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A bstract We construct the defining data of two-dimensional topological field theories (TFTs) enriched by non-invertible symmetries/topological defect lines. Simple formulae for the three-point functions and the lasso two-point functions are derived, and crossing symmetry is proven. The key ingredients are open-to-closed maps and a boundary crossing relation, by which we show that a diagonal basis exists in the defect Hilbert spaces. We then introduce regular TFTs, provide their explicit constructions for the Fibonacci, Ising and Haagerup ℋ 3 fusion categories, and match our formulae with previous bootstrap results. We end by explaining how non-regular TFTs are obtained from regular TFTs via generalized gauging.

A bstract We prove a recent conjecture by Harlow and Ooguri concerning a universal formula for the charged density of states in QFT at high energies for global symmetries associated with finite groups. An equivalent statement, based on the entropic order parameter associated with charged operators in the thermofield double state, was proven in a previous article by Casini, Huerta, Pontello, and the present author. Here we describe how the statement about the entropic order parameter arises, and how it gets transformed into the universal density of states. The use of the certainty principle, relating the entropic order and disorder parameters, is crucial for the proof. We remark that although the immediate application of this result concerns charged states, the origin and physics of such density can be understood by looking at the vacuum sector only. We also describe how these arguments lie at the origin of the so-called entropy equipartition in these type of systems, and how they generalize to QFT’s on non-compact manifolds.

A bstract It is believed that in SU( N ) Yang-Mills theory observables are N -branched functions of the topological θ angle. This is supposed to be due to the existence of a set of locally-stable candidate vacua, which compete for global stability as a function of θ . We study the number of θ vacua, their interpretation, and their stability properties using systematic semiclassical analysis in the context of adiabatic circle compactification on ℝ 3 × S 1 . We find that while observables are indeed N-branched functions of θ, there are only ≈ N/ 2 locally-stable candidate vacua for any given θ . We point out that the different θ vacua are distinguished by the expectation values of certain magnetic line operators that carry non-zero GNO charge but zero ’t Hooft charge. Finally, we show that in the regime of validity of our analysis YM theory has spinodal points as a function of θ , and gather evidence for the conjecture that these spinodal points are present even in the ℝ 4 limit.

We analyse the possibility of describing quark masses, mixing and CP violation in$$ {S}_4^{\\prime } $$S 4 ′ modular flavour models without flavons. We focus on the case where the closeness of the modulus to the point of residual$$ {\\mathbb{Z}}_3^{ST} $$ℤ 3 ST symmetry (the cusp) plays a role in generating quark mass hierarchies and discuss the role modular form normalisations play in such constructions. We find that fitting quark data requires explicit CP breaking, unless a second modulus is introduced.

Abstract We study the textures of SM fermion mass matrices and their mixings in a supersymmetric adjoint SU(5) Grand Unified Theory with modular S 4 being the horizontal symmetry. The Yukawa entries of both quarks and leptons are expressed by modular forms with lower weights. Neutrino sector has an adjoint SU(5) representation 24 as matter superfield, which is a triplet of S 4. The effective light neutrino masses is generated through Type-III and Type-I seesaw mechanism. The only common complex parameter in both charged fermion and neutrino sectors is modulus τ . Down-type quarks and charged leptons have the same joint effective operators with adjoint scalar in them, and their mass discrepancy in the same generation depends on Clebsch-Gordan factor. Especially for the first two generations the respective Clebsch-Gordan factors made the double Yukawa ratio 𝒴d𝒴μ/𝒴e𝒴s = 12, in excellent agreement with the experimental result. We reproduce proper CKM mixing parameters and all nine Yukawa eigenvalues of quarks and charged leptons. Neutrino masses and MNS parameters are also produced properly with normal ordering is preferred.

