Research interests

Mathematical physics is breathtakingly beautiful – it is an interplay between physics and pure mathematics with a tremendous power to model reality, as well as to create stunning worlds of its own. My research interests are at the interface of symplectic geometry, representation theory, algebraic topology, gauge theories, quantum field theories and string theory.
One of the main areas of activity is the study of extended objects in gauge theories. Recently, my colleagues and I developed a Wilson surface observable, a gauge invariant quantity defined on a 2-dimensional surface. We showed that this sigma-model is defined by the equivariant extension of the Kirillov symplectic form on the coadjoint orbit. We also gave a new path integral presentation of Wilson lines in terms of Poisson sigma-models. Later on, I introduced the description of a Wilson surface as a 2-dimensional topological quantum field theory. On a closed surface, the Wilson surface theory defines a topological invariant of the principal G-bundle, for G semisimple compact Lie group.
It was discovered that it can interact topologically with 2-dimensional Yang–Mills and BF theories modifying their partition functions.
My recent works with co-authors include the study of Virasoro coadjoint orbits, Schwarzian quantum mechanics, theory of holomorphic maps, enhancement of the Fukaya-Morse A∞ category. In particular, we demonstrated that a wide class of Virasoro coadjoint orbits admits a global Darboux chart, hence the associated physical systems, a priori rather complicated and nonlinear, actually turn into free theories. These research directions may seem relatively independent, but they are all united by the same logic: they describe mathematical structures inspired by or applied to understanding the laws of the Universe.