Extended supersymmetry in gauge theory, gravity and integrable models

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In the course of this project, we plan to consider the following main tasks:(1) Studying N=1 supersymmetric self-dual electrodynamics in the superspace formalism with chiral auxiliary superfields, including coset scalar superfields into this scheme with the purpose to extend U(N) duality to Sp(2N).(2) Revisiting the problem of constructing the N=2 Born-Infeld action with half-broken N=4 supersymmetry in the approach with chiral auxiliary superfields.(3) Constructing new harmonic superspace formulations of gauge theories in diverse dimensions, including d=3 superconformal Chern-Simons gauge theories and different versions of N=4 SYM theory, and investigating the quantum properties of four-dimensional N=2 supergravity in harmonic superspace.(4) Developing new methods for analyzing the quantum properties of Yang-Mills theories formulated in different admissible gauges and studying the two-loop structure of the low-energy effective action in d=3, N=2 supersymmetric non-Abelian gauge theory.(5) Finding new classical solutions to the Einstein-Chern-Simons equations coupled with matter fields in AdS_5 and identifying the corresponding holographically dual operators on the boundary.(6) Searching for new integrable and non-integrable cosmological and black-hole models which can fit into gauged extended supergavity. (7) Studying quantum mechanics with extended supersymmetry, including superconformal ones, revealing their relation to supersymmetric field theories in higher dimensions, AdS/CFT correspondence and black-hole physics.(8) Constructing new classical and quantum d=1 and d=2 models with deformed N=4 and N=8 supersymmetries in the superfield approach.(9) Constructing d=2 integrable models with a higher integral of motion (besides energy) within the polynomial approach and studying new solutions of the master equations by using symmetric variables. (10) Searching for invariant subvarieties of the full symmetric sl(n)-Toda flow on degenerate orbits, constructing the corresponding phase transition diagrams and generalizing it to Toda systems based on other Lie algebras and coset spaces.(11) Studying the harmonic spheres conjecture linking G-valued Yang-Mills fields on R^4 and harmonic maps of the Riemann sphere into the loop space over G.(12) Applying of non-commutative geometry to the magnetic Bloch theory and the quantum Hall effect as well as extending Dirac quantization to non-smooth observables.Most of these tasks are logical continuations of investigations that were carried out earlier.