# More Differential Operators on Almost Hermitian Manifolds

Samuel Hosmer  New York City Department of Health and Mental Hygiene
<EMAIL_ADDRESS>
###### Abstract.

In 1980 Michelsohn defined a differential operator on sections of the complex
Clifford bundle over a compact Kähler manifold $M$. This operator is a
differential and its Laplacian agrees with the Laplacian of the Dolbeault
operator on forms through a natural identification of differential forms with
sections of the Clifford bundle. Relaxing the condition that $M$ be Kähler, we
introduce two differential operators on sections of the complex Clifford
bundle over a compact almost Hermitian manifold which naturally generalize the
one introduced by Michelsohn. We show surprising Kähler-like symmetries of the
kernel of the Laplacians of these operators in the almost Hermitian and almost
Kähler settings, along with a correspondence of these operators to operators
on forms which are of present interest in almost complex geometry.

###### 2020 Mathematics Subject Classification:

###### Contents

  1. 1 Introduction
  2. 2 The Clifford Algebra
  3. 3 Almost Hermitian Dirac Identities
  4. 4 The Canonical Hermitian Connections
  5. 5 The Canonical Hermitian Dirac Operators
  6. 6 Almost Hermitian Identities via the Bismut Dirac Operator
  7. 7 Almost Kähler identities via the Riemannian Dirac Operator
  8. 8 Clifford Harmonics

##  1\. Introduction

Given a smooth manifold $M$ de Rham gave us a way of viewing the differential
forms on $M$ as a cochain complex computing the real cohomology of $M$ with
differential $d$, the exterior derivative on forms. The space of forms
$\Omega(M)$ is integrally graded as the space of sections of the exterior
algebra bundle on the dual of $TM$. Additionally $d$ is an elliptic
differential operator of degree 1. This graded differential algebra is a
fundamental object of study within geometry and topology, with Sullivan in
1960 showing that $\big{(}\Omega(M),d\big{)}$ holds the real homotopy type of
$M$.

In the presence of a metric on $M$ the operator $d$ and sections of the
exterior bundle $\Omega(M)=\Gamma(\Lambda(M))$ can be replaced with the
Riemannian Dirac operator $\widetilde{D}$ and sections of the Clifford bundle
$\Gamma(Cl(M))$, respectively. The bundle $Cl(M)$ is isomorphic to
$\Lambda(M)$ as a vector bundle, but the multiplication is not preserved by
the isomorphism. The operator $\widetilde{D}$ is an elliptic self adjoint
operator on sections of $Cl(M)$ which factorizes the Laplacian of $d$. That
is, $\widetilde{D}=d+d^{*}.$

There are some notable consequences of this replacement: $Cl(M)$ has no
natural integral grading and the Dirac operator is generically not a
differential. In effect, by this switch, we lose the cochain complex. What we
gain is Hodge theory.

Recent advances in almost complex geometry show that a valuable topological
insight into an almost complex manifold $M$ is given by the harmonic forms of
certain differential operators on $\Omega(M)$. On an almost complex manifold
$M$ of dimension $2n$ the exterior derivative $d$ over the complexification
$\Omega_{\mathbb{C}}(M)$ decomposes into 4 components in bidegree. Namely,

$d=\partial+\overline{\mu}+\overline{\partial}+\mu$

where $\partial,\overline{\mu},\overline{\partial},\mu$ are of bidegree
$(1,0),(-1,2),(0,1),(2,-1)$ respectively.

Given a compatible metric there is a real linear isomorphism induced from
complex conjugation $c$ and a complex linear isomorphism induced by the Hodge
star operator $\ast$ so that

$c:\Omega_{\mathbb{C}}(M)^{p,q}\xrightarrow{\sim}\Omega_{\mathbb{C}}(M)^{q,p}\hskip
21.68121pt$
$\ast:\Omega_{\mathbb{C}}(M)^{p,q}\xrightarrow{\sim}\Omega_{\mathbb{C}}(M)^{n-q,n-p}.$

Letting $\omega=\langle J\cdot,\cdot\rangle$, there is a classical $sl(2)$
representation on the bundle $\Omega(M)$ generated by the operators
$\big{(}L,\Lambda,H\big{)}$ where $L=\omega\wedge\cdot,\Lambda=L^{*},$ and
$H=[\Lambda,L]$. In the case the form $\omega$ is $d$ closed, there are local
commutation relations between the components of the exterior derivative and
these operators $L,\Lambda,H$ called the almost Kähler identities, which
translate over compact manifolds via Hodge theory to a representation on these
classical sl(2) operators on the space of certain harmonic forms. For $T$ an
elliptic differential operator on forms or sections of the Clifford bundle we
denote by $\bm{\mathcal{H}}_{T}=\ker(\Delta_{T})$ the space of $T$ harmonic
forms.

Cirici and Wilson show in [CW20] that over a compact almost Kähler manifold
$M$ the operators $(L,\Lambda,H)$ define a finite dimensional $sl(2)$
representation on

$\bigoplus_{p,q\geq
0}\bm{\mathcal{H}}_{\partial}^{p,q}\cap\bm{\mathcal{H}}_{\overline{\mu}}^{p,q}=\bigoplus_{p,q\geq
0}\bm{\mathcal{H}}_{d}^{p,q}$

where $\bm{\mathcal{H}}_{T}^{p,q}$ are the $T$ harmonic forms in a particular
bidegree $(p,q)$.

Tardini and Tomassini in [TT20] consider the operator
$\delta=\partial+\overline{\mu}$ and showed the corresponding $\mathbb{Z}$
graded result that $(L,\Lambda,H)$ define an $sl(2)$ representation on

$\bm{\mathcal{H}}_{\delta}=\bigoplus_{k}\bm{\mathcal{H}}_{\delta}^{k}$

where $\bm{\mathcal{H}}_{\delta}^{k}\subset\bm{\mathcal{H}}_{d}^{k}\textrm{
are the harmonic forms in degree }k$. Furthermore, in a particular bidegree,
they show

$\bm{\mathcal{H}}_{\partial}^{p,q}\cap\bm{\mathcal{H}}_{\overline{\mu}}^{p,q}=\bm{\mathcal{H}}_{d}^{p,q}=\bm{\mathcal{H}}_{\delta}^{p,q}.$

Following Michelsohn [M80] we introduce a bidegree decomposition of the
complex Clifford bundle
$\mathbb{C}l(M)\stackrel{{\scriptstyle\textrm{def}}}{{=}}Cl(M)\otimes\mathbb{C}$

$\mathbb{C}l(M)=\bigoplus_{r,s}\mathbb{C}l^{r,s}(M)$

and an $sl(2)$ representation on $\mathbb{C}l(M)$ generated by operators
$\mathcal{L},\overline{\mathcal{L}},\mathcal{H}$. Furthermore there is an
operator ✩ called the transpose map inducing complex linear isomorphisms

$\text{\char 73}:\mathbb{C}l^{r,s}(M)\xrightarrow{\sim}\mathbb{C}l^{r,-s}(M).$

Generalizing Michelsohn’s results in [M80], we introduce two elliptic
differential operators $\mathfrak{D},\mathfrak{B}$ and
$\mathfrak{B}^{\text{\char 73}}=\text{\char 73}\mathfrak{B}\text{\char 73}$ on
sections of $\mathbb{C}l(M)$. Additionally we define an operator
$\bm{\varepsilon}=\partial-i\rho_{\partial}$ on $\Omega_{\mathbb{C}}(M)$ where
$\rho_{\partial}$ is a zero order operator vanishing if and only if
$\partial\omega=\overline{\partial}\omega=0$. We prove local commutation
relations between these operators and the generators of the respective $sl(2)$
representations. We show

######  Proposition (Proposition 8.4).

Let $M$ be a compact almost Hermitian manifold. Complex conjugation $c$ on
$\mathbb{C}l(M)$ induces complex anti-linear isomorphisms

$c:\bm{\mathcal{H}}_{\mathfrak{B}}^{r,s}\cap\bm{\mathcal{H}}_{\mathfrak{B}^{\text{\char
73}}}^{r,s}\rightarrow\bm{\mathcal{H}}_{\mathfrak{B}}^{-r,-s}\cap\bm{\mathcal{H}}_{\mathfrak{B}^{\text{\char
73}}}^{-r,-s},$
$c:\bm{\mathcal{H}}_{\mathfrak{D}}^{r,s}\longrightarrow\bm{\mathcal{H}}_{\mathfrak{D}}^{-r,-s}.\hskip
70.10197pt$

######  Proposition (Proposition 8.5).

Let $M$ be a compact almost Hermitian manifold. The transpose map ✩ induces
complex linear isomorphisms

$\text{\char
73}:\bm{\mathcal{H}}_{\mathfrak{B}}^{r,s}\cap\bm{\mathcal{H}}_{\mathfrak{B}^{\text{\char
73}}}^{r,s}\rightarrow\bm{\mathcal{H}}_{\mathfrak{B}}^{r,-s}\cap\bm{\mathcal{H}}_{\mathfrak{B}^{\text{\char
73}}}^{r,-s},$ $\text{\char
73}:\bm{\mathcal{H}}_{\mathfrak{D}}^{r,s}\longrightarrow\bm{\mathcal{H}}_{\mathfrak{D}}^{r,-s}.\hskip
65.04256pt$

######  Theorem (Theorem 8.6).

Let $M$ be a compact almost Hermitian manifold. The Lie algebra generated by
$\big{(}\mathcal{H},\mathcal{L},\overline{\mathcal{L}}\big{)}$ on
$\Gamma\mathbb{C}l(M)$ defines a finite dimensional $sl(2)$ representation on
the space
$\bm{\mathcal{H}}_{\mathfrak{B}}\cap\bm{\mathcal{H}}_{\mathfrak{B}^{\text{\char
73}}}.$

######  Theorem (Theorem 8.7).

Let $M$ be a compact almost Hermitian manifold. Through the isomorphism
$\Gamma\mathbb{C}l(M)\cong\Omega_{\mathbb{C}}(M)$ we have

$\bm{\mathcal{H}}_{\mathfrak{B}}\cap\bm{\mathcal{H}}_{\mathfrak{B}^{\text{\char
73}}}\cong\bm{\mathcal{H}}_{\varepsilon}\cap\bm{\mathcal{H}}_{\overline{\varepsilon}}.$

######  Theorem (Theorem 8.8).

Let $M$ be a compact almost Kähler manifold. The Lie algebra generated by
$\big{(}\mathcal{H},\mathcal{L},\overline{\mathcal{L}}\big{)}$ on
$\Gamma\mathbb{C}l(M)$ defines a finite dimensional $sl(2)$ representation on
the space $\bm{\mathcal{H}}_{\mathfrak{D}}$.

######  Theorem (Theorem 8.9).

Let $M$ be a compact almost Kähler manifold. Through the isomorphism
$\Gamma\mathbb{C}l(M)\cong\Omega_{\mathbb{C}}(M)$ we have

$\bm{\mathcal{H}}_{\mathfrak{D}}\cong\bm{\mathcal{H}}_{\delta}.$

######  Corollary (Corollary 8.10).

On any compact almost Hermitian manifold $M$ there is a map $g$, called the
Hodge automorphism, inducing an isomorphism

$\bm{\mathcal{H}}_{\mathfrak{B}}^{q-p,n-p-q}\cap\bm{\mathcal{H}}_{\mathfrak{B}^{\text{\char
73}}}^{q-p,n-p-q}\stackrel{{\scriptstyle
g}}{{\cong}}\bm{\mathcal{H}}_{\varepsilon}^{p,q}\cap\bm{\mathcal{H}}_{\overline{\varepsilon}}^{p,q}.$

Moreover, if $M$ is almost Kähler we have that

$\bm{\mathcal{H}}_{\mathfrak{D}}^{q-p,n-p-q}\stackrel{{\scriptstyle
g}}{{\cong}}\bm{\mathcal{H}}_{\delta}^{p,q}.$

### Layout

In Section 2 we gather the necessary algebraic material on Clifford algebras,
following and expanding on [M80]. Section 3 elaborates on the multiplicative
interplay between Dirac operators defined using Hermitian connections and
Michelsohn’s $sl(2,\mathbb{C})$ operators. In Sections 4 and 5 we discuss
canonical Hermitian connections à la Gauduchon and establish some results
involving Dirac operators defined by these connections. Sections 7 and 8 prove
the theorems mentioned above and transfer the results to operators on forms.
The appendix provides an exposition of some fundamental results in almost
Hermitian geometry, following mostly [G97].

### Acknowledgements

This work constituted the author’s PhD thesis at the City University of New
York under the supervision of Luis Fernandez, whom he thanks for his patient
guidance.

##  2\. The Clifford Algebra

Let $V$ be a finite dimensional inner product space. We define the Clifford
Algebra of $V$, denoted by $Cl(V)$, to be the vector space $\Lambda(V)$, with
multiplication given by

$v\cdot\varphi=v\wedge\varphi-v\lrcorner\
\varphi=E_{v}(\varphi)-I_{v}(\varphi)$

for any $v\in V$ and $\varphi\in Cl(V)$, where $E_{v}^{*}=I_{v}$ is the
adjoint of left exterior multiplication by $v$.

Let $V^{\vee}$ be the dual space to $V$. There is an algebra isomorphism

$\Lambda(V)\cong\Lambda(V^{\vee})$

while, in general, the most one can hope for is a vector space isomorphism

$Cl(V)\cong\Lambda(V^{\vee}).$

###  2.1. Two Useful Involutions

The antipodal map $\alpha:V\rightarrow V$ given by $\alpha(v)=-v$ extends to
an algebra automorphism of $Cl(V)$ by $\alpha(v_{1}\cdots
v_{p})=\alpha(v_{1})\cdots\alpha(v_{p})$. This is made precise by noting that
$\alpha$ is an isometry of $V$, which are one-to-one with algebra
automorphisms of $Cl(V)$ preserving $V$.

The tranpose map is the algebra anti-automorphism ✩ of $Cl(V)$ reversing the
order of multiplication, so that $\text{\char 73}(v_{1}\cdots
v_{p})=v_{p}\cdots v_{1}$ and $\text{\char 73}(v)=v$ for $v\in V$. It is worth
noting that the Hodge star operator $\ast$ on forms (sections of $\Lambda(V)$)
takes the shape $\ast(\varphi)=\text{\char
73}\alpha(\varphi)\cdot\textrm{vol}$ where $\textrm{vol}=v_{1}\cdots
v_{\dim(V)}.$

###  2.2. Almost Complex Structures

An almost complex structure $J$ on a real vector space $V$ is an operator on
$V$ so that $J^{2}=-\textrm{Id}$. Such an operator bestows $V$ with a scalar
multiplication making it a complex vector space, and conversely every complex
vector space has a $J$ given by scalar multiplication by $i=\sqrt{-1}$. There
are two useful ways to extend an orthogonal $J$ as an endomorphism of $Cl(V)$.
One way is as an algebra automorphism

$J_{\textrm{alg}}(v_{1}\cdots v_{p})=J(v_{1})\cdots J(v_{p})\hskip 14.45377pt$

the other is as a derivation of the algebra

$J_{\textrm{der}}(v_{1}\cdots v_{p})=\sum_{j}v_{1}\cdots Jv_{j}\cdots v_{p}.$

These two extensions are related as follows: The family of orthogonal
transformations of $V$ given by $J_{t}=\cos(t)\textrm{Id}+\sin(t)J$ induce a
family of algebra automorphisms $J_{\textrm{alg}}(t)$ of $Cl(V)$.
Differentiating $J_{\textrm{alg}}(t)$ at the identity one obtains
$J_{\textrm{der}}$. Let $e_{1},\dots,e_{n},Je_{1},\dots,Je_{n}$ be an
orthonormal basis. The element $\omega\in\Lambda^{2}(V^{\vee})$ defined by

$\omega(v,w)=\langle Jv,w\rangle$

can be viewed as an element of $Cl(V)$ by writing

$\omega=\sum_{j}e_{j}\cdot Je_{j}$

sometimes referred to as the fundamental 2-form.

###  2.3. The Complex Clifford Algebra

We recall the usual orthogonal direct sum decomposition of the
complexification of $V$

$V\otimes\mathbb{C}=V^{1,0}\oplus V^{0,1}$

where $V^{1,0}$ is the $i$-eigenspace of $J$ complex linearly extended to
$V\otimes\mathbb{C}$ and $V^{0,1}=\overline{V^{1,0}}$. We define

$\mathbb{C}l(V)=Cl(V)\otimes\mathbb{C}.$

When $V$ has an orthogonal almost complex structure $J$, a $J$-adapted
orthonormal basis $e_{1},\dots,e_{n},Je_{1},\dots,Je_{n}$ of $V$ gives rise to
an orthogonal basis of $\mathbb{C}l(V)$

$\epsilon_{1},\dots,\epsilon_{n},\overline{\epsilon}_{1},\dots,\overline{\epsilon}_{n}$

where $\epsilon_{j}=\tfrac{1}{2}(e_{j}-iJe_{j})$ and
$\overline{\epsilon}_{j}=\tfrac{1}{2}(e_{j}+iJe_{j})$ for $j=1,\dots,n.$

All of these complex vectors anti-commute in $\mathbb{C}l(V)$ except for pairs
of the form $\epsilon_{k},\overline{\epsilon}_{k}$ where one has the remaining
relation

$\epsilon_{k}\overline{\epsilon}_{k}+\overline{\epsilon}_{k}\epsilon_{k}=-1$

###  2.4. Michelsohn’s Algebraic Results

Michelsohn defined the operators $\mathcal{L}$ and $\overline{\mathcal{L}}$
for any $\varphi\in\mathbb{C}l(V)$ by

$\mathcal{L}(\varphi)=-\sum_{k}\epsilon_{k}\cdot\varphi\cdot\overline{\epsilon}_{k}\hskip
36.135pt\textrm{and}\hskip
36.135pt\overline{\mathcal{L}}(\varphi)=-\sum_{k}\overline{\epsilon}_{k}\cdot\varphi\cdot\epsilon_{k}.$

and defined their commutator

$\mathcal{H}=[\mathcal{L},\overline{\mathcal{L}}].$

######  Theorem 2.5.

[M80] The operators $\mathcal{L},\overline{\mathcal{L}},$ and $\mathcal{H}$
satisfy the relations

$[\mathcal{L},\overline{\mathcal{L}}]=\mathcal{H},\ \
[\mathcal{H},\mathcal{L}]=2\mathcal{L},\ \textrm{ and
}[\mathcal{H},\overline{\mathcal{L}}]=-2\overline{\mathcal{L}}$

In particular the subalgebra of $\textrm{End}(\mathbb{C}l(V))$ generated by
the operators $\mathcal{L},\overline{\mathcal{L}},\mathcal{H}$ is an
$sl(2,\mathbb{C})$ representation on $\mathbb{C}l(V)$.

Letting $\omega_{0}=\tfrac{1}{2i}\omega$ and $\mathcal{J}=-iJ_{\textrm{der}}$
Michelsohn showed that for any $\varphi\in\mathbb{C}l(V)$

$\mathcal{H}(\varphi)=\omega_{0}\cdot\varphi+\varphi\cdot\omega_{0}$.

$\mathcal{J}(\varphi)=\omega_{0}\cdot\varphi-\varphi\cdot\omega_{0}$

in particular $\mathcal{J}$ and $\mathcal{H}$ commute. Furthermore since

$\displaystyle\mathcal{J}\mathcal{L}(\varphi)=-\sum_{j}\mathcal{J}(\epsilon_{j})\cdot\varphi\cdot\overline{\epsilon}_{j}+\mathcal{L}\mathcal{J}(\varphi)-\sum_{j}\epsilon_{j}\cdot\varphi\cdot\mathcal{J}(\overline{\epsilon}_{j})=\mathcal{L}\mathcal{J}(\varphi)$

we have that $\mathcal{J}$ commutes with all of the generators of the $sl(2)$
representation on $\mathbb{C}l(V)$.

We define

$\mathbb{C}l^{r,s}(V)=\left\\{\varphi\in\mathbb{C}l(V)\ |\
\mathcal{J}(\varphi)=r\varphi\textrm{ and
}\mathcal{H}(\varphi)=s\varphi\right\\}$

and there is a bigrading

$\displaystyle\mathbb{C}l(V)=\bigoplus_{r,s=-n}^{n}\mathbb{C}l^{r,s}(V).$

We state some facts regarding $\mathbb{C}l^{r,s}(V)$, the last of which we
state as a Theorem (All borne from Michelsohn’s article [M80])

  * •

$\mathbb{C}l^{r,s}(V)=\\{0\\}$ if $r,s>n$ or if $r,s<-n$

  * •

$\mathbb{C}l^{r,s}(V)=\\{0\\}$ for $r+s$ not congruent to
$n=\tfrac{1}{2}\dim(V)$ mod 2.

  * •

$J$ on $V$ induces $-J^{\vee}$ on $V^{\vee}$. Through
$Cl(V)\cong\Lambda(V^{\vee})$ we have
$J_{\textrm{der}}=-(J^{\vee})_{\textrm{der}}$

######  Theorem 2.6.

[M80] Through the isomorphism
$\mathbb{C}l(V)\cong\Lambda_{\mathbb{C}}(V^{\vee})$ induced by
complexification we have

$\displaystyle\bigoplus_{s=-n}^{n}\mathbb{C}l^{r,s}(V)\cong\bigoplus_{r=q-p}\Lambda_{\mathbb{C}}^{p,q}(V^{\vee}).$

Recall the operators $L,\Lambda,H$, defined on
$\Lambda_{\mathbb{C}}(V^{\vee})$ by

$L(\varphi)=\omega\wedge\varphi,\hskip 14.45377pt\Lambda=L^{*},\hskip
14.45377pt\textrm{and}\hskip
14.45377ptH(\varphi)=\sum_{p=0}^{2n}(n-p)\Pi_{p}(\varphi).$

Here $H=[\Lambda,L]$ and
$\Pi_{p}:\Lambda_{\mathbb{C}}(V^{\vee})\rightarrow\Lambda_{\mathbb{C}}^{p}(V^{\vee})$
is the usual projection.

######  Theorem 2.7.

[M80] Through the isomorphism
$\mathbb{C}l(V)\cong\Lambda_{\mathbb{C}}(V^{\vee})$ one has

$\mathcal{H}=i(\Lambda-L)\hskip
14.45377pt\mathcal{L}+\overline{\mathcal{L}}=\alpha H\hskip
14.45377pt\mathcal{L}-\overline{\mathcal{L}}=-i\alpha\big{(}\Lambda+L\big{)}$

where $\alpha$ is the antipodal involution.

