Topology-Dependent Phase Classification of Effective Potentials in Einstein–Cartan + Nieh–Yan Minisuperspace

We study Einstein–Cartan (EC) gravity supplemented with the Nieh–Yan (NY) term in a Euclidean–signature minisuperspace framework, and classify the resulting effective potential into distinct “phases. ” Based on the presence or absence of local minima and barriers in the effective potential, we define three types: (I) metastable well with barrier, (II) barrier-free rolling, and (III) unstable/boundary-attached configurations. While the Nieh–Yan density (a 4-form) can be written as an exact derivative, it is geometrically defined through the coframe and torsion. To facilitate meaningful comparison, we evaluate topology dependence under a unified ansatz. We adopt spatial sections that admit left-invariant coframes, enabling systematic description within a common minisuperspace framework. As concrete test beds, we consider: (i) S³, (ii) T³, and (iii) Nil³. For the NY term, we focus on the complete form (FULL) as the primary object, while also examining TT (torsion-torsion component only) and REE (the remaining component) as diagnostic comparisons to disentangle the contributions within FULL. Our investigation addresses two main questions: (1) How does the phase (Type I/II/III) of the effective potential depend on topology? (2) Can we identify, throughnumerical scanning, critical conditions corresponding to well formation/disappearance and barrier collapse, and organize their geometric dependence? Our scanning results suggest the following: The phase structure of the effective potential varies systematically with topology. Within the NY term, some contributions exhibit relative insensitivity to topology, while others appear selectively depending on geometric conditions.