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A single chiral fundamental has a nonzero local polynomial

Physical scope

A worked check of the formal one-loop polynomial for one positive-chirality (3,+1) species; it does not assert that its local cohomology class is nontrivial, that all counterterms fail, or that any global anomaly is present.

Example

Use the same flat-space background convention but take the spectrum to contain only one positive-chirality species $E = (3, + 1)$ . Its polynomial is

$I_{6} (E) = \frac{i}{24 π^{2}} (tr F_{A}^{3} + 3 i f \land tr F_{A}^{2} + 3 (i f)^{3}),$

where $F_{A}$ is the anti-Hermitian $S U (3)$ curvature, $f = d B$ is the $U (1)_{V}$ curvature, and the expression is evaluated as a degree-six polynomial on the formal extension used for descent. The mixed $U (1)_{V}$ – $S U (3)^{2}$ term and the pure $U (1)_{V}^{3}$ term have nonzero coefficients. This example verifies that the one-field polynomial is not the zero invariant polynomial; it does not claim that the associated relative BRST class is nontrivial modulo every admissible counterterm or that a global anomaly has been evaluated.

Example references

Bilal, §§7.1–7.2, 9.4, and 10.4

Alvarez, §§2.2–2.3

Facts & Assumptions

Given: The single positive-chirality species $E = (3, + 1)$ , its smooth background connection, and the source-backed one-loop polynomial convention.

[F1]

Flat-space chiral anomaly-matching data and regulator convention: for one species, the selected regulator assigns the degree-six polynomial from the same consistent descent convention as in the conjugate-pair model.

[F2]

Degree-six cubic trace polynomial for a chiral representation: $I_{6} (E) = \frac{i}{24 π^{2}} tr (F_{E}^{3})$ with $F_{E} = F_{A} + i f I_{3}$ .

Verification

1.1F1F2algebra

Expand $F_{E}^{3} = (F_{A} + i f I_{3})^{3}$ . Since $F_{A}$ is traceless and $tr (I_{3}) = 3$ , the result is the displayed sum of pure $S U (3)$ , mixed $U (1)_{V}$ – $S U (3)^{2}$ , and pure $U (1)_{V}^{3}$ terms.

2.1F2step 1.1algebra

The invariant polynomial is not identically zero: on a formal six-dimensional extension take $F_{A} = 0$ and $f = d x^{1} \land d x^{2} + d x^{3} \land d x^{4} + d x^{5} \land d x^{6}$ . Then $f^{3} = 6 d x^{1} \land d x^{2} \land d x^{3} \land d x^{4} \land d x^{5} \land d x^{6} \neq 0$ , so $I_{6} (E) = \frac{3 i}{24 π^{2}} (i f)^{3} \neq 0$ . This only checks the polynomial; it does not infer a nonzero relative cohomology class.

3.1givenstep 2.1∎

The scope is exactly the stated polynomial calculation. No conclusion about whether local counterterms remove its descent, or about large transformations, follows from steps 1.1–2.1.

A single chiral fundamental has a nonzero local polynomial

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources