where \( \Lambda_{j_0}\defeq \Lambda(j_0, \cdot,\cdot) \).
Moreover, for every \( A_0 \in \TBundle_{j_0} \SectionSpaceAbb{I}(V, \omega) \),

\begin{equation}
\label{kaehler_contraction_linear_second}
\int_0^1 \left(\bar{\Lambda}_j^* \, \Omega\right)_{(j_0, t)}
(\difp_t, A_0) \dif t = \frac{1}{4} \tr \bigl(\phi_{j_0}(j) A_0\bigr),
\end{equation}

where \( \bar{\Lambda}_j \defeq \Lambda(\cdot, j, \cdot) \).

Using the identities

\begin{align}
\tangent_j \phi_{j_0} (A) &= 2 (j+j_0)^{-1} A (j+j_0)^{-1} j_0,
\\
\tangent_S \phi_{j_0}^{-1} (C) &= 2 j_0 (1-S)^{-1} C (1-S)^{-1}
\end{align}

we find

\begin{equation}
\tangent_j \Lambda_{j_0} (\difp_t) = - 2 S (1+tS)^{-2} j_0
\end{equation}

\begin{equation}
\tangent_j \Lambda_{j_0}(A) =
4 t (1+tS)^{-1} (j+j_0)^{-1} A (j+j_0)^{-1} (1-tS)^{-1}.
\end{equation}

Here and in the following, we abbreviated \( S \defeq \phi_{j_0}(j) \).
Using these formulas
and (<ref>),
we calculate

\begin{equation}\begin{split}
\left(\Lambda_{j_0}^* \Omega\right)_{(j, t)} (\difp_t, A)
&= \Omega_{\Lambda_{j_0}(j,t)}\bigl(\tangent_j \Lambda_{j_0}
(\difp_t), \tangent_j \Lambda_{j_0}(A)\bigr)
\\
&= 2t \tr\left(S \bigl(1-t^2 S^2\bigr)^{-2} (j+j_0)^{-1} A (j+j_0)^{-1} \right).
\end{split}\end{equation}

\begin{equation}
\difFrac{}{t} \bigl(1-t^2 S^2\bigr)^{-1} = 2 t S^2 \bigl(1-t^2 S^2\bigr)^{-2},
\end{equation}

we obtain[Note that it is enough to verify this identity for invertible \( S \) by density of the invertible elements in \( \End(V) \).]

\begin{equation}\begin{split}
\int_0^1 \left(\Lambda_{j_0}^* \Omega\right)_{(j, t)} (\difp_t, A) \dif t
&= \tr \left(S \bigl(1-S^2\bigr)^{-1} (j+j_0)^{-1} A (j+j_0)^{-1}\right)  \\
&= \frac{1}{4} \tr \bigl(S A\bigr).
\end{split}\end{equation}

This establishes (<ref>).

The second identity (<ref>) follows
from a similar, but slightly more involved, calculation.
Indeed, the derivative of the map \( \bar{\Lambda}_{j}(j_0, t) \defeq
\Lambda(j_0, j, t) \) is given by

\begin{equation}
\tangent_{j_0} \bar{\Lambda}_j (A_0) =
(1-t) (1+tS)^{-1} (A_0 - t S A_0 S) (1-tS)^{-1}.
\end{equation}

Thus, we find

\begin{equation}
\left(\bar{\Lambda}_{j}^* \Omega\right)_{(j_0, t)} (\difp_t, A_0)
= \frac{1}{2} \tr \left(\bigl(1-t^2 S^2\bigr)^{-2} S \bigl(1-tS^2\bigr)
(1-t) A_0 \right).
\end{equation}

\begin{equation}
\difFrac{}{t} \bigl(-2 S^2 t+S^2 + 1\bigr)\bigl(1-t^2 S^2\bigr)^{-1}
= -2 S^2 (1-t) \bigl(1-t S^2\bigr) \bigl(1-t^2 S^2\bigr)^{-2},
\end{equation}

this expression can easily be integrated over \( t \) to obtain

\begin{equation}
\int_0^1 \left(\bar{\Lambda}_{j}^* \Omega\right)_{(j_0, t)}
(\difp_t, A_0) \dif t
= \frac{1}{4} \tr \bigl(S A_0\bigr).
\end{equation}

This verifies (<ref>) and finishes
the proof.

