The Wess—Zumino—Witten theory related to the GL(1|1) supergroup possesses some interesting features. On one hand, its structure is rather simple, but, on the other hand, it is an example of a so called logarithmic theory, i.e. a conformal field theory that contains fields whose correlation functions depend on distances logarithmically. The spectrum of conformal dimensions in this theory is continuous, and logarithmic operators appear at some degenerate points, including those of zero dimension. The free field representation is an effective tool to study models of the conformal field theory, and that of the GL(1|1) theory seems to be rather simple and well-studied in previous works. Nevertheless, on my opinion, not all advantages of this representation were used. In the present work, beside a more detailed calculation of the structure constants, the fusion and braiding matrices were studied. It was shown that in the vicinity of degenerate points it is possible to chose a basis of conformal blocks, which resolves degeneration. I show how this basis is related to the logarithmic operators of the theory.

It is shown by numerical simulations within a regularized Biot-Savart law that dynamical systems of two or three leapfrogging coaxial quantum vortex rings having a core width $\xi$ and initially placed near a torus of radii $R_0$ and $r_0$, can be three-dimensionally (quasi-)stable in some regions of parameters $\Lambda=\ln(R_0/\xi)$ and $W=r_0/R_0$. At fixed $\Lambda$, stable bands on $W$ are intervals between non-overlapping main parametric resonances for different (integer) azimuthal wave numbers $m$. The stable intervals are most wide ($\Delta W\sim$ 0.01--0.05) between $m$-pairs $(1,2)$ and $(2,3)$ at $\Lambda\approx$ 4--12 thus corresponding to micro/mesoscopic sizes of vortex rings in the case of superfluid $^4$He. With four and more rings, at least for $W>0.1$, resonances overlap for all $\Lambda$ and no stable domains exist.

We examine the question of the influence of sparse long-range communications on the synchronization in parallel discrete event simulations (PDES). We build a model of the evolution of local virtual times (LVT) in a conservative algorithm including several choices of local links. All network realizations belong to the small-world network class. We find that synchronization depends on the average shortest path of the network. The time profile dynamics are similar to the surface profile growth, which helps to analyze synchronization effects using a statistical physics approach. Without long-range links of the nodes, the model belongs to the universality class of the Kardar–Parisi–Zhang equation for surface growth. We find that the critical exponents depend logarithmically on the fraction of long-range links. We present the results of simulations and discuss our observations.