Gravitational, Mixed, Discrete, and Orientation-Reversing Anomalies

An anomaly depends on the full class of backgrounds on which the quantum theory is supposed to live. For fermions that class includes the tangent bundle, orientation, metric, Spin or Pin lift, internal bundle, and the way fermion parity is shared with other symmetries. Enlarging it can therefore reveal obstructions that a continuous gauge connection on an oriented spin manifold cannot see. Local curvature polynomials detect perturbative gauge, gravitational, and mixed terms; flat finite-group bundles and unorientable backgrounds can instead carry global or torsion phases.

The controlled four-dimensional result is especially sharp. Ordinary spin-$\tfrac12$ matter has no perturbative pure diffeomorphism or local-Lorentz anomaly in four dimensions, but a charged Weyl spectrum can have an independent mixed $U(1)$–gravity anomaly. Discrete and orientation-reversing symmetries then require separate global tests rather than a new reading of that same local coefficient.

Required background. Global and Torsion Anomalies supplies anomaly-line holonomy, mapping-torus tests, mod-two and $\eta$ diagnostics, and the ceiling of local polynomial tests.

Helpful background. Characteristic Classes and Chern–Weil Theory supplies Chern, Pontryagin, and Stiefel–Whitney classes and explains why real differential forms lose torsion. CPT: Hypotheses, Content, and Limits separates a CPT theorem from separately imposed parity or time-reversal symmetry.

Scientific evidence cutoff. Source versions, published corrections, and scope-refining results cited here were checked through 9 August 2026.

The full background data define the anomaly problem

Write a fermionic background schematically as

$$b=(X,g,\mathfrak s;P,\mathcal A,\mathfrak d).$$

Here $X$ is spacetime, $g$ is the metric, $\mathfrak s$ denotes the chosen tangential lift, $(P,\mathcal A)$ is the internal bundle with connection or finite holonomy data, and $\mathfrak d$ records the field domain and boundary conditions. A claimed symmetry must act consistently on every entry. The fermionic symmetry group must also record $(-1)^F$ and any quotient that identifies it with a central internal transformation.

This produces several independent axes:

• local versus global asks whether infinitesimal curvature or finite holonomy detects the obstruction;
• pure versus mixed asks which background transformations participate;
• continuous versus finite changes the available characteristic data;
• oriented versus unoriented changes the tangential structure; and
• background versus dynamical changes the physical verdict.

For a compact account of why local anomaly curvature and finite holonomy are complementary rather than competing definitions, see Monnier 2019, § 1, arXiv v2, pp. 1–2, Open PDF.

The following labels therefore answer different questions.

Four anomaly labels and the Ward question each one tests

Label	Transformation tested	Typical detector	Physical reading
Local gravitational	Infinitesimal diffeomorphism or local Lorentz rotation	Tangent-curvature polynomial and descent	Obstruction to preserving the corresponding geometric Ward identity
Mixed gauge–gravity	Gauge and geometric backgrounds together	Mixed characteristic term such as first Chern times first Pontryagin	One nontrivial class whose consistent representative can be reallocated
Weyl or trace	Local rescaling of the metric	Trace Ward identity and curvature scalars	Scale or conformal anomaly, not a diffeomorphism anomaly
Global gravitational	Large diffeomorphism or noncontractible geometric loop	Mapping-torus phase, reduced eta invariant, or bordism test	Finite holonomy left after local anomalies and counterterms are handled

A local counterterm can exchange the Einstein and local-Lorentz representatives of a gravitational anomaly, or move a mixed consistent anomaly between gauge and geometric Ward identities. It cannot erase the underlying nontrivial class while preserving every named symmetry. This distinction and the index/descent framework are summarized in Álvarez-Gaumé and Vázquez-Mozo 2024, §§ 1–3, arXiv v2, pp. 2–8 and 10, eqs. (1)–(11), (17)–(19), and (22), Open PDF.

