Based maps to Lagrangian Grassmannians, Quivers, and Bott Periodicity

We give a quiver description of the space of based algebraic maps from $\mathbb{P}^{1}$ to the Lagrangian Grassmannian (and its orthogonal counterpart). We show our descriptions lead to an algebro-geometric refinement of some of the homotopy equivalences in real Bott periodicity. In particular, we get an isomorphism in Larson and Vakil's ``naive algebro-geometric homotopy category'' whose topological realization (after specializing to $\mathbb{C}$) recovers the classical homotopy equivalences $\Omega^{2}(Sp/U) \simeq BO\times \mathbb{Z}$ and $\Omega^{2}(O/U) \simeq BSp\times \mathbb{Z}$.