# Equivariant L-infinity structure associated to a DGBV algebra

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.