Anomaly Polynomials and Inflow

An anomaly polynomial determines the local perturbative inflow class. If a $d$-dimensional boundary theory has a consistent anomaly obtained by descent from $I_{d+2}$, then a one-higher-dimensional Chern–Simons response has the opposite infinitesimal variation. The boundary and bulk can therefore form one invariant relative system.

That local cancellation is not yet a globally defined quantum phase. One must also fix the exponentiated normalization, orientation, charge lattice, tangential structure, allowed bundles, and either an extension or an intrinsic global refinement. A zero polynomial rules out the corresponding local curvature anomaly, but it does not rule out torsion or large-transformation phases.

Required background. Wess–Zumino Consistency and Descent supplies the descent bidegrees, representative shifts, and local sign convention used below. Coupling to Background Gauge Fields and Bundles supplies fixed connections on globally specified bundles, their patching data, and the distinction between probing a symmetry and gauging it.

Helpful background. Characteristic Classes and Chern–Weil Theory supplies invariant polynomials, relative transgression, and the distinction between a closed real form and an integral characteristic class.

The anomaly polynomial fixes the local inflow class

Let $X_d$ be an oriented Euclidean spacetime and let $Y_{d+1}$ be an oriented manifold with induced boundary

$$\partial Y_{d+1}=X_d.$$

The equality includes the outward-normal-first boundary orientation. Assume for the moment that the metric, tangential structure, gauge bundle, and background connection on $X_d$ extend over $Y_{d+1}$.

An anomaly polynomial is a universal invariant characteristic form $I_{d+2}$. It is evaluated on auxiliary curvature data in degree $d+2$; it is not an ordinary nonzero $(d+2)$-form on $X_d$. Locally one may choose a Chern–Simons representative and its first descendant,

$$I_{d+2}=\mathrm dQ_{d+1}^{(0)}, \qquad \delta_\Lambda Q_{d+1}^{(0)} =\mathrm dQ_d^{(1)}(\Lambda).$$

The superscript is ghost number and the subscript is form degree. The consistent boundary anomaly convention inherited from the prerequisite is

$$\delta_\Lambda W_{E,X} =-2\pi i\int_{X_d}Q_d^{(1)}(\Lambda).$$

This chain is the local content of the family-index polynomial and descent. The degree conventions and the four-dimensional Weyl normalization are developed in Álvarez-Gaumé and Vázquez-Mozo 2024, §§ 2–3, arXiv v2, pp. 3–8 and 10, eqs. (4)–(22), Open PDF. A compact derivation of the perturbative inflow variation from the same index density is given in Yonekura 2016, Appendix A, arXiv v1, pp. 26–28, eqs. (A.5)–(A.9), Open PDF.

Three manifolds play different roles. The distinction prevents an auxiliary filling from being mistaken for physical spacetime.

Boundary, inflow bulk, and auxiliary extension

Manifold	Role	Required datum	What can fail
Xd	Physical boundary QFT	Background bundle, connection, metric, and tangential structure	Its partition function can be line-valued rather than a number
Yd+1	Invertible inflow response	An extension of the allowed boundary data	A local Chern–Simons form may not patch to a global phase
Zd+2	Auxiliary filling used to test exponentiation	Extended data and an integral characteristic refinement	The filling or the background extension may not exist

The table does not assert that every $Y_{d+1}$ bounds or that every bundle extends. It states the data needed when an extension presentation is used.

Boundary orientation makes the variations cancel

In a local trivialization, define the Euclidean inflow action by

$$W_{E,\mathrm{inflow}}[Y] =+2\pi i\int_{Y_{d+1}}Q_{d+1}^{(0)}.$$

Stokes’ theorem and the induced orientation give

$$\begin{aligned} \delta_\Lambda W_{E,\mathrm{inflow}} &=+2\pi i\int_Y\mathrm dQ_d^{(1)}(\Lambda) \\ &=+2\pi i\int_XQ_d^{(1)}(\Lambda). \end{aligned}$$

Consequently,

$$\delta_\Lambda \bigl(W_{E,X}+W_{E,\mathrm{inflow}}\bigr)=0.$$

With the Euclidean convention $Z=e^{-W_E}$, the corresponding local bulk factor is

$$Z_{\mathrm{inflow}}[Y] =\exp\!\left(-2\pi i\int_YQ_{d+1}^{(0)}\right).$$

Reversing the orientation of $Y$ reverses the sign. So does exchanging the boundary theory for one of opposite chirality. These are orientation and chirality statements, not adjustable conventions once the prerequisite’s boundary variation has been fixed.

