Normalization of $U(1)$ gauge fields In G. W. Moore, “Introduction to Chern-Simons theories.” 2019 TASI School. [Online]. Available: https://www.physics.rutgers.edu/~gmoore/TASI-ChernSimons-StudentNotes.pdf the $U(1)$ gauge field has a normalization such that $F/2\pi$ has integral periods.
Assume you have a $U(1)$ gauge field and let $\Sigma$ be some closed 2-surface, say a sphere. Then we can slice $\Sigma$ into a northern hemisphere $N$ and a southern hemisphere $S$. Each is contractible and thus, on each the $U(1)$ bundle must be trivial. Accordingly, there must be sections of said bundle on each of these hemispheres allowing us to define the Yang-Mills fields $A_N$ on $N$ and $A_S$ on $S$. On each of these patches $F=dA_N$ and $F=dA_S$, and the Yang-Mills fields are related on any overlap between $N$ and $S$, say the equator $E$, by a gauge transformation $A_N=A_S+ig^ {-1}dg$, for some function $g:E\rightarrow U(1)$. Then we have
$$\int_\Sigma F=\int_N F+\int_S F=\int_N dA_N+\int_S dA_S=\int_E A_N-\int_E A_S=\int_E(A_N-A_S)=i\int_E g^{-1}dg.$$
All possible maps $g$ are of the form $g(\theta)=e^{i\varphi(\theta)}$ for some $\varphi:\mathbb{R}\rightarrow\mathbb{R}$ with a definite winding number $n$, i.e. $\varphi(2\pi)=\varphi(0)+2\pi n$. Then
$$\int_\sigma F=-\int_Ed\varphi=-2\pi n.$$
Thus $F/2\pi$ has integer periods. Where did I assume any convention on the normalization of the gauge field?
By the way, I know the above is only a proof for $E$ homeomorphic to a sphere. I don't know how to proof this in general. Any comments on this would be useful as well.
 A: You assumed the normalization when you wrote
$$
 A_N=A_S+ig^{-1}dg.
\tag{1}
$$
To make this more intuitive, consider a triangulation of the closed orientable 2-surface $\Sigma$. (I won't consider unorientable surfaces.) Let $T$ be the set of triangles, oriented so that whenever two triangles share an edge, the edge is oriented oppositely in the two triangles. Assign a group element $u(t)$ to each triangle $t\in T$ such that
$$
 \prod_{t\in T} u(t) = 1.
\tag{2}
$$
The quantities $u(t)$ are called plaquette variables. Given such an assignment, we can also choose a group element $u(\ell)$ for each edge $\ell$ such that
$$
 u(t)=\prod_{\ell\in t}u(\ell), 
\tag{3}
$$
where the edges are understood to be oriented monotonically around the triangle $t$, and where
$$
 u(-\ell)u(\ell) = 1
\tag{4}
$$
where $-\ell$ is the reversed-orientation version of $\ell$. The quantities $u(\ell)$ are link variables. The assignment $\ell\mapsto u(\ell)$ is not uniquely determined by the assignment $t\mapsto u(t)$, because we can replace
$$
 u(\ell)\to u^g(\ell)
\tag{5a}
$$
with
$$
 u^g(\ell)\equiv g^{-1}(v_1)u(\ell)g(v_2),
\tag{5b}
$$
where $v_1,v_2$ are the initial/final points of the orienged edge $\ell$, where $g(v)\in U(1)$ for each vertex $v$ in the triangulation. The plaquette variables (3) are invariant under the gauge transformation (5).
To relate this to the usual concept of a gauge field, write
$$
 u(\ell)=\exp\left(ic\int_\ell A\right)
\tag{6}
$$
for a one-form $A$, with a normalization constant $c$ that is the same for all links, and consider the limit as the size of the triangles in $T$ goes to zero. Such a one-form $A$ may not exist globally, and that's okay. The plaquette variables $u(t)$ can be written
$$
 u(t)\equiv \exp\left(ic\int_t F\right)
\tag{7}
$$
with $F=dA$. Using different one-forms $A$ in different patches corresponds to using a different choice of link variables $u(\ell)$ in different patches, related to each other by (5) where the patches overlap. Originally I used (5) to describe a gauge transformation, and here I'm using it to describe a transition function. To relate this to (1), write $v_2=v_1+\ell$ and consider
$$
 \omega \equiv \lim_{|\ell|\to 0}\frac{u^g(\ell)-1}{|\ell|}.
\tag{8}
$$
From the left-hand side of (5b), we get
$$
 \omega = ic A^g(v_1).
\tag{9}
$$
From the right-hand side of (5b), we get
$$
 \omega = icA(v_1) + g^{-1}(v_1)dg(v_1).
\tag{10}
$$
Altogether, $ic A^g(v_1) = icA(v_1) + g^{-1}(v_1)dg(v_1)$. Aside from the different sign-convention for $i$, this shows that (1) assumes the normalization $c=1$. We can't cancel the factor of $c$ by replacing $g\to g^c$, because replacing $g\to g^c$ is the same as applying a $c$-dependent gauge transformation. A gauge transformation is unable to change the product $\prod_{\ell\in L}u(\ell)$, or the integral $\oint_L A$ modulo $2\pi$, around any closed loop $L$.
The freedom to change the normalization factor $c$ comes from the fact that what really matters is the link variables $u(\ell)$, not the one-form $A$, because we're using a gauge theory based on the compact group $U(1)$, not on the non-compact group $\mathbb{R}$. In other words, $A$ and $F$ are always confined to the exponents in (6) and (7), so we can normalize $A$ arbitrarily and use the constant $c$ to compensate.
From this perspective, the proof of integer periods for a general closed orientable 2-surface $\Sigma$ is easy: (2) holds by construction if we write the plaquette variables in terms of link variables as in (3) and (4), which is the essence of the concept of a gauge field.
