Cartan calculi on the free loop spaces

A typical example of a Cartan calculus consists of the Lie derivative and the contraction with vector fields of a manifold on the derivation ring of the de Rham complex. In this manuscript, a second stage of the Cartan calculus is investigated. In a general setting, the stage is formulated with operators obtained by the Andr\'e-Quillen cohomology of a commutative differential graded algebra $A$ on the Hochschild homology of $A$ in terms of the homotopy Cartan calculus in the sense of Fiorenza and Kowalzig. Moreover, the Cartan calculus is interpreted geometrically with maps from the rational homotopy group of the monoid of self-homotopy equivalences on a space $M$ to the derivation ring on the loop cohomology of $M$. We also give a geometric description to Sullivan's isomorphism, which relates the geometric Cartan calculus to the algebraic one, via the $\Gamma_1$ map due to F\'elix and Thomas.