Read the index without overcounting zero modes

Work in Euclidean signature with a smooth gauge connection and a massless Dirac operator on a spin manifold. On a compact manifold without boundary, the chiral operator is elliptic and has finite-dimensional kernel and cokernel. Choose chirality orientation so $\nu=n_+-n_-$. The index theorem expresses this integer as the degree-four integral of $\widehat A(T)\operatorname{ch}_R(F)$. The tangent-bundle factor matters on curved backgrounds; keeping only the gauge density without checking geometry can miss a gravitational contribution.

For smooth finite-action SU(N) backgrounds on Euclidean space, not necessarily self-dual, use the standard compactified or square-integrable index setting with appropriate asymptotic behavior and the established fundamental trace normalization. With the gravitational contribution absent, the index in representation R is $2T(R)k$. The fundamental has $T=1/2$, whereas the adjoint has $T=N$. These representation factors count independent chiral modes and are unrelated to simply multiplying by the dimension of R. A different trace convention must be translated before inserting k.

Anticommutation with chirality maps a nonzero Dirac eigenmode into a partner. Zero modes can instead be chosen with definite chirality, so an imbalance survives in the regulated trace. Under smooth deformations within the same topological and domain class, the difference remains fixed. Additional opposite-chirality zero modes may appear or disappear together without changing it. Consequently, a nonzero index guarantees at least its absolute value in total zero modes, but does not prove equality unless extra properties of the background exclude paired zero modes.

This distinction is visible in a small real mass added to one complex Dirac fermion. For an anti-Hermitian massless Dirac operator, nonzero eigenvalues occur as $\pm i\lambda$. Their determinant factors combine into positive $m^2+\lambda^2$, while every actual zero mode contributes a factor m. The leading small-mass power is therefore the total kernel dimension, not the absolute index when extra zero pairs exist. Majorana or other reality constraints change determinant or Pfaffian counting and are not being silently included in this complex-Dirac formula.

A massless vacuum determinant vanishes in a sector with unsaturated fermion zero modes. Correlation functions can remain nonzero when their insertions supply the missing Grassmann variables. This produces the chiral selection rules represented schematically by an instanton-induced 't Hooft vertex. The index determines its net axial charge violation, while a semiclassical amplitude additionally depends on the instanton measure, nonzero-mode determinants, and the validity of the coupling expansion. Topology alone does not give a universal tunneling rate.

On a manifold with boundary, integration by parts and spectral pairing depend on the chosen boundary condition. Atiyah–Patodi–Singer boundary problems include a spectral asymmetry contribution, often described by an eta invariant, together with zero-mode terms. On noncompact space, normalizability and asymptotic conditions likewise matter. Finally, the mod-two invariant of a five-dimensional family used for the SU(2) global anomaly is a different object from this four-dimensional integer index. Their shared spectral language does not make them interchangeable.