G${}^2$WAM: World Action Models for Efficient and Geometry Grounded Image-Goal Navigation

World models capture environment dynamics and support counterfactual reasoning for planning. In image-goal navigation, where the target often lies beyond the current field of view, the agent must not only model environmental dynamics but also maintain a persistent spatial memory of the scene. Existing approaches either introduce spatial context by fine-tuning pretrained models on the target task, which underexploits their spatial priors, or rely on external 3D reconstruction, which is computationally expensive for real-world deployment. In this paper, we propose the Geometry Grounded World Action Model (G${}^2$WAM) for image-goal navigation. G${}^2$WAM encodes spatial cues through a learned scene-coordinate localization module that anchors the current, goal, and history frames in a common reference frame, yielding a stable and consistent spatial context that enables efficient navigation without explicit 3D reconstruction. Following the world-action model paradigm, G${}^2$WAM jointly predicts future states and actions, eliminating the need for costly CEM-style planning. In addition, G${}^2$WAM operates entirely in latent space, avoiding the computational burden of pixel-level reconstruction and substantially reducing inference cost. Across four trajectory prediction benchmarks, G${}^2$WAM reduces average ATE and RPE by 40.1% and 35.2%, respectively. In closed-loop navigation experiments, it improves the success rate from 54.0% to 63.0% and the success weighted by path length (SPL) from 36.4% to 59.2%, while maintaining high efficiency.