We review pedagogically non-Abelian discrete groups, which play an important role in the particle physics. We show group-theoretical aspects for many concrete groups, such as representations, their tensor products. We explain how to derive, conjugacy classes, characters, representations, and tensor products for these groups (with a finite number). We discussed them explicitly for $S_N$, $A_N$, $T'$, $D_N$, $Q_N$, $Σ(2N^2)$, $Δ(3N^2)$, $T_7$, $Σ(3N^3)$ and $Δ(6N^2)$, which have been applied for model building in the particle physics. We also present typical flavor models by using $A_4$, $S_4$, and $Δ(54)$ groups. Breaking patterns of discrete groups and decompositions of multiplets are important for applications of the non-Abelian discrete symmetry. We discuss these breaking patterns of the non-Abelian discrete group, which are a powerful tool for model buildings. We also review briefly about anomalies of non-Abelian discrete symmetries by using the path integral approach.
Non-Abelian discrete flavor symmetries of 10D SYM theory with magnetized extra dimensions
我们研究了基于磁化环面上紧化的10维超对称Yang-Mills（10 D SYM）理论的模型的离散味对称性。我们假设非零不可因式分解通量以及轨道投影。这些设置允许使用更多不同的风味结构构建模型。事实上，我们表明，存在各种类别的非阿贝尔离散味对称。特别地，我们发现$S_3$味对称性可以在磁化的10维SYM理论的框架下实现。
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…
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.
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 …
We combine two popular extensions of beyond the Standard Model physics within the framework of intersecting D6-brane models: discrete Zn symmetries and Peccei-Quinn axions. The underlying natural connection between both extensions is formed by the presence of massive U(1) gauge symmetries in D-brane model building. Global intersecting D6-brane models on toroidal orbifolds of the type T6/Z2N and T6/Z2xZ2M with discrete torsion offer excellent playgrounds for realizing these extensions. A generation-dependent Z2 symmetry is identified in a global Pati-Salam model, while global left-right symmetr…
While non-abelian groups are undoubtedly the cornerstone of Grand Unified Theories (GUTs), phenomenology shows that the role of abelian and discrete symmetries is equally important in model building. The latter are the appropriate tool to suppress undesired proton decay operators and various flavour violating interactions, to generate a hierarchical fermion mass spectrum, etc. In F-theory, GUT symmetries are linked to the singularities of the elliptically fibred K3 manifolds; they are of ADE type and have been extensively discussed in recent literature. In this context, abelian and discrete sy…
We study anomalies of non-Abelian discrete symmetries; which part of non-Abelian group is anomaly free and which part can be anomalous. It is found that the anomaly-free elements of the group $G$ generate a normal subgroup $G_0$ of $G$ and the residue class group $G/G_0$, which becomes the anomalous part of $G$, is isomorphic to a single cyclic group. The derived subgroup $D(G)$ of $G$ is useful to study the anomaly structure. This structure also constrains the structure of the anomaly-free subgroup; the derived subgroup $D(G)$ should be included in the anomaly-free subgroup. We study the deta…
We investigate a gauge theory realization of non-Abelian discrete flavor symmetries and apply the gauge enhancement mechanism in heterotic orbifold models to field-theoretical model building. Several phenomenologically interesting non-Abelian discrete symmetries are realized effectively from a $U(1)$ gauge theory with a permutation symmetry. We also construct a concrete model for the lepton sector based on a $U(1)^2 \rtimes S_3$ symmetry.
In F-theory compactifications, the abelian gauge sector is encoded in global structures of the internal geometry. These structures lie at the intersection of algebraic and arithmetic description of elliptic fibrations: While the Mordell--Weil lattice is related to the continuous abelian sector, the Tate--Shafarevich group is conjectured to encode discrete abelian symmetries in F-theory. In these notes we review both subjects with a focus on recent findings such as the global gauge group and gauge enhancements. We then highlight the application to F-theory model building.
We find new solutions to real cubic constraints on $N=1$ chiral superfields transforming under global abelian symmetries. These solutions describe the low-energy dynamics of a goldstino interacting with an axion (both belonging to the same chiral superfield) with non-linearly realized supersymmetry. We show the relation between our model and the approach of Komargodski and Seiberg for describing goldstino-axion dynamics which uses orthogonal nilpotent superfields.
We show that the subleading soft photon theorem in a $(d+2)$-dimensional massless abelian gauge theory gives rise to a Ward identity corresponding to divergent large gauge transformations acting on the celestial sphere at null infinity. We further generalize our analysis to $(d+2)$-dimensional non-abelian gauge theories and show that the leading and subleading soft gluon theorem give rise to Ward identities corresponding to asymptotic symmetries of the theory.
