I think the string field thing goes like this: If $\Phi(X) = \lim_{N \to \infty} \Phi(X_{\mu}(\sigma_1),...,X_{\mu}(\sigma_N))$ is a string field then
ignoring the limit and setting $N = 2$ in $(1,1)$ dimensions this apparently means, for example
$$\Phi(X) = \Phi(X_0(\sigma_1),X_1(\sigma_1);X_1(\sigma_2),X_2(\sigma_2)) = X_0(\sigma_1)^{44} + \frac{2}{3} X_0(\sigma_2)^3 + X_0(\sigma_1) X_0(\sigma_2) + X_1(\sigma_2)^3.$$
and if
\begin{align}
\mathcal{L}(\Phi,\partial \Phi) &= \frac{1}{2} (\frac{\partial \Phi}{\partial X_0(\sigma_1)} ) (\frac{\partial \Phi}{\partial X^0(\sigma_1)} ) + \frac{1}…