As a direct application, let us compute the momentum map \( J \) for the
action of \( \SpGroup(V, \omega) \) on \( \SectionSpaceAbb{I}(V, \omega) \).
According to <ref>, the unique momentum map
vanishing at \( j_0 \) is given by

\begin{equation}
\kappa\bigl(J(j), \xi \bigr) =
\frac{1}{4} \tr \bigl(\phi_{j_0}(j) (\xi \ldot j +
\xi \ldot j_0)\bigr) = \frac{1}{2} \tr \bigl((j-j_0)\xi\bigr).
\end{equation}

Thus, identifying \( \SpAlgebra(V, \omega)^* \) with
\( \SpAlgebra(V, \omega) \) using the pairing \( \kappa(\alpha, \xi)
= \frac{1}{2} \tr(\alpha \xi) \), we find \( J(j) = j - j_0 \).

We now finish the proof of <ref>.
In the nonlinear setting, the momentum map for the action of the
symplectomorphism group \( \DiffGroup(M, \omega) \) on
\( \SectionSpaceAbb{I} \equiv \SectionSpaceAbb{I}(M, \omega) \)
is computed in a very similar way, with the twist that the final
dualization involves integration by parts.
Let us discuss the details.
First, we extend the definition of the generalized Cayley transform
\( \phi \) and the contraction \( \Lambda \) to \( \SectionSpaceAbb{I} \)
by applying these maps pointwise.
The resulting map \( \Lambda: \SectionSpaceAbb{I} \times
\SectionSpaceAbb{I} \times [0,1] \to \SectionSpaceAbb{I} \) is smooth
in the \( \sFunctionSpace \)-topology by
[Theorem II.2.2.6]Hamilton1982.
Since \( \DiffGroup(M, \omega) \) acts on \( \SectionSpaceAbb{I} \)
by push-forward, the infinitesimal action is given by
\( \xi \ldot j = - \difLie_{\xi} j \), where \(\xi
\in  \VectorFieldSpace(M, \omega ) \defeq \set{\zeta \in \VectorFieldSpace(M) \given
\difLie_{\zeta} \omega =0} \).
By <ref>,
the momentum map \( \SectionSpaceAbb{J}: \SectionSpaceAbb{I} \to
\DiffFormSpace^{2n-1}(M) \slash \dif \DiffFormSpace^{2n-2}(M) \)
vanishing at \( j_0 \in \SectionSpaceAbb{I} \) is uniquely characterized by

\begin{equation}
\label{eq:kaehler:momentumMapIntegralFormula}
\kappa\bigl(\SectionSpaceAbb{J}(j), \xi \bigr) = -
\frac{1}{4} \int_M \tr \bigl((j+j_0)^{-1}(j-j_0)
\difLie_\xi (j + j_0)\bigr) \, \mu_\omega.
\end{equation}

In order to eliminate the derivative in \( \xi \)-direction, following
Garcia-PradaSalamonTrautwein2018, we fix a torsion-free
connection \( \nabla \) on \( M \) with \( \nabla \mu_\omega = 0 \) and
introduce \( \tau(j, A) \in \DiffFormSpace^1(M) \), for \( j \in
\SectionSpaceAbb{I} \) and \( A \in \TBundle_j \SectionSpaceAbb{I} \),

\begin{equation}
\label{eq:kaehler:tau}
\tau^\nabla(j, A)(Y) = \tr \bigl((\nabla A) Y\bigr) +
\frac{1}{2} \tr \left(A j \, \nabla_Y j\right)
= Y^r \nabla_p \tensor{A}{_r^p} +
\frac{1}{2} Y^r \tensor{A}{_s^p}\tensor{j}{_q^s} \, \nabla_r \tensor{j}{_p^q},
\end{equation}

for \( Y \in \VectorFieldSpace(M) \).
By [Theorem 2.6]Garcia-PradaSalamonTrautwein2018,
\( \tau^\nabla(j, A) \) does not depend on the connection \( \nabla \).
However, for the Levi-Civita connection of \( g_j \), the expression
of \( \tau^\nabla \) simplifies considerably.

Let \( j \in \SectionSpaceAbb{I} \) and
\( A \in \TBundle_j \SectionSpaceAbb{I} \).
Then, for the Levi-Civita connection \( \nabla \) of \( g_j \),
we have

\begin{equation}
\label{eq:kaehler:tauForLeviCivita}
\tau^\nabla(j, A)(Y) = \tr \bigl((\nabla A)Y\bigr) =
Y^r \nabla_p \tensor{A}{_r^p}.
\qedhere
\end{equation}

By taking the covariant derivative of \( g_j(j \cdot, j \cdot)
= g_j(\cdot, \cdot) \), we see that \( \nabla_Y j \) is
antisymmetric with respect to \( g_j \) for every
\( Y \in \VectorFieldSpace(M) \).
Thus, also \( j \nabla_Y j \) is antisymmetric.
But \( A \in \TBundle_j \SectionSpaceAbb{I} \) is a \( g_j \)-symmetric
tensor, so \( A j \nabla_Y j \) is trace-free.

