M-theory compactified on a $G_2$ manifold with resolved $E_8$ singularities realizes 4d $\mathcal{N} = 1$ supersymmetric gauge theories coupled to gravity with three families of Standard Model fermions. Beginning with one $E_8$ singularity, three fermion families emerge when $E_8$ is broken by geometric engineering deformations to a smaller subgroup with equal rank. In this paper, we use the local geometry of the theory to explain the origin of the three families and their mass hierarchy. We linearize the blowing-up of 2-cycles associated with resolving $E_8$ singularities. After imposing explicit constraints on the effectively stabilized moduli, we arrive at Yukawa couplings for the quarks and leptons. We fit the high scale Yukawa couplings approximately which results in the quark masses agreeing reasonably well with the observations, implying that the experimental hierarchy of the masses is achievable within this framework. The hierarchy separation of the top quark from the charm and up is a stringy effect, while the spitting of the charm and up also depends on the Higgs sector. The Higgs sector cannot be reduced to having a single vev; all three vevs must be non-zero.Three extra $U(1)$'s survive to the low scale but are not massless, so Z' states are motivated to occur in the spectrum, but may be massive.
We systematically study the possibility for realizing realistic values of lepton mass ratios and mixing angles by using only renormalizable Yukawa couplings derived from heterotic $Z_6$-I orbifold. We assume one pair of up and down sector Higgs fields. We consider both the Dirac neutrino mass scenario and the seesaw scenario with degenerate right-handed majorana neutrino masses. It is found that realistic values of the charged lepton mass ratios, $m_e/m_τ$ and $m_μ/m_τ$, the neutrino mass squared difference ratio, $Δm^2_{31}/Δm^2_{21}$, and the lepton mixing angles can be obtained in certain c…
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…
We study the Dirac equation of chiral fermions on a regularized version of the two-dimensional T^2/Z_2 orbifold, where the conical singularities are replaced by suitable spherical caps with constant curvature. This study shows how localized and bulk fermions arise in the orbifold as the resolved space approaches the orbifold limit. Our analysis also shows that not all possible fermion configurations on T^2/Z_2 admit such a simple resolution. We focus our study to a fermion coupled to a U(1) gauge field. It is explicitly shown how a resolution of the orbifold puts severe constraints on the allo…
In this article, written primarily for physicists and geometers, we survey several manifestations of a general localization principle for orbifold theories such as $K$-theory, index theory, motivic integration and elliptic genera.
Flavor symmetry has been widely studied for figuring out the masses and mixing angles of standard-model fermions. In this paper we present a framework for handling flavor symmetry breaking where the symmetry breaking is triggered by boundary conditions of scalar fields in extra-dimensional space. The alignment of scalar expectation values is achieved without referring to any details of scalar potential and its minimization procedure. As applications to non-abelian discrete flavor symmetries, illustrative lepton mass models are constructed where the S3 and A4 flavor symmetries are broken down t…
We study magnetized orbifold models. We assume the localized Fayet-Iliopoulos terms and the corresponding gauge background. Such terms lead to strong localization of zero-mode wavefunc- tions. In this setup, we compute quark mass matrices.
The large mixing in the lepton sector observed in the recent neutrino-oscillation experiments strongly suggest that nature of left-handed lepton doublets is very different from that of left-handed quark doublets. This means that there is a big disparity between the matter multiplets 5^*'s and 10's in the SU(5) unified theory. We show that this big difference can be explained in a six-dimensional spacetime compactified on the T^2/Z_3 orbifold. That is, we propose to put three families of 5^*'s on three equivalent fixed points of the orbifold and the three 10's in the two-dimensional bulk. We co…
We study the structure of Yukawa matrices derived from the supersymmetric Yang-Mills theory on magnetized orbifolds, which can realize the observed quark and charged lepton mass ratios as well as the CKM mixing angles, even with a small number of tunable parameters and without any critical fine-tuning. As a reason behind this, we find that the obtained Yukawa matrices possess a Froggatt-Nielsen like structure with Gaussian hierarchies, which provides a suitable texture for them favored by the experimental data.
We classify the combinations of parameters which lead three generations of quarks and leptons in the framework of magnetized twisted orbifolds on $T^2/Z_2$, $T^2/Z_3$, $T^2/Z_4$ and $T^2/Z_6$ with allowing nonzero discretized Wilson line phases and Scherk-Schwarz phases. We also analyze two actual examples with nonzero phases leading to one-pair Higgs and five-pair Higgses and discuss the difference from the results without nonzero phases studied previously.
