symmetric monoidal (∞,1)-category of spectra
A Batalin-Vilkovisky algebra or BV-algebra for short is
a Gerstenhaber algebra $(A, \cdot, [-,-])$
equipped with a unary linear operator $\Delta : A \to A$ of degree +1
such that
$\Delta$ is a derivation for $[-,-]$;
$[-,-]$ is the failure of $\Delta$ being a derivation for $\cdot$:
A $(n+1)$-BV algebra is a similar structure with a BV-operator being of degree $n$ if $n$ is odd, and of degree $n/2$ if it is even.
See (CohenVoronov, def. 5.3.1) for details.
The operad for BV-algebras is the homology of the framed little 2-disk operad.
This is due to (Getzler)
The operad for $(n+1)$-BV-algebras is the homology of the framed little n-disk operad.
This appears as (CohenVoronov, theorem 5.3.3).
Multivecotr field can be identified with Hochschild cohomology in good cases. So the next example is a generalization of the previous one.
BV-algebra
The identification o BV-algebras as algebras over the homology of the framed little disk operad is due to
The generalization to higher dimensional framed little disks is discussed in
The BV-algebra structure on multivector fields on an oriented smooth manifold is discussed for instance in section 2 of
and on p. 6 of
The BV-algebra structure on Hochschild cohomology is discussed for instance in