Higher brackets on cyclic and negative cyclic (co)homology

The purpose of this article is to embed the string topology bracket developed by Chas-Sullivan and Menichi on negative cyclic cohomology groups as well as the dual bracket found by de Thanhoffer de Volcsey-Van den Bergh on negative cyclic homology groups into the global picture of a noncommutative differential (or Cartan) calculus up to homotopy on the (co)cyclic bicomplex in general, in case a certain Poincare duality is given. For negative cyclic cohomology, this in particular leads to a Batalin-Vilkoviski (BV) algebra structure on the underlying Hochschild cohomology. In the special case in which this BV bracket vanishes, one obtains an -algebra structure on Hochschild cohomology. The results are given in the general and unifying setting of (opposite) cyclic modules over (cyclic) operads.