Remarks on Berry connection in QFT, anomalies, and applications

<jats:p>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 <jats:inline-formula><jats:alternatives><jats:tex-math>\Sigma^{(d)}× \mathbb{R}</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mi>Σ</mml:mi><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow></mml:msup><mml:mo>×</mml:mo><mml:mi>ℝ</mml:mi></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:tex-math>\Sigma^{(d)}</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>Σ</mml:mi><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow></mml:msup></mml:math></jats:alternatives></jats:inline-formula> is a <jats:inline-formula><jats:alternatives><jats:tex-math>d</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mi>d</mml:mi></mml:math></jats:alternatives></jats:inline-formula>-dimensional compact space and <jats:inline-formula><jats:alternatives><jats:tex-math>\mathbb{R}</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mi>ℝ</mml:mi></mml:math></jats:alternatives></jats:inline-formula> is time. Compactness of <jats:inline-formula><jats:alternatives><jats:tex-math>\Sigma^{(d)}</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>Σ</mml:mi><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow></mml:msup></mml:math></jats:alternatives></jats:inline-formula> 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 <jats:inline-formula><jats:alternatives><jats:tex-math>tt^*</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>t</mml:mi><mml:msup><mml:mi>t</mml:mi><mml:mo>*</mml:mo></mml:msup></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> equations), and when it is not. We also mention a relation to the boundary anomalies and boundary states on the Euclidean <jats:inline-formula><jats:alternatives><jats:tex-math>\Sigma^{(d)} × \mathbb{R}_{≥ 0}</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mi>Σ</mml:mi><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow></mml:msup><mml:mo>×</mml:mo><mml:msub><mml:mi>ℝ</mml:mi><mml:mrow><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>. We then work out the examples of a free 3D Dirac fermion and a 3D <jats:inline-formula><jats:alternatives><jats:tex-math>\mathcal{N}=2</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>𝒩</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> chiral multiplet. Finally, we consider 3D theories on <jats:inline-formula><jats:alternatives><jats:tex-math>\mathbb{T}^2× \mathbb{R}</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mi>𝕋</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>×</mml:mo><mml:mi>ℝ</mml:mi></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>, where the space <jats:inline-formula><jats:alternatives><jats:tex-math>\mathbb{T}^2</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>𝕋</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math></jats:alternatives></jats:inline-formula> 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. </jats:p>