$\displaystyle\begin{split}[\enspace,\enspace]_{\mathrm{G}}\colon\mathcal{C}^{m}\otimes\mathcal{C}^{n}&\to\mathcal{C}^{m+n},\\\
(\phi,\psi)&\mapsto[\phi,\psi]_{\mathrm{G}}:=\phi\circ\psi-(-1)^{mn}\psi\circ\phi.\end{split}$

######  Remark 4.2.39.

There is a more general notion of a _Gerstenhaber algebra_ [Ger63], which
describes the connection between _supercommutative rings_ [Var04] and _graded
Lie superalgebras_ [Kac77]. In particular, it consists of a graded commutative
algebra endowed with a Poisson bracket of degree $-1$. This structure plays
also a fundamental role in other constructions of theoretical physics, such as
in _DeDonder–Weyl theory_ [DeD30, Wey35] (a generalization of the Hamiltonian
formalism to field theory) and in the _Batalin–Vilkovisky formalism_ [BV77,
BV81, BV83] (a way of treating perturbative quantization of gauge theories).

######  Proposition 4.2.40.

The graded vector space $\mathcal{C}$ together with the Gerstenhaber bracket
(4.2.15) is a GLA, which we call _Hochschild GLA_.

###### Proof.

Clearly, the Gerstenhaber bracket is linear. Note that the sign $(-1)^{mn}$
ensures that it is also (graded) skew-symmetric, since

$[\phi,\psi]_{\mathrm{G}}=-(-1)^{mn}\bigg{(}\psi\circ\phi-(-1)^{mn}\phi\circ\psi\bigg{)}=-(-1)^{mn}[\psi,\phi]_{\mathrm{G}},\quad\forall\phi\in\mathcal{C}^{m},\psi\in\mathcal{C}^{n}.$

Next, we need to make sure that $[\enspace,\enspace]_{\mathrm{G}}$ satisfies
the (graded) Jacobi identity

(4.2.16)
$[\phi,[\psi,\chi]_{\mathrm{G}}]_{\mathrm{G}}=[[\phi,\psi]_{\mathrm{G}}]_{\mathrm{G}}+(-1)^{mn}[\psi,[\phi,\chi]_{\mathrm{G}}]_{\mathrm{G}},\quad\forall\phi\in\mathcal{C}^{m},\psi\in\mathcal{C}^{m},\chi\in\mathcal{C}^{p}.$

Note that the left hand side of (4.2.16) gives

(4.2.17)
$\bigg{(}\phi\circ\psi-(-1)^{mn}\psi\circ\phi\bigg{)}\circ\chi-(-1)^{(m+n)p}\chi\circ\bigg{(}\phi\circ\psi-(-1)^{mn}\psi\circ\phi\bigg{)}\\\
=\sum_{0\leq i\leq m\atop 0\leq k\leq
m+n}(-1)^{in+kp}(\phi\circ_{i}\psi)\circ_{k}\chi-\sum_{0\leq j\leq n\atop
0\leq k\leq m+n}(-1)^{m(j+n)+kp}(\psi\circ_{j}\phi)\circ_{k}\chi\\\
-\sum_{0\leq i\leq m\atop 0\leq k\leq
p}(-1)^{(m+n)(k+p)+in}\chi\circ_{k}(\phi\circ_{i}\psi)+\sum_{0\leq j\leq
n\atop 0\leq k\leq
p}(-1)^{(m+n)(k+p)+m(j+n)}\chi\circ_{k}(\psi\circ_{j}\phi).$

We can decompose the first sum according to the following rule

$(\phi\circ_{i}\psi)\circ_{k}\chi=\begin{cases}(\phi\circ_{k}\chi)\circ_{i}\psi,&k<i\\\
\phi\circ_{i}(\psi\circ_{k-i}\chi),&i\leq k\leq i+n\\\
(\phi\circ_{k-n}\chi)\circ_{i}\psi,&i+n<k\end{cases}$

into a term of the form

(4.2.18) $\sum_{0\leq i\leq m\atop i\leq k\leq
i+n}(-1)^{in+kp}\phi\circ_{i}(\psi\circ_{k-i}\chi)=\sum_{0\leq i\leq m\atop
0\leq k\leq n}(-1)^{(n+p)i+kp}\phi\circ_{i}(\psi\circ_{k}\chi).$

Note that the sign of (4.2.18) is the same as the one in the corresponding
term coming from $(\phi\circ\psi)\circ\chi$ on the left hand side of (4.2.17)
plus the terms in which the $i$-th and $k$-th composition commute. These then
cancel the corresponding terms coming from the expansion of the second term of
the right hand side of (4.2.16). Using the same approach for the remaining
terms, we get the proof. ∎

######  Remark 4.2.41.

Note that associative multiplications on $\mathcal{A}$ are elements in
$\mathcal{C}^{1}$. In particular, if we denote by
$\mathsf{m}\in\mathcal{C}^{1}$ a multiplication map, the associativity
condition is given by

$\mathsf{m}(\mathsf{m}(f\otimes g)\otimes
h)-\mathsf{m}(f\otimes\mathsf{m}(g\otimes h))=0.$

This is in fact equivalent to the condition that the Gerstenhaber bracket of
$\mathsf{m}$ with itself vanishes. Indeed, we have

$[\mathsf{m},\mathsf{m}]_{\mathrm{G}}(f\otimes g\otimes h)=\sum_{0\leq i\leq
1}(-1)^{i}(\mathsf{m}\circ_{i}\mathsf{m})(f\otimes g\otimes
h)-(-1)^{1}\sum_{0\leq i\leq
1}(-1)^{i}(\mathsf{m}\circ_{i}\mathsf{m})(f\otimes g\otimes h)\\\
=2\bigg{(}\mathsf{m}(\mathsf{m}(f\otimes g)\otimes
h)-\mathsf{m}(f\otimes\mathsf{m}(g\otimes h))\bigg{)}.$

Now, for an element $\phi$ of a (DG) Lie algebra $\mathfrak{g}$ (of degree
$\ell$), we get that

$\mathrm{ad}_{\phi}:=[\phi,\enspace]$

is a derivation (of degree $\ell$). Indeed, by the Jacobi identity, we have

$\mathrm{ad}_{\phi}[\psi,\xi]=[\mathrm{ad}_{\phi}\psi,\xi]+(-1)^{\ell
m}[\psi,\mathrm{ad}_{\phi}\xi],\quad\forall\psi\in\mathfrak{g}^{m},\xi\in\mathfrak{g}^{n}.$

######  Definition 4.2.42 (Hochschild differential).

The _Hochschild differential_ is given by

(4.2.19)
$\displaystyle\begin{split}{\mathrm{d}}_{\mathsf{m}}\colon\mathcal{C}^{i}&\to\mathcal{C}^{i+1}\\\
\psi&\mapsto{\mathrm{d}}_{\mathsf{m}}\psi:=[\mathsf{m},\psi]_{\mathrm{G}}.\end{split}$

######  Proposition 4.2.43.

The GLA $\mathcal{C}$ endowed with the Hochschild differential
${\mathrm{d}}_{\mathsf{m}}$ is a DGLA.

###### Proof.

The only thing we need to check is that the Hochschild differential indeed
squares to zero (i.e.
${\mathrm{d}}_{\mathsf{m}}\circ{\mathrm{d}}_{\mathsf{m}}=0$). This follows
immediately from the Jacobi identity and the associativity condition of
$\mathsf{m}$ expressed in terms of the Gerstenhaber bracket:

$({\mathrm{d}}_{\mathsf{m}}\circ{\mathrm{d}}_{\mathsf{m}})\psi=[\mathsf{m},[\mathsf{m},\psi]_{\mathrm{G}}]_{\mathrm{G}}=[[\mathsf{m},\mathsf{m}]_{\mathrm{G}},\psi]_{\mathrm{G}}-[\mathsf{m},[\mathsf{m},\psi]_{\mathrm{G}}]_{\mathrm{G}}\\\
=-[\mathsf{m},[\mathsf{m},\psi]_{\mathrm{G}}]_{\mathrm{G}}\Leftrightarrow{\mathrm{d}}_{\mathsf{m}}\circ{\mathrm{d}}_{\mathsf{m}}=0.$

∎

######  Remark 4.2.44.

In fact, we explicitly have

$({\mathrm{d}}_{\mathsf{m}}\psi)(f_{0}\otimes\dotsm\otimes
f_{n+1})=\sum_{0\leq i\leq n}(-1)^{i+1}\psi(f_{0}\otimes\dotsm\otimes
f_{i-1}\otimes\mathsf{m}(f_{i}\otimes f_{i+1})\otimes\dotsm\otimes f_{n+1})\\\
+\mathsf{m}(f_{0}\otimes\psi(f_{1}\otimes\dotsm\otimes
f_{n+1}))+(-1)^{n+1}\mathsf{m}(\psi(f_{0}\otimes\dotsm\otimes f_{n})\otimes
f_{n+1}).$

Let now $\mathcal{A}=C^{\infty}(M)$ and consider the and consider the
subalgebra $\widetilde{\mathcal{D}}(M)\subset\mathcal{C}$, which is the
(graded) vector space given as the collection

$\widetilde{\mathcal{D}}^{i}:=\bigoplus_{i}\widetilde{\mathcal{D}}^{i}$

of subspaces $\widetilde{\mathcal{D}}^{i}(M)\subset\mathcal{C}^{i}$ which
consist of differential operators on $C^{\infty}(M)$. One can check that
$\widetilde{\mathcal{D}}(M)$ is closed with respect to the Gerstenhaber
bracket $[\enspace,\enspace]_{\mathrm{G}}$ and the differential
${\mathrm{d}}_{\mathsf{m}}$ and hence is a DGL subalgebra555Note that we will
also allow differential operators of degree zero, i.e. operators which do not
“derive” anything. Thus, the associative product $\mathsf{m}$ is also an
element of $\widetilde{\mathcal{D}}^{1}(M)$. of $\mathcal{C}$.

Moreover, we want to restrict everything to differential operators which
vanish on constant functions. They will in fact form another DGL subalgebra
$\mathcal{D}(M)\subset\widetilde{\mathcal{D}}(M)$. Note that we want to have
this restriction because of the fact that the coefficients in the expansion of
a star product vanish on constant functions, i.e. $B_{i}(1,f)=0$ for all
$i\in\mathbb{N}$ and $f\in C^{\infty}(M)$. Note however that the Hochschild
differential ${\mathrm{d}}_{\mathsf{m}}$ is not inner derivation anymore for
$\widetilde{\mathcal{D}}(M)$ since the multiplication does not vanish on
constants.

Consider an element $B\in\mathcal{D}^{1}(M)$. Then we want to regard
$\mathsf{m}+B$ as a deformation of the original product. The associativity
condition for $\mathsf{m}+B$ implies

$[\mathsf{m}+B,\mathsf{m}+B]_{\mathrm{G}}=0,$

and by associativity of $\mathsf{m}$ we get

$[\mathsf{m},B]_{\mathrm{G}}=[B,\mathsf{m}]_{\mathrm{G}}={\mathrm{d}}_{\mathsf{m}}B.$

This does exactly lead to the Maurer–Cartan equation:

(4.2.20) ${\mathrm{d}}_{\mathsf{m}}B+\frac{1}{2}[B,B]_{\mathrm{G}}=0.$

######  Remark 4.2.45.

If we consider the _formal_ version of $\mathcal{D}(M)$, we can see that the
deformed product does exactly describe a star product since
$B\in\hbar\mathcal{D}^{1}(M)[\\![\hbar]\\!]$. Moreover, the gauge group is
given by formal differential operators and the action on the star product is
given by (4.1.5) since the adjoint action, by the definition of the
Gerstenhaber bracket, is exactly the composition of $D_{i}$ with $B_{j}$.

####  4.2.6. The Hochschild–Kostant–Rosenberg map

In [HKR62], Hochschild–Kostant–Rosenberg have shown that the cohomology of the
algebra of multidifferential operators and the algebra of multivector fields,
which is equal to its cohomology, are isomorphic. They established an
isomorphism

(4.2.21) $\mathrm{HKR}\colon
H(\widetilde{\mathcal{D}}(M))\xrightarrow{\sim}\widetilde{\mathcal{V}}(M)\cong
H(\widetilde{\mathcal{V}}(M)).$

In particular, this isomorphism is induced by the map

(4.2.22)
$\displaystyle\begin{split}U^{(0)}_{1}\colon\widetilde{\mathcal{V}}(M)&\to\widetilde{\mathcal{D}}(M),\\\
\xi_{0}\land\dotsm\land\xi_{n}&\mapsto\bigg{(}f_{0}\otimes\dotsm\otimes
f_{n}\mapsto\frac{1}{(n+1)!}\sum_{\sigma\in
S_{n+1}}\mathrm{sign}(\sigma)\xi_{\sigma(0)}(f_{0})\dotsm\xi_{\sigma(n)}(f_{n})\bigg{)}\end{split}$

######  Remark 4.2.46.

This is extended to vector fields of order zero by the identity map. However,
the map (4.2.22) fails to preserve the Lie structure. This can be easily
checked for order two. Indeed, given homogeneous bivector fields
$\chi_{1}\land\chi_{2}$ and $\xi_{1}\land\xi_{2}$, we get

$U^{(0)}_{1}\bigg{(}[\chi_{1}\land\chi_{2},\xi_{1}\land\xi_{2}]_{\mathrm{SN}}\bigg{)}\not=\left[U^{(0)}_{1}(\chi_{1}\land\chi_{2}),U^{(0)}_{1}(\xi_{1}\land\xi_{2})\right]_{\mathrm{G}}.$

The left hand side gives

$U^{(0)}_{1}\bigg{(}[\chi_{1},\xi_{1}]\land\chi_{2}\land\xi_{2}-[\chi_{1},\xi_{2}]\land\chi_{2}\land\xi_{1}-[\chi_{2},\xi_{1}]\land\chi_{1}\land\xi_{2}+[\chi_{2},\xi_{2}]\land\chi_{1}\land\xi_{1}\bigg{)}(f\otimes
g\otimes h)\\\
=\frac{1}{6}\bigg{(}\chi_{1}\big{(}\xi_{1}(f)\chi_{2}(g)\xi_{2}(h)\big{)}-\xi_{1}\big{(}\chi_{1}(f)\chi_{2}(g)\xi_{2}(h)\big{)}\\\
-\chi_{1}\big{(}\xi_{2}(f)\chi_{2}(g)\xi_{1}(h)\big{)}+\xi_{2}\big{(}\chi_{1}(f)\chi_{2}(g)\xi_{1}(h)\big{)}-\chi_{2}\big{(}\xi_{1}(f)\chi_{1}(g)\xi_{2}(h)\big{)}+\xi_{1}\big{(}\chi_{2}(f)\chi_{1}(g)\xi_{2}(h)\big{)}\\\
+\chi_{2}\big{(}\xi_{2}(f)\chi_{1}(g)\xi_{1}(h)\big{)}+\xi_{2}\big{(}\chi_{2}(f)\chi_{1}(g)\xi_{1}(h)\big{)}\bigg{)}+\mathrm{permutations}.$

The right hand side is given by

$\frac{1}{4}\bigg{(}\chi_{1}\big{(}\xi_{1}(f)\xi_{2}(g)\big{)}\chi_{2}(h)+\dotsm\bigg{)}.$

One can, however, still show that the difference of the two terms is given by
the image of a closed term in the cohomology of $\mathcal{D}(M)$. Thus, there
is some hope to control everything and still extend it to a Lie algebra
morphism whose first order approximation is given by (4.2.22). Again, to
resolve this problem, we will consider an $L_{\infty}$-morphism $U$.

####  4.2.7. The dual point of view

Let $V$ be a vector space and note that we can naturally identify polynomials
on $V$ as the _symmetric functions_ on the dual space $V^{*}$

$f(v):=\sum_{j}\frac{1}{j!}f_{j}(v\otimes\dotsm\otimes v),\quad\forall v\in
V,$

where $f_{j}\in\textnormal{Sym}^{j}(V^{*})$. We want to extend this to the
case when $V$ is a graded vector space. In this case, we need to consider the
exterior algebra instead. Note that, if we consider the _injective limit_ of
the $\textnormal{Sym}^{j}(V^{*})$ (resp. $\bigwedge^{j}V^{*}$) endowed with
the induced topology, we can define the corresponding completion
$\widehat{\textnormal{Sym}}(V^{*})$ (resp. $\widehat{\bigwedge V^{*}}$). Now
we can define a function in a formal neighborhood of $0$ to be given by the
formal Taylor expansion in the $\hbar$

$f(\hbar v):=\sum_{j}\frac{\hbar^{j}}{j!}f_{j}(v\otimes\dotsm\otimes
v),\quad\forall v\in V.$

Thus, a vector field $X$ on $V$ can be regarded as a derivation on
$\widehat{\bigwedge V^{*}}$ and the Leibniz rule makes sure that $X$ is
completely determined by its restriction on $V^{*}$. Similarly, if we have an
algebra morphism

$\varphi\colon\widehat{\bigwedge W^{*}}\to\widehat{\bigwedge V^{*}},$

it induces a map

$f:=\varphi^{*}\colon\widehat{\bigwedge V}\to\widehat{\bigwedge W}$

whose components $f_{j}$ are completely defined by their projection on $W$ as
the $\varphi_{j}$ are defined by their restriction on $W^{*}$.

######  Definition 4.2.47 ((Formal) pointed map).

A _(formal) pointed map_ is an algebra morphism between the reduced exterior
algebras

$\varphi\colon\widehat{\overline{\bigwedge
W^{*}}}\to\widehat{\overline{\bigwedge V^{*}}},$

where the overline means that we are considering the _reduced tensor algebra_
$\bar{T}(W^{*})$ (resp. $\bar{T}(V^{*})$) as in Example 4.2.21.

######  Definition 4.2.48 (Pointed vector field).

A _pointed vector field_ $X$ on a manifold $M$ is a vector field in
$\mathfrak{X}(M)$ which fixes the point zero, i.e.

$X(f)(0)=0,\quad\forall f\in C^{\infty}(M).$

######  Definition 4.2.49 (Cohomological pointed vector field).

A pointed vector field $X$ is called _cohomological_ (or _$Q$ -field_) if and
only if it commutes with itself, i.e. $X^{2}=\frac{1}{2}[X,X]=0$.

######  Definition 4.2.50 (Pointed $Q$-manifold).

A _pointed $Q$-manifold_ is a (formal) pointed manifold together with a
cohomological vector field.

Consider now a Lie algebra $\mathfrak{g}$. Note that the bracket
$[\enspace,\enspace]\colon\bigwedge^{2}\mathfrak{g}\to\mathfrak{g}$ induces a
linear map

$[\enspace,\enspace]^{*}\colon\mathfrak{g}^{*}\to\bigwedge^{2}\mathfrak{g}^{*}.$

This can be extended to the whole exterior algebra by a map

$\delta\colon\bigwedge\mathfrak{g}^{*}\to\bigwedge\mathfrak{g}^{*}[1]$

such that it satisfies the Leibniz rule and
$\delta\big{|}_{\mathfrak{g}^{*}}=[\enspace,\enspace]^{*}$.

######  Remark 4.2.51.

We can regard the exterior algebra as an _odd_ analogue of a manifold on which
$\delta$ takes the role of a (pointed) vector field. Note also that, since
$[\enspace,\enspace]$ satisfies the Jacobi identity, we have $\delta^{2}=0$
and thus $\delta$ defines a cohomological (pointed) vector field.

Consider two Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$ together with the
corresponding cohomological vector fields $\delta_{\mathfrak{g}}$ and
$\delta_{\mathfrak{h}}$ on the respective exterior algebras. Then a Lie
algebra morphism $\varphi\colon\mathfrak{g}\to\mathfrak{h}$ will correspond to
a chain map $\varphi^{*}\colon\mathfrak{h}^{*}\to\mathfrak{g}^{*}$ since

(4.2.23)
$\varphi\bigg{(}[\enspace,\enspace]_{\mathfrak{g}}\bigg{)}=[\varphi(\enspace),\varphi(\enspace)]_{\mathfrak{h}}\Longleftrightarrow\delta_{\mathfrak{g}}\circ\varphi^{*}=\varphi^{*}\circ\delta_{\mathfrak{h}}.$

######  Remark 4.2.52.

Equation (4.2.23) gives a first hint to the correspondence between
$L_{\infty}$-algebras and pointed $Q$-manifolds. Since a Lie algebra is a
particular case of a DGLA, which again can be endowed with an
$L_{\infty}$-structure, we can see that the map $\varphi$ satisfies exactly
the same condition for the first component of an $L_{\infty}$-morphism as in
(4.2.6) for $n=1$.

Let us extend this picture to the case of a graded vector space $Z$ over a
field $k$. Let us decompose the graded space $Z$ into its _even_ and _odd_
part, i.e.

