1) Boundary charges for solitons,
2) Vortex equations in anN=2 U(1)^M model,
3) Vibrational spectra of solitons, and
4) The mass gap problem of Yang-Mills theory"
This talk will contain four parts:
1) A brief introduction to the concepts of newly found boundary charges; they are integral identities and can be viewed as a generalization of the well-known Pohozaev identity. Apart from being mathematically interesting, they may in principle be used to extract non-perturbative information from physical systems.
2) Vortex equations in an extended supersymmetric model with M U(1) gauge groups and a charge matrix are studied. Interesting alignment between the Fayet-Iliopoulos parameters and the orientation of the vortex equations in SU(2)_R space is revealed. The relation in turn depends on the winding numbers in each gauge group.
3) Considerations in the Skyrme model of 3-dimensional solitons that are solutions to nonlinear PDEs show that a semiclassical approach to reproducing nuclear masses is impossible unless the quantization method is extended from rigid body quantization to including low-lying but massive modes of the solutions. Representation theory classifies the vibrational modes and the spin they can contribute to is in turn a result purely of representation theory.
4) The mass gap problem of Yang-Mills theory is briefly discussed.