I am trying to understand equivariant flat $U(1)$ bundles on a torus, $(S^1)^N$. By equivariant, I mean equivariance with respect to the natural action of $(U(1))^N$ on $(S^1)^N$.
$G$-equivariant bundles should satisfy the equivariant Bianchi identity, \begin{equation} \nabla\mu(X)=\iota(X_M)F, \end{equation} as given in page 211 of Berline, Getzler and Vergne's `Heat Kernels and Dirac Operators' (BGV). Here, $\mu$ is the moment, $X$ is an element of the Lie algebra of $G$ (which is $(U(1))^N$ in my case), $X_M$ is its corresponding Killing vector field on the base space $M$ of the bundle (which is $(S^1)^N$ in my case), and $F$ is the curvature of the bundle.
Since I am interested in a $U(1)$ bundle, I believe the above identity should reduce to \begin{equation} d\mu(X)=\iota(X_M)F \end{equation} (since $\nabla=d+[A,]$, where $A$ is the connection, and then since $U(1)$ has vanishing commutators, $\nabla=d$). Then, assuming that the bundle is flat reduces the identity to \begin{equation} d\mu(X)=0. \end{equation} In other words, it seems to me that the $\mu$, the moment, should be a constant. My question is, how do I determine this constant?
The following is my attempt at a solution. The formula for the moment, $\mu$ given in BGV is \begin{equation} \mu(X)=\mathcal{L}(X_M)-[\iota(X_M),\nabla], \end{equation} where in my case $M=(S^1)^N$.
As mentioned above, $\nabla$ reduces to $d$, so we have \begin{equation} \begin{aligned} \mu(X)&=\mathcal{L}(X_{(S^1)^N})-[\iota(X_{(S^1)^N}),d]\\ %&=2\mathcal{L}(X_{(S^1)^N}) \end{aligned} \end{equation}
Then, since $\mu(X)=\sum_a^N\mu^aX^a$, I think we should have \begin{equation} \begin{aligned} \mu^a&=\mathcal{L}(e^a_{(S^1)^N})-[\iota(e^a_{(S^1)^N}),d], \end{aligned} \end{equation} where $e^a_{(S^1)^N}$ are the generators of the $(U(1))^N$ action on ${(S^1)^N}$. I can't go beyond this, and am slightly confused since $\mu^a$ seems to be an operator in the expression above.
I believe that the constants given by $\mu^a$ should be proportional to the charges $Q^a$ of the $(U(1))^N$ action on ${(S^1)^N}$. Any help would be much appreciated.
UPDATE- I have found a solution in equation (3.27) of `Equivariant Cohomology and Localization of Path Integrals' by Szabo. There, the moment map for a $G$-equivariant $U(1)$ bundle is given as \begin{equation} \mu(X)=\mathcal{L}(X_M)-[\iota,\nabla]=\iota(X_M)A=X^{\mu}A_{\mu}, \end{equation} where $A$ is the $G$-invariant connection on the bundle, i.e., $F=dA$. For a $(U(1))^N$-equivariant flat $U(1)$ bundle, $X^{\mu}A_{\mu}$ is a constant as follows. $A_{\mu}$ is a constant since $F=dA=0$. The Killing vector field $X^{\mu}$ is constant, since the bundle is on $(S^1)^N$, and the Killing vector fields should just be proportional to the charges $Q^a_\mu$ of the $(U(1))^N$ action on $(S^1)^N$. However, I do not know how to show that \begin{equation} \mathcal{L}(X_M)-[\iota,\nabla]=\iota(X_M)A \end{equation}
Help with this computation together with corrections of any mistakes I've made above would be much appreciated, and will be accepted as the answer to this question.