$Z=Z_{[0]}\oplus Z_{[1]}$

where $Z_{[0]}$ denotes the even part, i.e. $Z_{[0]}:=\bigoplus_{i\in
2\mathbb{Z}}Z^{i}$ and $Z_{[1]}:=\bigoplus_{i\in 2\mathbb{Z}+1}Z^{i}$. Here we
have denoted by $[0],[1]$ the equivalence classes in $\mathbb{Z}/2$. For a
vector space $W$, denote by $\Pi W$ the (odd) space defined by a _parity
reversal_ on $W$, which we can also denote by $W[1]$ using the notation of
Definition 4.2.6. Let us define $V:=Z_{[0]}$ and $\Pi W:=Z_{[1]}$. Then
$Z=V\oplus\Pi W$ and functions can be identified by elements of

$\textnormal{Sym}(Z^{*}):=\textnormal{Sym}(V^{*})\otimes\bigwedge W^{*}.$

We can express the condition for a vector field
$\delta\colon\textnormal{Sym}(Z^{*})\to\textnormal{Sym}(Z^{*})[1]$ to be
_cohomological_ in terms of its coefficients

$\delta_{j}\colon\textnormal{Sym}^{j}(Z^{*})\to\textnormal{Sym}^{j+1}(Z^{*})$

by expanding the equation $\delta^{2}=0$. Hence, we get an infinite family of
equations

(4.2.24) $\displaystyle\delta_{0}\delta_{0}$ $\displaystyle=0,$ (4.2.25)
$\displaystyle\delta_{1}\delta_{0}+\delta_{0}\delta_{1}$ $\displaystyle=0,$
(4.2.26)
$\displaystyle\delta_{2}\delta_{0}+\delta_{1}\delta_{1}+\delta_{0}\delta_{2}$
$\displaystyle=0,$ $\displaystyle\vdots$

Let now $m_{j}:=(\delta_{j}|_{Z^{*}})^{*}$ be the dual coefficients and let
$\langle\enspace,\enspace\rangle\colon Z^{*}\otimes Z\to k$. Then we can
rewrite these conditions in terms of $m_{j}\colon\textnormal{Sym}^{j}(Z)\to
Z$. Note that the first Equation (4.2.24), which is now given by
$m_{0}m_{0}=0$, implies that $m_{0}$ is a differential on $Z$ and hence
induces a cohomology $H_{m_{0}}(Z)$. For $j=1$, i.e. for the second Equation
(4.2.25), we get

(4.2.27) $\langle\delta_{1}\delta_{0}f,x\otimes
y\rangle=\langle\delta_{0}f,m_{1}(x\otimes y)\rangle=\langle
f,m_{0}(m_{1}(x\otimes y))\rangle$

and

(4.2.28) $\langle\delta_{0}\delta_{1}f,x\otimes
y\rangle=\langle\delta_{1}f,m_{0}(x)\otimes
y\rangle+(-1)^{|x|}\langle\delta_{1}f,x\otimes m_{0}(y)\rangle\\\ =\langle
f,m_{1}(m_{0}(x)\otimes y)\rangle+(-1)^{|x|}\langle f,m_{1}(x\otimes
m_{0}(y))\rangle.$

If we put (4.2.27) and (4.2.28) together, we get that $m_{0}$ is a derivation
with respect to the multiplication defined by $m_{1}$.

######  Remark 4.2.53.

If we consider $Z:=\mathfrak{g}[1]$ and consider the exterior algebra by the
déclage isomorphism
($\textnormal{Sym}^{n}(\mathfrak{g}[1])\xrightarrow{\sim}\bigwedge^{n}\mathfrak{g}[n]$),
we can interpret $m_{1}$ as a bilinear skew-symmetric operator on
$\mathfrak{g}$.

Note that Equation (4.2.26) implies that $m_{1}$ is indeed a Lie bracket for
which the Jacobi identity holds up to terms containing $m_{0}$. Since $m_{0}$
is a differential, this means that $m_{1}$ is a Lie bracket up to _homotopy_.
Hence, considering all equations, we will get a strong homotopy Lie algebra
structure on $\mathfrak{g}$.

######  Remark 4.2.54.

Note that this will lead to a one-to-one correspondence between pointed
$Q$-manifolds and SHLAs. Since SHLAs are equivalent to $L_{\infty}$-algebras,
we get a one-to-one correspondence between pointed $Q$-manifolds and
$L_{\infty}$-algebras.

######  Definition 4.2.55 ($Q$-map).

A _$Q$ -map_ is a (formal) pointed map between two $Q$-manifolds $Z$ and
$\widetilde{Z}$ which commutes with the $Q$-fields, i.e. a map

(4.2.29)
$\displaystyle\begin{split}\varphi\colon\overline{\textnormal{Sym}}(\widetilde{Z}^{*})&\to\overline{\textnormal{Sym}}(Z^{*})\\\
\text{such
that}\qquad\varphi\circ\widetilde{\delta}&=\delta\circ\varphi.\end{split}$

We want to express the condition (4.2.6) more explicitly in this setting by
using $Q$-maps. Consider the vector field $\delta$ and its restriction to
$\widetilde{Z}$. We define the coefficients of the dual map as

$U_{j}:=\left(\varphi_{j}|_{Z^{*}}\right)^{*}\colon\textnormal{Sym}^{j}(Z)\to\widetilde{Z}.$

Then we can express the condition (4.2.29) of a $Q$-map on the dual
coefficients. The first equation gives

$\langle\varphi(\widetilde{\delta}f),x\rangle=\langle\delta\varphi(f),x\rangle\Rightarrow\langle\widetilde{\delta}_{0}f,U_{1}(x)\rangle=\langle\varphi(f),m_{0}(x)\rangle\\\
\Rightarrow\langle f,\widetilde{m}_{0}(U_{1}(x))\rangle=\langle
f,U_{1}(m_{0}(x))\rangle.$

This tells us that the first coefficient $U_{1}$ is a chain map with respect
to the differential defined by the first coefficient of the $Q$-structure:

$H(U_{1})\colon H_{m_{0}}(Z)\to H_{\widetilde{m}_{0}}(\widetilde{Z}).$

######  Remark 4.2.56.

Similarly, we can consider the equation for the next coefficient:

(4.2.30) $\widetilde{m}_{1}(U_{1}(x)\otimes
U_{1}(y))+\widetilde{m}_{1}(U_{2}(x\otimes y))\\\ =U_{2}(m_{0}(x)\otimes
y)+(-1)^{|x|}U_{2}(x\otimes m_{0}(y))+U_{1}(m_{1}(x\otimes y)).$

This shows that $U_{1}$ preserves the Lie structure induced by $m_{1}$ and
$\widetilde{m}_{1}$ up to terms containing $m_{0}$ and $\widetilde{m}_{0}$,
i.e. up to _homotopy_. Recall that this solves the problem that the map
$U^{(0)}_{1}$, defined in (4.2.22), is a chain map but not a DGLA morphism.
Namely, a $Q$-map $U$ (or equivalently an $L_{\infty}$-morphism) induces a map
$U_{1}$ which has the same property as $U^{(0)}_{1}$.

######  Definition 4.2.57 (Koszul sign).

Let $V$ be a vector space endowed with a graded commutative product $\otimes$.
The _Koszul sign_ $\varepsilon(\sigma)$ of a permutation $\sigma$ is the sign
defined by

$x_{1}\otimes\dotsm\otimes
x_{n}=\varepsilon(\sigma)x_{\sigma(1)}\otimes\dotsm\otimes x_{\sigma(n)},\quad
x_{i}\in V.$

######  Definition 4.2.58 (Shuffle permutation).

An _$(\ell,n-\ell)$ -shuffle permutation_ is a permutation $\sigma$ of
$(1,\ldots,n)$ such that $\sigma(1)<\dotsm<\sigma(\ell)$ and
$\sigma(\ell+1)<\dotsm<\sigma(n)$.

######  Remark 4.2.59.

The shuffle permutation associated to a partition $I_{1}\sqcup\dotsm\sqcup
I_{j}=\\{1,\ldots,n\\}$ is the permutation that takes at first all the
elements index by the subset $I_{1}$ in the given order, then those of $I_{2}$
and so on.

The $n$-th coefficient of $U$ satisfies the following condition:

(4.2.31) $\widetilde{m}_{0}(U_{n}(x_{1}\otimes\dotsm\otimes
x_{n}))+\frac{1}{2}\sum_{I\sqcup J=\\{1,\ldots,n\\}\atop
I,J\not=\varnothing}\,\,\varepsilon_{x}(I,J)\widetilde{m}_{1}(U_{|I|}(x_{I})\otimes
U_{|J|}(x_{J}))\\\ =\sum_{1\leq j\leq
n}\varepsilon^{j}_{x}U_{n}(m_{0}(x_{j})\otimes
x_{1}\otimes\dotsm\otimes\widehat{x}_{j}\otimes\dotsm\otimes x_{n})+\\\
+\frac{1}{2}\sum_{j\not=\ell}\varepsilon^{j\ell}_{x}U_{n-1}(m_{1}(x_{j}\otimes
x_{\ell})\otimes
x_{1}\otimes\dotsm\otimes\widehat{x}_{j}\otimes\dotsm\otimes\widehat{x}_{\ell}\otimes\dotsm\otimes
x_{n}),$

where $\varepsilon_{x}(I,J)$ is the _Koszul sign_ associated to the
$(|I|,|J|)$-shuffle permutation for the partition $I\sqcup J=\\{1,\ldots,n\\}$
and $\varepsilon^{j}_{x}$ (resp. $\varepsilon^{j\ell}_{x}$) for the particular
case $I=\\{j\\}$ (resp. $I=\\{j,\ell\\}$). Moreover, we have set
$x_{I}:=\bigotimes_{i\in I}x_{i}$.

######  Remark 4.2.60.

We will specialize (4.2.31) to an $L_{\infty}$-morphism where $Z$ is chosen to
be the DGLA $\mathcal{V}$ of multivector fields and $\widetilde{Z}$ the DGLA
$\mathcal{D}$ of multidifferential operators.

####  4.2.8. Formality of $\mathcal{D}$ and classification of star products on
$\mathbb{R}^{d}$

We want to analyze the relation between the formality of $\mathcal{D}$ and the
classification problem of all star products on $\mathbb{R}^{d}$. Recall that
the _associativity_ of the star product and the _Jacobi identity_ for a
bivector field are equivalent to certain Maurer–Cartan equations.

######  Definition 4.2.61 (Maurer–Cartan equation on (formal)
$L_{\infty}$-algebras).

The Maurer–Cartan equation on a (formal) $L_{\infty}$-algebra
$(\mathfrak{g}[\\![\hbar]\\!],Q)$ is given by

(4.2.32) $Q(\exp(\hbar x))=0,\qquad x\in\mathfrak{g}^{1}[\\![\hbar]\\!],$

where the exponential $\exp$ maps an element of degree $+1$ to a formal power
series in $\hbar\mathfrak{g}[\\![\hbar]\\!]$.

######  Remark 4.2.62.

Note that, from the dual point of view, condition (4.2.32) tells us that $x$
is a fixed point of the cohomological vector field $\delta$. This means that
for each $f\in\textnormal{Sym}(\mathfrak{g}^{*}[\\![\hbar]\\!][1])$ we have

(4.2.33) $\delta f(\hbar x)=0.$

Moreover, using that $(\delta f)_{j}=\delta_{j-1}f$, we can expand Equation
(4.2.33) into a formal Taylor series. In fact, using the pairing

$\langle\delta_{j-1}f,x\otimes\dotsm\otimes x\rangle=\langle
f,m_{j-1}(x\otimes\dotsm\otimes x)\rangle,$

we can write the Maurer–Cartan equation (4.2.32) as

(4.2.34) $\sum_{j\geq 1}\frac{\hbar^{j}}{j!}m_{j-1}(x\otimes\dotsm\otimes
x)=\hbar m_{0}(x)+\frac{\hbar^{2}}{2}m_{1}(x\otimes x)+O(\hbar^{3})=0.$

For two DGLAs $\mathfrak{g}$ and $\mathfrak{h}$, an $L_{\infty}$-morphism
$\varphi\colon\textnormal{Sym}(\mathfrak{h}^{*}[\\![\hbar]\\!][1])\to\textnormal{Sym}(\mathfrak{g}^{*}[\\![\hbar]\\!][1])$
preserves the Maurer–Cartan equation (4.2.34), similarly as a morphism of
DGLAs preserves the Maurer–Cartan equation (4.2.13). In particular, if $x$ is
a solution of (4.2.34) on $\mathfrak{g}[\\![\hbar]\\!]$, we get that

$U(\hbar x)=\sum_{j\geq 1}\frac{\hbar^{j}}{j!}U_{j}(x\otimes\dotsm\otimes x)$

is also a solution of (4.2.34) on $\mathfrak{h}[\\![\hbar]\\!]$.

######  Remark 4.2.63.

The action of the gauge group on solutions to the Maurer-Cartan equation
(4.2.13) can be generalized to the case of $L_{\infty}$-algebras. Namely, if
$x$ and $x^{\prime}$ are equivalent modulo this generalized action, then their
images under $U$ are still equivalent solutions.

######  Remark 4.2.64.

Again, we are especially interested in the case where
$\mathfrak{g}=\mathcal{V}$ and $\mathfrak{h}=\mathcal{D}$. Thus, we get an
$L_{\infty}$-morphism $U$ which gives as a formula of how to construct an
associative star product out of any (formal) Poisson bivector $\pi$, given by

(4.2.35) $U(\pi)=\sum_{j\geq
0}\frac{\hbar^{j}}{j!}U_{j}(\pi\land\dotsm\land\pi),$

where we insert the coefficient of degree zero to be the original non-deformed
product. Note that if $U$ is a quasi-isomorphism, the correspondence between
(formal) Poisson structures on a manifold $M$ and formal deformations of the
pointwise product on $C^{\infty}(M)$ are one-to-one. In particular, as soon as
we have a formality map, we have solved the problem of _existence_ and
_classification_ of star products on $M$.

###  4.3. Kontsevich’s star product

We want to construct an explicit expression of the formality map $U$. The idea
is to introduce graphs and to rewrite things in those terms. An easy example
is given by the Moyal product $f\star g$ for two smooth functions $f$ and $g$
on $(\mathbb{R}^{2n},\omega_{0})$. We can write $f\star g$ as the sum of
graphs in the following way:

$f$$g$$f$$g$$f$$g$$f$$g$+++$+\quad\ldots$ Figure 4.3.1. The Moyal product
$f\star g$ represented in terms of graphs. The gray vertices represent the
Poisson tensor $\pi$ induced by $\omega_{0}$. The term with $n$ Poisson
vertices in the sum represents the $n$-th term in the formal power series of
the Moyal product. For each gray vertex, the two outgoing arrows represent the
derivatives $\partial_{i}$ and $\partial_{j}$ acting on $f$ and $g$.

######  Remark 4.3.1.

This construction can be generalized by allowing more than two outgoing arrows
for gray vertices when considering general multivector fields and allowing
incoming arrows for gray vertices (i.e. derivations of the tensor coefficients
for the assigned multivector fields). Note that there are no incoming arrows
for the Moyal product since the Poisson structure is constant. In Kontsevich’s
star product, we assign to each graph $\Gamma$ a multidifferential operator
$B_{\Gamma}$ and a _weight_ $w_{\Gamma}$ such that the map $U$ that sends an
$n$-tuple of multivector fields to the corresponding weighted sum over all
possible graphs in this set of multidifferential operators is an
$L_{\infty}$-morphism.

####  4.3.1. Data for the construction

##### Admissible graphs

######  Definition 4.3.2 (Admissible graphs).

The set $\mathcal{G}_{n,\bar{n}}$ of _admissible graphs_ consists of all
connected graphs $\Gamma$ which satisfy the following properties:

  * •

The set of vertices $V(\Gamma)$ is decomposed in two ordered subsets
$V_{1}(\Gamma)$ and $V_{2}(\Gamma)$ isomorphic to $\\{1,\ldots,n\\}$
respectively $\\{\bar{1},\ldots,\bar{n}\\}$ whose elements are called vertices
of _first type_ respectively _second type_ ;

  * •

The following inequalities involving the number of vertices of the two types
are satisfied: $n\geq 0,\bar{n}\geq 0$ and $2n+\bar{n}-2\geq 0$;

  * •

The set of edges $E(\Gamma)$ is finite and does not contain _short loops_ ,
i.e. edges starting and ending at the same vertex;

  * •

All edges $E(\Gamma)$ are oriented and start from a vertex of the first type;

  * •

The set of edges starting at a given vertex $v\in V_{1}(\Gamma)$, which will
be denoted by $\mathrm{Star}(v)$, is ordered.

See Figure 4.3.2 and 4.3.3 for examples of admissible and non-admissible
graphs respectively.

$\bar{1}$$\bar{2}$$\bar{1}$$\bar{2}$$\bar{3}$$1$$2$$3$$1$$2$ Figure 4.3.2.
Examples of admissible graphs.
$\bar{1}$$\bar{2}$$\bar{1}$$\bar{2}$$1$$2$$1$$2$ Figure 4.3.3. Examples of
non-admissible graphs.

##### The multidifferential operators $B_{\Gamma}$

Let us consider pairs $(\Gamma,\xi_{1}\otimes\dotsm\otimes\xi_{n})$ consisting
of a graph $\Gamma\in\mathcal{G}_{n,\bar{n}}$ with $2n+\bar{n}-2$ edges and of
a tensor product of $n$ multivector fields on $\mathbb{R}^{d}$. We want to
understand how we can associate to such a pair a multidifferential operator
$B_{\Gamma}\in\mathcal{D}^{\bar{n}-1}$. This is done in the following way:

  * •

Associate to each vertex $v$ of first type with $k$ outgoing arrows the skew-
symmetric tensor $\xi^{j_{1}\dotsm j_{k}}_{i}$ corresponding to a given
$\xi_{i}$ via the natural identification.

  * •

Place a function at each vertex of second type.

  * •

Associate to the $\ell$-th arrow in $\mathrm{Star}(v)$ a partial derivative
with respect to the coordinate labeled by the $\ell$-th index of $\xi_{i}$
acting on the function or the tensor appearing at its endpoint.

  * •

Multiply such elements in the order prescribed by the labeling of the graph.

######  Example 4.3.3.

Denote by $\Gamma_{1}$ the first graph in Figure 4.3.2. Then, to a triple of
bivector fields $(\xi_{1},\xi_{2},\xi_{3})$ with
$\xi_{1}=\sum_{i_{1}<i_{2}}\xi_{1}^{i_{1}i_{2}}\partial_{i_{1}}\land\partial_{i_{2}}$,
$\xi_{2}=\sum_{j_{1}<j_{2}}\xi_{2}^{j_{1}j_{2}}\partial_{j_{1}}\land\partial_{j_{2}}$
and
$\xi_{3}=\sum_{\ell_{1}<\ell_{2}}\xi_{3}^{\ell_{1}\ell_{2}}\partial_{\ell_{1}}\land\partial_{\ell_{2}}$,
we associate the bidifferential operator

$U_{\Gamma_{1}}(\xi_{1}\land\xi_{2}\land\xi_{3})(f\otimes
g):=\xi_{2}^{j_{1}j_{2}}\partial_{j_{2}}\xi_{1}^{i_{1}i_{2}}\partial_{j_{2}}\xi_{3}^{\ell_{1}\ell_{2}}\partial_{i_{1}}\partial_{\ell_{1}}f\partial_{i_{2}}\partial_{\ell_{2}}g.$

######  Example 4.3.4.

Denote by $\Gamma_{2}$ the second graph in Figure 4.3.2. Then, to a tuple of
multivector fields $(\chi_{1},\chi_{2})$ with
$\chi_{1}=\sum_{i_{1}<i_{2}}\chi_{1}^{i_{1}i_{2}}\partial_{i_{1}}\land\partial_{i_{2}}$
and
$\chi_{2}=\sum_{j_{1}<j_{2}<j_{3}}\chi_{2}^{j_{1}j_{2}j_{3}}\partial_{j_{1}}\land\partial_{j_{2}}\land\partial_{j_{3}}$,
we associate the tridifferential operator

$U_{\Gamma_{2}}(\chi_{1}\land\chi_{2})(f\otimes g\otimes
h):=\chi_{1}^{i_{1}i_{2}}\partial_{i_{1}}\chi_{2}^{j_{1}j_{2}j_{3}}\partial_{j_{1}}f\partial_{j_{2}}g\partial_{j_{3}}\partial_{i_{2}}h.$

######  Remark 4.3.5.

In particular, for each admissible graph $\Gamma\in\mathcal{G}_{n,\bar{n}}$,
we get a linear map $U_{\Gamma}\colon\bigwedge^{n}\mathcal{V}\to\mathcal{D}$
which is equivariant with respect to the action of the symmetric group.
Kontsevich’s main construction was to choose _weights_
$w_{\Gamma}\in\mathbb{R}$ such that the linear combination

$U:=\sum_{n,\bar{n}\geq
0}\,\,\sum_{\Gamma\in\mathcal{G}_{n,\bar{n}}}w_{\Gamma}B_{\Gamma}$

becomes an $L_{\infty}$-morphism.

