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 Völcsey–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 Poincaré duality is given. For negative cyclic cohomology, this in particular leads to a Batalin–Vilkoviski˘ı (BV) algebra structure on the underlying Hochschild cohomology. In the special case in which this BV bracket vanishes, one obtains an e3-algebra structure on Hochschild cohomology. The results are given in the general and unifying setting of (opposite) cyclic modules over (cyclic) operads.