Coadjoint Orbits, Cocycles and Gravitational Wess–Zumino

About 30 years ago, in a joint work with L. Faddeev we introduced a geometric action on coadjoint orbits. This action, in particular, gives rise to a path integral formula for characters of the corresponding group [Formula: see text]. In this paper, we revisit this topic and observe that the geometric action is a 1-cocycle for the loop group [Formula: see text]. In the case of [Formula: see text] being a central extension, we construct Wess–Zumino (WZ) type terms and show that the cocycle property of the geometric action gives rise to a Polyakov–Wiegmann (PW) formula with boundary term given by the 2-cocycle which defines the central extension. In particular, we obtain a PW type formula for Polyakov’s gravitational WZ action. After quantization, this formula leads to an interesting bulk-boundary decoupling phenomenon previously observed in the WZW model. We explain that this decoupling is a general feature of the Wess–Zumino terms obtained from geometric actions, and that in this case, the path integral is expressed in terms of the 2-cocycle which defines the central extension. In memory of our teacher Ludwig Faddeev

Synchronization of Dissipative Dynamical Systems Driven by Non-Gaussian Lévy Noises

Dynamical systems driven by Gaussian noises have been considered extensively in modeling, simulation, and theory. However, complex systems in engineering and science are often subject to non-Gaussian fluctuations or uncertainties. A coupled dynamical system under a class of Lévy noises is considered. After discussing cocycle property, stationary orbits, and random attractors, a synchronization phenomenon is shown to occur, when the drift terms of the coupled system satisfy certain dissipativity and integrability conditions. The synchronization result implies that coupled dynamical systems share a dynamical feature in some asymptotic sense.