##### Weights of graphs

The weight $w_{\Gamma}$ associated to an admissible graph
$\Gamma\in\mathcal{G}_{n,\bar{n}}$ is defined by an integral of a differential
form $\omega_{\Gamma}$ over (a suitable compactification of) some
_configuration space_ $C^{+}_{n,\bar{n}}$. Namely, it is given by

(4.3.1) $w_{\Gamma}:=\prod_{1\leq k\leq
n}\frac{1}{|\mathrm{Star}(k)|!}\frac{1}{(2\pi)^{2n+\bar{n}-2}}\int_{\bar{C}^{+}_{n,\bar{n}}}\omega_{\Gamma}.$

This holds for the case when $\Gamma$ has exactly $2n+\bar{n}-2$ edges
otherwise the weight is just zero.

Let us look at the integral (4.3.1) more carefully.

##### Configuration spaces

Consider an embedding of the graph $\Gamma$ into the $2$-dimensional upper
half-space $\mathbb{H}^{2}$ (also called _upper half-plane_) as defined in
(1.4.7) where we want to identify $\mathbb{R}^{2}\cong\mathbb{C}$. Moreover,
denote by $\mathbb{H}^{2}_{+}:=\\{z\in\mathbb{H}\mid\Im(z)>0\\}$ where
$\Im(z)$ denotes the _imaginary part_ of $z$.

######  Definition 4.3.6 (Open configuration space).

Define the _open configuration space_ of the distinct $n+\bar{n}$ vertices of
an admissible graph $\Gamma\in\mathcal{G}_{n,\bar{n}}$ as the smooth manifold

(4.3.2)
$\mathrm{Conf}_{n,\bar{n}}:=\big{\\{}(z_{1},\ldots,z_{n},z_{\bar{1}},\ldots,z_{\bar{n}})\in\mathbb{C}^{n+\bar{n}}\mid
z_{i}\in\mathbb{H}^{2}_{+},\,z_{\bar{i}}\in\mathbb{R},\,z_{i}\not=z_{j}\,\,\text{for
$i\not=j$},\,z_{\bar{i}}\not=z_{\bar{j}}\,\,\text{for
$\bar{i}\not=\bar{j}$}\big{\\}}$

######  Remark 4.3.7.

More precisely, we need to consider everything up to _scaling_ and
_translation_. Hence, we consider the Lie group $G$ consisting of translations
in the horizontal direction and rescaling such that the action on
$z\in\mathbb{H}^{2}$ is given by

$z\mapsto az+b,\qquad a\in\mathbb{R}_{+},b\in\mathbb{R}.$

The action of $G$ is free whenever the number of vertices is $2n+\bar{n}-2\geq
0$. Thus the quotient space of $\mathrm{Conf}_{n,\bar{n}}$ with respect to the
$G$-action, which we will denote by $C_{n,\bar{n}}$, is again a smooth
manifold of (real) dimension $2n+\bar{n}-2$. If there are no vertices of
second type (i.e. $\bar{n}=0$), one can define the open configuration space as
a subset of $\mathbb{C}^{n}$ instead of $\mathbb{H}^{n}$ and one can consider
the Lie group which consists of rescaling and translation in any direction.
The corresponding quotient space $C_{n}$ for $n\geq 2$ still a smooth manifold
but now of dimension $2n-3$.

######  Definition 4.3.8 (Connected configuration space).

Define the _connected configuration space_ to be the subset of $C_{n,\bar{n}}$
given by

(4.3.3)
$C^{+}_{n,\bar{n}}:=\big{\\{}(z_{1},\ldots,z_{n},z_{\bar{1}},\ldots,z_{\bar{n}})\in
C_{n,\bar{n}}\mid z_{\bar{i}}<z_{\bar{j}}\,\,\text{for
$\bar{i}<\bar{j}$}\big{\\}}$

######  Remark 4.3.9.

One can show that $C^{+}_{n,\bar{n}}$ is again a smooth manifold which is now
connected.

Let us define an _angle map_

$\phi\colon C_{2,0}\to S^{1},$

which associates to each pair of distinct points
$z_{1},z_{2}\in\mathbb{H}^{2}$ the angle between the geodesics with respect to
the _Poincaré metric_ connecting $z_{1}$ to $\mathrm{i}\infty$ and to $z_{2}$,
in the counterclockwise direction (see Figure 4.3.4). Formally, we have

(4.3.4)
$\phi(z_{1},z_{2}):=\mathrm{arg}\left(\frac{z_{2}-z_{1}}{z_{2}-\bar{z}_{1}}\right)=\frac{1}{2\mathrm{i}}\log\left(\frac{(z_{2}-z_{1})(\bar{z}_{2}-z_{1})}{(z_{2}-\bar{z}_{1})(\bar{z}_{2}-\bar{z}_{1})}\right).$

$z_{1}$$z_{2}$$\mathbb{H}^{2}$$\mathrm{i}\infty$$\phi$ Figure 4.3.4.
Illustration of the angle map $\phi$.

The differential of $\phi$ is then a 1-form on $C_{2,0}$ which we can pull-
back to the configuration space including points in $\mathbb{R}$ by the
projection $p_{e}$ associated to each edge $e=(z_{i},z_{j})$ of $\Gamma$
defined by

$\displaystyle p_{e}\colon C_{n,\bar{n}}$ $\displaystyle\to C_{2,0},$
$\displaystyle(z_{1},\ldots,z_{n},z_{\bar{1}},\ldots,z_{\bar{n}})$
$\displaystyle\mapsto(z_{i},z_{j}).$

Hence we get a 1-form
${\mathrm{d}}\phi_{e}:=p_{e}^{*}{\mathrm{d}}\phi\in\Omega^{1}(C_{n,\bar{n}})$.
The differential form $\omega_{\Gamma}$ in (4.3.1) is then defined as

$\omega_{\Gamma}:=\bigwedge_{e\in E(\Gamma)}{\mathrm{d}}\phi_{e}.$

######  Remark 4.3.10.

Note that the ordering one these 1-forms is induced by the ordering on the set
of edges by the first-type vertices and the ordering on $\mathrm{Star}(v)$.
Moreover, note that we indeed obtain a top-form on $C_{n,\bar{n}}$ (resp.
$C_{n,\bar{n}}^{+}$) as long as we consider graphs with exactly $2n+\bar{n}-2$
edges since $\dim C_{n,\bar{n}}=\dim C^{+}_{n,\bar{n}}=2n+\bar{n}-2$.

######  Remark 4.3.11.

In order to make sense of the integral (4.3.1), we need to make sure that it
converges. Note that, by construction of $\phi$, we get that
${\mathrm{d}}\phi$ is not defined as soon two points approach each other. This
is the reason why we in fact need a suitable _compactification_
$\bar{C}^{+}_{n,\bar{n}}$ of the connected configuration space
$C^{+}_{n,\bar{n}}$.

######  Theorem 4.3.12 ([FM94, AS94]).

For any configuration space $C_{n,\bar{n}}$ (resp. $C_{n}$) there exists a
compactification $\bar{C}_{n,\bar{n}}$ (resp. $\bar{C}_{n}$) whose interior is
the open configuration space and such that the projections $p_{e}$, the angle
map $\phi$ (and thus the differential form $\omega_{\Gamma}$) extend smoothly
to the corresponding compactifications.

######  Remark 4.3.13.

This compactification is usually called _FMAS compactification_ for
_Fulton–MacPherson_ who proved a first result of this in the algebro-
geometrical setting for configuration spaces of points in non-singular
algebraic varieties [FM94] and _Axelrod–Singer_ who proved this in the smooth-
geometrical setting for configuration spaces of points in smooth manifolds
[AS94].

######  Remark 4.3.14.

The compactified configuration space is a compact smooth manifold with
_corners_.

######  Definition 4.3.15 (Manifold with corners).

A manifold with corners of dimension $m$ is a topological Hausdorff space $M$
which is locally homeomorphic to $\mathbb{R}^{m-n}\times\mathbb{R}^{n}_{+}$
with $n=0,\ldots,m$.

######  Remark 4.3.16.

The points $x\in M$ of a manifold with corners of dimension $m$ whose local
expression in some (and hence in any) chart has the form

$(x_{1},\ldots,x_{m-n},0,\ldots,0)$

are said to be of _type $n$_ and form submanifolds of $M$ which are called
_strata of codimension $n$_.

####  4.3.2. Proof of Kontsevich’s formula

To show that $U$ is indeed an $L_{\infty}$-morphism, we need to show the
following points:

  1. (1)

The first component of the restriction of $U$ to $\mathcal{V}$ is up to a
shift in degrees given by the natural map $U_{1}^{(0)}$ as defined in
(4.2.22).

  2. (2)

$U$ is a graded linear map of degree zero.

  3. (3)

$U$ satisfies the equations (4.2.6) for an $L_{\infty}$-morphism.

#####  $U_{1}$ coincides with $U_{1}^{(0)}$

######  Lemma 4.3.17.

The map

$U_{1}\colon\mathcal{V}\to\mathcal{D}$

is the natural map $U^{(0)}_{1}$ that identifies each multivector field with
the corresponding multidifferential operator.

###### Proof.

Consider the set $\mathcal{G}_{1,\bar{n}}$ of admissible graphs with one
vertex of first type and $\bar{n}$ vertices of second type. It is easy to see
that this set only contains one element $\Gamma_{\bar{n}}$ which has $2\cdot
1+\bar{n}-2=\bar{n}$ arrows outgoing from the single vertex of first type and
each arrow is incoming to a vertex of second type (see Figure 4.3.5).

$\bar{1}$$\bar{2}$$\bar{3}$$\bar{n}$$\dotsm$$1$ Figure 4.3.5. The single
element $\Gamma_{\bar{n}}\in\mathcal{G}_{1,\bar{n}}$.

Hence, for a multivector field $\xi$ of degree $k$ we associate the
multidifferential operator

$U_{\Gamma_{\bar{n}}}(\xi)\colon f_{\bar{1}}\otimes\dotsm\otimes
f_{\bar{n}}\mapsto w_{\Gamma_{\bar{n}}}\xi^{i_{\bar{1}}\dotsm
i_{\bar{n}}}\partial_{i_{\bar{1}}}f_{\bar{1}}\dotsm\partial_{i_{\bar{n}}}f_{\bar{n}}.$

One can easily check that

$\int_{\bar{C}_{1,\bar{n}}}\omega_{\Gamma_{\bar{n}}}=\int_{\bar{C}_{1,\bar{n}}}\,\,\bigwedge_{e\in
E(\Gamma_{\bar{n}})}{\mathrm{d}}\phi_{e}=(2\pi)^{\bar{n}}.$

Using (4.3.1), this will give us

$w_{\Gamma_{\bar{n}}}=\frac{1}{\bar{n}!}.$

This shows that $U_{1}$ is indeed the analogue for $U^{(0)}_{1}$ in (4.2.22)
which induces the HKR isomorphism (4.2.21). ∎

##### Checking the degrees

######  Lemma 4.3.18.

The $n$-th component

$U_{n}:=\sum_{\bar{n}\geq
1}\,\sum_{\Gamma\in\mathcal{G}_{n,\bar{n}}}w_{\Gamma}B_{\Gamma}$

has the correct degree for $U$ to be an $L_{\infty}$-morphism.

###### Proof.

Note that to a vertex $v_{i}$ with $|\mathrm{Star}(v_{i})|$ outgoing arrows,
we associate an element in
$\mathcal{V}^{|\mathrm{Star}(v_{i})|}=\widetilde{\mathcal{V}}^{|\mathrm{Star}(v_{i})|+1}$.
On the other hand, each graph $\Gamma$ with $\bar{n}$ vertices of second type
and $n$ multivector fields $\xi_{1},\ldots,\xi_{n}$ induce a multidifferential
operator $U_{n}(\xi_{1}\land\dotsm\land\xi_{n})$ of degree $s=\bar{n}-1$.
Since we are only considering graphs with exactly $2n+\bar{n}-2$ edges and
since

$|E(\Gamma)|=\sum_{1\leq i\leq n}|\mathrm{Star}(v_{i})|,$

we can write the degree of $U_{n}(\xi_{1}\land\dotsm\land\xi_{n})$ as

$s=(2n+\bar{n}-2)+1-n=|E(\Gamma)|+1-n=\sum_{1\leq i\leq
n}|\mathrm{Star}(v_{i})|+1-n.$

This is exactly the degree for the $n$-th component of an
$L_{\infty}$-morphism. ∎

##### Reformulation of the $L_{\infty}$-condition in terms of graphs

Next we discuss the geometric proof of the formality statement. We have to
extend the morphism $U$ with a degree zero component which represents the
usual multiplication between smooth maps. Therefore, we can consider the
special case of the $L_{\infty}$-condition (4.2.31) where
$m_{0},\widetilde{m}_{0}$ are given by the Taylor coefficients $U_{n}$:

(4.3.5) $\sum_{0\leq\ell\leq n}\,\,\sum_{-1\leq k\leq m}\,\,\sum_{0\leq i\leq
m-k}\varepsilon_{kim}\sum_{\sigma\in
S_{\ell,n-\ell}}\varepsilon_{\xi}(\sigma)U_{\ell}\bigg{(}\xi_{\sigma(1)}\land\dotsm\land\xi_{\sigma(\ell)}\bigg{)}\\\
\bigg{(}f_{0}\otimes\dotsm\otimes f_{i-1}\otimes
U_{n-\ell}\bigg{(}\xi_{\sigma(\ell+1)}\land\dotsm\land\xi_{\sigma(n)}\bigg{)}\big{(}f_{i}\otimes\dotsm\otimes
f_{i+k}\big{)}\otimes f_{i+k+1}\otimes\dotsm\otimes f_{m}\bigg{)}\\\
=\sum_{i\not=j=1}^{n}\varepsilon^{ij}_{\xi}U_{n-1}\bigg{(}\xi_{i}\circ\xi_{j}\land\xi_{1}\land\dotsm\land\widehat{\xi}_{i}\land\dotsm\land\widehat{\xi}_{j}\land\dotsm\land\xi_{n}\bigg{)}\big{(}f_{0}\otimes\dotsm\otimes
f_{n}\big{)},$

where $(\xi_{j})_{j=1,\ldots,n}$ are multivector fields, $f_{0},\ldots,f_{m}$
are smooth maps on which the multidifferential operator is acting,
$S_{\ell,n-\ell}$ is the subset of $S_{n}$ consisting of
$(\ell,n-\ell)$-shuffle permutations, the product $\xi_{i}\circ\xi_{j}$ is
defined in a way such that the Schouten–Nijenhuis bracket can be expressed in
terms of this composition by a formula similar to the one relating the
Gerstenhaber bracket to the analogous composition $\circ$ on $\mathcal{D}$ in
(4.2.14) and the signs involved are defined as follows:
$\varepsilon_{kim}:=(-1)^{k(m+i)}$, $\varepsilon_{\xi}(\sigma)$ is the Koszul
sign associated to the permutation $\sigma$ and $\varepsilon^{ij}_{\xi}$ is
defined as in (4.2.31).

######  Remark 4.3.19.

Note that (4.3.5) carries the formality condition since the left hand side
corresponds to the Gerstenhaber bracket of multidifferential operators and the
right hand side to (a part of) the Schouten–Nijenhuis bracket (since the
differential on $\mathcal{V}$ is identical to zero). Recall that the
differential on $\mathcal{D}$ is is given by the Hochschild differential
$[\mathsf{m},\enspace]_{\mathrm{G}}$ which in (4.3.5) is replaced by $U_{0}$.

We can now rewrite (4.3.5) in terms of admissible graphs and weights to prove
that it holds. Note that the difference between the left hand side and right
hand side of (4.3.5) can be formulated as a linear combination

(4.3.6)
$\sum_{\Gamma\in\mathcal{G}_{n,\bar{n}}}c_{\Gamma}U_{\Gamma}(\xi_{1}\land\dotsm\land\xi_{n})(f_{0}\otimes\dotsm\otimes
f_{n}).$

Recall that we want to consider admissible graphs $\Gamma$ with exactly
$2n+\bar{n}-2$ edges. In fact, (4.3.5) is satisfied if $c_{\Gamma}=0$ for all
such $\Gamma$. We will show that this is indeed the case.

##### The key is Stokes’ theorem

In order to prove that all these coefficients $c_{\Gamma}$ vanish, we will use
Stokes’ theorem for manifolds with corners. Similarly as for the usual Stokes’
theorem, we can replace an integral of an exact form over some manifold $M$ as
the integral of its primitive form over the boundary $\partial M$. In
particular, we have

(4.3.7)
$\int_{\partial\bar{C}^{+}_{n,\bar{n}}}\omega_{\Gamma}=\int_{\bar{C}^{+}_{n,\bar{n}}}{\mathrm{d}}\omega_{\Gamma}=0,\quad\forall\Gamma\in\mathcal{G}_{n,\bar{n}}$

since ${\mathrm{d}}\phi_{e}$ is closed and $\bar{C}^{+}_{n,\bar{n}}$ is
compact. Let us expand the left hand side of (4.3.7) in order to show that it
is exactly given by the coefficients $c_{\Gamma}$. Therefore, we want to take
a closer look on the manifold $\partial\bar{C}^{+}_{n,\bar{n}}$. Recall that
we have restricted the weights in (4.3.5) to be equal to zero if the form
degree does not match the dimension of the underlying manifold over which we
integrate. Hence, we only have to consider the codimension $1$ strata of
$\partial\bar{C}^{+}_{n,\bar{n}}$ which have dimension $2n+\bar{n}-3$. Note
that the dimension of $\partial\bar{C}^{+}_{n,\bar{n}}$ is equal to the number
of edges and thus of the 1-forms ${\mathrm{d}}\phi_{e}$.

######  Remark 4.3.20.

Intuitively, one can think of the boundary of $\bar{C}^{+}_{n,\bar{n}}$ to be
represented by the degenerate configuration in which some of the $n+\bar{n}$
points _collide with each other_.

##### Classification of boundary strata

Using Remark 4.3.20, we can classify the codimension $1$ strata of
$\partial\bar{C}^{+}_{n,\bar{n}}$ as follows:

  * •

(Strata of type S1) These are strata in which $i\geq 2$ points in
$\mathbb{H}_{+}^{2}$ collide together to a point which still lies in
$\mathbb{H}^{2}_{+}$. Points in such a stratum can be locally described by

(4.3.8) $C_{i}\times C_{n-i+1,\bar{n}}.$

The first term represents the relative position of the colliding points when
we look at them under a _magnifying glass_. The second term is the space of
all remaining points plus the point which occurs as the point on which the
first $i$ points have collapsed to.

  * •

(Strata of type S2) These are strata in which $i>0$ points in
$\mathbb{H}^{2}_{+}$ and $j>0$ points in $\mathbb{R}$ with $2i+j-2\geq 0$
collide to a single point on $\mathbb{R}$. Points in such a stratum can be
locally represented by

(4.3.9) $C_{i,j}\times C_{n-i,\bar{n}-j+1}.$

The two terms are similarly given as in (4.3.8).

See Figure 4.3.6 and 4.3.7 for an illustration of a stratum of type S1 and
type S2 respectively.

$\bar{1}$$\bar{2}$$\bar{3}$$\bar{n}$$\ell$$1$$n$$2$$\mathbb{H}^{2}$ Figure
4.3.6. Example of a stratum of type S1. Note that here $i=3$ points have
collapsed together to a point $\ell\in\mathbb{H}^{2}_{+}$.
$\bar{1}$$\bar{2}$$\bar{3}$$\bar{n}$$\bar{\ell}$$1$$n$$2$$\mathbb{H}^{2}$
Figure 4.3.7. Example of a stratum of type S2. Note that here $i=3$ and $j=2$
points have collapsed together to a point $\bar{\ell}\in\mathbb{R}$.

Now we can split the integral on the left hand side of (4.3.7) into the sum of
strata of type S1 and type S2. For the strata of type S1, we will distinguish
two subcases for the $i$ collapsing vertices. Since the integral vanishes if
the form degree is not equal to the dimension of the underlying manifold, one
can show that the only contributions come from graphs $\Gamma$ whose subgraphs
$\Gamma_{1}$, spanned by the collapsing vertices, contain exactly $2i-3$
edges.