The importance of \( \tau \) lies in the following integration
by parts identity [Theorem 2.6]Garcia-PradaSalamonTrautwein2018:

\begin{equation}
\label{eq:kaehler:integrationByPartsTau}
\frac{1}{2} \int_M \tr(A \difLie_\xi j) \, \mu_\omega
= -\int_M \tau^\nabla(j, Aj) \wedge (\xi \contr \mu_\omega)
= - \kappa\bigl(\tau^\nabla(j, Aj) \wedge \omega^{n-1}, \xi\bigr),
\end{equation}

for \( \xi \in \VectorFieldSpace(M, \omega) \) and
\( A \in \TBundle_j \SectionSpaceAbb{I} \).
Using this fact, we read-off from (<ref>)
that the momentum map is given by

\begin{equation}
\SectionSpaceAbb{J}(j) =
\frac{1}{2} \Bigl(\tau^\nabla\bigl(j, (j+j_0)^{-1}(j-j_0) j\bigr)
+ \tau^\nabla\bigl(j_0, (j+j_0)^{-1}(j-j_0) j_0\bigr)\Bigr) \wedge \omega^{n-1}.
\end{equation}

Now, for \( Y \in \VectorFieldSpace(M) \), we have by (<ref>)

\begin{equation}\begin{split}
\tau^\nabla\bigl(j, &(j+j_0)^{-1}(j-j_0) j\bigr)(Y) +
\tau^\nabla\bigl(j_0, (j+j_0)^{-1}(j-j_0) j_0\bigr)(Y)
\\
&= \tr \left(\nabla \bigl((j+j_0)^{-1}(j-j_0) (j+j_0)\bigr) Y
- \frac{1}{2} (j+j_0)^{-1}(j-j_0) \, \nabla_Y (j+j_0)\right)
\\
&= -\tr \bigl(\nabla (j-j_0) Y\bigr) -
\frac{1}{2} \tr \left((j+j_0)^{-1}(j-j_0) \,\nabla_Y (j+j_0)\right).
\end{split}\end{equation}

This finishes the proof of <ref>.

Alternatively, one can directly verify <ref>
using the integration by parts relation (<ref>)
and the following expression for the variation of the \( 1 \)-form
\( J(j_0, j) \).

For every \( j_0, j \in \SectionSpaceAbb{I} \) and \( A \in \TBundle_j
\SectionSpaceAbb{I} \), we have

\begin{equation}
\label{eq:kaehler:variationJ}
\tangent_j J(j_0, \cdot)(A) = - \frac{1}{2} \tau^\nabla(j, A)
- \frac{1}{4} \dif \tr\bigl(A \phi_{j_0}(j)\bigr).
\qedhere
\end{equation}

Continuing using the notation \( \phi_{j_0}(j) \) for the Cayley transform
introduced in (<ref>),
using (<ref>) we get

\begin{equation}\begin{split}
\tangent_j J(j_0, \cdot)(A)(Y)
&= -\frac{1}{2} \tr\bigl((\nabla A)Y\bigr) -
\frac{1}{2} \tr \bigl((j+j_0)^{-1} A (j+j_0)^{-1} j_0
\nabla_Y (j + j_0)\bigr) \\
&\qquad- \frac{1}{4} \tr \bigl(\phi_{j_0}(j) \nabla_Y A\bigr).
\end{split}\end{equation}

On the other hand,

\begin{equation}
\phi_{j_0}(j) jA = Aj - 2 (j+j_0)^{-1} A (j+j_0)^{-1} (j_0 + j).
\end{equation}

Since \( \phi_{j_0}(j) \), \( A \), and \( \nabla_Y j \) all anti-commute
with \( j \), we obtain

\begin{equation}
0 = \tr(\phi_{j_0}(j) jA) = \tr\bigl(Aj \nabla_Y j\bigr) -
2 \tr\bigl((j+j_0)^{-1} A (j+j_0)^{-1} (j_0 + j) \nabla_Y j \bigr)
\end{equation}

and thus

\begin{equation}\begin{split}
\tr\bigl(Aj \nabla_Y j\bigr) &- 2 \tr\bigl((j+j_0)^{-1} A
(j+j_0)^{-1} j_0 \nabla_Y (j+j_0) \bigr) \\
&= 2 \tr\left((j+j_0)^{-1} A (j+j_0)^{-1} \bigl(j \nabla_Y j
- j_0 \nabla_Y j_0 \bigr)\right) \\
&= 2 \tr\left(A (j+j_0)^{-1} \bigl(j \nabla_Y j -
j_0 \nabla_Y j_0 \bigr) (j+j_0)^{-1}\right) \\
&= -\tr(A \nabla_Y \phi_{j_0}(j)).
\end{split}\end{equation}

