Modular symmetry at level 6 and a new route towards finite modular groups

We propose to construct the finite modular groups from the quotient of two principal congruence subgroups as Γ(N′)/Γ(N″), and the modular group SL(2, ℤ) is ex- tended to a principal congruence subgroup Γ(N′). The original modular invariant theory is reproduced when N′ = 1. We perform a comprehensive study of $$ {\Gamma}_6^{\prime } $$ Γ 6 ′ modular symmetry corresponding to N′ = 1 and N″ = 6, five types of models for lepton masses and mixing with $$ {\Gamma}_6^{\prime } $$ Γ 6 ′ modular symmetry are discussed and some example models are studied numerically. The case of N′ = 2 and N″ = 6 is considered, the finite modular group is Γ(2)/Γ(6) ≅ T′, and a benchmark model is constructed.

Twisted 6d (2, 0) SCFTs on a circle

We study twisted circle compactification of 6d (2, 0) SCFTs to 5d $$ \mathcal{N} $$ N = 2 supersymmetric gauge theories with non-simply-laced gauge groups. We provide two complementary approaches towards the BPS partition functions, reflecting the 5d and 6d point of view respectively. The first is based on the blowup equations for the instanton partition function, from which in particular we determine explicitly the one-instanton contribution for all simple Lie groups. The second is based on the modular bootstrap program, and we propose a novel modular ansatz for the twisted elliptic genera that transform under the congruence subgroups Γ0(N) of SL(2, ℤ). We conjecture a vanishing bound for the refined Gopakumar-Vafa invariants of the genus one fibered Calabi-Yau threefolds, upon which one can determine the twisted elliptic genera recursively. We use our results to obtain the 6d Cardy formulas and find universal behaviour for all simple Lie groups. In addition, the Cardy formulas remain invariant under the twist once the normalization of the compact circle is taken into account.

Fermionic Rational Conformal Field Theories and Modular Linear Differential Equations

We define Modular Linear Differential Equations (MLDE) for the level-two congruence subgroups Γθ, Γ0(2) and Γ0(2) of SL2(ℤ). Each subgroup corresponds to one of the spin structures on the torus. The pole structures of the fermionic MLDEs are investigated by exploiting the valence formula for the level-two congruence subgroups. We focus on the first and second order holomorphic MLDEs without poles and use them to find a large class of ‘Fermionic Rational Conformal Field Theories’, which have non-negative integer coefficients in the q-series expansion of their characters. We study the detailed properties of these fermionic RCFTs, some of which are supersymmetric. This work also provides a starting point for the classification of the fermionic Modular Tensor Category.

Unique product groups and congruence subgroups

We prove that a uniform pro-[Formula: see text] group with no non-abelian free subgroups has a normal series with torsion-free abelian factors. We discuss this in relation to unique product groups. We also consider generalizations of Hantzsche–Wendt groups.