When $i=2$, there is only one edge $e$, an hence in the first integral of the
decomposition of (4.3.8) the differential ${\mathrm{d}}\phi_{e}$ is integrated
over $C_{2}\cong S^{1}$ and thus we get a factor of $2\pi$ which cancels the
coefficient in (4.3.1). The remaining integral represents the weight of the
corresponding quotient graph $\Gamma_{2}$ which is obtained from $\Gamma$
after the contraction of the edge $e$. In particular, to the vertex $j$ of
first type, which results from this contraction, we associate the
$j$-composition of the two multivector fields that were associated to the
endpoints of $e$. Hence, summing over all graphs and all strata of this
subtype, we get the right hand side of (4.3.5).

##### A trick using logarithms

When $i\geq 3$, the integral corresponding to this stratum involves the
product of $2i-3$ angle forms over $C_{i}$. According to a Lemma of
Kontsevich, this integral vanishes.

######  Lemma 4.3.21 (Kontsevich[Kon03]).

The integral over the configuration space $C_{n}$ of $n\geq 3$ points in the
upper half-plane of any $\dim C_{n}:=2n-3$ angle forms
${\mathrm{d}}\phi_{e_{i}}$ with $i=1,\ldots,n$ vanishes for $n\geq 3$.

$\bar{1}$$\bar{n}$$\mathbb{H}^{2}$ Figure 4.3.8. Example of a non-vanishing
term. $\bar{1}$$\bar{n}$$\mathbb{H}^{2}$ Figure 4.3.9. Example of a vanishing
term.

###### Proof of Lemma 4.3.21.

First we need to restrict the integral to an even number of angle forms. This
can be done by identifying $C_{n}$ with the subset of $\mathbb{H}^{n}$, where
one of the endpoints of $e_{1}$ is set to be the origin and the second is
forced to lie on the unit circle (note that this can always be done by
considering the action of the Lie group in the definition of $C_{n}$). Then
the integral decomposes into a product of integrals of ${\mathrm{d}}\phi_{e}$
over $S^{1}$ and the remaining $2n-4=:2N$ forms integrated over the resulting
complex manifold $U$ given by the isomorphism

$C_{n}\cong S^{1}\times U.$

Then the claim follows by the following computation:

(4.3.10)
$\displaystyle\begin{split}\int_{U}\bigwedge^{2N}_{j=1}{\mathrm{d}}\,\mathrm{arg}(f_{j})&=\int_{U}\bigwedge^{2N}_{j=1}{\mathrm{d}}\log(|f_{j}|)=\int_{\bar{U}}\mathcal{I}\bigg{(}{\mathrm{d}}\log(|f_{1}|)\bigwedge^{2N}_{j=2}{\mathrm{d}}\log(|f_{j}|)\bigg{)}\\\
&=\int_{\bar{U}}{\mathrm{d}}\bigg{(}\mathcal{I}\bigg{(}\log(|f_{1}|)\bigwedge^{2N}_{j=2}{\mathrm{d}}\log(|f_{j}|)\bigg{)}\bigg{)}=0.\end{split}$

Let us see what is happening here. First, note that the angle function
$\phi_{e_{j}}$ was expressed in terms of the argument of the (holomorphic)
function $f_{j}$. In particular, $f_{j}$ is just the difference of the
coordinates of the endpoints of $e_{j}$. The first equality follows by the
decompositions

$\displaystyle{\mathrm{d}}\,\mathrm{arg}(f_{j})$
$\displaystyle=\frac{1}{2\mathrm{i}}\big{(}{\mathrm{d}}\log(f_{j})-{\mathrm{d}}\log(\bar{f}_{j})\big{)},$
$\displaystyle{\mathrm{d}}\log(|f_{j}|)$
$\displaystyle=\frac{1}{2}\big{(}{\mathrm{d}}\log(f_{j})+{\mathrm{d}}\log(\bar{f}_{j})\big{)}.$

Thus, the product of $2N$ of these expressions is a linear combination of
products of $k$ holomorphic and $2N-k$ anti-holomorphic forms. By a result of
complex analysis, one can show that in the integration over the complex
manifold $U$, the only terms which do not vanish are the ones where $k=N$.
Then one can observe that the terms coming from the first decomposition are
equal to those coming from the second decomposition.

In the second equality the integral of the differential form is replaced by an
integral over a suitable 1-form with values in the space of _distributions_
over the compactification $\bar{U}$ of $U$. One can show (this is another
Lemma) that such a map $\mathcal{I}$, which sends usual 1-forms to
distributional ones, commutes with the differential and thus by Stokes’
theorem we get the claim. ∎

##### Last step: vanishing terms for type S2 strata

Finally, let us consider the strata of type S2. There we will have a similar
dimensional argument, i.e. it analogously restricts the possible non-vanishing
terms to the condition that the subgraph $\Gamma_{1}$, spanned by $i+j$
collapsing vertices of first and second type respectively, contains exactly
$2i+j-2$ edges. Similarly as before, for the quotient graph $\Gamma_{2}$
obtained by contracting $\Gamma_{1}$, the claim is that the only non-vanishing
contributions come from the graphs for which both graphs obtained from a given
$\Gamma$ are admissible. In this case we get a decomposition of the weight
$w_{\Gamma}$ into the product $w_{\Gamma_{1}}\cdot w_{\Gamma_{2}}$ which in
general, by the conditions on the number of edges of $\Gamma$ and
$\Gamma_{1}$, does not vanish.

The only remaining thing to check is that we do not have _bad edges_ by
contraction (see Figure 4.3.3). The only such possibility appears when
$\Gamma_{2}$ contains an edge which starts from a vertex of second type. In
this case the corresponding integral vanishes because it contains the
differential of an angle map evaluated on the pair $(z_{1},z_{2})$ where
$z_{1}$ is constrained to lie in $\mathbb{R}$ and such maps vanish for every
$z_{2}$ because the angle is measured with respect to the Poincaré metric
(recall Figure 4.3.4 for an intuitive picture).

$\bar{1}$$\bar{\ell}$$\overline{\ell+1}$$\bar{n}$$\mathbb{H}^{2}$ Figure
4.3.10. Example of an admissible quotient.
$\bar{1}$$\bar{\ell}$$\overline{\ell+1}$$\bar{n}$bad edge$\mathbb{H}^{2}$
Figure 4.3.11. Example of a non-admissible quotient. Such a term vanishes.

This means that the only non-vanishing terms correspond to the case when we
plug the differential operator associated to $\Gamma_{1}$ as the $k$-th
argument of the one associated to $\Gamma_{2}$, where $k$ is the vertex of the
second type emerging from the collapse. Summing over all these possibilities,
the strata of type S2 exactly corresponds to the left hand side of (4.3.5).

We have thus proven that $U$ is indeed an $L_{\infty}$-morphism and since the
first coefficients $U_{1}$ are given by $U^{(0)}_{1}$ it is also a quasi-
isomorphism and hence determines uniquely a star product for any bivector
field $\pi$ on $\mathbb{R}^{d}$ by the formula (4.2.35).

###  4.4. Globalization of Kontsevich’s star product

Kontsevich’s formula for the star product only gives a quantization for the
case when $M=\mathbb{R}^{d}$ for a general Poisson structure $\pi$, and thus
only describes a local description for the general case. Already in [Kon03],
Kontsevich gave a globalization method which was very briefly mentioned, but
it was explicitly constructed by Cattaneo, Felder and Tomassini in [CFT02a,
CFT02] in a similar way as Fedosov did for the symplectic case of the Moyal
product [Fed94] (see also Section 4.1.3). Their approach uses a flat
connection $\bar{D}$ on a vector bundle over $M$ such that the algebra of the
horizontal sections with respect to $\bar{D}$ is a quantization of
$C^{\infty}(M)$. Consider the vector bundle $E_{0}\to M$ of infinite jets of
functions endowed with a flat connection $D_{0}$. The fiber $(E_{0})_{x}$ over
$x\in M$ is naturally a commutative algebra and carries the Poisson structure
induced fiberwise by the Poisson structure on $C^{\infty}(M)$. The map which
associates to any globally defined function its infinite jet at each point
$x\in M$ is a Poisson isomorphism onto the Poisson algebra of horizontal
sections of $E_{0}$ with respect to $D_{0}$. Since the star product gives a
deformation of the pointwise product on $C^{\infty}(M)$, we want to have a
_quantum version_ of the vector bundle and the flat connection in order to get
an similar isomorphism. The vector bundle $E\to M$ is defined in terms of a
section $\phi^{\mathrm{aff}}$ of the fiber bundle $M^{\mathrm{aff}}\to M$,
where $M^{\mathrm{aff}}$ denotes the quotient of the manifold
$M^{\mathrm{coor}}$ of jets of coordinate systems on $M$ by the action of the
group $GL(d,\mathbb{R})$ of linear diffeomorphims given by
$E:=(\phi^{\mathrm{aff}})^{*}\widetilde{E}$, where $\widetilde{E}$ is the
bundle of $\mathbb{R}[\\![\hbar]\\!]$-modules
$M^{\mathrm{coor}}\times_{GL(d,\mathbb{R})}\mathbb{R}[\\![y^{1},\ldots,y^{d}]\\!][\\![\hbar]\\!]\to
M^{\mathrm{aff}}$. Note that the section $\phi^{\mathrm{aff}}$ can be realized
explicitly by a collection of jets at $0$ of maps
$\phi_{x}\colon\mathbb{R}^{d}\to M$ such that $\phi_{x}(0)=x$ for all $x\in M$
(modulo the action of $GL(d,\mathbb{R})$). Thus, we can assume for simplicity
that we have fixed a representative $\phi_{x}$ of the equivalence class for
each open set of a given covering, hence realizing a trivialization of the
bundle $E$. So from now on we will identify $E$ with the trivial bundle with
fiber given by $\mathbb{R}[\\![y^{1},\ldots,y^{d}]\\!][\\![\hbar]\\!]$.
Therefore, $E$ realizes the desired quantization, since it is isomorphic (as a
bundle of $\mathbb{R}[\\![\hbar]\\!]$-modules) to the bundle
$E_{0}[\\![\hbar]\\!]$ whose elements are formal power series with infinite
jets of functions as coefficients.

####  4.4.1. The multiplication, connection and curvature maps

To define the star product and the connection on $E$, one has to introduce new
objects whose existence and properties will be byproducts of the formality
theorem. For a Poisson structure $\pi$ and two vector fields $\xi$ and $\eta$
on $\mathbb{R}^{d}$, we define

(4.4.1) $\displaystyle P(\pi)$ $\displaystyle:=\sum_{j\geq
0}\frac{\hbar^{j}}{j!}U_{j}(\pi\land\dotsm\land\pi),$ (4.4.2) $\displaystyle
A(\xi,\pi)$ $\displaystyle:=\sum_{j\geq
0}\frac{\hbar^{j}}{j!}U_{j+1}(\xi\land\pi\land\dotsm\land\pi),$ (4.4.3)
$\displaystyle F(\xi,\eta,\pi)$ $\displaystyle:=\sum_{j\geq
0}\frac{\hbar^{j}}{j!}U_{j+2}(\xi\land\eta\land\pi\land\dotsm\land\pi).$

It is easy to show that the degree of the multidifferential operators on the
right hand sides of (4.4.1), (4.4.2) and (4.4.3) induce that $P(\pi)$ is a
(formal) bidifferential operator, $A(\xi,\pi)$ is a differential operator and
$F(\xi,\eta,\pi)$ is a function. Indeed, $P(\pi)$ is actually just the star
product associated to $\pi$. In particular, the maps $P,A$ and $F$ are
elements of degree $0,1$ and $2$ respectively of the Lie algebra cohomology
complex of (formal) vector fields with values in the space of local polynomial
maps, i.e. multidifferential operators which depend polynomially on $\pi$. An
element of degree $j$ of this complex is a map that sends
$\xi_{1}\land\dotsm\land\xi_{j}$ to a multidifferential operator
$S(\xi_{1}\land\dotsm\land\xi_{j}\land\pi)$. The differential $\delta$ of this
complex can then be defined as

$\delta S(\xi_{1}\land\dotsm\land\xi_{j+1}\land\pi):=\sum_{1\leq i\leq
j+1}(-1)^{i}\frac{\partial}{\partial
t}\Big{|}_{t=0}S\Big{(}\xi_{1}\land\dotsm\land\widehat{\xi}_{i}\land\dotsm\land\xi_{j+1}\land(\Phi^{\xi_{i}}_{t})_{*}\pi\Big{)}\\\
+\sum_{i<\ell}(-1)^{\ell}S\Big{(}[\xi_{i},\xi_{\ell}]\land\xi_{1}\land\dotsm\land\widehat{\xi}_{i}\land\dotsm\land\widehat{\xi}_{\ell}\land\dotsm\land\xi_{j+1}\land\pi\Big{)},$

where $\Phi^{\xi}_{t}$ denotes the flow of the vector field $\xi$. The
associativity of the star product can now be expressed by

$P\circ(P\otimes\mathrm{id}-\mathrm{id}\otimes P)=0.$

This follows from the formality theorem and hence the following equations do
hold in the a similar way:

(4.4.4) $P(\pi)\circ(A(\xi,\pi)\otimes\mathrm{id}+\mathrm{id}\otimes
A(\xi,\pi))=A(\xi,\pi)\circ P(\pi)+\delta P(\xi,\pi),$ (4.4.5)
$P(\pi)\circ(F(\xi,\eta,\pi)\otimes\mathrm{id}-\mathrm{id}\otimes
F(\xi,\eta,\pi))=-A(\xi,\pi)\circ A(\eta,\pi)+A(\eta,\pi)\circ
A(\xi,\pi)+\delta A(\xi,\eta,\pi),$ (4.4.6) $-A(\xi,\pi)\circ
F(\eta,\zeta,\pi)-A(\eta,\pi)\circ F(\zeta,\xi,\pi)-A(\zeta,\pi)\circ
F(\xi,\eta,\pi)=\delta F(\xi,\eta,\zeta,\pi)$

Equation (4.4.4) describes the fact that under coordinate transformation
induced by the vector field $\xi$, the star product $P(\pi)$ is changed to an
equivalent one up to higher order terms. Equations (4.4.5) and (4.4.6) will be
used for the construction of the relations between a connection 1-form $A$ and
its curvature $F_{A}$. For the explicit computation of the configuration space
integrals, which arise for the weight computation in the Taylor coefficients
of $U_{j}$, we can also describe the lowest order terms in the expansion of
$P,A$ and $F$ and their action on functions:

  1. (1)

$P(\pi)(f\otimes g)=fg+\hbar\pi({\mathrm{d}}f,{\mathrm{d}}g)+O(\hbar^{2})$,

  2. (2)

$A(\xi,\pi)=\xi+O(\hbar)$, where we identify $\xi$ with a first order
differential operator on the right hand side,

  3. (3)

$A(\xi,\pi)=\xi$, if $\xi$ is a linear vector field,

  4. (4)

$F(\xi,\eta,\pi)=O(\hbar)$,

  5. (5)

$P(\pi)(1\otimes f)=P(\pi)(f\otimes 1)=f$,

  6. (6)

$A(\xi,\pi)1=0$.

Equations $(1)$ and $(5)$ have already been introduced before as the defining
conditions for a star product. The equations involving the connection $A$ are
used to construct a connection $D$ on sections on $E$. A section
$f\in\Gamma(E)$ is locally given by a map $x\mapsto f_{x}$, where for every
$y$, $f_{x}(y)$ is a formal power series with coefficients given by infinite
jets. On this space we can introduce a deformed product $\star$ which will be
the desired star product on $C^{\infty}(M)$ after we have identified
horizontal section with ordinary functions. Let us denote analogously by
$\pi_{x}$ the push-forward by $\phi_{x}^{-1}$ of the Poisson bivector field
$\pi$ on $\mathbb{R}^{d}$. Then we can define the deformed product by the
formal bidifferential operator $P(\pi_{x})$ similarly as we did for the usual
product by $P(\pi)$:

$(f\star g)_{x}(y):=f_{x}(y)g_{x}(y)+\hbar\pi^{ij}_{x}(y)\frac{\partial
f_{x}(y)}{\partial y^{i}}\frac{\partial g_{x}(y)}{\partial
y^{j}}+O(\hbar^{2}).$

One can define the connection $D$ on $\Gamma(E)$ by

$(Df)_{x}:={\mathrm{d}}_{x}f+A^{M}_{x}f,$

where ${\mathrm{d}}_{x}f$ denotes the de Rham differential of $f$ regarded as
a function with values in
$\mathbb{R}[\\![y^{1},\ldots,y^{d}]\\!][\\![\hbar]\\!]$ and the formal
connection $1$-form is given through its action on a tangent vector $\xi$ by

$A^{M}_{x}(\xi):=A(\widehat{\xi}_{x},\pi_{x}),$

where $A$ is the operator defined as in (4.4.2) evaluated on the multivector
fields $\xi$ and $\pi$ given through the local coordinate system defined by
$\phi_{x}$. Note that since the coefficients $U_{j}$ of the formality map that
appear in the definition of $P$ and $A$ are polynomial in the derivatives of
the coordinate of the arguments $\xi$ and $\pi$, all results which hold for
$P(\pi)$ and $A(\xi,\pi)$ are inherited by their counterparts. In fact,
Equations $(1)$ and $(5)$ above ensure that $\star$ is an associative
deformation of the pointwise product on sections and Equations $(2)$ and $(3)$
can be used to prove that $D$ is independent of the choice of $\phi$ and hence
induces a global connection on $E$.

####  4.4.2. Construction of solutions for a Fedosov-type equation

We can extend $D$ and $\star$ by the (graded) Leibniz rule to the whole
complex of formal differential forms
$\Omega^{\bullet}(E):=\Omega^{\bullet}(M)\otimes_{C^{\infty}(M)}\Gamma(E)$ and
use (4.4.5) to get the following lemma.

######  Lemma 4.4.1.

Let $F^{M}$ be the $E$-valued 2-form given by $x\mapsto F^{M}_{x}$ where
$F_{x}^{M}(\xi,\eta):=F(\widehat{\xi}_{x},\widehat{\eta}_{x},\pi_{x})$ for any
pair of vector fields $\xi,\eta$. Then $F^{M}$ represents the curvature of $D$
and the two are related to each other and to the star product by the usual
identities:

  1. (1)

$D(f\star g)=D(f)\star g+f\star D(g)$,

  2. (2)

$D^{2}=[F^{M},\enspace]_{\star}$,

  3. (3)

$DF^{M}=0$.

###### Proof.

We can deduce these identities from the relations (4.4.4), (4.4.5) and
(4.4.6), in which the star commutator $[f,g]_{\star}:=f\star g-g\star f$ is
already implicitly defined, after identifying the complex of formal
multivector fields endowed with the differential $\delta$ with the complex of
formal multidifferential operators with the de Rham differential. Such an
isomorphism was explicitly given in [CFT02a]. ∎

######  Remark 4.4.2.

A connection $D$ which satisfies the above relation on a bundle $E$ of
associative algebras is called _Fedosov connection_ with _Weyl curvature_ $F$.
It is the kind of connection Fedosov introduced in order to give a global
construction for the symplectic case (Section 4.1.3) [Fed94].

The final step towards a globalization is to deform the connection $D$ into a
new connection $\bar{D}$ which has the same properties and moreover has
vanishing Weyl curvature, i.e. $\bar{D}^{2}=0$. Hence, we can define the
complex $H^{j}_{\bar{D}}(E)$ and thus the (sub)algebra of horizontal sections
$H^{0}_{\bar{D}}(E)$.

######  Lemma 4.4.3.

Let $D$ be a Fedosov connection on $E$ with Weyl curvature $F$ and $\gamma$ an
$E$-valued 1-form. Then

$\bar{D}:=D+[\gamma,\enspace]_{\star}$

is also a Fedosov connection with Weyl curvature
$\bar{F}=F+D\gamma+\gamma\star\gamma$.

