We know that there is Adler and Bell-Jackiw(ABJ) type anomalies for fermions. In some case, the ABJ anomaly affecs particle physics pheonomelogy, such as pion decays or kaon decays(in the case of pion, we still have a calculation on left/right chiral fermions running on the 1-loop triangle diagram). In some other case, there is 1+1D QED Schiwinger or axial anomaly for chiral fermions. The commutation of fermionic anomaly is usually done by, either a 1-loop Feynman diagram, or a Fujikawa path integral method.

The above may be some examples of anomalies for fermions.

Is there any example of quantum anomalies for bosons (pure bosonic systems)?

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    $\begingroup$ Naively, one wouldn't expect that for the following reason: When you use the prescription of dimensional regularization, you realize that analytically continuing $\gamma_5$ has subtleties and that is what gives an anomaly for a loop involving a chiral fermion. $\endgroup$
    – Siva
    Mar 21 '14 at 2:56
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    $\begingroup$ And this seems to have come up on arXiv recently -- arxiv.org/abs/1403.5256 $\endgroup$
    – Siva
    Mar 21 '14 at 3:05
  • $\begingroup$ Anomalies are more general the chiral fermions, as the answer below points out. It is just that chiral fermions are particularly "unnatural" (for example, they cannot be put on a lattice) and therefore in general any internal symmetry of chiral fermions will be anomalous. Whereas bosons, (especially spin-0 bosons) are very natural, and have an obvious lattice completion that respects internal symmetries. I should say I find thinking in terms of lattice completions helpful, although I understand it is not essential to do so. $\endgroup$ Mar 23 '14 at 20:39
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    $\begingroup$ @ BebopButUnsteady: Indeed the lattice thinking is helpful, but even for bosonic theory, there can be anomalous bosons which cannot regularized without anomalies. There are L and H types theory discussed here: arxiv.org/abs/1405.5858, you may be interested in. $\endgroup$
    – wonderich
    Jul 17 '14 at 22:26

The worldsheet Weyl anomaly in bosonic string theory is an example. More generally in any dimension you can have trace and Weyl anomalies that break scale or conformal invariance, even in systems with only bosons.

  • $\begingroup$ The conformal anomaly is what you calculate when you do the renormalization group, so whenever you have a non-trivial renormalization group flow you have an anomaly. $\endgroup$ Mar 23 '14 at 20:26
  • $\begingroup$ There are really two types of conformal anomaly. The effect you refer to is due to the introduction of a scale to regulate divergences, and occurs even in theories that are not classically conformal. The Weyl anomaly is an anomaly due to the presence of some scale of the background geometry (the anomaly terms are proportional to things like the Weyl tensor and Euler characteristic) and is more similar to the chiral anomaly in that it vanishes when the "background field" is trivial. $\endgroup$
    – Dan
    Mar 25 '14 at 2:27
  • $\begingroup$ @ Dan, would you mind to provide some Refs for your claim? many thanks. :) $\endgroup$
    – user32229
    Jul 17 '14 at 22:36

According to the theory of Symmetry Protected Topological (SPT) States: The underlying ultraviolet (UV) theory (at the cutoff/lattice scale) could be formed by fundamental bosons or fundamental fermions. For a $d+1$-dimensional gapped ground state that cannot be deformed to a trivial ground state under local unitary transformations when it is protected by global symmetry, we then have $d+1$-dimensional SPT states. The $d$-dimensional boundary of SPT states cannot be regularized and UV-complete in its own dimensions, unless the global symmetry is realized in an anomalous non-onsite manner. Accordingly, the anomalous global symmetry cannot be gauged, thus it is similar to the 't Hooft anomaly in $d$-dimensional spacetime -- the obstruction to gauge the global symmetry in an onsite manner.

In this sense, the conventional gauge anomalies (including 't Hooft anomaly) is actually the anomalies of realizing the global symmetries in the UV complete and in its own dimensions. And the "gauge" part of this "gauge" anomaly occurs (1) when you try to couple the anomalous global symmetry to a non-dynamical background gauge fields, or (2) to promote the anomalous global symmetry to a dynamical gauge theory.

This idea applies to both continuous symmetry or discrete finite symmetry, it applies to The systems formed by fundamental bosons are bosonic SPT states.

These are some examples of bosonic anomalies through SPT states:

Ref 1: Bosonic Anomalies in 1+1d PRB,

Ref 2: Anomalies and Cobordism of Oriented/non-Oriented in any dimensions, but non-spin manifold (thus non-fermionic) [arXiv only]

Ref 3: pure gauge and mixed gauge-gravitational anomalies in 0+1, 1+1, 2+1, 3+1, and any dimensions PRL,

Ref 4: Bosonic anomalies in any dimensions through extended Group Cohomology with an additional $SO(n)$ group PRB.

Some propose that classifying the SPT states are directly related to classifying the distinct anomalies of a group $G$, say related to an exact sequence:

$d$-dimensional gauge anomalies of gauge group G

$\to$ $d + 1$-dimensional SPT phases of symmetry group G

$\to$ 0.

Related phenomenon for bosonic anomalies:

  • Induced Fractional Quantum Numbers at domain walls: Jackiw-Rebbi and Goldstone-Wilczek (via bosonization/fermionization in 1+1d)

  • Degenerate Zero Modes at boundary/domain walls: Haldane chain., Kitaev chain, etc.


The only possible way Bosons admit anomalies in flat space (analogous to the case for fermions you have mentioned) is when they do not admit a covariant lagrangian. These are called by a special name, "Chiral Bosons". The usual Bose lagrangian can always be regulated to give an action free of anomalies. See section 8 of this paper. http://www.sciencedirect.com/science/article/pii/055032138490066X


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