######  Theorem 2.8.

[M80] Let $g=\textrm{exp}(\tfrac{-\pi
i}{\phantom{-}4}H)\textrm{exp}(\tfrac{\pi}{4}(\Lambda-L))\in
SL(2,\mathbb{C})$. Then we have the following identities on
$\mathbb{C}l(V)\cong\Lambda_{\mathbb{C}}(V^{\vee})$

$\mathcal{H}=gHg^{-1}\hskip 36.135pt\alpha\mathcal{L}=g\Lambda g^{-1}\hskip
21.68121pt\textrm{and }\alpha\overline{\mathcal{L}}=gLg^{-1}$

The operator $g$ she called the Hodge automorphism. By the previous Theorem we
also write

$g=\textrm{exp}\big{(}-\tfrac{\pi
i}{4}\alpha(\mathcal{L}+\overline{\mathcal{L}})\big{)}\textrm{exp}\big{(}-\tfrac{\pi
i}{4}\mathcal{H}\big{)}.$

######  Theorem 2.9.

The Hodge automorphism induces an isomorphism

$g^{-1}:\mathbb{C}l^{q-p,n-p-q}(V)\xrightarrow{\sim}\Lambda^{p,q}(V^{\vee}).$

###### Proof.

Suppose
$\varphi\in\mathbb{C}l^{r,s}(V)\subset\mathbb{C}l(V)\cong\Lambda_{\mathbb{C}}(V^{\vee})$
with $n\equiv r+s$ mod 2. $n-(r+s)=2p$ and $n-(s-r)=2q$ for some $p,q$. I.e.
$r=q-p$ and $s=n-p-q$. We have

$\begin{split}sg^{-1}\varphi&=g^{-1}\mathcal{H}\varphi=g^{-1}gHg^{-1}\varphi=Hg^{-1}\varphi\\\
rg^{-1}\varphi&=g^{-1}\mathcal{J}\varphi=\mathcal{J}g^{-1}\varphi=-\mathcal{J}^{\vee}g^{-1}\varphi\end{split}$

The result follows by observing

$\Lambda^{p,q}(V^{\vee})=\left\\{\psi\in\Lambda_{\mathbb{C}}(V^{\vee})\ |\
\mathcal{J}^{\vee}(\psi)=-r\psi\textrm{ and }H(\psi)=s\psi\right\\}.\qed$

######  Definition 2.10.

Let $M$ be a Riemannian manifold. The vector bundle $Cl(M)$ is the associated
bundle to the tangent bundle $TM$ with fibers $Cl(T_{x}(M))$ for every $x\in
M$. We define the complex Clifford bundle $\mathbb{C}l(M)$ by

$\mathbb{C}l(M)=Cl(M)\otimes\mathbb{C}.$

If $\nabla$ is an affine metric connection, $\nabla$ extends as a derivation
over $\Gamma(Cl(M))$ and $\Gamma\big{(}\Lambda(M))$.

###  2.11. Some Vector Valued 2-forms

Take $M$ to be an almost Hermitian manifold, i.e. $M$ is Riemannian with an
orthogonal $J$. We will be interested in affine metric connections $\nabla$ so
that $\nabla(J)=0$. Such connections are somewhat confusingly said to be
Hermitian.

The torsion of an affine connection $T=T_{\nabla}$ is given by

$T(X,Y)=\nabla_{X}(Y)-\nabla_{Y}(X)-[X,Y]$

and the Nijenhuis tensor, $N$, is defined as

$N(X,Y)=\tfrac{1}{4}([JX,JY]-J[JX,Y]-J[X,JY]-[X,Y]).$

###  2.12. The Riemannian Dirac Operator

######  Definition 2.13.

Let $v_{1},\dots,v_{2n}$ be a local orthonormal frame of $T(M)$ and let
$\widetilde{\nabla}$ be the Levi-Civita connection on $\Gamma(TM)$. We define
the Riemannian Dirac operator on any $\varphi\in\Gamma(Cl(M))$ by

$\widetilde{D}(\varphi)=\sum_{j}v_{j}\cdot\widetilde{\nabla}_{v_{j}}\varphi.$

Let $d$ be the exterior derivative on $\Omega(M)$.

######  Theorem 2.14.

Through the isomorphism $\Gamma(Cl(M))\cong\Omega(M)$ we have

$\widetilde{D}=d+d^{*}.$

Let $\pi_{p,q}:\Omega_{\mathbb{C}}(M)\rightarrow\Omega_{\mathbb{C}}^{p,q}(M)$
be the natural projection onto bidegree. Complex linearly extending the
exterior derivative $d$ to $\Omega_{\mathbb{C}}(M)$ we have a decomposition

$d=\partial+\overline{\mu}+\overline{\partial}+\mu$

where for $\displaystyle\varphi\in\Omega^{p,q}(M)$

$\displaystyle\partial\varphi=\pi_{p+1,q}\circ d(\varphi),\hskip
14.45377pt\overline{\partial}\varphi=\pi_{p,q+1}\circ d(\varphi),\hskip
14.45377pt\mu\varphi=\pi_{p+2,q-1}\circ d(\varphi),\hskip
14.45377pt\overline{\mu}\varphi=\pi_{p-1,q+2}\circ d(\varphi).$

Let $\delta=\partial+\overline{\mu}$ and
$\overline{\delta}=\overline{\partial}+\mu$ so that on the complexification
$\Gamma(\mathbb{C}l(M))\cong\Omega_{\mathbb{C}}(M)$ we have

$\widetilde{D}=\delta+\overline{\delta}+\delta^{*}+\overline{\delta}^{*}.$

######  Proposition 2.15.

[CW21] For an almost complex manifold $M$, the following relations among the
operators $\partial,\overline{\partial},\mu,\overline{\mu}$ along with their
adjoint identities obtain:

$\begin{split}\mu^{2}&=0\\\ \mu\partial+\partial\mu&=0\\\
\mu\overline{\partial}+\overline{\partial}\mu+\partial^{2}&=0\\\
\mu\overline{\mu}+\overline{\mu}\mu+\partial\overline{\partial}+\overline{\partial}\partial&=0\\\
\overline{\mu}\partial+\partial\overline{\mu}+\overline{\partial}^{2}&=0\\\
\overline{\mu}\overline{\partial}+\overline{\partial}\overline{\mu}&=0\\\
\overline{\mu}^{2}&=0.\end{split}$

##  3\. Almost Hermitian Dirac Identities

######  Definition 3.1.

Let $\nabla$ be a Hermitian connection. We define a Hermitian Dirac operator
on $\Gamma(Cl(M))$ by

$D(\varphi)=\sum_{j}v_{j}\cdot\nabla_{v_{j}}\varphi.$

For an operator $T$ on $\Gamma(Cl(M))\cong\Omega(M)$, we denote

$T_{c}\stackrel{{\scriptstyle\textrm{def}}}{{=}}J_{\textrm{alg}}^{-1}TJ_{\textrm{alg}}\hskip
7.22743pt\textrm{ and }T^{\text{\char
73}}\stackrel{{\scriptstyle\textrm{def}}}{{=}}\text{\char 73}T\text{\char
73}.$

On $\mathbb{C}l(M)$

$\mathcal{L}_{c}=\overline{\mathcal{L}},\hskip
10.84006pt\mathcal{H}_{c}=\mathcal{H},\hskip
10.84006pt\mathcal{J}_{c}=\mathcal{J},\hskip
10.84006pt\mathcal{J}^{\text{\char 73}}=\mathcal{J},\hskip
10.84006pt\mathcal{H}^{\text{\char 73}}=-\mathcal{H},\hskip
10.84006pt\mathcal{L}^{\text{\char 73}}=\overline{\mathcal{L}}.$

######  Theorem 3.2.

Let $D$ be a Hermitian Dirac operator. Then on any almost Hermitian manifold
$M$ we have the following Almost Hermitian Dirac identities on
$\Gamma(\mathbb{C}l(M))$

$\begin{split}[D,\mathcal{H}]=\phantom{-}[D,\mathcal{J}]\ \ =-iD_{c}\hskip
14.45377pt&\hskip
14.45377pt[D_{c},\mathcal{H}]=\phantom{-}[D_{c},\mathcal{J}]=\phantom{-}iD\\\
[D^{\text{\char 73}},\mathcal{H}]=-[D^{\text{\char
73}},\mathcal{J}]=\phantom{-}iD_{c}^{\text{\char 73}}\hskip 14.45377pt&\hskip
14.45377pt[D_{c}^{\text{\char 73}},\mathcal{H}]=-[D_{c}^{\text{\char
73}},\mathcal{J}]=-iD^{\text{\char 73}}.\end{split}$

###### Proof.

For any $\varphi\in\Gamma(\mathbb{C}l(M))$, we have

$[D,\mathcal{H}]\varphi=D(\omega_{0}\cdot\varphi)+D(\varphi\cdot\omega_{0})-\omega_{0}\cdot
D(\varphi)-D(\varphi)\cdot\omega_{0}.$

As $\nabla$ is metric and $J$ parallel, we have $\nabla(\omega_{0})=0$. Using
that $\nabla$ is a derivation, and that

$\omega_{0}\cdot v_{j}-v_{j}\cdot\omega_{0}=-iJv_{j}$

we obtain

$\begin{split}&D(\omega_{0}\cdot\varphi)=\sum_{j}e_{j}\cdot\omega_{0}\cdot\nabla_{e_{j}}(\varphi)+\sum_{j}Je_{j}\cdot\omega_{0}\cdot\nabla_{Je_{j}}(\varphi),\hskip
7.22743pt\textrm{ and similarly}\\\
&D(\varphi\cdot\omega_{0})=\sum_{j}e_{j}\cdot\nabla_{e_{j}}(\varphi)\cdot\omega_{0}+\sum_{j}Je_{j}\cdot\nabla_{Je_{j}}(\varphi)\cdot\omega_{0}=D(\varphi)\cdot\omega_{0}\end{split}$

which gives the first identity. The remaining identities are obtained by
conjugating with the operators ✩ and $J_{\textrm{alg}}$. ∎

######  Remark 3.3.

Recall
$J_{\textrm{der}}\stackrel{{\scriptstyle\textrm{def}}}{{=}}i\mathcal{J}$ is
the usual extension of $J$ to a derivation on the bundles
$\mathbb{C}l(M)\cong\Lambda_{\mathbb{C}}(M)$. We observe that
$[d,J_{\textrm{der}}]=d_{c}$ if and only if the manifold $M$ is complex.

One can check this by noting that

$[\mu,J_{\textrm{der}}]=-3i\mu\hskip 7.22743pt\textrm{ while }\hskip
7.22743ptJ_{\textrm{alg}}^{-1}\mu J_{\textrm{alg}}=\phantom{-}i\mu.$

Similarly

$[\overline{\mu},J_{\textrm{der}}]=\phantom{-}3i\overline{\mu}\hskip
7.22743pt\textrm{ while }\hskip
7.22743ptJ_{\textrm{alg}}^{-1}\overline{\mu}J_{\textrm{alg}}=-i\overline{\mu}.$

The expressions

$[\partial,J_{\textrm{der}}]=-i\partial=J_{\textrm{alg}}^{-1}\partial
J_{\textrm{alg}}\hskip 7.22743pt\textrm{ and }\hskip
7.22743pt[\overline{\partial},J_{\textrm{der}}]=i\overline{\partial}=J_{\textrm{alg}}^{-1}\overline{\partial}J_{\textrm{alg}}$

agree in general.

Contrarily, we’ve observed above that $[D,J_{\textrm{der}}]=D_{c}$ for any
almost-Hermitian manifold.

##  4\. The Canonical Hermitian Connections

Let $M$ be an almost Hermitian manifold of dimension $2n$. We denote by
$\Omega^{2}(TM)$ the space of vector valued $2$ forms on $M$. That is

$\Omega^{2}(TM)\stackrel{{\scriptstyle\textrm{def}}}{{=}}\Gamma\big{(}\textrm{Hom}(\Lambda^{2}(M),TM)\big{)}.$

This space has many important elements, including the torsion of a connection
and the Nijenhuis tensor, $N$. It is a well known theorem of Newlander and
Nirenberg [NN57] that an almost complex manifold is complex if and only if
$N=0$. We observe the following less well known fact, cf. [CW21],

######  Lemma 4.1.

The Nijenhuis tensor, $N$, complex bilinearly extended, is the dual of
$\mu+\overline{\mu}$ on 1-forms. That is $\mu+\overline{\mu}=N^{\vee}$.

The space $\Omega^{2}(TM)$ is often identified through the metric isomorphism
$TM\cong TM^{\vee}$ with the space of 3 tensors on $M$ which are skew-
symmetric in the last two entries. More explicitly if $\phi\in\Omega^{2}(TM)$
then for any $Y,Z\in TM$ we have $\phi(Y,Z)\in TM$. Through the metric
isomorphism $TM\cong TM^{\vee}$ we have
$\phi(Y,Z)\cong\langle\cdot,\phi(Y,Z)\rangle$. We identify

$\phi(X,Y,Z)=\langle X,\phi(Y,Z)\rangle.$

In particular, this identification through the metric allows us to view other
important objects as elements of $\Omega^{2}(TM)$. One special element of
interest in almost Hermitian geometry is the 3-form $d\omega(X,Y,Z)$ which
vanishes if and only if $M$ is almost Kähler. Another is
$(\widetilde{\nabla}\omega)(X,Y,Z)=\widetilde{\nabla}_{X}\omega(Y,Z)$ which
can be rewritten in the following form

$(\widetilde{\nabla}_{X}\omega)(Y,Z)=\left\langle(\widetilde{\nabla}_{X}J)Y,Z\right\rangle.$

There are two distinguished projections of $\Omega^{2}(TM)$ onto differential
forms of degree 1 and 3. There is the map
$\mathcal{P}:\Omega^{2}(TM)\rightarrow\Omega^{3}(M)$ defined by

$\mathcal{P}(\phi)(X,Y,Z)=\tfrac{1}{3}\big{(}\phi(X,Y,Z)+\phi(Y,Z,X)+\phi(Z,X,Y)\big{)}\hskip
50.58878pt$

and the map $r:\Omega^{2}(TM)\rightarrow\Omega^{1}(M)$ defined by

$r(\phi)(X)=\sum_{j}\phi(v_{j},v_{j},X)$

In fact, [G97] showed that the space $\Omega^{2}(TM)$ canonically decomposes
as

$\Omega^{2}(TM)\cong\Omega^{1}(M)\oplus\Omega^{3}(M)\oplus\big{(}\ker(r)\cap\ker(\mathcal{P})\big{)}.$

(See Proposition 8.21 for a proof.) We write the component of
$\phi\in\Omega^{2}(TM)$ both the kernel of $r$ and $\mathcal{P}$ as
$\phi_{0}$.

The space of 3-forms $\Omega^{3}(M)$ is decomposed into

$\ \ E^{+}=\textrm{Re}\big{(}\Omega^{2,1}(M)\oplus\Omega^{1,2}(M)\big{)},$
$E^{-}=\textrm{Re}\big{(}\Omega^{3,0}(M)\oplus\Omega^{0,3}(M)\big{)}$

so that $\Omega^{3}(M)=E^{-}\oplus E^{+}.$ For a 3-form
$\varphi\in\Omega^{3}(M)$ we set $\varphi^{+}$ and $\varphi^{-}$ to be its
components in $E^{+}$ and $E^{-}$ respectively.

######  Lemma 4.2.

Let $N$ be the Nijenhuis tensor. Then $r(N)=0$ and so
$N=\mathcal{P}(N)+N_{0}.$

###### Proof.

Let $v_{1},\dots,v_{2n}=e_{1},\dots,e_{n},Je_{1},\dots,Je_{n}$ be a $J$
adapted local orthonormal frame. Then for any $X\in TM$ we have

$r(N)(X)=\sum_{j=1}^{2n}N(v_{j},v_{j},X)=\sum_{j=1}^{2n}\langle
v_{j},N(v_{j},X)\rangle$

But

$\sum_{j=1}^{2n}\langle v_{j},N(v_{j},X)=\sum_{j=1}^{n}\langle
e_{j},N(e_{j},X)\rangle+\sum_{j=1}^{n}\langle Je_{j},N(Je_{j},X)\rangle$

Using that $N(Jv,w)=-JN(v,w)$ and that $J$ is orthogonal we obtain

$\sum_{j=1}^{n}\langle e_{j},N(e_{j},X)\rangle+\sum_{j=1}^{n}\langle
Je_{j},N(Je_{j},X)\rangle=\sum_{j=1}^{n}\langle
e_{j},N(e_{j},X)\rangle-\sum_{j=1}^{n}\langle e_{j},N(e_{j},X)\rangle=0.\qed$

Perhaps the most important element of $\Omega^{2}(TM)$, for our purposes, is
the potential $A^{\nabla}$ of an affine metric connection $\nabla$, defined by

$A^{\nabla}(X,Y,Z)=\big{\langle}\nabla_{X}(Y)-\widetilde{\nabla}_{X}(Y),Z\big{\rangle}.$

We shall drop the superscript $\nabla$ of the potential and torsion when the
referent connection is clear.

$A^{\nabla}$ is related to the torsion $T^{\nabla}$ by the following identity
(See Proposition 8.22)

$A^{\nabla}+T^{\nabla}=3\mathcal{P}(A^{\nabla})=\tfrac{3}{2}\mathcal{P}(T^{\nabla})$

Gauduchon [G97] defined an affine line of ‘canonical’ Hermitian connections on
an almost Hermitian manifold. This is the set of Hermitian connections,
$\nabla^{t}$, uniquely defined by their torsion $T^{t}=T^{\nabla^{t}}$
satisfying

(1)
$T^{t}=N+\tfrac{3t-1}{4}d_{c}\omega^{+}-\tfrac{t+1}{4}\mathcal{M}(d_{c}\omega^{+}).$

Where $\mathcal{M}$ is the operator on $\Omega^{2}(TM)$ defined by

$\mathcal{M}(\phi)(X,Y,Z)=\phi(X,JY,JZ).$

For example, a natural choice of Hermitian connection is the Chern connection
$\nabla^{\textrm{Ch}}$ characterized on a Hermitian manifold by the projection
onto the $(0,1)$ component agreeing with the Dolbeault Operator. It is
obtained by setting $t=1$.

In the case that $M$ is a Kähler manifold, we have the well known identity
$\nabla^{\textrm{ch}}=\widetilde{\nabla}$, which is true for all $\nabla^{t}$.
Moreover all of the canonical Hermitian connections agree in the case that
$d\omega=0$, despite being distinct from the Levi-Civita connection.

Another natural choice is the Bismut connection
$\nabla^{\textrm{Bm}}=\nabla^{-1}$ characterized on a Hermitian manifold by
its torsion – viewed as a $3$-tensor – being totally skew symmetric. For all
$t\in\mathbb{R}$ one has

$\nabla^{t}=\tfrac{1+t}{2}\nabla^{1}+\tfrac{1-t}{2}\nabla^{-1}.$

We denote the potential of a canonical Hermitian connection by $A^{t}$. By the
above relation between the torsion and the potential we observe

$A^{t}=-T^{t}+\tfrac{3}{2}\mathcal{P}T^{t}.$

We then have

######  Proposition 4.3.

[G97] Let $M$ be an almost Hermitian manifold and let $\nabla^{t}$ be a
canonical Hermitian connection in the sense of Gauduchon. Then

$A^{t}=-N+\tfrac{3}{2}\mathcal{P}N+\tfrac{t-1}{4}d_{c}\omega^{+}+\tfrac{t+1}{4}\mathcal{M}(d_{c}\omega^{+}).$

##  5\. The Canonical Hermitian Dirac Operators

######  Definition 5.1.

Given a local orthonormal frame $v_{1},\dots,v_{2n}$ of $TM$ and a
$t\in\mathbb{R}$ we define the canonical Hermitian Dirac operator $D_{t}$ by

$D_{t}=\sum_{j}v_{j}\cdot\nabla^{t}_{v_{j}}.$

For any $v\in TM$ and any $\varphi\in\Gamma(Cl(M))$ we define

$a^{t}_{v}(\varphi)\stackrel{{\scriptstyle\textrm{def}}}{{=}}\nabla^{t}_{v}(\varphi)-\widetilde{\nabla}_{v}(\varphi).$

Evidently $a^{t}_{v}$ is a derivation, and for vector fields $X,Y,Z$ we have
that, by definition

$A^{t}(X,Y,Z)=\left\langle a^{t}_{X}(Y),Z\right\rangle.$

For the time being, we suppress the parameter $t$, taking $A=A^{t}$ and
$a=a^{t}$.

We define the operator $D_{A}$ on $\Gamma(Cl(M))$ by

$D_{A}=\sum_{j}v_{j}\cdot a_{v_{j}}$

so that $D_{t}=\widetilde{D}+D_{A}$. The 1-form $r(A)$ is given, for any $v\in
TM$, by

$r(A)(v)=\sum_{j}A(v_{j},v_{j},v)=\sum_{j}\langle a_{v_{j}}(v_{j}),v\rangle$

so that $r(A)^{\sharp}=\sum_{j}a_{v_{j}}(v_{j})$. Lastly, for any vector $v\in
TM$ we define $L_{v}$ on $Cl(M)$ by

$L_{v}(\varphi)=v\cdot\varphi$

and we will write $L_{r(A)}$ to denote $L_{r(A)^{\sharp}}$.

###  5.2. A Result Concerning Self-Adjointness of the Canonical Hermitian
Dirac Operators

Since both $\nabla^{t}$ and $\widetilde{\nabla}$ are metric connections we
have, for any $v\in TM$, that the adjoint $a_{v}^{*}=-a_{v}$. Furthermore for
a unital vector $v$ we have $L_{v}$ is an isometry.

######  Proposition 5.3.

On any almost Hermitian manifold we have $D_{A}^{*}=D_{A}+L_{r(A)}.$

###### Proof.

For any $\varphi,\psi\in\Gamma(Cl(M))$ we have

$\begin{split}\langle
D_{A}(\varphi),\psi\rangle=\left\langle\sum_{j}v_{j}\cdot
a_{v_{j}}(\varphi),\psi\right\rangle&=-\sum_{j}\left\langle
a_{v_{j}}\varphi,v_{j}\cdot\psi\right\rangle\\\
&=\phantom{-}\sum_{j}\left\langle\varphi,a_{v_{j}}(v_{j}\cdot\psi)\right\rangle\\\
&=\phantom{-}\sum_{j}\left\langle\varphi,a_{v_{j}}(v_{j})\cdot\psi\right\rangle+\left\langle\varphi,v_{j}\cdot
a_{v_{j}}\psi\right\rangle\\\
&=\phantom{-}\left\langle\varphi,(D_{A}+L_{r(A)})(\psi)\right\rangle.\qed\end{split}$

We observe that
$r(A)=r\big{(}-N+\tfrac{3}{2}\mathcal{P}N+\tfrac{t-1}{4}d_{c}\omega^{+}+\tfrac{t+1}{4}\mathcal{M}(d_{c}\omega^{+})\big{)}$.
Since $r(N)=0$ by Section 4, and $r$ vanishes on any 3-form we obtain

(2) $r(A)=\tfrac{t+1}{4}r\big{(}\mathcal{M}(d_{c}\omega^{+})\big{)}.$

######  Definition 5.4.

We define the Lee form to be the $1$-form $\theta=\Lambda(d\omega)$.

Observe that, for bidegree reasons,
$\Lambda(\mu\omega)=\Lambda(\overline{\mu}\omega)=0$ on
$\Omega_{\mathbb{C}}(M)$, so that $\theta=\Lambda(d\omega^{+})$. Furthermore,
it is well known [G97] that $\theta=Jd^{*}\omega=\Lambda(d\omega)$ .