###### Proof.

Let $f$ be a section in $\Gamma(E)$. Then

$\displaystyle\bar{D}^{2}f$
$\displaystyle=[\bar{F},f]_{\star}+D[\gamma,f]_{\star}+[\gamma,Df]_{\star}+[\gamma,[\gamma,f]_{\star}]_{\star}$
$\displaystyle=[F,f]_{\star}+[D\gamma,f]_{\star}+[\gamma,[\gamma,f]_{\star}]_{\star}$
$\displaystyle=\left[F+D\gamma+\frac{1}{2}[\gamma,\gamma]_{\star},f\right]_{\star},$

where the last equality follows from the Jacobi identity of the star
commutator $[\enspace,\enspace]_{\star}$, since each associative product
induces a Lie bracket given by the commutator. if we apply $\bar{D}$ to the
obtained curvature, we get

$\displaystyle\bar{D}\left(F+D\gamma+\frac{1}{2}[\gamma,\gamma]_{\star}\right)$
$\displaystyle=D^{2}\gamma+\frac{1}{2}[D\gamma,\gamma]_{\star}-\frac{1}{2}[\gamma,D\gamma]_{\star}+[\gamma,F+D\gamma]_{\star}$
$\displaystyle=[F,\gamma]_{\star}+[\gamma,F]_{\star}$ $\displaystyle=0,$

where we again used the (graded) Jacobi identity and the (graded) skew-
symmetry of $[\enspace,\enspace]_{\star}$. ∎

######  Lemma 4.4.4.

Let $D$ be a Fedosov connection on a bundle $E=E_{0}[\\![\hbar]\\!]$ and $F$
its Weyl curvature and let

$D=D_{0}+\hbar D_{1}+\dotsm,\qquad F=F_{0}+\hbar F_{1}+\dotsm$

be their expansion as formal power series. If $F_{0}=0$ and the second
cohomology of $E_{0}$ with respect to $D_{0}$ is trivial, then there exists a
1-form $\gamma$ such that $\bar{D}$ has vanishing Weyl curvature.

###### Proof.

By Lemma 4.4.3, we can equivalently say that there exists a solution to the
equation

$\bar{F}=F+D\gamma+\frac{1}{2}[\gamma,\gamma]_{\star}=0.$

We can explicitly construct a solution by induction on the order of $\hbar$.
We start from $\gamma_{0}=0$ and assume that $\gamma^{(j)}$ is a solution
modulo $\hbar^{(j+1)}$. We can add to
$\bar{F}^{(j)}=F+D\gamma^{(j)}+\frac{1}{2}\left[\gamma^{(j)},\gamma^{(j)}\right]_{\star}$
the next term $\hbar^{j+1}D_{0}\gamma_{j+1}$ to get $\bar{F}^{j+1}$ modulo
higher terms. From the equation
$D\bar{F}^{(j)}+\left[\gamma^{(j)},\bar{F}^{(j)}\right]_{\star}=0$ and the
induction hypothesis $\bar{F}^{{j}}=0$ modulo $\hbar^{(j+1)}$, we get
$D_{0}\bar{F}^{(j)}=0$. Now since $H^{2}_{D_{0}}(E_{0})=0$, we can invert
$D_{0}$ in order to define $\gamma_{j+1}$ in terms of the lower order terms
$\bar{F}^{(j)}$ in a way such that $\bar{F}^{(j+1)}=0$ modulo $\hbar^{(j+2)}$.
This completes the induction step. ∎

######  Remark 4.4.5.

Note that in our case $D$ is a deformation of the natural flat connection
$D_{0}$ on sections of the bundle of infinite jets. Thus, the hypothesis of
Lemma 4.4.4 is indeed satisfied and we can find a flat connection $\bar{D}$
which is still a good deformation of $D_{0}$. In the setting of formal
geometry, a flat connection as $D_{0}$ is called a _Grothendieck connection_
[Gro68, Kat70] which is a certain generalization of the _Gauss–Manin
connection_ [Man58]. Another more technical lemma, which uses the notion of
_homological perturbation theory_ , actually gives an isomorphism between the
algebra of horizontal sections $H^{0}_{\bar{D}}(E)$ and its non-deformed
counterpart $H^{0}_{D_{0}}(E_{0})$ which is isomorphic to $C^{\infty}(M)$.
This implies the globalization procedure.

######  Remark 4.4.6.

In [Dol05], Dolgushev gave another proof of Kontsevich’s formality theorem for
general manifolds by using covariant tensors instead of $\infty$-jets of
multidifferential operators and multivector fields which is intrinsically
local. In particular, he also formulated the deformation quantization
construction on _Poisson orbifolds_.

###  4.5. Operadic approach to formality and Deligne’s conjecture

####  4.5.1. Operads and algebras

######  Definition 4.5.1 (Algebraic operad).

An (algebraic) _operad_ (of vector spaces) consists of the following;

  1. (1)

A collection of vector spaces $P(n)$, $n\geq 0$,

  2. (2)

An action of the symmetric group $S_{n}$ on $P(n)$ for all $n$,

  3. (3)

An identity element $\mathrm{id}_{P}\in P(1)$,

  4. (4)

Compositions $m_{n_{1},\ldots,n_{k}}$:

$P(k)\otimes(P(n_{1})\otimes\dotsm\otimes P(n_{k}))\to P(n_{1}+\dotsm+n_{k}),$

for all $k\geq 0$ and $n_{1},\ldots,n_{k}\geq 0$ satisfying a list of axioms.

######  Remark 4.5.2.

We want to obtain the list of axioms for an operad by looking at some
examples.

######  Example 4.5.3 (Endomorphism operad).

Consider the very simple operad $P(n):=\mathsf{Hom}(V^{\otimes n},V)$ for some
vector space $V$. The action of the symmetric group and the identity element
are obvious. The compositions are defined by

$(m_{n_{1}\ldots,n_{k}}(\phi\otimes(\psi_{1}\otimes\dotsm\otimes\psi_{k})))(v_{1}\otimes\dotsm\otimes
v_{n_{1}+\dotsm+n_{k}})\\\ :=\phi(\psi_{1}(v_{1}\otimes\dotsm\otimes
v_{n_{1}})\otimes\dotsm\otimes\psi_{k}(v_{n_{1}}+\dotsm+v_{n_{k-1}+1}\otimes\dotsm\otimes
v_{n_{1}+\dotsm+n_{k}})),$

where $\phi\in P(k)=\mathsf{Hom}(V^{\otimes k},V)$, $\psi_{i}\in
P(n_{i})=\mathsf{Hom}(V^{\otimes n_{i}},V)$ for $i=1,\ldots,k$. This operad is
called the _endomorphism operad_ of a vector space $V$.

######  Remark 4.5.4.

We can see from Example 4.5.3 that there should be an assoicativity axiom for
multiple compositions, various compatibilities for actions of symmetric
groups, and evident relations for compositions including the identity element.

######  Example 4.5.5 (Associative operad).

Consider the operad $P=\mathsf{Assoc}_{1}$. The $n$-th component
$P(n)=\mathsf{Assoc}_{1}(n)$ for $n\geq 0$ is defined as the collection of all
universal (functorial) $n$-linear operations $\mathcal{A}^{\otimes
n}\to\mathcal{A}$ of associative algebras $\mathcal{A}$ with unit. The space
$\mathsf{Assoc}_{1}(n)$ has dimension $n!$, and is spanned by the operations

$a_{1}\otimes\dotsm\otimes a_{n}\mapsto a_{\sigma(1)}\dotsm
a_{\sigma(n)},\quad\sigma\in S_{n}.$

We can identify $\mathsf{Assoc}_{1}(n)$ with the subspace of free associative
unital algebras in $n$ generators consisting of expressions which are
multilinear in each generator.

######  Definition 4.5.6 (Algebra over an operad).

An _algebra over an operad_ $P$ consists of a vector space $\mathcal{A}$ and a
collection of multilinear maps $f_{n}\colon P(n)\otimes\mathcal{A}^{\otimes
n}\to\mathcal{A}$ for all $n\geq 0$ satisfying the following axioms:

  1. (1)

The map $f_{n}$ is $S_{n}$-equivariant for any $n\geq 0$,

  2. (2)

We have $f_{1}(\mathrm{id}_{P}\otimes a)=a$ for all $a\in\mathcal{A}$,

  3. (3)

All compositions in $P$ map to compositions of multilinear operations on
$\mathcal{A}$.

######  Remark 4.5.7.

We often also call an algebra $\mathcal{A}$ over an operad $P$ a _$P$
-algebra_.

######  Example 4.5.8.

The algebra over the operad $\mathsf{Assoc}_{1}$ is an associative unital
algebra. If we replace the 1-dimensional space $\mathsf{Assoc}_{1}(0)$ by the
zero space $0$, we obtain an operad $\mathsf{Assoc}$ describing associative
algebras possibly without unit.

######  Example 4.5.9 (Lie operad).

There is an operad $\mathsf{Lie}$ such that $\mathsf{Lie}$-algebras are Lie
algebras. The dimension of the $n$-th component $\mathsf{Lie}(n)$ is $(n-1)!$
for $n\geq 0$ and zero for $n=0$.

######  Theorem 4.5.10.

Let $P$ be an operad and $V$ a vector space. Then the free $P$-algebra
$\mathsf{Free}_{P}(V)$ generated by $V$ is naturally isomorphic as a vector
space to

$\bigoplus_{n\geq 0}(P(n)\otimes V^{\otimes n})_{S_{n}},$

where the subscript $S_{n}$ denotes the quotient space of coinvariants for the
diagonal action of the symmetric group.

######  Remark 4.5.11.

The free algebra $\mathsf{Free}_{P}(V)$ is defined by the usual categorical
_adjunction property_ , i.e. the set
$\mathsf{Hom}_{P\text{-algebras}}(\mathsf{Free}_{P}(V),\mathcal{A})$ (i.e.
homomorphisms in the category of $P$-algebras) is naturally isomorphic to the
set $\mathsf{Hom}_{\text{vector spaces}}(V,\mathcal{A})$ for any $P$-algebra
$\mathcal{A}$.

####  4.5.2. Topological operads

We can replace vector spaces by _topological spaces_ in the definition of an
operad. The tensor product is then replaced by the Cartesian product.

######  Definition 4.5.12 (Topological operad).

A _topological operad_ consists of the following:

  1. (1)

A collection of topological spaces $P(n)$ for $n\geq 0$,

  2. (2)

A continuous action of the symmetric group $S_{n}$ on $P(n)$ for all $n$,

  3. (3)

An identity element $\mathrm{id}_{P}\in P(1)$,

  4. (4)

Compositions $m_{n_{1},\ldots,n_{k}}$:

$P(k)\times(P(n_{1})\times\dotsm\times P(n_{k}))\to P(n_{1}+\dotsm+n_{k}),$

which are continuous maps for all $k\geq 0$ and $n_{1},\ldots,n_{k}\geq 0$
satisfying a list of axioms similarly to the ones in Definition 4.5.1.

######  Example 4.5.13 (Topological version of endomorphism operad).

The analog version to the endopmorphism operad is the operad $P$ such that for
any $n\geq 0$, we get a topological space $P(n)$ which is the space of
continuous maps from $X^{n}\to X$, where $X$ is some compact topological
space.

######  Remark 4.5.14.

In general, one can define an operad and an algebra over an operad in any
arbitrary symmetric monoidal category $\mathscr{C}^{\otimes}$, i.e. a category
endowed with the functor

$\otimes\colon\mathscr{C}\times\mathscr{C}\to\mathscr{C},$

the identity element $1_{\mathscr{C}}\in\mathrm{Obj}(\mathscr{C})$ and various
coherence conditions for associativity, commutativity of $\otimes$ and so on.

In particular, we want to consider the symmetric monoidal category
$\mathsf{Complexes}$ of _$\mathbb{Z}$ -graded cochain complexes_ of abelian
groups (or vector spaces over some field). Such operads are called _dg-
operads_. Note that each component $P(n)$ is a cochain complex, i.e. a vector
space decomposed into a direct sum $P(n)=\bigoplus_{i\in\mathbb{Z}}P(n)^{i}$
and endowed with a differential ${\mathrm{d}}\colon P(n)^{i}\to P(n)^{i+1}$
which is of degree $+1$ satisfying ${\mathrm{d}}^{2}=0$.

######  Remark 4.5.15.

We can construct an operad of cochain complexes out of a topological operad by
considering a version of the _singular chain complex_. For a topological space
$X$ we denote by $\mathsf{Chains}(X)$ the complex concentrated in negative
degrees, whose $(-k)$-th component for $k\geq 0$ consists of the formal finite
additive combinations

$\sum_{1\leq i\leq N}n_{i}f_{i},\quad
n_{i}\in\mathbb{Z},\,N\in\mathbb{Z}_{\geq 0}.$

of continuous maps $f_{i}\colon[0,1]^{k}\to X$ (_singular cubes_ in $X$)
modulo the following relations:

  1. (1)

$f\circ\sigma=\mathrm{sign}(\sigma)f$ for any $\sigma\in S_{k}$ acting on the
standard curbe $[0,1]^{k}$ by permutations of coordinates,

  2. (2)

$f^{\prime}\circ\mathrm{pr}_{k\to k-1}=0$, where $\mathrm{pr}_{k\to
k-1}\colon[0,1]^{k}\to[0,1]^{k-1}$ is the projection onto the first $(k-1)$
coordinates and $f^{\prime}\colon[0,1]^{k-1}\to X$ is a continuous map.

The boundary operator is defined similarly as for singular chains. Note that,
in contrast to singular chains, for cubical chains we have an external product
map

$\bigotimes_{i\in I}\mathsf{Chains}(X_{i})\to\mathsf{Chains}\left(\prod_{i\in
I}X_{i}\right).$

If $P$ is a topological operad, then $\mathsf{Chains}(P(n))$ for $n\geq 0$ has
a natural operad structure in the category of complexes of abelian groups. The
compositions are then given in terms of the external tensor product of cubical
chains. On the level of cohomology, we obtain an operad $H(P)$ of
$\mathbb{Z}$-graded abelian groups which are complexes endowed with the zero
differential. This is called the _homology_ of the operad $P$.

####  4.5.3. The little disks operad

Let $d\geq 1$ and denote by $G_{d}$ the $(d+1)$-dimensional Lie group acting
on $\mathbb{R}^{d}$ by affine transformations $u\mapsto\lambda u+v$ where
$\lambda>0$ is a real number and $v\in\mathbb{R}^{d}$. This group acts simply
and transitively on the space of closed disks in $\mathbb{R}^{d}$ endowed with
the usual Euclidean metric. The disk with center $v$ and radius $\lambda$ is
given by a transformation of $G_{d}$ with parameters $(\lambda,v)$ of the
standard disk

$D_{0}:=\\{(x_{1},\ldots,x_{d})\in\mathbb{R}^{d}\mid
x_{1}^{2}+\dotsm+x_{d}^{2}\leq 1\\}.$

######  Definition 4.5.16 (Little disks operad).

The _little disks operad_ $C_{d}$ is a topological operad with the following
structure:

  1. (1)

$C_{d}(0)=\varnothing$,

  2. (2)

$C_{d}(1)=\\{\mathrm{id}_{C_{d}}\\}$,

  3. (3)

The space $C_{d}(n)$ is the space of configurations of $n$ disjoint disks
$(D_{i})_{1\leq i\leq n}$ inside the standard disk $D_{0}$ for $n\geq 2$.

The composition

$C_{d}(k)\times(C_{d}(n_{1})\times\dotsm\times C_{d}(n_{k}))\to
C_{d}(n_{1}+\dotsm+n_{k})$

is obtained by applying elements from $G_{d}$ associated with disks
$(D_{i})_{1\leq i\leq n}$ in the configuration in $C_{d}(k)$ to configurations
in all $C_{d}(n_{i})$, $i=1,\ldots,k$ and putting the resulting configurations
together. The action of the symmetric group $S_{n}$ on $C_{d}(n)$ is given by
renumerations of indices of disks $(D_{i})_{1\leq i\leq n}$ (see Figure
4.5.1).

$1$$2$$3$$3$$4$$1$$2$ Figure 4.5.1. Example for the compositions of little
disks.

######  Remark 4.5.17.

Note that $C_{d}(n)$ is homotopy equivalent to the configuration space
$\mathrm{Conf}_{n}(\mathbb{R}^{d})$ of $n$ pairwise distinct points in
$\mathbb{R}^{d}$. Obviously, we can consider the map
$C_{d}(n)\to\mathrm{Conf}_{n}(\mathbb{R}^{d})$ which takes a collection of
disjoint disks to the collection of the centers. Note that in particular, we
have that $\mathrm{Conf}_{2}(\mathbb{R}^{d})$ (and hence $C_{2}(n)$) is
homotopy equivalent to the $(d-1)$-dimensional sphere $S^{d-1}$. The homotopy
equivalence is given by the map

$(v_{1},v_{2})\mapsto\frac{v_{1}-v_{2}}{\|v_{1}-v_{2}\|}\in
S^{d-1}\subset\mathbb{R}^{d}.$

######  Remark 4.5.18.

Since we are abusing notation, one should not confuse the operad $C_{d}$ with
the configuration space $C_{n}$ as in Remark 4.3.7.

####  4.5.4. Deligne’s conjecture

Denote the Hochschild complex of an associative algebra $\mathcal{A}$
concentrated in positive degree by

$\mathcal{C}^{\bullet}(A,A):=\mathsf{Hom}_{\text{vector
space}}(\mathcal{A}^{\otimes n},\mathcal{A}),\quad n\geq 0$

and the Hochschild differential by ${\mathrm{d}}_{\mathrm{H}}$, given by the
formula

$({\mathrm{d}}_{\mathrm{H}}\phi)(a_{1}\otimes\dotsm\otimes
a_{n+1}):=a_{1}\phi(a_{2}\otimes\dotsm\otimes a_{n})+\sum_{1\leq i\leq
n}(-1)^{i}\phi(a_{1}\otimes\dotsm\otimes a_{i}a_{i+1}\otimes\dotsm\otimes
a_{n+1})+\dotsm\\\ \dotsm+(-1)^{n+1}\phi(a_{1}\otimes\dotsm\otimes
a_{n})a_{n+1},\quad\forall\phi\in\mathcal{C}^{n}(\mathcal{A},\mathcal{A}).$

######  Conjecture 4.5.19 (Deligne).

There exists a natural action of the operad $\mathsf{Chains}(C_{2})$ on the
Hochschild complex $\mathcal{C}^{\bullet}(\mathcal{A},\mathcal{A})$ for any
associative algebra $\mathcal{A}$.

######  Remark 4.5.20.

The proof of this conjecture was given by a combination of results of [Tam98,
Tam03] and (a higher version) also in [Kon99]. See also [MS99, KS00].

####  4.5.5. Formality of chain operads

######  Theorem 4.5.21 (Kontsevich–Tamarkin[Tam03, Kon99]).

The operad $\mathsf{Chains}(C_{d})\otimes\mathbb{R}$ of complexes of real
vector spaces is quasi-isomorphic to its cohomology operad endowed with the
zero differential.

##  Chapter 5 Quantum Field Theoretic Approach to Deformation Quantization

Kontsevich’s formula is in its nature a pure algebraic construction. However,
the concept of deformation quantization should give a physical quantization
procedure which opens the question whether it is possible to naturally extract
the star product out of a quantum field theory, i.e. a perturbative _Feynman
path integral quantization_ [Fey42, Fey49, Fey50, FH65, Pol05]. The appearance
of graphs in Kontsevich’s construction might already inspire to regard them as
Feynman diagrams of a certain field theory. Indeed, using the _Poisson sigma
model_ [Ike94, SS94, CF01], one can show that its perturbative quantization
produces the desired outcome. It is however necessary to formulate everything
in a suitable formalism since it appears to be a _gauge theory_ (i.e. the
theory has symmetries). It turns out that the Batalin–Vilkovisky (BV)
formalism [BV77, BV81, BV83] (see [Sch93] for a more mathematical formulation
and [Mne19] for a good introduction) is needed in order to deal with this
particular sigma model. In this chapter we will describe the mathematical
framework for functional integrals, construct the Moyal product out of a path
integral quantization, describe the Faddeev–Popov [FP67] and BRST [BRS74,
BRS75, BRS76, Tyu76] method to deal with gauge theories and introduce the
Poisson sigma model. Moreover, we will explicitly show how the perturbative
quantization of the Poisson sigma model for linear Poisson structures produces
Kontsevich’s star product (this only requires the BRST formalism). Finally, we
state the theorem for general Poisson structures on $\mathbb{R}^{d}$ (this
construction needs the BV formalism) [CF00, CF01]. This chapter is based on
[Zin94, Pol05, Cat+05, CF00, CF01, Kon94, Mne19, Fed96].

###  5.1. Functional integrals

####  5.1.1. Functional integrals and expectation values

For a field theory construction we want to consider a _space of fields_
$\mathcal{M}$ which in most cases is given by the space of sections for some
vector bundle and an _action function_ $S\colon\mathcal{M}\to\mathbb{R}$. The
action is usually given _locally_ (see also Definition 5.2.14), i.e., roughly,
as an integral over some Lagrangian density
$S(\phi)=\int\mathscr{L}(\phi,\partial\phi,\ldots,\partial^{N}\phi)$ for
$N\in\mathbb{Z}_{>0}$.

######  Definition 5.1.1 (Expectation value).

For a function $\mathcal{O}$ on $\mathcal{M}$ we define the _expectation
value_

$\langle\mathcal{O}\rangle:=\frac{\displaystyle{\int}_{\mathcal{M}}\exp\left(\mathrm{i}S/\hbar\right)\mathcal{O}}{\displaystyle{\int}_{\mathcal{M}}\exp\left(\mathrm{i}S/\hbar\right)}.$

######  Definition 5.1.2 (Observables).

An _observable_ is a function $\mathcal{O}$ on $\mathcal{M}$ whose expectation
value is well-defined.