This calculation proves only infinitesimal local cancellation. A Chern–Simons representative may exist only in local bundle charts, and its integral can change under a large transformation. Thus the formula above is best viewed as a local trivialization of the bulk response until the global tests in the next section have passed.

The infinitesimal parameter must also extend into $Y$ and preserve the chosen boundary conditions. A large or nonextending transformation is not covered by this Stokes calculation and must be tested against the intrinsic global response.

Once a compatible globally defined inverse anomaly theory is supplied, the combined partition function is a number even when the isolated boundary partition function is naturally a vector in a one-dimensional anomaly line: the bulk supplies the inverse line and a compatible pairing. In this sense the anomalous boundary is a relative theory, while its anomaly is encoded by an invertible theory in one higher dimension Freed 2014, §§ 2.2–2.3, arXiv v2, pp. 4–5, eqs. (2.5)–(2.10), Open PDF.

The exponentiated response needs global data

The cleanest extension-independence test uses a closed $(d+1)$-manifold $M_{d+1}$. Suppose $M$ and all its background data extend over $Z_{d+2}$ with $\partial Z=M$. Define

$$\mathcal Z_{\mathrm{bulk}}(M;Z) =\exp\!\left(-2\pi i\int_Z I_{d+2}\right).$$

If $Z'$ is a second extension, glue $Z$ to $-Z'$ along $M$. The ratio of the two definitions is

$$\frac{\mathcal Z_{\mathrm{bulk}}(M;Z)} {\mathcal Z_{\mathrm{bulk}}(M;Z')} =\exp\!\left[ -2\pi i\int_{Z\cup_M(-Z')}I_{d+2} \right].$$

The phase is independent of the filling if the integral on every allowed closed $(d+2)$-manifold is an integer:

$$\int_N I_{d+2}\in\mathbb Z.$$

The filling comparison, its gluing law, and the associated quantization test are developed in Witten and Yonekura 2021, § 3.1, arXiv v3, pp. 31–32, eqs. (3.1)–(3.5), Open PDF.

The word allowed carries real content. It includes the chosen oriented tangential structure, for example spin or spin$^c$; the global form of the symmetry group; the charge lattice and representation; and the admissible gauge bundles. Chern–Weil closedness alone does not imply this integrality.

For a fermionic anomaly the complete index density often has integral periods on closed manifolds of the required structure. Individual monomials in its expansion need not be integral separately: their fractional coefficients can combine into one integer index. Checking each term independently would give the wrong quantization test.

Even a successful filling test has a ceiling:

• a particular $M$ may not bound with the required tangential structure;
• the bundle may not extend across a chosen filling, and an extra condition such as flatness can obstruct an admissible extension;
• differential forms do not see flat torsion data; and
• a local polynomial records infinitesimal curvature, not all finite holonomies in background-field space.

An intrinsic global definition uses an appropriate generalized differential refinement, an $\eta$-invariant, or an equivalent invertible field theory rather than assuming a filling exists. Differential cohomology provides one natural language for such refinements because real forms retain curvature but lose integral and torsion information Freed 2000, Introduction and § 1, arXiv v2, pp. 1–5, Open PDF. For fermions, the relation between perturbative Chern–Simons data and the fuller exponentiated $\eta$-invariant, together with the possibility of a residual cobordism invariant when the polynomial vanishes, is developed in Witten and Yonekura 2021, Introduction, § 2.4, and §§ 3.1–3.3, arXiv v3, pp. 2–3, 21, and 31–38, eqs. (2.52), (3.1)–(3.5), and (3.10)–(3.11), Open PDF.

Thus a formal $\int Q_{d+1}^{(0)}$ on an arbitrary bundle is not a substitute for the intrinsic global analysis.

Counterterms move representatives without removing the class

A local change of Chern–Simons representative has the form

$$Q_{d+1}^{(0)} \longmapsto Q_{d+1}^{(0)}+\mathrm dB_d^{(0)}.$$

On $Y$ it changes the inflow action by a boundary term,

$$W_{E,\mathrm{inflow}} \longmapsto W_{E,\mathrm{inflow}} +2\pi i\int_XB_d^{(0)}.$$

The synchronized boundary redefinition

$$W_{E,X} \longmapsto W_{E,X}-2\pi i\int_XB_d^{(0)}$$

leaves the combined system unchanged. Infinitesimally this is the familiar shift of the consistent anomaly by the variation of a local counterterm. It moves the representative or reallocates a mixed anomaly among Ward identities; it does not erase a nontrivial total class while preserving every required symmetry.

