We study the flavor structures of zero-modes, which are originated from the modular symmetry on $T^2_1\times T^2_2$ and its orbifold with magnetic fluxes. We introduce the constraint on the moduli parameters by $τ_2=Nτ_1$, where $τ_i$ denotes the complex structure moduli on $T^2_i$. Such a constraint can be derived from the moduli stabilization. The modular symmetry of $T^2_1 \times T^2_2$ is $SL(2,\mathbb{Z})_{τ_1} \times SL(2,\mathbb{Z})_{τ_2} \subset Sp(4,\mathbb{Z})$ and it is broken to $Γ_0(N) \times Γ^0(N)$ by the moduli constraint. The wave functions represent their covering groups. We obtain various flavor groups in these models.
This paper proposes a concrete model of nonuniversal gaugino masses on the basis of higher-dimensional supersymmetric Yang-Mills theories compactified on a magnetized factorizable torus, and we estimate the gauge coupling constants and gaugino masses in the model. In the magnetized toroidal compactifications, the four-dimensional effective action can be obtained analytically identifying its dependence on moduli fields, where the magnetic fluxes are able to yield the flavor structure of the minimal supersymmetric standard model (MSSM). The obtained gauge kinetic functions contains multi moduli …
As the effective field theory of the superstring theory, ten-dimensional ${\cal N}=1$ supersymmetric Yang-Mills theory is induced. We consider the ten-dimensional space-time ${\cal M}_{10}$ as direct products of our four-dimensional space-time ${\cal M}_{4}$ and three two-dimensional compact spaces $X_i\ (i=1,2,3)$, i.e. ${\cal M}_{10} = {\cal M}_4 \times X_2 \times X_2 \times X_3$. In particular, we consider $X_i$ as a torus, the torus orbifold, and also the blow-up manifold of the torus orbifold with background magnetic fluxes. We discuss the modular symmetry in the magnetized torus and its …
We study discrete flavor symmetries of the models based on a ten-dimensional supersymmetric Yang-Mills (10D SYM) theory compactified on magnetized tori. We assume non-vanishing non-factorizable fluxes as well as the orbifold projections. These setups allow model-building with more various flavor structures. Indeed, we show that there exist various classes of non-Abelian discrete flavor symmetries. In particular, we find that $S_3$ flavor symmetries can be realized in the framework of the magnetized 10D SYM theory for the first time.
We study non-abelian discrete flavor symmetries, which can appear in magnetized brane models. For example, $D_4$, $Δ(27)$ and $Δ(54)$ can appear and matter fields with several representations can appear. We also study the orbifold background, where non-abelian flavor symmetries are broken in a certain way.
We study the modular invariance in magnetized torus models. Modular invariant flavor model is a recently proposed hypothesis for solving the flavor puzzle, where the flavor symmetry originates from modular invariance. In this framework coupling constants such as Yukawa couplings are also transformed under the flavor symmetry. We show that the low-energy effective theory of magnetized torus models is invariant under a specific subgroup of the modular group. Since Yukawa couplings as well as chiral zero modes transform under the modular group, the above modular subgroup (referred to as modular f…
We discuss the flavor structure in a resolutions of magnetized $T^2/\mathbb{Z}_2$ orbifold models. The flavor structure of Yukawa couplings among chiral zero-modes is sensitive to radii of the resolved fixed points of $T^2/\mathbb{Z}_2$ orbifold. We show in concrete models that the realistic mass hierarchy and mixing angles of the quark sector can be realized at certain values of blow-up radii, where the number of parameters in the blow-up model can be smaller than toroidal orbifold one.
We study the modular symmetry on magnetized toroidal orbifolds with Scherk-Schwarz phases. In particular, we investigate finite modular flavor groups for three-generation modes on magnetized orbifolds. The three-generation modes can be the three-dimensional irreducible representations of covering groups and central extended groups of $Γ_N$ for $N=3,4,5,7,8,16$, that is, covering groups of $Δ(6(N/2)^2)$ for $N=$ even and central extensions of $PSL(2,\mathbb{Z}_{N})$ for $N=$odd with Scherk-Schwarz phases. We also study anomaly behaviors.
