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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:  (pdf)
Date accepted: 2023-09-15
Date submitted: 2023-09-06 04:33
Submitted by: Dedushenko, Mykola
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


Berry connection has been recently generalized to higher-dimensional 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 old-fashioned Berry connection (for time-dependent 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 two-torus, 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.

List of changes

Questions 1,4,5,6 from the Report 1 (which also includes a comment from the Report 2) have been addressed.

1) New footnote 4 is meant to address the first question. This is also related to question 3. Indeed, in Section 3 the Berry connection is not proportional to volume, simply because it can be understood as descending from the topological term (CS term) in the 3d effective action. In other words, indeed, the example of Section 3 may be also seen as the higher Berry connection. In this paper, however, we study the "old-fashioned" Berry connection viewpoint, in the sense that we only allow parameters to vary in time, while remaining constant along the (compact) space. This mixes the"old" and "new" phenomena in the resulting 1d effective action, and a priori one does not assume any volume scaling.

2) As mentioned by the first referee, answering his question 2 is optional, since it refers to the paper 2305.06399 that appeared one year later than the current manuscript. We believe the burned is on the authors of 2305.06399 to make this comparison. Superficially, however, the connection is not immediate, since in Section 3 we look at the Chern-Simons term, which is different from the Thouless pump.

3) To address question 4, the first paragraph of Section 3 has been modified (which also includes a new footnote 9). Also, slightly modified the last paragraph of Subsection 3.1, as requested.

4) Missing reference added (as per comment 5 of the first referee and the comment of the second referee).

5) Paragraph after eqn. (3.48) modified to address the question 6.

Published as SciPost Phys. 15, 167 (2023)

Reports on this Submission

Report 1 by Lev Spodyneiko on 2023-9-6 (Invited Report)


The author addressed my questions in the revised version. I recommend publication.

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