Homotopy cyclic A-infinity algebras, potentials and related cohomology theories

Abstract

An A-infinity algebra has "associative up to homotopy" structure. For an A-infinity algebra $A$, we give a definition of strong homotopy inner products(if exist) which is the homotopy notion of cyclic inner products due to Kontsevich. From strong homotopy inner products we get several invariants which we call "potentials". We study their homotopy natures, gauge invariances etc. Also we find an explicit correspondence between cohomology elements of $A$ and isomorphism classes of strong homotopy inner products on $A$.