First, I will describe a new chain model of the (based and free) loop space of a path-connected topological space X, which can be viewed as a generalization of classical theorems of J. F. Adams and K-T Chen. Then I will combine this model with a Jones' type theorem on cyclic homology, as well as K. Irie's work on string topology, to describe chain level string topology operations in the S^1-equivariant setting, in particular, chain level string bracket (cyclic loop bracket). Finally, I will use this chain model to lift the Fukaya A-infinity algebra of a Lagrangian submanifold L to a Maurer-Cartan element in the dg Lie algebra of cyclic invariant chains on the free loop space of L, and discuss applications in symplectic topology.