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In this article, using an idea of the physics superselection principal, we study a modularity on vertex operator algebras arising from semisimple primary vectors. We generalizes the theta functions on vertex operator algebras and prove that the internal automorphisms do not change the genus one twisted conformal blocks.

The three-loop ladder diagram is a graph with six links and four cubic vertices that contributes to the $D^{12}{{ℛ}}^4$ amplitude at genus one in type II string theory. The vertices represent the insertion points of vertex operators on the toroidal worldsheet and the links represent scalar Green functions connecting them. By using the properties of the Green function and manipulating the various expressions, we obtain a modular invariant Poisson equation satisfied by this diagram, with source terms involving one-, two- and three-loop diagrams. Unlike the source terms in the Poisson equations for diagrams at lower orders in the momentum expansion or the Mercedes diagram, a particular source term involves a five-point function containing a holomorphic and a antiholomorphic worldsheet derivative acting on different Green functions. We also obtain simple equalities between topologically distinct diagrams, and consider some elementary examples.

We consider the $D^8{{ℛ}}^5$ and $D^{10}{{ℛ}}^5$ terms in the low momentum expansion of the five graviton amplitude in type IIB string theory at one loop. They involve integrals of various modular graph functions over the fundamental domain of $SL(2,\mathbb{Z})$. Unlike the graphs which arise in the four graviton amplitude or at lower orders in the momentum expansion of the five graviton amplitude where the links are given by scalar Green functions, there are several graphs for the $D^8{{ℛ}}^5$ and $D^{10}{{ℛ}}^5$ terms where each of these two links are given by a derivative of the Green function. Starting with appropriate auxiliary diagrams, we show that these graphs can be expressed in terms of those which do not involve any derivatives. This results in considerable simplification of the amplitude.

We review aspects of the modular invariance approach to the flavour problem. Harald Fritzsch was among the first to realise the existence and the fundamental nature of the quark and lepton flavour problems in particle physics, that symmetries may be the key to the solution(s) of these problems and to propose in 1978 and 1979 a solution to the quark flavour problem in the form of the Fritzsch quark mass matrices with texture zeros. After introducing the general ingredients of the modular invariance approach, we describe the formalism that allows to construct models in which fermion (charged-lepton and quark) mass hierarchies follow solely from the properties of the modular forms, avoiding the fine-tuning of the constant parameters present in the fermion mass matrices and the need to introduce extra fields. Focusing on the lepton sector, we show how the indicated formalism can be used in lepton flavour models to obtain the charged lepton mass hierarchies without parameter fine-tuning. Only a very limited number of models satisfy the non-fine-tuning requirement in explaining the observed fermion mass hierarchies. We review next the general conditions under which also the peculiar pattern of Pontecorvo, Maki, Nakagawa and Sakata (PMNS) lepton mixing consisting of two large and one small angles can be reproduced without fine-tuning. We then give an example of a phenomenologically viable “minimal” modular lepton flavour model in which both the lepton mixing and charged lepton mass hierarchies are generated without fine-tuning. We conclude by listing some of the open questions and problems in the modular invariant approach to the flavour problem.

The modular $A_4$ symmetry with three moduli is investigated. We assign different moduli to charged leptons, neutrinos, and quarks. We analyze these moduli at their fixed points where a residual symmetry exists. We consider two possibilities for right-handed neutrinos. First, they are assumed to be singlets under the modular symmetry. In this case, we show that the lepton masses and mixing can be obtained consistently with experimental observations. Second, they are assigned nontrivially under modular symmetry. We emphasize that a small deviation from their fixed point is required in this case. Finally, the quark masses and mixing are generated correctly around the fixed point of their modulus. In our analysis, we only consider the simple case of weight 2.

Dilaton and axion are ubiquitous in extended supergravities and closed superstrings. We propose new models of modular inflation in four-dimensional $N=1$ supergravity coupled to the chiral dilaton–axion–goldstino supermultiplet, which fit some necessary conditions of superstring cosmology. The model parameters are tuned to obey precision measurements of the cosmic microwave background radiation. We employ the modular-invariant superpotentials and asymptotically modular-invariant Kähler potentials, and achieve axion stabilization with high-scale supersymmetry breaking.

We solve the problem of constructing all chiral genus-one correlation functions from chiral genus-zero correlation functions associated to a vertex operator algebra satisfying the following conditions: (i) V_(n) = 0 for n < 0 and V₍₀₎ = ℂ1, (ii) every ℕ-gradable weak V-module is completely reducible and (iii) V is C₂-cofinite. We establish the fundamental properties of these functions, including suitably formulated commutativity, associativity and modular invariance. The method we develop and use here is completely different from the one previously used by Zhu and others. In particular, we show that the q-traces of products of certain geometrically-modified intertwining operators satisfy modular invariant systems of differential equations which, for any fixed modular parameter, reduce to doubly-periodic systems with only regular singular points. Together with the results obtained by the author in the genus-zero case, the results of the present paper solves essentially the problem of constructing chiral genus-one weakly conformal field theories from the representations of a vertex operator algebra satisfying the conditions above.

We review aspects of the modular invariance approach to the flavour problem. Harald Fritzsch was among the first to realise the existence and the fundamental nature of the quark and lepton flavour problems in particle physics, that symmetries may be the key to the solution(s) of these problems and to propose in 1978 and 1979 a solution to the quark flavour problem in the form of the Fritzsch quark mass matrices with texture zeros. After introducing the general ingredients of the modular invariance approach, we describe the formalism that allows to construct models in which fermion (charged-lepton and quark) mass hierarchies follow solely from the properties of the modular forms, avoiding the fine-tuning of the constant parameters present in the fermion mass matrices and the need to introduce extra fields. Focusing on the lepton sector, we show how the indicated formalism can be used in lepton flavour models to obtain the charged lepton mass hierarchies without parameter fine-tuning. Only a very limited number of models satisfy the non-fine-tuning requirement in explaining the observed fermion mass hierarchies. We review next the general conditions under which also the peculiar pattern of Pontecorvo, Maki, Nakagawa and Sakata (PMNS) lepton mixing consisting of two large and one small angles can be reproduced without fine-tuning. We then give an example of a phenomenologically viable “minimal” modular lepton flavour model in which both the lepton mixing and charged lepton mass hierarchies are generated without fine-tuning. We conclude by listing some of the open questions and problems in the modular invariant approach to the flavour problem.