UC Berkeley

This dissertation investigates ingredients necessary for the worldsheet formulation of string theory in nonequilibrium regimes, moving beyond the traditional framework of static, equilibrium target spaces. While conventional perturbative string theory has yielded profound mathematical insights into differential geometry and algebraic topology, these successes are overwhelmingly predicated on equilibrium dynamics and simplifying assumptions like supersymmetry. However, modeling the complex, time dependent evolution of realistic quantum systems from the rapid expansion of the early universe to driven quantum materials requires a complete generalization of string theory. The aim of this thesis is primarily to introduce the scientific reader to the problem of nonequilibrium string theory.To address this problem, we leverage the Schwinger Keldysh formalism alongside path integral localization. We suggest that from the perspective of the string worldsheet, the turnaround region of the Schwinger Keldysh closed time contour degenerates into a piecewise linear structure mathematically governed by tropical geometry. We explicitly show this in the simplified case of a topological sigma model by applying a tropical limit and through using the localization principle of cohomological path integrals, we formally construct tropological sigma models. These novel models do not require a nondegenerate metric or complex structure on the worldsheet, yet their path integrals successfully localize to tropical limits of pseudoholomorphic maps and yield a non relativistic string worldsheet foliation structure based on nilpotent worldsheet endomorphisms that we coin Jordan structures.Finally, the dissertation explores the extended dynamics of these models, including the introduction of boundary conditions that give rise to tropical branes. It further demonstrates that tropological sigma models can admit unusual higher dimensional target spaces structured as filtered geometries which further admit exotic kinematical symmetries given by Nilmanifolds. This forms an unusual connection between the geometric analysis of filtered geometries and Nilmanifolds with the real algebraic geometry given by tropical geometry.