Gerstenhaber algebra of an associative conformal algebra

We define a cup product on the Hochschild cohomology of an associative conformal algebra $A$, and show the cup product is graded commutative. We define a graded Lie bracket with the degree $-1$ on the Hochschild cohomology ${\text{HH}}^{\ast }(A)$ of an associative conformal algebra $A$, and show that the Lie bracket together with the cup product is a Gerstenhaber algebra on the Hochschild cohomology of an associative conformal algebra. Moreover, we consider the Hochschild cohomology of split extension conformal algebra $A\widehat{\oplus }M$ of $A$ with a conformal bimodule $M$, and show that there exist an algebra homomorphism from ${\text{HH}}^{\ast }(A\widehat{\oplus }M)$ to ${\text{HH}}^{\ast }(A)$.