Tangent curvature produces pure and mixed local classes

On an oriented spin background, the local fermion polynomial is built from

$$\widehat A(TX)\operatorname{ch}(E).$$

The pure gravitational contribution comes from the tangent-bundle factor alone. Pontryagin forms have degrees divisible by four, so the degree $d+2$ pure tangent term for a chiral spin-$\tfrac12$ field can occur in $d=4k+2$ dimensions. In four dimensions the anomaly polynomial has degree six, while $[\widehat A(TX)]_6=0$. This is the dimensional reason there is no perturbative pure diffeomorphism or local-Lorentz anomaly for ordinary four-dimensional spin-$\tfrac12$ matter.

That statement has three important limits. It does not remove mixed terms, does not remove global anomalies on a larger background category, and says nothing about the Weyl trace anomaly. That domain dependence is concrete: a 2024 analysis shows that generalized $\operatorname{Spin}_G$ backgrounds can support $\mathbb Z_2$ classes involving $w_2(TX)w_3(TX)$ even when the ordinary spin theory has no such obstruction Brennan and Intriligator 2024, § 1, arXiv v3, pp. 2–5, eqs. (1.1)–(1.3), Open PDF.

A four-dimensional Weyl spectrum separates the coefficients

Let $X_4$ be a closed oriented Euclidean spin manifold. Use the inherited Hermitian source connection $\mathcal A$, with

$$\mathcal F=\mathrm d\mathcal A-i\mathcal A^2, \qquad x=\frac{\mathcal F}{2\pi}, \qquad p_1(T)=-\frac{1}{8\pi^2} \operatorname{tr}_{\mathrm{vec}}(\mathcal R^2).$$

One physical Lorentzian left-handed Weyl fermion has negative Euclidean chirality under the site’s continuation. Its universal degree-six polynomial is

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

For the controlled Abelian application, take the exact direct-product background $\operatorname{Spin}(4)\times U(1)$: the spin structure and the $U(1)$ bundle are independent, rather than combined into a $\operatorname{Spin}^c$ structure. Let $a$ be a compact unit-charge $U(1)$ connection with global curvature $f$; only locally need $f=\mathrm da$. A field of charge $q$ has $\mathcal A=q a$, $\mathcal F=qf$, and $c_1=f/(2\pi)$. Retaining both chiralities in the bookkeeping, define

$$\begin{aligned} \kappa_3 &=\sum_{i\in L}q_i^3-\sum_{j\in R}q_j^3, & \kappa_1 &=\sum_{i\in L}q_i-\sum_{j\in R}q_j. \end{aligned}$$

Then

$$I_6 =-\frac{\kappa_3}{6}c_1^3 +\frac{\kappa_1}{24}c_1p_1(T).$$

The first coefficient is the pure $U(1)^3$ gauge anomaly; the second is the mixed $U(1)$–gravity anomaly. The two tests are independent.

Three left-handed charge spectra that separate the local coefficients

Charges	Cubic coefficient κ₃	Linear coefficient κ₁	Local verdict
1, 1, −2	−6	0	Mixed term cancels; cubic term remains
3, 4, 5, −6	0	6	Cubic term cancels; mixed term remains
q, −q	0	0	Both local coefficients cancel

For the second row, $3^3+4^3+5^3-6^3=0$ while $3+4+5-6=6$. Thus a flat-spacetime triangle check does not by itself test the universal mixed term. Conversely, the first row has zero linear sum but a nonzero cubic sum.

The normalization and the absence of a pure degree-six tangent term follow from Álvarez-Gaumé and Vázquez-Mozo 2024, § 3, arXiv v2, pp. 5–10, eqs. (11), (17)–(19), (22), and the discussion around (28), Open PDF. The polynomial fixes the anomaly class, not the Ward identity in which a chosen descent representative places its variation. The standard representative that preserves diffeomorphism and local-Lorentz covariance puts the mixed variation in the $U(1)$ Ward identity; Consistent and Covariant Anomalies develops that distinction. A Bardeen-type counterterm can move the mixed variation to a geometric Ward identity, but it cannot set a nonzero $\kappa_1$ class to zero.

