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

TL;DR

This paper bridges string theory dualities with rational homotopy theory by using Sullivan models and $L_$-algebras to mathematically formalize T-duality as a Fourier-Mukai transform.

Contribution

It introduces a novel formulation of topological T-duality within rational homotopy theory using $L_$-algebras, connecting string theory concepts with algebraic topology.

Findings

T-duality can be described as a Fourier-Mukai transform in rational homotopy theory.

$L_$-algebras provide a natural framework for formulating T-duality.

The approach unifies string theory dualities with algebraic models in topology.

Abstract

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.