Symplectic reduction to moduli space in Chern-Simons theory

You can write the symplectic form on the large phase space as

$$\Omega(\delta_i A,\delta_j A) = \frac{k}{4\pi} \int_\Sigma \langle\delta_i A \wedge \delta_j A\rangle.$$

Here $\langle, \rangle$ is the Killing form on $\mathfrak{g}$, where the $\delta_j A$ are valued. This formula is derived by studying the variational problem of the Chern-Simons action on a 3-manifold with boundary $\Sigma$. It's discussed here in a rigorous formulation (pdf) in section 4.2. They also go on to discuss Hamiltonian reduction to the finite dimensional phase space. The idea is that this small phase space is a $G$-quotient of a level set of the moment map $$\mu = \frac{-k}{2\pi} F$$ which generates the action of gauge transformations. You can use this to construct the symplectic form, since now you can simply pull back $\Omega$ to $\mu^{-1}(\epsilon)$ and evaluate it on $G$-invariant vector fields there, which are equivalent to vector fields on the true phase space $\mu^{-1}(\epsilon)/G$. So basically you use the same formula as $\Omega$.

Edit: As requested, if we're studying the moduli space of flat connections on a torus, we can think of this as a pair of commuting elements $g_1, g_2 \in G$ defined up to simultaneous conjugation $(g_1,g_2) \mapsto (hg_1h^{-1},hg_2h^{-1})$. This is a subquotient of $G \times G/G$, but the quotient $G$ acts by conjugation, so it's not the same as $G$, for example with $G = U(1)$ it's $U(1) \times U(1)$.

We can write $g_j = \exp(2\pi i t_j)$ and the connection form $A = t_1 dx_1 + t_2 dx_2$, where $x_1,x_2$ are $2\pi$ periodic coordinates on the torus. This way, $\exp i \int_{j} A = g_j$ and $dA + [A,A] = 0$.

The tangent space at $A$ is given by all 1-forms $\delta A$ such that $A + \epsilon\delta A$ is flat to order $\epsilon^2$. This means $d\delta A + [A,\delta A] = 0$. You can solve this equation using the structure constants of the Lie algebra. Then the symplectic form is evaluated as $\Omega$ above with $\Sigma = T^2$.

For instance, the tangent space at the zero sections is just the space of closed $\mathfrak{g}$-valued 1-forms, and we use the Killing form to evaluate $\Omega$.