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Lattice Regularization of Reduced Kähler-Dirac Fermions and Connections to Chiral Fermions
by Simon Catterall
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Submission summary
| Authors (as registered SciPost users): | Simon Catterall |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2311.02487v2 (pdf) |
| Date submitted: | 2024-02-14 20:44 |
| Submitted by: | Catterall, Simon |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We show how a path integral for reduced K\"{a}hler-Dirac fermions suffers from a phase ambiguity associated with the fermion measure that is an analog of the measure problem seen for chiral fermions. However, unlike the case of chiral symmetry, a doubler free lattice action exists which is invariant under the corresponding onsite symmetry. This allows for a clear diagnosis and solution to the problem using mirror fermions resulting in a unique gauge invariant measure. By introducing an appropriate set of Yukawa interactions which are consistent with 't Hooft anomaly cancellation we conjecture the mirrors can be decoupled from low energy physics. Moreover, the minimal such K\"{a}hler-Dirac mirror model yields a light sector which corresponds, in the flat space continuum limit, to the Pati-Salam GUT model.
Author comments upon resubmission
The only real issue I do not agree with is their final
comment concerning eqn 19 (in the previous draft) asking for me to add a discussion on the non-locality associated
with this transformation. This is incorrect - this gauge transformation and indeed the fermion operator are strictly local for the reduced KD fermion - there is no non-locality as appears in say an overlap construction for chiral fermions. There is
only the question of gauge invariance and an ambiguity in the measure associated with the presence of exact zero modes in the lattice KD equation which warrants the
addition of mirror fermions and new dofs. This is emphasized in various parts of the paper.
In the case of overlap fermions there are two sources of measure problems -- the presence of a non-local $\hat{\gamma}_5$ in the projector and the possibility of exact zero modes. In the KD case there is only the latter.
List of changes
1. Added definitions of $d,d^\dagger$ and additional text to improve the background.
2. Reordered the text as suggested for clarity. Now use the action rather than the equation of motion to discuss symmetries.
3. Again rewritten/reordered together with point 2 for better clarity/logical progression.
4.Fixed.
5. Added new definitions/notation concerning p-cochains and the action of boundary/co-boundary operators
on simplicial lattices throughout to conform to those in PRD paper they cite.
6. Agreed and changed $\omega_p\to \omega(C_p)$ throughout.
7. Agreed. $I(C_p,C_{P+1})$ only arises indirectly since in practice it is gotten by taking the transpose of $I(C_{p+1},C_p)$. I deleted all referencxe therefore to the latter.
8. Corrected eqn19 to agree with their suggestion.
9. See above. Eqn19 does not exhibit any non-locality - this is precisely why (reduced) KD fermions can be handled
in a more complete way than chiral fermions. One one has fixed issues of gauge invariance , remaining measure problem for reduced KD fermions only arises because of the presence of exact zero modes - not because of non-localities in the fermion operator or projector.
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 4) on 2024-4-2 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2311.02487v2, delivered 2024-04-02, doi: 10.21468/SciPost.Report.8807
Report
The author has address all issues raised in my previous report thoroughly and satisfactorily. I must partially disagree only on one point, i.e. on the issue of locality.
I agree on the fact that both gauge transformation and the fermion operator are strictly local. However, as pointed out by the author, the matrix $\mathcal{K}$ introduced in eq. 19 is rectangular. If one integrates exp(-action) with the local fermionic measure one simply gets zero. In fact the author points out that, in order to obtain the partition function in eq. 24, the zero modes need to be ignored. This meant that the integration measure is defined by restricting the integration domain to the complement of the zero modes, and this operation is non-local (since it cannot be expressed as a local contraint). Therefore, truncating the zero modes, introduced not only a phase-ambiguity and possible breaking of gauge invariance, but also a non-locality.
In a sense this comment does not change the essence of the paper, since the mirror-fermion strategy solves potentially (assuming that the decoupling works) all issues at once, including the non-locality issue. However I believe that this issue should be addresses. In the off-chance that I am mistaken and the operation of throwing away the zero modes can be in fact implemented as a local constraint, then this would be highly non-trivial and should be discussed as well.