Vol 187, No 3 (2016)

We show that the current interaction of massless fields in four dimensions breaks the sp(8) symmetry of free massless equations of arbitrary spin down to the conformal symmetry su(2, 2). This breaking agrees with the form of the nonlinear higher-spin field equations.

We show that the non-Abelian Hirota difference equation is directly related to a commutator identity on an associative algebra. Evolutions generated by similarity transformations of elements of this algebra lead to a linear difference equation. We develop a special dressing procedure that results in an integrable non-Abelian Hirota difference equation and propose two regular reduction procedures that lead to a set of known equations, Abelian or non-Abelian, and also to some new integrable equations.

We demonstrate that statistics for several types of set partitions are described by generating functions arising in the theory of integrable equations.

We derive a simple formula for one-loop logarithmic divergences on the background of a two-dimensional curved space–time for theories in which the second variation of the action is a nonminimal second-order operator with small nonminimal terms. In particular, this formula allows calculating terms that are integrals of total derivatives. As an application of the result, we obtain one-loop divergences for higher-spin fields on a constant-curvature background in a nonminimal gauge that depends on two parameters. By an explicit calculation, we demonstrate that with the considered accuracy, the result is gauge independent and, moreover, spin independent for spins s ≥ 3.

We consider an exactly solvable model of nonautonomous W*-dynamics driven by repeated harmonic interaction. The dynamics is Hamiltonian and quasifree. Because of inelastic interaction in the large-time limit, it leads to relaxation of initial states to steady states. We derive the explicit entropy production rate accompanying this relaxation. We also study the evolution of different subsystems to elucidate their eventual correlations and convergence to equilibriums. In conclusion, we prove that the W*-dynamics manifests a universal stationary behavior in a short-time interaction limit.

Three characteristics of two-dimensional uniform spanning trees are nontrivially related to one another: the average density of a sandpile, the looping constant of a square lattice, and the return probability of a loop-erased random walk. We briefly trace the long history of the discovery of their unexpected rational values.