Anomalies as Relative and Invertible Field Theories

An anomaly can be encoded by an invertible $(d+1)$-dimensional field theory $\alpha$, while the anomalous $d$-dimensional theory $F$ is relative to its boundary truncation. Consequently, a closed $d$-manifold receives not an invariant complex number but a vector in the line $\alpha(M)$; a choice of anomaly cancellation is a coherent trivialization of the entire invertible theory, not a counterterm chosen independently in each coordinate chart. Bulk inflow records the same transformation law geometrically. This formulation unifies local and global anomaly data without identifying them and without claiming that every invertible bulk admits the desired dynamical boundary.

Required background. Invertible field theories supply the Picard-valued bulk; anomaly polynomials and inflow supply the local descent calculation; and global anomalies, determinant lines, and eta invariants supply the global phase obstruction.

Helpful background. Quantum-master-equation obstructions identify the perturbative BV representative that this functorial description must reproduce.

Relative theories and anomaly lines

Let $\alpha$ be an invertible $(d+1)$-dimensional extended field theory, including the chosen background fields and tangential structure. Its truncation $\tau_{\le d}\alpha$ retains values in dimensions at most $d$. A $d$-dimensional theory relative to $\alpha$ is a morphism of field theories

$$F:\mathbb 1\longrightarrow\tau_{\le d}\alpha.$$

For a closed $d$-manifold $M$ with background $A$, this means

$$Z_F(M,A)\in\alpha(M,A),$$

where $\alpha(M,A)$ is a one-dimensional complex line. Only after choosing an isomorphism $\alpha\simeq\mathbb 1$ does $Z_F$ become an ordinary complex-valued partition function. On a closed $(d-1)$-manifold, the anomalous state space is correspondingly twisted by the higher line assigned by $\alpha$. Freed states this locality-compatible formulation in Freed 2014, §2.3, equations (2.8)–(2.10), pp. 5–6.

The familiar “phase under a gauge transformation” is a trivialized shadow of this geometry. A gauge transformation $g$ determines parallel transport

$$T_g:\alpha(M,A)\longrightarrow\alpha(M,A^g).$$

If local bases have been selected, transport is represented by a phase $\mathcal A(g;A)$ and

$$Z_F(M,A^g)=\mathcal A(g;A)\,Z_F(M,A).$$

Changing the bases changes this cocycle by a coboundary, but it cannot remove nontrivial holonomy of the anomaly line. The invariant question is whether the line with its connection and all higher gluing data is coherently trivial.

Inflow and the two anomaly tests

Suppose $M=\partial Y$ and the background extends over $Y$. The bulk amplitude $Z_\alpha(Y)$ is a vector in the dual line $\alpha(M)^\vee$, so the contraction

$$Z_\alpha(Y)\,Z_F(M)$$

is a number. Under a boundary transformation, the two factors transform inversely. This is anomaly inflow: the combined bulk–boundary system is absolute even though the boundary factor alone is relative.

For a local perturbative anomaly, descent begins from a characteristic form $I_{d+2}$ and yields

$$I_{d+2}=\mathrm d I_{d+1}^{(0)}, \qquad \delta_\lambda I_{d+1}^{(0)} =\mathrm d I_d^{(1)}(\lambda).$$

The boundary variation is the integral of $I_d^{(1)}$. A global anomaly instead appears as holonomy around a loop of backgrounds, often computed by an eta invariant or an index on the mapping torus. A vanishing local polynomial does not force that holonomy to be trivial. Conversely, a single trivial loop does not prove global triviality over the whole background groupoid. The anomaly-theory viewpoint packages both tests into one invertible functor while preserving their distinct computations; Freed’s explicit discussion of anomaly lines, holonomy, and extended locality is in Freed 2014, §§2.1–2.3, pp. 2–6.

Chiral fermion as a relative partition function

For the exact first application, take a two-dimensional chiral fermion coupled to a background gauge field. Its regularized determinant is naturally a section of a determinant line over the background space rather than a function. The associated three-dimensional invertible anomaly theory assigns that line to the boundary surface. On a three-dimensional mapping cylinder or mapping torus, its exponentiated eta invariant supplies the parallel-transport phase. In the local limit, the same transformation is reproduced by Chern–Simons inflow whose exterior derivative is the anomaly polynomial.

Thus a bulk filling changes the representative but not the relative datum:

$$Z_{\mathrm{combined}}(Y,M;A) =Z_\alpha(Y,A)\,Z_F(M,A)$$

is gauge invariant when the bulk and boundary normalizations match. The calculation is passed to Standard Model anomaly cancellation as the low-dimensional model for how a chiral partition function transforms and how inflow compensates it. The Standard Model page owns its four-dimensional representation arithmetic; this page owns the line-valued and relative-theory mechanism.

An independent gluing check composes two bordisms that transport the background by $g_1$ and $g_2$. Functoriality requires

$$T_{g_2g_1}=T_{g_2}T_{g_1}.$$

In local bases this becomes the anomaly cocycle condition. Failure signals either inconsistent regularization or omitted bulk data.

The adversarial failure trivializes the determinant line on one chart and declares the theory absolute. A loop that crosses charts can still return a vector multiplied by nontrivial holonomy. No change of local basis removes it. The strongest surviving claim is a locally trivial relative theory; a global cancellation requires a coherent functorial trivialization $\alpha\simeq\mathbb 1$.

Exercises

Explain why adding a local counterterm changes an anomaly cocycle by a coboundary.

Solution

A counterterm multiplies a chosen local partition function by a phase $e^{iC(A)}$. Under $g$, the anomaly phase is multiplied by $e^{iC(A^g)-iC(A)}$, which is precisely a groupoid coboundary. It can remove a cohomologically trivial cocycle but not nontrivial line holonomy.

Why is an inflow-cancelled boundary not automatically realizable by a standalone $d$-dimensional regulator?

Solution

Inflow proves consistency of the combined relative system. A standalone regulator would amount to a compatible trivialization of the bulk anomaly together with locality, positivity, and continuum control. Those additional structures do not follow from the existence of the bulk theory.

References

• Dai, Xianzhe, and Daniel S. Freed. “Eta-Invariants and Determinant Lines.” Journal of Mathematical Physics 35 (1994): 5155–5194. DOI; Open PDF.
• Freed, Daniel S. “Anomalies and Invertible Field Theories.” Proceedings of Symposia in Pure Mathematics 88 (2014): 25–45. DOI; Open PDF.
• Witten, Edward. “Global Gravitational Anomalies.” Communications in Mathematical Physics 100 (1985): 197–229. DOI.