Hence we conclude

\begin{equation}\begin{split}
\tangent_j J(j_0, \cdot)(A)(Y)
&= -\frac{1}{2} \tr\bigl((\nabla A)Y\bigr) -
\frac{1}{4} \tr\bigl(Aj \nabla_Y j\bigr) -
\frac{1}{4}\tr(A \nabla_Y \phi_{j_0}(j)) \\
&\qquad- \frac{1}{4} \tr \bigl(\phi_{j_0}(j) \nabla_Y A\bigr)\\
&= - \frac{1}{2} \tau^\nabla(j, A)(Y) -
\frac{1}{4} \nabla_Y \tr\bigl(A \phi_{j_0}(j)\bigr).
\end{split}\end{equation}

This finishes the proof.

In order to give a geometric interpretation of the \( 1 \)-form, and
thereby of the momentum map \( \SectionMapAbb{J} \), we recall a few
basic facts from almost complex geometry; see
GarciaPradaSalamon2020,Gauduchon2017 for more details.
For every almost complex structure \( j \), the Levi-Civita
connection \( \nabla \) associated with the Riemannian metric
\( g_j = \omega(\cdot, j \cdot) \) induces the Chern connection
(also called the second canonical Hermitian connection)

\begin{equation}
\tilde{\nabla}_X Y = \nabla_X Y - \frac{1}{2} j \, (\nabla_X j) Y.
\end{equation}

This is the unique connection that preserves the metric \( g_j \),
the complex structure \( j \), and the symplectic form \( \omega \),
and whose torsion is the Nijenhuis tensor \( N_j \).
The Chern–Ricci form is defined by

\begin{equation}
\mathrm{Ric}_j(X, Y) \defeq
\frac{1}{2} \tr \left(R^{\tilde{\nabla}}(X, Y) j\right),
\end{equation}

where \( R^{\tilde{\nabla}} \) is the curvature of the Chern connection
(on \( \TBundle M \)). With these conventions, \( \I \, \mathrm{Ric}_j \)
is the curvature of the induced Chern connection on the anti-canonical bundle
\( \KBundle^{-1}_j M = \ExtBundle^{0,n} (\CotBundle M) \) and
\( \frac{1}{2\pi}\mathrm{Ric}_j \) represents
\( c_1(M) = c_1 \bigl(\KBundle^{-1}_j M\bigr) \in \sCohomology^2(M, \Z) \).
Moreover, \( S_j \defeq \tr_\omega \mathrm{Ric}_j \) is
the Chern scalar curvature.
In the following, we also need the normalized version

\begin{equation}
\label{eq:kaehler:scalarCurvatureNormalized}
\bar{S}_{j} \defeq S_{j}
- \frac{1}{\vol_{\mu_\omega}(M)} \int_M S_{j} \, \mu_\omega \, .
\end{equation}

Since the space \( \SectionSpaceAbb{I} \) of almost complex structures
compatible with \( \omega \) is contractible, the anti-canonical bundles
\( \KBundle^{-1}_j M \) are all isomorphic as \( j \) varies
in \( \SectionSpaceAbb{I} \).
For every \( j_0, j \in \SectionSpaceAbb{I} \), choose an isomorphism
\( \KBundle^{-1}_{j_0} M \isomorph \KBundle^{-1}_j M \), and
let \( \widebar{J}(j_0, \cdot) \) be the difference of the
Chern connections on \( \KBundle^{-1}_{j_0} M \) and
\( \KBundle^{-1}_j M \) (under this isomorphism).
Choosing a different isomorphism of the anti-canonical bundles changes
\( \widebar{J}(j_0, \cdot) \) by an exact \( 1 \)-form. This gives
rise to a well-defined map

\begin{equation}
\widebar{J}(j_0, \cdot): \SectionSpaceAbb{I} \to
\DiffFormSpace^1(M) \slash \dif \DiffFormSpace^0(M).
\end{equation}

