Only four   <math> <msup> <mi>T</mi> <mn>2</mn> </msup> <mo>/</mo> <msub> <mi>ℤ</mi> <mi>K</mi> </msub> </math>  $$ {\mathbbm{T}}^2/{\mathbb{Z}}_K $$  orbifold building blocks are admissible in heterotic string compactifications. We investigate the flavor properties of all of these building blocks. In each case, we identify the traditional and modular flavor symmetries, and determine the corresponding representations and (fractional) modular weights of the available massless matter states. The resulting finite flavor symmetries include Abelian and non-Abelian traditional symmetries, discrete R symmetries, as well as the double-covered finite modular groups (S 3 × S 3) ⋊ ℤ 4, T ′, 2D 3 and S 3 × T ′. Our findings provide restrictions for bottom-up model building with consistent ultraviolet embeddings.

Germany

Mexico

Alexander Baur

Hans Nilles

Saúl Ramos–Sánchez

Andreas Trautner

Patrick Vaudrevange

The eclectic flavor symmetries of   <math> <msup> <mi>T</mi> <mn>2</mn> </msup> <mo>/</mo> <msub> <mi>ℤ</mi> <mi>K</mi> </msub> </math>  $$ {\mathbbm{T}}^2/{\mathbb{Z}}_K $$  orbifolds

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/ℤ3 × ℤ3 orbifold compactification of the heterotic string that gives rise to an Ω(2) eclectic flavor symmetry. We find that the interplay of flavon alignment and the localization of the modulus in the vicinity of a symmetry-enhanced point leads to naturally protected fermion mass hierarchies, favoring normal-ordered neutrino masses arising from a see-saw mechanism. We show that our model can reproduce all observables in the lepton sector with a small number of parameters and deliver predictions for so far undetermined neutrino observables. Furthermore, we extend the fit to quarks and find that Kähler corrections are instrumental in obtaining a successful simultaneous fit to the quark and lepton sectors.

Saúl Ramos-Sánchez

The first string-derived eclectic flavor model with realistic phenomenology

We present a detailed analysis of the eclectic flavor structure of the two-dimensional ℤ2 orbifold with its two unconstrained moduli T and U as well as SL(2, ℤ) T  × SL(2, ℤ) U  modular symmetry. This provides a thorough understanding of mirror symmetry as well as the R-symmetries that appear as a consequence of the automorphy factors of modular transformations. It leads to a complete picture of local flavor unification in the (T, U) modulus landscape. In view of applications towards the flavor structure of particle physics models, we are led to top-down constructions with high predictive power. The first reason is the very limited availability of flavor representations of twisted matter fields as well as their (fixed) modular weights. This is followed by severe restrictions from traditional and (finite) modular flavor symmetries, mirror symmetry,   <math> <mi>CP</mi> </math>  $$ \mathcal{CP} $$  and R-symmetries on the superpotential and Kähler potential of the theory.

Moritz Kade

Completing the eclectic flavor scheme of the ℤ2 orbifold

Modular symmetries naturally combine with traditional flavor symmetries and   <math> <mi>CP</mi> </math>  $$ \mathcal{CP} $$ , giving rise to the so-called eclectic flavor symmetry. We apply this scheme to the two-dimensional ℤ2 orbifold, which is equipped with two modular symmetries SL(2, ℤ) T  and SL(2, ℤ) U  associated with two moduli: the Kähler modulus T and the complex structure modulus U. The resulting finite modular group is ((S 3 × S 3) ⋊ ℤ4) × ℤ2 including mirror symmetry (that exchanges T and U) and a generalized   <math> <mi>CP</mi> </math>  $$ \mathcal{CP} $$ -transformation. Together with the traditional flavor symmetry (D 8 × D 8)/ℤ2, this leads to a huge eclectic flavor group with 4608 elements. At specific regions in moduli space we observe enhanced unified flavor symmetries with as many as 1152 elements for the tetrahedral shaped orbifold and   <math> <mfenced> <mi>T</mi> </mfenced> <mo>=</mo> <mfenced> <mi>U</mi> </mfenced> <mo>=</mo> <mo>exp</mo> <mfenced> <mfrac> <mrow> <mi>π</mi> <mi>i</mi> </mrow> <mn>3</mn> </mfrac> </mfenced> </math>  $$ \left\langle T\right\rangle =\left\langle U\right\rangle =\exp \left(\frac{\pi \mathrm{i}}{3}\right) $$ . This rich eclectic structure implies interesting (modular) flavor groups for particle physics models derived form string theory.

