Orbifold Compactifications Research Articles

Light fermion masses in partially deconstructed models

Considering a theory space consisting of a large number of five-dimensional Dirac fermion field theories including background abelian gauge fields, we can construct a theory similar to a continuous six-dimensional theory compactified with two-dimensional manifolds with and without magnetic flux or orbifolds as extra dimensions. This method, called dimensional deconstruction, can be used to construct a model with one-dimensional discrete space, which represents general graph structures. In this paper, we propose the models with two extra dimensions, which resemble two-dimensional tori, cylinders, and rectangular regions, as continuum limits. We also try to build a model that mimics one with the two-dimensional orbifold compactification.

A Note on Modular Invariant Species Scale and Potentials

AbstractThe species scale provides an upper bound for the ultraviolet cutoff of effective theories of gravity coupled to a number of light particle species. Modular invariant (super‐)potentials provide a simple and computable expression of the species scale as a function of the moduli in toroidal orbifold compactifications of type II and heterotic string are pointed out. Due t o modular symmetry, these functions are valid over all moduli space and not only in asymptotic regions. Additive logarithmic corrections to the species scale arise from the requirement that the latter be modular invariant are observed. The moduli‐dependent expression of the species scale in terms of the gravitino mass or the scalar potential of these models are recasted and swampland conjectures such as the anti‐de Sitter distance conjecture and the gravitino conjecture are connected.

Index theorem on magnetized blow-up manifold of T2/ZN

We investigate blow-up manifolds of ${T}^{2}/{\mathbb{Z}}_{N}(N=2,3,4,6)$ orbifolds with magnetic flux. First, we construct the blow-up manifolds and zero-mode wave functions on them more precisely. In particular, through an appropriate singular gauge transformation, winding numbers of wave functions on ${T}^{2}/{\mathbb{Z}}_{N}$ can be replaced with localized curvature and localized flux at orbifold fixed points. In addition, since the blow-up manifolds have no singularities, we apply the Atiyah-Singer index theorem to them; the chiral zero-mode number is given by the total magnetic flux. It can be also applied for ${T}^{2}/{\mathbb{Z}}_{N}$ orbifolds through the blow-up process, and then we find that it is consistent with the zero-mode counting formula in M. Sakamoto et al. [Zero-mode counting formula and zeros in orbifold compactifications, Phys. Rev. D 102, 025008 (2020).]. Furthermore, the Atiyah-Singer index theorem shows that an additional degree of freedom of localized flux gives new chiral zero modes. We study their wave functions and then we find that they correspond to localized modes at the orbifold singular points. We also calculate their Yukawa couplings.

Top-down anatomy of flavor symmetry breakdown

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 model building, it seems to be well suited to provide a realistic flavor pattern for quarks and leptons.

Fractionally charged BPS particles in M-string on 3D orbifolds

In this paper, we study the M-string realization of chiral [Formula: see text]-super-conformal field theory in 6 dimensions and its orbifold compactification down to three-dimensional (3D). We analyze its fractionally charged BPS particle spectrum in connection with effective 3D Chern–Simons gauge theory and the supersymmetric fractional quantum Hall effect in [Formula: see text] dimensions. We construct the set of underlying fractionally charged BPS particles in the ground state of the compactified M string and find that it contains 144 BPS states that are generated by four basic quasi-particles (two bosonic-like and two fermionic like) and their CPT conjugate. Two representations of the gauge bosons and the gauginos as condensates of the basic quasiparticles are found and explicit realizations are also given. Other features concerning generalizations are also discussed.

Toroidal & Orbifold Compactifications at Large D and D‐Duality

AbstractIn this paper I will further investigate the spectrum of quantum gravity or string theories at large number of dimensions. We will see that volumes of certain orbifolds shrink at large D. It follows that the mass spectra of the leading Kaluza‐Klein towers and also of wrapped brane states on these orbifolds possess a non‐trivial dependence on D: KK modes become heavy at large D, whereas wrapped branes become light. This observation can be used either to apply the Large Dimension Conjecture or, as we will do, to investigate the possibility of a D‐duality symmetry, which relates in gravity compactifications of different dimensions. We will set up the general rules for D‐duality in higher dimensional gravity. However due to existence of the critical dimensions, D‐duality is quite restricted in string theory. As simple tests for D‐duality in string theory, we will discuss the duality between M‐theory on a two‐dimensional orbifold, namely the Möbius strip, and an IIB S‐fold on a circle, which corresponds to the heterotic CHL string, as well the duality between a (truncated) 12‐dimensional theory on a three‐dimensional orbifold and another IIB S‐fold compactification. Finally I also comment on the possible existence of exotic theories at large D.