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…
We show that the observed structure of quark and lepton masses and mixing angles can arise entirely geometrically from superstring constructions, at the renormalizable level. The model we consider is a $Z_3$ orbifold compactification of heterotic string with two Wilson lines, where three families of particles of $SU(3)_c\times SU(2)_L\times U(1)_Y$, including Higgses, are automatically present. In orbifold models, Yukawa couplings can be calculated explicitly, and it is known that they get exponential supression factors depending on the distance between the fixed points to which the fields are…
M-theory compactified on a $G_2$ manifold with resolved $E_8$ singularity is a promising candidate for a unified theory. The experimentally observed masses of quarks and charged leptons put a restriction on the moduli of the $G_2$ manifold. These moduli in turn uniquely determine the Dirac interactions of the neutrinos. In the paper, we explicitly compute the Dirac terms for neutrino mass matrix using the moduli from a localized model with resolved $E_8$ singularities on a $G_2$ manifold. This is a novel approach as the Dirac terms are not assumed but derived from the structure of quarks' and …
We develop a mathematical framework for the computation of open orbifold Gromov-Witten invariants of [C^3/Z_n], and provide extensive checks with predictions from open string mirror symmetry. To this aim we set up a computation of open string invariants in the spirit of Katz-Liu, defining them by localization. The orbifold is viewed as an open chart of a global quotient of the resolved conifold, and the Lagrangian as the fixed locus of an appropriate anti-holomorphic involution. We consider two main applications of the formalism. After warming up with the simpler example of [C^3/Z_3], where we…
We note that orbifold boundary conditions that produce chiral fermion zero modes in compactified higher dimensional theories may distort scalar field vacuum expectation values, giving rise to nontrivial dependence on the extra dimensions. We illustrate this in a simple five dimensional model which has chiral fermion zero-modes stuck to fat branes. The model could provide a simple and explicit realization of the separation of quarks and leptons in the fifth dimension. We discuss the KK expansion in some detail. We find that there are in general non-zero-mode states stuck to the brane, like the …
Discrete symmetries are studied in warped space-times with one extra dimension. In particular, we analyze the compatibility of five- and four-dimensional charge conjugation, parity, time reversal and the orbifold symmetry Z_2 with localization of fermions on the four-dimensional brane-world and Lorentz invariance. We then show that, when a suitable topological scalar field (the ``kink'') is included, fermion localization is a consequence of (five-dimensional) CPT invariance.
The capability of string theories to reproduce at low energy the observed pattern of quark and lepton masses and mixing angles is examined, focusing the attention on orbifold constructions, where the magnitude of Yukawa couplings depends on the values of the deformation parameters which describe the size and shape of the compactified space. A systematic exploration shows that for $Z_3$, $Z_4$, $Z_6$--I and possibly $Z_7$ orbifolds a correct fit of the physical fermion masses is feasible. In this way the experimental masses, which are low--energy quantities, select a particular size and shape o…
We consider six-dimensional supergravity with gauge group $SO(10)\times U(1)_A$, compactified on the orbifold $T^2 / \mathbb{Z}_2$. Three quark-lepton generations arise as zero modes of a bulk $\bf16$-plet due to magnetic flux of the anomalous $U(1)_A$. Boundary conditions at the four fixed points break $SO(10)$ to subgroups whose intersection is the Standard Model gauge group. The gauge and Higgs sector consist of "split" $SO(10)$ multiplets. As consequence of the $U(1)_A$ flux, squarks and sleptons are much heavier than gauge bosons, Higgs bosons, gauginos and higgsinos. We thus obtain a pic…
Motivated by orbifold grand unified theories, we construct a class of three-family Pati-Salam models in a Z6 abelian symmetric orbifold with two discrete Wilson lines. These models have marked differences from previously-constructed three-family models in prime-order orbifolds. In the limit where one of the six compactified dimensions (which lies in a Z2 sub-orbifold) is large compared to the string length scale, our models reproduce the supersymmetry and gauge symmetry breaking pattern of 5d orbifold grand unified theories on an S1/Z2 orbicircle. We find a horizontal 2+1 splitting in the chir…
