Loop homology of moment-angle complexes in the flag case

We develop a general homological approach to presentations of connected graded associative algebras, and apply it to the loop homology of moment-angle complexes $Z_K$ that correspond to flag simplicial complexes $K$. For arbitrary coefficient ring, we describe generators of the Pontryagin algebra $H_{\ast }(\mathrm{\Omega }{Z}_K)$ and defining relations between them. We prove that such moment-angle complexes are coformal over $\mathbb{Q},$ give a necessary condition for rational formality, and compute their homotopy groups in terms of homotopy groups of spheres.