Mohsen2003 showed that the derivative of this map is given by

\begin{equation}
\tangent_j \bigl(\widebar{J}(j_0, \cdot)\bigr)(A) =
-\frac{1}{2} \tau^\nabla(j, A) \mod \dif \DiffFormSpace^0(M)
\end{equation}

for the Levi-Civita connection \( \nabla \); see also
[Proposition 9.5.1]Gauduchon2017[Proposition 9]Vernier2020.
Clearly, \( \widebar{J}(j_0, \cdot) \) vanishes at \( j_0 \), and
hence, by <ref>, the maps
\( j \mapsto \widebar{J}(j_0, j) \) and
\( j \mapsto J(j_0, j) \mod \dif \DiffFormSpace^0(M) \)
have to coincide.
In other words, the \( 1 \)-form \( J(j_0, j) \) is the difference of
the Chern connections on the anti-canonical bundles
\( \KBundle^{-1}_{j_0} M \) and \( \KBundle^{-1}_j M \) under the isomorphism
\( \KBundle^{-1}_{j_0} M \isomorph \KBundle^{-1}_j M \) induced by
the generalized Cayley transform \( \Lambda \).
Based on this discussion, an equivalent restatement of
<ref>, enlightening the geometric meaning,
is the following.

For every \( j_0, j \in \SectionSpaceAbb{I} \), let \( \widebar{J}(j_0, j)
\in \DiffFormSpace^{1}(M) \slash \dif \DiffFormSpace^{0}(M) \) be the
difference of the Chern connections on the anti-canonical
bundles \( \KBundle^{-1}_{j_0} M \)
and \( \KBundle^{-1}_j M \) under an arbitrary isomorphism
\( \KBundle^{-1}_{j_0} M \isomorph \KBundle^{-1}_j M \).
Then the unique momentum map \( \SectionMapAbb{J}: \SectionSpaceAbb{I}
\to \DiffFormSpace^{2n-1}(M) \slash \dif \DiffFormSpace^{2n-2}(M) \)
for the action of \( \DiffGroup(M, \omega) \) on \( \SectionSpaceAbb{I} \)
satisfying \( \SectionMapAbb{J}(j_0) = 0 \) is given by
\( \SectionMapAbb{J}(j) = \widebar{J}(j_0, j) \wedge \omega^{n-1} \).

In DiezRatiuAutomorphisms, we have investigated the action
of \( \DiffGroup(M, \omega) \) on \( \SectionSpaceAbb{I} \) and showed
that it admits a so-called group-valued momentum map.
Let us briefly outline the construction. Assume that \( \omega \) has
integral periods so that there exists a prequantum bundle \( L \to M \).
Let \( \KBundle_j M \) be the canonical bundle induced by
\( j \in \SectionSpaceAbb{I} \), and consider the map

\begin{equation}
\skew{3}{\tilde}{\SectionMapAbb{J}}: \SectionSpaceAbb{I} \to
\csCohomology^{2n}(M, \UGroup(1)), \qquad j \mapsto \KBundle_j M \star L^{n-1},
\end{equation}

where \( \csCohomology^{k}(M, \UGroup(1)) \) is the group of Cheeger-Simons
differential characters, and \( \star: \csCohomology^{k}(M, \UGroup(1)) \times
\csCohomology^{l}(M, \UGroup(1)) \to \csCohomology^{k+l}(M, \UGroup(1)) \)
is the natural ring structure; see, , BaerBecker2013.
By construction, \( \KBundle^{-1}_j M \star L^{n-1} \) can be viewed as a
higher bundle with connection whose curvature is
\( \mathrm{Ric}_j \wedge \omega^{n-1} = \frac{S_j}{2n} \omega^n \).
By [Theorem 4.10]DiezRatiuAutomorphisms,
\( \skew{3}{\tilde}{\SectionMapAbb{J}} \) is a group-valued momentum
map for the action of the group of symplectomorphisms in the sense
that the left logarithmic derivative
\( \difLog \skew{3}{\tilde}{\SectionMapAbb{J}} \in
\DiffFormSpace^1\bigl(\SectionSpaceAbb{I},
\DiffFormSpace^{2n-1}(M) \slash \dif \DiffFormSpace^{2n-2}(M)\bigr) \)

\begin{equation}
\xi^* \contr \Omega + \kappa\bigl(\difLog \skew{3}{\tilde}{\SectionMapAbb{J}},
\xi\bigr) = 0,
\end{equation}

where \( \xi^* \) is the fundamental vector field on \( \SectionSpaceAbb{I} \)
induced by the action of \( \xi \in \VectorFieldSpace(M, \omega) \).

