Sheu, Chao-Hsiang2025-01-282025-01-282024-08https://hdl.handle.net/11299/269594University of Minnesota Ph.D. dissertation. August 2024. Major: Physics. Advisor: Mikhail Shifman. 1 computer file (PDF); viii, 130 pages.Sigma models with inhomogeneous target manifolds exhibit richer structures in many perspectives compared to traditional models with homogeneous spaces. The more complex geometric structures lead to additional features and anomalies as quantum corrections are considered. Moreover, the associated theory in the infrared, under renormalization group flows, can be notably non-trivial. In this thesis, we present a thorough investigation of two sigma models within this category. We begin with a classic example, the weighted projective model 𝕎ℂℙ(N, Ñ), which is well-studied in the physics and mathematics literature. Our investigation focuses on the geometry and related quantum effects of the Ɲ = (0,2) supersymmetrized version of the weighted projective model. The quantum modifications induce an emergent structure that contrasts with the standard cases of homogeneous spaces, where corrections are proportional to the metric tensor and only governed by the renormalization of the coupling constant. In addition, due to the existence of two equivalent formulations of the present model, we identify an apparent mismatch in the predicted infrared behavior. This issue is resolved by numerically studying the corresponding renormalization group flow with the exact twisted superpotential. The results show that both models flow to a non-trivial conifold. Then we focus on the Lie-algebraically deformed sigma models. This class of models is derived by deforming the principal chiral models along specific directions associated with a subset of su(2) Lie algebra generators. We examine both the supersymmetric and non-supersymmetric versions of this model. Additionally, we reveal the quasi-exact solvability of the associated quantum mechanics under a particular compactification scheme. Lastly, we address the renormalon ambiguity in the supersymmetric O(N) model. We compute all-order contributions in the coupling constants up to the next-to-leading order in large N. Our results demonstrate that the ambiguities arising from coefficient functions cancel with those from condensates. This cancellation not only links the perturbative data with the non-perturbative data but also confirms the consistency of the theory.enSigma modelsSupersymmetryThesis or Dissertation