We study the flavor landscape of particle physics models based on a ten-dimensional super Yang-Mills theory compactified on magnetized tori preserving four-dimensional ${\cal N}=1$ supersymmetry. Recently, we constructed a semi-realistic model which contains the minimal supersymmetric standard model (MSSM) using an Ansatz of magnetic fluxes and orbifolding projections. However, we can consider more various configurations of magnetic fluxes and orbifolding projections preserving four-dimensional ${\cal N}=1$ supersymmetry. We research systematically such possibilities for leading to MSSM-like m…
In certain class of flux compactification, moduli mediated supersymmetry (SUSY) breaking preserves flavor and CP at leading order in the perturbative expansion controlled by the vacuum expectation value of the messenger modulus. Nevertheless there still might be dangerous flavor or CP violation induced by higher order Kähler potential. We examine the constraints on such SUSY breaking scheme imposed by low energy flavor and/or CP violating observables. It is found that all phenomenological constraints can be satisfied even for generic form of higher order Kähler potential and sparticle spectra …
We study the modular symmetry in magnetized D-brane models on $T^2$. Non-Abelian flavor symmetry $D_4$ in the model with magnetic flux $M=2$ (in a certain unit) is a subgroup of the modular symmetry. We also study the modular symmetry in heterotic orbifold models. The $T^2/Z_4$ orbifold model has the same modular symmetry as the magnetized brane model with $M=2$, and its flavor symmetry $D_4$ is a subgroup of the modular symmetry.
We present a calculating method for the quark and lepton mixing angles. After a general discussion in field theoretic models, we present a working model from a string compactification through $Z_{12-I}$ orbifold compactification. It is beyond presenting just three families of the standard model but is the first example from string compactification successfully fitting to the observed data. Assuming that all Yukawa couplings from string compactification are real, we also comment on a relation between the CP phases in the Jarlskog determinants obtained from the CKM and PMNS matrices. The flipped…
In string compactifications, frequently there appears the anomalous U(1) gauge symmetry which belonged to E8$\times$E8 of the heterotic string. This anomalous U(1) gauge boson obtains mass at the compactification scale, just below $10^{18\,}$GeV, by absorbing one pseudoscalar (corresponding to the model-independent axion) from the second rank anti-symmetric tensor field $B_{MN}$. Below the compactification scale, there results a global symmetry U(1)$_{\rm anom}$ whose charge $Q_{\rm anom}$ is the original gauge U(1) charge. This is the most natural global symmetry, realizing the "invisible" ax…
We study quark and lepton mass matrices derived from magnetized $T^2/\mathbb{Z}_2$ orbifold models. Quark and lepton masses have a large hierarchy. In addition, mixing angles are large in the lepton sector, while those are small in the quark sector. We find that this structure can be realized in certain flavor models, which are identified by the zero points of the zero-mode wave functions of fermions and Higgs modes. We classify such realistic flavor models. Fixed points $τ=i$, $e^{2πi/3}$ and $i\infty$ of the modulus $τ$ play a role in realizing a large mass hierarchy through our scenario, wh…
Heterotic orbifolds can explain the origin of flavor symmetries and the flavor representations of matter fields in particle physics as a result of the geometric properties of the associated string states in the compact space. After a review of the method to obtain flavor symmetries in these models, we determine the most frequent non-Abelian flavor symmetries appearing in promising Abelian heterotic orbifolds. Interestingly, these symmetries correspond only to D_4, Delta(54) and products of these symmetries and Abelian factors. A large set of promising models exhibits purely Abelian flavor symm…
A Z_2 orbifold compactification of the heterotic string is considered. The resulting 6D GUT groups can be SO(16) or E_7 times SU(2) plus some hidden sector groups. The N=4 supersymmetry is reduced to N=2. In particular, the SO(16) 6D model with one spinor representation 128 can reduce to the previous 5D SO(16) or SO(14) family unification models after compactifying the sixth dimension. To obtain one spinor, we have to take into account the left-over center of SO(16). We also comment on the E_7 times SU(2) model.
