Chiral Current on a Compact Spacetime

Question

This is a basic question I haven't see answered anywhere and I can't seem to figure out.

The usual statement of the 1+1D chiral anomaly Ward identity is that the divergence of the chiral current is the background field strength:

$\partial_\mu \langle j^\mu\rangle = \epsilon^{\mu \nu} F_{\mu \nu}/2\pi.$

I want to rewrite this in terms of the covariant chiral current $J^\mu = \epsilon^{\mu \nu}\langle j_\nu\rangle$. I believe it says $dJ = F/2\pi$. I am worried about this expression on a compact spacetime, however, since $F/2\pi$ may have a nonzero surface integral, while the integral of a divergence over a closed surface is zero. Must it be that somehow the covariant chiral current $J$ is not gauge invariant? I don't see a mechanism for this to happen though.

Thanks!

Comments

I don't remember all the story, but it is discussed, say, in Nakahara "Geometry, Topology and Physics". Check Ch. 13, and, in particular, formula (13.35).

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I am aware of the index theorem and that reference, thanks. Maybe I should clarify that equation 13.35 is inconsistent with equation 13.33 because the integral of 13.33 is zero on the LHS.

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Or rather, that whole part of Nakahara's discussion seems like nonsense because the integral of dj is zero unless j is somehow not gauge invariant. There is no gauge anomaly though, so this interpretation seems unlikely. Somehow I think that equation 13.33 only makes sense in flat (in particular contractible) space.

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Hmm, as far as I understand, on the curved manifold the equation should have the form $D \star j=(d+\omega) \star j=\frac{1}{2 \pi} F$, where $\omega$ is a spin connection, because the Dirac operator must be modified, and generically $\int\limits_{M}D... \neq \int\limits_{\partial M}...$. Does it make sense?

P.S. Moreover, the right hand side must also be modified by the term quadratic in curvature. See, (45) here: https://arxiv.org/pdf/hep-th/0509097v1.pdf.

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I agree those terms should be there, but adding the metric dependence doesn't really help. I can ask my question on a flat torus and have the same issue. The Chern number of the gauge bundle and the A-hat genus are independent.

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@RyanThorngren Hmm, right. It seems that in order for this to be correct, $\star j$ must transform under the gauge transformation in the same way as $A$ does.

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I am now thinking that the Ward identity as written only holds for flat space. All the derivations of it I can find use the LSZ techniques, which require asymptotic plane wave states and the Fourier transform. As you say, we would need $\star j$ to transform like A does, but there is no gauge anomaly. The classical current and the path integral measure are both invariant under vector gauge transformations. Thanks for your thoughts on this!