Unified Structural Embedding of Orbifold Sigma Models

This paper introduces a unified structural embedding framework for orbifold sigma models that systematically incorporates twisted sectors, singularities, and smooth regions into a single algebraic object. Traditional approaches to orbifold theories often treat sectors separately, requiring ad hoc regularizations near singularities and failing to capture inter-sector interactions under renormalization group flow. The framework presented herein resolves these limitations through the construction of a unified orbifold algebra $\mathcal{A}(X/G)$ that decomposes canonically into idempotent-projected components corresponding to conjugacy classes of the finite group $G$ acting on the target space $X$. A generalized Laplacian operator is defined on this algebra, enabling the construction of a one-parameter family of renormalization group compatible endomorphisms that preserve the sector structure while implementing scale transformations. The formalism is shown to recover conventional sigma model results in the smooth limit where $G$ approaches the trivial group, with the internal renormalization group derivation reducing to the standard one-loop beta function proportional to the Ricci tensor. Several examples demonstrate the framework's utility, including explicit calculations for the $\mathbb{C}/\mathbb{Z}_2$ orbifold that exhibit the decomposition into untwisted and twisted field contributions. This approach provides a mathematically rigorous foundation for quantum field theories on singular spaces while maintaining compatibility with established results for smooth manifolds.