't Hooft Anomaly Matching

An exact global symmetry carries the same anomaly in the ultraviolet and the infrared. Renormalization can change fields, couplings, and the way the symmetry is realized, but it can change the anomalous transformation law only by the variation of admissible local terms. It therefore cannot erase a nontrivial anomaly class. A proposed infrared description must reproduce the ultraviolet class through gapless fields, Goldstone and Wess–Zumino terms, symmetry-compatible topological order, a fixed inflow bulk, or a combination of these mechanisms.

This is a necessary constraint, not a phase-selection theorem. A mismatch rules out an infrared proposal, while a match does not prove that the proposed phase, composite spectrum, or duality is dynamically realized. The discussion below concerns exact global symmetries probed by fixed backgrounds. An anomaly of a dynamical gauge redundancy must cancel in the complete theory rather than merely reappear at low energy.

Required background. What Is an Anomaly? supplies the counterterm quotient and the distinction between a background global symmetry and a dynamical gauge redundancy. Anomaly Polynomials and Inflow supplies the local descent convention, the relative bulk–boundary formulation, and the global completion problem.

Helpful background. Global and Torsion Anomalies supplies the holonomy and mapping-torus data that a polynomial does not see. Symmetry Realization and Order Parameters distinguishes exact, spontaneously broken, explicitly broken, and emergent symmetries.

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

Matching equates anomaly classes, not spectra

Let a $d$-dimensional unitary local QFT have an exact global symmetry $G$. Denote by $b$ the complete fixed background: the $G$-bundle and connection, metric, tangential structure, fermion-parity quotient, boundary domain, and any other datum needed to define the theory. For an admissible transformation $g$, a nonzero local trivialization of the quantum functional obeys

$$Z_{\mathcal T}[g\!\cdot\! b] = e^{i\Theta_{\mathcal T}(g;b)}Z_{\mathcal T}[b].$$

When zero modes make the numerical partition function vanish, the invariant statement is parallel transport in the determinant or anomaly line rather than this ratio. A local counterterm $C[b]$ shifts the phase by

$$\Theta_{\mathcal T}(g;b) \longmapsto \Theta_{\mathcal T}(g;b)+C[g\!\cdot\! b]-C[b].$$

The anomaly is the resulting equivalence class $[\Theta_{\mathcal T}]$. Suppose an RG flow is generated by $G$-invariant deformations and retains the same background category. Then ’t Hooft matching is the equality

$$\boxed{ [\Theta_{\mathrm{UV}}] = [\Theta_{\mathrm{IR}}] }$$

for the complete infrared theory, including decoupled topological sectors, Goldstone terms, and any physical inflow bulk. In the local perturbative sector this implies

$$[I_{d+2}^{\mathrm{UV}}] = [I_{d+2}^{\mathrm{IR}}],$$

where the brackets again mean equality modulo allowed representative shifts. It does not say that the UV and IR spectra, correlation functions, or individual triangle diagrams are equal. Nor is it the statement that Weyl trace-anomaly coefficients such as $a$ and $c$ are constant along a generic flow.

The original matching condition was formulated for chiral flavor anomalies in ’t Hooft 1980, §§ III10–III12, pp. 149–151. The classic four-dimensional massless-composite argument and its physical scope are reviewed in Harvey 2005, § 2.3, arXiv v1, pp. 19–20, Open PDF. The modern statement that the entire global anomaly is invariant under RG flow, rather than only a triangle coefficient, is summarized in Bhardwaj et al. 2024, Introduction and § 4 opening, arXiv v2, pp. 3 and 59, Open PDF.

If the infrared symmetry is larger, with a homomorphism $\iota:G_{\mathrm{UV}}\to G_{\mathrm{IR}}$, only the pullback is fixed:

$$\iota^*[\Theta_{\mathrm{IR}}] = [\Theta_{\mathrm{UV}}].$$

Anomalies involving genuinely accidental IR generators are not fixed by UV anomaly matching without additional data or dynamical hypotheses that relate the IR symmetry generators and charged operators to UV operator content. A 2026 analysis shows how mixed anomalies of an emergent symmetry can constrain non-genuine UV operators when those extra hypotheses hold Gu, Pei, and Yu 2026, § 1, arXiv v2, pp. 1–3, and § 6, pp. 17–22, especially eqs. (6.4)–(6.14), Open PDF.

