Lattice Regularization of Reduced Kähler-Dirac Fermions and Connections to Chiral Fermions

The content of this paper has significant overlap with PhysRevD.107.014501 (by the same author). In particular, the following topics are already covered in PhysRevD.107.014501:
- the anomaly of the global twisted chiral symmetry of Kähler-Dirac fermions in curved space,
- the gauging of the residual Z4 symmetry,
- the discussion of a possible mechanism to decouple the mirror sector
- the presentation of a model which, assuming that the decoupling mechanism works as expected, reduces to the Pati-Salam model.
The discussion in Section 3 is the new contribution of this paper: the author discusses how gauge anomalies are produced in the reduced Kähler-Dirac fermions in generic curved space. In particular, he shows that the underlying mechanism is similar to what happens e.g. with Ginsparg-Wilson fermions in topologically non-trivial gauge fields: the Dirac operator is a rectangular matrix and reasonable attempt to generalise the fermionic determinant produce effective actions that are generically non gauge-invariant and non-local. The discussion is surely interesting and illuminating, and it is worth publication. However three major shortcomings should be addressed.

1- The notation used throughout the paper is confusing and sometimes ambiguous (or at least not completely explained). I notice that the author uses a much more rigorous notation in his PhysRevD.107.014501 paper, and I would recommend to apply the same standards here.

2- Eq. (19) seems problematic to me. The goal of the author is "to gauge the $U(1)$ symmetry given in eq. (8)". The author defines $\omega_p = e^{i \alpha_p \Gamma}$ and states that "$\omega_p$ varies for each element of the $p$-cochain. If this is the case, the notation is confusing since $\omega_p$ suggests a dependence on the order of the cochain only. If I interpret correctly, the gauge transformation is really meant to be defined as
$$
\omega(C_p) = e^{i \alpha(C_p) (-1)^p}
$$
where I have used the observation that $\Gamma$ acts as $(-1)^p$ on cochains of order $p$. More explicitly, I guess this means that the fields $\Phi$ and $\bar{\Phi}$ (with a generic charge $q$) are supposed to transform like
$$
\Phi^\omega(C_p) = e^{i q \alpha(C_p) (-1)^p} \Phi(C_p)
$$
$$
\bar{\Phi}^\omega(C_p) = \bar{\Phi}(C_p) e^{i q \alpha(C_p) (-1)^p}
$$
These equations reduce to eq. 8 if $q=1$ and $\alpha$ is constant. This observation makes me think that I am on the right track. Now, if I interpret eq. (19) as
$$
W(C_p,C_{p-1}) \to \omega(C_p) W(C_p,C_{p-1}) \omega(C_{p-1})^\dagger \ ,
$$
I get the following explicit expression for the gauge transformed field
$$
W^\omega(C_p,C_{p-1})
=
e^{i [ \alpha(C_p) + \alpha(C_{p-1}) ] (-1)^p}
W(C_p,C_{p-1})
$$
However this is not consistent with the idea of gauging the $U(1)$ symmetry given in eq. (8): when $\alpha$ is constant, the gauge field $W$ should not transform. It seems to me that the correct transformation should be
$$
W^\omega(C_p,C_{p-1})
= \omega(C_p)^\dagger W(C_p,C_{p-1}) \omega(C_{p-1})^\dagger \ ,
$$
which can be written explicitly as
$$
W^\omega(C_p,C_{p-1})
=
e^{-i [ \alpha(C_p) - \alpha(C_{p-1}) ] (-1)^p}
W(C_p,C_{p-1})
$$
This transformation guarantees that the $W$ does not transform under a global transformation. Moreover, it yields
$$
K[W^\omega] \Phi^\omega = \omega^{-q} K[W] \Phi
$$
which is what one needs to guarantee invariance of the action. If this is correct, then under a gauge transformation, $K \to \omega^{-q} K \omega^{-q}$ which implies $U \to \omega^{-q} U$ and $V \to \omega^{+q} V$. This is at odds with what stated in the paper (in the case of $q=1$).

I see two possibilities here: either eq. (19) is not correct and should be fixed (together with what comes after), or it is correct but then there is not enough explanation in the paper of why this should be the case.

3- The author fails to comment on the non-locality of eq. (19), which is one the crucial obstructions in the non-perturbative construction of chiral gauge theories. Even when the anomaly cancellation condition is satisfied, the fermionic measure is defined only up to a phase that depends on the gauge field. It is not obvious that this ambiguity can be used to restore the locality of the resulting quantum field theory.

1- In order to make the paper self-contained, it would be good to write explicitly the definitions of the operators $d$ and $d^\dagger$ appearing in eq. (1).

2- The sentence which includes eq. (4) should be clarified: in fact eq. (4) does not follow from anything written above. I guess that the author means something along the lines of: "It can be easily shown that, the equations of motion obtained from the action in eq. (1) are equivalent to:". However, since the paper deals with a quantum theory, my recommendation is to leave out the equations of motion are reformulate the sentence in terms of the action. The sentence may look something like this:
"In terms of the variables introduced in (3), the action (eq. 1) can be equivalently written as..."

3- Above eq. (5), it says: " transform only under a diagonal subgroup of the Lorentz and flavor symmetries." The statement is correct, but puzzling at this point, since the flavour symmetry is introduced only after. This could be rearranged in order to improve readability.

4- The transformation given in eq. (8) is show to generate an anomaly in sec. 4. In view of this fact, the statement
"This is a symmetry of the massless theory and will be very important in our later discussion of anomalies."
is misleading. The author should say clearly that this is a symmetry of the action, but not of the quantum theory in curved space.

5- In general, the formulae would benefit from a more precise notation of the sums. For instance, the sum in eq. (10) is really a sum over all cochains. The sum in eq. (11) is not a sum over the index $p$, but a sum over all (p-1)-cochains. I recommend using $\sum_{C_{p-1}}$ instead of $\sum_{p-1}$. Eq. (12) should be written with a sum rather than an integral. These are just examples, and the notation should be made more precise throughout the paper.

6- Gauge transformations should be denoted by $\omega(C_p)$, instead of $\omega_p$, as they depend on the cochain, not only on its order.

7- Around eq. (19), the symbols $I(C_p,C_{p\pm 1})$ and $W(C_p,C_{p\pm 1})$ are used several times. However $I(C_p,C_{p+1})$ and $W(C_p,C_{p+1})$ are never defined, nor used. So all $\pm$ occurrences should be replaced by $-$.

8- Eq. (19) and the following discussion should be improved, in order to address point 2 in the report.

9- The role of locality should be discussed (see point 3 of the report).

1- the paper is clearly written and well motivated
2-the problem with the ambiguous phase is new and interesting
3-the proposed resolution, to use mirror fermions and attempt decoupling is interesting
4-opens a new arena of future research

1-the only thing I can criticize is that it requires some background knowledge

I think this is a very interesting paper. I was not aware of the phase issues in theories with gauged reduced Kahler Dirac fermions. The proposal is interesting and worthy of further pursuit. Thus, I recommend publication.