Multiple anomalous U(1)s in heterotic blow-ups

The existence of multiple anomalous U(1)s is demonstrated explicitly in a blow-up version of a heterotic Z_3 orbifold. Another blow-up of the same orbifold supports further evidence for the type-I/heterotic duality in four dimensions. It has a single anomalous U(1) which does not factorize universally. As multiple anomalous U(1)s as well as non-universal factorization have never been established on heterotic orbifolds explicitely, these findings might appear contradictory at first sight. Possible inconsistencies are avoided by reinterpreting a charged twisted state as a second non-universal localized axion. The mismatch between the charges of the orbifold and blow-up spectra is resolved by suitable field redefinitions. The anomaly of the field redefinitions corresponds to the difference of blow-up and heterotic orbifold anomalies.


Introduction and results
For phenomenological applications of heterotic string compactifications it is important to know the number of anomalous U(1)s [1,2]. It has been generally accepted that heterotic orbifold models [3][4][5][6] contain at most a single anomalous U (1). Recent publications [7,8] argued that the low energy limit of the heterotic string, i.e. super Yang-Mills (YM) theory coupled to supergravity, on general smooth Calabi-Yau(CY)s with U(1) bundle backgrounds multiple anomalous U(1)s are possible. Assuming that such CYs have continuous singular orbifold limits, the existence of multiple anomalous U(1)s seems puzzling. In 6D matching between heterotic orbifolds and blow-ups seems always possible [9]. To investigate this issue in 4D, explicit blow-ups of the orbifold fixed points would be very instructive.
Since for the C 3 /Z 3 orbifold the blow-up and its U(1) bundles have recently been constructed [10] (see also [11]), we would like to focus on anomalous U(1)s in such blow-ups.
The following obstructions seem to prevent a straightforward identification of heterotic orbifold models and their blow-up counterparts: 1. In contrast to heterotic orbifolds, blow-ups can have more than one anomalous U(1).
2. While orbifold theories seem to contain at most a single axion relevant for Green-Schwarz(GS) anomaly cancellation, blow-up models can have multiple axions.
3. In blow-up models with a single anomalous U(1) the anomalies do not factorize universally as they do in heterotic orbifold models.
4. The spectra of orbifolds and their corresponding blow-ups often do not match.
5. Even when the non-Abelian spectra agree, the U(1) charges of twisted states seem never to be identical.
We argue that all these discrepancies can be explained by suitable field redefinitions. In particular, we reinterpret a charged twisted singlet on the orbifold, that drives the blow-up by taking Vacuum Expectation Value(VEV), as a localized axion. It takes part in a non-universal GS anomaly cancellation, very much like twisted RR-axions in type-I models. The anomaly in these field redefinitions precisely corresponds to the difference between blow-up and heterotic orbifold anomalies. We first describe this procedure for C 3 /Z 3 blow-up models with U(1) bundles in general. After that we inspect two concrete blow-up models of a specific heterotic Z 3 orbifold to illustrate some details. The orbifold model has been considered before [12,13] as evidence for heterotic/type-I duality [14] in 4D. The second example constitutes a different blow-up of the same orbifold that has two anomalous U(1)s.

Multiple Anomalous U(1)s in Blow-Up
In [10] the blow-up M 3 of the C 3 /Z 3 orbifold and its line bundles are described. Using these results we decompose the 10D GS 2-form B 2 into 4D perturbations as Both the U(1) bundle field strength iF 2 and the Kähler form ω 2 are harmonic 2-forms. (For the Kähler form this follows automatically, while for the U(1) field strength this is guaranteed by the Hermitean YM equations.) Aside from the 4D anti-symmetric tensor b 2 , dual to the universal axion a uni , the 2-form B 2 is decomposed into two 4D scalars b 0 and B 0 . The B 0 is interpreted as the 6D internal part of B 2 , i.e. the nine untwisted B ij , i, j = 1, 2, 3, because ω 2 reduces to the orbifold Kähler form in the blow down limit. The state b 0 is localized near the orbifold singularity, since iF becomes strongly peaked there in that limit [10]. The expansion (1) is complete: As each of the nine untwisted B ij states contributes 1/27 at a Z 3 fixed point [15], the blow-up has h 11 = 9/27 + 1 = 4/3, which indeed equals half its Euler number [10]. Which of the b 2 , b 0 and B 0 mediate GS mechanisms is determined by their anomalous variations, that leave the 3-form field strength H 3 of B 2 invariant. Expanding H 3 in these 4D perturbations gives The usual anomalous variations of b 2 are found by restricting the YM and gravitational Chern-Simons and Ω L 3 , to 4D. Since the expansion of 10D YM CS also contains a linear term in The H I are the Cartan generators of 10D gauge group and V I are numbers characterizing the U(1) bundle [10]. Contrary B 0 never has an anomalous variation, because neither CS 3-forms have decompositions proportional to ω 2 .
The 4D anomaly I 6 is obtained from the factorized 10D anomaly polynomial I 12 = X 4 X 8 , where X 4 = tr(iF 2 ) 2 − tr R 2 and X 8 is given in [16,17]. By integrating over the blow-up we get where the first (second) index specifies the number of 6D internal (4D Minkowski) indices of the forms. A third possible contribution, X 4,0 X 2,6 , integrates to zero due to the background Bianchi identity. The integral of the first term can give rise to more than one term in general. However, for the U(1) bundles on the blow-up of C 3 /Z 3 we get which ensure that this gives a single factor. Factorization in heterotic orbifolds only occurs through the last term in (3). Hence the blow-up of C 3 /Z 3 can support at most two anomalous U(1)s, and we can interpret b 0 = a non as a non-universal axion. Together with the universal axion a uni (4D dual to b 2 ) it will take part in the GS mechanism of anomaly cancellation.

Spectral matching
Above we argued that on the blow-up of C 3 /Z 3 orbifold there can be two anomalous U(1)s. In this section we explain why there is no contradiction with the general finding that heterotic orbifolds have at most a single anomalous U(1). Consider a heterotic orbifold model and a proposal for a corresponding blow-up. Because the heterotic orbifold has an anomalous U(1), the vacuum is unstable and some twisted fields need to get non-vanishing VEVs. Depending on which fields attain VEVs this is accompanied by further symmetry breaking to the blow-up gauge group. Some twisted fields decouple by VEV induced superpotential mass terms; this makes the non-Abelian spectrum of the heterotic orbifold equal to that of the blow-up model [12,13]. But this does not resolve why charges of the fields present on both sides can be different.
A possible mismatch of U(1) charges can be understood by analyzing the consequences of fields with non-vanishing VEVs. Let Ψ q be a twisted singlet chiral superfield, w.r.t. the unbroken blowup gauge group with U(1) charges q = (q 1 , . . . , q n ), and Φ Q another twisted chiral superfield, not necessarily a singlet, with charges Q. A non-vanishing VEV v of Ψ q means that in the quantum theory this field can be represented as an exponential, and we can redefine Φ Q with an arbitrary power r of this exponential: The new superfields T and Φ ′ Q ′ transform under the U(1) gauge transformations as: where ϕ i are the gauge parameters of the U(1)s. Hence the imaginary part a T of T is related to the non-universal axion a non of the blow-up theory. In addition, we have obtained a superfield Φ ′ Q ′ with U(1) charges Q ′ = Q − rq . We claim that it is always possible to perform such field redefinitions to make the charges of the non-decoupled twisted states equal to their blow-up counterparts. Moreover, the fields that decouple can be given gauge invariant superpotential mass terms using field redefinitions.
Next we explain how these field redefinitions help to get agreement between the anomalies of the heterotic orbifold and the blow-up models. Upto this point a T only has anomalous gauge variations (6) but no anomalous couplings; still at this stage only one axion is involved in the GS mechanism. We call this the heterotic axion a het . Its anomalous couplings are determined by the anomaly polynomial, that factorizes universally I het 6 = X het 2 X 4 . Here the GS 4-form X 4 is restricted to 4D, and X het 2 , via the descent equations, determines the anomalous variation of a het . The anomalous couplings for the axion a T arises because the path integral measure is not invariant under the anomalous field redefinitions (5). The resulting couplings can be deduced from the anomaly polynomial I red This equation reflects that after the field redefinitions the chiral spectra of the heterotic orbifold and the blow-up models become identical. Finally we would like to understand the precise relation between the heterotic axion a het and the axion a T obtained from the superfield T used in the rescaling (5), and the two blow-up axions, a uni (the 4D dual of b µν ) and a non = b 0 . The anomaly polynomials I uni 6 and I non 6 of the latter two are determined by the anomalous couplings of the zero modes of B 2 They factorize as where X non  Table 1: The spectra of the heterotic Z 3 orbifold and two blow-ups are displayed: The spectrum of "TypeI" equals the type-I Z 3 orbifold, and "U(1) 2 " has two anomalous U(1)s.
Hence each term in (7) can be computed independently, providing a consistency check. The relation between the axions is now fixed by noting that the anomalous variations of the axions a het , a T , a uni and a non are determined from X het 2 , q i iF i 2 , X uni 2 and X non 2 , respectively, via the standard descent equations. Therefore, the sum relation for the anomalies (7) implies that the interactions of the axions with gauge and gravitational fields are related via As the unbroken gauge group typically consists of two or more group factors, this leads to an over constrained system of linear equations relating (a uni , a non ) to (a het , a T ). Because both a het and a non multiply X 4 , we find where c and d are model dependent constants.
To summarize we have shown that the chiral spectra of the heterotic orbifold and blow-up models are identical upon using field redefinitions that allow one to modify U(1) charges. In particular, a charged twisted singlet on the heterotic orbifold is reinterpreted as a localized axion in the blow-up theory. The difference between their anomalies is precisely canceled by the anomalous variation of this localized axion.

A heterotic SO(32) Z 3 orbifold
We illustrate our general findings by considering two U(1) bundle blow-ups of the heterotic SO(32) C 3 /Z 3 orbifold model with gauge shift 1 3 (0 4 , 1 8 , -2 4 ). The gauge group and the spectrum of this heterotic model [18][19][20] are given in the first row of table 1. Since a multiplicity factor of 1 9 signals untwisted modes [15], the model contains two twisted states: a singlet (1, 1) 4 and a SO(8) spinor (8 + , 1) -2 with charges 4 and -2, respectively. As usual for heterotic models the anomaly polynomial factorizes universally where F 1 , F 12 and F 8 denote the gauge field strengths of U(1) , U(12) and SO (8) , respectively. The blow-ups are determined by VEVs of the twisted states that satisfy the D-and F-flatness conditions.
The D-flatness implies that at least the singlet has a VEV to cancel the one-loop Fayet-Iliopolous term [1,2]. (In addition, if the spinor takes a VEV, two of its components are non-vanishing.) Assuming that none of the untwisted states have VEVs, the relevant part of the superpotential reads When both (1, 1) 4 and (8 + , 1) -2 have VEVs, F-flatness is assured through the presence of higher superpotential terms. The superpotential (being a complex function of these fields) allows solutions with vanishing F-terms (and D-terms) at isolated points in parameter space. This heterotic orbifold model has received quite some attention in the past because this model was suggested to have a type-I Z 3 orbifold model dual [12,13,21,22]. However, because the GS anomaly cancellation in both models is mediated by different fields, it had been questioned whether these models can really be dual to each other [23]. Applying the general formalism we developed here, the duality is realized in all fine print.

Type-I blow-up model
If we only give a VEV to (1, 1) 4 , the gauge group of the blow-up model remains the same as on the heterotic orbifold. We identify this case with blow-up model characterized by V = (0 4 , 1 12 ) defined in ref. [10]. The spectra of the blow-up model and the type-I Z 3 orbifold model [21,22] are identical, and given in the second row of table 1. Even though there is just a single U(1), it does not factorize universally: Because only the singlet takes a VEV, we make the following field redefinitions of the twisted states The superpotential (14) gives the state (8 + , 1) ′ 0 a regular mass. This transformation takes a twisted singlet to a localized axion in the blow-up theory; from the type-I point of view this is the twisted RR-axion. By computing the various anomaly polynomials we derive the identification of the axions Hence the type-I Z 3 orbifold model coincides with a blow-up of a heterotic Z 3 orbifold.