More generally, a UV zero-form symmetry can act nonfaithfully on local IR fields or transmute into a higher-form symmetry. Matching is then expressed by a map of background fields,

$$[\Theta_{\mathrm{UV}}(A_{\mathrm{UV}})] = [\Theta_{\mathrm{IR}}(\Phi(A_{\mathrm{UV}}))],$$

not necessarily by a homomorphism of ordinary groups. This current refinement and its background-field formulation are developed in Seiberg and Seifnashri 2025, §§ 1.1–1.2 and 7, arXiv v2, pp. 3–7 and 50–52, especially eqs. (1.1)–(1.4) and (7.1)–(7.2), Open PDF.

Locality is the reason the class cannot disappear

Choose a scale $\mu$ below the modes that have been integrated out, retain every light and topological sector, and couple both descriptions to the same slowly varying background $b$. The complete IR theory reproduces the low-energy nonlocal response. Once that common response is included, the remaining scheme dependence of the anomalous transformation law is the variation of an admissible local functional. At the level of anomaly phases this gives

$$\Theta_{\mathrm{UV}}(g;b) - \Theta_{\mathrm{IR}}(g;b) = C_\mu[g\!\cdot\! b]-C_\mu[b] \pmod{2\pi}.$$

The right-hand side is precisely a trivial representative. If the UV class is nontrivial, no choice of local $C_\mu$ can make the complete IR class trivial. This is why anomaly matching is insensitive to whether the useful infrared variables are elementary fields, composites, Goldstones, or topological degrees of freedom.

The argument has sharp hypotheses. The exact group, its faithful global form, charge lattice, embedding into any larger group, tangential structure, allowed bundles, and boundary conditions must be held fixed. If a deformation explicitly breaks $G$ to $H$, only the restricted $H$ anomaly must match. Spontaneous breaking is different: the theory still has exact $G$, so its anomaly survives in a nonlinear realization. On a boundary, the invariant object is the complete boundary-plus-fixed-bulk system; silently dropping the bulk changes the theory being compared.

An anomaly can equivalently be represented by an invertible theory in one higher dimension, with the anomalous QFT understood as relative to it. Under RG flow that inverse-anomaly theory is unchanged even though a local Chern–Simons representative may move. This line-valued formulation is given in Freed 2014, §§ 2.2–2.3, arXiv v2, pp. 4–6, eqs. (2.5)–(2.10), Open PDF.

A four-dimensional Weyl flow matches gauge and gravity terms

Consider a closed oriented Euclidean spin four-manifold $X_4$ and the exact direct-product background $\operatorname{Spin}(4)\times U(1)$. Let $a$ be a compact unit-charge $U(1)$ connection with global curvature $f$; only locally need $f=\mathrm da$. Define

$$c_1=\frac{f}{2\pi}, \qquad p_1(T)=-\frac{1}{8\pi^2} \operatorname{tr}_{\mathrm{vec}}(\mathcal R^2).$$

With the site’s continuation, a physical Lorentzian left-handed Weyl fermion has negative Euclidean chirality. For left-handed charges $q_i$, set

$$\kappa_3=\sum_i q_i^3, \qquad \kappa_1=\sum_i q_i.$$

Its local anomaly polynomial is

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

Take two left-handed Weyl fields $\psi_1,\psi_2$ of charges $1,2$ and a complex scalar $\Phi$ of charge $-3$. The Yukawa interaction

$$y\Phi\psi_1\psi_2+\mathrm{h.c.}$$

is $U(1)$-invariant. The scalar does not contribute a chiral anomaly, while the fermions give

$$\kappa_3=1^3+2^3=9, \qquad \kappa_1=1+2=3.$$

Therefore

$$\begin{aligned} I_6^{\mathrm{UV}} &=-\frac{3}{2}c_1^3+\frac18c_1p_1(T) =c_1K_4, \\ K_4 &=-\frac32c_1^2+\frac18p_1(T). \end{aligned}$$

This normalization follows from the degree-six index polynomial in Álvarez-Gaumé and Vázquez-Mozo 2024, § 3, arXiv v2, pp. 5–8, eqs. (11), (17)–(19), and (22), Open PDF. In the representative that preserves diffeomorphism and local-Lorentz covariance and places the mixed term in the $U(1)$ Ward identity,

$$\delta_\alpha W_{E,\mathrm{UV}} = -i\int_{X_4}\alpha K_4.$$

Now choose a symmetry-broken infinite-volume pure phase, or its phase-selected semiclassical EFT; the closed $X_4$ below is a compact background probe of that EFT, not a claim of spontaneous breaking in a generic finite-volume state. In a smooth Goldstone patch write

$$\Phi=(v+\rho)e^{i\vartheta}, \qquad \delta_\alpha\vartheta=-3\alpha.$$

The radial mode and the two Weyl fermions become massive, but the exact continuous symmetry has been spontaneously—not explicitly—broken. Define

$$\mathcal J_4 = \left[\widehat A(T)\operatorname{ch}(L)\right]_4 = \frac12c_1^2-\frac1{24}p_1(T), \qquad K_4=-3\mathcal J_4.$$