Choose \( j_0 \in \SectionSpaceAbb{I} \).
Since the Chern class of the anti-canonical bundle \( \KBundle^{-1}_j M \)
is independent
of the almost complex structure \( j \), there exists a map
\( \tilde{J}(j_0, \cdot): \SectionSpaceAbb{I} \to \DiffFormSpace^{1}(M) \slash
\clZDiffFormSpace^{1}(M) \) such that

\begin{equation}
\label{eq:kaehler:momentumMap:asDifferenceCanonicalBundle}
\KBundle^{-1}_j M = \KBundle^{-1}_{j_0} M -
\iota\bigl(\tilde{J}(j_0, j)\bigr),
\end{equation}

where \( \iota: \DiffFormSpace^{k}(M) \slash \clZDiffFormSpace^{k}(M)
\to \csCohomology^{k+1}(M, \UGroup(1)) \) is the inclusion of topologically
trivial characters and \( \clZDiffFormSpace^{k}(M) \) is the space of
closed forms with integral periods. This identity states that
\( \KBundle^{-1}_j M \) and \( \KBundle^{-1}_{j_0} M \) are isomorphic,
and \( \tilde{J}(j_0, j) \)
is the difference of the Chern connections on these bundles up to gauge
transformations. Since \( \SectionSpaceAbb{I} \) is contractible, there
exists a lift \( \widebar{J}(j_0, \cdot): \SectionSpaceAbb{I}
\to \DiffFormSpace^{1}(M) \slash \dif \DiffFormSpace^{0}(M) \) of
\( \tilde{J}(j_0, \cdot) \) covering the projection
\( \pr: \DiffFormSpace^{k}(M) \slash \dif \DiffFormSpace^{k-1}(M)
\to \DiffFormSpace^{k}(M) \slash \clZDiffFormSpace^{k}(M) \). Thus,

\begin{equation}
\skew{3}{\tilde}{\SectionMapAbb{J}}(j) =
\skew{3}{\tilde}{\SectionMapAbb{J}}(j_0) +
\iota \circ \pr \bigl(\widebar{J}(j_0, j) \wedge \omega^{n-1} \bigr).
\end{equation}

Hence, the logarithmic derivative is given by
\( \difLog \skew{3}{\tilde}{\SectionMapAbb{J}} =
\bigl(\tangent \widebar{J}(j_0, \cdot)\bigr) \wedge \omega^{n-1} \).
This shows that the map

\begin{equation}
\SectionSpaceAbb{I} \ni j \mapsto \widebar{J}(j_0, j) \wedge \omega^{n-1}
\in \DiffFormSpace^{2n-1}(M) \slash \dif \DiffFormSpace^{2n-2}(M)
\end{equation}

is a momentum map for the action of \( \DiffGroup(M, \omega) \)
on \( \SectionSpaceAbb{I} \). Clearly, it vanishes at \( j_0 \) and
thus by uniqueness has to coincide with the momentum map
\( \SectionMapAbb{J} \).
In this way, we recover
Note that the group-valued momentum map \( \skew{3}{\tilde}{\SectionMapAbb{J}} \)
is equivariant under the action of \( \DiffGroup(M, \omega) \), but this
equivariance is broken for \( \SectionMapAbb{J} \) by choosing a reference
complex structure. We study this non-equivariance in more detail below
and will see that it has a topological character.

<Ref> implies that the Chern scalar
curvature is the momentum map for the action of the subgroup of Hamiltonian
In this way we recover the result of Fujiki1992,Donaldson1997
that the Chern scalar curvature is the momentum map for the action of the
subgroup of Hamiltonian diffeomorphisms.

The action of the group of Hamiltonian diffeomorphisms on
\( \SectionSpaceAbb{I} \) has a momentum map

\begin{equation}
\SectionMapAbb{J}_{\HamDiffGroup}(j) =
\frac{1}{2} \bigl(S_{j} - S_{j_0}\bigr) \, \mu_\omega
\end{equation}

relative to the integration pairing of \( \sFunctionSpace_0(M) \)
and \( \dif \DiffFormSpace^{2n-1}(M) \).
The non-equivariance one-cocycle

\begin{equation}
\HamDiffGroup(M, \omega) \to \dif \DiffFormSpace^{2n-1}(M),
\qquad \phi \mapsto \frac{1}{2} \bigl(S_{j_0} \circ
\phi^{-1} - S_{j_0}\bigr) \, \mu_\omega
\end{equation}

is a coboundary.

Consider the isomorphism of the space
\(\HamVectorFields(M, \omega)\) of Hamiltonian vector fields with the
space \( \sFunctionSpace_0(M) \) of smooth functions on \( M \) with zero
mean given by the map \( f \mapsto X_f \).
By Hodge theory, the natural integration pairing gives a non-degenerate
pairing of \( \sFunctionSpace_0(M) \) and \( \dif \DiffFormSpace^{2n-1}(M) \).
The following calculation, for
\( \alpha \in \DiffFormSpace^{2n-1}(M) \) and
\( f \in \sFunctionSpace_0(M) \),