The word local is not enough. The counterterm must be globally defined on the allowed bundles, compatible with the boundary domain, and exponentiated with the required quantization. A polynomial written in one gauge potential on one patch need not be an admissible counterterm. On a manifold with an additional boundary or corner, the displayed total derivative produces a new surface term and requires boundary data rather than silent deletion. The quantized-counterterm and extension qualifications are made explicit in Witten and Yonekura 2021, § 3.3, arXiv v3, pp. 36–38, eqs. (3.10)–(3.11), Open PDF.

A four-dimensional Weyl fermion fixes the normalization

Let $X_4$ be a closed oriented Euclidean spin four-manifold carrying a fixed background gauge bundle. Use the site’s Hermitian source connection $\mathcal A$, including couplings and charges, with

$$\mathcal F=\mathrm d\mathcal A-i\mathcal A^2, \qquad x=\frac{\mathcal F}{2\pi}.$$

For one physical Lorentzian left-handed Weyl fermion in representation $R$, the accepted Wick continuation gives negative Euclidean chirality. Its universal degree-six polynomial is therefore

$$I_6^L =-\frac{1}{6}\operatorname{tr}_R(x^3) +\frac{p_1(T)}{24}\operatorname{tr}_R(x),$$

Here

$$p_1(T) =-\frac{1}{8\pi^2} \operatorname{tr}_{\mathrm{vec}}(\mathcal R^2).$$

This index normalization and its chirality reversal follow from Álvarez-Gaumé and Vázquez-Mozo 2024, § 3, arXiv v2, pp. 5–8 and 10, eqs. (11), (17)–(19), and (22), Open PDF.

Here $T$ is the tangent bundle of the manifold on which the universal characteristic form is being evaluated: $TX$ for boundary curvature data and $TZ$ after extension to the auxiliary six-manifold.

The first term is the pure gauge contribution. The second is mixed gauge–gravity; it vanishes for a semisimple factor because $\operatorname{tr}_R\mathcal F=0$, but it can be nonzero for $U(1)$. Four-dimensional spin-$\tfrac12$ matter has no perturbative pure diffeomorphism/local-Lorentz anomaly of the type classified here; this is distinct from the Weyl trace anomaly Álvarez-Gaumé and Vázquez-Mozo 2024, § 3, arXiv v2, pp. 9–10, text around eq. (28), Open PDF.

If a closed spin five-manifold $M_5$ and its bundle extend over a spin six-manifold $Z_6$ with boundary $M_5$, the local-polynomial bulk phase is

$$\mathcal Z_{\mathrm{bulk}}(M_5;Z_6) =\exp\!\left(-2\pi i\int_{Z_6}I_6^L\right).$$

Two fillings $Z_6$ and $Z_6'$ glue to a closed spin six-manifold $N_6=Z_6\cup_{M_5}(-Z_6')$. On this closed comparison manifold, the full integral is an index,

$$\int_{N_6}I_6^L =-\operatorname{ind} \bigl(\mathcal D_{N_6}^{+}\otimes E_R\bigr) \in\mathbb Z.$$

The same family-index density and its integral interpretation are stated in Yonekura 2016, Appendix A, arXiv v1, pp. 26–27, eqs. (A.5)–(A.6), Open PDF.

This is the extension-independence check. It is the full $[\widehat A(TN)\operatorname{ch}(E_R)]_6$ combination that is integral. The equation does not assert that every spin five-manifold with every bundle admits such a filling.

Compact U(1) gauge and gravitational backgrounds

Let $a$ be a compact unit-charge background connection, so a charge $q\in\mathbb Z$ has holonomy $e^{iq\oint a}$. Its curvature $f$ is global, although $f=\mathrm da$ only in a local trivialization. Set

$$\mathcal A=q a, \qquad c_1=\frac{f}{2\pi}.$$

For one left-handed charge-$q$ fermion,

$$I_6^L =-\frac{q^3}{6}c_1^3 +\frac{q}{24}c_1p_1(T).$$

On a trivializing patch, write

$$K_4 =-\frac{q^3}{6}c_1^2 +\frac{q}{24}p_1(T).$$

Then one local descent representative is

$$Q_5^{(0),L}=\frac{a}{2\pi}K_4, \qquad Q_4^{(1),L}(\lambda)=\frac{\lambda}{2\pi}K_4,$$

Here $\delta_\lambda a=\mathrm d\lambda$.