The local smooth-Goldstone representative

$$W_{E,\mathrm{WZ}}^{\mathrm{loc}} = -i\int_{X_4}\vartheta\,\mathcal J_4$$

has exactly the missing variation,

$$\delta_\alpha W_{E,\mathrm{WZ}}^{\mathrm{loc}} = -i\int_{X_4}\alpha K_4 = \delta_\alpha W_{E,\mathrm{UV}}.$$

Its coefficient passes the periodicity check because

$$\int_{X_4}\mathcal J_4 = \operatorname{ind}\!\left(D_X^+\!\otimes L\right) \in\mathbb Z.$$

Thus $\vartheta\sim\vartheta+2\pi$ changes the Euclidean action by an integer multiple of $-2\pi i$. The condensate leaves an exact $\mathbb Z_3$ subgroup; the two fermion charges become $1$ and $2\equiv-1$ modulo $3$, so their mass is vectorlike under the unbroken subgroup. The exact untwisted $\operatorname{Spin}\times\mathbb Z_3$ congruences discussed on Gravitational, Mixed, Discrete, and Orientation-Reversing Anomalies also vanish:

$$20(1^3+2^3)=180\equiv0\pmod{18}, \qquad 2(1+2)=6\equiv0\pmod3.$$

This is a genuine matching mechanism: the massive fermions no longer appear as light particles, yet their cubic and mixed gauge–gravity anomaly survives in the Goldstone functional. The general construction and its global conditions are developed in Yonekura 2021, Introduction and §§ 2–4, arXiv v3, pp. 1–12, especially eqs. (3.8)–(3.11), Open PDF.

The displayed $\vartheta\mathcal J_4$ term is not a functional on every $U(1)$ background. The charge-$-3$ field is a section of $L^{-3}$; a nowhere-zero section trivializes $L^{-3}$, equivalently reducing the principal $U(1)$ bundle to its $\mathbb Z_3$ subgroup. A general bundle may force vortices or zeros where a single smooth phase does not exist. Matching on arbitrary allowed backgrounds then requires the globally gauged Wess–Zumino construction, its inflow completion, and the defect sectors. The local calculation fixes the coefficient and variation; it does not waive those global conditions Yonekura 2021, § 4, arXiv v3, pp. 10–12, Open PDF.

For contrast, a constant invariant mass can pair left-handed charges $q$ and $-q$. That pair has $\kappa_3=\kappa_1=0$, so integrating it out leaves no anomaly deficit and requires no Wess–Zumino remnant.

The infrared has several matching mechanisms

A nontrivial anomaly rules out a unique, symmetry-preserving, trivially gapped standalone infrared theory. It does not by itself decide which nontrivial alternative occurs. The complete IR anomaly is the sum of every sector and every required relative contribution. Read each row of the table from the candidate carrier to the check it must pass and then to the conclusion that matching still does not license.

Ways an infrared theory can reproduce one fixed ultraviolet anomaly

Infrared realization	Carrier of the anomaly	Required check	What matching does not prove
Gapless fields or a CFT	Elementary or composite operators and their background functional	All local and global anomaly data agree	That the proposed spectrum or fixed point is dynamically realized
Spontaneous breaking, G → H	Goldstone Wess–Zumino term plus the remaining H-sector	The H anomaly matches and the Goldstone term is globally quantized	That Goldstones alone suffice on every background
Symmetric topological order	A TQFT with a specified anomalous symmetry action	Its background coupling reproduces the same anomaly theory	That such a unitary TQFT exists for an arbitrary anomaly
Discrete symmetry breaking	Vacua, domain walls, junctions, and any wall theories	The full symmetry action and defect anomaly are included	That vacuum degeneracy alone completes the match
Relative boundary theory	Boundary response paired with a fixed invertible bulk	Bulk and boundary variations cancel on the same backgrounds	That the isolated boundary is an absolute anomaly-free QFT

For breaking $G\to H$, the non-Goldstone IR sector must reproduce the pullback of the UV anomaly to $H$. A Goldstone term supplies the remaining nonlinear $G$ variation only when the anomaly admits the required transgression and global quantization. If it does not, additional gapless or topological degrees of freedom are compulsory.

