I want to find some algebraic models relating string topology to Hochschild and cyclic homologies. If the space $X$ is simply-connected and we are working over rational numbers, we can use Sullivan model or Quillen model of $X$ to do the job. See nLab page or this paper.

My question is how about non-simply conncted spaces? We still have Sullivan model. But how to relate them to free loop spaces? Can we still use Hochschild and cyclic homologies to compute loop homology and string homology?

For example, the torus $T^{2}$. Its Sullivan model is just $\wedge (x,\,y)$ with zero differential where $|x|=|y|=1$, and it is isomorphic to the Chevalley-Eilenberg cochain algebra of 2 dimensional abelian Lie algebra. But I don't know how to use this to compute loop and string homologies.

Any help would be very appreciated.