Since my question is related to sign convention, I want to define everything from the very beginning. $T_{poly}^k(M)=\Gamma(\wedge^{k+1} TM)$ are the multi vector fields with shifted degree and with the Schouten bracket uniquely determined by \begin{align*} [\![f,g]\!] =0\ \text{, } [\![X,f]\!]=X(f) \text{ and } [\![X,Y]\!]=[X,Y] \end{align*} for $X,Y\in T_{poly}^0(M)$ and $f,g\in T_{poly}^{-1}(M)$. Hence $(T_{poly}^\bullet(M),d=0, [\![-,-]\!] )$ is a DGLA.

Let us denote by \begin{align*} D_{poly}^k= \begin{cases} \mathrm{DiffOp}(C^{\infty}(M)^{\otimes(k+1)},C^{\infty}(M)),\text{ for }k\geq 0 \\ C^\infty(M), \text{ for } k=-1\\ 0, \text{ else } \end{cases} \end{align*} the multi differential operators with shifted degree by $1$. For $f_i\in C^{\infty}(M)$, $\phi\in D_{\mathrm{poly}}^k(M) $ and $\psi\in D_{\mathrm{poly}}^l(M)$, we define \begin{align*} (\phi\circ_i \psi)(f_0,\dots,f_{k+l})=\phi(f_0,\dots, \psi(f_i,\dots, f_{i+l}),\dots,f_{k+l}) \end{align*} for $k,l\geq 0$ and \begin{align*} (\phi\circ_i \psi)(f_0,\dots,f_{k+l})=\phi(f_0,\dots,f_{i-1}, \psi,f_{i}, \dots,f_{k}) \end{align*} for $k\geq 0$ and $l=-1$. The Gerstenhaber product is defined by \begin{align*} \phi\circ \psi=\sum_{i=0}^{\deg \phi}(-1)^{i\deg\psi} \phi\circ_i\psi \end{align*} and finally \begin{align*} [\phi,\psi]=\phi\circ\psi -(-1)^{\deg\phi\deg\psi}\psi\circ\phi. \end{align*} The point-wise multiplication of functions $m_0\in D_{poly}^1(M)$ fulfills $[m_0,m_0]=0$ and hence $\delta(x)=[m_0,x]$ is a differential, which makes $(D_{poly}^\bullet(M),\delta,[-,-])$ into a DGLA. As far as I got, Kontesevich proved the existence of an $L_\infty$-morphism $F$ between $T^\bullet_{poly}(M)$ and $D^\bullet_{poly}(M)$ uniquely determined by is Taylor coefficients \begin{align*} F_k\colon \wedge^k T^\bullet_{poly}(M)[1]\to D_{poly}^\bullet(M)[1-k] \end{align*} where \begin{align*} F_1(X_1\cdots X_k)(f_1,\dots,f_k)= \frac{1}{k!}\sum_{\sigma\in S_k}X_{\sigma(1)}(f_1)\cdots X_{\sigma(k)}(f_k) \end{align*} is the Hochschild-Kostant-Rosenberg map which is a chain map. Since $F$ is a $L_\infty$-morphism, we know that for a Poisson structure $\pi\in T^1_{poly}(M)$ the element $\hbar\pi$ is a Maurer-Cartan element in $T_{poly}^\bullet(M)[[\hbar]]$ and so \begin{align*} \hbar m_\star:= \sum_{k=1}^\infty \frac{\hbar^k}{k!}F_k(\pi\wedge\dots \wedge \pi) \end{align*} is a Maurer-Cartan element and hence $m_0+\hbar m_\star$ is a star product. Moreover, we have then two new DGLA's \begin{align*} (T^\bullet_{poly}(M)[[\hbar]], d_\pi=[\![\hbar\pi,-]\!], [\![-,-]\!]) \text{ and } (D^\bullet_{poly}(M)[[\hbar]], \delta_\star=[m_0+\hbar m_\star,-],[-,-]) \end{align*} and a $L_\infty$-morphism $F^\pi$ uniquely determined by \begin{align*} F^\pi_l (X_1\wedge\dots\wedge X_l)=\sum_{i=0}^{\infty} \frac{\hbar^i}{i!}F_{i+l}(\pi\wedge\dots\wedge\pi\wedge X_1\wedge\dots\wedge X_l) \end{align*} Since $F^\pi$ is an $L_\infty$-morphism we know again that $F^\pi_1$ is a chain map. Let us denote by $X_f=[\![\pi,f]\!]$, then we have \begin{align*} \hbar \mathcal{L}_{X_f}&= \hbar F_1(X_f)=F_1(d_\pi(f))\\& =\sum_{i=0}^\infty \frac{\hbar^i}{i!} F_{i+1}(\pi\wedge\dots\wedge\pi\wedge d_\pi(f))\mod\hbar^2\\& =F_1^\pi(d_\pi(f))\mod\hbar^2\\& =\delta_\star F_1^\pi(f) \mod\hbar^2\\& =[f,-]_\star \mod\hbar^2\\& =\hbar F_1(\pi)(f,-)\mod\hbar^2 \end{align*} where I used that for a function $f$ the differential $d_\star $ is just the insertion in the first argument of the (deformed) product minus the insertion in the second argument. But now we have that $F_1(\pi)(f,-)=\mathcal{L}_{\pi^\sharp(df)}=\mathcal{L}_{-[\![\pi,f]\!]}=-\mathcal{L}_{X_f}$, which clearly contradicts the left hand side of the equation.

Now, my questions:

(1) Are my computations with the defined DGLA structures correct?

(2) If no, where exactly is the mistake?

(3) If yes, which is the right DGLA structure on $D_{poly}^\bullet(M)$, such that the $F_k$'s are an $L_\infty$-morphism. (I am assuming that $D_{poly}^\bullet(M)$ has a wrong sign convention, since in the literature the Schouten bracket is usually defined the same ways, but in the definition of the structures on the Hochschild complex, there are many different conventions.)


Welcome to mathoverflow!

There is actually a whole paper (in French) about choices of signs for Kontsevich formality: https://arxiv.org/pdf/math/0003003.pdf

For instance, they define the Hochschild coboundary operator as $\delta(x)=-[m_0,x]$ (see also their Remark on page 20).

I hope this will help.

  • $\begingroup$ Thanks for your answer. I took a look in this paper and they even use a different sign convention for the Schouten bracket. Moreover, the additional minus sign of the differential makes things for my purposes a bit more complicated: Maurer-Cartan elements have to be subtracted from the original product to obtain a star product. I havve found this article: arxiv.org/abs/0909.4259. At least the sign convention in there does not give the same contradiction as in my post. $\endgroup$ – Heinz Doofenschmirtz Feb 14 at 16:57

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