Symmetry-preserving RG anomaly matching

Use natural units $c=\mathrm{\hslash }=1$. Assume Countable Choice: every countable family of nonempty sets admits a choice function. Fix a finite-regulator model on a compact, closed, oriented Riemannian spin four-manifold $M$, with its actual boundary-condition data, fields and jets, regulator, measure, and admissible local counterterms at every scale. Let $G$ be a compact finite-dimensional real Lie group serving as the internal global symmetry with fixed global form, let $P\to M$ be a smooth principal right $G$-bundle with supplied smooth connection $A$, let $V_R$ be a finite-dimensional Hermitian complex vector space, and let $R:G\to U(V_R)$ be a smooth group homomorphism. The admitted background groupoid has these nondynamical connections as objects and the selected background gauge transformations as arrows; the symmetry is exact and unbroken along the flow and is not a dynamical gauge redundancy. Fix a smooth family of Fredholm chiral Dirac operators $D_{A,R}^{+}$ and its inverse determinant line with the Quillen metric and Bismut–Freed unitary connection, with precisely the spin/Clifford, associated-bundle, induced-connection, Sobolev-domain, and admissible spectral-cut/regularization conventions of the determinant-line definition. On each background patch, the finite-regulator partition functional at scale $\mu$ is the local scalar representative of a section of that line, and its transition functions and endpoint gauge actions are retained.

Also fix a Minkowski coordinate region $U\subseteq {\mathbb{R}}^{1,3}$ with its orientation, spin structure, boundary-term convention, fields and jet domains, and identify it with the Euclidean local-variation problem by a specified Wick-rotation map. Here that map means an isomorphism between the complexified local background finite-jet algebras on the two patches, mapping the admitted background fields and horizontal forms and intertwining the horizontal derivatives and BRST differentials while preserving the declared boundary-term convention. Use the Minkowski patch's background-only finite-jet BRST complex, with nilpotent differential $s$ and horizontal derivative $d_H$; its local anomaly class is the relative cohomology class of ghost-number-one horizontal four-form cocycles modulo $d_H$ and the declared four-dimensional background counterterms. Write ${\mathcal{C}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$ for the background-only finite-jet local density whose integral on this patch is the counterterm $C$. Assume an exact symmetry-preserving Wilsonian comparison from ${\mu }_{\mathrm{U}\mathrm{V}}$ to ${\mu }_{\mathrm{I}\mathrm{R}}$ retains all remaining low-energy and topological sectors and satisfies $W_{{\mu }_{\mathrm{U}\mathrm{V}}}[A]-W_{{\mu }_{\mathrm{I}\mathrm{R}}}[A]=C[A]$ on each common local branch, for one admissible, single-valued, dimensionless local counterterm.

For the local anomaly conclusion, impose the following additional compatibility hypothesis. In the same regularization and boundary-term convention at both endpoints, the infinitesimal BRST variation of each branch $W_{\mu }$ is local in finite jets and is represented by the corresponding $a_{\mu ,4}\in {\mathcal{A}}_{\mathrm{b}\mathrm{g},U}^{1,4}$ used in the local-anomaly definition: ${\delta }_s{W}_{\mu }[A]={\int }_U{a}_{\mu ,4}$ modulo the fixed boundary term. The specified Wick-rotation map carries these Euclidean variations to that BRST complex and commutes with its differential. The representative assignment is linear on the common branches, and the local counterterm assignment is compatible with it, so replacing a branch by that branch plus $C[A]={\int }_U{\mathcal{C}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$ changes its representative by $s{\mathcal{C}}_{\mathrm{l}\mathrm{o}\mathrm{c}}+d_H{q}_3$ with ${\mathcal{C}}_{\mathrm{l}\mathrm{o}\mathrm{c}}\in \mathcal{C}$ and $q_3\in \mathcal{N}$, where the same declared counterterm and boundary spaces $\mathcal{C},\mathcal{N}$ are used at both endpoints. This is model and regularization data; it is not inferred from the scalar partition equality alone.

For the determinant-line comparison, take $C$ in the unitary phase convention so $e^{-C[A]}\in U(1)$. Assume the line-valued comparison induces a unitary isomorphism of the endpoint determinant-line families which, in local frames, is multiplication by the phase $e^{-C[A]}$ and which preserves the specified unitary connections and endpoint gauge identifications. Then the ultraviolet and complete infrared four-dimensional local BRST anomaly classes agree modulo the declared local counterterms, and the determinant-line holonomies agree on every admitted loop of background connections modulo gauge transformations. This is a conditional matching constraint only; it determines no infrared spectrum or gap and asserts no continuum limit or empirical result.

Harvey, TASI 2003 Lectures on Anomalies, §2.3

Tong, Gauge Theory, §3.5