The higher-loop-space conjecture on iterated integrals over membranes

Let $X$ be a smooth complex quasi-projective algebraic variety. For each $n$, let

$$Maps((T^n,\dot{T}^n),(X,x_1,\dots,x_n))$$

denote the corresponding quotient of the space of maps from the infinite-dimensional torus, and let $Maps((T^\infty,\dot{T}^\infty),(X,\{x_n\}))$ be the space of maps that factor through a finite-dimensional torus. Iterated integrals over membranes are the differential forms constructed from iterated integrals on these toroidal membranes.

Higher-loop-space conjecture. (a) The space $Maps((T^\infty,\dot{T}^\infty),(X,\{x_n\}))$, together with all its quotients $Maps((T^n,\dot{T}^n),(X,x_1,\dots,x_n))$, captures the homotopy type of $X$ together with its Postnikov tower. (b) Iterated integrals over membranes on $X$ capture the rational homotopy type of $X$ together with its Postnikov tower after tensoring with the rational numbers.

The conjecture proposes that higher loop spaces and their de Rham complexes retain both integral homotopy-theoretic information and its rational model for smooth complex quasi-projective varieties. The paper does not provide a resolution of these assertions.