Chiral Gauge Theories and Standard Model Construction

Perturbative anomaly cancellation is necessary for the Standard Model, but it is not a nonperturbative construction. A full result must specify the global gauge group and Weyl representation, remove local and global anomalies, regulate the chiral fermion measure without unwanted mirrors, prove locality and exact gauge invariance, take infinite-volume and continuum limits, recover a positive physical Hilbert space, and identify the intended gauge-invariant observables. Each item is independent enough to fail while the familiar triangle sums still vanish.

Required background. The quantum master equation and anomaly obstructions supplies the perturbative cohomology. Perturbative gauge-QFT scope fixes formal-series conclusions. What is an anomaly? separates local and global obstructions, and existence, uniqueness, and equivalence fixes the construction target. Helpful background. Ginsparg–Wilson symmetry and the lattice index, chiral gauge theories on the lattice, and the Nielsen–Ninomiya obstruction supply the regulator problem.

One Standard Model generation

Write every fermion as a left-handed Weyl field. One generation, without a sterile neutrino, is

$$Q:(\mathbf3,\mathbf2)_{1/6},\quad u^c:(\overline{\mathbf3},\mathbf1)_{-2/3},\quad d^c:(\overline{\mathbf3},\mathbf1)_{1/3},\quad L:(\mathbf1,\mathbf2)_{-1/2},\quad e^c:(\mathbf1,\mathbf1)_1.$$

The mixed non-Abelian and Abelian anomaly coefficients cancel:

$$\begin{aligned} SU(3)^2U(1):&\quad 2\frac16\frac12-\frac23\frac12+\frac13\frac12=0,\\ SU(2)^2U(1):&\quad 3\frac16\frac12-\frac12\frac12=0. \end{aligned}$$

Including multiplicities, $\sum Y=0$ and $\sum Y^3=0$. There are four $SU(2)$ doublets—three colored copies of $Q$ and one $L$—so the original mod-two $SU(2)$ global anomaly is absent. Witten’s theorem shows why this last parity check is genuinely global and is not encoded by a perturbative triangle Witten 1982, pp. 324–328.

These arithmetic checks are the first application at Standard Model anomaly cancellation. They establish consistency of the representation against the displayed anomalies. The precise global form, often a quotient of $SU(3)\times SU(2)\times U(1)$, and the allowed bundle sectors can introduce further global questions. More importantly, cancellation gives no regulator-removal estimate.

Why a chiral regulator is difficult

The Nielsen–Ninomiya theorem shows that under its translation-invariant, local, Hermitian lattice hypotheses, fermion zeros come with compensating chirality Nielsen and Ninomiya 1981, Theorem 1, pp. 20–28. Successful approaches must change an assumption or encode chirality through a higher-dimensional/domain-wall, overlap, Ginsparg–Wilson, or interacting-mirror construction. Evading the no-go theorem is an entrance step, not proof of the target continuum theory.

For a Ginsparg–Wilson operator,

$$\gamma_5D+D\gamma_5=aD\gamma_5D, \qquad \widehat P_-=\frac12\left[1-\gamma_5(1-aD)\right],$$

the chiral subspace depends on the gauge field. A basis choice for that subspace determines a fermion-measure phase. Gauge invariance requires a globally integrable, local measure current whose curvature reproduces the local anomaly and whose holonomies vanish on admissible loops. Lüscher constructs exactly gauge-invariant anomaly-free abelian chiral lattice theories under admissibility, locality, integrability, and sector conditions Lüscher 1999, §§5–7, pp. 310–329. This is a theorem for the stated abelian setting, not a construction of the full non-Abelian Standard Model.

The 2026 symmetry-disentangler result

Thorngren, Preskill, and Fidkowski introduce constant-depth symmetry disentanglers that turn certain not-on-site symmetries into on-site, gaugeable ones. They give exactly solvable Hamiltonians for a broad anomaly-free class of $1+1$-dimensional $U(1)$ chiral theories and construct relevant anomaly-canceling symmetry data in $3+1$ dimensions. For one Standard Model generation with a sterile neutrino, they explain how the hypercharge assignments fit their grouped charge construction 2026, §1.1, pp. 6–7.

The boundary is explicit in the source. A complete local Hamiltonian realization of the full Standard Model is not obtained. In the $3+1$-dimensional hypercharge application, an exactly solvable Hamiltonian producing the four Weyl modes is still missing; the proposed trivial gapped interface is sketched and left for verification Thorngren, Preskill, and Fidkowski 2026, §5, pp. 27–28. The result is therefore a substantive construction route and exact lower-dimensional theorem, not closure of the four-dimensional Standard Model problem.

The remaining construction chain

A complete claim must still prove, for the chosen regulator:

1. the intended chiral spectrum and decoupling of every mirror mode;
2. exact gauge invariance over all included topological sectors;
3. locality bounds uniform as $a\to0$;
4. reflection positivity or another route to a positive physical Hilbert space;
5. tuning and convergence of gauge and matter correlations in the continuum and volume limits;
6. recovery of the full non-Abelian interactions, Yukawa sector, Higgs sector, and declared global gauge form.

Perturbative BRST restoration can establish Ward identities coefficient by coefficient when the local anomaly class vanishes. It gives a formal series, not convergence at physical coupling. A Hamiltonian lattice model at fixed spacing is nonperturbative in a different sense, but it still needs the continuum theorem.

Failure test: anomaly-free plus vectorlike

Take a vectorlike lattice regulator, note that Standard Model anomaly sums vanish, and declare that the unwanted mirror sector must decouple. Nothing in the arithmetic supplies a symmetric mass gap for the mirrors, proves that no topological order remains, or identifies the infrared chiral theory. The strongest surviving statement is an anomaly-free target representation embedded in a regulated vectorlike system.

An independent check is to compute local anomaly sums, the $SU(2)$ mod-two count, and any bordism/global obstruction before inspecting the regulator. Passing all of them removes known consistency obstructions; it does not prove existence.

Exercises

Verify the cubic hypercharge cancellation for the displayed generation.

Solution

Including color and weak multiplicities gives $6(1/6)^3+3(-2/3)^3+3(1/3)^3+2(-1/2)^3+1^3$. With denominator $36$ this is $1-32+4-9+36=0$. The multiplicities are part of the representation and cannot be dropped.

References

• Lüscher, Martin. “Abelian Chiral Gauge Theories on the Lattice with Exact Gauge Invariance.” Nuclear Physics B 549 (1999): 295–334. DOI; Open PDF.
• Nielsen, Holger Bech, and Masao Ninomiya. “Absence of Neutrinos on a Lattice: I. Proof by Homotopy Theory.” Nuclear Physics B 185 (1981): 20–40. DOI.
• Thorngren, Ryan, John Preskill, and Łukasz Fidkowski. “Chiral Lattice Gauge Theories from Symmetry Disentanglers.” arXiv:2601.04304 (2026). arXiv.
• Witten, Edward. “An $SU(2)$ Anomaly.” Physics Letters B 117 (1982): 324–328. DOI.