Pair production and gravity as the weakest force

The Weak Gravity Conjecture (WGC) is usually formulated in terms of the stability of extremal black-holes or in terms of long distance Coulomb/Newton potentials. However one can think of other physical processes to compare the relative strength of gravity versus other forces. We argue for an alternative formulation in terms of particle pair production at threshold or, equivalently, pair annihilation at rest. Imposing that the production rate by any force mediator (photon or scalar) of pairs of charged particles be larger or equal to graviton production, we recover known conditions for the U(1) WGC and its extensions. Unlike other formulations though, threshold pair production is sensitive to short range couplings present in scalar interactions and gives rise to a Scalar WGC. Application to moduli scalars gives rise to specific conditions on the trilinear and quartic couplings which involve first and second derivatives of the WGC particle mass with respect to the moduli. Some solutions saturating equations correspond to massive states behaving like BPS, KK and winding states which feature duality invariance and are in agreement with the Swampland distance conjecture. Conditions for N = 2 BPS states saturate our bounds and we discuss specific examples of BPS states which become massless at large Kahler moduli in Type IIA N=2, D=4 CY and orbifold compactifications. We study possible implications for potentials depending on moduli only through WGC massive states. For some simple classes of potentials one recovers constraints somewhat similar but not equivalent to a Swampland dS conjecture.

Revisiting modular symmetry in magnetized torus and orbifold compactifications

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 $\Gamma \equiv SL(2,\mathbb{Z})$, $\widetilde{\Gamma} \equiv \widetilde{SL}(2,\mathbb{Z})$. Each wavefunction on $T^2$ with the magnetic flux $M$ transforms under $\widetilde{\Gamma}(2|M|)$, which is the normal subgroup of $\widetilde{SL}(2,\mathbb{Z})$. Then, $|M|$ zero-modes are representations of the quotient group $\widetilde{\Gamma}'_{2|M|} \equiv \widetilde{\Gamma}/\widetilde{\Gamma}(2|M|)$. We also study the modular symmetry on twisted and shifted orbifolds $T^2/\mathbb{Z}_N$. Wavefunctions are decomposed into smaller representations by eigenvalues of twist and shift. They provide us with reduction of reducible representations on $T^2$.

Modular symmetry by orbifolding magnetized T2\u2009\xd7\u2009T2: realization of double cover of \u0393N

We study the modular symmetry of zero-modes on {T}_1^2times {T}_2^2 and orbifold compactifications with magnetic fluxes, M1, M2, where modulus parameters are identified. This identification breaks the modular symmetry of {T}_1^2times {T}_2^2 , SL(2, ℤ)1× SL(2, ℤ)2 to SL(2, ℤ) ≡ Γ. Each of the wavefunctions on {T}_1^2times {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 M1 and M2. Then, zero-modes transform each other under the modular symmetry as multiplets of double covering groups of ΓN such as the double cover of S4.

Scotogenic dark matter in an orbifold theory of flavor

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 ℤ3 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.

Loop Fayet-Iliopoulos terms in T2/Z2 models: Instability and moduli stabilization

We study Fayet-Iliopoulos (FI) terms of six-dimensional supersymmetric Abelian gauge theory compactified on a $T^2/Z_2$ orbifold. Such orbifold compactifications can lead to localized FI-terms and instability of bulk zero modes. We study 1-loop correction to FI-terms in more general geometry than the previous works. We find induced FI-terms depend on the complex structure of the compact space. We also find the complex structure of the torus can be stabilized at a specific value corresponding to a self-consistent supersymmetric minimum of the potential by such 1-loop corrections, which is applicable to the modulus stabilization.

Eclectic flavor scheme from ten-dimensional string theory – I. Basic results

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, traditional flavor, CP and R) share a common origin in string theory: on a technical level, they are given by outer automorphisms of the Narain space group. The eclectic top-down approach leads to a very restrictive scheme with high predictive power. It remains a challenge to connect this with existing bottom-up constructions of modular flavor symmetry.

Zero-mode counting formula and zeros in orbifold compactifications

We thoroughly analyze the number of independent zero modes and their zero points on the toroidal orbifold ${T}^{2}/{\mathbb{Z}}_{N}(N=2,3,4,6)$ with magnetic flux background, inspired by the Atiyah-Singer index theorem. We first show a complete list for the number ${n}_{\ensuremath{\eta}}$ of orbifold zero modes belonging to ${\mathbb{Z}}_{N}$ eigenvalue $\ensuremath{\eta}$. Since it turns out that ${n}_{\ensuremath{\eta}}$ quite complicatedly depends on the flux quanta $M$, the Scherk-Schwarz twist phase $({\ensuremath{\alpha}}_{1},{\ensuremath{\alpha}}_{2})$, and the ${\mathbb{Z}}_{N}$ eigenvalue $\ensuremath{\eta}$, it seems hard that ${n}_{\ensuremath{\eta}}$ can be universally explained in a simple formula. We, however, succeed in finding a single zero-mode counting formula ${n}_{\ensuremath{\eta}}=(M\ensuremath{-}{V}_{\ensuremath{\eta}})/N+1$, where ${V}_{\ensuremath{\eta}}$ denotes the sum of winding numbers at the fixed points on the orbifold ${T}^{2}/{\mathbb{Z}}_{N}$. The formula is shown to hold for any pattern.