Abstract In this paper, we incorporate the modular S 4 flavor symmetry into the supersymmetric version of the minimal type-I seesaw model, in which only two right-handed neutrino singlets are introduced to account for tiny Majorana neutrino masses, and explore its implications for the lepton mass spectra, flavor mixing and CP violation. The basic idea is to assign two right-handed neutrino singlets into the unique two-dimensional irreducible representation of the modular S 4 symmetry group. Moreover, we show that the matter-antimatter asymmetry in our Universe can be successfully explained via the resonant leptogenesis mechanism working at a relatively-low seesaw scale ΛSS ≈ 107 GeV, with which the potential problem of the gravitino overproduction can be avoided. In this connection, we emphasize that the observed matter-antimatter asymmetry may lead to a stringent constraint on the parameter space and testable predictions for low-energy observables.

Abstract We discuss the modular A 4 invariant model of leptons combining with the generalized CP symmetry. In our model, both CP and modular symmetries are broken spontaneously by the vacuum expectation value of the modulus τ. The source of the CP violation is a non-trivial value of Re[τ] while other parameters of the model are real. The allowed region of τ is in very narrow one close to the fixed point τ = i for both normal hierarchy (NH) and inverted ones (IH) of neutrino masses. The CP violating Dirac phase δ CP is predicted clearly in [98°, 110°] and [250°, 262°] for NH at 3 σ confidence level. On the other hand, δ CP is in [95°, 100°] and [260°, 265°] for IH at 5 σ confidence level. The predicted ∑m i is in [82, 102] meV for NH and ∑m i = [134, 180] meV for IH. The effective mass 〈m ee 〉 for the 0νββ decay is predicted in [12.5, 20.5] meV and [54, 67] meV for NH and IH, respectively.

Abstract We classify all non-invertible Kramers-Wannier duality defects in the E 8 lattice Vertex Operator Algebra (i.e. the chiral (E 8)1 WZW model) coming from ℤ m symmetries. We illustrate how these defects are systematically obtainable as ℤ2 twists of invariant sub-VOAs, compute defect partition functions for small m, and verify our results against other techniques. Throughout, we focus on taking a physical perspective and highlight the important moving pieces involved in the calculations. Kac’s theorem for finite automorphisms of Lie algebras and contemporary results on holomorphic VOAs play a role. We also provide a perspective from the point of view of (2+1)d Topological Field Theory and provide a rigorous proof that all corresponding Tambara-Yamagami actions on holomorphic VOAs can be obtained in this manner. We include a list of directions for future studies.

Abstract We study a flavor model with A 4 symmetry which originates from S 4 modular group. In S 4 symmetry, Z 2 subgroup can be anomalous, and then S 4 can be violated to A 4. Starting with a S 4 symmetric Lagrangian at the tree level, the Lagrangian at the quantum level has only A 4 symmetry when Z 2 in S 4 is anomalous. We obtain modular forms of two singlets and a triplet representations of A 4 by decomposing S 4 modular forms into A 4 representations. We propose a new A 4 flavor model of leptons by using those A 4 modular forms. We succeed in constructing a viable neutrino mass matrix through the Weinberg operator for both normal hierarchy (NH) and inverted hierarchy (IH) of neutrino masses. Our predictions of the CP violating Dirac phase δ CP and the mixing sin2 θ 23 depend on the sum of neutrino masses for NH.

Abstract We present minimalist constructions for lepton masses and mixing based on flavour symmetry under the modular group Γ N of lowest level N = 2. As opposed to the only existing model of Γ2 ≅ S 3 formulated in a SUSY framework, the only non-SM field is the modulus τ, and a generalised CP symmetry is implemented. Charged-leptons masses are reproduced through symmetry arguments, without requiring fine-tuning of the free parameters. As a result, all lepton observables (masses and mixing) are reproduced within 1σ experimental range using a minimum of nine free real parameters (including the real and imaginary parts of the modulus). A normal ordering for the neutrino masses is predicted. We also obtain predictions for the CP violating phases: the Dirac CP phase is predicted around 1.6π, the Majorana phases lie in narrow regions near ±π. The sum of neutrino masses is within the current bound at ∼ 0.09 eV. Furthermore, we provide predictions for the neutrinoless double beta decay and tritium decay effective masses, around 20 meV. Given the reduced number of free input parameters as compared to the existing literature on modular S 3, this work renews interest for a unified predictive model of quark-lepton sectors based on Γ2 ≅ S 3.