######  Lemma 5.5.

$r\big{(}\mathcal{M}(d_{c}\omega^{+})\big{)}=2\theta.$

###### Proof.

Let $v_{1},\dots,v_{2n}=e_{1},\dots,e_{n},Je_{1},\dots Je_{n}$ be a
$J$-adapted local orthonormal frame. For any $X\in TM$ we have

$\begin{split}r\big{(}\mathcal{M}(d_{c}\omega^{+})\big{)}(X)&=\phantom{-2}\sum_{j=1}^{2n}\mathcal{M}(d_{c}\omega^{+})(v_{j},v_{j},X)=\phantom{.}\sum_{j=1}^{2n}(d_{c}\omega)^{+}(v_{j},Jv_{j},JX)\\\
&=\phantom{-2}\sum_{j=1}^{n}(d_{c}\omega)^{+}(e_{j},Je_{j},JX)-\phantom{2}\sum_{j=1}^{n}(d_{c}\omega)^{+}(Je_{j},e_{j},JX)\\\
&=\phantom{-}2\sum_{j=1}^{n}(d_{c}\omega)^{+}(e_{j},Je_{j},JX)=-2\sum_{j=1}^{n}(d\omega)^{+}(Je_{j},e_{j},X)\\\
&=-2\big{(}\omega\lrcorner\ d\omega^{+}(X)\big{)}\hskip 7.22743pt=\hskip
7.22743pt2\Lambda(d\omega^{+})(X)\hskip 7.22743pt=\hskip
7.22743pt2\theta(X).\qed\end{split}$

Hence, by the identity (2) we have that

$r(A)=\tfrac{t+1}{2}\theta.$

Together with Proposition 5.3 we obtain

$D_{t}^{*}-\widetilde{D}^{*}=D_{A}^{*}=D_{A}+L_{r(A)}=D_{t}-\widetilde{D}+\tfrac{t+1}{2}L_{\theta}.$

As the Riemannian Dirac operator $\widetilde{D}$ is self-adjoint we conclude
the

######  Proposition 5.6.

For any $t\in\mathbb{R}$ the adjoint of the canonical Hermitian Dirac operator
$D_{t}$ is given by

$D_{t}^{*}=D_{t}+\tfrac{t+1}{2}L_{\theta}.$

Following [M82] we recall a condition on the metric determined by requiring
$d^{*}\omega=0$. Such metrics are said to be ‘balanced’. We note that the
balanced condition obtains if and only if $Jd^{*}\omega=\theta=0$. Thus
Proposition 5.6 implies the following

######  Corollary 5.7.

For all $t\neq-1$ we have $D_{t}$ is self-adjoint if and only if the metric is
balanced.

For $t=-1$ the Dirac operator $D_{-1}$ is self-adjoint for any almost
Hermitian manifold, sans any requirement on the metric. This directly
corresponds to an observation made by Bismut [B89], [G97] regarding Hermitian
Dirac operators on the spinor bundle induced by the connection $\nabla^{-1}$
being self-adjoint. Presently, the connection $\nabla^{-1}$ is widely referred
to as the Bismut connection. We define

$B\stackrel{{\scriptstyle\textrm{def}}}{{=}}D_{-1}.$

We also define the operator $\mathfrak{d}_{t}$ on
$\Gamma\big{(}\mathbb{C}l(M)\big{)}$ by

(3)
$\mathfrak{d}_{t}\stackrel{{\scriptstyle\textrm{def}}}{{=}}\tfrac{1}{2}\big{(}D_{t}+i(D_{t})_{c}\big{)}=2\sum_{j}\epsilon_{j}\cdot\nabla^{t}_{\overline{\epsilon}_{j}}$

We observe (cf. [M80]) that the operator $\mathfrak{d}_{t}$ is of Clifford
bidegree $(1,1)$ as left Clifford multiplication by an element of $T^{1,0}(M)$
is of Clifford bidegree $(1,1)$ and any Hermitian connection is both
$\mathcal{H}$ and $\mathcal{J}$ parallel.

In her seminal paper [M82] Michelsohn showed that for a Hermitian manifold $M$
the operator $\mathfrak{d}_{1}$ has adjoint $\overline{\mathfrak{d}_{1}}$ if
and only if the metric is balanced (Proposition 2.3 in that paper). We extend
this result to the almost Hermitian setting and to all $\mathfrak{d}_{t}$. By
the above

$\mathfrak{d}_{t}^{*}=\tfrac{1}{2}\left(D_{t}^{*}-i(D_{t})_{c}^{*}\right)=\overline{\mathfrak{d}_{t}}+\tfrac{t+1}{4}L_{\small\tfrac{\theta+iJ\theta}{2}.\normalsize}$

Since $\tfrac{1}{2}(\theta+iJ\theta)$ is the projection of $\theta\in
T_{\mathbb{C}}(M)$ to $T^{0,1}(M)$ we conclude

######  Proposition 5.8.

Let $M$ be a compact almost Hermitian manifold. The operator
$\mathfrak{B}\stackrel{{\scriptstyle\textrm{def}}}{{=}}\mathfrak{d}_{-1}$ is
conjugate self-adjoint on $\Gamma(\mathbb{C}l(M))$. Moreover, for all
$t\neq-1$ $\mathfrak{d}_{t}$ is conjugate self adjoint if and only if the
metric is balanced.

###  5.9. Bochner Identities

Using only that $\nabla^{t}$ is Hermitian, and the local expression (3) of
$\mathfrak{d}_{t}$ via a local frame of $T_{\mathbb{C}}(M)$ we have

$\tfrac{1}{4}\mathfrak{d}_{t}^{2}=\sum_{j,k}\epsilon_{j}\cdot\epsilon_{k}\cdot\big{(}\nabla^{t}_{\overline{\epsilon}_{j}}\nabla^{t}_{\overline{\epsilon}_{k}}-\nabla^{t}_{\nabla^{t}_{\overline{\epsilon}_{j}}\overline{\epsilon}_{k}}\big{)}$

which we rewrite as

$\tfrac{1}{4}\mathfrak{d}_{t}^{2}=\sum_{j<k}\epsilon_{j}\cdot\epsilon_{k}\cdot\big{(}\nabla^{t}_{\overline{\epsilon}_{j},\overline{\epsilon}_{k}}-\nabla^{t}_{\overline{\epsilon}_{k},\overline{\epsilon}_{j}}\big{)}$

where

$\nabla_{X,Y}\stackrel{{\scriptstyle\textrm{def}}}{{=}}\nabla_{X}\nabla_{Y}-\nabla_{\nabla_{X}Y}$

is the second covariant derivative with respect to an affine connection
$\nabla$. The curvature tensor $R^{\nabla}$ defined by

$R^{\nabla}(X,Y)=[\nabla_{X},\nabla_{Y}]-\nabla_{[X,Y]}$

is related to the second covariant derivative and the torsion of $\nabla$ by

$R^{\nabla}(X,Y)=\nabla_{X,Y}-\nabla_{Y,X}-\nabla_{T(X,Y)}$

so that

$\tfrac{1}{4}\mathfrak{d}_{t}^{2}=\sum_{j<k}\epsilon_{j}\cdot\epsilon_{k}\cdot\big{(}R^{t}(\overline{\epsilon}_{j},\overline{\epsilon}_{k})+\nabla^{t}_{T^{t}(\overline{\epsilon}_{j},\overline{\epsilon}_{k})}\big{)}.$

Complex bilinearly extending the metric, we observe

$T^{t}(\overline{\epsilon}_{j},\overline{\epsilon}_{k})=\sum_{i}T^{t}(\epsilon_{i},\overline{\epsilon}_{j},\overline{\epsilon}_{k})\overline{\epsilon}_{i}+T^{t}(\overline{\epsilon}_{i},\overline{\epsilon}_{j},\overline{\epsilon}_{k})\epsilon_{i}$

so that by the definition of a canonical Hermitian connection (1)

$T^{t}(\overline{\epsilon}_{j},\overline{\epsilon}_{k})=\sum_{i}N(\overline{\epsilon}_{i},\overline{\epsilon}_{j},\overline{\epsilon}_{k})\epsilon_{i}+\tfrac{3t-1}{4}d_{c}\omega^{+}(\epsilon_{i},\overline{\epsilon}_{j},\overline{\epsilon}_{k})\overline{\epsilon}_{i}-\tfrac{t+1}{4}\mathcal{M}(d_{c}\omega^{+})(\epsilon_{i},\overline{\epsilon}_{j},\overline{\epsilon}_{k})\overline{\epsilon}_{i}$

where we use that $N$ sends elements of $\Lambda^{0,2}TM$ to $T^{1,0}M$ and
that $d_{c}\omega^{+}$ vanishes on sections of
$\Lambda^{3,0}TM\oplus\Lambda^{0,3}TM$ (by definition of the $+$ component of
a 3-form). Furthermore, by definition of $\mathcal{M}$ we observe

$\mathcal{M}(d_{c}\omega^{+})(\epsilon_{i},\overline{\epsilon}_{j},\overline{\epsilon}_{k})=d_{c}\omega^{+}(\epsilon_{i},J\overline{\epsilon}_{j},J\overline{\epsilon}_{k})=-d_{c}\omega^{+}(\epsilon_{i},\overline{\epsilon}_{j},\overline{\epsilon}_{k})$

and thus

$T^{t}(\overline{\epsilon}_{j},\overline{\epsilon}_{k})=\sum_{i}N(\overline{\epsilon}_{i},\overline{\epsilon}_{j},\overline{\epsilon}_{k})\epsilon_{i}+td_{c}\omega^{+}(\epsilon_{i},\overline{\epsilon}_{j},\overline{\epsilon}_{k})\overline{\epsilon}_{i}.$

We conclude

######  Proposition 5.10.

Let $M$ be an almost Hermitian manifold. On $\Gamma(\mathbb{C}l(M))$ we have

$\tfrac{1}{4}\mathfrak{d}_{t}^{2}=\sum_{j<k}\epsilon_{j}\cdot\epsilon_{k}\cdot
R^{t}(\overline{\epsilon}_{j},\overline{\epsilon}_{k})+\epsilon_{j}\cdot\epsilon_{k}\cdot\big{(}\sum_{i}N(\overline{\epsilon}_{i},\overline{\epsilon}_{j},\overline{\epsilon}_{k})\nabla^{t}_{\epsilon_{i}}+td_{c}\omega^{+}(\epsilon_{i},\overline{\epsilon}_{j},\overline{\epsilon}_{k})\nabla^{t}_{\overline{\epsilon}_{i}}\big{)}.$

Let $t=1$ and $N=0$, the curvature tensor $R=R^{1}$ of the Chern connection is
of type (1,1) and hence $R(\overline{\epsilon}_{j},\overline{\epsilon}_{k})=0$
for all $j,k$. In this case, the operator $\mathfrak{d}_{1}$ is a differential
if and only if $d_{c}\omega=d_{c}\omega^{+}=0$, i.e. if and only if $M$ is
Kähler. This was first shown by Michelsohn ([M82] Proposition 2.1). Evidently
if $M$ is Kähler then $\mathfrak{d}_{t}=\mathfrak{d}_{1}$ and so
$\mathfrak{d}_{t}$ is a differential.

A more involved, but completely analogous computation to the above reveals

$\mathfrak{d}_{t}\overline{\mathfrak{d}_{t}}+\overline{\mathfrak{d}_{t}}\mathfrak{d}_{t}=-\sum_{j}\nabla^{t}_{\overline{\epsilon}_{j},\epsilon_{j}}+\sum_{j,k}\overline{\epsilon}_{k}\cdot\epsilon_{j}\big{(}R^{t}(\epsilon_{k},\overline{\epsilon}_{j})-\nabla^{t}_{T^{t}(\epsilon_{k},\overline{\epsilon}_{j})}\big{)}.$

Following Michelsohn [M80] we define the operators

$\bm{\nabla^{*}\nabla}_{t}=-\sum_{j}\nabla^{t}_{\overline{\epsilon}_{j},\epsilon_{j}}\textrm{
and }\mathcal{R}_{t}=\sum_{j,k}\overline{\epsilon}_{k}\cdot\epsilon_{j}\cdot
R^{t}(\epsilon_{k},\overline{\epsilon}_{j})$

so that

$\mathfrak{d}_{t}\overline{\mathfrak{d}_{t}}+\overline{\mathfrak{d}_{t}}\mathfrak{d}_{t}=\bm{\nabla^{*}\nabla}_{t}+\mathcal{R}_{t}-\tfrac{t-1}{2}\sum_{i,j,k}\overline{\epsilon}_{k}\cdot\epsilon_{j}\cdot\big{(}d_{c}\omega^{+}(\epsilon_{i},\epsilon_{k},\overline{\epsilon}_{j})\nabla^{t}_{\overline{\epsilon}_{i}}+d_{c}\omega^{+}(\overline{\epsilon}_{i},\epsilon_{k},\overline{\epsilon}_{j})\nabla^{t}_{\epsilon_{i}}\big{)}.$

For $t=-1$ this produces an elegant description of the Laplacian of
$\mathfrak{B}$ over a compact almost Hermitian manifold, while for $t=1$ the
terms involving $d_{c}\omega^{+}$ vanish, yielding Michelsohn’s Bochner
identity in [M80] in the almost Hermitian setting.

###  5.11. The Canonical Operators on Forms

Proceeding with our study of canonical Hermitian Dirac operators, we begin
with the following

######  Definition 5.12.

Let $v_{1},\dots,v_{2n}$ be a local orthonormal frame. We define the operator
$d_{A}$ on $\Omega(M)$ by

$d_{A}=\sum_{j}v_{j}\wedge a_{v_{j}}.$

Since for any $v\in TM$ we have $a_{v}$ is a derivation on $\Omega(M)$ it
follows that $d_{A}$ is a graded derivation on $\Omega(M)$.

Furthermore, we observe for any $Y\in\Gamma(TM)\cong\Omega^{1}(M)$

$d_{A}(Y)=\sum_{j}v_{j}\wedge a_{v_{j}}(Y)=\sum_{j,k}\langle
a_{v_{j}}(Y),v_{k}\rangle v_{j}\wedge
v_{k}=\sum_{j,k}A(v_{j},Y,v_{k})v_{j}\wedge v_{k}\hskip 43.36243pt$

and so by Proposition 4.3 we see that

(4)
$d_{A}(Y)=\sum_{j,k}\Big{(}\textrm{-}(N-\tfrac{3}{2}\mathcal{P}N)(v_{j},Y,v_{k})+\tfrac{t-1}{4}d_{c}\omega^{+}(v_{j},Y,v_{k})+\tfrac{t+1}{4}\mathcal{M}(d_{c}\omega^{+})(v_{j},Y,v_{k})\Big{)}v_{j}\wedge
v_{k}.$

With regard to the adjoint of $d_{A}$ we have the following

######  Lemma 5.13.

Let $M$ be an almost Hermitian manifold. For any $\varphi\in\Omega(M)$ the
adjoint of $d_{A}$ is given by

$d_{A}^{*}(\varphi)=-r(A)\lrcorner\ \varphi-\sum_{j}v_{j}\lrcorner\
a_{v_{j}}(\varphi).$

###### Proof.

Let $\varphi\in\Omega(M)$. As interior multiplication by a vector is the
adjoint of exterior multiplication, and $a_{v}$ is anti-self-adjoint we have
that

$d_{A}^{*}(\varphi)=-\sum_{j}a_{v_{j}}(v_{j}\lrcorner\ \varphi).$

Using that, for any $v\in TM$, $a_{v}$ is a $C^{\infty}$-linear derivation, it
is also a derivation over interior multiplication by vectors. That is

$a_{v}(v\lrcorner\ \varphi)=v\lrcorner\ a_{v}(\varphi)+a_{v}(v)\lrcorner\
\varphi.$

Hence

$\sum_{j}a_{v_{j}}(v_{j}\lrcorner\ \varphi)=\sum_{j}v_{j}\lrcorner\
a_{v_{j}}(\varphi)+\sum_{j}a_{v_{j}}(v_{j})\lrcorner\ \varphi$

and the result follows. ∎

######  Proposition 5.14.

For any $\varphi\in\Gamma(Cl(M))\cong\Omega(M)$ we have

$D_{t}(\varphi)-\widetilde{D}(\varphi)=D_{A}(\varphi)\cong
d_{A}(\varphi)+d_{A}^{*}(\varphi)+r(A)\lrcorner\ \varphi.$

###### Proof.

For any $\varphi\in\Gamma(Cl(M))\cong\Omega(M)$ we have

$D_{A}(\varphi)=\sum_{j}v_{j}\cdot a_{v_{j}}(\varphi)\cong\sum_{j}v_{j}\wedge
a_{v_{j}}(\varphi)-\sum_{j}v_{j}\lrcorner\
a_{v_{j}}(\varphi)=d_{A}(\varphi)+d_{A}^{*}(\varphi)+r(A)\lrcorner\
\varphi.\qed$

We now define a host of operators on forms. Most of these operators will
likely be familiar to the reader, with one exception. We begin by defining the
operator $\lambda^{+}:\Omega(M)\rightarrow\Omega(M)$ for any
$\varphi\in\Omega(M)$ by

$\lambda^{+}(\varphi)=d\omega^{+}\wedge\varphi$

and the operator $\tau^{+}$ on $\Omega(M)$ by

$\tau^{+}=[\Lambda,\lambda^{+}].$

Lastly we define the novel operator $\rho^{+}:\Omega(M)\rightarrow\Omega(M)$,
for any $\varphi=\varphi_{1}\wedge\cdots\wedge\varphi_{k}\in\Omega^{k}(M)$, by

$\rho^{+}(\varphi)=\sum_{j=1}^{k}(-1)^{j}\varphi_{1}\wedge\cdots\wedge(\varphi_{j}\lrcorner\
d\omega^{+})\wedge\cdots\wedge\varphi_{k}.$

Note that, as usual, by $\varphi_{j}\lrcorner\ d\omega$ we mean
$\varphi_{j}^{\sharp}\lrcorner\ d\omega$ where $\sharp:T^{\vee}(M)\rightarrow
T(M)$ is the metric isomorphism. Letting $v_{1},\dots,v_{2n}$ be a $J$-adapted
local orthornormal frame of $TM$, we can also express

$\rho^{+}(\varphi)=-\sum_{j}(v_{j}\lrcorner\
d\omega^{+})\wedge(v_{j}\lrcorner\ \varphi)\hskip 14.45377pt\textrm{ for any
}\varphi\in\Omega(M).$

Complex linearly extending $\rho^{+}$ we restrict to components of
$d\omega^{+}=\partial\omega+\overline{\partial}\omega$ and define for
$\varphi=\varphi_{1}\wedge\cdots\wedge\varphi_{k}\in\Omega_{\mathbb{C}}^{p,q}(M)\subset\Omega_{\mathbb{C}}^{k}(M)$

$\rho_{\gamma}(\varphi)=\sum_{j=1}^{k}(-1)^{j}\varphi_{1}\wedge\cdots\wedge\left(\varphi_{j}\lrcorner\
\gamma\omega\right)\wedge\cdots\wedge\varphi_{k}\textrm{ for
}\gamma=\partial,\overline{\partial}.$

In the complexification by $\varphi_{j}\lrcorner\ d\omega$ we mean
$\overline{\varphi_{j}^{\sharp}}\lrcorner\ d\omega$. We also define the
complex linearly extended operators $\lambda_{\gamma}$ and $\tau_{\gamma}$ for
$\gamma=\partial,\overline{\partial}$ by

$\lambda_{\gamma}(\varphi)\stackrel{{\scriptstyle\textrm{def}}}{{=}}\gamma(\omega)\wedge\varphi\hskip
36.135pt\textrm{ and }\hskip
36.135pt\tau_{\gamma}(\varphi)\stackrel{{\scriptstyle\textrm{def}}}{{=}}[\Lambda,\lambda_{\gamma}]\varphi.$

It is quick to check that $\rho_{\partial},\ \rho_{\overline{\partial}}$ have
bidegree $(1,0)$ and $(0,1)$ respectively, and that
$\rho_{\overline{\partial}}=\overline{\rho_{\partial}}$. Moreover, for
$\varphi\in\Omega^{p,q}(M)$ we have

$J_{\textrm{alg}}^{-1}\rho_{\partial}J_{\textrm{alg}}(\varphi)=\frac{i^{p-q}}{i^{p+1-q}}\rho_{\partial}(\varphi)=-i\rho_{\partial}(\varphi)\hskip
7.22743pt\textrm{ and }\hskip
28.90755ptJ_{\textrm{alg}}^{-1}\overline{\rho_{\partial}}J_{\textrm{alg}}(\varphi)=i\overline{\rho_{\partial}}(\varphi).\hskip
50.58878pt$

Note that as operators on $\Omega_{\mathbb{C}}(M)$ we have

$\rho^{+}=\rho_{\partial}+\overline{\rho_{\partial}}\hskip 36.135pt\textrm{
and }\hskip 36.135pt\tau^{+}=\tau_{\partial}+\overline{\tau_{\partial}}.$

By the last two remarks we also have

$\rho^{+}_{c}=J_{\textrm{alg}}^{-1}\rho^{+}J_{\textrm{alg}}=i(\overline{\rho_{\partial}}-\rho_{\partial}).$

The following result allows us to express our canonical Hermitian Dirac
operators in terms of the above operators on forms

######  Theorem 5.15.

For any almost Hermitian manifold $M$ we have through the canonical vector
bundle isomorphism $\Omega_{\mathbb{C}}(M)\cong\Gamma(\mathbb{C}l(M))$ that

$D_{t}=\partial+\overline{\partial}+\partial^{*}+\overline{\partial}^{*}+\tfrac{t+1}{4}(\tau_{\partial}+\overline{\tau_{\partial}}+\tau_{\partial}^{*}+\overline{\tau_{\partial}}^{*}-E_{\theta}+I_{\theta})+\tfrac{3t-1}{4}i(\rho_{\partial}-\overline{\rho_{\partial}}-\rho_{\partial}^{*}+\overline{\rho_{\partial}}^{*}).$

In order to prove the theorem, we will require a few lemmas, starting with

######  Lemma 5.16.

For $Y\in\Gamma(T_{\mathbb{C}}M)\cong\Omega_{\mathbb{C}}^{1}(M)$ we have

$\sum_{j,k}(N-\tfrac{3}{2}\mathcal{P}N)(v_{j},Y,v_{k})v_{j}\wedge
v_{k}=(\mu+\overline{\mu})(Y).$

###### Proof.

By Section 4 we observe $N=\mathcal{P}N+N_{0}$ and so

$N-\tfrac{3}{2}\mathcal{P}N=\mathcal{P}N+N_{0}-\tfrac{3}{2}\mathcal{P}N=N_{0}-\tfrac{1}{2}\mathcal{P}N.$

Thus for $Y\in\Gamma(TM)\cong\Omega^{1}(M)$ we have

$\sum_{j,k}(N-\tfrac{3}{2}\mathcal{P}N)(v_{j},Y,v_{k})v_{j}\wedge
v_{k}=\sum_{j,k}(N_{0}-\tfrac{1}{2}\mathcal{P}N)(v_{j},Y,v_{k})v_{j}\wedge
v_{k}.\hskip 28.90755pt$

By definition $\mathcal{P}N_{0}=0$, so that

$\sum_{j,k}N_{0}(v_{j},Y,v_{k})v_{j}\wedge
v_{k}=\sum_{j,k}N_{0}(Y,v_{j},v_{k})v_{j}\wedge
v_{k}-\sum_{j,k}N_{0}(v_{j},Y,v_{k})v_{j}\wedge v_{k}$

and thus

$\sum_{j,k}N_{0}(v_{j},Y,v_{k})v_{j}\wedge
v_{k}=\tfrac{1}{2}\sum_{j,k}N_{0}(Y,v_{j},v_{k})v_{j}\wedge v_{k}.\hskip
101.17755pt$

Finally using that $\mathcal{P}N$ is a 3-form we conclude

$\sum_{j,k}(N-\tfrac{3}{2}\mathcal{P}N)(v_{j},Y,v_{k})v_{j}\wedge
v_{k}=\tfrac{1}{2}\sum_{j,k}(N_{0}+\mathcal{P}N)(Y,v_{j},v_{k})v_{j}\wedge
v_{k}=\tfrac{1}{2}\sum_{j,k}N(Y,v_{j},v_{k})v_{j}\wedge v_{k}.$

Observing that $N^{\vee}=\mu+\overline{\mu}$ on $\Omega_{\mathbb{C}}(M)$ gives
the result.∎

######  Lemma 5.17.

For $Y\in\Gamma(TM)\cong\Omega^{1}(M)$ we have

$\sum_{j,k}\mathcal{M}(d_{c}\omega^{+})(v_{j},Y,v_{k})v_{j}\wedge
v_{k}=\tfrac{1}{2}\sum_{j,k}\big{(}d_{c}\omega^{+}(v_{j},Y,v_{k})v_{j}\wedge
v_{k}+d\omega^{+}(v_{j},JY,v_{k})v_{j}\wedge v_{k}\big{)}.$

###### Proof.

We have for any $\varphi^{+}\in E^{+}$

$\varphi^{+}(X,Y,Z)=\varphi^{+}(JX,JY,Z)+\varphi^{+}(X,JY,JZ)+\varphi^{+}(JX,Y,JZ)$

(see Lemma 8.34 for a proof). Thus

$\begin{split}\sum_{j,k}\mathcal{M}(d_{c}\omega^{+})(v_{j},Y,v_{k})v_{j}\wedge
v_{k}&=-\sum_{j,k}d\omega^{+}(Jv_{j},Y,v_{k})v_{j}\wedge v_{k}\\\
&=-\sum_{j,k}\big{(}d\omega^{+}(Jv_{j},JY,Jv_{k})-d\omega^{+}(v_{j},Y,Jv_{k})-d\omega^{+}(v_{j},JY,v_{k})\big{)}v_{j}\wedge
v_{k}.\\\ \end{split}$

Subtracting the expression
$\displaystyle\sum_{j,k}d\omega^{+}(v_{j},Y,Jv_{k})v_{j}\wedge v_{k}$ from
both sides of the last equality we obtain

$\begin{split}\sum_{j,k}\mathcal{M}(d_{c}\omega^{+})(v_{j},Y,v_{k})v_{j}\wedge
v_{k}&=\tfrac{1}{2}\sum_{j,k}\big{(}-d\omega^{+}(Jv_{j},JY,Jv_{k})+d\omega^{+}(v_{j},JY,v_{k})\big{)}v_{j}\wedge
v_{k}\\\
&=\tfrac{1}{2}\sum_{j,k}\big{(}d_{c}\omega^{+}(v_{j},Y,v_{k})v_{j}\wedge
v_{k}+d\omega^{+}(v_{j},JY,v_{k})v_{j}\wedge v_{k}\big{)}.\qed\end{split}$

######  Lemma 5.18.

For $Y\in\Gamma(TM)\cong\Omega^{1}(M)$ we have

$\tfrac{1}{2}\sum_{j,k}d\omega^{+}(v_{j},JY,v_{k})v_{j}\wedge
v_{k}=\tau^{+}(Y)-\theta\wedge Y.$

###### Proof.

Observe that for $Y\in\Gamma(TM)\cong\Omega^{1}(M)$

$\tau^{+}(Y)=[\Lambda,\lambda^{+}](Y)=-\Lambda(Y\wedge
d\omega^{+})=\sum_{j=1}^{n}e_{j}\lrcorner\ Je_{j}\lrcorner\ (Y\wedge
d\omega^{+}).$

Using that interior multiplication is a graded derivation we obtain

$\begin{split}\sum_{j=1}^{n}e_{j}\lrcorner\ Je_{j}\lrcorner\ (Y\wedge
d\omega^{+})&=\sum_{j}e_{j}\lrcorner\ \big{(}\langle Y,Je_{j}\rangle
d\omega^{+}-Y\wedge(Je_{j}\lrcorner\ d\omega^{+})\big{)}\\\
&=\sum_{j}\big{(}\langle Y,Je_{j}\rangle e_{j}\lrcorner\ d\omega^{+}-\langle
Y,e_{j}\rangle Je_{j}\lrcorner\ d\omega^{+}+Y\wedge(e_{j}\lrcorner\
Je_{j}\lrcorner\ d\omega^{+})\big{)}\\\ &=\sum_{j}\big{(}-\langle
JY,e_{j}\rangle e_{j}\lrcorner\ d\omega^{+}-\langle JY,Je_{j}\rangle
Je_{j}\lrcorner\ d\omega^{+}\big{)}+Y\wedge\sum_{j}e_{j}\lrcorner\
Je_{j}\lrcorner\ d\omega^{+}\\\ &=-JY\lrcorner\
d\omega^{+}+\Lambda(d\omega^{+})\wedge Y.\end{split}$

As $\Lambda(d\omega^{+})=\Lambda(d\omega)=\theta$, the result follows. ∎

######  Lemma 5.19.

For $Y\in\Gamma(TM)\cong\Omega^{1}(M)$ we have

$\tfrac{1}{2}\sum_{j,k}d_{c}\omega^{+}(v_{j},Y,v_{k})v_{j}\wedge
v_{k}=-\rho_{c}^{+}(Y).$

###### Proof.

For $Y\in\Gamma(TM)\cong\Omega^{1}(M)$ we have

$\rho^{+}(Y)=\tfrac{1}{2}\sum_{j,k}d\omega^{+}(v_{j},Y,v_{k})v_{j}\wedge
v_{k}$

and hence

$\begin{split}\rho_{c}^{+}(Y)&=\tfrac{1}{2}J_{\textrm{alg}}^{-1}\big{(}\sum_{j,k}d\omega^{+}(v_{j},JY,v_{k})v_{j}\wedge
v_{k}\big{)}\\\
&=\phantom{-}\tfrac{1}{2}\sum_{j,k}d\omega^{+}(v_{j},JY,v_{k})Jv_{j}\wedge
Jv_{k}\\\
&=\phantom{-}\tfrac{1}{2}\sum_{j,k}d\omega^{+}(Jv_{j},JY,Jv_{k})v_{j}\wedge
v_{k}\\\ &=-\tfrac{1}{2}\sum_{j,k}d_{c}\omega^{+}(v_{j},Y,v_{k})v_{j}\wedge
v_{k}.\qed\end{split}$

Combining lemmas lemma 5.17, lemma 5.18 and lemma 5.19 we obtain

######  Proposition 5.20.

For $Y\in\Gamma(TM)\cong\Omega^{1}(M)$ we have

$\sum_{j,k}\mathcal{M}(d_{c}\omega^{+})(v_{j},Y,v_{k})v_{j}\wedge
v_{k}=\tau^{+}(Y)-\theta\wedge Y-\rho_{c}^{+}(Y).$

We now prove Theorem 5.15:

###### Proof.

(of Theorem 5.15) Complex linearly extending $d_{A}$ to
$\Omega_{\mathbb{C}}(M)$, we recall that for any
$Y\in\Omega_{\mathbb{C}}^{1}(M)$

$d_{A}(Y)=\sum_{j,k}\Big{(}\textrm{-}(N-\tfrac{3}{2}\mathcal{P}N)(v_{j},Y,v_{k})+\tfrac{t-1}{4}d_{c}\omega^{+}(v_{j},Y,v_{k})+\tfrac{t+1}{4}\mathcal{M}(d_{c}\omega^{+})(v_{j},Y,v_{k})\Big{)}v_{j}\wedge
v_{k}.$

By Proposition 5.20, Lemma 5.16, Lemma 5.19 we have

$d_{A}(Y)=-\big{(}\mu+\overline{\mu}\big{)}(Y)+\tfrac{t+1}{4}\big{(}\tau_{\partial}(Y)+\overline{\tau_{\partial}}(Y)-\theta\wedge
Y\big{)}+\tfrac{3t-1}{4}i\big{(}\rho_{\partial}(Y)-\overline{\rho_{\partial}}(Y)\big{)}.$

As $d_{A}$ is a graded derivation, vanishing on smooth functions on $M$, we
have that for any $\varphi\in\Omega_{\mathbb{C}}(M)$

$d_{A}(\varphi)=-\big{(}\mu+\overline{\mu}\big{)}(\varphi)+\tfrac{t+1}{4}\big{(}\tau_{\partial}(\varphi)+\overline{\tau_{\partial}}(\varphi)-\theta\wedge\varphi\big{)}+\tfrac{3t-1}{4}i\big{(}\rho_{\partial}(\varphi)-\overline{\rho_{\partial}}(\varphi)\big{)}.$

The result then follows by Proposition Proposition 5.14 and the identity
$r(A)=\tfrac{t+1}{2}\theta$. ∎

##  6\. Almost Hermitian Identities via the Bismut Dirac Operator

We have shown above that the canonical Hermitian and Riemannian Dirac
operators on $\Gamma\mathbb{C}l(M)$ agree up to a tensorial expression. In
particular we have by Theorem 5.15

$D_{t}-\widetilde{D}\cong-(\mu+\overline{\mu}+\mu^{*}+\overline{\mu}^{*})+\tfrac{t+1}{4}(\tau+\overline{\tau}+\tau^{*}+\overline{\tau}^{*}-E_{\theta}+I_{\theta})+\tfrac{3t-1}{4}i(\rho_{\partial}-\overline{\rho_{\partial}}-\rho_{\partial}^{*}+\overline{\rho_{\partial}}^{*}).$

Since $\nabla^{t}=\widetilde{\nabla}$ if and only if $d\omega=0=N$ we have
$\widetilde{D}=D_{t}$ if and only if the almost Hermitian manifold is Kähler.

We recall that for any Dirac operator $D$ defined in terms of a Hermitian
connection we have by Proposition Theorem 3.2

$[D,\mathcal{H}]=-iD_{c}.$

Moreover since $D_{t}$ is a Hermitian Dirac operator defined in terms of a
Gauduchon connection we obtain the following

######  Proposition 6.1.

Let $M$ be an almost Hermitian manifold. We have the following identities

$\begin{split}[D_{t},\mathcal{H}]=-i(D_{t})_{c}\hskip 14.45377pt&\hskip
14.45377pt[(D_{t})_{c},\mathcal{H}]=\phantom{-}iD_{t}\\\ [D_{t}^{\text{\char
73}},\mathcal{H}]=\phantom{-}i(D_{t})_{c}^{\text{\char 73}}\hskip
14.45377pt&\hskip 14.45377pt[(D_{t})_{c}^{\text{\char
73}},\mathcal{H}]=-iD_{t}^{\text{\char 73}}.\end{split}$

Using the expression previously obtained for the adjoint of $D_{t}$ in
Proposition 5.6 we observe that

$\left(D_{t}\right)_{c}=\left(D_{t}\right)_{c}^{*}+\tfrac{t+1}{4}L_{J\theta}$

and thus

$[D_{t},\mathcal{H}]=-i\left(D_{t}\right)_{c}^{*}-\tfrac{i(t+1)}{4}L_{J\theta}.$

In particular for any almost Hermitian manifold $M$ we have the Dirac operator
$B\stackrel{{\scriptstyle\textrm{def}}}{{=}}D_{-1}$ is self-adjoint and so

$[B,\mathcal{H}]=-iB_{c}=-iB_{c}^{*}.$

######  Definition 6.2.

We define the operators $\varepsilon$ and $\overline{\varepsilon}$ on
$\Omega_{\mathbb{C}}(M)$ by

$\varepsilon=\partial-i\rho_{\partial}\hskip 7.22743pt\textrm{ and }\hskip
7.22743pt\overline{\varepsilon}=\overline{\partial}+i\overline{\rho_{\partial}}.$

Following [TT20] we also define the differential operators $\delta$ and
$\overline{\delta}$ on $\Omega_{\mathbb{C}}(M)$ by

$\delta=\partial+\overline{\mu}\hskip 7.22743pt\textrm{ and }\hskip
7.22743pt\overline{\delta}=\overline{\partial}+\mu$

where $d=\delta+\overline{\delta}$ is the exterior derivative on
$\Omega_{\mathbb{C}}(M).$

We note that through the canonical isomorphism
$\Gamma\mathbb{C}l(M)\cong\Omega_{\mathbb{C}}(M)$ we have

(5)
$\widetilde{D}=\delta+\overline{\delta}+\delta^{*}+\overline{\delta}^{*}\hskip
14.45377pt\textrm{ and }\hskip
14.45377ptB=\varepsilon+\overline{\varepsilon}+\varepsilon^{*}+\overline{\varepsilon}^{*}.$

The zero-order term $\mu$ vanishes if and only if $M$ is integrable, and
symmetrically we show below in Corollary 6.5 that $\rho_{\partial}$ vanishes
if and only if $\partial\omega=\overline{\partial}\omega=0.$

Note that $J\mapsto-J^{\vee}$ by the metric isomorphism $T(M)\cong
T^{\vee}(M)$. This has the effect of flippling the sign through the
isomorphism $\Gamma(\mathbb{C}l(M))\cong\Omega_{\mathbb{C}}(M)$ when
conjugating by $J_{\textrm{alg}}$. From this we obtain the

######  Lemma 6.3.

On any almost Hermitian manifold we have

$\widetilde{D}_{c}\cong-
d_{c}-d_{c}^{*}=i(\delta-\overline{\delta}+\overline{\delta}^{*}-\delta^{*})$

and

$B_{c}\cong
i(\varepsilon-\overline{\varepsilon}+\overline{\varepsilon}^{*}-\varepsilon^{*}).$

###### Proof.

The first identity is clear from the above remark. The second identity follows
from the observation
$J_{\textrm{alg}}^{-1}\rho_{\partial}J_{\textrm{alg}}=-i\rho_{\partial}$ so
that
$\varepsilon_{c}=\partial_{c}-i\rho_{\partial}{{}_{c}}=-i(\partial-i\rho_{\partial})=-i\varepsilon$
and similarly $\overline{\varepsilon}_{c}=i\overline{\varepsilon}$. ∎

The relation $[B,\mathcal{H}]=-iB_{c}$ implies the following identities of
operators on $\Omega(M).$

######  Proposition 6.4.

Let $M$ be an almost Hermitian manifold. We obtain the following identities of
operators on $\Omega_{\mathbb{C}}(M)$:

$\begin{split}[\Lambda,\overline{\varepsilon}]=-i\varepsilon^{*}\hskip
14.45377pt&\hskip 14.45377pt[L,\varepsilon^{*}]=i\overline{\varepsilon}\\\
[\Lambda,\varepsilon]=i\overline{\varepsilon}^{*}\ \ \hskip 14.45377pt&\hskip
14.45377pt[L,\overline{\varepsilon}^{*}]=-i\varepsilon\\\
0=[\Lambda,\varepsilon^{*}]=[\Lambda,\overline{\varepsilon}^{*}]\hskip
7.22743pt&\hskip
18.06749pt[L,\varepsilon]=[L,\overline{\varepsilon}]=0.\end{split}$

###### Proof.

The identity $[B,\mathcal{H}]=-iB_{c}$ implies

$\left[\varepsilon+\overline{\varepsilon}+\varepsilon^{*}+\overline{\varepsilon}^{*},i(\Lambda-L)\right]=\varepsilon-\overline{\varepsilon}+\overline{\varepsilon}^{*}-\varepsilon^{*}.$

That is

$i[\varepsilon,\Lambda]+i[\overline{\varepsilon},\Lambda]+i[\varepsilon^{*},\Lambda]+i[\overline{\varepsilon}^{*},\Lambda]-i[\varepsilon,L]-i[\overline{\varepsilon},L]-i[\varepsilon^{*},L]-i[\overline{\varepsilon}^{*},L]=\varepsilon-\overline{\varepsilon}+\overline{\varepsilon}^{*}-\varepsilon^{*}.$

The result then follows by comparing bidegree of the operators in the above
equality. For example, $[\overline{\varepsilon}^{*},L]$ has bidegree $(1,0)$,
as does $\varepsilon$ and so the equality implies
$i\varepsilon=[\overline{\varepsilon}^{*},L]$.∎

######  Corollary 6.5.

Let $M$ be an almost Hermitian manifold. Then $\rho_{\partial}=0$ if and only
if $\partial\omega=\overline{\partial}\omega=0$.

###### Proof.

Evidently if $\partial\omega=0$ then $\rho_{\partial}=0$, and by conjugating
by complex conjugation $\partial\omega=0$ if and only if
$\overline{\partial}\omega=\overline{\partial\overline{\omega}}=\overline{\partial\omega}=0$.
Conversely, by the identity

$[L,\varepsilon]=0$

we have

$[L,\partial]=i[L,\rho_{\partial}].$

Since both $\partial$ and $\rho_{\partial}$ are graded derivations, by
evaluating at $1$, we see $i\rho_{\partial}(\omega)=\partial\omega$. Hence if
$\rho_{\partial}=0$ we have $\partial\omega=0.$∎

Conjugating our Dirac operators by the transpose map ✩ we obtain the following
correspondences to operators on forms. For the antipodal involution $\alpha$
on $Cl(M)$ we have $\alpha\nabla=\nabla\alpha$ for any metric connection
$\nabla$. In particular $B\alpha=-\alpha B$ and
$\widetilde{D}\alpha=-\alpha\widetilde{D}$. Evidently the operators
$\varepsilon$ and $\delta$ also anticommute with $\alpha$.

######  Lemma 6.6.

On any almost Hermitian manifold we have

$\widetilde{D}^{\text{\char 73}}\cong
d\alpha-d^{*}\alpha=(\delta+\overline{\delta}-\delta^{*}-\overline{\delta}^{*})\alpha$

and

$B^{\text{\char
73}}\cong(\varepsilon+\overline{\varepsilon}-\varepsilon^{*}-\overline{\varepsilon}^{*})\alpha.$

###### Proof.

Let $\varphi\in\Gamma(Cl(M))\cong\Omega(M)$. Recall that for any $v\in TM$ we
have

$v\cdot\varphi\cong v\wedge\varphi-v\lrcorner\ \varphi\textrm{ and
}\varphi\cdot v\cong\big{(}v\wedge\alpha\varphi+v\lrcorner\
\alpha\varphi\big{)}.$

Since $\displaystyle d=\sum_{k}v_{k}\wedge\widetilde{\nabla}_{v_{k}}\textrm{
and }d^{*}=-\sum_{k}v_{k}\lrcorner\ \widetilde{\nabla}_{v_{k}}$ on $\Omega(M)$
we see that

$\widetilde{D}^{\text{\char
73}}(\varphi)=\sum_{k}\widetilde{\nabla}_{v_{k}}(\varphi)\cdot
v_{k}\cong\sum_{k}v_{k}\wedge\widetilde{\nabla}_{v_{k}}(\alpha\varphi)+\sum_{k}v_{k}\lrcorner\
\widetilde{\nabla}_{v_{k}}(\alpha\varphi)=d(\alpha\varphi)-d^{*}(\alpha\varphi).$

Moreover

$D_{A}^{\text{\char 73}}(\varphi)=\sum_{j}a_{v_{j}}(\varphi)\cdot
v_{j}\cong\sum_{j}v_{j}\wedge a_{v_{j}}(\alpha\varphi)+\sum_{j}v_{j}\lrcorner\
a_{v_{j}}(\alpha\varphi)$

and thus

$B^{\text{\char 73}}(\varphi)-\widetilde{D}^{\text{\char 73}}(\varphi)\cong
d_{A}(\alpha\varphi)-d_{A}^{*}(\alpha\varphi).$

Since for $t=-1$, by the proof of Theorem 5.15 we have

$d_{A}=-\mu-\overline{\mu}-i\rho_{\partial}+i\overline{\rho_{\partial}}\hskip
10.84006pt\textrm{ and hence }\hskip
10.84006ptd_{A}^{*}=-\mu^{*}-\overline{\mu}^{*}+i\rho_{\partial}^{*}-i\overline{\rho_{\partial}}^{*},$

we conclude $B^{\text{\char
73}}(\varphi)\cong\varepsilon(\alpha\varphi)+\overline{\varepsilon}(\alpha\varphi)-\varepsilon^{*}(\alpha\varphi)-\overline{\varepsilon}^{*}(\alpha\varphi)$
as desired.∎

For operators $A,B$ on $\Gamma(\mathbb{C}l(M))\cong\Omega_{\mathbb{C}}(M)$ we
define the anti-commutator

$\left\\{A,B\right\\}=AB+BA.$

######  Remark 6.7.

By the bracket $[\cdot,\cdot]$ of operators on sections of the Clifford or
exterior algebra bundles we will always mean the vanilla (ungraded)
commutator. Much of the discussion can be rewritten via a $\mathbb{Z}_{2}$
graded commutator, though there are some inconveniences one is forced to
address when adopting this approach.

######  Proposition 6.8.

On any almost Hermitian manifold $M$ we have the following identities

$\begin{split}\left\\{\mathcal{L}+\overline{\mathcal{L}},B\right\\}&=B^{\text{\char
73}}\hskip 36.135pt\left\\{\mathcal{L}-\overline{\mathcal{L}},B\right\\}\
=-iB_{c}^{\text{\char 73}}\\\
\left\\{\mathcal{L}+\overline{\mathcal{L}},B^{\text{\char
73}}\right\\}&=B\hskip 36.135pt\
\left\\{\mathcal{L}-\overline{\mathcal{L}},B^{\text{\char
73}}\right\\}=\phantom{-}iB_{c}\\\
\left\\{\mathcal{L}+\overline{\mathcal{L}},B_{c}\right\\}&=B_{c}^{\text{\char
73}}\hskip
36.135pt\left\\{\mathcal{L}-\overline{\mathcal{L}},B_{c}\right\\}=\phantom{-}iB^{\text{\char
73}}\\\ \end{split}$

###### Proof.

By Section 2.4 we have through the identification
$\Gamma(\mathbb{C}l(M))\cong\Omega_{\mathbb{C}}(M)$ that

$\begin{split}\left\\{\mathcal{L}+\overline{\mathcal{L}},B\right\\}&=\left\\{\alpha
H,\varepsilon+\overline{\varepsilon}+\varepsilon^{*}+\overline{\varepsilon}^{*}\right\\}\\\
&=\left[\varepsilon+\overline{\varepsilon}+\varepsilon^{*}+\overline{\varepsilon}^{*},H\right]\alpha\\\
&=\left[\varepsilon+\overline{\varepsilon},H\right]\alpha+\left[\varepsilon^{*}+\overline{\varepsilon}^{*},H\right]\alpha\\\
&=(\varepsilon+\overline{\varepsilon}-\varepsilon^{*}-\overline{\varepsilon}^{*})\alpha\\\
&=B^{\text{\char 73}}\end{split}$

while by Proposition 6.4 and Section 2.4 we observe

$\begin{split}\left\\{\mathcal{L}-\overline{\mathcal{L}},B\right\\}&=i\left[\Lambda+L,\varepsilon+\overline{\varepsilon}+\varepsilon^{*}+\overline{\varepsilon}^{*}\right]\alpha\\\
&=i\left[\Lambda+L,\varepsilon+\overline{\varepsilon}\right]\alpha+i\left[\Lambda+L,\varepsilon^{*}+\overline{\varepsilon}^{*}\right]\alpha\\\
&=(\varepsilon-\overline{\varepsilon}+\varepsilon^{*}-\overline{\varepsilon}^{*})\alpha\\\
&=-iB_{c}^{\text{\char 73}}.\end{split}$

The remaining identities are obtained by conjugating with the operators
$J_{\textrm{alg}}$ and ✩. By the identities $\mathcal{L}_{c}=\mathcal{L}$ and
$\overline{\mathcal{L}}_{c}=\overline{\mathcal{L}}$ we see

$\left\\{\mathcal{L}+\overline{\mathcal{L}},B_{c}\right\\}=J_{\textrm{alg}}^{-1}\left\\{\mathcal{L}+\overline{\mathcal{L}},B\right\\}J_{\textrm{alg}}=J_{\textrm{alg}}^{-1}B^{\text{\char
73}}J_{\textrm{alg}}=B_{c}^{\text{\char 73}}.\qed$

Summing the above identities gives rise to new objects of interest. For
example

$\left\\{\mathcal{L},B^{\text{\char
73}}\right\\}=\tfrac{1}{2}\big{(}\left\\{\mathcal{L}+\overline{\mathcal{L}},B^{\text{\char
73}}\right\\}+\left\\{\mathcal{L}-\overline{\mathcal{L}},B^{\text{\char
73}}\right\\}\big{)}=\tfrac{1}{2}(B+iB_{c}).$

We define

$\mathfrak{B}=\tfrac{1}{2}(B+iB_{c})\hskip
72.26999pt\overline{\mathfrak{B}}=\tfrac{1}{2}(B-iB_{c})$

and

$\mathfrak{D}=\tfrac{1}{2}(\widetilde{D}+i\widetilde{D}_{c})\hskip
72.26999pt\overline{\mathfrak{D}}=\tfrac{1}{2}(\widetilde{D}-i\widetilde{D}_{c}).$

More generally we recall the operator $\mathfrak{d}_{t}$ on
$\Gamma\big{(}\mathbb{C}l(M)\big{)}$ defined by

$\mathfrak{d}_{t}=\tfrac{1}{2}\big{(}D_{t}+i(D_{t})_{c}\big{)}.$

Notice that for any $\varphi\in\Gamma(\mathbb{C}l(M))$ and for any $J$-adapted
local orthonormal frame
$v_{1},\dots,v_{2n}=e_{1},\dots,e_{n},Je_{1},\dots,Je_{n}$ of $TM$ we have

$\mathfrak{d}_{t}(\varphi)=\tfrac{1}{2}\big{(}\sum_{j=1}^{2n}v_{j}\cdot\nabla^{t}_{v_{j}}(\varphi)-iJv_{j}\cdot\nabla^{t}_{v_{j}}(\varphi)\big{)}=2\sum_{j=1}^{n}\epsilon_{j}\cdot\nabla^{t}_{\overline{\epsilon}_{j}}(\varphi).$

We observe (cf. [M80]) that the operator $\mathfrak{d}_{t}$ is of Clifford
bidegree $(1,1)$ as left Clifford multiplication by an element of $T^{1,0}(M)$
is of Clifford bidegree $(1,1)$ and any Hermitian connection is both
$\mathcal{H}$ and $\mathcal{J}$ parallel. Contrarily the operator
$\mathfrak{D}$ on sections of $\mathbb{C}l(M)$ is not generically an operator
of pure Clifford bidegree.

Focusing attention once again on our operators $\mathfrak{D}$ and
$\mathfrak{B}$, we state the below immediate consequence of the identity
(Equation 5) along with Lemmas lemma 6.3 and lemma 6.6.

######  Proposition 6.9.

On any almost Hermitian manifold $M$ we have the following correspondences of
operators through the isomorphism
$\Gamma\mathbb{C}l(M)\cong\Omega_{\mathbb{C}}(M)$

$\begin{split}\mathfrak{D}\phantom{i}=\overline{\delta}+\delta^{*}\hskip
34.68929pt&\hskip
24.57153pt\overline{\mathfrak{D}}\phantom{i}=\delta+\overline{\delta}^{*}\\\
\mathfrak{D}^{\text{\char 73}}=\overline{\delta}\alpha-\delta^{*}\alpha\hskip
21.68121pt&\hskip 21.68121pt\overline{\mathfrak{D}}^{\text{\char
73}}=\delta\alpha-\overline{\delta}^{*}\alpha\\\
\mathfrak{B}\phantom{i}=\overline{\varepsilon}+\varepsilon^{*}\hskip
34.68929pt&\hskip
23.12692pt\overline{\mathfrak{B}}\phantom{i}=\varepsilon+\overline{\varepsilon}^{*}\\\
\mathfrak{B}^{\text{\char
73}}=\overline{\varepsilon}\alpha-\varepsilon^{*}\alpha\hskip
21.68121pt&\hskip 21.68121pt\overline{\mathfrak{B}}^{\text{\char
73}}=\varepsilon\alpha-\overline{\varepsilon}^{*}\alpha\\\ \end{split}$

###### Proof.

We observe that by the identity (5) we have on
$\Gamma\mathbb{C}l(M)\cong\Omega_{\mathbb{C}}(M)$ that

$\widetilde{D}=\delta+\overline{\delta}+\delta^{*}+\overline{\delta}^{*}$

and by Lemma 6.3 we have
$\widetilde{D}_{c}=i(\delta-\overline{\delta}+\overline{\delta}^{*}-\delta^{*})$
so that

$\mathfrak{D}=\tfrac{1}{2}(D+iD_{c})=\overline{\delta}+\delta^{*}.$

The other identities are similarly obtained by use of (eq. 5) along with
Lemmas lemma 6.3 and lemma 6.6.∎

######  Lemma 6.10.

Let $M$ be an almost Hermitian manifold. Then we have

$[\widetilde{D},\widetilde{D}^{\text{\char
73}}]=[\widetilde{D}_{c},\widetilde{D}_{c}^{\text{\char 73}}]=0.$

Furthermore if $M$ is Kähler then

$[B,B^{\text{\char 73}}]=[B_{c},B_{c}^{\text{\char 73}}]=0.$

###### Proof.

We observe

$\widetilde{D}\widetilde{D}^{\text{\char
73}}\cong(d+d^{*})(d\alpha-d^{*}\alpha)=d^{*}d\alpha-
dd^{*}\alpha=(d\alpha-d^{*}\alpha)(d+d^{*})\cong\widetilde{D}^{\text{\char
73}}\widetilde{D}.$

It follows that $[\widetilde{D},\widetilde{D}^{\text{\char 73}}]=0$.
Conjugating this identity by $J_{\textrm{alg}}$ gives
$[\widetilde{D}_{c},\widetilde{D}_{c}^{\text{\char 73}}]=0$.

If $M$ is Kähler then $B=\widetilde{D}$ and hence

$[B,B^{\text{\char 73}}]=[B_{c},B_{c}^{\text{\char 73}}]=0.\qed$

######  Definition 6.11.

For an operator $T$, we define the Laplacian of $T$ by

$\Delta_{T}=TT^{*}+T^{*}T.$

######  Proposition 6.12.

On any almost Hermitian manifold $M$ we have
$\Delta_{\mathfrak{D}}=\Delta_{\mathfrak{D}^{\text{\char 73}}}$. Furthermore
$\Delta_{\mathfrak{D}}=\Delta_{\delta}+\Delta_{\overline{\delta}}$ through the
isomorphism $\Gamma(\mathbb{C}l(M))\cong\Omega_{\mathbb{C}}(M)$.

###### Proof.

The identity $\Delta_{\mathfrak{D}}=\Delta_{\mathfrak{D}^{\text{\char 73}}}$
can be checked by observing

$\begin{split}\Delta_{\mathfrak{D}}&=(\overline{\delta}+\delta^{*})(\delta+\overline{\delta}^{*})+(\delta+\overline{\delta}^{*})(\overline{\delta}+\delta^{*})\\\
&=\Delta_{\delta}+\Delta_{\overline{\delta}}+\left\\{\delta,\overline{\delta}\right\\}+\left\\{\delta^{*},\overline{\delta}^{*}\right\\}\\\
\end{split}$

while

$\begin{split}\Delta_{\mathfrak{D}^{\text{\char
73}}}&=(\overline{\delta}\alpha-\delta^{*}\alpha)(\delta\alpha-\overline{\delta}^{*}\alpha)+(\delta\alpha-\overline{\delta}^{*}\alpha)(\overline{\delta}\alpha-\delta^{*}\alpha)\\\
&=\Delta_{\delta}+\Delta_{\overline{\delta}}-\left\\{\delta,\overline{\delta}\right\\}-\left\\{\delta^{*},\overline{\delta}^{*}\right\\}.\\\
\end{split}$

By Proposition 2.15 we see that

$\left\\{\delta,\overline{\delta}\right\\}=\left\\{\partial,\overline{\partial}\right\\}+\left\\{\mu,\overline{\mu}\right\\}+\left\\{\partial,\mu\right\\}+\left\\{\overline{\partial},\overline{\mu}\right\\}=\left\\{\partial,\overline{\partial}\right\\}-\left\\{\partial,\overline{\partial}\right\\}+0+0=0$

and so $\Delta_{\mathfrak{D}}=\Delta_{\mathfrak{D}^{\text{\char 73}}}$. ∎

Repeating the above argument for $\Delta_{\mathfrak{B}}$ and
$\Delta_{\mathfrak{B}^{\text{\char 73}}}$ we subsequently obtain the

######  Proposition 6.13.

On any almost Hermitian manifold $M$ we have
$\Delta_{\mathfrak{B}}+\Delta_{\mathfrak{B}^{\text{\char 73}}}\cong
2\left(\Delta_{\varepsilon}+\Delta_{\overline{\varepsilon}}\right)$ through
the isomorphism $\Gamma(\mathbb{C}l(M))\cong\Omega_{\mathbb{C}}(M)$.
Furthermore $\Delta_{\mathfrak{B}}=\Delta_{\mathfrak{B}^{\text{\char 73}}}$ if
$M$ is Kähler.

###### Proof.

By the identities in Proposition 6.9 we compute

$\begin{split}\Delta_{\mathfrak{B}}\phantom{i}&=\Delta_{\varepsilon}+\Delta_{\overline{\varepsilon}}+\left\\{\varepsilon,\overline{\varepsilon}\right\\}+\left\\{\varepsilon^{*},\overline{\varepsilon}^{*}\right\\}\textrm{
and }\\\ \Delta_{\mathfrak{B}^{\text{\char
73}}}&=\Delta_{\varepsilon}+\Delta_{\overline{\varepsilon}}-\left\\{\varepsilon,\overline{\varepsilon}\right\\}-\left\\{\varepsilon^{*},\overline{\varepsilon}^{*}\right\\}.\end{split}$

Hence $\Delta_{\mathfrak{B}}-\Delta_{\mathfrak{B}^{\text{\char 73}}}=0$ when
$\varepsilon=\partial$ as then
$\left\\{\varepsilon,\overline{\varepsilon}\right\\}=\left\\{\partial-i\rho_{\partial},\overline{\partial}+i\overline{\rho_{\partial}}\right\\}=\left\\{\partial,\overline{\partial}\right\\}=0$.∎

We recall the operator of complex conjugation
$c:\mathbb{C}l^{r,s}(M)\rightarrow\mathbb{C}l^{-r,-s}(M)$ (see [M80]). As
$\widetilde{D}$ and $B$ are real operators, they commute with $c$. Thus we
have the following

######  Lemma 6.14.

On an almost Hermitian manifold $M$ we have

$c\ \mathfrak{D}c=\overline{\mathfrak{D}},\hskip 21.68121ptc\
\mathfrak{B}c=\overline{\mathfrak{B}},\hskip 21.68121ptc\
\mathfrak{B}^{\text{\char 73}}c=\overline{\mathfrak{B}^{\text{\char 73}}}.$

###### Proof.

As ✩ and $J_{\textrm{alg}}$ are also real operators, both commute with $c$.
Hence

$c\
\mathfrak{D}c=\tfrac{1}{2}\big{(}c\widetilde{D}c-icJ_{\textrm{alg}}^{-1}\widetilde{D}J_{\textrm{alg}}c\big{)}=\tfrac{1}{2}\big{(}\widetilde{D}-i\widetilde{D}_{c}\big{)}=\overline{\mathfrak{D}}.$

The remaining identities are obtained similarly. ∎

We now turn to investigating the relationship between the intrinsic $sl(2)$
operators on $\mathbb{C}l(M)$ with our operators $\mathfrak{B}$ and
$\mathfrak{D}$.

######  Proposition 6.15.

On any almost Hermitian manifold $M$ the following identities obtain

$\begin{split}\left\\{\mathfrak{B},\mathcal{L}\right\\}=0\hskip
36.135pt&\hskip
36.135pt\left\\{\overline{\mathfrak{B}},\overline{\mathcal{L}}\right\\}=0\\\
\left\\{\mathfrak{B},\overline{\mathcal{L}}\right\\}=\mathfrak{B}^{\text{\char
73}}\hskip 28.90755pt&\hskip
36.135pt\left\\{\overline{\mathfrak{B}},\mathcal{L}\right\\}=\overline{\mathfrak{B}}^{\text{\char
73}}\\\ \hskip 36.135pt[\mathcal{H},&\mathfrak{B}]=\mathfrak{B}.\end{split}$

###### Proof.

We observe that by Proposition 6.1,

$\begin{split}[\mathcal{H},\mathfrak{B}]&=[\mathcal{H},\tfrac{1}{2}(B+iB_{c})]=\tfrac{1}{2}([\mathcal{H},B]+i[\mathcal{H},B_{c}])\\\
&=\tfrac{1}{2}(B+iB_{c})=\mathfrak{B}\end{split}$

giving the last identity. For the first identity we observe by Proposition 6.8

$\begin{split}\left\\{\mathfrak{B},\mathcal{L}\right\\}&=\tfrac{1}{2}\big{(}(B+iB_{c})\mathcal{L}+\mathcal{L}(B+iB_{c})\big{)}=\tfrac{1}{2}\big{(}\left\\{B,\mathcal{L}\right\\}+i\left\\{B_{c},\mathcal{L}\right\\}\big{)}\\\
&=\tfrac{1}{4}\big{(}(B^{\text{\char 73}}-iB_{c}^{\text{\char
73}})+i(B_{c}^{\text{\char 73}}+iB^{\text{\char 73}})\big{)}=0.\end{split}$

A similar argument, again using Proposition 6.8 yields
$\left\\{\mathfrak{B},\overline{\mathcal{L}}\right\\}=\mathfrak{B}^{\text{\char
73}}$. The remaining identities are obtained by conjugating by complex
conjugation. For example by Lemma 6.14 we have

$\left\\{\overline{\mathfrak{B}},\mathcal{L}\right\\}=\left\\{c\
\mathfrak{B}c,c\
\overline{\mathcal{L}}c\right\\}=c\left\\{\mathfrak{B},\overline{\mathcal{L}}\right\\}c=c\
\mathfrak{B}^{\text{\char 73}}c=\overline{\mathfrak{B}}^{\text{\char
73}}.\qed$

######  Proposition 6.16.

On any almost Hermitian manifold $M$ we have

$\begin{split}[\mathcal{H},\Delta_{\mathfrak{B}}]&=0\hskip
49.14307pt[\mathcal{H},\Delta_{\mathfrak{B}^{\text{\char 73}}}]=0\\\
[\mathcal{L},\Delta_{\mathfrak{B}}]&=[\overline{\mathfrak{B}}^{\text{\char
73}},\mathfrak{B}]\hskip
21.68121pt[\mathcal{L},\Delta_{\mathfrak{B}^{\text{\char
73}}}]=[\mathfrak{B},\overline{\mathfrak{B}}^{\text{\char 73}}]\\\
[\overline{\mathcal{L}},\Delta_{\mathfrak{B}}]&=[\mathfrak{B}^{\text{\char
73}},\overline{\mathfrak{B}}]\hskip
21.68121pt[\overline{\mathcal{L}},\Delta_{\mathfrak{B}^{\text{\char
73}}}]=[\overline{\mathfrak{B}},\mathfrak{B}^{\text{\char 73}}].\end{split}$

###### Proof.

We compute by Proposition 6.15

$\begin{split}[\mathcal{H},\Delta_{\mathfrak{B}}]&=\mathcal{H}\mathfrak{B}\overline{\mathfrak{B}}+\mathcal{H}\overline{\mathfrak{B}}\mathfrak{B}-\mathfrak{B}\overline{\mathfrak{B}}\mathcal{H}-\overline{\mathfrak{B}}\mathfrak{B}\mathcal{H}+\mathfrak{B}\mathcal{H}\overline{\mathfrak{B}}-\mathfrak{B}\mathcal{H}\overline{\mathfrak{B}}\\\
&=(\mathcal{H}\mathfrak{B}-\mathfrak{B}\mathcal{H})\overline{\mathfrak{B}}+\mathfrak{B}(\mathcal{H}\overline{\mathfrak{B}}-\overline{\mathfrak{B}}\mathcal{H})+\mathcal{H}\overline{\mathfrak{B}}\mathfrak{B}-\overline{\mathfrak{B}}\mathfrak{B}\mathcal{H}\\\
&=\mathcal{H}\overline{\mathfrak{B}}\mathfrak{B}-\overline{\mathfrak{B}}\mathfrak{B}\mathcal{H}\\\
&=\mathcal{H}\overline{\mathfrak{B}}\mathfrak{B}-\overline{\mathfrak{B}}\mathfrak{B}\mathcal{H}+\overline{\mathfrak{B}}\mathcal{H}\mathfrak{B}-\overline{\mathfrak{B}}\mathcal{H}\mathfrak{B}\\\
&=(\mathcal{H}\overline{\mathfrak{B}}-\overline{\mathfrak{B}}\mathcal{H})\mathfrak{B}+\overline{\mathfrak{B}}(\mathcal{H}\mathfrak{B}-\mathfrak{B}\mathcal{H})=0.\end{split}$

Similarly

$\begin{split}[\mathcal{L},\Delta_{\mathfrak{B}}]&=\mathcal{L}\mathfrak{B}\overline{\mathfrak{B}}+\mathcal{L}\overline{\mathfrak{B}}\mathfrak{B}-\mathfrak{B}\overline{\mathfrak{B}}\mathcal{L}-\overline{\mathfrak{B}}\mathfrak{B}\mathcal{L}\\\
&=\mathcal{L}\mathfrak{B}\overline{\mathfrak{B}}+\mathcal{L}\overline{\mathfrak{B}}\mathfrak{B}-\mathfrak{B}\overline{\mathfrak{B}}\mathcal{L}-\overline{\mathfrak{B}}\mathfrak{B}\mathcal{L}+\mathfrak{B}\mathcal{L}\overline{\mathfrak{B}}-\mathfrak{B}\mathcal{L}\overline{\mathfrak{B}}\\\
&=\big{(}(\mathcal{L}\mathfrak{B}+\mathfrak{B}\mathcal{L})\overline{\mathfrak{B}}-\mathfrak{B}(\overline{\mathfrak{B}}\mathcal{L}+\mathcal{L}\overline{\mathfrak{B}})+\mathcal{L}\overline{\mathfrak{B}}\mathfrak{B}-\overline{\mathfrak{B}}\mathfrak{B}\mathcal{L}\big{)}\\\
&=\big{(}-\mathfrak{B}\overline{\mathfrak{B}}^{\text{\char
73}}+(\mathcal{L}\overline{\mathfrak{B}}+\overline{\mathfrak{B}}\mathcal{L})\mathfrak{B}-\overline{\mathfrak{B}}(\mathfrak{B}\mathcal{L}+\mathcal{L}\mathfrak{B})\big{)}\\\
&=[\overline{\mathfrak{B}}^{\text{\char 73}},\mathfrak{B}].\end{split}$

The remaining identities are obtained by conjugating with ✩ and complex
conjugation. ∎

##  7\. Almost Kähler identities via the Riemannian Dirac Operator

######  Lemma 7.1.

Let $M$ be an almost Kähler manifold. For any $v\in TM$ and any
$\varphi\in\Gamma\mathbb{C}l(M)$ we have

$\left[\widetilde{\nabla}_{v},\mathcal{J}\right]\varphi=i\left(J_{\textrm{alg}}^{-1}\widetilde{\nabla}_{Jv}(J_{\textrm{alg}}\varphi)-\widetilde{\nabla}_{Jv}\varphi\right).$

###### Proof.

We recall that $\mathcal{J}=-iJ_{\textrm{der}}$ on $\mathbb{C}l(M)$. As the
commutator of two derivations is a derivation, and conjugating a derivation by
an algebra automorphism is a derivation we observe
$[\widetilde{\nabla}_{v},J_{\textrm{der}}]+J_{\textrm{alg}}^{-1}\widetilde{\nabla}_{Jv}\circ
J_{\textrm{alg}}$ is a derivation. Thus, it suffices to check the identity

$[\widetilde{\nabla}_{v},J_{\textrm{der}}]+J_{\textrm{alg}}^{-1}\widetilde{\nabla}_{Jv}\circ
J_{\textrm{alg}}=\widetilde{\nabla}_{Jv}$

on functions and vector fields. Noticing that $J_{\textrm{der}}$ vanishes on
functions, and that $J_{\textrm{alg}}$ is the identity map on functions, we
see that for a smooth function $f$ on $M$ we have

$[\widetilde{\nabla}_{v},J_{\textrm{der}}](f)+J_{\textrm{alg}}^{-1}\widetilde{\nabla}_{Jv}(J_{\textrm{alg}}f)=J_{\textrm{alg}}^{-1}\widetilde{\nabla}_{Jv}(J_{\textrm{alg}}f)=\widetilde{\nabla}_{Jv}(f).$

If $X\in\Gamma(TM)$ then

$[\widetilde{\nabla}_{v},J_{\textrm{der}}](X)+J_{\textrm{alg}}^{-1}\widetilde{\nabla}_{Jv}(J_{\textrm{alg}}X)=\widetilde{\nabla}_{v}JX-J\widetilde{\nabla}_{v}X+J^{-1}\widetilde{\nabla}_{Jv}JX=\widetilde{\nabla}_{v}JX-J\widetilde{\nabla}_{v}X-J\widetilde{\nabla}_{Jv}JX.$

When $d\omega=0$ we have (see, for instance, [KN63]) the identity
$(\widetilde{\nabla}_{v}J)=J(\widetilde{\nabla}_{Jv}J)$. That is

$\widetilde{\nabla}_{v}JX-J\widetilde{\nabla}_{v}X=J\big{(}\widetilde{\nabla}_{Jv}JX-J\widetilde{\nabla}_{Jv}X\big{)}$

and hence

$[\widetilde{\nabla}_{v},J_{\textrm{der}}](X)+J_{\textrm{alg}}^{-1}\widetilde{\nabla}_{Jv}(J_{\textrm{alg}}X)=\widetilde{\nabla}_{v}JX-J\widetilde{\nabla}_{v}X-J\widetilde{\nabla}_{Jv}JX=-J^{2}\widetilde{\nabla}_{Jv}X=\widetilde{\nabla}_{Jv}X.$

The result follows by substituting the identity
$\mathcal{J}=-iJ_{\textrm{der}}$ on the complex Clifford bundle
$\mathbb{C}l(M).$ ∎

######  Proposition 7.2.

On any almost Kähler manifold $M$ we have the following identities

$\begin{split}[\widetilde{D},\mathcal{H}]=-i\widetilde{D}_{c}\hskip
14.45377pt&\hskip
14.45377pt[\widetilde{D}_{c},\mathcal{H}]=\phantom{-}i\widetilde{D}\\\
[\widetilde{D}^{\text{\char
73}},\mathcal{H}]=\phantom{-}i\widetilde{D}_{c}^{\text{\char 73}}\hskip
14.45377pt&\hskip 14.45377pt[\widetilde{D}_{c}^{\text{\char
73}},\mathcal{H}]=-i\widetilde{D}^{\text{\char 73}}.\end{split}$

###### Proof.

Let $v_{1},\dots,v_{2n}$ be a $J$-adapted orthonormal frame of $TM$. We first
compute $[\widetilde{D},\mathcal{J}]$. For any
$\varphi\in\Gamma(\mathbb{C}l(M))$ we have

$\begin{split}[\widetilde{D},\mathcal{J}](\varphi)&=\sum_{j}\big{(}v_{j}\cdot\widetilde{\nabla}_{v_{j}}\mathcal{J}\varphi-\mathcal{J}(v_{j}\cdot\widetilde{\nabla}_{v_{j}}\varphi)\big{)}\\\
&=\sum_{j}\big{(}v_{j}\cdot\widetilde{\nabla}_{v_{j}}\mathcal{J}\varphi+iJv_{j}\cdot\widetilde{\nabla}_{v_{j}}\varphi-
v_{j}\cdot\mathcal{J}(\widetilde{\nabla}_{v_{j}}\varphi)\big{)}\\\
&=\sum_{j}\big{(}v_{j}\cdot\left[\widetilde{\nabla}_{v_{j}},\mathcal{J}\right](\varphi)+iJv_{j}\cdot\widetilde{\nabla}_{v_{j}}\varphi\big{)}.\\\
\end{split}$

By the identity
$\left[\widetilde{\nabla}_{v},\mathcal{J}\right]\varphi=i\left(J_{\textrm{alg}}^{-1}\widetilde{\nabla}_{Jv}J_{\textrm{alg}}\varphi-\widetilde{\nabla}_{Jv}\varphi\right)$
above, we have

$\begin{split}[\widetilde{D},\mathcal{J}](\varphi)&=i\sum_{j}\big{(}v_{j}\cdot
J_{\textrm{alg}}^{-1}\widetilde{\nabla}_{Jv_{j}}J_{\textrm{alg}}\varphi-
v_{j}\cdot\widetilde{\nabla}_{Jv_{j}}\varphi+Jv_{j}\cdot\widetilde{\nabla}_{v_{j}}\varphi\big{)}\\\
&=i\widetilde{D}_{c}(\varphi)-2i\sum_{j}v_{j}\cdot\widetilde{\nabla}_{Jv_{j}}\varphi.\end{split}$

Let
$\widetilde{\rotatebox[origin={c}]{180.0}{D}}(\varphi)=\sum_{j}v_{j}\cdot\widetilde{\nabla}_{Jv_{j}}\varphi$
and let $L_{\omega_{0}}$ be left-multiplication by $\omega_{0}$. Clearly
$\mathcal{H}+\mathcal{J}=2L_{\omega_{0}}$ and
$[\widetilde{D},\mathcal{J}]=i\widetilde{D}_{c}-2i\widetilde{\rotatebox[origin={c}]{180.0}{D}}$.
Furthermore, for any $\varphi\in\Gamma(\mathbb{C}l(M))$

$\begin{split}\widetilde{D}(\omega_{0}\cdot\varphi)&=\sum_{j}\big{(}v_{j}\cdot(\widetilde{\nabla}_{v_{j}}\omega_{0})\cdot\varphi+v_{j}\cdot\omega_{0}\cdot\widetilde{\nabla}_{v_{j}}\varphi\big{)}\\\
&=\widetilde{D}(\omega_{0})\cdot\varphi+\sum_{j}\big{(}(\omega_{0}\cdot
v_{j}+iJv_{j})\cdot\widetilde{\nabla}_{v_{j}}(\varphi)\big{)}\\\
&=\omega_{0}\cdot\widetilde{D}(\varphi)-i\widetilde{\rotatebox[origin={c}]{180.0}{D}}(\varphi).\end{split}$

Thus we have that
$[\widetilde{D},L_{\omega_{0}}]=-i\widetilde{\rotatebox[origin={c}]{180.0}{D}}$.
But then

$[\widetilde{D},\mathcal{H}]+[\widetilde{D},\mathcal{J}]=2[\widetilde{D},L_{\omega_{0}}]=-2i\widetilde{\rotatebox[origin={c}]{180.0}{D}}$

and so $[\widetilde{D},\mathcal{H}]=-i\widetilde{D}_{c}$. The remaining
identities are obtained by conjugating with $J_{\textrm{alg}}$ and ✩ as in the
proof of Theorem 3.2. ∎

######  Lemma 7.3.

Let $M$ be an almost Kähler manifold. Then
$[\widetilde{D}_{c},\widetilde{D}^{\text{\char
73}}]=[\widetilde{D},\widetilde{D}_{c}^{\text{\char 73}}]$.

###### Proof.

By the identites in Proposition 7.2 we have

$[\widetilde{D}_{c},\widetilde{D}^{\text{\char
73}}]=i[[\widetilde{D},\mathcal{H}],\widetilde{D}^{\text{\char 73}}]$

and

$[\widetilde{D},\widetilde{D}_{c}^{\text{\char
73}}]=-i[\widetilde{D},[\widetilde{D}^{\text{\char 73}},\mathcal{H}]].$

By use of Lemma 6.10 it is quickly checked that the difference

$[\widetilde{D}_{c},\widetilde{D}^{\text{\char
73}}]-[\widetilde{D},\widetilde{D}_{c}^{\text{\char
73}}]=i[[\widetilde{D},\widetilde{D}^{\text{\char 73}}],\mathcal{H}]=0.\qed$

As a consequence of Proposition 7.2 we recover the almost Kähler identities.
(See [CW20], [TT20]).

######  Proposition 7.4.

Let $M$ be an almost Kähler manifold. We have the following generalizations of
the Kähler identities:

$\begin{split}[\Lambda,\overline{\delta}]=-i\delta^{*}\hskip 14.45377pt&\hskip
14.45377pt[L,\delta^{*}]=i\overline{\delta}\\\
[\Lambda,\delta]=i\overline{\delta}^{*}\ \ \hskip 14.45377pt&\hskip
14.45377pt[L,\overline{\delta}^{*}]=-i\delta\\\
0=[\Lambda,\delta^{*}]=[\Lambda,\overline{\delta}^{*}]\hskip 7.22743pt&\hskip
18.06749pt[L,\delta]=[L,\overline{\delta}]=0.\end{split}$

###### Proof.

By the identity $[\widetilde{D},\mathcal{H}]=-i\widetilde{D}_{c}$ we have that
$[d+d^{*},i(\Lambda-L)]=i(d_{c}+d_{c}^{*})$ or equivalently

$i[d,\Lambda]+i[d^{*},\Lambda]-i[d,L]-i[d^{*},L]=id_{c}+id_{c}^{*}.$

By considering degree of each operator in the above equality we obtain

$[d,\Lambda]=d_{c}^{*}\hskip 14.45377pt[d^{*},L]=-d_{c}\hskip
14.45377pt\textrm{ and }\hskip 7.22743pt[d,L]=[d^{*},\Lambda]=0.$

Furthermore since

$[\Lambda,d]=[\Lambda,\delta+\overline{\delta}]=[\Lambda,\partial+\overline{\mu}+\overline{\partial}+\mu]=[\Lambda,\partial]+[\Lambda,\overline{\mu}]+[\Lambda,\overline{\partial}]+[\Lambda,\mu]$

the above implies
$[\Lambda,d]=-id_{c}^{*}=i(\overline{\delta}^{*}-\delta^{*})=i(\overline{\partial}^{*}+\mu^{*}-\partial^{*}-\overline{\mu}^{*})$
as well. The argument is again completed by considering the effect of each
operator in the equality on bidegree. ∎

It is quick to check, using the identities of Proposition 7.4 that
$\Delta_{\delta}=\Delta_{\overline{\delta}}$ in the almost Kähler setting.
(See, for instance, [TT20, Proposition 6.2]). Hence by Proposition 6.12 we
have

######  Corollary 7.5.

For an almost Kähler manifold $M$ we have through the identification
$\Gamma(\mathbb{C}l(M))\cong\Omega_{\mathbb{C}}(M)$ that

$\Delta_{\mathfrak{D}}=2\Delta_{\delta}.$

######  Proposition 7.6.

On any almost Kähler manifold $M$ we have the following identities

$\begin{split}\left\\{\mathcal{L}+\overline{\mathcal{L}},\widetilde{D}\right\\}&=\widetilde{D}^{\text{\char
73}}\hskip
36.135pt\left\\{\mathcal{L}-\overline{\mathcal{L}},\widetilde{D}\right\\}\
=-i\widetilde{D}_{c}^{\text{\char 73}}\\\
\left\\{\mathcal{L}+\overline{\mathcal{L}},\widetilde{D}^{\text{\char
73}}\right\\}&=\widetilde{D}\hskip 36.135pt\
\left\\{\mathcal{L}-\overline{\mathcal{L}},\widetilde{D}^{\text{\char
73}}\right\\}=\phantom{-}i\widetilde{D}_{c}\\\
\left\\{\mathcal{L}+\overline{\mathcal{L}},\widetilde{D}_{c}\right\\}&=\widetilde{D}_{c}^{\text{\char
73}}\hskip
36.135pt\left\\{\mathcal{L}-\overline{\mathcal{L}},\widetilde{D}_{c}\right\\}=\phantom{-}i\widetilde{D}^{\text{\char
73}}.\\\ \end{split}$

###### Proof.

We have through the identification
$\Gamma(\mathbb{C}l(M))\cong\Omega_{\mathbb{C}}(M)$ that

$\begin{split}\left\\{\mathcal{L}+\overline{\mathcal{L}},\widetilde{D}\right\\}&=\alpha
H(d+d^{*})+(d+d^{*})\alpha H\\\ &=\alpha[H,d+d^{*}]=\alpha([H,d]+[H,d^{*}])\\\
&=d\alpha-d^{*}\alpha\\\ &=\widetilde{D}^{\text{\char 73}}\end{split}$

and

$\begin{split}\left\\{\mathcal{L}-\overline{\mathcal{L}},\widetilde{D}\right\\}&=-i\left(\alpha(\Lambda+L)(d+d^{*})+(d+d^{*})\alpha(\Lambda+L)\right)=-i\alpha[(\Lambda+L),d+d^{*}]\\\
&=-i\alpha[\Lambda,d]-i\alpha[L,d^{*}]=i\alpha d_{c}^{*}-i\alpha d_{c}\\\
&=-i(d_{c}^{*}\alpha-d_{c}\alpha).\end{split}$

Hence
$\left\\{\mathcal{L}-\overline{\mathcal{L}},\widetilde{D}\right\\}=-i\widetilde{D}_{c}^{\text{\char
73}}$. The remaining identities are again obtained by conjugating with the
operators $J_{\textrm{alg}}$ and $\text{\char 73}.$ ∎

######  Proposition 7.7.

On any almost Kähler manifold $M$ the following identities obtain

$\begin{split}\left\\{\mathfrak{D},\mathcal{L}\right\\}=0\hskip
36.135pt&\hskip
36.135pt\left\\{\overline{\mathfrak{D}},\overline{\mathcal{L}}\right\\}=0\\\
\left\\{\mathfrak{D},\overline{\mathcal{L}}\right\\}=\mathfrak{D}^{\text{\char
73}}\hskip 28.90755pt&\hskip
36.135pt\left\\{\overline{\mathfrak{D}},\mathcal{L}\right\\}=\overline{\mathfrak{D}}^{\text{\char
73}}\\\ \hskip 36.135pt[\mathcal{H},&\mathfrak{D}]=\mathfrak{D}.\end{split}$

###### Proof.

We observe that by Proposition 7.6

$\begin{split}\left\\{\mathfrak{D},\mathcal{L}\right\\}&=\tfrac{1}{2}\big{(}(\widetilde{D}+i\widetilde{D}_{c})\mathcal{L}+\mathcal{L}(\widetilde{D}+i\widetilde{D}_{c})\big{)}=\tfrac{1}{2}\big{(}\left\\{\widetilde{D},\mathcal{L}\right\\}+i\left\\{\widetilde{D}_{c},\mathcal{L}\right\\}\big{)}\\\
&=\tfrac{1}{4}\big{(}(\widetilde{D}^{\text{\char
73}}-i\widetilde{D}_{c}^{\text{\char 73}})+i(\widetilde{D}_{c}^{\text{\char
73}}+i\widetilde{D}^{\text{\char 73}})\big{)}=0.\end{split}$

The argument
$\left\\{\mathfrak{D},\overline{\mathcal{L}}\right\\}=\mathfrak{D}^{\text{\char
73}}$ follows similarly. By Proposition 7.2

$\begin{split}[\mathcal{H},\mathfrak{D}]&=[\mathcal{H},\tfrac{1}{2}(\widetilde{D}+i\widetilde{D}_{c})]=\tfrac{1}{2}([\mathcal{H},\widetilde{D}]+i[\mathcal{H},\widetilde{D}_{c}])\\\
&=\tfrac{1}{2}(\widetilde{D}+i\widetilde{D}_{c})=\mathfrak{D}.\end{split}$

The remaining identities follow by conjugating by complex conjugation and
Lemma 6.14. ∎

######  Proposition 7.8.

On any almost Kähler manifold $M$ we have

$[\mathcal{H},\Delta_{\mathfrak{D}}]=0,\hskip
18.06749pt[\mathcal{L},\Delta_{\mathfrak{D}}]=0\hskip 7.22743pt\textrm{ and
}\hskip 7.22743pt[\overline{\mathcal{L}},\Delta_{\mathfrak{D}}]=0.$

###### Proof.

Using Proposition 7.7 and an identical argument to the proof of Proposition
6.16 we obtain

$[\mathcal{H},\Delta_{\mathfrak{D}}]=0\textrm{ and
}[\mathcal{L},\Delta_{\mathfrak{D}}]=[\overline{\mathfrak{D}}^{\text{\char
73}},\mathfrak{D}].$

By definition of $\mathfrak{D}$ and $\overline{\mathfrak{D}}^{\text{\char
73}}$ we see that

$[\overline{\mathfrak{D}}^{\text{\char
73}},\mathfrak{D}]=\tfrac{1}{4}\left([\widetilde{D}^{\text{\char
73}},\widetilde{D}]+[\widetilde{D}_{c}^{\text{\char
73}},\widetilde{D}_{c}]-i[\widetilde{D}_{c}^{\text{\char
73}},\widetilde{D}]+i[\widetilde{D}^{\text{\char
73}},\widetilde{D}_{c}]\right)$

and so the result follows by Lemmas 6.10 and 7.3. That is

$[\widetilde{D},\widetilde{D}^{\text{\char 73}}]=0\hskip 14.45377pt\textrm{
and }\hskip 14.45377pt[\widetilde{D}_{c},\widetilde{D}^{\text{\char
73}}]-[\widetilde{D}_{c}^{\text{\char 73}},\widetilde{D}]=0.$

∎

##  8\. Clifford Harmonics

Let $M$ be a compact almost Hermitian manifold. For any finite collection of
operators $T_{j}$, for $j=1,\dots,n$, on
$\Gamma\mathbb{C}l(M)\cong\Omega_{\mathbb{C}}(M)$ we have

$\displaystyle\varphi\in\ker\left(\sum_{j=1}^{n}\Delta_{T_{j}}\right)\hskip
3.61371pt\textrm{ if any only if }\hskip
7.22743pt0=\displaystyle\sum_{j=1}^{n}||T_{j}\varphi||^{2}+||T_{j}^{*}\varphi||^{2}.\hskip
115.63243pt$

I.e. $\displaystyle\varphi\in\ker\left(\sum_{j=1}^{n}\Delta_{T_{j}}\right)$ if
and only if
$\displaystyle\varphi\in\bigcap_{j=1}^{n}\ker(T_{j})\cap\ker(T_{j}^{*})$.

######  Definition 8.1.

For an elliptic differential operator $T$ on
$\mathcal{A}=\Gamma\mathbb{C}l(M),\hskip 0.72229pt\Omega_{\mathbb{C}}(M)$ we
define

$\bm{\mathcal{H}}_{T}=\ker(\Delta_{T}).$

Furthermore we define

$\bm{\mathcal{H}}_{T}^{r,s}=\bm{\mathcal{H}}_{T}\cap\mathcal{A}^{r,s}.$

Recall from Theorem 2.8 the Hodge automorphism $g=\textrm{exp}(\tfrac{-\pi
i}{\phantom{-}4}H)\textrm{exp}(\tfrac{\pi}{4}(\Lambda-L))$. By Section 2.4 we
rewrite

$g=\textrm{exp}\big{(}-\tfrac{\pi
i}{4}\alpha(\mathcal{L}+\overline{\mathcal{L}})\big{)}\textrm{exp}\big{(}-\tfrac{\pi
i}{4}\mathcal{H}\big{)}.$

######  Proposition 8.2.

Let $M$ be a compact almost Hermitian manifold. The action of the Hodge
automorphism $g$ on $\Gamma(\mathbb{C}l(M))$ preserves
$\bm{\mathcal{H}}_{\mathfrak{B}}\cap\bm{\mathcal{H}}_{\mathfrak{B}^{\text{\char
73}}}$.

###### Proof.

By Proposition 6.16 we have

$\begin{split}[\alpha(\mathcal{L}+\overline{\mathcal{L}}),\Delta_{\mathfrak{B}}]=\alpha[\mathfrak{B}^{\text{\char
73}},\overline{\mathfrak{B}}]+\alpha[\overline{\mathfrak{B}}^{\text{\char
73}},\mathfrak{B}],\hskip 14.45377pt&\hskip
14.45377pt[\alpha(\mathcal{L}+\overline{\mathcal{L}}),\Delta_{\mathfrak{B}^{\text{\char
73}}}]=-\alpha[\mathfrak{B}^{\text{\char
73}},\overline{\mathfrak{B}}]-\alpha[\overline{\mathfrak{B}}^{\text{\char
73}},\mathfrak{B}]\\\ \textrm{ and }\hskip
10.84006pt[\mathcal{H},\Delta_{\mathfrak{B}}]&=0.\\\ \end{split}$

The first two equalities imply
$[\alpha(\mathcal{L}+\overline{\mathcal{L}}),\Delta_{\mathfrak{B}}+\Delta_{\mathfrak{B}^{\text{\char
73}}}]=0$. Since $\Delta_{\mathfrak{B}}+\Delta_{\mathfrak{B}^{\text{\char
73}}}$ commutes with $\alpha(\mathcal{L}+\overline{\mathcal{L}})$ and
$\mathcal{H}$, and as $g$ is the product of the exponentials of these
operators, we have

$g\big{(}\Delta_{\mathfrak{B}}+\Delta_{\mathfrak{B}^{\text{\char
73}}}\big{)}=\big{(}\Delta_{\mathfrak{B}}+\Delta_{\mathfrak{B}^{\text{\char
73}}}\big{)}g.$

This gives the result. ∎

######  Proposition 8.3.

Let $M$ be a compact almost Kähler manifold. The action of the Hodge
automorphism $g$ on $\Gamma(\mathbb{C}l(M))$ preserves
$\bm{\mathcal{H}}_{\mathfrak{D}}$.

###### Proof.

By Proposition 7.8 we have

$[\alpha(\mathcal{L}+\overline{\mathcal{L}}),\Delta_{\mathfrak{D}}]=0\hskip
21.68121pt\textrm{ and }\hskip
10.84006pt[\mathcal{H},\Delta_{\mathfrak{D}}]=0.\hskip 158.99377pt$

Since $\Delta_{\mathfrak{D}}$ commutes with
$\alpha(\mathcal{L}+\overline{\mathcal{L}})$ and $\mathcal{H}$, and as $g$ is
the product of the exponentials of these operators, we have

$g(\Delta_{\mathfrak{D}})=(\Delta_{\mathfrak{D}})g.$

This again gives the result. ∎

######  Proposition 8.4.

Let $M$ be a compact almost Hermitian manifold. Complex conjugation $c$ on
$\mathbb{C}l(M)$ induces isomorphisms

$c:\bm{\mathcal{H}}_{\mathfrak{B}}^{r,s}\cap\bm{\mathcal{H}}_{\mathfrak{B}^{\text{\char
73}}}^{r,s}\rightarrow\bm{\mathcal{H}}_{\mathfrak{B}}^{-r,-s}\cap\bm{\mathcal{H}}_{\mathfrak{B}^{\text{\char
73}}}^{-r,-s},$
$c:\bm{\mathcal{H}}_{\mathfrak{D}}^{r,s}\longrightarrow\bm{\mathcal{H}}_{\mathfrak{D}}^{-r,-s}.\hskip
70.10197pt$

###### Proof.

We recall that by Lemma 6.14

$c\ \mathfrak{D}c=\overline{\mathfrak{D}},\hskip 21.68121ptc\
\mathfrak{B}c=\overline{\mathfrak{B}},\hskip 21.68121ptc\
\mathfrak{B}^{\text{\char 73}}c=\overline{\mathfrak{B}^{\text{\char 73}}}.$

Hence $c\ \big{(}\Delta_{\mathfrak{D}}\big{)}c=\Delta_{\mathfrak{D}}$ and $c\
\big{(}\Delta_{\mathfrak{B}}+\Delta_{\mathfrak{B}^{\text{\char
73}}}\big{)}c=\Delta_{\mathfrak{B}}+\Delta_{\mathfrak{B}^{\text{\char 73}}}$.
That is

$c:\Gamma(\mathbb{C}l^{r,s}(M))\rightarrow\Gamma(\mathbb{C}l^{-r,-s}(M))$

descends to complex antilinear isomorphisms

$c:\bm{\mathcal{H}}_{\mathfrak{D}}^{r,s}\rightarrow\bm{\mathcal{H}}_{\mathfrak{D}}^{-r,-s}\textrm{
and }\hskip
7.22743ptc:\bm{\mathcal{H}}_{\mathfrak{B}}^{r,s}\cap\bm{\mathcal{H}}_{\mathfrak{B}^{\text{\char
73}}}^{r,s}\rightarrow\bm{\mathcal{H}}_{\mathfrak{B}}^{-r,-s}\cap\bm{\mathcal{H}}_{\mathfrak{B}^{\text{\char
73}}}^{-r,-s}.\qed$

######  Proposition 8.5.

Let $M$ be a compact almost Hermitian manifold. The transpose map ✩ induces
isomorphisms

$\text{\char
73}:\bm{\mathcal{H}}_{\mathfrak{B}}^{r,s}\cap\bm{\mathcal{H}}_{\mathfrak{B}^{\text{\char
73}}}^{r,s}\rightarrow\bm{\mathcal{H}}_{\mathfrak{B}}^{r,-s}\cap\bm{\mathcal{H}}_{\mathfrak{B}^{\text{\char
73}}}^{r,-s},$ $\text{\char
73}:\bm{\mathcal{H}}_{\mathfrak{D}}^{r,s}\longrightarrow\bm{\mathcal{H}}_{\mathfrak{D}}^{r,-s}.\hskip
65.04256pt$

###### Proof.

We observe $\text{\char 73}\big{(}\Delta_{\mathfrak{D}}\big{)}\text{\char
73}=\Delta_{\mathfrak{D}^{\text{\char 73}}}$ and that $\text{\char
73}\big{(}\Delta_{\mathfrak{B}}+\Delta_{\mathfrak{B}^{\text{\char
73}}}\big{)}\text{\char
73}=\Delta_{\mathfrak{B}}+\Delta_{\mathfrak{B}^{\text{\char 73}}}$. By
Proposition 6.12 we have
$\Delta_{\mathfrak{D}}=\Delta_{\mathfrak{D}^{\text{\char 73}}}$ and so the
complex algebra anti-automorphism

$\text{\char
73}:\Gamma(\mathbb{C}l^{r,s})\rightarrow\Gamma(\mathbb{C}l^{r,-s})$

descends to complex linear isomorphisms

$\text{\char
73}:\bm{\mathcal{H}}_{\mathfrak{D}}^{r,s}\rightarrow\bm{\mathcal{H}}_{\mathfrak{D}}^{r,-s}\textrm{
and }\hskip 7.22743pt\text{\char
73}:\bm{\mathcal{H}}_{\mathfrak{B}}^{r,s}\cap\bm{\mathcal{H}}_{\mathfrak{B}^{\text{\char
73}}}^{r,s}\rightarrow\bm{\mathcal{H}}_{\mathfrak{B}}^{r,-s}\cap\bm{\mathcal{H}}_{\mathfrak{B}^{\text{\char
73}}}^{r,-s}.\qed$

######  Theorem 8.6.

Let $M$ be a compact almost Hermitian manifold. The Lie algebra generated by
$\big{(}\mathcal{H},\mathcal{L},\overline{\mathcal{L}}\big{)}$ on
$\Gamma\mathbb{C}l(M)$ defines a finite dimensional $sl(2)$ representation on
the space
$\bm{\mathcal{H}}_{\mathfrak{B}}\cap\bm{\mathcal{H}}_{\mathfrak{B}^{\text{\char
73}}}.$

###### Proof.

By Proposition 6.16 we have

$[\mathcal{L},\Delta_{\mathfrak{B}}+\Delta_{\mathfrak{B}^{\text{\char
73}}}]=0,\hskip
7.22743pt[\overline{\mathcal{L}},\Delta_{\mathfrak{B}}+\Delta_{\mathfrak{B}^{\text{\char
73}}}]=0,\hskip 7.22743pt\textrm{ and
}[\mathcal{H},\Delta_{\mathfrak{B}}+\Delta_{\mathfrak{B}^{\text{\char
73}}}]=0.$

Hence each of the operators $\mathcal{L},\overline{\mathcal{L}},\mathcal{H}$
preserve
$\bm{\mathcal{H}}_{\mathfrak{B}}\cap\bm{\mathcal{H}}_{\mathfrak{B}^{\text{\char
73}}}$. ∎

######  Theorem 8.7.

Let $M$ be a compact almost Hermitian manifold. Through the isomorphism

$\Gamma\mathbb{C}l(M)\cong\Omega_{\mathbb{C}}(M)$

we have

$\bm{\mathcal{H}}_{\mathfrak{B}}\cap\bm{\mathcal{H}}_{\mathfrak{B}^{\text{\char
73}}}\cong\bm{\mathcal{H}}_{\varepsilon}\cap\bm{\mathcal{H}}_{\overline{\varepsilon}}.$

###### Proof.

By Proposition 6.13 we have
$2\big{(}\Delta_{\varepsilon}+\Delta_{\overline{\varepsilon}}\big{)}=\Delta_{\mathfrak{B}}+\Delta_{\mathfrak{B}^{\text{\char
73}}}$. ∎

######  Theorem 8.8.

Let $M$ be a compact almost Kähler manifold. The Lie algebra generated by
$\big{(}\mathcal{H},\mathcal{L},\overline{\mathcal{L}}\big{)}$ on
$\Gamma\mathbb{C}l(M)$ defines a finite dimensional $sl(2)$ representation on
the space $\bm{\mathcal{H}}_{\mathfrak{D}}$.

###### Proof.

By Proposition 7.8 we have

$[\mathcal{L},\Delta_{\mathfrak{D}}]=0,\hskip
7.22743pt[\overline{\mathcal{L}},\Delta_{\mathfrak{D}}]=0,\hskip
7.22743pt\textrm{ and }[\mathcal{H},\Delta_{\mathfrak{D}}]=0.$

Hence $\mathcal{L},\overline{\mathcal{L}},\mathcal{H}$ preserve
$\bm{\mathcal{H}}_{\mathfrak{D}}.$ ∎

######  Theorem 8.9.

Let $M$ be a compact almost Kähler manifold. Through the isomorphism

$\Gamma\mathbb{C}l(M)\cong\Omega_{\mathbb{C}}(M)$

we have

$\bm{\mathcal{H}}_{\mathfrak{D}}\cong\bm{\mathcal{H}}_{\delta}.$

###### Proof.

By Corollary 7.5 we have $2\Delta_{\delta}=\Delta_{\mathfrak{D}}$. ∎

######  Corollary 8.10.

On any compact almost Hermitian manifold $M$ we have that the Hodge
automorphism induces an isomorphism

$\bm{\mathcal{H}}_{\mathfrak{B}}^{q-p,n-p-q}\cap\bm{\mathcal{H}}_{\mathfrak{B}^{\text{\char
73}}}^{q-p,n-p-q}\cong\bm{\mathcal{H}}_{\varepsilon}^{p,q}\cap\bm{\mathcal{H}}_{\overline{\varepsilon}}^{p,q}.$

Moreover, on any compact almost Kähler manifold $M$ we have that

$\bm{\mathcal{H}}_{\mathfrak{D}}^{q-p,n-p-q}\cong\bm{\mathcal{H}}_{\delta}^{p,q}.$

###### Proof.

By Theorem 2.8 we have

$\Gamma(\mathbb{C}l^{q-p,n-p-q}(M))\stackrel{{\scriptstyle
g}}{{\cong}}\Omega_{\mathbb{C}}^{p,q}(M).$

The result then follows from Proposition 8.3 and the previous two theorems. ∎

######  Example 8.11.

Let $M$ be the Kodaira-Thurston manifold. Let $x,y,z,w$ be an orthonormal
left-invariant coframe on $M$ with $dz=x\wedge y$ and $dx=dy=dw=0$. We define
an almost complex structure $J$ by $Jy=x$ and $Jz=w$. We compute
$d\omega=x\wedge y\wedge w\neq 0$ and $N_{J}=0$ so that this almost complex
structure is integrable and not an almost Kähler structure on $M$.

Recalling the operators $\varepsilon=\partial-i\rho_{\partial}\hskip
3.61371pt\textrm{,}\hskip
3.61371pt\overline{\varepsilon}=\overline{\partial}+i\overline{\rho_{\partial}}\hskip
3.61371pt$ and the spaces

$\bm{\mathcal{H}}_{\varepsilon}^{p,q}(M)\cap\bm{\mathcal{H}}_{\overline{\varepsilon}}^{p,q}(M)=\ker(\varepsilon)\cap\ker(\overline{\varepsilon})\cap\ker(\varepsilon^{*})\cap\ker(\overline{\varepsilon}^{*})\cap\Omega_{\mathbb{C}}^{p,q}(M),$

we compute the “Hodge diamond” of their dimensions to be:

${\mathbf{1}}$${\mathbf{1}}$${\mathbf{1}}$${\mathbf{0}}$${\mathbf{2}}$${\mathbf{0}}$${\mathbf{1}}$${\mathbf{1}}$${\mathbf{1}}$

## Appendix: Hermitian Connections and Dirac Operators

For the convenience of the reader we give here an exposition, in our notation,
of the parts of Gauduchon’s paper [G97] relevant for us.

######  Lemma 8.12.

For any vector fields $X,Y,Z$ on an almost Hermitian manifold $M$ we have

$(\widetilde{\nabla}_{X}\omega)(Y,Z)=\left\langle(\widetilde{\nabla}_{X}J)Y,Z\right\rangle.$

###### Proof.

For vector fields $X,Y,Z\in\Gamma(TM)$ we observe

$\begin{split}(\widetilde{\nabla}_{X}\omega)(Y\wedge Z)&=X(\omega(Y\wedge
Z))-\omega(\widetilde{\nabla}_{X}(Y\wedge Z))\\\ &=X\big{(}\langle
JY,Z\rangle\big{)}-\omega\big{(}(\widetilde{\nabla}_{X}Y)\wedge
Z+Y\wedge(\widetilde{\nabla}_{X}Z)\big{)}\\\
&=\langle\widetilde{\nabla}_{X}JY,Z\rangle+\langle
JY,\widetilde{\nabla}_{X}Z\rangle-\langle
J\widetilde{\nabla}_{X}Y,Z\rangle-\langle JY,\widetilde{\nabla}_{X}Z\rangle\\\
&=\langle\widetilde{\nabla}_{X}JY,Z\rangle-\langle
J\widetilde{\nabla}_{X}Y,Z\rangle=\left\langle(\widetilde{\nabla}_{X}J)Y,Z\right\rangle.\qed\end{split}$

In fact, the elements $N,d\omega$ and $\widetilde{\nabla}\omega$ of
$\Omega^{2}(TM)$ are related by the following well known fundamental identity
of almost Hermitian geometry (see for instance [KN63] where disparities in the
conventional definitions of $N$ and the exterior derivative $d$ affect the
coefficients in the statement)

######  Lemma 8.13.

[KN63] On any almost Hermitian manifold $M$ we have

$\tfrac{1}{2}\widetilde{\nabla}\omega(X,Y,Z)=\tfrac{1}{4}d\omega(X,Y,Z)-\tfrac{1}{4}d\omega(X,JY,JZ)+N(JX,Y,Z).\hskip
36.135pt$

######  Definition 8.14.

Let $\nabla$ be an affine metric connection. We define the potential
$A^{\nabla}\in\Omega^{2}(TM)$ of $\nabla$ by

$A^{\nabla}(X,Y,Z)=\big{\langle}\nabla_{X}(Y)-\widetilde{\nabla}_{X}(Y),Z\big{\rangle}.$

Note that an affine connection $\nabla$ is metric if and only if
$A^{\nabla}\in\Omega^{2}(TM)$. We write $A=A^{\nabla}$ when the referent
connection is apparent.

######  Proposition 8.15.

[G97] An affine metric connection $\nabla$ on $M$ is Hermitian if and only if

$A(X,JY,Z)+A(X,Y,JZ)=-(\widetilde{\nabla}\omega)(X,Y,Z).$

###### Proof.

By Lemma 8.12 we have for all vector fields $X,Y,Z$ that

$(\widetilde{\nabla}\omega)(X,Y,Z)=\big{\langle}(\widetilde{\nabla}_{X}J)Y,Z\big{\rangle}.$

Using that $J$ is orthogonal and the definition of $A$ we observe

$A(X,JY,Z)+A(X,Y,JZ)=\big{\langle}(\nabla_{X}J)Y,Z\big{\rangle}-\big{\langle}(\widetilde{\nabla}_{X}J)Y,Z\big{\rangle}.$

Consequently

$A(X,JY,Z)+A(X,Y,JZ)+(\widetilde{\nabla}\omega)(X,Y,Z)=0\textrm{ if and only
if }\big{\langle}(\nabla_{X}J)Y,Z\big{\rangle}=0$

for all vector fields $X,Y,Z$. That is, if and only if $(\nabla J)=0.$ ∎

######  Definition 8.16.

[G97] The subspaces $\Omega^{2,0}(TM),\ \Omega^{0,2}(TM),\ \Omega^{1,1}(TM)$
of $\Omega^{2}(TM)$ are defined by

$\begin{split}\phi&\in\Omega^{2,0}(TM)\textrm{ if and only if
}\phi(X,JY,Z)=-\phi(JX,Y,Z).\\\ \phi&\in\Omega^{0,2}(TM)\textrm{ if and only
if }\phi(X,JY,Z)=\phantom{-}\phi(JX,Y,Z).\\\ \phi&\in\Omega^{1,1}(TM)\textrm{
if and only if }\phi(X,JY,JZ)=\phi(X,Y,Z).\end{split}$

There are projections

$p_{j,k}:\Omega(TM)\rightarrow\Omega^{j,k}(TM)\textrm{ with
}j,k\in\left\\{0,1\right\\}\textrm{ and }j+k=2$

defined for any $\phi\in\Omega(TM)$ by

$\begin{split}p_{2,0}(\phi)(X,Y,Z)&=\tfrac{1}{4}\big{(}\phi(X,Y,Z)-\phi(X,JY,JZ)+\phi(JX,Y,JZ)+\phi(JX,JY,Z)\big{)},\\\
p_{0,2}(\phi)(X,Y,Z)&=\tfrac{1}{4}\big{(}\phi(X,Y,Z)-\phi(X,JY,JZ)-\phi(JX,Y,JZ)-\phi(JX,JY,Z)\big{)},\\\
p_{1,1}(\phi)(X,Y,Z)&=\tfrac{1}{2}\big{(}\phi(X,Y,Z)+\phi(X,JY,JZ)\big{)},\end{split}$

so that the space $\Omega^{2}(TM)$ is endowed with a natural orthogonal
splitting

$\Omega^{2}(TM)=\Omega^{2,0}(TM)\oplus\Omega^{1,1}(TM)\oplus\Omega^{0,2}(TM).$

As an example, the Nijenhuis tensor $N\in\Omega^{0,2}(TM)$ as
$N(X,JY,Z)=N(JX,Y,Z)$.

######  Definition 8.17.

For $\phi\in\Omega^{2}(TM)$ we set $p_{j,k}(\phi)=\phi^{j,k}.$

######  Lemma 8.18.

[G97] For any $\phi\in\Omega^{1,1}(TM)$ we have $\phi(X,JY,Z)+\phi(X,Y,JZ)=0.$
Consequently

$(\widetilde{\nabla}\omega)^{1,1}=0.$

###### Proof.

We have $\phi(X,JY,Z)=\phi(X,J^{2}Y,JZ)=-\phi(X,Y,JZ)$ giving the first
result. Writing $A=A^{2,0}+A^{1,1}+A^{0,2}$, Proposition 8.15 implies

$0=A^{1,1}(X,JY,Z)+A^{1,1}(X,Y,JZ)=-(\widetilde{\nabla}\omega)^{1,1}(X,Y,Z).\qed$

The space $\Omega^{2}(TM)$ also has two other distinguished projections which,
apriori, do not involve the almost complex structure $J$.

######  Definition 8.19.

[G97] We define the operator $\mathcal{P}$ on $\Omega^{2}(TM)$ by

$\mathcal{P}(\phi)(X,Y,Z)=\tfrac{1}{3}\big{(}\phi(X,Y,Z)+\phi(Y,Z,X)+\phi(Z,X,Y)\big{)}\hskip
50.58878pt$

and the operator $\mathcal{Q}$, defined by

$\mathcal{Q}(\phi)(X,Y,Z)=\tfrac{1}{2n-1}\big{(}\sum_{j}\phi(v_{j},v_{j},Z)\langle
X,Y\rangle-\sum_{j}\phi(v_{j},v_{j},Y)\langle Z,X\rangle\big{)}$

where $v_{1},\dots,v_{2n}$ is a local orthonormal frame.

Evidently $\mathcal{P}$ is the identity on the restriction to
$\Omega^{3}(M)\subset\Omega^{2}(TM)$ and so $\Omega^{3}(M)$ can be identified
with the image of $\mathcal{P}$ on $\Omega^{2}(TM)$. We also note that the map
$\mathcal{Q}$ is actually the composition of the map onto $1$ forms,
$r:\Omega^{2}(TM)\rightarrow\Omega^{1}(M)$ defined by

$r(\phi)(X)=\sum_{j}\phi(v_{j},v_{j},X)$

with the map $i:\Omega^{1}(M)\rightarrow\Omega^{2}(TM)$ defined by

$i(\varphi)(X,Y,Z)=\tfrac{1}{2n-1}\big{(}\varphi(Z)\langle
X,Y\rangle-\varphi(Y)\langle Z,X\rangle\big{)}$

so that $i\circ r=\mathcal{Q}$. It is also easy to see that $r\circ i$ is the
identity map on $\Omega^{1}(M)$, so that $i$ is an inclusion of $1$-forms into
$\Omega(TM)$ and $r$ is a retraction of $\Omega^{2}(TM)$ onto $1$-forms. We
conclude $\mathcal{P}^{2}=\mathcal{P}$ and
$\mathcal{Q}^{2}=(ir)(ir)=i(ri)r=ir=\mathcal{Q}$ so that $\mathcal{P}$ and
$\mathcal{Q}$ are projections.

######  Lemma 8.20.

$\mathcal{P}\mathcal{Q}=\mathcal{Q}\mathcal{P}=0.$

###### Proof.

Evidently $\mathcal{Q}\mathcal{P}=0$ as $\mathcal{Q}$ vanishes on elements of
$\Omega^{3}(M)$. That $\mathcal{P}\mathcal{Q}=0$ follows from

$\left(\mathcal{P}\circ\mathcal{Q}\right)(\phi)(X,Y,Z)=\mathcal{Q}(\phi)(X,Y,Z)+\mathcal{Q}(\phi)(Y,Z,X)+\mathcal{Q}(\phi)(Z,X,Y)$

and applying the definition of $\mathcal{Q}(\phi)$. ∎

Given two projections $\mathcal{P},\mathcal{Q}$ on a vector space $V$,
satisfying $\mathcal{P}\mathcal{Q}=\mathcal{Q}\mathcal{P}=0$ one can conclude
$V=\textrm{Im}(\mathcal{P})\oplus\textrm{Im}(\mathcal{Q})\oplus\big{(}\ker(\mathcal{Q})\cap\ker(\mathcal{P})\big{)}$.
Thus any $\phi\in\Omega(TM)$ can be uniquely written as

$\phi=\mathcal{P}\phi+\mathcal{Q}\phi+\phi_{0}$

where $\phi_{0}$ is the component of $\phi$ in
$\ker(\mathcal{P})\cap\ker(\mathcal{Q})$. Moreover we have the following
identification

######  Proposition 8.21.

[G97]
$\Omega^{2}(TM)\cong\Omega^{1}(M)\oplus\Omega^{3}(M)\oplus\big{(}\ker(\mathcal{Q})\cap\ker(\mathcal{P})\big{)}.$

###### Proof.

This follows from the above remarks, along with the identifications
$\Omega^{1}(M)\cong\textrm{Im}(\mathcal{Q})$ and
$\Omega^{3}(M)\cong\textrm{Im}(\mathcal{P})$. ∎

######  Proposition 8.22.

[G97] Let $T$ be the torsion of an affine metric connection and $A$ the
potential. Then $A+T=3\mathcal{P}(A)=\tfrac{3}{2}\mathcal{P}(T).$

###### Proof.

We observe

$T(X,Y,Z)=\langle X,T(Y,Z)\rangle=\langle
X,\nabla_{Y}Z-\nabla_{Z}Y-\widetilde{\nabla}_{Y}Z+\widetilde{\nabla}_{Z}Y\rangle=A(Y,Z,X)+A(Z,X,Y)$

so that

$A(X,Y,Z)+T(X,Y,Z)=A(X,Y,Z)+A(Y,Z,X)+A(Z,X,Y)=3\mathcal{P}(A)(X,Y,Z).\hskip
25.29494pt$

Using again that $T(X,Y,Z)=A(Y,Z,X)+A(Z,X,Y)$ and the definition of
$\mathcal{P}$ one obtains $3\mathcal{P}(A)=\tfrac{3}{2}\mathcal{P}(T)$. ∎

Let $J$ be an almost complex structure on the complexified cotangent bundle
$T_{\mathbb{C}}M^{\vee}$ and let $E_{\lambda}(J_{\textrm{der}})$ denote the
$\lambda$ eigenspace of $J_{\textrm{der}}$ where $J_{\textrm{der}}$ is $J$
extended as a derivation on forms. We recall for $k=0,\dots,2n$ we have

$E_{ik}(J_{\textrm{der}})=\bigoplus_{k=p-q}\Omega^{p,q}(M).$

We define the real subspaces $E^{+}$ and $E^{-}$ of $\Omega^{3}(M)$ by

$E^{+}=\textrm{Re}\big{(}E_{i}(J_{\textrm{der}})\oplus
E_{-i}(J_{\textrm{der}})\big{)}\ =\
\textrm{Re}\big{(}\Omega^{2,1}(M)\oplus\Omega^{1,2}(M)\big{)},$
$E^{-}=\textrm{Re}\big{(}E_{3i}(J_{\textrm{der}})\oplus
E_{-3i}(J_{\textrm{der}})\big{)}=\textrm{Re}\big{(}\Omega^{3,0}(M)\oplus\Omega^{0,3}(M)\big{)}$

so that the space of real 3-forms $\Omega^{3}(M)=E^{-}\oplus E^{+}.$

######  Definition 8.23.

[G97] For a 3-form $\varphi\in\Omega^{3}(M)$ we set $\varphi^{+}$ and
$\varphi^{-}$ to be its components in $E^{+}$ and $E^{-}$ respectively.