######  Remark 5.1.3.

Integrals as in Definition 5.1.1 are called _functional integrals_ or _path
integrals_. One can consider a perturbative evaluation of such integrals by
expanding $S$ around a nondegenerate critical point and defining the integral
as a formal power series in $\hbar$ with coefficients given by Gaussian
expectation values. If $S$ carries certain _symmetries_ , the critical points
will always be degenerate. In this case we speak of a _gauge theory_. There
are different methods to deal with such theories such a the _Faddeev–Popov
ghost method_ [FP67], the _BRST method_ [BRS74, BRS75, Tyu76] and the
_Batalin–Vilkovisky method_ [BV77, BV81, BV83]. Interestingly, it was shown
that the field theoretic construction of Kontsevich’s star product for general
Poisson structures requires the Batalin–Vilkovisky formalism to deal with the
gauge theory given by the Poisson sigma model [CF00].

####  5.1.2. Gaussian integrals

Let $A$ be a positive-definite symmetric matrix on $\mathbb{R}^{n}$ which we
assume to be endowed with the Lebesgue measure ${\mathrm{d}}^{n}x$ and the
Euclidean inner product $\langle\enspace,\enspace\rangle$ (note that $n$ has
to be even). Then

$I(\lambda):=\int_{\mathbb{R}^{n}}\exp\left(-\frac{\lambda}{2}\langle
x,Ax\rangle\right){\mathrm{d}}^{n}x=\frac{(2\pi)^{\frac{n}{2}}}{\lambda^{\frac{n}{2}}}\frac{1}{\sqrt{\det
A}},\quad\lambda>0.$

If we continue $I$ to the whole complex plane without the negative real axis,
we get

$I(-\mathrm{i})=(2\pi)^{\frac{n}{2}}\exp\left(\frac{\mathrm{i}\pi
n}{4}\right)\frac{1}{\sqrt{\det A}},\qquad
I(\mathrm{i})=(2\pi)^{\frac{n}{2}}\exp\left(\frac{-\mathrm{i}\pi
n}{4}\right)\frac{1}{\sqrt{\det A}}.$

Thus, when $A$ is negative-definite, we can define the integral by

$\int_{\mathbb{R}^{n}}\exp\left(\frac{\mathrm{i}}{2}\langle
x,Ax\rangle\right){\mathrm{d}}^{n}x=\int_{\mathbb{R}^{n}}\exp\left(-\frac{\mathrm{i}}{2}(-\langle
x,Ax\rangle)\right){\mathrm{d}}^{n}x=(2\pi)^{\frac{n}{2}}\exp\left(\frac{-\mathrm{i}\pi
n}{4}\right)\frac{1}{\sqrt{|\det A|}}.$

For the case when $A$ is nondegenerate (not necessarily positive- or negative-
definite), we get

$\int_{\mathbb{R}^{n}}\exp\left(\frac{\mathrm{i}}{2}\langle
x,Ax\rangle\right){\mathrm{d}}^{n}x=(2\pi)^{\frac{n}{2}}\exp\left(\frac{\mathrm{i}\pi\,\mathrm{sign}\,A}{4}\right)\frac{1}{\sqrt{|\det
A|}},$

where $\mathrm{sign}\,A$ denotes the _signature_ of $A$.

We want to compute expectation values with respect to a Gaussian distribution.
Let us denote such an expectation value by
$\langle\enspace,\enspace\rangle_{0}$. Define first the generating function

$Z(J):=\int_{\mathbb{R}^{n}}\exp\left(\frac{\mathrm{i}}{2}\langle
x,Ax\rangle+\frac{\mathrm{i}}{2}\langle J,x\rangle\right){\mathrm{d}}^{n}x\\\
=(2\pi)^{\frac{n}{2}}\exp\left(\frac{\mathrm{i}\pi\,\mathrm{sign}\,A}{4}\right)\frac{1}{\sqrt{|\det
A|}}\exp\left(\frac{\mathrm{i}}{2}\langle J,A^{-1}J\rangle\right).$

Then we get

$\langle x^{i_{1}}\dotsm
x^{i_{k}}\rangle_{0}=\frac{\displaystyle{\int}_{\mathbb{R}^{n}}\exp\left(\frac{\mathrm{i}}{2}\langle
x,Ax\rangle\right)x^{i_{1}}\dotsm
x^{i_{k}}{\mathrm{d}}^{n}x}{\displaystyle{\int}_{\mathbb{R}^{n}}\exp\left(\frac{\mathrm{i}}{2}\langle
x,Ax\rangle\right){\mathrm{d}}^{n}x}\\\ =\frac{\frac{\partial}{\partial
J^{i_{1}}}\dotsm\frac{\partial}{\partial
J^{i_{k}}}Z(J)|_{J=0}}{Z(0)}=\frac{\partial}{\partial
J^{i_{1}}}\dotsm\frac{\partial}{\partial
J^{i_{k}}}\exp\left(\frac{\mathrm{i}}{2}\langle
J,A^{-1}J\rangle\right)\bigg{|}_{J=0}.$

######  Remark 5.1.4.

Note that $\langle x^{i_{1}}\dotsm x^{i_{k}}\rangle_{0}$ vanishes if $k$ is
odd and is a sum of products of matrix elements of the inverse of $A$ if $k$
is even. For example, if $k=2$ we have $\langle
x^{i}x^{j}\rangle_{0}=\mathrm{i}(A^{-1})^{ij}$ and if $k=2s$, we get

$\langle x^{i_{1}}\dotsm x^{i_{2s}}\rangle_{0}=\mathrm{i}^{s}\sum_{\sigma\in
S_{2s}}\frac{1}{2^{s}s!}(A^{-1})^{i_{\sigma(1)}i_{\sigma(2)}}\dotsm(A^{-1})^{i_{\sigma(2s-1)}i_{\sigma(2s)}}.$

######  Theorem 5.1.5 (Wick).

Denote by $P(s)$ the set of _pairings_ , i.e. permutations $\sigma\in S_{2s}$
with the property that $\sigma(2i-1)<\sigma(2i)$ for $i=1,\ldots,s$ and
$\sigma(1)<\sigma(3)<\dotsm<\sigma(2s-3)<\sigma(2s-1)$. Then we have

(5.1.1) $\langle x^{i_{1}}\dotsm
x^{i_{2s}}\rangle_{0}=\mathrm{i}^{s}\sum_{\sigma\in
P(s)}(A^{-1})^{i_{\sigma(1)}i_{\sigma(2)}}\dotsm(A^{-1})^{i_{\sigma(2s-1)}i_{\sigma(2s)}}.$

##### Infinite-dimensional case

By (5.1.1), the infinite-dimensional case only makes sense if $A$ is
invertible. However, usually $A$ will be a differential operator and thus
$G:=A^{-1}$ will denote the distributional kernel of its inverse, i.e. its
_Green function_. So if e.g. $A$ is a differential operator on functions on
some manifold $\Sigma$, we get

(5.1.2)
$\begin{split}\langle\phi(x_{1})\dotsm\phi(x_{2s})\rangle_{0}=\frac{\displaystyle{\int}\exp\left(\frac{\mathrm{i}}{2\hbar}\int_{\Sigma}\phi
A\phi\right)\phi(x_{1})\dotsm\phi(x_{2s})\mathscr{D}[\phi]}{\displaystyle{\int}\exp\left(\frac{\mathrm{i}}{2\hbar}\int_{\Sigma}\phi
A\phi\right)\mathscr{D}[\phi]}\\\ :=(\mathrm{i}\hbar)^{s}\sum_{\sigma\in
P(s)}G\big{(}x_{\sigma(1)},x_{\sigma(2)}\big{)}\dotsm
G\big{(}x_{\sigma(2s-1)},x_{\sigma(2s)}\big{)},\end{split}$

where $\phi$ denotes a function on $\Sigma$, $\mathscr{D}[\phi]$ denotes the
_formal Lebesgue measure_ on the space of functions and $x_{1},\ldots,x_{2s}$
are distinct points in $\Sigma$.

######  Remark 5.1.6 (Normal ordering).

Equation (5.1.2) is usually extended to the singular case when points coincide
by restricting the sum to pairings with the property that
$x_{\sigma(2i-1)}\not=x_{\sigma(2i)}$ for all $i$. This is called _normal
ordering_ since it corresponds to the usual normal ordering in the operator
formulation of Gaussian field theories.

######  Remark 5.1.7 (Propagator).

In Gaussian integrals all expectation values are given in terms of the
expectation value of the quadratic monomials. These are usually called
_2-point functions_ or _propagators_. For the case of a Gaussian field theory
defined by a differential operator, propagator is just another name for the
Green function.

######  Remark 5.1.8 (Derivatives).

One can extend the definition of expectation values to derivatives of fields
by linearity. In particular, for multi-indices $I_{1},\ldots,I_{2s}$ we set

(5.1.3)
$\langle\partial_{I_{1}}\phi(x_{1})\dotsm\partial_{I_{2s}}\phi(x_{2s})\rangle_{0}=(\mathrm{i}\hbar)^{s}\frac{\partial^{|I_{1}|}}{\partial
x_{1}^{I_{1}}}\dotsm\frac{\partial^{|I_{2s}|}}{\partial
x_{2s}^{I_{2s}}}\sum_{\sigma\in
P(s)}G\big{(}x_{\sigma(1)},x_{\sigma(2)}\big{)}\dotsm
G\big{(}x_{\sigma(2s-1)},x_{\sigma(2s)}\big{)},$

where the derivatives on the right hand side are meant in the distributional
sense.

####  5.1.3. Integration of Grassmann variables

Let $V$ be a vector space. Recall that the exterior algebra $\bigwedge V^{*}$
is regarded as the algebra of functions on the odd vector space $\Pi V$.
Choosing a basis and orientation, we can identify
$\bigwedge^{\mathrm{top}}V^{*}$ with $\mathbb{R}$. The composition of this
isomorphism with the projection $\bigwedge
V^{*}\to\bigwedge^{\mathrm{top}}V^{*}$ gives a map $\bigwedge
V^{*}\to\mathbb{R}$ which will be denoted by $\int_{\Pi V}$ and we call it the
_integral on $\Pi V$_.

Let $B\in\operatorname{End}(V)$ and regard it as an element of $V^{*}\otimes
V$ and thus as a function on

$\Pi V^{*}\times\Pi V:=\Pi(V^{*}\oplus V).$

There is a natural identification of $\bigwedge^{\mathrm{top}}(V^{*}\oplus V)$
with $\mathbb{R}$. Thus, we have

$\int_{\Pi V^{*}\times\Pi V}\exp(B)=\det B.$

If $B$ is nondegenerate, we define the expectation value of a function $f$ on
$\Pi V^{*}\times\Pi V$ by

$\langle f\rangle_{0}:=\frac{\displaystyle{\int}_{\Pi V^{*}\times\Pi
V}\exp(B)f}{\displaystyle{\int}_{\Pi V^{*}\times\Pi V}\exp(B)}.$

Let $\\{e_{i}\\}$ be a basis of $V$ and denote by $\\{\bar{e}^{i}\\}$ the
corresponding dual basis. Then $\bigwedge(V^{*}\oplus V)$ can be identified
with the Grassmann algebra generated by the anticommuting _coordinate
functions_ $\bar{e}^{i}$ and $e_{j}$. Functions on $\Pi V^{*}\times\Pi V$ are
then linear combinations of monomials $e_{j_{1}}\dotsm
e_{j_{r}}\bar{e}^{i_{1}}\dotsm\bar{e}^{i_{s}}$. To $B$ we associate the
function

$\langle\bar{e},Be\rangle=\bar{e}^{j}B^{i}_{j}e_{i},$

where $\langle\enspace,\enspace\rangle$ denotes the canonical pairing between
$V^{*}$ and $V$. Hence, we can write

$\int\exp\left(\langle\bar{e},Be\rangle\right)=\det B,$

and

$\langle e_{j_{1}}\dotsm
e_{j_{r}}\bar{e}^{i_{1}}\dotsm\bar{e}^{i_{s}}\rangle_{0}=\frac{\displaystyle{\int}\exp\left(\langle\bar{e},Be\rangle\right)e_{j_{1}}\dotsm
e_{j_{r}}\bar{e}^{i_{1}}\dotsm\bar{e}^{i_{s}}}{\displaystyle{\int}\exp\left(\bar{e}^{j}B^{i}_{j}e_{i}\right)}.$

It is easy to see that $\langle e_{j_{1}}\dotsm
e_{j_{r}}\bar{e}^{i_{1}}\dotsm\bar{e}^{i_{s}}\rangle_{0}$ vanishes if
$r\not=s$ and that

$\langle e_{j_{1}}\dotsm
e_{j_{s}}\bar{e}^{i_{1}}\dotsm\bar{e}^{i_{s}}\rangle_{0}=\sum_{\sigma\in
S_{s}}\mathrm{sign}(\sigma)(B^{-1})^{i_{\sigma(1)}}_{j_{1}}\dotsm(B^{-1})^{i_{\sigma(s)}}_{j_{s}}.$

######  Remark 5.1.9 (Odd vector fields).

A vector field on $\Pi V$ is by definition a graded derivation of $\bigwedge
V^{*}$. In particular, an endomorphism $X$ of $\bigwedge V^{*}$ is a vector
field of degree $|X|$ if

$X(fg)=X(f)g+(-1)^{|X|r}fX(g),\quad\forall f\in\bigwedge^{r}V^{*},\forall
g\in\bigwedge V^{*},\forall r.$

A _right_ vector field $X$ of degree $|X|$ is an endomorphism $f\mapsto(f)X$
of $\bigwedge V^{*}$ that satisfies

$(fg)X=f(g)X+(-1)^{|X|s}(f)Xg,\quad\forall f\in\bigwedge V^{*},\forall
g\in\bigwedge^{s}V^{*},\forall s.$

We can identify the vector space of all vector fields on $\Pi V$ with
$\bigwedge V^{*}\otimes V$. Note that the elements of $V$ can be regarded as
constant vector fields. Integration gives us

$\int_{\Pi V}X(f)=0,\quad\forall f,$

if $X$ is a constant vector field. In general, one can define the _divergence_
$\operatorname{\textnormal{div}}X$ of $X$ by

$\int_{\Pi V}X(f)=\int_{\Pi V}\operatorname{\textnormal{div}}Xf,\quad\forall
f.$

If $X=g\otimes v$ for $g\in\bigwedge^{s}V^{*}$ and $v\in V$, we get

$\operatorname{\textnormal{div}}X=(-1)^{s+1}\iota_{v}g.$

###  5.2. The Moyal product as a path integral quantization

Consider the symplectic manifold $T^{*}\mathbb{R}^{n}$ endowed with the
canonical symplectic form $\omega_{0}=\sum_{1\leq i\leq
n}{\mathrm{d}}p_{i}\land{\mathrm{d}}q^{i}$, where
$(q^{1},\ldots,q^{n},p_{1},\ldots,p_{n})\in
T^{*}\mathbb{R}^{n}\cong\mathbb{R}^{2n}$. Denote by $\alpha=\sum_{1\leq i\leq
n}p_{i}{\mathrm{d}}q^{i}$ the Liouville 1-form such that
$\omega_{0}={\mathrm{d}}\alpha$. Consider a path $\gamma\colon I\to
T^{*}\mathbb{R}^{n}$, where $I$ is a 1-dimensional manifold, and define an
action $S(\gamma)=\int_{I}\gamma^{*}\alpha$. Let us write
$\gamma(t)=(Q(t),P(t))$ for $t\in I$. Then we can rewrite the action as

(5.2.1) $S(Q,P)=\int_{t\in
I}P_{i}\frac{{\mathrm{d}}}{{\mathrm{d}}t}Q^{i}{\mathrm{d}}t.$

Given a Hamiltonian function $H$, one can deform the action to

(5.2.2) $S_{H}(Q,P)=\int_{t\in
I}\left(P_{i}\frac{{\mathrm{d}}}{{\mathrm{d}}t}Q^{i}+H(Q(t),P(t),t)\right){\mathrm{d}}t.$

Let now $I=S^{1}$ and, in order to make the quadratic form nondegenerate, we
choose a basepoint $\infty\in S^{1}$. Let

$\mathcal{M}:=\\{(Q,P)\in C^{\infty}(S^{1},T^{*}\mathbb{R}^{n})\\}$

and

$\mathcal{M}(q,p):=\\{(Q,P)\in C^{\infty}(S^{1},T^{*}\mathbb{R}^{n})\mid
Q(\infty)=q,\,P(\infty)=p\\}.$

The path integral can then be defined by imposing Fubini’s theorem:

(5.2.3) $\int_{\mathcal{M}}(\dotsm)=\int_{(q,p)\in
T^{*}\mathbb{R}^{n}}\mu(q,p)\int_{\mathcal{M}(q,p)}(\dotsm)$

where $\mu$ denotes a measure on $T^{*}\mathbb{R}^{n}$. One can check that the
quadratic form in $S$ is nondegenerate when restricted to $\mathcal{M}(q,p)$.
Hence, we can compute

$\langle\mathcal{O}\rangle_{0}(q,p):=\frac{\displaystyle{\int}_{\mathcal{M}(q,p)}\exp\left(\mathrm{i}S/\hbar\right)\mathcal{O}}{\displaystyle{\int}_{\mathcal{M}(q,p)}\exp\left(\mathrm{i}S/\hbar\right)},$

where $\mathcal{O}$ is some function on $\mathcal{M}$ that is either
polynomial or a formal power series in $Q$ and $P$. Using the fact that the
denominator is constant (infinite though), we can use (5.2.3) to write

$\langle\mathcal{O}\rangle_{0}:=\frac{\displaystyle{\int}_{(q,p)\in
T^{*}\mathbb{R}^{n}}\mu(q,p)\langle\mathcal{O}\rangle_{0}(q,p)}{\displaystyle{\int}_{(q,p)\in
T^{*}\mathbb{R}^{n}}\mu(q,p)}$

whenever the functions $\langle\mathcal{O}\rangle_{0}(q,p)$ and $1$ are
integrable. The latter condition prevents us from choosing $\mu$ to be the
Liouville volume $\frac{\omega_{0}^{n}}{n!}$. However, at this point one can
also forget about the denominator and define

$\langle\mathcal{O}\rangle_{0}^{\prime}:=\int_{(q,p)\in
T^{*}\mathbb{R}^{n}}\mu(q,p)\langle\mathcal{O}\rangle_{0}(q,p)$

such that the Liouville volume is allowed. In fact, we could also choose $\mu$
to be the delta measure peaked at a point $(q,p)\in T^{*}\mathbb{R}^{n}$. In
this case we have

$\langle\mathcal{O}\rangle_{0}=\langle\mathcal{O}\rangle_{0}^{\prime}=\langle\mathcal{O}\rangle_{0}(q,p).$

If we have fixed $(q,p)\in T^{*}\mathbb{R}^{n}$, we can can use the _change of
variables_ $Q=q+\tilde{Q}$ and $P=p+\tilde{P}$, where $(\tilde{Q},\tilde{P})$
is a map from $S^{1}$ to $T^{*}\mathbb{R}^{n}$ which vanishes at $\infty$.
This is the same as a map $\mathbb{R}\to T^{*}\mathbb{R}^{n}$ which vanishes
at $\infty$. The action is then given by

$S(Q,P)=S(q+\tilde{Q},p+\tilde{P})=\int_{\mathbb{R}}\tilde{P}_{i}\frac{{\mathrm{d}}}{{\mathrm{d}}t}\tilde{Q}^{i}{\mathrm{d}}t.$

Expectation values are given by

$\langle\mathcal{O}(Q,P)\rangle_{0}(q,p)=\left\langle\mathcal{O}(q+\tilde{Q},p+\tilde{P})\right\rangle^{\sim}_{0}:=\frac{\displaystyle{\int}\exp\left(\mathrm{i}S/\hbar\right)\mathcal{O}(q+\tilde{Q},p+\tilde{P})\mathscr{D}[\tilde{P}]\mathscr{D}[\tilde{Q}]}{\displaystyle{\int}\exp\left(\mathrm{i}S/\hbar\right)\mathscr{D}[\tilde{P}]\mathscr{D}[\tilde{Q}]}.$

####  5.2.1. The propagator

We need to find the Green function for the differential operator
$\frac{{\mathrm{d}}}{{\mathrm{d}}t}$. It is given in terms of the _sign
function_ :

$\left(\frac{{\mathrm{d}}}{{\mathrm{d}}t}\right)^{-1}(u,v)=\theta(u-v)=\frac{1}{2}\mathrm{sig}(u-v):=\begin{cases}\frac{1}{2},&u>v\\\
-\frac{1}{2},&u<v\end{cases}$

As a consequence, we get

$\left\langle\tilde{P}_{i}(u)\tilde{Q}^{j}(v)\right\rangle^{\sim}_{0}=\mathrm{i}\hbar\theta(v-u)\delta^{j}_{i}$

and more generally

$\left\langle\tilde{P}_{i_{1}}(u_{1})\dotsm\tilde{P}_{i_{s}}(u_{s})\tilde{Q}^{j_{1}}(v_{1})\dotsm\tilde{Q}^{j_{s}}(v_{s})\right\rangle^{\sim}_{0}=(\mathrm{i}\hbar)^{s}\sum_{\sigma\in
S_{s}}\theta(v_{\sigma(1)}-u_{1})\dotsm\theta(v_{\sigma(s)}-u_{s})\delta^{j_{\sigma(1)}}_{i_{1}}\dotsm\delta^{j_{\sigma(s)}}_{i_{s}}.$

The normal ordering can be implemented by setting $\theta(0)=0$ which is
compatible with the skew-symmetry of $\frac{{\mathrm{d}}}{{\mathrm{d}}t}$.