We quantitatively analyze a quark-lepton flavour model derived from a six-dimensional supersymmetric theory with $SO(10)\times U(1)$ gauge symmetry, compactified on an orbifold with magnetic flux. Two bulk $\mathbf{16}$-plets charged under the $U(1)$ provide the three quark-lepton generations whereas two uncharged $\mathbf{10}$-plets yield two Higgs doublets. At the orbifold fixed points mass matrices are generated with rank one or two. Moreover, the zero modes mix with heavy vectorlike split multiplets. The model possesses no flavour symmetries. Nevertheless, there exist a number of relations…
In a (0,1) supersymmetric (SUSY) six-dimensional gauge theory, a gauge fermion gives rise to box anomalies. These anomalies are completely canceled by assuming a vector multiplet of (1,1) SUSY. With a T^2/Z_3 orbifold compactification of the extra two dimensional space, the theory provides three chiral multiplets and three equivalent fixed points. We regard them as the origin of the three families of quarks and leptons. Quasi anarchy structure in the SU(5)-5^* representation sector and hence the bimaximal mixing in the neutrino oscillation are explained quite naturally in this framework. We al…
The location of matter fields and the pattern of gauge symmetry in extra dimensions are crucial ingredients for string model building. We analyze realistic MSSM models from the heterotic Z6 Mini-Landscape and extract those properties that are vital for their success. We find that Higgs bosons and the top quark are not localized in extra dimensions and live in the full D=10 dimensional space-time. The first two families of quarks and leptons, however, live at specific fixed points in extra dimensional space and exhibit a (discrete) family symmetry. Within a newly constructed Z2XZ4 orbifold fram…
We construct a three family flipped SU(5) model from the heterotic string theory compactified on the $\Z_{12-I}$ orbifold with one Wilson line. The gauge group is $\rm SU(5)\times U(1)_X\times U(1)^2\times[SU(2)\times SO(10)\times U(1)^2]^\prime$. This model does not derive any nonabelian group except SU(5) from $E_8$, which is possible only for two cases in case of one shift $V$, one in ${\bf Z}_{12-I}$ and the other in ${\bf Z}_{12-II}$. We present all possible Yukawa couplings. We place the third quark family in the twisted sectors and two light quark families in the untwisted sector. From …
We apply a recently proposed mechanism for predicting the weak mixing angle to theories with TeV-size dimensions. "Reconstruction" of the associated moose (or quiver) leads to theories which unify the electroweak forces into a five dimensional SU(3) symmetry. Quarks live at an orbifold fixed point where SU(3) breaks to the electroweak group. A variety of theories -- all sharing the same successful prediction of the weak mixing angle -- emerges; they differ primarily by the spatial location of the leptons and the absence or presence of supersymmetry. A particularly interesting theory puts lepto…
Localization properties of fields in compact extra dimensions are crucial ingredients for string model building, particularly in the framework of orbifold compactifications. Realistic models often require a slight deviation from the orbifold point, that can be analyzed using field theoretic methods considering (singlet) fields with nontrivial vacuum expectation values. Some of these fields correspond to blow-up modes that represent the resolution of orbifold singularities. Improving on previous analyses we give here an explicit example of the blow-up of a model from the heterotic Mini-landscap…
A multiplicity of quark-lepton families can naturally arise as zero-modes in flux compactifications. The flavour structure of quark and lepton mass matrices is then determined by the wave function profiles of the zero-modes. We consider a supersymmetric $SO(10)\times U(1)$ model in six dimensions compactified on the orbifold $T^2/{\mathbb Z_2}$ with Abelian magnetic flux. A bulk $\mathbf{16}$-plet charged under the $U(1)$ provides the quark-lepton generations whereas two uncharged $\mathbf{10}$-plets yield two Higgs doublets. Bulk anomaly cancellation requires the presence of additional $\math…
We propose a unification scenario for supersymmetric intersecting brane models. The quarks and leptons are embedded into adjoint representations of SO(32), which are obtained and break by type I string compactified on orbifolds. Its single unified gauge coupling can give rise to different gauge couplings below the unification scale, due to effects of magnetic fluxes. The crucial mechanism is brane recombination preserving supersymmetry.
