Kurianovych, Yevhen2024-01-052024-01-052023-10https://hdl.handle.net/11299/259751University of Minnesota Ph.D. dissertation.October 2023. Major: Physics. Advisor: Mikhail Shifman. 1 computer file (PDF); vii, 102 pages.We study various applications of sigma models to heterotic supersymmetry and topological defects. We start with (0,1) minimal O(N) model where only left-moving fermions are present. Then we consider n connected copies of O(N) models in non-minimal model where both left-moving and right-moving fermions are present, but supersymmetry is broken for the right-movers. We study the non-minimal supersymmetric heterotically deformed (0, 2) sigma model with the Grassmannian target space. To develop the appropriate superfield formalism, we begin with a simplified model with flat target space, find its beta function up to two loops, and prove a non-renormalization theorem. Then we generalize the results to the full model with the Grassmannian target space. Using the geometric formulation, we calculate the beta functions and discuss the ’t Hooft and Veneziano limits. In the second part we study topological defects localized on other topological defects. We start with a simple model allowing the existence of domain walls with orientational moduli localized on them. We observe an O(3) sigma model on the wall world volume in the low-energy limit. We solve numerically classical static equations of motion and find the wall profile functions. We consider adding a spin-orbit interaction in the bulk, which gives rise to an entanglement between rotational and translational moduli, and calculate the corresponding low-energy Lagrangian. Within the O(3) sigma wall we obtain baby skyrmions localized on domain walls, and provide a solution for a skyrmion configuration, based on an analogy with instantons. We discuss the existence of one-dimensional domain walls localized on two-dimensional ones, and construct the corresponding effective action. We perform symmetry analysis of the initial model and of the low-energy theory on the domain wall world volume. In the end we give field-theoretic description of U(1) defects localized on the domain lines on thin films that were recently discovered experimentally. We describe topology of our model and solve this model in the adiabatic approximation. It turns out that such a model naturally provides periodic structure observed in experiment. The effective theory turns out to be the sine-Gordon model, but unlike the previous theoretical considerations we argue that in this case it is favorable for sine-Gordon kinks to merge into one defect with a uniform winding. We consider a system of adjacent domain lines and anti-lines and explain the experimental fact that the appearance of defects on a domain line prevents defect creation on the adjacent anti-lines. We also quantize the model and investigate possible effects of finite transverse dimension of the film.enDomain wallsHeterotic supersymmetryQFTSolitonsSupersymmetryTopological defectsThesis or Dissertation