If $a$ is a fixed source for an exact global symmetry, nonzero coefficients are ’t Hooft-anomaly data. If $a$ is integrated over, the complete dynamical gauge anomaly must cancel. The same distinction applies if gravity is made dynamical: changing the Ward identity in which a mixed anomaly appears is not a cancellation mechanism.

In this exact ordinary-spin domain the bounded test also closes the fermionic global question:

$$\Omega^{\operatorname{Spin}}_5(\mathrm{pt})=0, \qquad \Omega^{\operatorname{Spin}}_5(BU(1))=0.$$

Consequently, when $\kappa_3=\kappa_1=0$, no additional Dai–Freed anomaly remains for the declared $\operatorname{Spin}(4)\times U(1)$ backgrounds. This conclusion stops immediately if the faithful symmetry is instead $\operatorname{Spin}^c$, a diagonal quotient, or another twisted tangential structure García-Etxebarria and Montero 2019, § 3.3, arXiv v3, pp. 21–23, eqs. (3.19)–(3.20), and Appendix B, pp. 64–65, table (B.1), Open PDF.

Large diffeomorphisms use the mapping-torus test

Let $\varphi:X\to X$ be an admissible orientation-preserving large diffeomorphism, and use the active convention $b^\varphi=(\varphi^{-1})^*b$. Choose a path $b_t$ with $b_0=b$ and $b_1=b^\varphi$, then glue the endpoints to form

$$T_\varphi =\frac{X\times[0,1]} {(x,0)\sim(\varphi(x),1)}.$$

The internal-bundle isomorphism, spin lift, and field domain must be glued too, including the choice of an additional $(-1)^F$ lift around the circle. These two mapping-circle lifts are distinguished in Witten 2016, § 3.3, arXiv v2, pp. 47–48, Open PDF. These formulas assume $\partial X=\varnothing$. If $\partial X\ne\varnothing$, one must instead specify an elliptic or Fredholm boundary problem, outward orientation, and the symmetry preserved by the boundary conditions. APS conditions are nonlocal and are not interchangeable with a physical local boundary condition; only the correctly combined boundary-plus-inflow system has the claimed invariance Witten 2016, § 4.4, arXiv v2, pp. 62–64, especially eq. (4.10), Open PDF.

After the total local anomaly has canceled, define the reduced spectral asymmetry of the total mapping-torus operator by

$$\bar\eta_{\mathrm{tot}}(T_\varphi) =\frac{\eta(\mathcal D_{T_\varphi,\mathrm{tot}};0) +\dim\ker\mathcal D_{T_\varphi,\mathrm{tot}}}{2} \pmod{\mathbb Z}.$$

The corresponding holonomy of the total complex chiral determinant line is

$$\operatorname{Hol}_\varphi(\mathcal L_{\mathrm{tot}}) =\exp\!\bigl(-2\pi i\,\bar\eta_{\mathrm{tot}}(T_\varphi)\bigr).$$

For real or pseudoreal fermions, a Pfaffian-line phase uses the corresponding real-eta or mod-two normalization; it is not obtained by simply relabeling this complex determinant formula.

At zero modes this formula denotes line transport, not a ratio of two nonzero partition-function numbers. Opposite chirality complex-conjugates the phase. If the total local polynomial is nonzero, the eta factor alone is not a bordism invariant; the complete regulated Dai–Freed or inflow phase is required. Witten’s original large-diffeomorphism construction and its mapping-torus form are given in Witten 1985, Introduction and §§ II, IV, pp. 198–199, 203–205, and 212–218, eqs. (17), (24), and (43), Open PDF. The modern determinant-line and inflow formulation, including zero modes and counterterms, is Witten and Yonekura 2021, §§ 2.1–2.5 and 3.1–3.3, arXiv v3, pp. 4–29 and 31–38, especially eqs. (2.13), (3.1)–(3.5), and (3.9)–(3.11), Open PDF.

One nontrivial phase proves an anomaly. A trivial result for one $T_\varphi$ clears only that diffeomorphism and spin lift; it is not a classification of every bordism class.

