On Higher Topological T-duality Functors

We propose a higher version of the Topological T-duality Functor P of Bunke and co-workers {BunkeS1}. We begin with an introduction to T-duality in string theory and Topological T-duality. We then study morphisms in the category of pairs X over X defined in Ref.\ {BunkeS1}. For every X we naturally construct a simplicial complex $G(X)$ from X and study its homotopy properties. We calculate all the homotopy groups of $G(X).$ We show that P is naturally π₀(G(X)) P₀(X) and define Pᵢ as the higher homotopy groups of $G(X).$ We argue that the Pᵢ are invariant under the action of Topological T-duality on X. We argue that $G(X)$ is a way of labelling toroidally compactified vacua in string field theory with X as the space of uncompactified directions. We also argue that P₀, P₁ may be described naturally in terms of string field theory. We study some properties of the functors P₀,P₁. We show the connection between these functors and topological T-duality for triples studied in Ref.\ {Pan2}. We also show that this approach to Topological T-duality gives us a natural definition of T-folds {HullT}.