The string landscape satisfies interesting finiteness properties imposed by supersymmetry and string-theoretical consistency conditions. We study N=1 supersymmetric compactifications of Type IIB string theory on smooth elliptically fibered Calabi-Yau threefolds at large volume with magnetized D9-branes and D5-branes. We prove that supersymmetry and tadpole cancellation conditions imply that there is a finite number of such configurations. In particular, we derive an explicitly computable bound on the number of magnetic flux quanta, as well as the number of D5-branes, which is independent of th…
We study 10D super Yang-Mills E8 theory on the 6D torus compactification with magnetic fluxes. We study systematically the possibilities for realizing 4D supersymmetric standard models with three generations of quarks and leptons. We also study quark mass matrices.
We study the modular symmetry in $T^2$ and orbifold comfactifications with magnetic fluxes. There are $|M|$ zero-modes on $T^2$ with the magnetic flux $M$. Their wavefunctions as well as massive modes behave as modular forms of weight $1/2$ and represent the double covering group of $Γ\equiv SL(2,\mathbb{Z})$, $\widetildeΓ \equiv \widetilde{SL}(2,\mathbb{Z})$. Each wavefunction on $T^2$ with the magnetic flux $M$ transforms under $\widetildeΓ(2|M|)$, which is the normal subgroup of $\widetilde{SL}(2,\mathbb{Z})$. Then, $|M|$ zero-modes are representations of the quotient group $\widetildeΓ'_{2…
We analyze the action of the target-space modular group in toroidal type IIB orientifold compactifications with magnetized D-branes and continuous Wilson lines. The transformation of matter fields agree with that of twisted fields in heterotic compactifications, constituting a check of type I/heterotic duality. We identify the holomorphic N = 1 variables for these compactifications. Matter fields and closed string moduli are both redefined by open string moduli. The redefinition of matter fields can be read directly from the perturbative Yukawa couplings, whereas closed string moduli redefinit…
We study the modular symmetry anomaly in magnetized orbifold models. The non-perturbative effects such as D-brane instanton effects can break tree-level symmetry. We study which part of the modular symmetry is broken explicitly by Majorana mass terms with three generations of neutrinos. The modular weight of neutrino mass terms does not match with other coupling terms in tree-level Lagrangian. In addition, the $Z_N$ symmetry of the modular flavor symmetry is broken and a certain normal subgroup of the modular flavor symmetry remains in neutrino mass terms.
We study the photon self-energy for the in-medium photon in a three-flavor color superconductor in the presence of a magnetic field. At strong magnetic field, the quark dynamics becomes $(1+1)$-dimensional and the self-energy tensor only has longitudinal components. In this approximation there is no Debye or Meissner screenings at zero temperature, but the electric susceptibility is nonzero and highly anisotropic. In the direction transverse to the applied field, the electric susceptibility is the same as in vacuum, while in the longitudinal direction it depends on the magnitude of the magneti…
A two-flavor color superconducting (2SC) Nambu--Jona-Lasinio (NJL) model is introduced at finite temperature T, chemical potential mu and in the presence of a constant magnetic field eB. The effect of (T,mu,eB) on the formation of chiral and color symmetry breaking condensates is studied. The complete phase portrait of the model in T-mu, mu-eB, and T-eB phase spaces for various fixed eB, T, and mu is explored. A threshold magnetic field eB_t~ 0.5 GeV^2 is found above which the dynamics of the system is solely dominated by the lowest Landau level (LLL) and the effects of T and mu are partly com…
F-theory is one of frameworks in string theory where supersymmetric grand unification is accommodated, and all the Yukawa couplings and Majorana masses of right-handed neutrinos are generated. Yukawa couplings of charged fermions are generated at codimension-3 singularities, and a contribution from a given singularity point is known to be approximately rank 1. Thus, the approximate rank of Yukawa matrices in low-energy effective theory of generic F-theory compactifications are minimum of either the number of generations N_gen = 3 or the number of singularity points of certain types. If there i…
The center-flavor symmetry of a gauge theory specifies the global form of consistent gauge and flavor bundle background field configurations. For 6d gauge theories which arise from a tensor branch deformation of a superconformal field theory (SCFT), we determine the global structure of such background field configurations, including possible continuous Abelian symmetry and R-symmetry bundles. Proceeding to the conformal fixed point, this provides a prescription for reading off the global form of the continuous factors of the zero-form symmetry, including possible non-trivial mixing between fla…
