Unified Structural Embedding of Orbifold Sigma Models

This study introduces a new unified structural framework for orbifold sigma models that incorporates twisted sectors, singularities, and smooth regions into a single algebraic object. Traditional approaches to orbifold theories often treat such sectors separately, requiring ad hoc regularizations near singularities and failing at capturing inter-sector interactions under renormalization group flow. Therefore, the scope of this study aims at resolving these limitations through the construction of a unified orbifold algebra $\mathcal{A}(X/G)$ that decomposes into idempotent-projected components corresponding to conjugacy classes of the finite group $G$ acting on the target space $X$. 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. Examples demonstrate the applicability, including explicit calculations for the $\mathbb{C}/\mathbb{Z}_2$ orbifold that exhibit the decomposition into untwisted and twisted field contributions.