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A mixed-background counterterm shift along RG

Physical scope

Background-functional illustration of the four-dimensional local counterterm quotient on a closed four-torus with trivial U(1) bundles and small gauge transformations; no microscopic matter model, global-bundle conclusion, or continuum limit is asserted.

Example

Let $M = T^{4}$ be a closed flat Euclidean four-torus. Fix three nondynamical $U (1)$ background connections $A_{1}, A_{2}, A_{3}$ , each a globally defined real 1-form on the trivial bundle, and allow only gauge transformations $A_{i} \mapsto A_{i} + d λ_{i}$ with smooth single-valued periodic dimensionless real functions $λ_{i}$ . Use natural units $c = ℏ = 1$ , so $[A_{i}] =$ mass, $[F_{i}] =$ mass squared, and the functional below is dimensionless. For interpreting endpoint anomaly representatives, assume the model specifies the same Wick-rotation isomorphism between complexified Minkowski and Euclidean local background finite-jet algebras as in the symmetry-preserving RG matching theorem; it maps these fields and horizontal forms and intertwines the BRST differentials and the fixed boundary-term convention. Assume also that the same regularized local BRST-variation assignment at both endpoints identifies each endpoint anomaly representative with the variation of its own $W_{μ}$ branch in that finite-jet complex, is linear on common branches, and sends a local counterterm density to its BRST variation modulo a $d_{H}$ -exact form. This is additional model data; the scalar RG identity and the calculation below do not supply that identification or compatibility. In a declared counterterm scheme, take the allowed local background counterterm, with $F_{i} : = d A_{i}$ and $d$ the exterior derivative on $T^{4}$ .

C_{k} [A_{1}, A_{2}, A_{3}] = \frac{i k}{2 π} \int_{T^{4}} A_{1} \land A_{2} \land F_{3},

where $k \in R$ is a fixed dimensionless coefficient. In the common local scheme require $C_{k} = \frac{i k}{2 π} A_{1} \land A_{2} \land F_{3} \in C$ and require the boundary primitive $q_{3}$ supplied by the endpoint BRST-variation identification to belong to the same space $N$ at both endpoints; the closed-torus convention discards its total derivative. Suppose an exact background-coupled RG comparison has $W_{U V} [A_{1}, A_{2}, A_{3}] = W_{I R} [A_{1}, A_{2}, A_{3}] + C_{k} [A_{1}, A_{2}, A_{3}]$ on this background domain. Then the gauge variation of the counterterm is

δ C_{k} = \frac{i k}{2 π} \int_{T^{4}} (- λ_{1} F_{2} \land F_{3} + λ_{2} F_{1} \land F_{3}), F_{i} = d A_{i},

Here $s$ and $d_{H}$ are the nilpotent differential and horizontal derivative of the background-only finite-jet BRST complex in the local perturbative ’t Hooft anomaly definition; $H_{C, N}^{1, 4} (s ∣ d_{H})$ is its ghost-number-one four-form cocycle quotient by total derivatives and the declared counterterms. The integrand in the displayed gauge variation is the gauge-parameter evaluation of the local counterterm variation. The common regularized BRST assignment in the preceding hypothesis lifts this variation to the ghost-number-one local representative modulo $d_{H}$ , so the endpoint representatives define the same local anomaly class; the gauge-variation formula alone does not make that identification.

Example references

Harvey, TASI 2003 Lectures on Anomalies, §2.3

Tong, Gauge Theory, §3.5

Verification

Given: The closed four-torus, globally defined background 1-forms, smooth periodic gauge parameters, declared local counterterm, and exact RG relation in the Example.

[F1] Symmetry-preserving RG anomaly matching: the local-class conclusion is conditional on the common Wick-rotation and endpoint BRST-variation identification. The Example states that extra condition and computes the counterterm's local variation directly.

[F2] Exact background-coupled RG comparison: the counterterm is single-valued on the admitted background domain, and the same background groupoid is used at both scales.

[A1] The model uses only trivial $U (1)$ bundles and gauge transformations connected to the identity; no large-gauge quantization condition is asserted for $k$ .

[A2] The same branchwise finite-jet BRST-variation assignment, its local-counterterm compatibility, the Wick-rotation isomorphism, and the total-derivative boundary convention are used at both RG endpoints as required by the theorem, with $C_{k} \in C$ and the identified boundary primitive in $N$ . The torus calculation establishes only the gauge variation of $C_{k}$ ; it does not independently identify the endpoint anomaly representatives or infer their BRST-density relation from scalar partition equality.

1.1A1givenalgebra

Since $F_{3}$ is unchanged by every allowed gauge transformation, expanding $C_{k} [A_{1} + d λ_{1}, A_{2} + d λ_{2}, A_{3} + d λ_{3}] - C_{k} [A_{1}, A_{2}, A_{3}]$ gives $\frac{i k}{2 π} \int_{T^{4}} (d λ_{1} \land A_{2} \land F_{3} + A_{1} \land d λ_{2} \land F_{3} + d λ_{1} \land d λ_{2} \land F_{3})$ . The final term is exact because $d λ_{1} \land d λ_{2} \land F_{3} = d (λ_{1} d λ_{2} \land F_{3})$ , using $d F_{3} = 0$ , so its integral vanishes on closed $T^{4}$ . Integration by parts in the first two terms gives $- \int_{T^{4}} λ_{1} F_{2} \land F_{3} + \int_{T^{4}} λ_{2} F_{1} \land F_{3}$ , which is exactly the displayed variation.

2.1F1F2A2step 1.1algebra∎

By [F2], the same $C_{k}$ is the admissible local term at both endpoints. Under the common local BRST-variation assignment [A2], the RG relation identifies the difference of the evaluations of the endpoint anomaly representatives on the allowed gauge parameters with the gauge variation computed in step 1.1. The local-density compatibility assumption of the theorem lifts this equality to the BRST representative of the counterterm density $C_{k} = \frac{i k}{2 π} A_{1} \land A_{2} \land F_{3}$ , modulo a $d_{H}$ -exact term, so [F1] places the difference in zero class in the four-dimensional anomaly quotient. The representatives may differ, but their local perturbative anomaly classes agree.

A mixed-background counterterm shift along RG

Example

Example references

Verification

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