Hypercharge Flux, Exotics, and Anomaly Cancellation in F-theory GUTs
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Hypercharge Flux, Exotics, and Anomaly Cancellation in F-theory GUTs
Joseph Marsano
Enrico Fermi Institute, University of Chicago5640 S Ellis Ave, Chicago, IL 60637, USA
We sharpen constraints related to hypercharge flux in F-theory GUTs that possess U (1) sym-metries and argue that they arise as a consequence of 4-dimensional anomaly cancellation. Thisgives a physical explanation for all restrictions that were observed in spectral cover models whiledemonstrating that the phenomenological implications for a well-motivated set of models are nottied to any particular formalism. I. INTRODUCTION
The vastness of the string landscape presents a seriousobstacle for studying particle physics in string theory.To make progress, it is often helpful to adopt a bottom-up approach [1] that mirrors the successful techniquesof effective field theory. Type II string theories providea natural setting for this since the charged degrees offreedom can localize on branes that probe only a smallpart of the compactification geometry. The low energyphysics associated to these branes is captured by a non-Abelian gauge theory whose bare coupling constants atthe compactification scale are determined by local geo-metric data.This approach is particularly appealing for the con-struction of Grand Unified Theories (GUTs) [2–4] as thecharged sector is engineered on a single stack of branes.The volume of the internal cycle wrapped by the branesintroduces a new scale into the problem that can helpto realize the small observed hierarchy between M GUT and M Planck . In this setting, the large top Yukawa cou-pling suggests an underlying exceptional group structure[5] that motivates the study of nonperturbative type IIconfigurations described by M-theory or F-theory. Thelatter has received significant attention over the past fewyears in large part because powerful techniques of alge-braic geometry are available to simplify the analysis.Most approaches to F-theory GUTs make crucial useof two important ingredients. The first is the presenceof U (1) symmetries that can be used to protect againstproton decay [5–9] or to motivate scenarios for how super-symmetry breaking is mediated to the Standard Model[10]. The second important ingredient is “hyperchargeflux”, which provides an elegant mechanism for break-ing the GUT group while addressing the doublet-tripletsplitting problem [3]. In explicit constructions based onspectral cover techniques [11], these two ingredients ap-pear to be interrelated [6, 8]; spectral cover models witha particular set of U (1) symmetries tend to exhibit tightconstraints on how “hypercharge flux” can be distributedamong the matter curves where charged fields localize [6].This, in turn, has a striking impact on the 4-dimensionalphysics of all F-theory GUT models built to date.The goal of this letter is to understand the nature andsource of these constraints. Because of the dramatic phe-nomenological implications [6], it is crucial to understand if the relationship between U (1) symmetries and “hyper-charge flux” represents a limitation of our current model-building toolbox or a more general lesson with an intrin-sic physical origin. One indication of the latter can befound in a recent paper of Dudas and Palti [12], who no-ticed a simple pattern in the distribution of “hyperchargeflux” in a set of spectral cover models. It is not hard toprove their relations for generic (suitably nondegenerate)spectral cover models and we do this in the upcoming pa-per [13]. More intriguing, however, is that we can rewritethe original Dudas-Palti observation in a simple way thatdoes not make explicit reference to spectral covers at all X mattercurves, a q a Z Σ ( a ) ω Y = X mattercurves, i q i Z Σ ( i ) ω Y (1)Here, q a denotes the common U (1) charge of or fields that localize along curves Σ ( x ) in the compactifi-cation and ω Y is a “hypercharge flux” that is chosen toensure that the U (1) Y gauge boson remains massless. Arelation this simple should have a physical origin and, inthis letter, we will demonstrate that it is a consequenceof 4-dimensional anomaly cancellation. In addition toclarifying the physics of all known constraints of spectralcover models, this observation allows us to derive a gen-eralization of (1) that must be satisfied by any F-theoryGUT that combines U (1) symmetries and “hyperchargeflux” regardless of how it is constructed. Among themany implications for phenomenology, our results implythat any U (1) symmetry in a model that combines “hy-percharge flux” with the flavor scenario of [14] must be U (1) B − L , which cannot address µ or dimension 5 protondecay. Insisting on the existence of a U (1) P Q symmetryto deal with these necessarily introduces charged exoticsinto the spectrum.
II. F-THEORY GUTS AND ANOMALYCANCELLATIONA. Spectrum and “Hypercharge Flux”
The charged sector of an F-theory GUT model is de-scribed by the 8-dimensional worldvolume theory thatdescribes the physics of a stack of 7-branes. This the-ory, which we take to have gauge group SU (5) GUT , iscompactified on a complex surface S GUT and can be UVcompleted by embedding that surface into a consistentF-theory compactification. Adjoint-valued fields propa-gate throughout the 8-dimensional worldvolume but themodel contains additional degrees of freedom in the and representations (and their conjugates) that localizeon holomorphic ”matter curves” in S GUT . Determiningthe 4-dimensional spectrum requires a dimensional reduc-tion in either case and can be influenced by introducingsuitable fluxes into the model.While most of these fluxes descend from the bulk of thecompactification, worldvolume flux plays an importantrole. An internal flux of the U (1) Y gauge field can break SU (5) GUT down to the MSSM gauge group and, whenchosen correctly, remove unwanted degrees of freedomlike Higgs triplets and leptoquarks [3]. In general, the netchirality of leptoquarks that descend from the SU (5) GUT adjoint is determined by an index theorem [3] n ( , ) − / − n ( , ) +5 / = Z S GUT c ( S GUT ) ∧ c ( L / Y )where L Y is a line bundle that specifies the “hyperchargeflux”. The spectrum on a matter curve Σ, on the otherhand, is computed as [3] n R − n R = Z Σ c (cid:16) V Σ ⊗ L Y R Y (cid:17) = Z Σ h c ( V Σ ) + M Σ c ( L Y R Y ) i where V Σ is a bundle of rank M Σ that roughly encodesthe ”bulk” fluxes and Y R is the U (1) Y charge of fields inthe representation R . The bundle V Σ and its rank M Σ are intrinsic properties of the matter curve Σ but thecharges Y R can differ among the various MSSM multi-plets contained in the SU (5) GUT multiplet that localizesthere. In this way, a nontrivial “hypercharge flux” canbe used to generate incomplete GUT-multiplets, whichis very useful for obtaining Higgs doublets without theirtriplet partners. The ranks M Σ are all 1 for spectralcover models that are suitably nondegenerate but can belarger in more general constructions [4, 15]. B. Constraints on “Hypercharge Flux” fromMSSM Gauge Anomalies
When building models, we need some freedom to dis-tribute “hypercharge flux” among the matter curves thatare present. This freedom must be limited, though, be-cause “hypercharge flux” induces a chiral spectrum withrespect to the MSSM gauge groups that generically leadsto anomalies. The SU (3) anomaly, for instance, is pro-portional to3 X mattercurves, i M Σ ( i ) Z Σ ( i ) c ( L Y ) − X mattercurves, a M Σ ( a ) Z Σ ( a ) c ( L Y )+ 5 Z S GUT c ( S GUT ) ∧ c ( L Y ) Since this must cancel regardless of how we choose c ( L Y ), we see that the matter curves of any consistentF-theory GUT model should satisfy3 X mattercurves, i M Σ ( i ) [Σ ( i ) ] − X mattercurves, a M Σ ( a ) [Σ ( a ) ] + 5[ c ] = 0where [ c ] is the anti-canonical curve of S GUT . This rela-tion is well-known [11, 16] for constructions with M Σ ( i ) = M Σ ( a ) = 1 and has been derived using a “stringy”anomaly cancellation argument [11]. It is amusing tosee, however, that it can be understood already as a con-sequence of anomaly cancellation in 4-dimensions.Cancellation of mixed gauge anomalies involving U (1) Y is not guaranteed for generic choices of L Y be-cause, in most cases, the hypercharge gauge boson islifted through an induced coupling to RR fields [3]. Theconditions that L Y must satisfy in order to prevent thisare known in F-theory and correspond to constructing L Y from a (1 , ω Y ∼ c ( L Y ), that trivializes inthe full compactification. Any “hypercharge flux” of thistype will necessarily be constrained; at the very least,its distribution among the matter curves must guaranteethat all MSSM gauge anomalies are cancelled. This leadsto the conditions0 = X mattercurves, i M Σ ( i ) Z Σ ( i ) c ( L Y ) = X mattercurves, a M Σ ( a ) Z Σ ( a ) c ( L Y )(2)that are easy to verify in generic F-theory GUT models[16] with a massless U (1) Y . C. Implications of Mixed Gauge Anomalies
We would now like to ask if a “hypercharge flux” ω Y that doesn’t lift U (1) Y exhibits any additional proper-ties in a geometry that engineers bulk U (1) symmetriesin addition to SU (5) GUT [17]. To address this, let usconsider what happens when we turn on this flux and noother fluxes . Our flux will induce a nontrivial spectrumbut, because all U (1)’s remain massless, it cannot giverise to any gauge anomalies [18]. Of particular interestto us are mixed anomalies with insertions of both MSSMand U (1) currents since these only get contributions fromthe chiral fields that localize on matter curves in S GUT .We will see that the Dudas-Palti relations (1) for spectralcover models simply express a set of nontrivial relationsthat the (1 , ω Y must satisfy in order for these4-dimensional mixed gauge anomalies to cancel.To make things completely explicit, we use ω Y to definea line bundle L Y on the GUT 7-branes that defines anontrivial U (1) Y background. We further normalize thatbackground so that all charged fields on matter curvesare sections of the integer quantized gauge bundles listedbelow SU (5) SU (3) × SU (2) × U (1) Y Bundle ( , ) +1 L Y ( , ) +1 / L Y ( , ) − / L − Y ( , ) +1 / L Y ( , ) − / L − Y (3)We now determine the contributions to mixed gaugeanomalies that arise from the chiral spectrum on ageneric or matter curve. To obtain (1) and itsgeneralization beyond spectral cover models, it will besufficient to consider anomalies of the type G SM × U (1),where G SM denotes a Standard Model gauge group.Consider first the contribution from fields that local-ize on a curve, Σ ( a ) , which carry a U (1) charge q a .Denoting the M Σ ( a ) -weighted U (1) Y flux there by N a N a = M Σ ( a ) Z Σ ( a ) c ( L ) = M Σ ( a ) Z Σ ( a ) ω Y (4)the contributions to mixed G SM × U (1) anomalies areMultiplet Chir SU (3) U (1) SU (2) U (1) U (1) Y U (1)( , ) +1 N a q a N a ( , ) +1 / N a q a N a q a N a q a N a / , ) − / − N a − q a N a − q a N a / − q a N a q a N a q a N a / U (1) charge. We now do the same thing for fields ona ( i ) curve that carry U (1) charge q i . Letting N i denotethe M Σ ( i ) -weighted U (1) Y flux N i = M Σ ( i ) Z Σ ( i ) c ( L ) = M Σ ( i ) Z Σ ( i ) ω Y (5)we findMultiplet Chir SU (3) U (1) SU (2) U (1) U (1) Y U (1)( , ) +1 / N i q i N i q i N i / , ) − / − N i − q i N i − q i N i / q i N i − q i N i − q i N i / G SM × U (1)anomalies implies that ω Y must satisfy X mattercurves, a q a M Σ ( a ) Z Σ ( a ) ω Y = X mattercurves, i q i M Σ ( i ) Z Σ ( i ) ω Y (6)which, for M Σ ( a ) = M Σ ( i ) = 1, is nothing other than theDudas-Palti relations (1). We refer to (6) as the gen-eralized Dudas-Palti relations, which must hold for any ω Y that can be used to construct “hypercharge flux” inan SU (5) GUT
F-theory GUT model with an extra U (1)symmetry. It is easy to see that other mixed anoma-lies, as well as the U (1) anomaly, vanish without givingrise to any additional constraints. Though the story isless constrained than in 6-dimensions [19], it would beinteresting to pursue a more general analysis of anomalycancellation in 4-dimensional F-theory compactificationsin the future. III. IMPLICATIONS OF THE GENERALIZEDDUDAS-PALTI RELATIONS
The first question to ask about (6) and (2) is whetherthey represent all of the nontrivial constraints on the dis-tribution of “hypercharge flux” in F-theory GUTs. Inthe case of spectral cover models, we suspect that theydo because it appears that one can use spectral covers toconstruct, at least in principle, all distributions of “hy-percharge flux” that satisfy them [13]. Based on this,it is natural to conjecture that, even for more generalclasses of F-theory GUTs, (6) and (2) represent the onlyconstraints.In light of this, we should correct some misstatementsthat were made in [6]. There, it was claimed that thepresence of “hypercharge flux” on matter curves auto-matically implied that “hypercharge flux” must threadsome matter curves as well. The DP relations (1) donot forbid a configuration in which “hypercharge flux”threads only curves, though, and it is possible to con-struct spectral covers that do precisely this [13].Finally, let us comment on implications of the general-ized Dudas-Palti relations (6) for F-theory model build-ing. While several approaches to flavor have been sug-gested in the past few years [20], the mechanism of wavefunction overlaps is particularly attractive [14]. Thismechanism requires all three generations of the tolocalize on one matter curve and similar for all threegenerations of the . The Higgs fields then lie on dis-tinct matter curves, Σ ( H u ) and Σ ( H d ) , which must have M Σ ( Hu ) = M Σ ( Hd ) = 1 and carry +1 and -1 units of“hypercharge flux”, respecitvely, to lift the triplets [3].Crucial to this scenario is that “hypercharge flux” notbe allowed to thread any curve Σ other than Σ ( H u ) andΣ ( H d ) ; if it did, we would obtain massless matter fieldson Σ that do not comprise a complete GUT multiplet.As one assumes that the Standard Model fields are en-gineered as complete GUT multiplets, the threading of“hypercharge flux” through such a Σ necessarily intro-duces new charged exotics into the spectrum [6].If we wish to combine this scenario with a U (1) symme-try, the generalized Dudas-Palti relations (6) imply thatthe charges q H u and q H d associated to the matter curvesΣ ( Hu ) and Σ ( Hd ) must satisfy q H u − q H d = 0 (7)The doublet H u comes from a rather than a , though,so its charge is actually − q H u . Writing (7) in terms ofthe actual H u and H d charges we get Q ( H u ) + Q ( H d ) = 0 (8)What type of U (1) symmetry can this be? Because all ’s ( ’s) are engineered on a single curve, all of themmust carry a common charge. The only U (1) symmetryof this type that commutes with SU (5), satisfies (8), andpreserves the MSSM superpotential is the famous U (1) χ ,which is the linear combination of U (1) Y and U (1) B − L that enters naturally in SO (10) unification models. Wesee that P Q symmetries, broadly defined as U (1)’s forwhich (8) does not hold, cannot be combined with thedesired distribution of hypercharge flux. If we insist onrealizing all 3 generations of ’s ( ’s) on a single mat- ter curve, the presence of U (1) P Q implies the existenceof additional charged matter fields that do not come incomplete GUT multiplets [6].
Acknowledgements:
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