What is the total space of this infinite-dimensional vector bundle?

Question

In McDuff-Salamon, the moduli space $\mathcal{M}(A,\Sigma;J)$ of $J$-holomorphic curves $u:(\Sigma,j)\to(M,j)$ satisfying $u_*[\Sigma]=A\in H_2(M;\mathbb Z)$ is thought of as the zero set of a section of some infinite-dimensional vector bundle. In particular, the base space is $\mathcal B=\{u\in C^\infty(\Sigma,M):u_*[\Sigma]=A\}$, and the vector bundle is defined by setting the fiber at $u$ to be the space $\mathcal E_u=\Omega^{0,1}(\Sigma,u^*TM)$. What I don't understand is how this fully determines the vector bundle. We probably set the vector bundle so that the map $u\mapsto(u,\overline{\partial}_J(u))$ is a smooth section. But in general, knowing the fibers and a single global section isn't enough to determine the manifold structure of the total space. I'm not sure if this is because it isn't important in this context, or if there is another natural condition on $\mathcal E$ that should tell me what $\mathcal E$ really "is."