Detect a sign invisible to local anomaly polynomials

Consider an ordinary SU(2) gauge theory on spin four-manifolds, with left-handed Weyl fermions in specified representations. The fundamental doublet is pseudoreal, and its symmetrized cubic trace vanishes. Therefore its perturbative four-dimensional gauge anomaly is zero. Nevertheless, gauge transformations on compactified spacetime $S^4$ have disconnected classes classified by $\pi_4(SU(2))=\mathbb Z_2$. The nontrivial class cannot be reached by exponentiating a continuously shrinking sequence of infinitesimal transformations.

Transporting the chiral fermion functional integral around the associated loop can reverse its sign. For one doublet, the sign is minus one. The gauge-related configurations should describe the same physical situation, so an uncanceled sign prevents a gauge-invariant quantum definition. The issue is often described using the Pfaffian or the sign of a square root of a Dirac determinant, with the precise language depending on the fermion formulation. It is not ordinary symmetry breaking or an arbitrary phase convention that can be chosen independently at every background.

An interpolation between a gauge field and its large-gauge transform gives a family of four-dimensional Dirac operators. The parity of eigenvalue-pair crossings determines the sign change. Equivalently, a five-dimensional mapping-torus construction expresses it through a mod-two index. This is a different spectral invariant from the integer four-dimensional index controlling a local axial selection rule. The extra dimension organizes the family and its holonomy; it is not an additional physical spacetime dimension required in the original four-dimensional model.

For an isospin-j representation, the ordinary obstruction is controlled by the parity of twice its quadratic index. Normalize the doublet index to one half. Then $2T(j)=2j(j+1)(2j+1)/3$ is an integer, and contributions add modulo two. Representations of isospin one half, five halves, and so on can contribute an odd value. Pseudoreality alone therefore does not guarantee global consistency even though it removes the perturbative cubic trace.

A quark doublet with three color components contributes three weak doublets to this SU(2) test. A weak singlet contributes none, regardless of its Abelian charge. This explains why a local hypercharge sum and a global weak-doublet parity test are genuinely different calculations. Replacing SU(2) by a quotient group, admitting different bundles, or changing the allowed spacetime structure can alter the complete anomaly problem. The Lie algebra and perturbative Feynman rules alone do not specify those global choices.

The word global here refers to large transformations and topology, not necessarily to a global rather than gauged symmetry. A global anomaly of a dynamical gauge redundancy must cancel in a stand-alone theory. A corresponding obstruction for an external background global symmetry can instead be meaningful anomaly data constraining the infrared. A higher-dimensional inflow system can supply compensating variation for a boundary theory, but that changes the complete physical system and does not magically make the isolated anomalous four-dimensional gauge theory consistent.