T-duality in rational homotopy theory via $L_\infty$-algebras

We combine Sullivan models from rational homotopy theory with Stasheff's $L_\infty$-algebras to describe a duality in string theory. Namely, what in string theory is known as topological T-duality between $K^0$-cocycles in type IIA string theory and $K^1$-cocycles in type IIB string theory, or as Hori's formula, can be recognized as a Fourier-Mukai transform between twisted cohomologies when looked through the lenses of rational homotopy theory. We show this as an example of topological T-duality in rational homotopy theory, which in turn can be completely formulated in terms of morphisms of $L_\infty$-algebras.