Topological order is likewise a conditional option, not a universal escape. Some anomaly classes admit symmetry-preserving gapped boundaries, while other classes obey obstructions that no unitary symmetry-preserving TQFT can satisfy. For the discrete generalized-symmetry systems and mapping-torus backgrounds treated there, a precise necessary obstruction and explicit saturating examples are given in Córdova and Ohmori 2020, §§ 1.2–1.5 and Appendix A, arXiv v2, pp. 5–11 and 27–29, especially eq. (1.7), Open PDF.

For a fixed inflow bulk, the inherited signs are

$$\delta_\Lambda W_{E,\mathrm{IR}} =-2\pi i\int_XQ_d^{(1)}, \qquad \delta_\Lambda W_{E,\mathrm{bulk}} =+2\pi i\int_XQ_d^{(1)}.$$

Their sum is invariant. This is matching for a relative system, not a way to turn the isolated anomalous boundary into a standalone gauge theory.

Global and torsion data must match too

Equality of anomaly polynomials is only the free, local part of the test. If $\mathcal L_{\mathrm{UV}}$ and $\mathcal L_{\mathrm{IR}}$ are the anomaly lines over the same background space, then every admissible loop $\gamma$ must have the same counterterm-quotiented holonomy:

$$\operatorname{Hol}_\gamma(\mathcal L_{\mathrm{UV}}) = \operatorname{Hol}_\gamma(\mathcal L_{\mathrm{IR}}).$$

For example, one ordinary-spin $SU(2)$ Weyl doublet has vanishing local cubic polynomial but acquires a minus sign under the nontrivial class in $\pi_4(SU(2))$. Any IR description of the same exact background symmetry must reproduce that sign. Matching every perturbative trace while missing this mod-two phase is still a failed match. The primary example is Witten 1982, pp. 324–328; the mapping-torus and mod-two diagnostic is developed on Global and Torsion Anomalies.

Conversely, checking one mapping torus does not establish equality of the full anomaly theories. The global form of the symmetry, allowed bundles, tangential structure, fermion-parity quotient, and non-mapping-torus bordism classes remain part of the comparison.

What matching rules out—and what it leaves open

The theorem should be applied in the following order:

1. State the exact microscopic symmetry, faithful global form, and complete background category.
2. Compute the UV anomaly modulo globally admissible local counterterms.
3. State which symmetry is exact in the proposed IR and how the UV group maps into it.
4. Include every gapless, Goldstone, topological, defect, and inflow sector.
5. Compare both local and global anomaly data on the same backgrounds.

A mismatch is decisive evidence against the proposal. A match is only a consistency check. It does not calculate a mass gap, prove confinement, select a vacuum, establish operator completeness, or prove a duality.

Several apparent exceptions are changes of question:

• An explicitly $G$-breaking deformation removes the $G$ matching theorem; only its exact subgroup remains constrained.
• Spontaneous breaking keeps $G$ exact, so the anomaly is carried by the nonlinear Goldstone, defect, and residual-sector data.
• Gauging the anomalous background field demands cancellation of the complete gauge anomaly. A matched nonzero class is not sufficient.
• An accidental infrared enlargement is constrained only after pulling its anomaly back to the exact microscopic subgroup.
• Changing a quotient, charge lattice, spin or Pin structure, boundary condition, or inflow bulk changes the anomaly problem and requires a new comparison.

Concrete strong-coupling phase constraints are developed in Anomaly and Generalized-Symmetry Constraints on Infrared Phases. For the detailed operator, chiral-ring, and coefficient checks across $\mathcal N=1$ Seiberg duality, continue to Operator Dictionaries, Chiral Rings, and Anomaly Matching.

Common pitfalls

Matching only massless fermion traces. Light fermions are one possible carrier. Goldstone terms, TQFTs, defects, and inflow contributions belong in the same total anomaly functional.

Treating spontaneous breaking as an exception. The state breaks the symmetry, but the theory does not. The nonlinear realization must still match the UV anomaly, including its restriction to the unbroken subgroup.

Assuming any topological order can absorb any anomaly. The symmetry action on the TQFT is extra structure and may itself be obstructed. One must construct and test it rather than cite “topological order” as a placeholder.

Using perturbative equality as a global proof. Equal anomaly polynomials do not compare torsion phases or determinant-line holonomy. Global data need their own tests.

Calling a match proof of a phase or duality. Matching supplies a necessary condition. Independent dynamical and operator-level evidence remains essential.

