The free loop space equivariant cohomology algebra of some formal spaces

Let \mathbbK{\mathbb{K}} be a field of characteristic p > 0 and S ¹ the unit circle. We construct a model for the negative cylic homology of a commutative cochain algebra with two stages Sullivan minimal model. Using the notion of shc-formality introduced in Bitjong and Thomas (Topology 41:85–106), the main result of Bitjong and El Haouari (Math Ann 338:347–354) and techniques of Vigué-Poirrier (J Pure Appl Algebra 91:347–354) we compute the S ¹-equivariant cohomology algebras of the free loop spaces of the infinite complex projective space \mathbbCP(￥){\mathbb{CP}(\infty)} and the odd spheres S ^2q+1.