\begin{equation}
(n-1)! \, \kappa\bigl(\equivClass{\alpha}, X_f \bigr)
= \int_M \alpha \wedge (X_f \contr \omega)
= - \int_M \alpha \wedge \dif f
= -\int_M f \dif \alpha
= \dualPair{-\dif \alpha}{f},
\end{equation}

shows that the adjoint of the map \( f \mapsto X_f \) is essentially
given by the exterior differential \( \dif: \DiffFormSpace^{2n-1}(M)
\slash \dif \DiffFormSpace^{2n-2}(M) \to \dif \DiffFormSpace^{2n-1}(M) \).
Thus, the momentum map for the action of \( \HamDiffGroup(M, \omega) \)
on \( \SectionSpaceAbb{I} \) is given by

\begin{equation}
\SectionMapAbb{J}_{\HamDiffGroup}(j) =
\frac{-1}{(n-1)!} \dif \SectionMapAbb{J}(j) =
-\dif J(j_0, j) \wedge \frac{\omega^{n-1}}{(n-1)!}.
\end{equation}

\begin{equation}
\label{eq:kaehler:difOfJForm}
\dif J(j_0, j) = \mathrm{Ric}_{j_0} - \mathrm{Ric}_{j}.
\end{equation}

This identity follows either from a direct calculation using identities
of [Proof of Theorem 2.6]Garcia-PradaSalamonTrautwein2018 or
from the identification of \( J(j_0, j) \) as the difference of the
Chern connections on the anti-canonical bundle by recalling
that \( \I \, \mathrm{Ric}_j \) is
the curvature of the Chern connection on \( K^{-1}_j M \).
Thus, invoking (<ref>) and the
definition of the scalar curvature in terms of the Ricci form, we find

\begin{equation}
\SectionMapAbb{J}_{\HamDiffGroup}(j) =
\bigl(\mathrm{Ric}_{j} - \mathrm{Ric}_{j_0}\bigr) \wedge
\frac{\omega^{n-1}}{(n-1)!} = \frac{1}{2} (S_{j} - S_{j_0}) \, \mu_\omega.
\end{equation}

The expression for the non-equivariance cocycle follows directly from
the definition (<ref>).
This finishes the proof.

§.§ Central extension and quasimorphism of Diff(M, w)

In contrast to the momentum map \( \SectionMapAbb{J}_{\HamDiffGroup} \) for the
subgroup of Hamiltonian diffeomorphisms, the momentum map \( \SectionMapAbb{J} \)
for the full group of symplectomorphisms is not equivariant, in general.
The different equivariance properties of these momentum maps can be succinctly
captured using the work of Vizman2006, who used the exact sequence

0 →[r]
(M, ω) →[r]
(M, ω) →[r]
^1(M) →[r]

to show that the second continuous Lie algebra cohomology
of \( \VectorFieldSpace(M, \omega) \) consists of sums of extensions of
certain \( 2 \)-cocycles on \( \HamVectorFields(M, \omega) \) and pull-backs
of elements of \( \ExtBundle^2 {\deRCohomology^1(M)}^* \); see
[Corollary 4.4]Vizman2006.

In order to describe the corresponding decomposition of the non-equivariance
cocycle of \( \SectionMapAbb{J} \), we need the following description of a
cocycle associated with a closed \( 2 \)-form.

Let \( (M, \omega) \) be a closed \( 2n \)-dimensional symplectic manifold
with \( n \geq 2 \), and \( \lambda \) be a closed \( 2 \)-form on \( M \).
Then the associated Lichnerowicz \( 2 \)-cocycle
This is a slight abuse of conventions since, usually, the
name Lichnerowicz cocycle refers to the cocycle defined on
the Lie algebra of volume-preserving vector fields. However, we are
mainly interested in its restriction to the subalgebra of symplectic
vector fields.

\begin{equation}
\lambda_c (\xi, \eta) \defeq \int_M \lambda(\xi, \eta) \, \mu_\omega
\end{equation}

on \( \VectorFieldSpace(M, \omega) \) is cohomologous to the pull-back
by the map
\( \VectorFieldSpace(M, \omega) \ni \xi \mapsto  [\xi \contr \omega]
\in \deRCohomology^1(M) \) of the bilinear form

\begin{equation}
\label{eq:kaehler:LichnerowiczCocyclePullback:onH1}
\lambda_c^H\bigl(\equivClass{\alpha}, \equivClass{\beta}\bigr)
= \int_M \Biggl(\frac{\mathrm{Av}_\omega (\lambda)}{2 (n-1)}  \, \omega
- \lambda\Biggr) \wedge \alpha \wedge \beta \wedge
\frac{\omega^{n-2}}{(n-2)!}
\end{equation}

on \( \deRCohomology^1(M) \), where \( \mathrm{Av}_\omega (\lambda) \defeq
\frac{1}{\vol_{\mu_\omega}(M)} \int_M \tr_\omega (\lambda) \, \mu_\omega \) and
\( \tr_\omega (\lambda) = \tensor{\lambda}{_i^i} = \varpi^{ij} \lambda_{ij}\);
see (<ref>).
In particular, the restriction of \( \lambda_c \) to
\( \HamVectorFields(M, \omega) \) is trivial in Lie algebra cohomology.

