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As far as I understand, the Nielsen-Ninomiya theorem states that (under mild conditions) the number of left and right-handed chiral fermions must be equal on the lattice, while the chiral gauge anomaly is the statement that the $U(1)$ gauge symmetry is violated if the number of left and right-handed chiral fermions are not equal (in the continuum).

Is it correct to say that the Nielsen-Ninomiya is equivalent to (a lattice version of) ensuring the chiral gauge anomaly vanishes? Or are there subtleties in moving from the lattice to the continuum?

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  • $\begingroup$ Upon reading the original paper more carefully, indeed it argues that Nielson-Ninomiya theorem forbids putting the SM on the lattice though it has no gauge anomaly. Definitely there is a distinction between the two for non-abelian gauge symmetries. However, are they equivalent in the case where we only allow for $U(1)$ gauge symmetries? I at least learned that Nielsen-Ninomiya generically applies for all chiral theories, not just the abelian ones! $\endgroup$
    – Aaron
    Nov 26, 2018 at 17:24

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I think the Nielsen-Ninomiya theorem is more closely related to a gravitational anomaly than a $U(1)$ anomaly. For instance in 1+1d, two left-moving fermions of charge +3 and +4 and a single right-moving fermion of charge +5 has a vanishing $U(1)$ anomaly, since $3^2+4^2 =5^2$. However, it still cannot be put on a lattice, because it has a chiral central charge.

Here is a recent paper which addresses why a 1+1d chiral theory (in the central charge sense) cannot be put on a lattice, from the perspective of energy currents: https://arxiv.org/abs/1904.05491 . The up-shot is that a local lattice model cannot have a nonvanishing energy current in the ground state, which is exactly what a chiral CFT has.

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