Anomalies Revisited

Some of the talks at Strings 2015 have now appeared online, and one of them I found quite fascinating, Witten’s Anomalies Revisited. Some of his motivation comes from string perturbation theory and M-theory, but the questions he addresses are fundamental, deep questions about quantum field theory (and not just any quantum field theory, but exactly the sort of qft that appears in the SM, spinors chirally-coupled to gauge-fields/metrics).

The fundamental issue is that these are theories where the path integral does not determine the phase of the partition function. Part of story here is the well-known story of anomalies, perturbative and global. One interesting point Witten makes is that vanishing of these anomalies is not sufficient to be able to consistently choose the phase of the partition function, and he gives a conjecture for a necessary condition that is stronger than anomaly cancellation.

The standard story of QFT textbooks is that once a Lagrangian is chosen, the corresponding QFT is well-defined. But quantization really is a lot more subtle than that, and the anomaly phenomenon is just one indication of the problem. In earlier parts of my career I spent a lot of time thinking about anomalies; the connections to some of the deepest mathematics around (K-theory, index theory and much more) are truly remarkable. In recent years I’ve been thinking about other things, but Witten’s talk is a strong encouragement to go back and revisit the anomaly story (some of his best work has “revisited” in the title…)

In particular, Witten emphasizes a particular case I’d never paid much attention to: the 3d massless Majorana fermion. One would think that this is among the simplest quantum field theories around, but Witten explains how this is an example of a theory with a potential inconsistency (no way to consistently choose the sign of the partition function), even when the anomaly vanishes. This theory also appears in a hot topic in condensed matter physics, the theory of topological superconductors. One reference Witten gives to related work is to this recent paper (there’s a typo in his reference, should be 1406.7329 not 1407.7329).

Witten ends with:

I hope I have at least succeeded today in giving an overview of the tools that are needed to study the subtle fermion integrals that frequently arise in string/M-theory. A detailed analysis of a specific problem would really require a different lecture. Write-ups of some of the problems I’ve mentioned – and some similar ones – will appear soon.

I look forward to those write-ups, with the theories he’s talking about of a lot more interest than just their role in string/M-theory calculations.

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9 Responses to Anomalies Revisited

  1. Dave Miller in Sacramento says:


    From time to time, I have seen claims that the anomaly problem amounts to nothing more than the breaking of conformal invariance by renormalization. Can you tell me if this is true, false, part of the story, or not even wrong?

    And, what is the simplest (but correct) exposition of the issues that you can recommend (i.e., for someone who took QFT a long time ago)?



  2. Peter Woit says:

    That’s just a small part of the story. More generally, the anomaly refers to a subtlety in quantization: a symmetry of the classical theory does not work in the expected way in the quantum theory. You already see this in the phenomenon of the one-half energy of the ground state in the harmonic oscillator. You can get rid of this by redefining the Hamiltonian, but that changes how the symmetries of the classical system are implemented in the quantum system. For a finite number of degrees of freedom, you can work with either Hamiltonian, but in QFT, with an infinite number of degrees of freedom, you don’t have a finite shift and this causes the anomaly.

    Put differently, the anomaly is due to the fact that normal-ordering is needed in QFT, and this sometimes changes how classical symmetries appear after quantization. Maybe the simplest example is the U(1) global chiral symmetry in the theory of 1+1 d fermions. I wrote something about the general phenomenon in the book I’m writing which is online, would like to add a section working out the 1+1 d fermion story, but that may end up on the list of things I have to drop for lack of time/space.

  3. Chris Austin says:

    Witten argues that a partition function or functional integral can sometimes have an undefined phase factor exp(i \alpha), where \alpha is a constant. But in quantum field theory, we always in practice calculate a ratio such as Z[J]/Z[0], where J are some sources or background fields. This is necessary in order to cancel the vacuum bubble diagrams, (or equivalently, if the interactions are turned on and off adiabatically, as in the Gell-Mann – Low derivation of the Feynman diagram expansion, it is necessary in order to get a stable limit as the adiabatic switching is removed).

    Witten’s constant phase factor will always cancel between the numerator and denominator of Z[J]/Z[0] if the manifolds and gauge fields in the numerator and denominator are topologically equivalent, so does it really matter? From pages 83 and 84, he is concerned about consistency of his phase factor between different manifolds. Does this mean it is now possible to have topologically distinct manifolds or gauge fields in the numerator and denominator?

    This seems to raise the issue of when one should or should not sum over some topologies in Z[0], which Witten doesn’t address. We have to sum over Yang-Mills instantons, so do we also have to sum over all compactification topologies in the M-theory Z[0]?

    Witten must presumably know that the condition he proposes on page 89 is satisfied for D = 11 Majorana gravitinos, gravitons, and 3-forms on all 12-manifolds Y, (so that the condition doesn’t say that M-theory is inconsistent), but I can’t find anywhere in the talk where he says so. (Nor in Freed and Moore, which Witten cites on page 124.)

    Since spinors are involved, should it be all spin D + 1-manifolds, at least in the sense that w_2 vanishes? Page 101 seems to suggest yes. On page 144, Witten invokes RP^4, which is Pin^+ in the sense that w_2 vanishes although w_1 doesn’t, (according to pages 45 to 46 of Milnor and Stasheff). What about D + 1-manifolds that are Pin^-, in the sense that w_2 + w_1^2 vanishes, although w_2 doesn’t, for example RP^2, RP^6, RP^10? What about D + 1-manifolds for which w_1 vanishes but w_2 doesn’t, for example RP^5, RP^9?

  4. Tim Nguyen says:


    “You can get rid of this by redefining the Hamiltonian, but that changes how the symmetries of the classical system are implemented in the quantum system.”

    I don’t immediately understand what this means. Could you please elaborate? My impression is that you meant something involving Poisson brackets (classical) or commutators (quantum), both of which annihilate constants.

  5. Peter Woit says:

    It”s an intricate story, but here’s quite a lot of detail about this in

    One basic point is that, even classically, if you have a symplectic action of a Lie group G on a phases space, there’s an ambiguity of a constant in how you define the moment map, and that’s an origin of the problem. In some sense what you have is a central extension of G that is acting, and it may be non-trivial.

  6. Anon says:


    The phase doesn’t cancel for all physically relevant operators in the QFT. The typical example is the energy-momentum tensor in classically conformally invariant theories. In this case the overall phase factor comes from the definition of the path integral measure, which no matter how you cut it introduces an unavoidable extra dependence on the metric (besides the explicit occurrence of the metric in the Lagrangian), which may then cause the trace of the EM-tensor (which is the generator of conformal transformations) to be nonzero in the quantum theory.

  7. Alex says:

    I’m confused by your comment. The example you gave seems to indicate that the phase factor depends on the metric, hence is not a constant. In the example in Witten’s lecture the partition function can be determined up to an undetermined phase factor exp(i\alpha) where \alpha is a constant. Please explain.

  8. Anon says:


    In the case I’m most familiar with (two-dimensional conformal theories) the coefficient alpha is a real number that can be expressed as an integral over the manifold of a density depending on the metric (something like alpha = ∫ sqrt(g) R, where R is the curvature and the integral is over the whole manifold). This is a constant for fixed background, but can vary if you vary the background metric (which is what you do to determine the EM-tensor, and contributes something nontrivial even to the EM-tensor on a fixed flat background).

  9. Sakura-chan says:

    Off-topic to this thread but not to the blog…

    Looks like NIST has formally removed the Dual_EC_DRBG from its recommended methods for generating pseudorandom numbers.

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