Chiral Gauge Perturbative Construction: Open Obligations

For a chiral gauge theory, cancellation of the local anomaly is the entrance condition to perturbative quantization, not the endpoint of construction. Dimensional renormalization can restore local Ward identities order by order, while a Ginsparg–Wilson lattice formulation can define an exactly gauge-invariant fermion measure in certain abelian sectors. Neither statement alone supplies a positive continuum Hilbert space. A concrete anomaly-free $U(1)$ multiplet makes the separation testable.

Required background. Anomaly cancellation versus Hilbert-space existence separates algebraic and analytic obligations; perturbative gauge-QFT constructions fix formal-series scope; and Standard-Model anomaly cancellation provides the charge-trace tests.

Helpful background. Chiral lattice gauge theories explain Ginsparg–Wilson fermions and measure integrability, while chiral gauge-theory anomaly constraints give further representation-theoretic examples.

A genuinely chiral anomaly-free U(1) multiplet

Take five left-handed Weyl fermions of integer charges

$$(q_1,q_2,q_3,q_4,q_5)=(-9,-5,-1,7,8).$$

There is no opposite-charge pair, so the spectrum is genuinely chiral. Direct arithmetic gives

$$\sum_i q_i=-9-5-1+7+8=0,$$

and

$$\sum_i q_i^3 =-729-125-1+343+512=0.$$

The first equality cancels the mixed gauge–gravitational anomaly, and the second cancels the local cubic $U(1)^3$ anomaly. This is the exact first application returned to Standard-Model anomaly cancellation: the same trace conditions are evaluated, but here the toy spectrum is small enough to compare two proposed construction routes explicitly.

In a dimensional-renormalization scheme for chiral fermions, $\gamma_5$ and evanescent operators require a prescription. At a fixed loop order, the broken Slavnov–Taylor identity is a local insertion. If its Wess–Zumino class vanishes, finite local counterterms can restore the identity at that order. Repeating the cohomological step defines a formal perturbative theory. The conclusion is local and coefficientwise; the argument gives no estimate on the loop expansion and no measure at finite coupling.

The Ginsparg–Wilson route and its domain

On a Euclidean lattice of spacing $a$, choose a local, gauge-covariant Dirac operator satisfying

$$\gamma_5D+D\gamma_5=aD\gamma_5D.$$

Then

$$\widehat\gamma_5=\gamma_5(1-aD), \qquad \widehat\gamma_5^2=1, \qquad \widehat P_- =\frac{1-\widehat\gamma_5}{2}$$

defines a gauge-field-dependent chiral subspace. The dependence is the central issue: choosing a basis of $\operatorname{im}\widehat P_-$ defines the phase of the fermion measure, and infinitesimal changes produce a connection on the measure bundle. Gauge invariance requires its curvature to match the local anomaly and its holonomy to be trivial on admissible loops. The Ginsparg–Wilson relation is the algebraic input that connects the projector, index, and anomaly Ginsparg and Wilson 1982, pp. 2649–2657.

Lüscher proves an exact gauge-invariant construction for anomaly-free abelian chiral multiplets on a finite periodic lattice with an admissibility condition, locality assumptions on $D$, and a reconstructed measure current. His theorem includes all topological sectors only with an additional global condition: for every odd absolute charge, the number of fermions carrying that absolute charge must be even Lüscher 1999, §§5–7, pp. 310–329. Our charge set has one fermion of each odd magnitude $1$, $5$, $7$, and $9$, so it fails that all-sector condition. It is therefore a valid vacuum-sector illustration of local anomaly cancellation, not an example to which the full all-sector conclusion may be applied.

This failure is instructive rather than cosmetic. The sums $\sum q_i$ and $\sum q_i^3$ see the local curvature of the measure connection. The parity condition detects a global integrability obstruction over the finite-lattice gauge-field space. A perturbative calculation near the trivial connection cannot certify that global trivialization.

Matching results and unresolved obligations

Within the vacuum sector and at weak field, both routes should reproduce the same local Ward identities and anomaly polynomial. One can check the continuum expansion of the lattice measure curvature against the dimensional-renormalization descent representative, allowing local counterterm changes. Agreement is a regulator-matching check; it does not show that the lattice theory converges to the formal continuum series or that either defines the desired nonperturbative theory.

The remaining obligations are concrete. One needs a gauge-invariant measure on every intended topological sector; locality uniform in the cutoff; existence and universality of the continuum limit; control of the infinite-volume limit and massless modes; reconstruction of a positive physical Hilbert space; unitary time evolution; and nontrivial gauge-invariant observables. For nonabelian chiral theories, the global measure problem is harder and the abelian theorem cannot simply be imported.

An independent check has three parts: recompute both charge sums; evaluate the odd-charge multiplicities before claiming an all-sector lattice theorem; and inspect whether any proposed continuum statement supplies cutoff-uniform estimates. The adversarial failure is to combine triangle cancellation with a formal BRST charge and announce a nonperturbative physical Hilbert space. The missing regulator-removal and positivity theorems are not consequences of either premise.

Exercises

Verify that the charge set is chiral and locally anomaly-free.

Solution

No charge is the negative of another, so no pair forms a gauge-invariant Dirac mass. The displayed linear and cubic sums both vanish, cancelling the mixed gravitational and cubic abelian local anomalies.

Why does this multiplet not satisfy Lüscher’s all-sector parity condition?

Solution

For each odd magnitude $1$, $5$, $7$, and $9$, exactly one Weyl fermion occurs. Each multiplicity is odd, whereas the theorem requires an even multiplicity for every odd absolute charge. Local anomaly cancellation therefore does not remove this particular global obstruction.

References

• Ginsparg, Paul H., and Kenneth G. Wilson. “A Remnant of Chiral Symmetry on the Lattice.” Physical Review D 25 (1982): 2649–2657. DOI.
• Hollands, Stefan. “Renormalized Quantum Yang–Mills Fields in Curved Spacetime.” Reviews in Mathematical Physics 20 (2008): 1033–1172. DOI; Open PDF.
• Lüscher, Martin. “Abelian Chiral Gauge Theories on the Lattice with Exact Gauge Invariance.” Nuclear Physics B 549 (1999): 295–334. DOI; Open PDF.