The generation of exponential flavor hierarchies from extra-dimensional wavefunction overlaps is re-examined. We find, surprisingly, that coexistence of anarchic fermion mass matrices with such hierarchies is intrinsic and natural to this setting. The salient features of charged fermion and neutrino masses and mixings can thereby be captured within a single framework. Both Dirac and Majorana neutrinos can be realized. The minimal phenomenological consequences are discussed, including the need for a fundamental scale far above the weak scale to adequately suppress flavor-changing neutral curren…
We revisit the flavor symmetries arising from compactifications on tori with magnetic background fluxes. Using Euler's Theorem, we derive closed form analytic expressions for the Yukawa couplings that are valid for arbitrary flux parameters. We discuss the modular transformations for even and odd units of magnetic flux, M, and show that they give rise to finite metaplectic groups the order of which is determined by the least common multiple of the number of zero-mode flavors involved. Unlike in models in which modular flavor symmetries are postulated, in this approach they derive from an under…
Discovering a selection principle and the origin of flavor symmetries from an ultraviolet completion of particle physics is an interesting open task. As a step in this direction, we classify all possible flavor symmetries of 4D massless spectra emerging from supersymmetric Abelian orbifold compactifications, including roto-translations and non-factorizable compact spaces, for generic moduli values. Although these symmetries are valid in all string theories, we focus on the E8 x E8 heterotic string. We perform the widest known search of E8 x E8 Abelian orbifold compactifications, yielding over …
F-theory is one of the frameworks where up-type Yukawa couplings of SU(5) unified theories are naturally generated. As charged matter fields have localized zero modes in F-theory, a study of flavor structure could be easier in F-theory than in Heterotic string theory. In a study of flavor structure in the lepton sector, however, an important role is played by right-handed neutrinos, which are not charged under the SU(5) unified gauge group. It is therefore solicited to find out what right-handed neutrinos are in F-theory compactifications and how their Majorana mass terms are generated togethe…
Eclectic flavor groups arising from string compactifications combine the power of modular and traditional flavor symmetries to address the flavor puzzle. This top-down scheme determines the representations and modular weights of all matter fields, imposing strict constraints on the structure of the effective potential, which result in controlled corrections. We study the lepton and quark flavor phenomenology of an explicit, potentially realistic example model based on a T6/Z3xZ3 orbifold compactification of the heterotic string that gives rise to an Omega(2) eclectic flavor symmetry. We find t…
We study a scenario to derive four-dimensional modular flavor symmetric models from higher dimensional theory by assuming the compactification consistent with the modular symmetry. In our scenario, wavefunctions in extra dimensional compact space are modular forms. That leads to constraints on combinations between modular weights and $Γ_N$ ($Γ_N')$ representations of matter fields. We also present illustrating examples.
I discuss the correspondence between realistic four dimensional free fermionic models and $Z_2\times Z_2$ orbifold compactification. %I illustrate this correspondence for one particular example. I discuss the properties of the $Z_2\times Z_2$ orbifold that are reflected in the realistic free fermionic models. I argue that the properties of the realistic free fermionic models arise due to the underlying $Z_2\times Z_2$ orbifold compactification with nontrivial background fields. I suggest that three generation is a natural outcome of $Z_2\times Z_2$ orbifold compactification with ``standard emb…
We discuss E3-brane instantons in N=1 F-theory compactifications to four dimensions and clarify the structure of E3-E3 zero modes for general world-volume fluxes. We consistently incorporate SL(2,Z) monodromies and highlight the relation between F-theory and perturbative IIB results. We explicitly show that world-volume fluxes can lift certain fermionic zero-modes, whose presence would prevent the generation of non-perturbative superpotential terms, and we discuss in detail the geometric interpretation of the zero-mode lifting mechanism. We provide a IIB derivation of the index for generation …
An important issue in supersymmetry phenomenology is the suppression of squarks contributions to Flavor Changing Neutral Currents (FCNC). Recently it was noted that in some free fermionic three generation models the anomalous U(1) is family universal. It was further shown that if the $D$--term of the $U(1)_A$ is the dominant source of supersymmetry breaking, the squark masses are indeed approximately degenerate. In this paper I discuss the properties of the superstring models that give rise to the flavor universal anomalous U(1). The root cause for the universal $U(1)_A$ is the cyclic permutat…
We study the modular symmetry of zero-modes on $T_1^2 \times T_2^2$ and orbifold compactifications with magnetic fluxes, $M_1,M_2$, where modulus parameters are identified. This identification breaks the modular symmetry of $T^2_1 \times T^2_2$, $SL(2,\mathbb{Z})_1 \times SL(2,\mathbb{Z})_2$ to $SL(2,\mathbb{Z})\equivΓ$. Each of the wavefunctions on $T^2_1 \times T^2_2$ and orbifolds behaves as the modular forms of weight 1 for the principal congruence subgroup $Γ$($N$), $N$ being 2 times the least common multiple of $M_1$ and $M_2$. Then, zero-modes transform each other under the modular symm…
We study the mediation of supersymmetry breaking in string compactifications whose moduli are stabilized by nonperturbative effects. We begin with a critical review of arguments for sequestering in supergravity and in string theory. We then show that geometric isolation, even in a highly warped space, is insufficient to achieve sequestering: in type IIB compactifications, nonperturbative superpotentials involving the Kahler moduli introduce cross-couplings between well-separated visible and hidden sectors. The scale of the resulting soft terms depends on the moduli stabilization scenario. In t…
We study the CP-violating phase of the quark sector in the $U(8)$ flavor model on $T^2/Z_N \, (N=2,3,4,6)$ with non-vanishing magnetic fluxes, where properties of possible origins of the CP violation are investigated minutely. In this system, a non-vanishing value is mandatory in the real part of the complex modulus parameter $τ$ of the two-dimensional torus. On $T^2$ without orbifolding, underlying discrete flavor symmetries severely restrict the form of Yukawa couplings and it is very difficult to reproduce the observed pattern in the quark sector including the CP-violating phase $δ_{\rm CP}…
A top-down approach to the flavor puzzle leads to eclectic flavor groups which include modular and traditional flavor symmetries. Based on examples of semirealistic T2/Z3 orbifold compactifications of heterotic string theory, we discuss the breakdown patterns of the eclectic flavor group via the interplay of vacuum expectation values (vevs) of moduli and flavon fields. This leads to an attractive flavor scheme with various possibilities to obtain "flavor hierarchies" through the alignment of these vevs. Despite the fact that the top-down approach gives strong restrictions for bottom-up flavor …
We study Kähler moduli stabilizations in semi-realistic magnetized D-brane models based on $ Z_2\times Z_2'$ toroidal orbifolds. In type IIB compactifications, 3-form fluxes can stabilize the dilaton and complex structure moduli fields, but there remain some massless closed string moduli fields, Kähler moduli. The magnetic fluxes generate Fayet-Iliopoulos terms, which can fix ratios of Kähler moduli. On top of that, we consider D-brane instanton effects to stabilize them in concrete D-brane models and investigate the brane configurations to confirm that the moduli fields can be stabilized succ…
We study the realization of non-Abelian discrete gauge symmetries in 4d field theory and string theory compactifications. The underlying structure generalizes the Abelian case, and follows from the interplay between gaugings of non-Abelian isometries of the scalar manifold and field identifications making axion-like fields periodic. We present several classes of string constructions realizing non-Abelian discrete gauge symmetries. In particular, compactifications with torsion homology classes, where non-Abelianity arises microscopically from the Hanany-Witten effect, or compactifications with …
We suggest a common origin for dark matter, neutrino mass and family symmetry within the orbifold theory proposed in [arXiv:1910.05605v2,arXiv:2004.06735v2]. Flavor physics is described by an $A_4$ family symmetry that results naturally from compactification. WIMP Dark matter emerges from the first Kaluza-Klein excitation of the same scalar that drives family symmetry breaking and neutrino masses through the inverse seesaw mechanism. In addition to the "golden" quark-lepton mass relation and neutrino predictions for $0νββ$ decay, the model provides a good global description of all flavor obser…
