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Local anomaly cancellation for a conjugate chiral pair

Physical scope

A conditional one-loop result for infinitesimal background transformations of the stated flat-space SU(3)×U(1) global symmetry in one common large-mass Pauli–Villars subtraction convention. It proves only vanishing of the perturbative local class for the conjugate pair; it does not test global anomalies, establish a dynamical gauge theory, or construct a continuum QFT.

Statement

Use the selected flat-space anomaly-matching model on oriented $M = R^{1, 3}$ with metric $η = diag (-, +, +, +)$ and natural units $c = ℏ = 1$ . Its nondynamical global-symmetry backgrounds are a smooth compactly supported anti-Hermitian $S U (3)$ connection $A$ and a smooth compactly supported real $U (1)_{V}$ connection $B$ on the trivial bundles. The finite field spectrum consists of two massless positive-chirality Weyl species, one in $E = (3, + 1)$ and one in its dual $E^{*} = (\overset{ˉ}{3}, - 1)$ . The background is not integrated over. Use one common large-mass Pauli–Villars subtraction scheme and the consistent effective-action convention fixed in the model data.

Adopt the source-backed one-loop input that each positive-chirality species in this regulator has consistent local variation given by descent from $I_{6} (R) = 2 π [{tr}_{R} \exp (- i F_{R} / (2 π))]_{6}$ , with the same linear descent for every species: $I_{6} = d I_{5}^{(0)}$ , $s I_{5}^{(0)} + d_{H} I_{4}^{(1)} = 0$ , and $s I_{4}^{(1)} + d_{H} I_{3}^{(2)} = 0$ , and $δ_{ϵ, α} Γ = \sum_{R} \int_{M} E_{ϵ, α}^{4} (I_{4}^{(1)} (R))$ . Here $s$ is the component BRST differential extended with $s (d x^{μ} \land a) = - d x^{μ} \land s (a)$ . The linear map $E_{ϵ, α}^{4}$ writes each ghost-number-one form with its four $d x$ factors on the left, then replaces its single $s u (3)$ or real $U (1)_{V}$ ghost jet by the corresponding ordinary parameter jet; it is not an algebra substitution on products of ghosts. The local anomaly class is the class of $I_{4}^{(1)}$ in $H^{1, 4} (s ∣ d_{H})$ : local ghost-number-one four-forms satisfying the second descent equation, modulo $s$ -variations of admitted local counterterms and total derivatives.

Then the total perturbative local anomaly class of this conjugate pair vanishes. This conclusion concerns only the identity component of the background-transformation group in the adopted one-loop convention; it asserts no absence of large-transformation or torsion anomalies and no consistency result for a dynamical gauge redundancy.

Statement references

Bilal, §§5.2, 7.1–7.2, 9.4, and 10.4

Alvarez, §§2.1–2.4

Facts & Assumptions

Given: The selected flat-space background model, its two positive-chirality species $E = (3, + 1)$ and $E^{*} = (\overset{ˉ}{3}, - 1)$ , compactly supported infinitesimal transformations, and the common consistent descent convention in the Statement.

[F1]

Flat-space chiral anomaly-matching data and regulator convention: the background group is the fixed global form $S U (3) \times U (1)_{V}$ ; the connections are anti-Hermitian in the stated representations; the common large-mass Pauli–Villars subtraction assigns, after the regulator masses are sent to infinity, each positive-chirality species the consistent descent of its degree-six polynomial and adds species linearly.

[F2]

Local perturbative ’t Hooft anomaly modulo counterterms: the local anomaly of a nondynamical global-symmetry background is the class in $H^{1, 4} (s ∣ d_{H})$ modulo the model’s admitted local counterterms and total derivatives; the definition makes no global-anomaly claim.

[F3]

A conjugate representation reverses the cubic anomaly polynomial: for the dual background connection, $I_{6} (E^{*}) = - I_{6} (E)$ and the canonical Chern–Simons transgressions also cancel.

[F4]

A local counterterm changes only the representative of a local anomaly class: adding an admitted local counterterm changes a local anomaly representative only by an $s$ -coboundary and a total derivative, so its relative cohomology class is unchanged.

[A1]

Adopt the source-backed one-loop determinant law in [F1] as a premise for this regulator. The proof below does not derive the chiral determinant variation from a loop integral or an index theorem.

Proof

1.1givenA1F1algebra

By [A1] and [F1], the total consistent background variation is the common linear descent of $I_{6} (E) + I_{6} (E^{*})$ . Because the two species use the same regulator and trace normalization, additivity of the determinant variation identifies its ghost-number-one representative with the descent of this sum; its physical variation is the stated linear evaluation of that representative.

2.1F3step 1.1algebra

By [F3], $I_{6} (E) + I_{6} (E^{*}) = 0$ and the chosen transgressions have zero sum. Applying the same linear BRST and exterior-derivative descent to this zero transgression gives $I_{4}^{(1)} (E) + I_{4}^{(1)} (E^{*}) = 0$ as a local representative.

3.1F2step 2.1algebra

The representative in step 2.1 is the zero four-form, whose class in $H^{1, 4} (s ∣ d_{H})$ is zero by the quotient definition in [F2]. The linear map $E_{ϵ, α}^{4}$ sends this zero representative to zero ordinary variation. Hence this specified chiral spectrum has no perturbative local ’t Hooft anomaly under the stated infinitesimal background transformations.

4.1F2F4step 3.1∎

By [F4], any admitted local counterterm changes only the representative by a relative coboundary; it cannot create a nonzero class from this zero class. The result remains a local one-loop statement and does not test transformations outside the stated identity component.

Local anomaly cancellation for a conjugate chiral pair

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources