From Projective Representations to Anomalies Preview

A projective representation gives a consistent action of a group on quantum rays. An anomaly concerns a quantum symmetry’s compatibility with additional structures, especially local background fields or a dynamical gauge field. The ideas are related, but a phase in a multiplication law is not by itself evidence of a failed conservation law.

Projective Symmetry and Its Cocycle

For unitary representatives of a group $G$, write

$$U(g)U(h)=\omega(g,h)U(gh),\qquad |\omega(g,h)|=1.$$

Associativity requires

$$\omega(g,h)\omega(gh,k)=\omega(h,k)\omega(g,hk).$$

Changing representatives to $U'(g)=e^{i\beta(g)}U(g)$ changes the phase by

$$\omega'(g,h)=e^{i[\beta(g)+\beta(h)-\beta(gh)]}\omega(g,h).$$

A phase that cannot be removed this way defines a nontrivial projective class. Antiunitary transformations require a modified cocycle rule because they conjugate phases; the formulas here concern unitary symmetries.

A spin-half system is consistent under rotations: it carries a projective representation of $SO(3)$ and an ordinary representation of its double cover $SU(2)$. The sign under a $2\pi$ rotation does not violate angular-momentum conservation. Projective Representations develops the quantum-mechanical construction.

Three Different Anomaly Questions

Does a classical current remain conserved? The axial, or Adler–Bell–Jackiw, anomaly is an example of quantum current nonconservation. For a massless charged Dirac field in four spacetime dimensions, a gauge-invariant definition of the vector current leaves an axial divergence proportional to $F_{\mu\nu}\widetilde F^{\mu\nu}$. The coefficient depends on the specified charge, current, and dual-tensor normalization. Parallel electric and magnetic fields provide a setting where axial charge can change. This is a physical quantum effect, distinct from adding an explicit fermion mass.

Can an exact global symmetry be gauged? A ‘t Hooft anomaly obstructs a fully invariant, local coupling to background gauge fields, even though the global symmetry can remain exact without those backgrounds. It constrains the infrared realization of that symmetry. It does not automatically make the ungauged theory inconsistent or require the symmetry to be broken. This distinction is explained in Tong, Section 3.5.

Is a proposed dynamical gauge theory consistent? A gauge anomaly obstructs the redundancy needed to remove unphysical descriptions. A standalone gauge theory must cancel this obstruction; sometimes additional degrees of freedom or a bulk inflow mechanism cancel it in a larger system. Calling it small does not repair the redundancy.

These are physical roles, not mutually exclusive diagram labels. A mixed anomaly can involve several symmetries, and which fields are dynamical matters.

Global Anomalies Are Not Simply Anomalies of Global Symmetries

“Perturbative” anomalies can be seen in infinitesimal transformations and local Ward identities. A global anomaly instead detects a failure under transformations or topological configurations that such an infinitesimal calculation misses. It can affect a dynamical gauge symmetry.

Witten’s four-dimensional $SU(2)$ example has an odd number of left-handed Weyl doublets. The perturbative triangle gauge anomaly vanishes, yet a large gauge transformation changes the fermion functional integral by a sign. The uncanceled sign makes that gauge theory inconsistent. Thus checking the Lie algebra and triangle diagrams is insufficient.

This use of “global” refers to the global structure of the transformation problem. It must not be substituted for “physical global symmetry.”

How Projective Representations Fit the Anomaly Language

The connection becomes precise in quantum mechanics regarded as a $0+1$-dimensional theory. A nontrivial projective action of a specified group $G$ can represent a ‘t Hooft anomaly for gauging that $G$. The quantum system remains consistent as a system with a global ray action; the obstruction appears when demanding the additional gauging structure.

For spin half, distinguishing $SO(3)$ from $SU(2)$ therefore matters. Passing to the cover makes the representation linear, but changes the group whose gauging is being discussed. This does not turn all higher-dimensional gauge anomalies into ordinary central extensions. One must specify the dimension, symmetry group, allowed background fields, and what consistency condition is obstructed.

The same caution applies to magnetic translations: a flux-dependent projective multiplication law is a well-defined quantum symmetry algebra. Calling it an anomalous current divergence would add a conclusion not contained in that law.

Topological Phases and Inflow

An anomalous boundary theory can sometimes be consistently combined with a bulk whose action has the opposite variation. The total bulk-plus-boundary system is invariant; the boundary alone need not admit that symmetry as a standalone local theory with arbitrary backgrounds. Berry Phase and Topological Terms introduces the action phases used in such descriptions.

Exercises

1. Derive the cocycle condition by associating $U(g)U(h)U(k)$ in two ways.

Solution

The left-associated product is $\omega(g,h)\omega(gh,k)U(ghk)$; the right-associated product is $\omega(h,k)\omega(g,hk)U(ghk)$. Their equality gives the stated condition. An arbitrary assignment of pairwise phases need not define a consistent projective representation.

2. Does the spinor sign under a $2\pi$ rotation prove nonconservation of angular momentum?

Solution

No. It is compatible with an exact rotational symmetry and conserved generators. In the more specific $0+1$-dimensional question of gauging $SO(3)$, the nontrivial projective class can be an anomaly. The conservation question and the gauging question are different.

3. If a global symmetry has a ‘t Hooft anomaly, must it fail to be a symmetry of the quantum theory?

Solution

No. It can remain exact. The obstruction concerns its consistent coupling to background gauge fields, and hence constrains possible infrared descriptions. This differs from the axial current’s nonzero divergence in a dynamical gauge theory.

4. Why is a vanishing perturbative gauge anomaly insufficient in the $SU(2)$ doublet example?

Solution

Infinitesimal transformations do not detect the sign acquired under the relevant large transformation. Each Weyl doublet contributes a sign; an odd number leaves a minus sign, whereas an even number cancels this particular obstruction.

References

• S. L. Adler, “Axial-vector vertex in spinor electrodynamics,” Physical Review 177, 2426–2438 (1969).
• V. Bargmann, “On unitary ray representations of continuous groups,” Annals of Mathematics 59, 1–46 (1954).
• J. S. Bell and R. Jackiw, “A PCAC puzzle: $\pi^0\to\gamma\gamma$ in the sigma model,” Nuovo Cimento A 60, 47–61 (1969).
• K. Fujikawa, “Path-integral measure for gauge-invariant fermion theories,” Physical Review Letters 42, 1195–1198 (1979).
• G. ‘t Hooft, “Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking,” in Recent Developments in Gauge Theories, Plenum Press (1980).
• D. Kapec, R. Mahajan, and D. Stanford, “Matrix ensembles with global symmetries and ‘t Hooft anomalies from 2d gauge theory,” Journal of High Energy Physics 2020(04), 186 (2020). Author preprint.
• D. Tong, Gauge Theory, University of Cambridge lecture notes (2018), Section 3. Anomalies.
• E. Witten, “An SU(2) anomaly,” Physics Letters B 117, 324–328 (1982). DOI.