Check your understanding

1. A vectorlike massive pair

Two left-handed Weyl fermions have charges $q$ and $-q$ under a fixed $U(1)$ background. Show that a constant mass can remove them without an anomaly remnant.

Solution

The bilinear has charge $q-q=0$, so the mass preserves $U(1)$. The pair gives

$$\kappa_3=q^3+(-q)^3=0, \qquad \kappa_1=q+(-q)=0.$$

It contributes neither the cubic nor the mixed gauge–gravity coefficient. Integrating it out therefore leaves no anomaly deficit.

2. The Goldstone variation

In the charge-$(1,2)$ example, verify the variation of the local Wess–Zumino term.

Solution

Because $\delta_\alpha\vartheta=-3\alpha$ and $K_4=-3\mathcal J_4$,

$$\delta_\alpha W_{E,\mathrm{WZ}}^{\mathrm{loc}} =-i\int_{X_4}(-3\alpha)\mathcal J_4 =-i\int_{X_4}\alpha K_4.$$

This equals the UV consistent anomaly in the declared representative.

3. A claimed trivial symmetric infrared

A theory has a nontrivial global anomaly, but a proposed IR phase has a unique vacuum, a gap, unbroken symmetry, no topological order, and no physical inflow bulk. What does matching say?

Solution

The proposal is impossible under the stated assumptions. Its complete low-energy background functional would be local and could carry only a counterterm-trivial representative, contradicting the nontrivial UV class. At least one assumption—gaplessness, symmetry realization, topological sector, relative bulk, or even the claimed exact symmetry—must change.

4. Explicit versus spontaneous breaking

Why does a $G$-breaking relevant operator change the matching theorem while a $G$-invariant flow into a $G\to H$ broken phase does not?

Solution

The first deformation changes the theory and leaves only $H$ exact, so only the restricted $H$ anomaly is protected. In spontaneous breaking the action and full quantum theory remain $G$-symmetric; the selected state is not. Goldstone, defect, and residual-$H$ sectors must therefore reproduce the full $G$ anomaly nonlinearly.

5. A polynomial match with a sign mismatch

Two proposed descriptions have identical local anomaly polynomials, but one has holonomy $-1$ and the other $+1$ around an admissible mapping-torus loop. Do they match?

Solution

No. The polynomial compares only local curvature data. The unequal holonomies are distinct global anomaly classes, so the proposed IR theory fails the complete matching condition.

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.
• Bhardwaj, Lakshya, Lea E. Bottini, Ludovic Fraser-Taliente, Liam Gladden, Dewi S. W. Gould, Arthur Platschorre, and Hannah Tillim. “Lectures on Generalized Symmetries.” Physics Reports 1051 (2024): 1–87. DOI. Open PDF, arXiv v2.
• Córdova, Clay, and Kantaro Ohmori. “Anomaly Obstructions to Symmetry Preserving Gapped Phases.” Physical Review D 102, no. 2 (2020): 025011. DOI. Open PDF, arXiv v2.
• Freed, Daniel S. “Anomalies and Invertible Field Theories.” Proceedings of Symposia in Pure Mathematics 88 (2014): 25–46. DOI. Open PDF, arXiv v2.
• Gu, Wei, Du Pei, and Xingyang Yu. “On Mixed ’t Hooft Anomalies of Emergent Symmetries.” Journal of High Energy Physics 2026, no. 7 (2026): 195. DOI. Open PDF, arXiv v2.
• Harvey, Jeffrey A. “TASI 2003 Lectures on Anomalies.” arXiv:hep-th/0509097v1, 2005. Stable record. Open PDF.
• ’t Hooft, Gerard. “Naturalness, Chiral Symmetry, and Spontaneous Chiral Symmetry Breaking.” In Recent Developments in Gauge Theories, edited by Gerard ’t Hooft, C. Itzykson, A. Jaffe, H. Lehmann, P. K. Mitter, I. M. Singer, and R. Stora, 135–157. NATO Advanced Study Institutes Series B, vol. 59. New York: Plenum Press, 1980. DOI. Open PDF.
• Seiberg, Nathan, and Sahand Seifnashri. “Symmetry Transmutation and Anomaly Matching.” Journal of High Energy Physics 2025, no. 9 (2025): 014. DOI. Open PDF, arXiv v2.
• Witten, Edward. “An SU(2) Anomaly.” Physics Letters B 117, no. 5 (1982): 324–328. DOI.
• Yonekura, Kazuya. “General Anomaly Matching by Goldstone Bosons.” Journal of High Energy Physics 2021, no. 3 (2021): 057. DOI. Open PDF, arXiv v3.