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This question is motivated by 2-dimensional CFTs where the Classical conformal group (defined by the Witt algebra) is modified to the Virasoro algebra in the quantum theory. In this question, it was pointed out to me that the extra term in the Virasoro algebra is due to the conformal anomaly (for example, when quantizing the theory on a curved surface, the Ricci scalar explicitly introduces a length scale). This led me to wonder about this:

Consider a field theory has a classical symmetry with algebra $G$. Is it true that if this symmetry is anomalous in the quantum theory the algebra $G$ is modified? If so, how?

For example, Yang-Mills theory with a massless Dirac fermion, has a classical symmetry group $G_{gauge} \times G_{axial}$, where $G$ is the gauge group. In the quantum theory, $G_{axial}$ is anomalous. Does this somehow also imply that the algebra of $G_{axial}$ is modified somehow?

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  • $\begingroup$ Nice question. I don't know the answer off hand, but I would be tempted to say that rather than the algebra being modified, the theory just doesn't have any true representations of the $G_{axial}$. To explore this I would suggest looking at the simplest possible model first: the Schwinger model on a circle. There you have an exact solution, explicit construction of the full Fock space etc. The issue that might crop up is that it might make a difference whether $G$ is abelian or not. $\endgroup$ – Michael Brown Sep 5 '13 at 23:50
  • $\begingroup$ @MichaelBrown - I found this interesting reference - arxiv.org/abs/hep-th/9903141 which states "Anomalies, for example, which manifest themselves through the appearance of central terms at the right-hand side of some commutators....." $\endgroup$ – Prahar Sep 6 '13 at 1:53
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    $\begingroup$ It seems this question has essentially put forward what I am asking - physics.stackexchange.com/q/33195 $\endgroup$ – Prahar Sep 6 '13 at 1:59
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I'm pretty sure the answer is "no." There's a conformal anomaly in any even number of dimensions. For instance, in 4 dimensions it's the statement that $T^\mu_\mu$ is nonzero for a CFT on a general curved background, equal to coefficients $a$ and $c$ times the Euler density and Weyl$^2$ terms. But there's no central extension of the conformal algebra the way there is in two dimensions.

(You'll notice in the 2D case that the SL(2) part of the algebra is unaffected by the central extension, so in some sense it's only the conformal generators that don't generalize to higher dimensions that have a modified algebra.)

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The existence of anomalies is almost always accompanied by an extension of the gauge group commutation relations. The case of non-Abelian axial anomaly is may be the most known case.

The abstract gauge group algebra:

$ [G_a(x), G_b(y)] = if_{ab}^{c} \delta^N(x-y)$

($N$ is the number of dimensions), is not realized at the quantum field level

When $N=1$, one gets the central Kac-Moody extension:

$ [G_a(x), G_b(y)] = if_{ab}^{c} \delta(x-y) + \frac{ik}{2\pi} \delta_{ab} \delta^{'}(x-y)$

For odd $N>1$, one gets an (non-central) Abelian extension. For $N=3$, the extension is called the: "The Mickelsson-Faddeev" extension (sometimes the Mickelsson-Faddeev Shatashvili extension) (Mickelsson's original article , Faddeev's original article ):

$ [G_a(x), G_b(y)] = if_{ab}^c \delta^3(x-y) + \frac{ik}{24 \pi^2}d_{abc} \epsilon_{ijk} \partial_i A_{cj}(x) \partial_k \delta^{3}(x-y)$

($d_{abc} = \mathrm{tr}(T_a \{T_b, T_c\})$). As can be seen this extension depends explicitly on a background gauge field $A_j(x) $ and it is not central (does not commute with the gauge group generators) because the gauge field is not gauge invariant:

$ [G_a(x), A_{bi}(y)] = if_{ab}^{c} A_{ci}(x) \delta^3(x-y) + i \delta_{ab} \partial_i \delta^{3}(x-y)$

These extensions are born because of the requirement of renormalization, making products of operators ill defined. However, the extensions do not depend upon the type of the adopted regularization.

In the Kac-Moody (and also the Virasoro) case, it is well known that the extension is essential in getting unitary irreducible positive energy representations of the algebra, which allows quantum mechanical interpretation of the spectra. In the Mickelsson-Faddeev case, however, no such representations are known. In fact, there is a theorem by Pickrell, which almost slams the door on finding such representations.

There is a series of works by Juoko Mickelsson, trying to understand this extension by means of an infinite dimensional representation theory, please, see this work by Mickelsson and references therein.

The non-Abelian chiral anomaly is very well understood algebraically and topoplogically, please see the following review by R. A. Bertlmann. In the example following equation (38) in the article, all the quantities associated with the non-Abelian chiral anomaly are algebraically computed (without solving Feynman diagrams) through what is called the Stora-Zumino descent equations.

These equations give on the first level the Chern-Simons term, on the second level, the anomaly (divergence of the current), on the third level, the extension in the gauge group commutation relations and on the fourth level, the associator causing the violation of the Jacobi-identity, thus resulting a non associative algebra (please see the following related physics stack exchange question).

I mentioned, the descent equations, because there is a modern concept of gerbes trying to find geometrical realization of these equations (please see also the following Mickelsson's review). This direction of research has the potential of providing deeper understanding what are the quantum structures that we must associate to these algebras (interpreted as classical algebras of Poisson brackets) because the usual Hilbert spaces and unitary representations do not seem to work. The Mickelsson-Faddeev algebra was extensively analyzed within the theory of gerbes, please see for example this work by Hekmati, Murray, Stevenson and Vozzo (and also the above Micklsson's reference).

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