A Clifford-index approach connecting boundary Dirac fermions to sigma-model topological terms in the periodic table of topological insulators and superconductors

Abstract

The periodic table of topological insulators and superconductors emerges from two distinct approaches: boundary Anderson localization and momentum-space K-theory. Connecting these approaches physically requires deriving the boundary sigma-model topological terms from the underlying fermion theories. We develop an index-theoretic framework for minimal massless boundary Dirac models in the eight real Altland--Zirnbauer classes. Its central ingredients are an elliptic cylinder operator interpolating between two sigma-field textures, a quotient index defined by the cylinder's graded Clifford symmetry algebra, and an index theorem identifying its protected zero-mode index with the relative homotopy class of the endpoint textures. To establish this correspondence, we deform the cylinder texture while preserving the kernel's index class and localize the obstruction to a Clifford defect. An exact oscillator-kernel calculation fixes the generator normalization. A field-independent congruence and a common regularization provide a real antisymmetric Pfaffian representative whose sign transport is related to the cylinder mod-two index by Witten's spectral-flow argument. This yields the nontrivial $\mathbb{Z}_2$ theta weight for one minimal physical module. Combining complex-parent anomalies with physical Clifford multiplicities and real-target periods also gives the real-class Wess--Zumino terms with unit spherical level. The framework thus provides an operator-level connection between boundary Dirac theories and the sigma-model topological terms associated with the periodic table.