symmetric monoidal (∞,1)-category of spectra
For the actual relation to BV-complexes see at relation between BV and BD.
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 the same degree as the bracket
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 (Cohen-Voronov, def. 5.3.1, theorem 2.1.3) for details.
The operad for BV-algebras is the homology of the framed little 2-disk operad.
This is due to (Getzler)
The homology of an algebra over an operad over the framed little n-disk operad has a natural structure of an $(n+1)$-BV-algebra.
This appears as (CohenVoronov, theorem 5.3.3). The full homology of the framed little n-disk operad is described by (SalvatoreWahl, theorem 5.4).
Multivector fields may be identified with Hochschild cohomology in good cases (the Hochschild-Kostant-Rosenberg theorem). So the next example is a generalization of the previous one.
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
Ralph Cohen, Alexander Voronov, Notes on string topology (arXiv:math/0503625)
Paolo Salvatore, Nathalie Wahl, Framed discs operads and Batalin-Vilkovisky algebras (pdf)
There are examples coming from Lagrangian intersection theory
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
Luc Menichi, Batalin-Vilkovisky algebra structures on Hochschild cohomology (pdf)
Thomas Tradler, The BV Algebra on Hochschild Cohomology Induced by Infinity Inner Products, Annales de l’institut Fourier (2008) Volume: 58, Issue: 7, page 2351-2379 (arXiv:math/0210150)
There is a prominent class of examples coming from Lie-Rinehart algebras