This representative preserves diffeomorphism and local-Lorentz covariance and places the mixed anomaly in the $U(1)$ Ward identity. An admissible local counterterm can reallocate that mixed consistent anomaly, but cannot erase the total class.

The two variations are

$$\delta_\lambda W_{E,X} =-i\int_X\lambda K_4, \qquad \delta_\lambda W_{E,\mathrm{inflow}} =+i\int_X\lambda K_4.$$

This controlled gauge-plus-gravity application fixes the chirality, trace, orientation, curvature, charge, and Euclidean phase conventions. For several fermions one sums their full index polynomials.

If $a$ is a fixed source for an exact global symmetry, the nontrivial class is ’t Hooft-anomaly data and the bulk–boundary pair is a valid relative system. If the same connection is integrated over as a dynamical gauge field, the total anomaly of the complete system must cancel; writing an external inflow factor does not by itself make an inconsistent standalone boundary gauge theory well defined.

Three-dimensional Chern–Simons inflow

The same mechanism appears one dimension lower. Let $a$ now be a fixed compact $U(1)$ connection on $Y_3$ with $\partial Y_3=X_2$, and take the formal polynomial

$$I_4=\frac{k}{2}c_1^2.$$

In a trivialization,

$$Q_3^{(0)}=\frac{k}{8\pi^2}a\,\mathrm da, \qquad W_{E,\mathrm{inflow}} =\frac{ik}{4\pi}\int_{Y_3}a\,\mathrm da.$$

Under $a\mapsto a+\mathrm d\lambda$,

$$\delta_\lambda W_{E,\mathrm{inflow}} =\frac{ik}{4\pi}\int_{X_2}\lambda\,\mathrm da.$$

A two-dimensional chiral boundary variation with the opposite sign is therefore canceled. This is the local bulk-to-boundary transgression. It deliberately assumes that $k$, the charge lattice, and the spin or non-spin structure make the exponentiated background response well defined. More explicitly, $k\in\mathbb Z$ passes the extension test on closed spin four-manifolds because $\int c_1^2$ is even; without spin, $k$ must be even unless extra structure changes the quantization law Belov and Moore 2005, Introduction, arXiv v1, pp. 3–4, eqs. (1.1)–(1.3), Open PDF. For global level tests, see Chern–Simons Actions and Level Quantization and, for the compatible BV–BFV equations, see Bulk–Boundary Master Equations and Anomaly Inflow.

Local inflow is not a global anomaly classification

The implications established here are one-way:

• descent plus the induced boundary orientation gives local infinitesimal cancellation;
• integral periods on all allowed closed extensions give independence of an extension presentation;
• a global differential or spectral refinement can define the response when no convenient filling exists; and
• the bulk pairing turns the anomalous boundary into a relative system.

None of these statements says that the local polynomial is the complete anomaly. Flat torsion phases, large transformations, determinant-line holonomy, the global form of the symmetry group, and nonbounding backgrounds can survive when $I_{d+2}=0$. Those questions continue in Global and Torsion Anomalies. Once the full exact symmetry has been identified, ’t Hooft Anomaly Matching explains what the class constrains along renormalization-group flow.

The later When Is a Topological Term Well Defined? and Background Responses and Invertible Phases pages develop quantized response actions, stacking, and the contrast with a non-invertible topological theory. Evidence Programs for Holographic Duality uses anomaly matching among the distinct checks relevant to holographic and quantum-gravity applications. String and brane realizations require their own worldvolume, normal-bundle, and charge-quantization data and are not derived here.

Common pitfalls

Treating a local primitive as a global action. The equation $I_{d+2}=\mathrm dQ_{d+1}^{(0)}$ is generally patchwise. A global exponentiated response needs patching and integrality data or an intrinsic refinement.

Assuming every background extends. A filling is a presentation when it exists, not a theorem that all manifolds and bundles bound. Nonbounding data must be handled intrinsically or retained as an obstruction.

Checking the monomials instead of the index density. Fractional gauge and mixed gauge–gravity terms can combine into an integral Dirac index. The allowed closed-manifold test applies to the complete polynomial.