We study proton decay in a six-dimensional orbifold GUT model with gauge group $SO(10)\times U(1)_A$. Magnetic $U(1)_A$ flux in the compact dimensions determines the multiplicity of quark-lepton generations, and it also breaks supersymmetry by giving universal GUT scale masses to scalar quarks and leptons. The model can successfully account for quark and lepton masses and mixings. Our analysis of proton decay leads to the conclusion that the proton lifetime must be close to the current experimental lower bound. Moreover, we find that the branching ratios for the decay channels $p \rightarrow e…
Localization approach to $\mathcal N=2$ superconformal $SU(N) \times SU(N)$ quiver theory leads to a non-Gaussian two-matrix model representation for the expectation value of BPS circular $SU(N)$ Wilson loop $\langle\mathcal W\rangle$. We study the subleading $1/N^2$ term in the large $N$ expansion of $\langle\mathcal W\rangle$ at weak and strong coupling. We concentrate on the case of the symmetric quiver with equal gauge couplings which is equivalent to the $\mathbb Z_{2}$ orbifold of the $SU(2N)$ $\mathcal N=4$ SYM theory. This orbifold gauge theory should be dual to type IIB superstring in…
We construct eight-stack intersecting D5-brane models, with an orbifold transverse space, that yield the (non-supersymmetric) standard model up to vector-like leptons. The matter includes right-chiral neutrinos and the models have the renormalisable Yukawa couplings to tachyonic Higgs doublets needed to generate mass terms for {\it all} matter, including the vector-like leptons. The models are constrained by the requirement that twisted tadpoles cancel, that the gauge boson coupled to the weak hypercharge $U(1)_Y$ does not get a string-scale mass via a generalised Green-Schwarz mechanism, and …
We study 4-dimensional $\mathcal{N}=2$ superconformal quiver gauge theories obtained with an orbifold projection from $\mathcal{N}=4$ SYM, and compute the 2- and 3-point correlation functions among chiral/anti-chiral single-trace scalar operators and the corresponding structure constants. Exploiting localization, we map the computation to an interacting matrix model and obtain expressions for the correlators and the structure constants that are valid for any value of the 't Hooft coupling in the planar limit of the theory. At strong coupling, these expressions simplify and allow us to extract …
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…
In this paper, we study the implementation of brane worlds in type II string theory. Starting with the NS/NS sector of type II string, we first compactify the $(D+d_{+} + d_{-})$-dimensional spacetime, and reduce the corresponding action to a D-dimensional effective action, where the topologies of $M_{d_{+}}$ and $M_{d_{-}}$ are arbitrary. We further compactify one of the $(D-1)$ spatial dimensions on an $S^{1}/Z_{2}$ orbifold, and derive the gravitational and matter field equations both in the bulk and on the branes. Then, we investigate two key issues in such a setup: (i) the radion stabilit…
We propose a bottom-up approach to the building of particle physics models from string theory. Our building blocks are Type II D-branes which we combine appropriately to reproduce desirable features of a particle theory model: 1) Chirality ; 2) Standard Model group ; 3) N=1 or N=0 supersymmetry ; 4) Three quark-lepton generations. We start such a program by studying configurations of D=10, Type IIB D3-branes located at singularities. We study in detail the case of Z_N, N=1,0 orbifold singularities leading to the SM group or some left-right symmetricextension. In general, tadpole cancellation c…
We present a Z_6 orbifold compactification of the E_8xE_8 heterotic string which leads to the (supersymmetric) Standard Model gauge group and matter content. The quarks and leptons appear as three 16-plets of SO(10), whereas the Higgs fields do not form complete SO(10) multiplets. The model has no exotic states at low energies and is consistent with gauge coupling unification. The top quark Yukawa coupling arises from gauge interactions and is of the order of the gauge couplings, whereas the other Yukawa couplings are suppressed.
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…
We construct brane solutions in 6 dimensional Einstein-Skyrme systems. A class of baby skyrmion solutions realizes warped compactification of the extra dimensions and gravity localization on the brane for negative bulk cosmological constant. Coupling of the fermions with the brane skyrmions lead to the brane localized fermions. In terms of the level crossing picture, emergence of the massive localized modes as well as the zero mode are observed. Nonlinear nature of the skyrmions brings richer information for the fermions level structure. The level comprises doubly degenerate lowest plus single…
Using supersymmetric localization, we consider four-dimensional $\mathcal{N}=2$ superconformal quiver gauge theories obtained from $\mathbb{Z}_n$ orbifolds of $\mathcal{N}=4$ Super Yang-Mills theory in the large $N$ limit at weak coupling. In particular, we show that: 1) The partition function for arbitrary couplings can be constructed in terms of universal building blocks. 2) It can be computed in perturbation series, which converges uniformly for $|λ_I|<π^2$, where $λ_I$ are the 't Hooft coupling of the gauge groups. 3) The perturbation series for two-point functions can be explicitly comput…
We present a systematical study of brane worlds in string theory on $S^{1}/Z_{2}$. Starting with the toroidal compactification of the NS/NS sector in (D+d) dimensions, we first obtain an effective $D$-dimensional action, and then compactify one of the $(D-1)$ spatial dimensions by introducing two orbifold branes as its boundaries. By combining the Gauss-Codacci and Lanczos equations, we write down explicitly the general gravitational field equations on each of the two branes, while using distribution theory we express the matter field equations on the branes in terms of the discontinuities of …
