Berry phase and Wilson loop According to the definition, the Wilson loop is
\begin{equation}
W[\mathcal{C}] =\operatorname{Tr}\left[\mathcal{P} \exp\left\{i\oint _{\mathcal{C}} A_{\mu } dx^{\mu }\right\}\right]
\end{equation}
where $\mathcal{P} $ is the path ordering,  $A_{\mu }$ is the gauge field.
And closed-path Berry phase is defined as
\begin{equation*}
\gamma _{n} =\oint _{c} A\cdot dR
\end{equation*}
where A is the Berry connection
\begin{equation}
A=i\ \langle n(R)| \nabla _{R}| n(R)\rangle 
\end{equation}
$| n(R)\rangle $ is the eigenstate of Hamiltonian $\hat{H}(R)$, which depends on parameter $R$.
Since they can both be viewed as holonomy , my question is under what condition could they be identical? Or when could Berry connection be interpreted as gauge field in Yang-Mills theory?
 A: *

*Both the usual gauge fields ($A_\text{QFT}$) and the Berry connection ($A_$) are connections on a principal bundle.

*In usual gauge theory in QFT you have a spacetime manifold, $S$ and a Lie algebra $\mathfrak{g}$. The gauge fields are $\mathfrak{g}$-valued one-forms on $S$, $A_\text{QFT}\in\Omega^1(S;\mathfrak{g})$.

*In the Berry phase business you have a quantum mechanical theory defined on some space with a parameter space, $P$. Moreover, the underlying Lie algebra is $\mathfrak{u}(1)$. The Berry connection takes values in $A_ \in\Omega^1(P;\mathfrak{u}(1))$.

*They could be, therefore, formally identical, if the parameter space of your quantum mechanics is the spacetime manifold of an abelian gauge theory.

*A difference between them is that the Berry connection is a background gauge field; namely fixed by the system, while in gauge theories you need to path-integrate over all possible connections. Hence the word formally, above.

*You can always build an abelian gauge theory with the Berry connection as its gauge connection but you should not expect that it would be a very physical theory.

*Finally, another thing you can do, if you have a map $f:M\to P$, and a QFT defined on $M$, is that you can pull back the Berry connection along $f$, so that $f^*A_\in\Omega^1(M;\mathfrak{u}(1))$. Now $f^*A_$ behaves like a usual abelian background gauge field, which you can now couple to your physical fields.

