions for Regularization
The terms we must cancel come from linear, logarithmic, and quadratic divergences. It is helpful to organize these terms by forming subsets based on whether terms depend on nonabelian gauge interactions, nonanomalous Abelian gauge interactions, anomalous Abelian gauge interactions, or none of the above. We will refer to these groupings as nonabelian divergences, U(1) a divergences, U(1) X divergences, and modular divergences, respectively. As an overview, the divergences come from the terms where for our PV fields defined above. The PV fields involved in this procedure are numerous. We take all of the PV fields described in sections 3 and 4 of [1] and supplement them with further fields. However, to satisfy the divergences above, we need only focus on theẎ andφ fields of [1]. We now group all the terms in the above expressions with our organizational scheme.

Modular Divergences
To cancel all the modular divergences, we require Note that only fields that haveQ X = 0 will contribute to Eq. (A.29).

Nonabelian Divergences
To cancel the nonabelian divergences, we need where T a is a generator of a nonabelian gauge group factor.
In all of the above sets, we have assumed that the modular weights of all PV fields satisfy sum rules reminiscent of those satisfied by the light sector, (2.21). Indeed, this will be baked directly into our choice of PV fields. We have also used the total derivative identities (4.19). In addition to the above conditions, we must enforce the sum rules of [1]:

A.3 Conditions for Anomaly Matching
By drawing an analogy with the calculation of [3], we infer that in four dimensions the anomaly polynomial for the FIQS model has the form [13] In the above, we have implicitly assumed wedge products in the multiplication of differential forms.
To get the 4D anomaly from the 6-form anomaly polynomial, one goes through the usual descent equations: For example, under a modular transformation, Z i → Z i + dIm(F i ) so that the modular-gravitygravity anomaly has the form which is precisely what one would expect if one considers the modular-gravity-gravity anomaly to have the same form as a U (1) Focusing on the second term of Eq. (A.64) , we again break up terms based on whether they contribute to the U(1) X related anomalies or the pure modular anomaly.

U(1) X Anomaly Conditions
To match the anomalies involving U(1) X , we require Note that the last term is fixed by cancellation of the linear divergence term Eq.(A.29).

B Solution to the Pauli-Villars Regularization Conditions
We will now elucidate a solution to the system described above. The solution consists of sets S a , a = 1, 2, . . . of PV fields that address each of the divergence and anomaly sets of conditions more or less separately. For example, it is possible to introduce PV fields that cancel only the nonabelian divergences and contribute to no other conditions. We will try to follow the same strategy for all the sets of conditions described above. It is not entirely possible to do so -for example, fields that solve the modular anomaly conditions will generically contribute to modular divergences. Of course, this is far from the only way to tackle the system, but it is straightforward method to illustrate that a solution can be found. To this end, we define the notion of clone fields for PV fields. For a given pair of PV fields Φ P , Φ P , we define clone fields Φ P cl , Φ P cl that have almost the same parameters (α, β, q n , . . .) and quantum numbers as the original pair but with negative signature. We say almost here because this notion is only useful if the Φ P , Φ P have quantum numbers different from the clones so that the two sets cancel each other's contributions to some subset of the conditions, but not all conditions. As a concrete example, which will be described below, one can introduce PV fields with nonabelian gauge interactions to eliminate divergences associated with those same interactions. One can then introduce clone PV fields without gauge interactions that exactly cancel the contributions of the gauge charged PV fields to all other terms. The primary advantage of this technique is tidiness.

B.1 PV Fields for U(1) X Anomaly Matching
The fields described here will satisfy Eqs. (A.67)-(A.80) and will contribute to some of the U(1) X divergence conditions (A.24)-(A.34). In particular, only PV fields withQ X = 0 contribute to Eq. (A.29), so this condition will be taken care of by this sector only. The sets of PV fields we need are • S 1 : A set of PV fields with modular invariant masses,α 1 = α 1 =γ 1 = 1/2, andq (1) n = 0 and modular weights of the form (q (1) ) C m = q P (1) δ n m and clone fields with no U(1) X .
We then place the following conditions on the parameters for these fields: (1) Once again, the first condition is a linear divergent term that can only be cancelled by fields with masses that are noninvariant under U(1) X . This in turn forces the correct coefficient for the pure U(1) X anomaly in the last condition. While the second set must satisfy The first condition here comes from Eq. (A.85) and potentially can be relaxed.
These fields will also contribute to the modular divergence conditions, as outlined below. We also have to consider theφ fields of [1] here since they have noninvariant masses under a modular transformation. These fields have no β or modular weight parameters but do have fφ =αK. then the conditions the S 3 , S 4 , andφ fields must satisfy are • S 5 : A set of pairs of PV fields withγ 5 = 1 2 and (q (5) ) C n = 0 with (q (5) ) C m = (q (5) ) P δ n m . Then the conditions we must satisfy are We include an explicit P in the modular weights simply to remind ourselves that we sum over the "P" index and not the "n" index since C = (P,n).

