NOT because I wanted you guys to unknowingly click on my paper!

LOL, I don’t care much even if you did Thanks for the link!

In the mean time, I found that Moshe Rozali posted a link to Luscher lectures, and I quickly went though them. There is a complelling picture of chiral gauge theory appearance by dimensional reduction from 5d gauge theory with massive Dirac fermions(anomaly-free).

Cheers,
Dmitry.

Although I haven’t worked the whole thing out yet. But given the above understanding, I am quite confident that similar argument holds true in odd dimensions, namely, naive discretization of Dirac operator necessarily fails to reproduce the parity anomaly and therefore there must be fermion doubling issues. It must be that in the continuum limit, the theory contains many copies, each is the parity transformation of the other.

Thanks again for Dmitry’s very insightful question.

[1] L. Alvarez-Gaume, S. Della Pietra and G. W. Moore, “Anomalies And Odd Dimensions,” Annals Phys. 163, 288 (1985).
[2] E. Poppitz and M. Unsal, “Index theorem for topological excitations on R^3 * S^1 and Chern-Simons theory,” arXiv:0812.2085 [hep-th].

I have to admit that I haven’t thought about this at all. Our excuse is we were interested in defining chiral gauge theories on lattice. In odd dimensions, the spinor representation doesn’t reduce to left-handed and right-handed ones. There is only one irreducible spinor representation and the theory has to be “vector-like” (in the sense that psi and psi-bar are the same fields). So in this sense our problem doesn’t exist there.

But you might be interested in whether naive discretized Dirac operator on an odd dimensional lattice could cause fermion doubling issues, because my argument seemed to rely on the existence of anomaly which disappears in odd dimensions. I don’t really know the answer but my guess would be yes. The real question, however, is how serious this problem is. Because even for the bosonic theories, a naive discretization could cause species doubled. But even if it happens, it would just be trivial repetition of the same theory and can be cured by just properly rescale all the fields, which one would have to do while taking the continuum limit anyway. It’s only a real problem in fermionic theories because the “doublers” must carry opposite chiralities to cancel the anomlay. With this requirement seemingly gone in odd dimensions, I guess odd dimensions would not suffer from the same level of difficulties as even dimensions.

Again, I haven’t thought about odd dimensions carefully at all and would love to hear more experts explaining it. A quick google search led me to the key word “parity anomaly” which claims that in odd dimensions, while the partition function will not suffer from the anomaly related to its complex phase, there still is something that can cause its sign not well defined. But there seem to exist contradicting claims and I would love to know if someone else understand those things well. This may also be totally irrelevant to the topic here.

thanks a lot for the very nice post which nicely fits to the “Open problems” series

Let me ask you a naive question – what happens in 3d (or, say, 5d) where anomalies are absent?

Cheers,
Dmitry.