On the cohomology algebra of free loop spaces

Let X be a simply connected space and K be any field. The normalized singular cochains N∗(X; K) admit a natural strongly homotopy commutative algebra structure, which induces a natural product on the Hochschild homology HH∗N∗X of the space X. We prove that, endowed with this product, HH∗N∗X is isomorphic to the cohomology algebra of the free loop space of X with coefficients in K. We also show how to construct a simpler Hochschild complex which allows direct computation.