Global Anomalies, Determinant Lines, and Eta Invariants

A global fermion anomaly is the failure to choose the phase of a determinant or Pfaffian consistently over the full parameter space of gauge fields and metrics. Local anomaly polynomials measure the curvature of the corresponding line bundle; eta invariants and mod-two indices measure its holonomy around noncontractible loops. Flatness therefore does not imply triviality. The result depends on dimension, representation, spin or pin structure, and the exact class of gauge transformations.

Required background. Determinant lines and global obstructions supply the family-index geometry; the quantum master equation and anomaly classes supply the local obstruction; and global and torsion anomalies supply the physical large-transformation test.

Helpful background. Dirac index theorems explain spectral flow; gravitational, mixed, and orientation anomalies extend the symmetry types; and spin, gauge, and gravitational anomaly response fixes curved-background currents.

Determinant lines over parameter space

Let $\mathcal B$ be a specified space of background connections and metrics modulo gauge transformations connected to the identity, and let $D_b$ be a chiral Dirac operator for $b\in\mathcal B$. With one common convention, its determinant line is

$$\operatorname{Det}D_b= \det(\ker D_b)^*\otimes\det(\operatorname{coker}D_b).$$

The fermion partition function is not initially a complex number; it is a section of the line bundle $\operatorname{Det}D\to\mathcal B$. Real or pseudoreal fermions can instead produce a Pfaffian line. A quantum theory requires a symmetry-compatible trivialization, including a rule through zero modes. A local anomaly is the curvature of the natural connection on this line. If that curvature vanishes, the connection is flat, but a flat line bundle can retain nontrivial holonomy.

For a loop $\gamma$ in $\mathcal B$, form its mapping torus $Y_\gamma$ by taking $M\times[0,1]$ and identifying the two ends using the terminal gauge or diffeomorphism transformation. The self-adjoint Dirac operator on $Y_\gamma$ has reduced eta invariant

$$\xi(D_{Y_\gamma})= \frac{\eta(D_{Y_\gamma})+\dim\ker D_{Y_\gamma}}{2}.$$

With convention-dependent complex conjugation and sign, the determinant-line holonomy is the adiabatic limit of $\exp(2\pi i\xi)$. Dai and Freed prove the determinant-line interpretation, covariant variation, gluing, and holonomy formula in Dai and Freed 1994, Theorems 1.9 and 5.3, §§1 and 5. The kernel term is essential: $\eta$ alone jumps at spectral crossings, whereas the exponentiated reduced invariant varies smoothly.

This construction is directional. A nontrivial holonomy obstructs a globally defined fermion phase. Trivial holonomy for one loop says nothing about other components of the gauge group, other bordism classes, or a different spin structure.

The SU(2) sign anomaly

In four dimensions, a left-handed Weyl fermion in the fundamental representation of $SU(2)$ has no perturbative cubic gauge anomaly: the representation is pseudoreal and the symmetric cubic invariant vanishes. Nevertheless

$$\pi_4(SU(2))\cong\mathbb Z_2 .$$

Let $g$ represent the nontrivial class and interpolate from a connection $A$ to $A^g$. Along this path, an odd number of eigenvalue pairs crosses zero. Equivalently, the five-dimensional mapping-torus Dirac operator has nonzero mod-two index. The four-dimensional Pfaffian changes by

$$\operatorname{Pf}(D_{A^g}) =(-1)^{\operatorname{ind}_2D_{Y_g}} \operatorname{Pf}(D_A).$$

For a single doublet the sign is $-1$, so no gauge-invariant choice of Pfaffian exists. Two doublets give the square of the sign and pass this particular test. Witten derives the spectral-flow and five-dimensional interpretation in Witten 1982, pp. 324–328.

This is the exact first application returned to Standard-Model anomaly cancellation: analyze one $SU(2)$ Weyl doublet with the mod-two mapping-torus index and contrast its global sign with the vanishing perturbative cubic anomaly. The Standard Model has an even number of $SU(2)$ doublets per generation when color multiplicity is counted, but that arithmetic is only one item in the full cancellation analysis.

Eta phases, bordism, and scope

For complex fermions, the exponentiated eta invariant can yield a general $U(1)$ phase rather than a sign. When the local index density vanishes, the phase may descend to a bordism invariant of manifolds with the required tangential and gauge structure. Modern anomaly field theories package this as an invertible theory in one higher dimension. This packaging clarifies gluing and inflow, but it does not turn every anomaly question into an ordinary cohomology calculation. Torsion information can be invisible to de Rham characteristic forms.

An independent check deforms the metric or connection without closing the spectral gap. The eta phase changes according to the local index density; if that density cancels, the residual phase is constant on the deformation class. A second check composes two loops and verifies multiplication of holonomies. These checks test the line-bundle structure, not the existence of the interacting gauge theory.

The adversarial failure is to declare a theory anomaly-free after finding $I_6=0$. The single $SU(2)$ doublet passes that local test and fails the mapping-torus test. The strongest valid conclusion from $I_6=0$ is absence of the corresponding perturbative local anomaly polynomial. Global consistency remains open until determinant/Pfaffian holonomy is trivialized on every relevant loop or bordism class.

Exercises

Explain why zero curvature does not force zero holonomy.

Solution

A flat connection has locally constant parallel transport, but on a space with nontrivial fundamental group it defines a representation $\pi_1(\mathcal B)\to U(1)$. That representation can be nontrivial even though the curvature two-form vanishes everywhere.

Compute the $SU(2)$ sign for $N$ identical Weyl doublets.

Solution

Each doublet contributes $-1$ around the nontrivial loop, so the combined Pfaffian transforms by $(-1)^N$. This particular global anomaly cancels exactly when $N$ is even.

References

• Dai, Xianzhe, and Daniel S. Freed. “Eta-Invariants and Determinant Lines.” Journal of Mathematical Physics 35 (1994): 5155–5194. DOI; Open PDF.
• Witten, Edward. “An SU(2) Anomaly.” Physics Letters B 117 (1982): 324–328. DOI.
• Witten, Edward. “Fermion Path Integrals and Topological Phases.” Reviews of Modern Physics 88 (2016): 035001. DOI; Open PDF.