B.4 PV Fields for the Regulation of U(1) X Divergences
Here we introduce fields that cancel the contributions to Eqs. (A.24)-(A.34) from theẎ , S 1 , and S 2 . Note that we will omit Eq. (A.29) since has been taken care of above. We introduce the following set: • S 6 : A set of pairs of PV fields with Q (6) X andq (6) n = 0 and clone fields without U(1) X charge.
Then the conditions we must satisfy are

B.5 PV Fields for the Regulation of Nonabelian Divergences
Here we introduce fields to cancel Eqs. (A.35)-(A.37). We introduce a separate PV set for each of the nonabelian factors of the FIQS gauge group as follows • S 7 : A set of pairs of PV fields in the fundamental of SU(3) ( anti-fundamental for the primed fields) with no modular weights, uniform constants, and clone fields with no gauge charges. By uniform coefficients, we mean that α C and β C are independent of index within the set: α C = α and β C = β.
• S 8 : A set of pairs of PV fields in the fundamental of SU (2) with no modular weights, uniform constants, and clone fields with no gauge charges.
• S 9 : A set of pairs of PV fields in the 16 (and 16 for primed fields) of SO(10) and a set of pairs of PV fields in the 10 of SO(10), all with no modular weights, uniform constants, and clone fields with no gauge charges.
• S 10 : A set of PV fields with γ = γ = 1/2, zero modular weights, a nonzero trace U(1) X charge matrix, and charged under the nonabelian gauge groups in the same reps as the light fields and clone fields without nonabelian gauge charges.
Let us discuss this choice briefly. First we need to check the number of fields in a given representation. This is because we care about the quantity sinceα +β = 1. The overall sign is the sum of the signatures. Cancellation then requires provided that the PV fields have no modular weights. The first sum is over all PV fields whereas the second is over PV pairs. Both of our potential solutions can work here since we have either 1 or 2 free parameters in the γ's. In the list of sets of PV fields above, we opted for the combination of PV fields in the 10 and 16 of SO (10). For the last nonabelian divergence, Eq. (A.36), we explicitly write out the contribution from theẎ so that is takes the form where C m G is the Casimir of the representation of the matter fields. If we consider fields from the set S 10 , then this becomes The fields in S 10 contribute to Eq. (A.35) but not to Eq. (A.37) since we have restricted their γ parameters to be γ = 1 2 . Their contribution to Eq. (A.35) is not an issue since we can simply include more fields in the other sets described in this section to cancel their contribution. Finally, the clone fields ensure that none of the sets described in this section contribute to other conditions.

B.6 PV Fields for the Regulation of Abelian Divergences
Here we satisfy the conditions Eqs. (A.40)-(A.47). TheẎ contribute here, and to cancel them we will need to introduce fields withq n = 0, which is different from all other fields considered thus far. This would alter some of the expressions we have used above, but we will not consider these alterations since we will employ clone fields that cancel contributions to previously considered terms from the fields introduced here. Specifically, we consider • S 11 : A set of pairs of PV fields such that the unprimed fields have the same abelian gauge charges as the light fields (including U(1) X ), α P 11 =α, β P 11 =β, q where again a subscript or superscript (L) implies a trace over the corresponding values of the light fields. Note that we have omitted some conditions that are automatically zero. There are also terms in the above that vanish for the choice of U(1) charges defined in this paper but do not vanish for other choices. If one substitutes the parameters of S 11 and S 12 as per the discussion above, one sees that all the remaining conditions above are satisfied.

C The FIQS spectrum
The FIQS model was described in [15,16,4,18,19]. The modular weights in this model are simple: the fields in the ith untwisted sector have q i n = δ i n , and the twisted sector fields have q n = 2 3 , except for the Y i with q i n = δ i n + 2 3 . Here we will focus in particular on the U(1) charges of the low-energy matter spectrum. The U(1) charge generators arising from the Cartan subalgebra of the E 8 × E 8 and the corresponding charges were worked out in [18,19]. Table 2 of [16] lists the charges of the massless spectrum. However, the linear combinations of generators given in [4] have a mixed anomaly: Tr(Q 6 Q 7 Q X ) = 1296. (C.1) To avoid this, one should re-define Q 6 and Q 7 . The fix is very simple: Below we produce a table of the new charge designations.