Discrete backgrounds can carry curvature-free phases

A finite internal symmetry has no Lie algebra connection to vary infinitesimally. Its background is a principal finite-group bundle, or equivalently a classifying map $a:X\to BG$. Such a bundle is locally flat but can have nontrivial transition functions and holonomy. Zero de Rham curvature is therefore not a zero-anomaly test.

There is no universal rule that simply reduces a continuous anomaly coefficient modulo the group order. A useful bounded formula illustrates why the exact total symmetry matters. For $n\ge2$ and the untwisted product $\operatorname{Spin}(4)\times\mathbb Z_n$, choose integer lifts $s$ of the $\mathbb Z_n$ charges and define

$$\Delta s_k =\sum_{i\in L}s_i^k-\sum_{j\in R}s_j^k.$$

The four-dimensional chiral-fermion anomaly vanishes precisely when

$$\begin{aligned} (n^2+3n+2)\Delta s_3 &\equiv0\pmod{6n}, \\ 2\Delta s_1 &\equiv0\pmod n. \end{aligned}$$

These congruences already combine the cubic and mixed gravitational data with the allowed charge lifts and counterterms. They are not the conditions for

$$\operatorname{Spin}^{\mathbb Z_{2m}}(4) =\frac{\operatorname{Spin}(4)\times\mathbb Z_{2m}} {\mathbb Z_2^{\mathrm{diag}}}, \qquad \mathbb Z_2^{\mathrm{diag}} =\left\langle\bigl((-1)^F,m\bigr)\right\rangle,$$

where $m$ is the order-two element of additive $\mathbb Z_{2m}$. Fermion charges allowed by this quotient are odd modulo $2m$. The quotient changes the allowed manifolds, representations, bordism problem, and anomaly conditions. Both cases, including the exact formulas and their symmetry-extension ceiling, are analyzed in Hsieh 2018, §§ 2.1–2.3 and 3.1–3.2, arXiv v1, pp. 4–21, eqs. (2.31)–(2.32) and (2.50), Open PDF.

A current preprint rederivation recovers and refines these finite anomaly coordinates while retaining the distinction between direct products and fermion-parity quotients Wan 2025, §§ I.C–I.D and IV.C–IV.D, arXiv v5, pp. 3–5 and 13–15, especially eqs. (11), (93), (97), and (102), Open PDF.

When the local polynomial vanishes, a residual finite phase can be tested as a character on the appropriate bordism classes. A nonzero bordism group only permits such phases; it does not prove that a specified fermion spectrum realizes one. Full finite-group, differential-cohomology, and bordism classification is beyond this reference.

Orientation reversal requires Pin data

An orientation-reversing symmetry changes the manifold category. In Euclidean signature its transition functions can reverse orientation, so an ordinary oriented spin bundle is no longer enough. With $w_i=w_i(TX)$,

$$\begin{aligned} X\text{ is orientable} &\Longleftrightarrow w_1=0, \\ X\text{ admits }\operatorname{Pin}^{+} &\Longleftrightarrow w_2=0, \\ X\text{ admits }\operatorname{Pin}^{-} &\Longleftrightarrow w_2+w_1^2=0. \end{aligned}$$

These tangent-bundle obstruction conditions are stated explicitly in Bais 2025, § 2, p. 5, Open PDF.

These signs use the tangent-bundle convention; normal-bundle conventions can interchange the labels. The theory must also specify how the reflection acts on internal quantum numbers and whether its square is $1$ or $(-1)^F$. In the standard continuation used below, Lorentzian $T^2=(-1)^F$ corresponds to $\operatorname{Pin}^{+}$, while $T^2=1$ corresponds to $\operatorname{Pin}^{-}$ Witten 2016, § 1.3 and Appendix A.2, arXiv v2, pp. 9 and 72–74, Open PDF.

A single oriented four-dimensional Weyl multiplet does not automatically admit this extension: reflection reverses chirality, so the field content and internal representation must close under the proposed operation before an orientation-reversing anomaly can even be tested. CPT under its theorem hypotheses does not supply separate $P$ or $T$ symmetry and does not choose a pin lift.

