Defects in supersymmetric field theories and string theory

Supersymmetry allows us to calculate non-perturbative quantities exactly and analytically in various models of QFT and string theory.We will consider applications of these non-perturbative methods and describe properties (quantum numbers, wave functions, scattering amplitudes) of Bogomolny-Prasad- Sommerfeld (BPS) states. Despite these states are more symmetric with respect to supersymmetry, nevertheless they are authentic quantum states in theories at strong coupling. One could think of them as defects: instantons, solitons, quasi-particles etc. Therefore, BPS states in supersymmetric theories represent a nice playground to test our understanding of behavior patterns for strongly correlated systems. In the talk we will consider examples of applications to systems of D-branes on toric Calabi-Yau three-folds and construct BPS algebras. Also, we will consider the role of BPS states in the problem of Chern-Simons knot invariant categorification.