In this article we describe Hecke operators on the differential algebra of geometric quasi-modular forms. As an application for each natural number d we construct a vector field in six dimensions which determines uniquely the polynomial relations between the Eisenstein series of weight 2, 4 and 6 and their transformation under multiplication of the argument by d, and in particular, it determines uniquely the modular curve of degree d isogenies between elliptic curves.

A Calabi–Yau threefold is called of type K if it admits an ´etale Galois covering by the product of a K3 surface and an elliptic curve. In our previous paper [16], based on Oguiso–Sakurai’s fundamental work [24], we have provided the full classification of Calabi–Yau threefolds of type K and have studied some basic properties thereof. In the present paper, we continue the study, investigating them from the viewpoint of mirror symmetry. It is shown that mirror symmetry relies on duality of certain sublattices in the second cohomology of the K3 surface appearing in the minimal splitting covering. The duality may be thought of as a version of the lattice duality of the anti-symplectic involution on K3 surfaces discovered by Nikulin [23]. Based on the duality, we obtain several results parallel to what is known for Borcea–Voisin threefolds. Along the way, we also investigate the Brauer groups of Calabi–Yau threefolds of type K.

We show that there exists an infinite tower of fermionic symmetries in pure d=4, N=1 supergravity on an asymptotically flat background. The Ward identities associated with these symmetries are equivalent to the soft limit of the gravitino and to the statement of supersymmetry at every angle. Additionally, we show that these charges commute into charges associated with the (unextended) BMS group, providing a supersymmetrization of the BMS translations.

We study chiral algebras associated with Argyres-Douglas theories engineered from M5 brane. For the theory engineered using 6d (2,0) type J theory on a sphere with a single irregular singularity (without mass parameter), its chiral algebra is the minimal model of W algebra of J type. For the theory engineered using an irregular singularity and a regular full singularity, its chiral algebra is the affine Kac-Moody algebra of J type. We can obtain the Schur index of these theories by computing the vacua character of the corresponding chiral algebra.

We initiate the study of M-strings in the thermodynamic limit. In this limit the BPS partition function of M5 branes localizes on configurations with a large number of strings which leads to a reformulation of the partition function in terms of a matrix model. We solve this matrix model and obtain its spectral curve which can be interpreted as the Seiberg-Witten curve associated to the compactified M5 brane theory.