In the framework of the ANR project "Superconformal Yang-Mills theories" (E, Sokatchev, Annecy, G. Korchemsky, University Paris-Sud, and A. Smilga, SUBATECH) we studied different problems of the dynamics of conformal supersymmetric gauge theories and/or theories with higher derivatives. In particular, we have shown that a certain supersymmetric higher-derivative model involving nontrivial interactions is unitary in spite of the presence of the "ghosts" (i.e. in spite of the absence of the ground state in the spectrum).

We also studied the so called "crypto-Hermitian" models (the models where the Hamiltonian seems not to be Hermitian, but for which all the eigenvalues are still real). In another work, we have checked  the validity of the string-gauge duality in Maldacena spirit for a certain supersymmetric quantum mechanical gauge model with 16 real supercharges. Still another interesting study concerns  the Witten index in 3-dimensional supersymmetric Chern-Simons-Yang-Mills theories. We calculated it by a direct microscopic calculation and reproduced the result obtained earlier by Witten with conformal theory methods.

In the framework of the collaboration IN2P3-Dubna on  supersymmetric quantum mechanical (SQM) models which
involves E.Ivanov, S. Fedoruk, and B. Zupnik from Dubna and A.Smilga from SUBATECH (in 2008-2012, it involved also the PhD student M. Konyushikhin who defended his thesis and works now at CERN), we constructed a new interesting model which describes the motion on conformally flat manifolds with self-dual electromagnetic field. We also generalized this model to non-Abelian gauge fields.