####  5.2.2. Expectation values

Let us consider the observable on $\mathcal{M}$ which is given by evaluation
of a smooth function $f$ on $T^{*}\mathbb{R}^{n}$ at some point $u$ in the
path. Define

$\mathcal{O}_{f,u}(Q,P):=f(Q(u),P(u)),\quad f\in
C^{\infty}(T^{*}\mathbb{R}^{n}),\,u\in S^{1}\setminus\\{\infty\\}.$

in order to compute the expectation value of this observable, we need to
introduce some notation. Let $I=(i_{1},\ldots,i_{r})$ be a multi-index and set
$|I|=r$, $p_{I}:=p_{i_{1}}\dotsm p_{i_{r}}$, $q^{I}:=q^{i_{1}}\dotsm
q^{i_{r}}$ (similarly for $\tilde{P}_{I}$ and $\tilde{Q}^{I}$). Moreover,
define

$\partial_{I}:=\frac{\partial}{\partial
q^{i_{1}}}\dotsm\frac{\partial}{\partial
q^{i_{r}}},\quad\partial^{I}:=\frac{\partial}{\partial
p_{i_{1}}}\dotsm\frac{\partial}{\partial p_{i_{r}}}.$

We extend everything to the case of $|I|=0$ by setting $p_{I}=q^{I}=1$ and
$\partial_{I}=\partial^{I}=\mathrm{id}$. If we use the change of variables as
before and use the Taylor expansion of $f$, we get

$f(Q(u),P(u))=f(q,\tilde{Q}(u),p+\tilde{P}(u))=\sum_{r,s\geq
0}\frac{1}{r!s!}\sum_{|I|=r\atop|J|=s}\tilde{P}_{I}(u)\tilde{Q}^{J}(u)\partial^{I}\partial_{J}f(q,p).$

Hence

$\langle\mathcal{O}_{f,u}\rangle_{0}(q,p)=\left\langle\mathcal{O}_{f,u}(q+\tilde{Q},p+\tilde{P})\right\rangle_{0}^{\sim}=f(q,p).$

If $f$ is integrable we can also define

$\langle\mathcal{O}_{f,u}\rangle_{0}^{\prime}=\int_{T^{*}\mathbb{R}^{n}}\mu(q,p)f(q,p).$

######  Remark 5.2.1.

Note that these expectation values do not depend on the point $u$.

Consider now the observable given by

(5.2.4) $\mathcal{O}_{f,g;u,v}=f(Q(u),P(u))g(Q(v),P(v)),\quad f,g\in
C^{\infty}(T^{*}\mathbb{R}^{n}),\,u,v\in
S^{1}\setminus\\{\infty\\}\cong\mathbb{R},\,u<v.$

Its expectation value can then be computed as

$\langle\mathcal{O}_{f,g;u,v}\rangle_{0}(q,p)=\left\langle
f(q+\tilde{Q}(u),p+\tilde{P}(u))g(q+\tilde{Q}(v),p+\tilde{P}(v))\right\rangle^{\sim}_{0}\\\
=\sum_{r_{1},s_{1},r_{2},s_{2}\geq
0}\frac{1}{r_{1}!s_{1}!r_{2}!s_{2}!}\sum_{|I_{1}|=r_{1}\atop|J_{1}|=s_{1}}\sum_{|I_{2}|=r_{2}\atop|J_{2}|=s_{2}}\left\langle\tilde{P}_{I_{1}}(u)\tilde{Q}^{J_{1}}(u)\tilde{P}_{I_{2}}(v)\tilde{Q}^{J_{2}}(v)\right\rangle^{\sim}_{0}\partial^{I_{1}}\partial_{J_{1}}f(q,p)\partial^{I_{2}}\partial_{J_{2}}g(q,p)\\\
=\sum_{r,s\geq
0}\frac{1}{r!s!}\left(\frac{\mathrm{i}\hbar}{2}\right)^{r+s}(-1)^{s}\sum_{|I|=r\atop|J|=s}\partial^{I}\partial_{J}f(q,p)\partial^{J}\partial_{I}g(q,p)=f\star
g(q,p).$

Here we have denoted by $\star$ the Moyal product induced by the Poisson
bivector field $\frac{\partial}{\partial p_{i}}\land\frac{\partial}{\partial
q^{i}}$.

######  Remark 5.2.2.

Note also here that the expectation is independent of $u$ and $v$.

If $f\star g$ is integrable, we can compute

$\langle\mathcal{O}_{f,g;u,v}\rangle_{0}^{\prime}=\int_{T^{*}\mathbb{R}^{n}}\mu
f\star g.$

If $\mu$ is given by the Liouville volume $\frac{\omega_{0}^{n}}{n!}$, we can
define a trace (see also [Fed94, NT95, Fed96])

$\operatorname{Tr}(f\star
g):=\int_{T^{*}\mathbb{R}^{n}}\frac{\omega_{0}^{n}}{n!}f\star
g=\int_{T^{*}\mathbb{R}^{n}}\frac{\omega_{0}^{n}}{n!}fg,$

where the second equality follows from using integration by parts in the
correction terms to the commutative product (see also [CF10] for a similar
trace formula for the Poisson manifold $\mathbb{R}^{d}$ endowed with any
Poisson structure and [Mos19] for the formulation on general Poisson
manifolds). More generally, we can define an observable (see Figure 5.2.1)

$\mathcal{O}_{f_{1},\ldots,f_{k};u_{1},\ldots,u_{k}}=f_{1}(Q(u_{1}),P(u_{1}))\dotsm
f_{k}(Q(u_{k}),P(u_{k})),\\\ \ f_{1},\ldots,f_{k}\in
C^{\infty}(T^{*}\mathbb{R}^{n}),\,u_{1},\ldots,u_{k}\in
S^{1}\setminus\\{\infty\\}\cong\mathbb{R},\,u_{1}<\dotsm<u_{k}.$

######  Exercise 5.2.3.

Show that

$\langle\mathcal{O}_{f_{1},\ldots,f_{k};u_{1},\ldots,u_{k}}\rangle_{0}(q,p)=f_{1}\star\dotsm\star
f_{k}(q,p).$

$u_{1}$$u_{2}$$u_{3}$$u_{4}$$\vdots$$u_{k}$$\infty$$S^{1}$ Figure 5.2.1. The
points $u_{1},\ldots,u_{k}$ on $S^{1}$

####  5.2.3. Divergence of vector fields

The normal ordering condition implies that local vector fields are in fact
divergence free. Consider a vector at
$(\tilde{Q},\tilde{P})\in\mathcal{M}(q,p)$ which can be identified with a
smooth map $\mathbb{R}\to T^{*}\mathbb{R}^{n}$ that vanishes at $\infty$. Note
that a vector field $X$ on $\mathcal{M}(q,p)$ is an assignment of a map
$X(\tilde{Q},\tilde{P})$ to each path $(\tilde{Q},\tilde{P})$. The vector
field is said to be _local_ if $X(\tilde{Q},\tilde{P})(t)$ is a function of
$\tilde{Q}(t)$ and $\tilde{P}(t)$ for all $t\in\mathbb{R}$.

Formally, we want to express the divergence of a vector field in terms of path
integrals similar to the description using ordinary integrals. We would like
to have

$\int_{\mathcal{M}(q,p)}X\left(\exp\left(\mathrm{i}S/\hbar\right)\mathcal{O}\right)=\int_{\mathcal{M}(q,p)}\operatorname{\textnormal{div}}X\exp\left(\mathrm{i}S/\hbar\right)\mathcal{O},$

for all observables $\mathcal{O}$.

######  Definition 5.2.4 (Divergence).

If there exists an observable $\operatorname{\textnormal{div}}X$ such that for
any observable $\mathcal{O}$ we have

$\langle X(S)\mathcal{O}\rangle_{0}(q,p)=\mathrm{i}\hbar\langle
X(\mathcal{O})\rangle_{0}(q,p)-\mathrm{i}\hbar\langle\operatorname{\textnormal{div}}X\mathcal{O}\rangle_{0}(q,p),$

then we say that $\operatorname{\textnormal{div}}X$ is the _divergence_ of
$X$.

######  Lemma 5.2.5.

If $X$ is a local vector field, then $\operatorname{\textnormal{div}}X=0$.

###### Proof.

Write
$X(\tilde{Q},\tilde{P})(t)=(X_{q}(\tilde{Q}(t),\tilde{P}(t)),X_{p}(\tilde{Q}(t),\tilde{P}(t)))$.
Then

$X(S)=\int\left(X_{p}\frac{{\mathrm{d}}}{{\mathrm{d}}t}\tilde{Q}-X_{q}\frac{{\mathrm{d}}}{{\mathrm{d}}t}\tilde{P}\right){\mathrm{d}}t.$

By the normal ordering and the locality of $X$, when computing $\langle
X(S)\mathcal{O}\rangle_{0}(q,p)$ we only have to contract the $\tilde{P}$’s
($\tilde{Q}$’s) in $\mathcal{O}$ with the $\tilde{Q}$’s ($\tilde{P}$’s) in
$X(S)$ and replace each pair by a propagator. Whenever a $\tilde{P}$
($\tilde{Q}$) in $\mathcal{O}$ is contracted with the
$\frac{{\mathrm{d}}}{{\mathrm{d}}t}\tilde{Q}$
($\frac{{\mathrm{d}}}{{\mathrm{d}}t}\tilde{P}$) in $X(S)$, we get the identity
operator times $\mathrm{i}\hbar$ ($-\mathrm{i}\hbar$). Summing up the various
terms, we get that this is the same as taking the expectation value of
$X(\mathcal{O})$ multiplied with $\mathrm{i}\hbar$. This completes the proof.
∎

We can also consider the divergence of vector fields on $\mathcal{M}$. In
fact, a local vector field $X$ on $\mathcal{M}$ can be uniquely written as the
sum of a vector field $X_{\infty}$ on $T^{*}\mathbb{R}^{n}$ and a section
$\tilde{X}$ of local vector fields (i.e. an assignment of a local vector field
$\tilde{X}(q,p)$ on $\mathcal{M}(q,p)$ to each $(q,p)\in
T^{*}\mathbb{R}^{n}$). By Lemma 5.2.5 we get

$\langle X(S)\mathcal{O}\rangle_{0}^{\prime}=\mathrm{i}\hbar\langle
X(\mathcal{O})\rangle_{0}^{\prime}-\mathrm{i}\hbar\langle\operatorname{\textnormal{div}}_{\mu}X_{\infty}\mathcal{O}\rangle_{0}^{\prime},$

where $\operatorname{\textnormal{div}}_{\mu}X_{\infty}$ denotes the ordinary
divergence of the vector field $X_{\infty}$ with respect to the measure $\mu$.
One can then define $\operatorname{\textnormal{div}}_{\mu}X_{\infty}$ to be
the _divergence_ of the local vector field $X$ on $\mathcal{M}$.

######  Exercise 5.2.6.

Using the local vector field $X(\tilde{Q},\tilde{P})(t):=(\tilde{P}(t),0)$,
show that

$\hbar\frac{{\mathrm{d}}}{{\mathrm{d}}\hbar}(f\star
g)=(p_{i}\partial^{i}f)\star
g+f\star(p_{i}\partial^{i}g)-p_{i}\partial^{i}(f\star g).$

####  5.2.4. Independence of evaluation points

Note that in the previous computations we have considered observables whose
definition depends on the choice of points in $S^{1}\setminus\\{\infty\\}$.
However, computing the expectation values we have seen that they are in fact
independent of these points. In particular, we have the following Proposition:

######  Proposition 5.2.7.

The expectation values are invariant under (pointed) diffeomorphism of the
source manifold $S^{1}$ on which the field theory is defined.

######  Remark 5.2.8.

A quantum field theory with such a property is usually called _topological
quantum field theory (TQFT)_.

Given $f\in C^{\infty}(T^{*}\mathbb{R}^{n})$ and two points
$a,b\in\mathbb{R}$, we have

$f(Q(b),P(b))-f(Q(a),P(a))=\int_{a}^{b}\left(\partial_{i}f(Q(t),P(t))\frac{{\mathrm{d}}}{{\mathrm{d}}t}Q^{i}(t)+\partial^{i}f(Q(t),P(t))\frac{{\mathrm{d}}}{{\mathrm{d}}t}P_{i}(t)\right){\mathrm{d}}t\\\
=\lim_{r\to\infty}\int_{-\infty}^{+\infty}\left(\partial_{i}f(Q(t),P(t))\frac{{\mathrm{d}}}{{\mathrm{d}}t}Q^{i}(t)+\partial^{i}f(Q(t),P(t))\frac{{\mathrm{d}}}{{\mathrm{d}}t}P_{i}(t)\right)\lambda_{r}(t){\mathrm{d}}t\\\
=\lim_{r\to\infty}\tilde{X}_{f,r}(S),$

where $(\lambda_{r})$ is a sequence of smooth, compactly supported functions
that converges almost everywhere to the characteristic function of the
interval $[a,b]$ and $\tilde{X}_{f,r}$ is the local vector field

$\tilde{X}_{f,r}(t)=(-\partial^{i}f(Q(t),P(t)),\partial_{i}f(Q(t),P(t)))\lambda_{r}(t).$

######  Remark 5.2.9.

Recall that $\mathcal{M}$ is a space of maps. So a vector field on the target
$T^{*}\mathbb{R}^{n}$ generates a local vector field on $\mathcal{M}$. Let
$\tilde{X}_{f}$ be the local vector field corresponding to the Hamiltonian
vector field $X_{f}$ of $f$. Then $\tilde{X}_{f,r}$ is given by multiplying
$\tilde{X}_{f}$ by $\lambda_{r}$.

If we restrict $\tilde{X}_{f,r}$ to $\mathcal{M}(q,p)$, we can observe by
Lemma 5.2.5 that it is divergence-free. Hence, we get

$\langle(f(Q(b),P(b))-f(Q(a),P(a)))\mathcal{O}\rangle_{0}(q,p)=\lim_{r\to\infty}\left\langle\tilde{X}_{f,r}(S)\mathcal{O}\right\rangle_{0}(q,p)=\mathrm{i}\hbar\lim_{r\to\infty}\left\langle\tilde{X}_{f,r}(\mathcal{O})\right\rangle_{0}(q,p),$

which vanishes if $\mathcal{O}$ does not depend on
$(\tilde{Q}(t),\tilde{P}(t))$ for $t\in[a,b]$. This means that we can move the
evaluation point at least as long we do not meet another evaluation point.

####  5.2.5. Associativity

Let us prove the associativity of the Moyal product by using the path integral
description considering the expectation value of the observable
$\mathcal{O}_{f,g,h;u,v,w}$. There will be three propagators appearing which
correspond to the three different ways of pairing $u,v,w$. Note that the
function $\theta$ will not see the difference, since

$\theta(w-u)=\theta(w-v)=\theta(v-u)=\frac{1}{2}.$

We group the propagators by considering first only those between $u$ and $v$
and only thereafter the other ones. This will imply that

$\langle\mathcal{O}_{f,g,h;u,v,w}\rangle_{0}=(f\star g)\star h.$

If we group the propagators such that we first consider those between $v$ and
$w$ and then the others, we get

$\langle\mathcal{O}_{f,g,h;u,v,w}\rangle_{0}=f\star(g\star h).$

This proves associativity of the Moyal product.

######  Remark 5.2.10.

We can also prove this result by using the independence of evaluation points.
This then implies

$\lim_{v\to u^{+}}\langle\mathcal{O}_{f,g,h;u,v,w}\rangle_{0}(q,p)=\lim_{v\to
w^{-}}\langle\mathcal{O}_{f,g,h;u,v,w}\rangle_{0}(q,p).$

The left hand side corresponds to evaluating first the expectation value of
$\mathcal{O}_{f,g;u,v}$ then placing the result at $u$ and computing the
expectation value of
$\mathcal{O}_{\langle\mathcal{O}_{f,g;u,v}\rangle_{0},h;w}$. This will give
the result $(f\star g)\star h$. Repeating this computation on the right hand
side, we get associativity.

####  5.2.6. The evolution operator

Let us consider the deformed action $S_{H}$ as in (5.2.2). We restrict
ourselves to a time-independent Hamiltonian $h$ and let it act from the
instant $a$ to the instant $b$ with $a<b$. Then we consider the action $S_{H}$
with

(5.2.5) $H(q,p,t)=h(q,p)\chi_{[a,b]}(t),$

where $\chi_{[a,b]}$ denotes the characteristic function on the interval
$[a,b]$. The aim is to compute the _evolution operator_

$U(q,p,T):=\frac{\displaystyle{\int}_{\mathcal{M}(q,p)}\exp\left(\mathrm{i}S_{H}/\hbar\right)}{\displaystyle{\int}_{\mathcal{M}(q,p)}\exp\left(\mathrm{i}S/\hbar\right)},$

where $T=b-a$. Observe that

$U(q,p,T)=\left\langle\exp\left(\frac{\mathrm{i}}{\hbar}\int
H(Q(t),P(t),t){\mathrm{d}}t\right)\right\rangle_{0}(q,p)\\\
=\left\langle\exp\left(\frac{\mathrm{i}}{\hbar}\int_{a}^{b}h(Q(t),P(t),t){\mathrm{d}}t\right)\right\rangle_{0}(q,p).$

Now we can express the integral in terms of Riemann sums as

$\int_{a}^{b}h(Q(t),P(t),t){\mathrm{d}}t=\lim_{N\to\infty}\frac{T}{N}\sum_{1\leq
r\leq N}h(Q(a+rT/N),P(a+rT/N)).$

Hence, we get

$U(q,p,T)=\lim_{N\to\infty}\left\langle\prod_{1\leq r\leq
N}\exp\left(\frac{\mathrm{i}}{\hbar}\frac{T}{N}h(Q(a+rT/N),P(a+rT/N))\right)\right\rangle_{0}(q,p)\\\
=\lim_{N\to\infty}\left\langle\mathcal{O}_{\exp\left(\frac{\mathrm{i}}{\hbar}\frac{T}{N}h\right),\ldots,\exp\left(\frac{\mathrm{i}}{\hbar}\frac{T}{N}h\right);a+\frac{T}{N},a+2\frac{T}{N},\ldots,a+T}\right\rangle_{0}(q,p)\\\
=\lim_{N\to\infty}\left(\exp\left(\frac{\mathrm{i}}{\hbar}\frac{T}{N}h\right)\right)^{\star
N}(q,p)=\exp_{\star}\left(\frac{\mathrm{i}}{\hbar}h\right)(q,p).$

######  Remark 5.2.11.

Note that the previous result also includes negative powers of $\hbar$.
However, everything is well-defined since each term in the power series
expansion of $\exp_{\star}$ is actually a _Laurent series_ in $\hbar$.

