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We review the construction of the supersymmetric sigma model for unitary maps, using the color- flavor transformation. We then illustrate applications by three case studies in quantum chaos. In two of these cases, general Floquet maps and quantum graphs, we show that universal spectral fluctuations arise provided the pertinent classical dynamics are fully chaotic (ergodic and with decay rates sufficiently gapped away from zero). In the third case, the kicked rotor, we show how the existence of arbitrarily long-lived modes of excitation (diffusion) precludes universal fluctuations and entails quantum localization.
We investigate the chaotic behaviour of multiparticle systems, in particular DNA and graphene models, by applying methods of nonlinear dynamics. Using symplectic integration techniques, we present an extensive analysis of chaos in the Peyrard-Bishop-
In four-dimensional N=1 Minkowski superspace, general nonlinear sigma models with four-dimensional target spaces may be realised in term of CCL (chiral and complex linear) dynamical variables which consist of a chiral scalar, a complex linear scalar
The transformation of a system from one state to another is often mediated by a bottleneck in the systems phase space. In chemistry these bottlenecks are known as emph{transition states} through which the system has to pass in order to evolve from re
A gauge PDE is a natural notion which arises by abstracting what physicists call a local gauge field theory defined in terms of BV-BRST differential (not necessarily Lagrangian). We study supergeometry of gauge PDEs paying particular attention to glo
Recently the phase space structures governing reaction dynamics in Hamiltonian systems have been identified and algorithms for their explicit construction have been developed. These phase space structures are induced by saddle type equilibrium points