The eclectic flavor symmetry of the ℤ2 orbifold

We show that in a wide class of string derived models of particle physics, heavy string modes with masses around the GUT scale can serve as a viable dark matter candidate. These heavy string modes wind around specific cycles in the extra-dimensional space, closely related to the fundamental group . As a consequence of a non-trivial , there is an exact discrete symmetry that stabilizes such winding strings. The dark matter candidate couples to the Standard Model via gravity and via the exchange of heavy string states. We find that, for reasonable values of the string coupling, our dark matter candidate can be produced in sizable amounts via freeze-in. Our scheme applies to many string constructions, including Calabi-Yau compactifications, and can be tested against constraints from the CMB.

Andreas Mütter

String scale interacting dark matter from

The simultaneous study of top-down and bottom-up approaches to modular flavor symmetry leads necessarily to the concept of eclectic flavor groups. These are non-trivial products of modular and traditional flavor symmetries that exhibit the phenomenon of local flavor enhancement in moduli space. We develop methods to determine the eclectic flavor groups that can be consistently associated with a given traditional flavor symmetry. Applying these methods to a large family of prominent traditional flavor symmetries, we try to identify potential candidates for realistic eclectic flavor groups and show that they are relatively rare. Model building with finite modular flavor symmetries thus appears to be much more restrictive than previously thought.

Eclectic flavor groups

We systematically develop the formalism necessary for ensuring that boundary conditions of flavon fields in extra dimensions are consistent with heterotic string theory. Having developed a set of consistency conditions on the boundary conditions, we explore a series of examples of orbifolds in various dimensions to see which ones can satisfy them. In addition we impose the further phenomenological requirements of having non-trivial flavon vacuum alignments and also of having quarks and leptons located appropriately in extra dimensions. The minimal successful case seems to be a 10d theory with a SU(3)fl gauged flavour symmetry, where the six-dimensional torus is compactified on a 𝕋6 /∆(54) orbifold. We construct a realistic SU(5) grand unified theory along these lines, leading to tribimaximal-reactor lepton mixing, which we show to be consistent with current neutrino data.

Francisco Anda

United Kingdom

Stephen King

Elena Perdomo

Flavon alignments from orbifolding: SU(5) × SU(3) model with 𝕋6 /∆(54)

It is well-known that the space group selection rule constrains the interactions of closed strings on orbifolds. For some examples, this rule has been described by an effective Abelian symmetry that combines with a permutation symmetry to a non-Abelian flavor symmetry like D 4 or Δ(54). However, the general case of the effective Abelian symmetries was not yet fully understood. In this work, we formalize the computation of the Abelian symmetry that results from the space group selection rule by imposing two conditions only: (i) well-defined discrete charges and (ii) their conservation. The resulting symmetry, which we call the space group flavor symmetry D  S , is uniquely specified by the Abelianization of the space group. For all Abelian orbifolds with   <math> <mi>N</mi> <mo>=</mo> <mn>1</mn> </math>  $$ \mathcal{N}=1 $$  supersymmetry we compute D  S  and identify new cases, for example, where D  S  contains a ℤ 2 dark matter-parity with charges 0 and 1 for massless and massive strings, respectively.

Note on the space group selection rule for closed strings on orbifolds

To obtain a unified framework for symmetric and asymmetric heterotic orbifold constructions we provide a systematic study of Narain compactifications orbifolded by finite order T -duality subgroups. We review the generalized vielbein that parametrizes the Narain moduli space (i.e. the metric, the B -field and the Wilson lines) and introduce a convenient basis of generators of the heterotic T -duality group. Using this we generalize the space group description of orbifolds to Narain orbifolds. This yields a unified, crystallographic description of the orbifold twists, shifts as well as Narain moduli. In particular, we derive a character formula that counts the number of unfixed Narain moduli after orbifolding. More-over, we develop new machinery that may ultimately open up the possibility for a full classification of Narain orbifolds. This is done by generalizing the geometrical concepts of and affine classes from the theory of crystallography to the Narain case. Finally, we give a variety of examples illustrating various aspects of Narain orbifolds, including novel T -folds.