Let $V$ be a differential graded Batalin-Vilkovisky (DGBV) algebra over say $\mathbb{C}$. This means that $V$ is a commutative differential graded algebra (CDGA), with differential $\partial$ of degree $+1$, equipped with a BV operator $\Delta: V \rightarrow V$ of degree $-1$ satisfying the following conditions:

- $\Delta^2 = 0$
- $\partial \Delta + \Delta \partial = 0$
- $[x,y\cdot z] = [x,y]\cdot z + (-1)^{(|x|-1)|y|}y\cdot [x,z]$, where $[x,y] := (-1)^{|x|}(\Delta(xy) - (\Delta x)\cdot y - (-1)^{|x|}x\cdot (\Delta y))$.

In particular, the differential, product, and bracket give $V$ the structure of a differential graded Gerstenhaber algebra, and the differential and bracket give it the structure of a differential graded Lie algebra.

The differential and BV operator also give $V$ the structure of an $S^1$-complex, so we can define the equivariant chain complex by putting $V_{S^1} := V \otimes \mathbb{C}[u^{-1}]$, with equivariant differential $\partial_{S^1} := \partial + u\Delta$. Here $\mathbb{C}[u^{-1}]$ is a shorthand for $\left(\mathbb{C}[[u]][u^{-1}]\right)/\left(u\mathbb{C}[[u]]\right)$. There is also a *homology level* Lie bracket on $V_{S^1}$ of degree $-2$, given by $[x,y]_{S^1} := (\Delta x_0) \cdot (\Delta y_0)$, for $x = x_0 + u^{-1}x_1 + ... + u^{-k}x_k$ and $y = y_0 + u^{-1}y_1 + ... + u^{-l}y_l$.

Question: Does the homology level bracket $[-,-]_{S^1}$ naturally come from an $\mathcal{L}_\infty$ structure $(V_{S^1},\ell_{S^1}^1 = \partial_{S^1},\ell_{S^1}^2,\ell_{S^1}^3,...)$, and are there explicit formulas for these operations? Here $\ell_{S^1}^j$ should have degree $4-3j$.

Note: prominent examples of $V$ include Hochschild cochains of an associative algebra and chains on the free loop space of a smooth manifold. In the latter case, $V_{S^1}$ gives equivariant chains on the free loop space, and the homology level bracket $[-,-]_{S^1}$ is the string bracket.