Calling the boundary anomaly canceled in isolation. Inflow makes the combined bulk–boundary system invariant. The boundary partition function remains relative to the bulk response.

Equating a zero polynomial with complete anomaly freedom. It removes the corresponding local perturbative class. Torsion and finite phases require the next set of tests.

Check your understanding

1. Starting from the two descent equations, derive the sign of the inflow variation when $\partial Y=X$.

Solution

Using the outward-normal-first orientation,

$$\begin{aligned} \delta_\Lambda W_{E,\mathrm{inflow}} &=2\pi i\int_Y\delta_\Lambda Q_{d+1}^{(0)} \\ &=2\pi i\int_Y\mathrm dQ_d^{(1)} =2\pi i\int_XQ_d^{(1)}. \end{aligned}$$

This is opposite to $\delta_\Lambda W_{E,X}=-2\pi i\int_XQ_d^{(1)}$.

2. Two fillings $Z$ and $Z'$ define a bulk phase for the same closed $M$. What is their ratio, and what condition makes it one?

Solution

Gluing gives the closed manifold $N=Z\cup_M(-Z')$, so

$$\frac{\mathcal Z_{\mathrm{bulk}}(M;Z)} {\mathcal Z_{\mathrm{bulk}}(M;Z')} =\exp\!\left(-2\pi i\int_NI_{d+2}\right).$$

It equals one when the full characteristic refinement has integral periods on every allowed closed $N$ with the specified bundles and tangential structure.

3. Show that $Q_{d+1}^{(0)}\mapsto Q_{d+1}^{(0)}+\mathrm dB_d^{(0)}$ changes no combined bulk–boundary quantity when the boundary counterterm is shifted coherently.

Solution

The bulk action gains $+2\pi i\int_XB_d^{(0)}$. The boundary action is redefined by the negative of this term. Their sum and its variation are unchanged. The move changes a representative, not the total anomaly class.

4. For the charge-$q$ Weyl fermion, why is it incorrect to demand separately that $-q^3c_1^3/6$ and $q c_1p_1/24$ have integer periods?

Solution

The index theorem quantizes their sum,

$$\int_{N_6} \left(-\frac{q^3}{6}c_1^3 +\frac{q}{24}c_1p_1\right) =-\operatorname{ind}(\mathcal D_{N_6}^{+}\otimes L^q).$$

Here $N_6$ is closed. The individual rational terms need not be integers; only the complete index density has the required integrality on closed spin six-manifolds.

5. A matter spectrum has $I_6=0$. State the strongest conclusion licensed by this calculation.

Solution

Its corresponding local perturbative gauge and mixed gauge–gravity anomaly polynomial vanishes, so no local Chern–Simons inflow is required for that class. The calculation has not tested torsion phases, large transformations, global group form, nonbounding backgrounds, or other nonperturbative anomaly data.

References

• Álvarez-Gaumé, Luis, and Miguel Á. Vázquez-Mozo. “Anomalies and the Green–Schwarz Mechanism.” In Handbook of Quantum Gravity, edited by Cosimo Bambi, Leonardo Modesto, and Ilya L. Shapiro, 2241–2284. Singapore: Springer, 2024. DOI. Open PDF, arXiv v2.

• Belov, Dmitriy M., and Gregory W. Moore. “Classification of Abelian Spin Chern–Simons Theories.” arXiv:hep-th/0505235v1, 2005. Stable record. Open PDF, arXiv v1.

• Freed, Daniel S. “Anomalies and Invertible Field Theories.” Proceedings of Symposia in Pure Mathematics 88 (2014): 25–46. DOI. Open PDF, arXiv v2.

• Freed, Daniel S. “Dirac Charge Quantization and Generalized Differential Cohomology.” In Surveys in Differential Geometry VII, 129–194. Somerville, MA: International Press, 2000. DOI. Open PDF, arXiv v2.

• Witten, Edward, and Kazuya Yonekura. “Anomaly Inflow and the $\eta$-Invariant.” In Memorial Volume for Shoucheng Zhang, edited by Biao Lian, Chao-Xing Liu, Eugene Demler, Steven Kivelson, and Xiao-Liang Qi, 283–352. Singapore: World Scientific, 2021. DOI. Open PDF, arXiv v3.

• Yonekura, Kazuya. “Dai–Freed Theorem and Topological Phases of Matter.” Journal of High Energy Physics 2016, no. 9 (2016): 022. DOI. Open PDF, arXiv v1.