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Remarks on Berry Connection in QFT, Anomalies, and Applications
by Mykola Dedushenko
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Authors (as registered SciPost users):  Mykola Dedushenko 
Submission information  

Preprint Link:  https://arxiv.org/abs/2211.15680v2 (pdf) 
Date submitted:  20230422 09:40 
Submitted by:  Dedushenko, Mykola 
Submitted to:  SciPost Physics 
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Academic field:  Physics 
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Approach:  Theoretical 
Abstract
Berry connection has been recently generalized to higherdimensional QFT, where it can be thought of as a topological term in the effective action for background couplings. Via the inflow, this term corresponds to the boundary anomaly in the space of couplings, another notion recently introduced in the literature. In this note we address the question of whether the oldfashioned Berry connection (for timedependent couplings) still makes sense in a QFT on $\Sigma^{(d)}\times \mathbb{R}$, where $\Sigma^{(d)}$ is a $d$dimensional compact space and $\mathbb{R}$ is time. Compactness of $\Sigma^{(d)}$ relieves us of the IR divergences, so we only have to address the UV issues. We describe a number of cases when the Berry connection is well defined (which includes the $tt^*$ equations), and when it is not. We also mention a relation to the boundary anomalies and boundary states on the Euclidean $\Sigma^{(d)} \times \mathbb{R}_{\geq 0}$. We then work out the examples of a free 3D Dirac fermion and a 3D $\mathcal{N}=2$ chiral multiplet. Finally, we consider 3D theories on $\mathbb{T}^2\times \mathbb{R}$, where the space $\mathbb{T}^2$ is a twotorus, and apply our machinery to clarify some aspects of the relation between 3D SUSY vacua and elliptic cohomology. We also comment on the generalization to higher genus.
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Anonymous Report 2 on 202382 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2211.15680v2, delivered 20230802, doi: 10.21468/SciPost.Report.7594
Report
This work studied the question whether the conventional Berry connection in quantum mechanics is welldefined when we compactify a higherdimensional QFT on a compact space down to a quantum mechanics. It addressed the question by classifying possible counterterms that can potentially spoil the welldefineness of the Berry connection. Then it discussed the applications of Berry connections in various examples.
Overall, it is a solid work that clarifies the use of Berry connections in higherdimensional QFTs and points out interesting relations between Berry connections and various old results in the literature. The referee recommends the publication of this manuscript.
Requested changes
One minor comment:
1. A citation in the first paragraph on page 18 is missing.
Report 1 by Lev Spodyneiko on 2023710 (Invited Report)
 Cite as: Lev Spodyneiko, Report on arXiv:2211.15680v2, delivered 20230710, doi: 10.21468/SciPost.Report.7469
Report
The paper discussed the definition and applications of Berry connection in the framework of QFT. The author thoroughly studies complications and ambiguities related to it. The discussion smoothly evolves from a general QFT to more specific examples of supersymmetric theories. The paper provides a novel and synergetic link between different research areas and thus satisfies the publication criteria. I have some comments/suggestions/questions:
1. The IR divergences are claimed to be fixed by considering the theory on a compact space $\Sigma$. However, the effective action (say of the example in Sec. 3) does not scale with the volume of $\Sigma$. This is counterintuitive because one naively would expect that it should scale with the number of degrees of freedom.
2. A recent paper (https://arxiv.org/abs/2305.06399v1) discussed an interpretation of a higher Thouless pump in 2+1D. They have shown that it gives the Berry connection of a mode localized on fluxon insertion. Namely, the Berry curvature of the 2+1 D system is infinite as well as the Berry curvature of the same system with fluxon insertion. However, their difference is finite and welldefined. The discussion of Sec 2 and 3 of the present paper seem to be similar because Berry curvature couples only to the holonomy of the fields on $\Sigma$. I think it is a good idea to discuss the relationship. (The mentioned paper was published after the present one, so I can't insist on it.)
3. Points 1 and 2 above, make me think that the author actually discusses higher Berry curvatures. Namely, I would call "oldfashioned" Berry curvature the one which scales with the volume. If it does not, I would expect that it is actually the Berry curvature of a single mode, or more precisely the difference of Berry curvature of the whole system with and without the mode.
4. I think it is a good idea to explicitly discuss how different issues raised in Sec 2 affect the simple example of Sec 3. For example, how the counterterm of eq. (2.6) appears in the analysis of Sec 3 when $F\ne0$? As far as I understand the other counterterm (2.9) is discussed in the last paragraph of Sec 3.1. If it is, then the author should clarify this by explicit reference to this equation.
And some minor corrections:
5. There is an empty reference [] in the first paragraph of Sec 2.4.
6. $B^{(p)}_i$ in eq. (3.48) is not defined.
Requested changes
Please address 1,4,5,6 from above, and I would appreciate the author's thoughts on 2,3.