We show that for a particular choice of gauge the vector potential of any non-radiating source is spatially localized along with its electric and magnetic fields. Important on its own, this special property of non-radiating sources dramatically simplifies the analysis of their quantitative aspects, and enables the interpretation of non-radiating sources as distributions of the elementary dynamic anapoles. Using the developed approach we identify and discuss a possible scenario for observing the time-dependent version of the Aharonov-Bohm effect in such systems.

The Alday–Gaiotto–Tachikawa relations reduce S-duality to the modular transformations of conformal blocks. It was recently conjectured that, for the four-point conformal block, the modular transform up to the non-perturbative contributions can be written in the form of the ordinary Fourier transform when β ≡ −ϵ1/ϵ2 = 1. Here I extend this conjecture to general values of ϵ1, ϵ2. Namely, I argue that, for a properly normalized four-point conformal block the S-duality is perturbatively given by the Fourier transform for arbitrary values of the deformation parameters ϵ1, ϵ2. The conjecture is based on explicit perturbative computations in the first few orders of the string coupling constant g2 ≡ −ϵ1ϵ2 and hypermultiplet masses.

The complete dynamic multipole expansion of electromagnetic sources contains more types of multipole terms than it is conventionally perceived. The toroidal multipoles are one of the examples of such contributions that have been widely studied in recent years. Here we inspect more closely the other type of commonly overlooked terms known as the mean-square radii. In particular, we discuss both quantitative and qualitative aspects of the mean-square radii and provide a general geometrical framework for their visualization. We also consider the role of the mean-square radii in expanding the family of non-trivial non-radiating electromagnetic sources.