Emergent gauge flux in mixed QED$_3$ with flavor chemical potential: application to magnetized U(1) Dirac spin liquids
The University of Hong Kong（香港大学）；HK Institute of Quantum Science & Technology（香港量子科学院）；State Key Laboratory of Optical Quantum Materials（光学量子材料国家重点实验室）；University of Cologne（科隆大学）；University of Utah（犹他大学）；Kavli Institute for Theoretical Physics（卡弗里理论物理研究所）；University of California, Santa Barbara（加州大学圣塔芭芭拉分校）；French American Center for Theoretical Science（法美理论科学中心）；CNRS（法国国家科学研究中心）；KITP（卡弗里理论物理研究所）；Canadian Institute for Advanced Research（加拿大高级研究院）
We propose a graph-based approach to 5d superconformal field theories (SCFTs) based on their realization as M-theory compactifications on singular elliptic Calabi--Yau threefolds. Field-theoretically, these 5d SCFTs descend from 6d $\mathcal{N}=(1,0)$ SCFTs by circle compactification and mass deformations. We derive a description of these theories in terms of graphs, so-called Combined Fiber Diagrams, which encode salient features of the partially resolved Calabi--Yau geometry, and provides a combinatorial way of characterizing all 5d SCFTs that descend from a given 6d theory. Remarkably, thes…
We propose a flavour theory in which the family symmetry results naturally from a six-dimensional orbifold compactification. "Diracness" of neutrinos is a consequence of the spacetime dimensionality, and the fact that right-handed neutrinos live in the bulk. Dark matter is incorporated in a scotogenic way, as a result of an auxiliary $\mathbb{Z}_3$ symmetry, and its stability is associated to the conservation of a "dark parity" symmetry. The model leads naturally to a "golden" quark-lepton mass relation.
We derive the potential modular symmetries of heterotic string theory. For a toroidal compactification with Wilson line modulus, we obtain the Siegel modular group $\mathrm{Sp}(4,\mathbb{Z})$ that includes the modular symmetries $\mathrm{SL}(2,\mathbb{Z})_T$ and $\mathrm{SL}(2,\mathbb{Z})_U$ (of the "geometric" moduli $T$ and $U$) as well as mirror symmetry. In addition, string theory provides a candidate for a CP-like symmetry that enhances the Siegel modular group to $\mathrm{GSp}(4,\mathbb{Z})$.
In a consistent top-down approach based on orbifold compactifications, modular and traditional flavor symmetries combine nontrivially to the so-called eclectic flavor symmetry. We extend this scheme from two extra dimensions, discussed previously, to the six extra dimensions of string theory. By doing so, new insights on the nature of CP and its spontaneous breaking emerge. Moreover, we identify a new interpretation of R-symmetries as unbroken remnants from modular symmetries that are associated with geometrically stabilized complex structure moduli. Hence, all symmetries (i.e. modular, tradit…
String theory leads to a flavor scheme where modular symmetries play a crucial role. Together with the traditional flavor symmetries they combine to an eclectic flavor group, which we determine via outer automorphisms of the Narain space group. Unbroken flavor symmetries are subgroups of this eclectic group and their size depends on the location in moduli space. This mechanism of local flavor unification allows a different flavor structure for different sectors of the theory (such as quarks and leptons) and also explains the spontaneous breakdown of flavor- and CP-symmetries (via a motion in m…
We have performed the leading order perturbative calculation to obtain the equation of state (EoS) of the strange quark matter (SQM) at zero temperature under the magnetic field B = 10^18 G. The SQM comprises two massless quark flavors (up and down) and one massive quark flavor(strange). Consequently, we have used the obtained EoS to calculate the maximum gravitational mass and the corresponding radius of the magnetized strange quark star (SQS). We have employed two approaches, including the regular perturbation theory (RPT) and the background perturbation theory (BPT). In RPT the infrared (IR…
We study the presence of discrete flavor symmetries in D-brane models of particle physics. By analyzing the compact extra dimensions of these models one can determine when such symmetries exist both in the context of intersecting and magnetized D-brane constructions. Our approach allows to distinguish between approximate and exact discrete symmetries, and it can be applied to compactification manifolds with continuous isometries or to manifolds that only contain discrete isometries, like Calabi-Yau three-folds. We analyze in detail the class of rigid D-branes models based on a Z_2 x Z'_2 toroi…
Instituto de Física, Universidad Nacional Autónoma de México（墨西哥国立自治大学物理研究所）；Facultad de Ciencias Físico-Matemáticas, Universidad Autónoma de Coahuila（科阿韦拉自治大学理学院）
