T-Duality with H-Flux for 2d \(\sigma \)-Models

In this paper, we establish graded T-duality for 2d \(\sigma \) $\sigma$ -models with H-flux after localization. This establishes the most general version of T-duality for Type II String Theory. The graded T-duality map, which we call graded Hori morphism, is compatible with the Jacobi property of the graded fields, that was earlier studied in (Adv Theor Math Phys 25: 1235–1266, 2021). Also included are some open problems/conjectures.