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We define a dg-category of looped diagrams which we use to constructoperations on the Hochschild complex of commutative Frobenius dg-algebras. Weshow that we recover the operations known for symmetric Frobenius dg-algebrasconstructed using Sullivan chord diagrams as well as all formal operations forcommutative algebras (including Loday's lambda operations) and prove that thereis a chain level version of a suspended Cactus operad inside the complex oflooped diagrams. This recovers the suspended BV algebra structure on theHochschild homology of commutative Frobenius algebras defined by Abbaspour andproves that it comes from an action on the Hochschild chains.

Natural operations on the Hochschild complex of commutative Frobenius algebras via the complex of looped diagrams