A modified theory of general relativity is proposed, where the gravitational constant is replaced by a dynamical variable in space-time. The dynamics of the gravitational coupling is described by a family of parametrized null geodesics, implying that the gravitational coupling at a space-time point is determined by solving transport equations along all null geodesics through this point. General relativity with dynamical gravitational coupling (DGC) is introduced. We motivate DGC from general considerations and explain how it arises in the context of causal fermion systems. The underlying physical idea is that the gravitational coupling is determined by microscopic structures on the Planck scale which propagate with the speed of light. In order to clarify the mathematical structure, we analyze the conformal behaviorn and prove local existence and uniqueness of the time evolution. The differences to Einstein’s theory are worked out in the examples of the Friedmann-RobertsonWalker model and the spherically symmetric collapse of a shell of matter. Potential implications for the problem of dark matter and for inflation are discussed. It is shown that the effects in the solar system are too small for being observable in present-day experiments.

Recently, a number of new Ward identities for large gauge transformations and large diffeomorphisms have been discovered. Some of the identities are reinterpretations of previously known statements, while some appear to be genuinely new. We use Noether’s second theorem with the path integral as a powerful way of generating these kinds of Ward identities. We reintroduce Noether’s second theorem and discuss how to work with the physical remnant of gauge symmetry in gauge fixed systems. We illustrate our mechanism in Maxwell theory, Yang–Mills theory, p-form field theory, and Einstein–Hilbert gravity. We comment on multiple connections between Noether’s second theorem and known results in the recent literature. Our approach suggests a novel point of view with important physical consequences.

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 prove that if a curve of a non-isotrivial family of abelian varieties over a curve contains infinitely many isogeny orbits of a finitely generated subgroup of a simple abelian variety then it is special. This result fits into the context of Zilber-Pink conjecture and partially generalizes a result of Faltings. Moreover by using the polyhedral reduction theory we give a new proof of a result of Bertrand.

We construct SYZ mirrors of the local Calabi–Yau manifolds of type à by developing an equivariant SYZ theory for the toric Calabi–Yau manifolds of infinite-type. The equations for the SYZ mirrors involve the Riemann theta functions and generating functions of the open Gromov–Witten invariants. We obtain explicit formulae for the generating functions which are open analogs of the Yau–Zaslow formula in dimension 2, and show that they have nice modular properties. We also relate the SYZ mirror pairs with mirror symmetry for the abelian varieties and hypersurfaces therein.