####  5.2.7. Perturbative evaluation of integrals

Let $\mathcal{M}$ be an $n$-dimensional manifold and $S\in
C^{\infty}(\mathcal{M})$. We want to understand how we can compute an integral
of the form $Z:=\int_{\mathcal{M}}\exp(\mathrm{i}S/\hbar){\mathrm{d}}^{n}x$.
Consider first the case where $S$ has a unique critical point
$x_{0}\in\mathcal{M}$ which is nondegnerate. Then we can write

$S(x_{0}+\sqrt{\hbar}x)=S(x_{0})+\frac{\hbar}{2}{\mathrm{d}}_{x_{0}}^{2}S(x)+R_{x_{0}}(\sqrt{\hbar}x),$

where $R_{x_{0}}$ is a formal powers series given by the Taylor expansion of
$S$ starting with the cubic term in $\sqrt{\hbar}x$ with $x\in
T_{x_{0}}\mathcal{M}$. Let $A$ be the _Hessian_ of $S$ at $x_{0}$ with respect
to the Euclidean metric $\langle\enspace,\enspace\rangle$. In particular, we
write

${\mathrm{d}}_{x_{0}}^{2}S(x)=\langle x,Ax\rangle.$

######  Theorem 5.2.12 (Stationary phase expansion).

The _stationary phase expansion_ (or also _saddle-point approximation_) of the
integral $Z:=\int_{\mathcal{M}}\exp(\mathrm{i}S/\hbar){\mathrm{d}}^{n}x$ is
given by

$Z=\hbar^{\frac{n}{2}}\exp(\mathrm{i}S(x_{0})/\hbar)\int_{\mathcal{M}}\exp\left(\frac{\mathrm{i}}{2}\langle
x,Ax\rangle\right)\sum_{r\geq
0}\frac{1}{r!}R^{r}_{x_{0}}(\sqrt{\hbar}x){\mathrm{d}}^{n}x\\\
=(2\pi\hbar)^{\frac{n}{2}}\exp(\mathrm{i}S(x_{0})/\hbar)\exp\left(\frac{\mathrm{i}\pi\,\mathrm{sign}\,A}{4}\right)\frac{1}{\sqrt{|\det
A|}}\sum_{r\geq 0}\frac{1}{r!}\left\langle
R^{r}_{x_{0}}(\sqrt{\hbar}x)\right\rangle_{0},$

where $\langle\enspace\rangle_{0}$ denotes the Gaussian expectation value with
respect to the nondegenerate symmetric matrix $A$.

Expectation values as in Definition 5.1.1 are then given by

(5.2.6) $\langle\mathcal{O}\rangle=\frac{\displaystyle{\sum}_{r\geq
0}\frac{1}{r!}\left\langle\mathcal{O}(x_{0}+\sqrt{\hbar}x)R_{x_{0}}^{r}(\sqrt{\hbar}x)\right\rangle_{0}}{\displaystyle{\sum}_{r\geq
0}\frac{1}{r!}\left\langle R^{r}_{x_{0}}(\sqrt{\hbar}x)\right\rangle_{0}}.$

######  Remark 5.2.13.

Note that the denominator is of the form $1+O(\hbar)$ and thus the expectation
value can be computed as a formal power series in $\hbar$. One can consider
the computation in a graphical way by associating a vertex of valence $k$ to
the term of degree $k$ in $R_{x_{0}}$ and a vertex of valence $\ell$ to the
term of degree $\ell$ in $\mathcal{O}$. By Wick’s theorem (Theorem 5.1.5) one
has to connect these vertices in pairs in all possible ways the half-edges
emanating from each vertex. The result is a collection of _Feynman diagrams_
with a weight associated to each of them. The expectation value is then a sum
over all Feynman diagrams of their weights. A Feynman diagram with vertices
coming only from $R_{x_{0}}$, i.e. a Feynman diagram appearing only in the
denominator of (5.2.6), is called a _vacuum diagram_. A combinatorial fact of
(5.2.6) is that an expectation value is given by the sum over graphs which do
not have a connected component which is a vacuum diagram (see [Zin94, Pol05]
for more details on Feynman diagrams).

####  5.2.8. Infinite dimensions

Let us look at the situation of computing (5.2.6) in the case when
$\mathcal{M}$ is infinite-dimensional. We want to recall the following
definition.

######  Definition 5.2.14 (Local function).

A function on a space of fields on some manifold $M$ is _local_ if it is the
integral on $M$ of a function that depends at each point on finite jets of
fields at that point.

######  Remark 5.2.15.

If the action is a local function, $A$ will be a differential operator. For
our purpose, we will need its Green function (the propagator) and in the
Gaussian expectation values of $\mathcal{O}R^{r}$ and $R^{r}$ we will use
(5.1.3). Instead of summing over indices we will have to integrate over the
Cartesian products of the source manifold. The normal ordering will exclude
all the graphs with edges having the same endpoints (these edges are called
_tadpoles_ or _short loops_) and thus integration is restricted to the
configuration space of the source manifold. However, this is usually not
enough to make the integrals converge as the Green functions are very singular
when the arguments approach each other. The general procedure to get rid of
divergencies is called _renormalization_. Anyway, for our case (i.e. the case
of a TQFT), the configuration-space integrals associated to Feynman diagrams
without tadpoles do indeed converge.

######  Remark 5.2.16.

If the action $S$ has more critical points and all of them are nondegenerate,
the asymptotic expansion is obtained by computing the stationary phase
expansion (Theorem 5.2.12) around each critical point and then summing all
these terms together. The expression for expectation value is then no longer
given by (5.2.6). It can happen that one of the critical points dominates the
others so that one can forget them. In physical theories this is usually done
by regarding $\exp(\mathrm{i}S/\hbar)$ as the analytic continuation of the
exponential of minus a poisitive-definite function, called _Euclidean action_
, so that the dominating critical point is the absolute minimum. There are
still examples where there is no way to do it as all critical points appear as
saddle points. In such a case, one can consider another option which consists
of selecting one particular critical point (called the _sector_) and expanding
only around this point. In this case, (5.2.6) is again the correct expression
for the infinite-dimensional extension.

As already mentioned, it can often happen that the critical points are
degenerate. A simple case is given when critical points are parametrized by a
finite-dimensional manifold $\mathcal{M}_{\mathrm{crit}}$. Then, using
Fubini’s theorem for the finite-dimensional case, we can rewrite the integral
similarly as in (5.2.3) as

$\int_{\mathcal{M}}(\dotsm)=\int_{x_{0}\in\mathcal{M}_{\mathrm{crit}}}\mu(x_{0})\int_{\mathcal{M}(x_{0})}(\dotsm)$

where $\int_{\mathcal{M}(x_{0})}$ denotes the asymptotic expansion of the
integral in the complement to $T_{x_{0}}\mathcal{M}_{\mathrm{crit}}$ of a
formal neighborhood of $x_{0}$, while $\mu$ denotes a measure on
$\mathcal{M}_{\mathrm{crit}}$ (which is determined in the finite-dimensional
case and has to be chosen in the infinite-dimensional case as part of the
definition). If the Hessian is constant on $\mathcal{M}_{\mathrm{crit}}$, we
can express the expectation value as

$\langle\mathcal{O}\rangle=\frac{\displaystyle{\int}_{x_{0}\in\mathcal{M}_{\mathrm{crit}}}\mu(x_{0})\displaystyle{\sum}_{r\geq
0}\frac{1}{r!}\left\langle\mathcal{O}(x_{0}+\sqrt{\hbar}x)R_{x_{0}}^{r}(\sqrt{\hbar}x)\right\rangle_{0}}{\displaystyle{\int}_{x_{0}\in\mathcal{M}_{\mathrm{crit}}}\mu(x_{0})\displaystyle{\sum}_{r\geq
0}\frac{1}{r!}\left\langle R_{x_{0}}^{r}(\sqrt{\hbar}x)\right\rangle_{0}},$

where $\langle\enspace\rangle_{0}(x_{0})$ denotes the Gaussian expectation
value which is computed by expanding around $x_{0}$ orthogonally to
$T_{x_{0}}\mathcal{M}_{\mathrm{crit}}$. A possible choice for the measure
$\mu$ is the delta measure peaked at some point $x_{0}$. In this case, as we
already did before, the expectation value will be denoted by
$\langle\enspace\rangle(x_{0})$.

####  5.2.9. A simple generalization

Note that we have only considered the perturbative expansion with formal
expansion parameter $\hbar$. It might happen that there is another small
expansion parameter, or some coefficient in $S$ which is much smaller than
$\hbar$ or in our setting is an element of
$\hbar^{2}\mathbb{R}[\\![\hbar]\\!]$. In such a case, we can define the
Gaussian part by using the quadratic $\hbar$-independent term of $S/\hbar$ and
consider all the other terms as the perturbation $R$. It can also happen that
the perturbation term $R$ contains quadratic and linear terms as well. In
particular, we can have an action function of the form

(5.2.7) $S(y,z)=\langle y,Bz\rangle+f(y,z),$

where $B$ is a nondegenerate matrix and $f$ is a function quadratic in $z$. If
we work around the critical point $y=z=0$, we can rescale $z$ by $\hbar$ to
get

$S(y,\hbar z)=\hbar\langle y,Bz\rangle+\hbar f(y,z)$

and consider $f$ as the perturbation to the Gaussian theory defined by $B$.

##### Quantum mechanics

Recall that the topological action (5.2.1) defines the Moyal product in terms
of the expectation value of the observable (5.2.4), which is independent of
the evaluation points. Consider a Hamiltonian $H$ as in (5.2.5) and define the
expectation

$f\hat{\star}_{a,b;u,v}g(q,p):=\frac{\displaystyle{\int}_{\mathcal{M}(q,p)}\exp(\mathrm{i}S_{H})/\hbar)\mathcal{O}_{f,g;u,v}}{\displaystyle{\int}_{\mathcal{M}(q,p)}\exp(\mathrm{i}S_{H}/\hbar)}=\frac{\left\langle\exp\left(\frac{\mathrm{i}}{\hbar}\displaystyle{\int}_{a}^{b}h{\mathrm{d}}t\right)\mathcal{O}_{f,g;u,v}\right\rangle_{0}(q,p)}{\left\langle\exp\left(\frac{\mathrm{i}}{\hbar}\displaystyle{\int}_{a}^{b}h{\mathrm{d}}t\right)\right\rangle_{0}(q,p)}$

which will no longer be independent to $u$ and $v$ ($a<u<v<b$) nor will it
define an associative product. By the computations before, we get

$f\hat{\star}_{a,b;u,v}g=\frac{\exp_{\star}\left(\frac{\mathrm{i}}{\hbar}(u-a)h\right)\star
f\star\exp_{\star}\left(\frac{\mathrm{i}}{\hbar}(v-u)h\right)\star
g\star\exp_{\star}\left(\frac{\mathrm{i}}{\hbar}(b-v)h\right)}{\exp_{\star}\left(\frac{\mathrm{i}}{\hbar}(b-a)h\right)}$

Let us consider the perturbative computation for the Hamiltonian

$h(q,p)=\frac{1}{2}G^{ij}(q)p_{i}p_{j},$

at $p=0$ and for $f$ and $g$ only depending on $q$. By setting
$Q(t)=q+\tilde{Q}(t)$ and $P(t)=p+\tilde{P}(t)$, we get

$S_{H}=\hbar\int_{-\infty}^{+\infty}\tilde{P}(t)\frac{{\mathrm{d}}}{{\mathrm{d}}t}\tilde{Q}(t){\mathrm{d}}t+\frac{\hbar^{2}}{2}\int_{a}^{b}G^{ij}(q+\tilde{Q}(t))\tilde{P}_{i}(t)\tilde{P}_{j}(t){\mathrm{d}}t.$

Hence, $f\hat{\star}_{a,b;u,v}g(q,0)$ is the ratio of the expectation value of
$\exp\left(\frac{\mathrm{i}\hbar}{2}\int_{a}^{b}G^{ij}(q+\tilde{Q})\tilde{P}_{i}\tilde{P}_{j}{\mathrm{d}}t\right)\mathcal{O}_{f,g;u,v}$
and the expectation value of
$\exp\left(\frac{\mathrm{i}\hbar}{2}\int_{a}^{b}G^{ij}(q+\tilde{Q})\tilde{P}_{i}\tilde{P}_{j}{\mathrm{d}}t\right)$.
The propagator pairs the $\tilde{P}$’s to $\tilde{Q}$’s, so it is better to
consider oriented graphs. The Feynman diagrams will be oriented graphs with

  1. (1)

a vertex at $u$ with no outgoing arrows,

  2. (2)

a vertex at $v$ with no outgoing arrows,

  3. (3)

vertices between $a$ and $b$ with exactly two outgoing arrows.

Moreover, graphs containing tadpoles or vacuum subgraphs will not be allowed.
See Figures 5.2.2, 5.2.3 and 5.2.4 for examples of allowed and not allowed
Feynman diagrams. Note that to a vertex of first type with $r$ incoming arrows
we associate an $r$-th derivative of $f$, to a vertex of second type with $r$
incoming arrows we associate an $r$-th derivative of $g$ and to a vertex of
third type with $r$ incoming arrows we associate an $r$-th derivative of $G$.
The order in $\hbar$ is given by the number of vertices of the third type. So,
at order zero, we have $f$ placed at $u$ and $g$ placed at $v$. This yields
the pointwise product of $f$ and $g$. At order $1$, we have the graphs of
Figure 5.2.2. Hence, we get

$f\hat{\star}_{a,b;u,v}g(q,0)=f(q)g(q)-\frac{\mathrm{i}\hbar}{4}(b-a+2u-2v)G^{ij}(q)\partial_{i}f(g)\partial_{j}g(q)\\\
-\frac{\mathrm{i}\hbar}{4}(b-a)G^{ij}(q)[\partial_{i}\partial_{j}f(q)g(q)+f(q)\partial_{i}\partial_{j}g(q)]+O(\hbar^{2}).$

######  Exercise 5.2.17.

Compute some of the further orders.

$a$$u$$v$$b$$a$$u$$v$$b$$a$$u$$v$$b$ Figure 5.2.2. Feynman diagrams of order
one. $a$$u$$v$$b$$a$$u$$v$$b$$a$$u$$v$$b$ Figure 5.2.3. Feynman diagrams of
order two. $a$$u$$v$$b$$a$$u$$v$$b$$a$$u$$v$$b$ Figure 5.2.4. Feynman diagrams
which are not allowed: tadpoles, vacuum subdiagrams and both.

###  5.3. Symmetries and the BRST formalism

If the action $S$ has symmetries encoded in a free action of a Lie group $G$
(gauge theory with gauge group $G$), the critical points will never be
nondegenerate since they always will appear in $G$-orbits. This is also the
case for the Poisson sigma model for linear Poisson structures and hence we
need to understand how to deal with these type of theories.

####  5.3.1. The main construction: Faddeev–Popov ghost method

Let $\mathcal{M}$ be a finite-dimensional manifold together with a measure
$\mu$. Let $G$ be a compact Lie group which is endowed with an invariant
measure (Haar measure) with a measure-preserving free action on $\mathcal{M}$.
Moreover, we assume that $\mathcal{M}/G$ is a manifold. Then an invariant
function $f$ is the pullback of a function $\underline{f}\in\mathcal{M}/G$ and

$I:=\frac{1}{\mathrm{Vol}(G)}\int_{\mathcal{M}}f\mu=\int_{\mathcal{M}/G}\underline{f}\underline{\mu},$

where $\underline{\mu}$ is the measure induced by $\mu$ on the quotient. If we
consider a section of a principal bundle
$\pi\colon\mathcal{M}\to\mathcal{M}/G$, we can rewrite $I$ as an integral on
the image of this section. Assume that this integral is locally given by the
zero set of a function $F\colon\mathcal{M}^{\prime}\to\mathfrak{g}$ where
$\mathcal{M}^{\prime}\subset\mathcal{M}$ is such that
$\pi(\mathcal{M}^{\prime})=\mathcal{M}/G$ and $\mathfrak{g}$ denotes the Lie
algebra of $G$. In the physics literature the condition $F=0$ is called _gauge
fixing_ and $F$ is called _gauge-fixing function_. For
$x\in\mathcal{M}^{\prime}$, let $A(x)$ be the differential ${\mathrm{d}}F(x)$
restricted to the vertical tangent space at $x$. This can be identified with
the Lie algebra $\mathfrak{g}$ and thus we can regard $A(x)$ as an
endomorphism of $\mathfrak{g}$ and denote by $J$ its determinant. In the
physics literature $J$ is called _Faddeev–Popov determinant_. Then we get

$I=\int_{\mathcal{M}^{\prime}}f\delta_{0}(F)J\mu,$

where $\delta_{0}$ denotes the delta function at $0\in\mathfrak{g}$. Let us
now rewrite $I$ such that it is suitable for the stationary phase expansion
formula. We need to write $\delta_{0}(F)$ and $J$ in exponential form.
Therefore, we use the Fourier transform of the delta function and the
Grassmann (odd) integration as in 5.1.3. Denote by
$\langle\enspace,\enspace\rangle$ the canonical pairing between
$\mathfrak{g}^{*}$ and $\mathfrak{g}$. Then we have

$I=C\int
f(x)\mu(x)\exp\left(\frac{\mathrm{i}}{\hbar}\langle\lambda,F(x)\rangle\right)\omega(\lambda)\exp\left(\frac{\mathrm{i}}{\hbar}\langle\bar{c},A(x)c\rangle\right),$

where the integral is over $x\in\mathcal{M}^{\prime}$,
$\lambda\in\mathfrak{g}^{*}$, $c\in\Pi\mathfrak{g}$,
$\bar{c}\in\Pi\mathfrak{g}^{*}$ and where $C$ is a constant depending on
$\hbar$ and on the choice of the top form
$\omega\in\bigwedge^{\mathrm{top}}\mathfrak{g}^{*}$. Note that we have added
the prefactors $\frac{\mathrm{i}}{\hbar}$ for later purposes.

######  Remark 5.3.1.

In the physics literature $c$ is called _ghost_ and $\bar{c}$ is called
_antighost_. $\lambda$ is usually called _Lagrange multiplier_.

Note that if we choose a basis $\\{c^{i}\\}$ of $\mathfrak{g}^{*}$, then
$\bigwedge\mathfrak{g}^{*}$ can be identified with the algebra generated by
the $c^{i}$s with the relations

$c^{i}c^{j}=-c^{j}c^{i},\qquad\forall i,j.$

The generators $c^{i}\in\mathfrak{g}^{*}$ are often called _ghost variables_.
Similarly, one can introduce _antighost variables_
$\bar{c}_{i}\in\mathfrak{g}$.

######  Remark 5.3.2.

The determinant of $A$ can also be obtained in terms of an ordinary Gaussian
integral over $\mathfrak{g}\times\mathfrak{g}^{*}$. However, then one needs to
put $A^{-1}$ into the exponential. In the infinite-dimensional case we may use
the techniques of Feynman diagrams only when dealing with local functions. If
the action of the group is local, $A$ will be a differential operator, so
$\langle\bar{c},Ac\rangle$ will be a local function, in contrary to the
quadratic function defined in terms of the Green function $A^{-1}$.

The expectation value of an invariant function $g$ with respect to a given
invariant action $S$ can be written as

(5.3.1) $\langle
g\rangle=\frac{\displaystyle{\int}_{\mathcal{M}}\exp(\mathrm{i}S/\hbar)g\mu}{\displaystyle{\int}_{\mathcal{M}}\exp(\mathrm{i}S/\hbar)\mu}=\frac{\displaystyle{\int}_{\widetilde{\mathcal{M}}}\exp(\mathrm{i}S_{F}/\hbar)g\widetilde{\mu}}{\displaystyle{\int}_{\widetilde{\mathcal{M}}}\exp(\mathrm{i}S/\hbar)\widetilde{\mu}}=:\langle
g\rangle_{F},$

where

$\displaystyle\widetilde{\mathcal{M}}$
$\displaystyle:=\mathcal{M}^{\prime}\times\Pi\mathfrak{g}\times\mathfrak{g}^{*}\times\Pi\mathfrak{g}^{*},$
$\displaystyle S_{F}$
$\displaystyle:=S+\langle\lambda,F\rangle-\langle\bar{c},A,c\rangle,$

and $\widetilde{\mu}:=\mu\omega$. The function $S_{F}$ is usually called the
_gauge-fixed action_.

####  5.3.2. The BRST formalism

Note that, by construction, the right hand side of (5.3.1) is independent of
$F$. Moreover, the assumptions we had are quite restrictive, namely, we want
to have a measure-preserving action on $\mathcal{M}$ by a compact Lie group
$G$ and we have to assume that the principal bundle
$\mathcal{M}\to\mathcal{M}/G$ is trivial. On the other hand, to define

(5.3.2) $\langle
g\rangle_{F}=\frac{\displaystyle{\int}_{\widetilde{\mathcal{M}}}\exp(\mathrm{i}S_{F}/\hbar)g\widetilde{\mu}}{\displaystyle{\int}_{\widetilde{\mathcal{M}}}\exp(\mathrm{i}S/\hbar)\widetilde{\mu}}$

we only need the infinitesimal action

$\displaystyle X\colon\mathfrak{g}$
$\displaystyle\to\mathfrak{X}(\mathcal{M})$ $\displaystyle\gamma$
$\displaystyle\mapsto X_{\gamma}$

of a Lie algebra $\mathfrak{g}$ on $\mathcal{M}$. In this case $A$ is simply
given by

$A_{\gamma}=L_{X_{\gamma}}F,\quad\gamma\in\mathfrak{g}.$

Moreover, we want to relax the condition that $F^{-1}(0)$ defines a section an
rather require that $A(x)$ should be nondegenerate for all