The standard bounded comparator is the $2+1$-dimensional Majorana boundary of a $3+1$-dimensional topological superconductor with $T^2=(-1)^F$. For a closed $\operatorname{Pin}^{+}$ four-manifold $M_4$, Witten’s normalization gives

$$Z_\nu(M_4) =\exp\!\left(-\frac{\nu\pi i}{2}\eta(M_4)\right).$$

Here $\eta(M_4)$ is the APS eta invariant in Witten’s real/Majorana $\operatorname{Pin}^{+}$ normalization, not the reduced $\bar\eta$ used for the preceding complex-Weyl determinant-line formula.

On $\mathbb{RP}^4$ the two pin structures give conjugate primitive phases,

$$Z_\nu(\mathbb{RP}^4) =\exp\!\left(\pm\frac{2\pi i\nu}{16}\right),$$

so interacting stacking identifies $\nu$ modulo $16$. This is a global orientation-reversing anomaly and inflow example, not a pure local four-dimensional gravitational anomaly and not a universal classification of time-reversal systems Witten 2016, Introduction and §§ 4.5–4.6, arXiv v2, pp. 7–9 and 64–66, Open PDF.

Each background sector selects a detector

Background data determine which anomaly detector is licensed

Sector	Additional datum	First detector	What a zero result does not establish
Continuous, oriented, local	Gauge and tangent connections on a declared spin background	Degree d+2 characteristic polynomial and descent	Absence of finite holonomy or torsion phases
Large diffeomorphism	Endpoint diffeomorphism, bundle lift, spin lift, and domain	Mapping torus and full eta or Dai–Freed phase	Triviality on other loops or non-mapping-torus bordisms
Finite internal symmetry	Exact finite group, global form, charge lattice, and fermion-parity quotient	Finite holonomy, eta invariant, or appropriate bordism character	The result for a different product or quotient symmetry
Orientation reversing	Unoriented background, Pin choice or twisted lift, and anti-linear action	Pin eta phase, mapping manifold, or bordism test	Anomaly freedom for the other Pin choice or other symmetry extension

A practical order is therefore: state the exact symmetry and whether it is background or dynamical; declare the manifold and tangential structure; run the local polynomial test; run the licensed finite/global tests; quotient by globally admissible counterterms; and only then state the physical verdict. Changing any entry restarts the problem rather than merely changing notation.

What this reference can and cannot decide

This taxonomy identifies the data and detector needed for a controlled anomaly claim. It does not classify all tangential structures, prove the Dai–Freed theorem, enumerate every finite-group bordism invariant, or derive transport and stress-tensor response.

For curved-space current and stress Ward identities, continue to Spin, Gauge, and Gravitational-Anomaly Responses. For determinant lines and $\eta$ theorems, see Global Anomalies, Determinant Lines, and Eta Invariants; for the full geometric background categories, see Tangential Structures: Oriented, Spin, and Framed Theories. Time-Reversal-Invariant Z2 Topological Insulators and Crystalline and Higher-Order Topological Matter develop time-reversal and crystalline applications. The next anomaly question is what survives renormalization-group flow, addressed by ’t Hooft Anomaly Matching.

Common pitfalls

Calling every curvature term gravitational. A term $c_1p_1(T)$ is mixed gauge–gravity. A pure gravitational term contains only tangent-bundle data.

Equating a gravitational anomaly with a trace anomaly. The first is a failure of a diffeomorphism or local-Lorentz Ward identity. The second is a failure of Weyl rescaling and can be present even when diffeomorphism invariance is exact.

Treating a counterterm as cancellation. A Bardeen counterterm can move a mixed consistent variation between Ward identities. The nontrivial total class remains unless another sector cancels it.

Using curvature to test a finite bundle. A flat discrete background can have nontrivial holonomy and torsion data. Its zero de Rham curvature is not evidence of anomaly freedom.

