Local perturbative ’t Hooft anomaly modulo counterterms

Fix a four-dimensional smooth spacetime region $U\subseteq {\mathbb{R}}^{1,3}$ with a stated orientation, spin structure, boundary conditions, and unit system. A selected model specifies a continuous global symmetry group $H$, its action on the fields and observables, the nondynamical background connections, the field and jet domains, a regulator and measure or formal perturbative generating functional, and the admissible local counterterms. A background connection is nondynamical model data: it transforms under local changes of background trivialization but is not integrated over.

Let ${\mathcal{A}}_{\mathrm{b}\mathrm{g},U}^{g,p}$ be the finite-jet horizontal $p$-forms of ghost number $g$ generated by the nondynamical background connections and their BRST ghost jets, without matter-field jets. Fix their coefficient field $\mathbb{K}$ as part of the model and restrict its nilpotent BRST differential $s$ and horizontal exterior derivative $d_H$ to this background subalgebra, with $s^2=d_H^2=s{d}_H+d_{H}s=0$. The model supplies two $\mathbb{K}$-linear subspaces

$\mathcal{C}\subseteq {\mathcal{A}}_{\mathrm{b}\mathrm{g},U}^{0,4},\mathcal{N}\subseteq {\mathcal{A}}_{\mathrm{b}\mathrm{g},U}^{1,3}\mathrm{.}$

Here $\mathcal{C}$ is exactly the space of admissible changes of local background counterterm density, with coefficients and units giving dimensionless integrated action changes. The space $\mathcal{N}$ is exactly the space of boundary primitives whose $d_H$-derivatives are discarded in the infinitesimal variation under the stated support or boundary conditions. This convention must hold on every admitted background and ghost input; a surface contribution that is not discarded is not an element of $d_H\mathcal{N}$. Linearity of these two spaces is explicit model input, not a consequence of locality or nilpotence. No linear span of a restricted admissibility set is taken implicitly. If quantization or other restrictions instead make the admissible changes nonlinear, this vector-space quotient does not apply to them without a separately specified equivalence structure.

The space of compatible Wess–Zumino cocycles and the space of admissible changes are

$Z^{1,4}:=\{a\in {\mathcal{A}}_{\mathrm{b}\mathrm{g},U}^{1,4}:sa+d_{H}b=0\text{ for some }b\in {\mathcal{A}}_{\mathrm{b}\mathrm{g},U}^{2,3}\},B_{\mathcal{C},\mathcal{N}}:=s\mathcal{C}+d_H\mathcal{N}\mathrm{.}$

Both are linear spaces, and $B_{\mathcal{C},\mathcal{N}}\subseteq Z^{1,4}$: for $a=sc+d_{H}n$ choose the ghost-number-two three-form $b=sn$, giving $sa+d_{H}b=s^{2}c+s{d}_{H}n+d_{H}sn=0$. More generally, if $sa+d_{H}b=0$, then $a^{\mathrm{\prime }}=a+sc+d_{H}n$ obeys $s{a}^{\mathrm{\prime }}+d_H(b+sn)=0$, so every admitted representative change preserves consistency. The witness space for $b$ has bidegree $(2,3)$ and is distinct from the boundary-ambiguity space $\mathcal{N}$ of bidegree $(1,3)$.

An infinitesimal local anomaly representative is a specified $a_4\in Z^{1,4}$ for the model's quantum background variation. The local perturbative ’t Hooft anomaly is its class in

$H_{\mathcal{C},\mathcal{N}}^{1,4}(s\mid d_H):=Z^{1,4}\mathrm{/}{B}_{\mathcal{C},\mathcal{N}},a_4^{\mathrm{\prime }}\sim a_4⟺a_4^{\mathrm{\prime }}-a_4=sc+d_{H}n\text{ for some }c\in \mathcal{C},n\in \mathcal{N}\mathrm{.}$

The subspace property makes this an equivalence relation and a quotient vector space. Its class is zero exactly when $a_4=sc+d_{H}n$ for some such $c,n$; adding the admissible counterterm density $-c$ then leaves only the discarded term $d_{H}n$. Conversely, if adding an admissible density $c^{\mathrm{\prime }}$ leaves a discarded term, $a_4+s{c}^{\mathrm{\prime }}=d_{H}n$, then $a_4=s(-c^{\mathrm{\prime }})+d_{H}n$ has zero class. This is precisely cancellation of the infinitesimal local background variation in the declared convention. The ordinary unrestricted relative cohomology $H^{1,4}(s\mid d_H)$ is the special case $\mathcal{C}={\mathcal{A}}_{\mathrm{b}\mathrm{g},U}^{0,4}$ and $\mathcal{N}={\mathcal{A}}_{\mathrm{b}\mathrm{g},U}^{1,3}$ with a boundary convention discarding all these total derivatives; it is not substituted for a restricted model quotient.

Any prescribed noninvariance of the classical action or selected couplings is an explicit-symmetry-breaking insertion and is separated before assigning this quantum anomaly class. It is not, by itself, a measure anomaly. A claim of anomaly matching therefore assumes the stated classical symmetry, or carries such breaking insertions separately.

This definition applies to a global symmetry whose background is nondynamical. If the same local class concerns a field integrated over as a gauge redundancy, it is a local gauge anomaly, not a ’t Hooft anomaly; an uncancelled gauge anomaly obstructs the proposed gauging unless a separately specified cancellation mechanism is included. The definition addresses only infinitesimal transformations and makes no claim about large or disconnected transformations, determinant-line holonomy, torsion phases, or a physical continuum limit.

Group-like global symmetry and nondynamical backgrounds

Finite-jet local cohomology modulo total derivatives

Bilal, §§9.1–9.4 and 10.4

Alvarez, §§2.1–2.4