We note [G97] that on the space $\Omega^{3}(M)$ the above decompositions are
related as follows. For any 3-form $\varphi\in\Omega^{3}(M)$ one can check
that

$\varphi^{-}=\varphi^{0,2},\hskip 21.68121pt$
$\varphi^{+}=\varphi^{2,0}+\varphi^{1,1}.$

To illustrate this, if $\varphi\in E^{-}$ then
$J_{\textrm{der}}^{2}(\varphi)=-9\varphi$. Expanding
$J_{\textrm{der}}^{2}(\varphi)$ one gets

$J_{\textrm{der}}^{2}\varphi(X,Y,Z)=-3\varphi(X,Y,Z)+2\varphi(JX,JY,Z)+2\varphi(X,JY,JZ)+2\varphi(JX,Y,JZ)$

and so

$\varphi(X,Y,Z)=\tfrac{1}{4}\big{(}\varphi(X,Y,Z)-\varphi(X,JY,JZ)-\varphi(JX,Y,JZ)-\varphi(JX,JY,Z)\big{)}=\varphi^{0,2}(X,Y,Z).$

We now state and prove two useful lemmas involving the Nijenhuis tensor which
we will require

######  Lemma 8.24.

$\mathcal{P}N=\tfrac{1}{3}(d_{c}\omega)^{-}.$

###### Proof.

For $X\in T_{\mathbb{C}}(M)$ consider $g_{X}=\left(X,\cdot\right)$ where
$\left(\cdot,\cdot\right)$ is the metric complex bilinearly extended to
$T_{\mathbb{C}}(M)$. We observe that $N(X,Y,Z)$ (and hence
$\mathcal{P}N(X,Y,Z)$) is non-zero only if the vectors $X,Y,Z$ are all of pure
type $(1,0)$ (or of type $(0,1)$) in the direct sum decomposition
$T_{\mathbb{C}}(M)=T^{1,0}(M)\oplus T^{0,1}(M)$. We have

$\begin{split}3\mathcal{P}N(X,Y,Z)&=g_{X}(N(Y,Z))+g_{Y}(N(Z,X))+g_{Z}(N(X,Y))\\\
&=N^{\vee}(g_{X})(Y,Z)+N^{\vee}(g_{Y})(Z,X)+N^{\vee}(g_{Z})(X,Y)\\\
&=\big{(}\mu+\overline{\mu}\big{)}(g_{X})(Y,Z)+\big{(}\mu+\overline{\mu}\big{)}(g_{Y})(X,Z)+\big{(}\mu+\overline{\mu}\big{)}(g_{Z})(X,Y)\\\
&=-g_{X}\big{(}[Y,Z]\big{)}-g_{Y}\big{(}[Z,X]\big{)}-g_{Z}\big{(}[X,Y]\big{)}\end{split}$

where in the last equality we use that
$d\varphi(X,Y)=X(\varphi(Y))-Y(\varphi(X))-\varphi[X,Y]$ for any
$\varphi\in\Omega_{\mathbb{C}}^{1}(M)$, and that $\left(X,Y\right)=0$ for
$X,Y$ both of type $(1,0)$ or type $(0,1)$. Furthermore on vectors $X,Y,Z$ all
of type $(1,0)$ or $(0,1)$ we have

$\begin{split}d\omega(X,Y,Z)&=X(\omega(Y,Z))-Y(\omega(X,Z))+Z(\omega(X,Y))\\\
&-\omega([X,Y],Z)+\omega([X,Z],Y)-\omega([Y,Z],X)\hskip 79.49744pt\\\
&=g_{JX}([Y,Z])+g_{JY}([Z,X])+g_{JZ}([X,Y]).\end{split}$

Noticing that

$d_{c}\omega(X,Y,Z)=-d\omega(JX,JY,JZ)$

and again using that $X,Y,Z$ are all of pure type we obtain

$(d_{c}\omega)^{-}(X,Y,Z)=-g_{X}\big{(}[Y,Z]\big{)}-g_{Y}\big{(}[Z,X]\big{)}-g_{Z}\big{(}[X,Y]\big{)}=3\mathcal{P}N(X,Y,Z).\hskip
36.135pt\qed$

######  Proposition 8.25.

[G97] The map $\mathcal{P}$ is invariant on $\Omega^{0,2}(TM)$. Furthermore
the restriction of $\mathcal{P}$ to $\Omega^{0,2}(TM)$ has image $E^{-}$. That
is, the restriction is a surjective map

$\mathcal{P}:\Omega^{0,2}(TM)\rightarrow E^{-}.$

###### Proof.

We observe $\phi\in\Omega^{0,2}(TM)$ if and only if

$\phi(X,Y,Z)=\phi^{0,2}(X,Y,Z)=\tfrac{1}{4}\big{(}\phi(X,Y,Z)-\phi(X,JY,JZ)-\phi(JX,Y,JZ)-\phi(JX,JY,Z)\big{)}$

or equivalently

$\phi(X,Y,Z)=-\tfrac{1}{3}\big{(}\phi(X,JY,JZ)+\phi(JX,Y,JZ)+\phi(JX,JY,Z)\big{)}.$

We compute

$\begin{split}3\mathcal{P}\phi(X,Y,Z)&=\phi(X,Y,Z)+\phi(Y,Z,X)+\phi(Z,X,Y)\\\
&=-\tfrac{1}{3}\Big{(}\phi(X,JY,JZ)+\phi(JX,Y,JZ)+\phi(JX,JY,Z)\\\ &\hskip
16.62178pt+\phi(Y,JZ,JX)+\phi(JY,Z,JX)+\phi(JY,JZ,X)\\\ &\hskip
16.62178pt+\phi(Z,JX,JY)+\phi(JZ,X,JY)+\phi(JZ,JX,Y)\Big{)}\\\
&=-\mathcal{P}\phi(X,JY,JZ)-\mathcal{P}\phi(JX,Y,JZ)-\mathcal{P}\phi(JX,JY,Z)\\\
\end{split}$

and thus $\mathcal{P}\phi\in\Omega^{0,2}(TM)$. As
$\mathcal{P}\phi\in\Omega^{3}(M)$ as well we have $\mathcal{P}\phi\in E^{-}$.
Finally if $\varphi\in E^{-}$ then $\varphi=\varphi^{0,2}$ and
$\mathcal{P}(\varphi^{0,2})=\mathcal{P}(\varphi)=\varphi$ so that the
restriction is surjective. ∎

######  Definition 8.26.

[G97] We define the subspace $\Omega_{s}^{1,1}(TM)\subset\Omega^{1,1}(TM)$ by

$\Omega_{s}^{1,1}(TM)=\ker(\mathcal{P})\cap\Omega^{1,1}(TM).$

Furthermore we define $\Omega_{a}^{1,1}(TM)$ to be its orthogonal complement
in $\Omega^{1,1}(TM)$.

######  Definition 8.27.

[G97] We define the operator $\mathcal{M}$ on $\Omega^{2}(TM)$ by

$\mathcal{M}(\phi)(X,Y,Z)=\phi(X,JY,JZ).$

With this operator, the proof of the following two propositions can also be
checked directly [G97].

######  Proposition 8.28.

[G97] The map $\mathcal{P}$ restricted to $\Omega^{2,0}(TM)$ is an isomorphism

$\mathcal{P}:\Omega^{2,0}(TM)\xrightarrow{\sim}E^{+}.$

Furthermore for any $\phi\in\Omega^{2,0}(TM)$ we have
$\phi=\tfrac{3}{2}(\mathcal{P}\phi-\mathcal{M}\mathcal{P}\phi).$

######  Proposition 8.29.

[G97] The map $\mathcal{P}$ restricted to $\Omega_{a}^{1,1}(TM)$ is an
isomorphism

$\mathcal{P}:\Omega_{a}^{1,1}(TM)\xrightarrow{\sim}E^{+}.$

Furthermore for any $\phi\in\Omega_{a}^{1,1}(TM)$ we have
$\phi=\tfrac{3}{4}(\mathcal{P}\phi+\mathcal{M}\mathcal{P}\phi).$

######  Lemma 8.30.

For any $\phi\in\Omega^{2}(TM)$ we have
$\mathcal{P}(\phi_{a}^{1,1})+\mathcal{P}(\phi^{2,0})=(\mathcal{P}\phi)^{+}$.

###### Proof.

Evidently $\phi=\phi^{1,1}+\phi^{2,0}+\phi^{0,2}$ and so

$\mathcal{P}\phi=\mathcal{P}(\phi^{1,1})+\mathcal{P}(\phi^{2,0})+\mathcal{P}(\phi^{0,2}).$

Thus, by Propositions 8.28 and 8.29 above we have

$(\mathcal{P}\phi)^{+}=\mathcal{P}(\phi_{a}^{1,1})+\mathcal{P}(\phi^{2,0}).\qed$

######  Lemma 8.31.

$\mathcal{P}(\widetilde{\nabla}\omega)=\tfrac{1}{3}d\omega$. In particular

$\mathcal{P}(\widetilde{\nabla}\omega^{2,0})=\tfrac{1}{3}d\omega^{+}\hskip
7.22743pt\textrm{ and }\hskip
7.22743pt\mathcal{P}(\widetilde{\nabla}\omega^{0,2})=\tfrac{1}{3}d\omega^{-}.\hskip
7.22743pt$

###### Proof.

The identity $\mathcal{P}(\widetilde{\nabla}\omega)=\tfrac{1}{3}d\omega$
follows at once from the observation that

$d\varphi(v_{1},\dots,v_{k+1})=\sum_{j}(-1)^{j+1}\widetilde{\nabla}_{v_{j}}\varphi(v_{1},\dots,\widehat{v_{j}},\dots,v_{k+1})\textrm{
for any }k\textrm{ form }\varphi.$

The remaining identities follows from the isomorphisms in Propositions 8.28
and 8.25 and the observation, proven in Lemma 8.18, that
$(\widetilde{\nabla}\omega)^{1,1}=0$. ∎

######  Lemma 8.32.

$(d\omega)^{+}=3\mathcal{P}\mathcal{M}((d\omega)^{+}).$

###### Proof.

We have by Proposition 8.28 and Lemma 8.31 that

$(\widetilde{\nabla}\omega)^{2,0}=\tfrac{3}{2}(\mathcal{P}(\widetilde{\nabla}\omega^{2,0})-\mathcal{M}(\mathcal{P}(\widetilde{\nabla}\omega^{2,0}))=\tfrac{1}{2}((d\omega)^{+}-\mathcal{M}(d\omega)^{+}).$

Applying $\mathcal{P}$ on both sides of the equality, and using Lemma 8.31
once more, we conclude

$\tfrac{1}{3}(d\omega)^{+}=\mathcal{P}\mathcal{M}(d\omega^{+}).\qed$

######  Lemma 8.33.

$(d\omega)^{-}(X,Y,Z)=-3(\mathcal{P}T)^{-}(JX,Y,Z).$

###### Proof.

Observe that by Proposition 8.15 we have

$-(\widetilde{\nabla}\omega)^{0,2}(X,Y,Z)=A^{0,2}(X,JY,Z)+A^{0,2}(X,Y,JZ)$

so that

$\begin{split}-3\mathcal{P}\big{(}(\widetilde{\nabla}\omega)^{0,2}\big{)}(X,Y,Z)&=A^{0,2}(X,JY,Z)+A^{0,2}(X,Y,JZ)\\\
&+A^{0,2}(Y,JZ,X)+A^{0,2}(Y,Z,JX)\\\
&+A^{0,2}(Z,JX,Y)+A^{0,2}(Z,X,JY).\end{split}$

Using the identity $T(X,Y,Z)=A(Y,Z,X)+A(Z,X,Y)$ obtained in the proof of
Proposition 8.22, we have

$\begin{split}-3\mathcal{P}\big{(}(\widetilde{\nabla}\omega)^{0,2}\big{)}(X,Y,Z)&=T^{0,2}(JY,Z,X)+T^{0,2}(JZ,X,Y)+T^{0,2}(JX,Y,Z)\\\
&=T^{0,2}(Y,Z,JX)+T^{0,2}(Z,JX,Y)+T^{0,2}(JX,Y,Z)\end{split}$

where in the last equality we use that $\phi(JX,Y,Z)=\phi(X,JY,Z)$ for any
$\phi\in\Omega^{0,2}(TM)$. We conclude

$-3\mathcal{P}\big{(}(\widetilde{\nabla}\omega)^{0,2}\big{)}(X,Y,Z)=3\mathcal{P}\big{(}T^{0,2}\big{)}(JX,Y,Z)=3(\mathcal{P}T)^{-}(JX,Y,Z)$

and the result follows by Lemma 8.31.

∎

######  Lemma 8.34.

If $\varphi\in E^{+}$ then
$\varphi(X,Y,Z)=\varphi(JX,JY,Z)+\varphi(X,JY,JZ)+\varphi(JX,Y,JZ).$

###### Proof.

For any $\varphi\in E^{+}$ we have $J_{\textrm{der}}^{2}\varphi=-\varphi$. But

$J_{\textrm{der}}^{2}\varphi(X,Y,Z)=-3\varphi(X,Y,Z)+2\varphi(JX,JY,Z)+2\varphi(X,JY,JZ)+2\varphi(JX,Y,JZ)$

which gives the result. ∎

######  Proposition 8.35.

[G97] An affine metric connection $\nabla$ on $M$ is Hermitian if and only if

$\begin{split}T^{2,0}-\tfrac{3}{2}\big{(}\mathcal{P}(T^{1,1})-\mathcal{M}\mathcal{P}(T^{1,1})\big{)}&=\tfrac{1}{2}((d_{c}\omega)^{+}-\mathcal{M}(d_{c}\omega)^{+})\hskip
7.22743pt\textrm{ and }\\\ T^{0,2}&=N.\end{split}$

###### Proof.

By Proposition 8.15 we have that $\nabla$ is hermitian if and only if

$A(X,JY,Z)+A(X,Y,JZ)=-(\widetilde{\nabla}\omega)(X,Y,Z).$

Using Proposition 8.22 we rewrite the identity in Proposition 8.15 as

$T(X,JY,Z)+T(X,Y,JZ)-\tfrac{3}{2}\big{(}\mathcal{P}T(X,JY,Z)+\mathcal{P}T(X,Y,JZ)\big{)}=(\widetilde{\nabla}\omega)(X,Y,Z).$

We decompose $T=T^{1,1}+T^{2,0}+T^{0,2}$ and collecting the $(0,2)$ parts of
the expression (Proof.), by Proposition 8.25 we obtain

(6)
$2T^{0,2}(JX,Y,Z)-3(\mathcal{P}T)^{-}(JX,Y,Z)=(\widetilde{\nabla}\omega)^{0,2}(X,Y,Z).$

We observe that by Lemma 8.33 we have

$2T^{0,2}(JX,Y,Z)+(d\omega)^{-}(X,Y,Z)=(\widetilde{\nabla}\omega)^{0,2}(X,Y,Z).$

By Lemma 8.13, we see that

$2T^{0,2}(JX,Y,Z)=(\widetilde{\nabla}\omega)^{0,2}(X,Y,Z)-d\omega^{-}(X,Y,Z)=2N^{0,2}(JX,Y,Z)=2N(JX,Y,Z)$

and we conclude equation (6) holds if and only if $T^{0,2}=N$.

We recall that by Lemma 8.18 we have

$A^{1,1}(X,JY,Z)+A^{1,1}(X,Y,JZ)=-(\widetilde{\nabla}\omega)^{1,1}(X,Y,Z)=0.$

Collecting the $(1,1)$ parts of the expression (Proof.), we observe, as in the
proof of Proposition 8.15, that $T^{1,1}(X,JY,Z)+T^{1,1}(X,Y,JZ)=0$ and so

$\mathcal{P}(T^{1,1})(X,JY,Z)+\mathcal{P}(T^{1,1})(X,Y,JZ)=0.$

Thus, by Lemma 8.30 we have

$(\mathcal{P}T)^{+}(X,JY,Z)+(\mathcal{P}T)^{+}(X,Y,JZ)=\mathcal{P}(T^{2,0})(X,JY,Z)+\mathcal{P}(T^{2,0})(X,Y,JZ).$

Finally, collecting the $(2,0)$ parts of the expression we have

$-2T^{2,0}(JX,Y,Z)-\tfrac{3}{2}\big{(}(\mathcal{P}T)^{+}(X,JY,Z)+(\mathcal{P}T)^{+}(X,Y,JZ)\big{)}=(\widetilde{\nabla}\omega)^{2,0}(X,Y,Z).$

By Lemma 8.34 we rewrite (Proof.) as

$-2T^{2,0}(JX,Y,Z)+\tfrac{3}{2}\big{(}(\mathcal{P}T)^{+}(JX,Y,Z)-(\mathcal{P}T)^{+}(JX,JY,JZ)\big{)}=(\widetilde{\nabla}\omega)^{2,0}(X,Y,Z).$

Notice that by Lemma 8.30 and Proposition 8.28 we have
$(\mathcal{P}T)^{+}=\mathcal{P}(T^{2,0})+\mathcal{P}(T^{1,1})$ and
$T^{2,0}=\tfrac{3}{2}\big{(}\mathcal{P}(T^{2,0})-\mathcal{M}\mathcal{P}(T^{2,0})\big{)}$.
We rewrite (Proof.) again as

$-T^{2,0}(JX,Y,Z)+\tfrac{3}{2}\big{(}\mathcal{P}T^{1,1}(JX,Y,Z)-\mathcal{M}\mathcal{P}T^{1,1}(JX,Y,Z)\big{)}=(\widetilde{\nabla}\omega)^{2,0}(X,Y,Z).$

By Lemma 8.31 and Proposition 8.28 we see

$(\widetilde{\nabla}\omega)^{2,0}=\tfrac{3}{2}\big{(}\mathcal{P}((\widetilde{\nabla}\omega)^{2,0})-\mathcal{M}(\mathcal{P}((\widetilde{\nabla}\omega)^{2,0}))\big{)}=\tfrac{1}{2}\big{(}(d\omega)^{+}-\mathcal{M}(d\omega)^{+}\big{)}$

so that (Proof.) holds if and only if

$T^{2,0}-\tfrac{3}{2}\big{(}\mathcal{P}(T^{1,1})-\mathcal{M}\mathcal{P}(T^{1,1})\big{)}=\tfrac{1}{2}((d_{c}\omega)^{+}-\mathcal{M}(d_{c}\omega)^{+}).$

Evidently $\nabla$ is hermitian if and only if (Proof.), which holds if and
only if both (6) and (Proof.). ∎

######  Lemma 8.36.

[G97] Let $\nabla$ be a Hermitian connection on $M$. Then

$\mathcal{P}(T^{2,0}-T_{a}^{1,1})=\tfrac{1}{3}d_{c}\omega^{+}.$

###### Proof.

By the proof of Proposition 8.35 we have

$T^{2,0}-\tfrac{3}{2}\big{(}\mathcal{P}T^{1,1}-\mathcal{M}(\mathcal{P}T^{1,1})\big{)}=(\nabla\omega)^{2,0}(J\cdot,\cdot,\cdot).$

Applying $\mathcal{P}$ to both sides of the equation we obtain by Lemma 8.31

$\mathcal{P}(T^{2,0})-\tfrac{3}{2}\big{(}\mathcal{P}(T^{1,1})-\mathcal{P}\mathcal{M}\mathcal{P}(T^{1,1})\big{)}=\tfrac{1}{3}d_{c}\omega^{+}.$

Using that $T^{1,1}(X,JY,Z)+T^{1,1}(X,Y,JZ)=0$ the identity
$\mathcal{P}(T^{1,1})=3\mathcal{P}\mathcal{M}\mathcal{P}(T^{1,1})$ is readily
verified. This gives the result. ∎

Combining Lemmas 8.30 and 8.36 we see that whenever $T$ is the torsion of a
Hermitian connection on an almost Hermitian manifold $M$ we have the equations

$\begin{split}\mathcal{P}(T_{a}^{1,1})&=\tfrac{1}{2}(\mathcal{P}T^{+}-\tfrac{1}{3}(d_{c}\omega)^{+}),\\\
\mathcal{P}(T^{2,0})&=\tfrac{1}{2}(\mathcal{P}T^{+}+\tfrac{1}{3}(d_{c}\omega)^{+}).\end{split}$

This leads us to the following

######  Theorem 8.37.

[G97] On any almost Hermitian manifold $M$ the torsion of any Hermitian
connection $\nabla$ on $M$ is given by

$T=N+\tfrac{9}{8}\mathcal{P}T^{+}+\tfrac{1}{8}d_{c}\omega^{+}-\tfrac{3}{8}\mathcal{M}(\mathcal{P}T^{+})-\tfrac{3}{8}\mathcal{M}(d_{c}\omega^{+})+T^{1,1}_{s}.$

###### Proof.

Let $T$ be the torsion of a Hermitian connection. We have

$\begin{split}T&=T^{0,2}+T^{1,1}+T^{2,0}\\\
&=N+T_{a}^{1,1}+T^{2,0}+T_{s}^{1.1}\\\
&=N+\tfrac{3}{4}\big{(}\mathcal{P}(T^{1,1})+M\mathcal{P}(T^{1,1})\big{)}+\tfrac{3}{2}\big{(}\mathcal{P}(T^{1,1})-\mathcal{M}\mathcal{P}(T^{1,1})\big{)}+\tfrac{1}{2}(d_{c}\omega^{+}-\mathcal{M}(d_{c}\omega^{+}))+T_{s}^{1,1}\\\
&=N+\tfrac{3}{8}\big{(}3\mathcal{P}T^{+}-d_{c}\omega^{+}-\mathcal{M}(\mathcal{P}T^{+})+\tfrac{1}{3}\mathcal{M}(d_{c}\omega^{+})\big{)}+\tfrac{1}{2}\big{(}d_{c}\omega^{+}-\mathcal{M}(d_{c}\omega^{+})\big{)}+T_{s}^{1,1}\\\
&=N+\tfrac{9}{8}\mathcal{P}T^{+}+\tfrac{1}{8}d_{c}\omega^{+}-\tfrac{3}{8}\mathcal{M}(\mathcal{P}T^{+})-\tfrac{3}{8}\mathcal{M}(d_{c}\omega^{+})+T^{1,1}_{s}.\qed\end{split}$

Therefore we obtain the

######  Corollary 8.38.

[G97] A Hermitian connection $\nabla$ is uniquely determined by choice of
$T_{s}^{1,1}$ and $\mathcal{P}T^{+}$.

From the above Theorem and Corollary, Gauduchon [G97] defines an affine line
of ‘canonical’ Hermitian connections on an almost Hermitian manifold. This is
the set of Hermitian connections, $\nabla^{t}$, with torsion $T^{t}=T$
satisfying

$T_{s}^{1,1}=0\hskip 7.22743pt\textrm{ and }\hskip
7.22743pt\mathcal{P}T^{+}=\tfrac{2t-1}{3}(d_{c}\omega)^{+}\textrm{ for any
}t\in\mathbb{R}.$

Theorem 8.37 then yields the following elegant characterization of any
canonical Hermitian connection. Namely the torsion of $\nabla^{t}$ is given by

$T^{t}=N+\tfrac{3t-1}{4}d_{c}\omega^{+}-\tfrac{t+1}{4}\mathcal{M}(d_{c}\omega^{+}).$

For example, a natural choice of Hermitian connection is the Chern connection
$\nabla^{\textrm{ch}}$ defined by requiring $T^{1,1}=0$ or equivalently $t=1$.
It is readily seen that, in the case of a Kähler manifold, we have the well
known identity $\nabla^{\textrm{ch}}=\widetilde{\nabla}$ as all torsion
components of $\nabla^{\textrm{ch}}$ vanish.

What is more, evidently all of the canonical Hermitian connections agree in
the case that $d\omega=0$. We denote the potential of a canonical hermitian
connection by $A^{t}$. By Proposition 8.22 we observe

$A^{t}=-T^{t}+\tfrac{3}{2}\mathcal{P}T^{t}.$

We then have

######  Proposition 8.39.

[G97] Let $M$ be an almost Hermitian manifold and let $\nabla^{t}$ be a
canonical Hermitian connection in the sense of Gauduchon. Then

$A^{t}=-N+\tfrac{3}{2}\mathcal{P}N+\tfrac{t-1}{4}d_{c}\omega^{+}+\tfrac{t+1}{4}\mathcal{M}(d_{c}\omega^{+}).$

###### Proof.

The result follows by substituting the identity (Appendix: Hermitian
Connections and Dirac Operators) in
$A^{t}=-T^{t}+\tfrac{3}{2}\mathcal{P}T^{t}$. We observe that

$\mathcal{P}T^{t}=\mathcal{P}N+\tfrac{3t-1}{4}d_{c}\omega^{+}-\tfrac{t+1}{4}\mathcal{P}\mathcal{M}(d_{c}\omega^{+})=\mathcal{P}N+\tfrac{2t-1}{3}d_{c}\omega^{+},$

where we use that
$\mathcal{P}\mathcal{M}(d_{c}\omega^{+})=\tfrac{1}{3}d_{c}\omega^{+}$ which
follows by Lemma 8.32. ∎

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