Probing the predictions of an orbifold theory of flavor

We examine the implications of a recently proposed theory of fermion masses and mixings in which an ${A}_{4}$ family symmetry emerges from orbifold compactification. We analyze two variant schemes concerning their predictions for neutrino oscillations, neutrinoless double-beta decay, and the golden quark-lepton unification mass relation. We find that upcoming experiments DUNE as well as LEGEND and nEXO offer good chances of exploring a substantial region of neutrino parameters.

Flavour and CP predictions from orbifold compactification

We propose a theory for fermion masses and mixings in which an A4 family symmetry arises naturally from a six-dimensional spacetime after orbifold compactification. The flavour symmetry leads to the successful “golden” quark-lepton unification formula. The model reproduces oscillation parameters with good precision, giving sharp predictions for the CP violating phases of quarks and leptons, in particular δℓ≃+268∘. The effective neutrinoless double-beta decay mass parameter is also sharply predicted as 〈mββ〉≃2.65meV.

U(1)′ coupling constant at low energies from heterotic orbifolds

Additional Abelian gauge interactions are generic to string compactifications. In heterotic string models, gauge coupling unification of such forces and other gauge interactions is natural due to their common origin. In this letter we study systematically the 1-loop running of the coupling constants in effective vacua emerging from Z8 heterotic orbifold compactifications that provide the matter spectrum of the MSSM plus some vectorlike exotics, restricting to vacua that yield a non-anomalous U(1)′ symmetry, gauge coupling unification and the observed values of known gauge couplings. We determine the low-energy value of the U(1)′ coupling constant for different scales of supersymmetry breakdown. We find that the U(1)′ coupling constant is quite restricted in string models to lie in the range 0.46–0.7 for low-scale supersymmetry or 0.44–0.6 in other cases. We argue that the phenomenology of these string vacua should be further explored to solve some extant issues, such as the stability of the Higgs vacuum.

Modular symmetries and the swampland conjectures

Recent string theory tests of swampland ideas like the distance or the dS conjectures have been performed at weak coupling. Testing these ideas beyond the weak coupling regime remains challenging. We propose to exploit the modular symmetries of the moduli effective action to check swampland constraints beyond perturbation theory. As an example we study the case of heterotic 4d mathcal{N}=1 compactifications, whose non-perturbative effective action is known to be invariant under modular symmetries acting on the Kähler and complex structure moduli, in particular SL(2, Z) T-dualities (or subgroups thereof) for 4d heterotic or orbifold compactifications. Remarkably, in models with non-perturbative superpotentials, the corresponding duality invariant potentials diverge at points at infinite distance in moduli space. The divergence relates to towers of states becoming light, in agreement with the distance conjecture. We discuss specific examples of this behavior based on gaugino condensation in heterotic orbifolds. We show that these examples are dual to compactifications of type I’ or Horava-Witten theory, in which the SL(2, Z) acts on the complex structure of an underlying 2-torus, and the tower of light states correspond to D0-branes or M-theory KK modes. The non-perturbative examples explored point to potentials not leading to weak coupling at infinite distance, but rather diverging in the asymptotic corners of moduli space, dynamically forbidding the access to points with global symmetries. We perform a study of general modular invariant potentials and find that there are dS maxima and saddle points but no dS minima, and that all examples explored obey the refined dS conjecture.

Leptonic CP violation in flipped SU(5) GUT from Z12−I orbifold compactification

We obtain a phenomenologically acceptable PMNS matrix in a flipped SU(5) model, possessing the Z4R discrete symmetry, from the compactification of heterotic string E8×E8′. To analyze the Jarlskog determinant efficiently, we include the simple Kim-Seo form for the Pontecorbo-Maki-Nakagawa-Sakata matrix. We also noted that |δPMNS|≲63o for the normal hierarchy of neutrino masses with the PDG book parametrization.

Runaway quintessence, out of the swampland

We propose a simple, well-motivated and robust alternative to a metastable de Sitter vacuum in string theory, consistent with current observations of dark energy and naturally satisfying conjectured swampland constraints. Inflation ends in a supersymmetric Minkowski minimum, with a flat direction that is protected by non-renormalisation theorems. At some scale non-perturbative effects kick in, inducing a runaway scalar potential. The tail of this runaway potential cannot sustain a late-time dominating, slow-roll quintessence. However, the potential always contains a dS maximum. If the modulus starts close to the hilltop, it remains frozen there by Hubble friction for much of the cosmological history, at first sourcing a tiny classical vacuum energy and then constituting a rolling quintessence, with observable consequences. So long as the modulus is localised away from the Standard Model in the extra dimensions, there are no observable fifth forces nor is there time-variation of fundamental constants, and the modulus mass is protected from radiative corrections. We revisit concrete string models based on heterotic orbifold compactifications, and show that their de Sitter extrema satisfy the recently refined dS Swampland Conjecture.

Charting the flavor landscape of MSSM-like Abelian heterotic orbifolds

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 121,000 models with MSSM-like features. About 75.4% of these models exhibit flavor symmetries containing D4 factors and only nearly 1.2% have Delta(54) factors. The remaining models are furnished with purely Abelian flavor symmetries. Our findings suggest that, should particle phenomenology arise from a heterotic orbifold, it could accommodate only one of these flavor symmetries.