Determinant Lines, Global Obstructions, and Orientations

For a family of chiral or real fermion operators, the determinant is naturally a section of a line bundle over gauge-field parameter space, not globally a complex number. Local anomaly cancellation controls the curvature of this line. A consistent fermion measure additionally requires the relevant determinant or Pfaffian line to be trivialized with compatible gauge equivariance; nontrivial holonomy around a large-gauge loop is a global obstruction.

Required background. Principal-bundle sectors and large transformations supplies loops in gauge-field orbit space. Moduli stacks and derived geometry supplies families with automorphisms. Trace ideals and Fredholm determinants supplies finite-rank regularization and Fredholm index. Helpful background. Global and torsion anomalies supplies anomaly holonomy. Dirac index and zero modes supplies the families-index input.

The determinant line of a Fredholm family

Let $D_b:H_b^+\to H_b^-$ be a smooth family of complex Fredholm operators parametrized by $b\in B$. Fix the convention

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

Even when $\dim\ker D_b$ jumps, finite spectral cutoffs patch these one-dimensional spaces into a smooth line bundle $\operatorname{Det}D\to B$. The regularized determinant is a canonical section; it vanishes where zero modes occur. A choice of basis at each $b$ is not enough, because the choices must glue smoothly and respect gauge transformations.

Freed gives the patching construction, Quillen metric, and natural connection for a family of Dirac operators in Freed 1987, §1, pp. 3–10, especially Theorem 1.12. Quillen’s original construction for Cauchy–Riemann operators is Quillen 1985, pp. 31–34. Some physics conventions use the dual line because the Grassmann integral transforms contragrediently; all curvature and holonomy statements must be dualized consistently if that convention is chosen.

The determinant-line connection has two distinct invariants. Its curvature is a local differential form on $B$, computed by the degree-two part of a families index density. Its holonomy around a loop $\gamma:S^1\to B$ is global and is expressed by an eta invariant on the associated mapping torus. Zero curvature makes the connection flat; it does not force a flat line bundle to have trivial holonomy.

Pfaffian lines and orientation

For a real or skew-adjoint fermionic family, the functional integral is a Pfaffian. Under the appropriate reality and dimension hypotheses there is a line $\operatorname{Pf}(D)$ with

$$\operatorname{Pf}(D)^{\otimes2}\cong\operatorname{Det}(D), \qquad \operatorname{Pfaff}(D)^{\otimes2}=\det D.$$

Freed states the square-root construction, its hypotheses, curvature, and holonomy in Freed 1987, §3, Theorem 3.1 and equation (3.3), pp. 20–23. In a real formulation, orienting this line chooses the sign of the fermion measure. Along a loop, an odd mod-two spectral flow reverses that orientation. Squaring removes the sign, which is why inspecting only the determinant can miss a Pfaffian obstruction.

This statement has boundaries. A Pfaffian square root is not automatic for every Fredholm family; the operator symmetry, dimension, and real structure are hypotheses. A vanishing ordinary index does not by itself orient the line, and a local polynomial calculation does not determine torsion holonomy.

A large-gauge loop and the SU(2) sign

The first application is the consistency test in Standard Model Anomaly Cancellation. Consider a four-dimensional left-handed fermion in the $SU(2)$ doublet. A path of gauge fields $A_t$, $0\leq t\leq1$, whose endpoints differ by a gauge transformation in the nontrivial class

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

becomes a loop in orbit space. The family of chiral Dirac operators along this loop has odd mod-two spectral flow for one doublet, so the Pfaffian changes sign. Witten’s argument and its odd-doublet inconsistency are given in Witten 1982, pp. 324–328.

For $N$ identical doublets, the holonomy is $(-1)^N$. An even number cancels this particular global anomaly. The perturbative cubic $SU(2)$ anomaly vanishes because the fundamental representation is pseudoreal, so the example proves the central logical point: cancellation of the infinitesimal anomaly polynomial does not imply trivial Pfaffian holonomy.

The independent check is multiplicativity. Tensoring $N$ Pfaffian lines multiplies their loop holonomies, giving $(-1)^N$. Two anomalous doublets therefore have trivial combined sign even though each factor is nontrivial. This check does not establish cancellation of other gauge, mixed, gravitational, or newer cobordism-valued anomalies; each requires its own line or invertible-field-theory invariant.

Failure test

Cancel every local anomaly coefficient and then declare the fermion measure globally defined. Transport its sign around the nontrivial $SU(2)$ gauge loop. For one doublet it returns with a minus sign, contradicting single-valuedness. The omitted hypothesis was trivial holonomy, or equivalently an appropriate equivariant trivialization of the Pfaffian line.

The converse boundary matters too: nontrivial connection holonomy signals an obstruction to the chosen standalone theory, but it may be canceled by adding matter or a precisely matched bulk inflow theory. The line data determine what must cancel; they do not prescribe a unique completion.

Exercises

Assume one $SU(2)$ Weyl doublet has Pfaffian holonomy $-1$ around the generator of $\pi_4(SU(2))$. Determine the holonomy for $N$ doublets and the consistency condition for this anomaly.

Solution

The total Pfaffian is the tensor product of the individual Pfaffians, so holonomies multiply: $\operatorname{hol}_N=(-1)^N$. This particular obstruction vanishes exactly for even $N$. The result is only a test of Witten’s original $SU(2)$ global anomaly; other anomaly conditions remain independent.

References

• Freed, Daniel S. “On Determinant Line Bundles.” In Mathematical Aspects of String Theory, edited by Shing-Tung Yau, 189–238. Singapore: World Scientific, 1987. Open PDF.
• Quillen, Daniel. “Determinants of Cauchy–Riemann Operators over a Riemann Surface.” Functional Analysis and Its Applications 19 (1985): 31–34. DOI.
• Witten, Edward. “An SU(2) Anomaly.” Physics Letters B 117 (1982): 324–328. DOI.