We derive the off-shell nilpotent symmetries of the two (1+1)-dimensional (2D) non-Abelian 1-form gauge theory by using the theoretical techniques of the geometrical superfield approach to Becchi-Rouet-Stora-Tyutin (BRST) formalism. For this purpose, we exploit the augmented version of superfield approach (AVSA) and derive theoretically useful nilpotent (anti-)BRST, (anti-)co-BRST symmetries and Curci-Ferrari (CF) type restrictions for the self-interacting 2D non-Abelian 1-form gauge theory (where there is no interaction with matter fields). The derivation of the (anti-)co-BRST symmetries and …
We determine the unitary and anti-unitary Lagrangian and quantum symmetries of arbitrary abelian Chern-Simons theories. The symmetries depend sensitively on the arithmetic properties (e.g. prime factorization) of the matrix of Chern-Simons levels, revealing interesting connections with number theory. We give a complete characterization of the symmetries of abelian topological field theories and along the way find many theories that are non-trivially time-reversal invariant by virtue of a quantum symmetry, including $U(1)_k$ Chern-Simons theory and $(\mathbb Z_k)_\ell$ gauge theories. For examp…
A spinor fields classification with non-Abelian gauge symmetries is introduced, generalizing the the U(1) gauge symmetries-based Lounesto's classification. Here, a more general classification, contrary to the Lounesto's one, encompasses spinor multiplets, corresponding to non-Abelian gauge fields. The particular case of SU(2) gauge symmetry, encompassing electroweak and electromagnetic conserved charges, is then implemented by a non-Abelian spinor classification, now involving 14 mixed classes of spinor doublets. A richer flagpole, dipole, and flag-dipole structure naturally descends from this…
We study the behaviour of abelian gauge symmetries in six-dimensional N=(1,0) theories upon decoupling gravity and investigate abelian flavour symmetries in the context of 6d N=(1,0) SCFTs. From a supergravity perspective, the anomaly cancellation mechanism implies that abelian gauge symmetries can only survive as global symmetries as gravity is decoupled. The flavour symmetries obtained in this way are shown to be free of ABJ anomalies, and their 't Hooft anomaly polynomial in the decoupling limit is obtained explicitly. In an F-theory realisation the decoupling of abelian gauge symmetries im…
The presence of non-Abelian discrete gauge symmetries in four-dimensional F-theory compactifications is investigated. Such symmetries are shown to arise from seven-brane configurations in genuine F-theory settings without a weak string coupling description. Gauge fields on mutually non-local seven-branes are argued to gauge both R-R and NS-NS two-form bulk axions. The gauging is completed into a generalisation of the Heisenberg group with either additional seven-brane gauge fields or R-R bulk gauge fields. The former case relies on having seven-brane fluxes, while the latter case requires tors…
We provide the first explicit example of Type IIB string theory compactification on a globally defined Calabi-Yau threefold with torsion which results in a four-dimensional effective theory with a non-Abelian discrete gauge symmetry. Our example is based on a particular Calabi-Yau manifold, the quotient of a product of three elliptic curves by a fixed point free action of Z_2 x Z_2. Its cohomology contains torsion classes in various degrees. The main technical novelty is in determining the multiplicative structure of the (torsion part of) the cohomology ring, and in particular showing that the…
We investigate the asymptotic symmetry group of a scalar field minimally-coupled to an abelian gauge field using the Hamiltonian formulation. This extends previous work by Henneaux and Troessaert on the pure electromagnetic case. We deal with minimally coupled massive and massless scalar fields and find that they behave differently insofar as the latter do not allow for canonically implemented asymptotic boost symmetries. We also consider the abelian Higgs model and show that its asymptotic canonical symmetries reduce to the Poincaré group in an unproblematic fashion.
Efficient quantum simulation protocols for any quantum theories demand efficient protection protocols for its underlying symmetries. This task is nontrivial for gauge theories as it is involves local symmetry/invariance. For non-Abelian gauge theories, protecting all the symmetries generated by a set of mutually non-commuting generators, is particularly difficult. In this letter, a global symmetry-protection protocol is proposed. Using the novel loop-string-hadron formalism of non-Abelian lattice gauge theory, we numerically demonstrate that all of the local symmetries get protected even for l…
Asymptotic symmetries of gauge theories are known to encode infrared properties of radiative fields. In the context of tree-level Yang-Mills theory, the leading soft behavior of gluons is captured by large gauge symmetries with parameters that are $O(1)$ in the large $r$ expansion towards null infinity. This relation can be extended to subleading order provided one allows for large gauge symmetries with $O(r)$ gauge parameters. The latter, however, violate standard asymptotic field fall-offs and thus their interpretation has remained incomplete. We improve on this situation by presenting a rel…
Symmetries of three-dimensional topological field theories are naturally defined in terms of invertible topological surface defects. Symmetry groups are thus Brauer-Picard groups. We present a gauge theoretic realization of all symmetries of abelian Dijkgraaf-Witten theories. The symmetry group for a Dijkgraaf-Witten theory with gauge group a finite abelian group $A$, and with vanishing 3-cocycle, is generated by group automorphisms of $A$, by automorphisms of the trivial Chern-Simons 2-gerbe on the stack of $A$-bundles, and by partial e-m dualities. We show that transmission functors naturall…