Using Pin$^+$ and Pin$^-$ interchangeably. They impose different Stiefel–Whitney conditions and encode different reflection squares. The normal/tangent convention and the action of $(-1)^F$ must be fixed first.

Applying a reflection to field content that it does not preserve. A single chiral representation may be mapped to a missing opposite-chirality or conjugate multiplet. That is failure to define the claimed symmetry, before the anomaly question begins.

Promoting one global test to a classification. One nontrivial phase is a witness. One trivial mapping torus does not rule out other diffeomorphisms, bundles, pin structures, or bordism classes.

Check your understanding

1. Why is there no pure local gravitational term in the anomaly polynomial of an ordinary four-dimensional Weyl fermion?

Solution

The local polynomial has degree six. Pure tangent-bundle contributions come from $\widehat A(T)$, whose Pontryagin terms have degrees divisible by four, so $[\widehat A(T)]_6=0$. Gauge curvature can supply the missing degree: $c_1p_1(T)$ is therefore allowed, but it is mixed rather than pure.

2. Evaluate the two coefficients for left-handed charges $(3,4,5,-6)$.

Solution

$$\kappa_3=27+64+125-216=0, \qquad \kappa_1=3+4+5-6=6.$$

The pure cubic $U(1)^3$ anomaly cancels while the mixed $U(1)$–gravity anomaly remains.

3. A Bardeen counterterm moves the mixed variation from the $U(1)$ Ward identity to the gravitational Ward identity. Has the anomaly canceled?

Solution

No. The counterterm changes the consistent representative and the allocation among Ward identities. It does not remove the total characteristic class or make both symmetries simultaneously exact.

4. Why can a flat $\mathbb Z_n$ background still detect an anomaly?

Solution

Flatness removes local curvature but not transition functions, finite holonomy, or torsion characteristic data. The anomaly can therefore appear as a finite eta or bordism phase. One must also specify whether the symmetry is a direct product with spin or a quotient sharing fermion parity.

5. A theory is defined on oriented spin manifolds and is invariant under CPT. Does that establish an anomaly-free time-reversal symmetry on unorientable manifolds?

Solution

No. One must first define a separate time-reversal action on the fields, state its square relative to $(-1)^F$, choose the corresponding Pin or twisted structure, and verify that the multiplets close under it. Only then is an unorientable global anomaly test meaningful.

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.

• Bais, Valentina. “$\operatorname{Pin}^{\pm}$-Structures on Non-Oriented 4-Manifolds via Lefschetz Fibrations.” Proceedings of the Royal Society of Edinburgh Section A: Mathematics, First View (2025): 1–20. DOI. Open PDF.

• Brennan, T. Daniel, and Kenneth Intriligator. “Anomalies of 4d $\operatorname{Spin}_G$ Theories.” Journal of High Energy Physics 2024, no. 7 (2024): 157. DOI. Open PDF, arXiv v3.

• García-Etxebarria, Iñaki, and Miguel Montero. “Dai–Freed Anomalies in Particle Physics.” Journal of High Energy Physics 2019, no. 8 (2019): 003. DOI. Open PDF, arXiv v3.

• Hsieh, Chang-Tse. “Discrete Gauge Anomalies Revisited.” arXiv:1808.02881v1 [hep-th], 2018. Stable record. Open PDF, arXiv v1.

• Monnier, Samuel. “A Modern Point of View on Anomalies.” Fortschritte der Physik 67, nos. 8–9 (2019): 1910012. DOI. Open PDF, arXiv v2.

• Wan, Zheyan. “Anomaly of 4d Weyl Fermions with Discrete Symmetries.” arXiv:2506.19710v5 [hep-th], 2025; revised 28 July 2026. Stable record. Open PDF, arXiv v5.

• Witten, Edward. “Fermion Path Integrals and Topological Phases.” Reviews of Modern Physics 88, no. 3 (2016): 035001. DOI. Open PDF, arXiv v2.

• Witten, Edward. “Global Gravitational Anomalies.” Communications in Mathematical Physics 100, no. 2 (1985): 197–229. DOI. Open PDF.

• 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.