Moreover, \( \lambda_c^H \) vanishes if \( \lambda \wedge \omega^{n-2} \)
is exact. If the cup product yields an isomorphism
\( \ExtBundle^2 {\deRCohomology^1(M)} = \deRCohomology^2(M) \), then
exactness of \( \lambda \wedge \omega^{n-2} \) is also necessary for
\( \lambda_c^H \) to vanish.

Note that the bilinear form \( \lambda_c^H \) factors through the
cup product \( \deRCohomology^1(M) \times \deRCohomology^1(M) \to
\deRCohomology^2(M) \), and as such is closely related to the
skew-structures on \( \pi_1(M) \) studied by JohnsonRees1991.

By construction, \( \bar{\lambda} =
\tr_\omega (\lambda) - \mathrm{Av}_\omega (\lambda) \)
has average value zero. Thus, there exists
\( \tau \in \DiffFormSpace^{2n-1}(M) \) such
that \( \dif \tau = \bar{\lambda} \mu_\omega \).
Now, the calculation

\begin{equation}
\int_M \bar{\lambda} \omega(\xi, \eta) \, \mu_\omega
= \int_M \omega(\xi, \eta) \dif \tau
= - \int_M \difLie_{\eta}(\xi \contr \omega) \wedge \tau
= \int_M (\commutator{\xi}{\eta} \contr \omega) \wedge \tau
\end{equation}

shows that \( \lambda_c \) is cohomologous to the cocycle

\begin{equation}
\tilde{\lambda}_c(\xi, \eta) = \int_M \left(\lambda -
\frac{\bar{\lambda}}{2} \omega\right) (\xi, \eta) \, \mu_\omega \, .
\end{equation}

On the other hand, using (<ref>),
we find for every \( 2 \)-form \( \alpha \) that

\begin{equation}\begin{split}
\alpha(\xi, \eta) \, \mu_\omega
&= (\xi \contr \omega) \wedge (\eta \contr \alpha)
\wedge \frac{\omega^{n-1}}{(n-1)!}
\\
&= \alpha \wedge \frac{\omega^{n-1}}{(n-1)!} \, \omega(\xi, \eta)
- \lambda \wedge (\xi \contr \omega) \wedge (\eta \contr \omega) \wedge
\frac{\omega^{n-2}}{(n-2)!}
\\
&= \frac{1}{2} \tr_\omega(\alpha) \, \omega(\xi, \eta) \, \mu_\omega -
\alpha \wedge (\xi \contr \omega) \wedge (\eta \contr \omega)
\wedge \frac{\omega^{n-2}}{(n-2)!}.
\end{split}\end{equation}

The first and second identities follow from contracting
\( (\xi \contr \alpha) \wedge \omega^n = 0 \) and
\( (\xi \contr \omega) \wedge \alpha \wedge \omega^{n-1} = 0 \)
with \( \eta \), respectively.
Hence, using this relation for \( \alpha = \lambda \)
and in the second line for \( \alpha = \omega \), we obtain

\begin{equation}\begin{split}
\tilde{\lambda}_c(\xi, \eta)
&= \int_M \frac{1}{2} \mathrm{Av}_\omega (\lambda) \,
\omega (\xi, \eta) \, \mu_\omega -
\lambda \wedge (\xi \contr \omega) \wedge
(\eta \contr \omega) \wedge \frac{\omega^{n-2}}{(n-2)!}
\\
&= \int_M \Biggl(\frac{1}{2 (n-1)} \mathrm{Av}_\omega(\lambda) \, \omega
- \lambda\Biggr) \wedge (\xi \contr \omega) \wedge
(\eta \contr \omega) \wedge \frac{\omega^{n-2}}{(n-2)!}\,.
\end{split}\end{equation}

From this expression, it is evident that \( \tilde{\lambda}_c \) is the
pull-back from \( \deRCohomology^1(M) \) of \( \lambda_c^H \).

Finally, if \( \lambda \wedge \omega^{n-2} \) is exact, then the average
of \( \lambda \) vanishes by (<ref>),
and thus also \( \lambda_c^H = 0 \). Conversely, assume that cup product
identifies \( \ExtBundle^2 {\deRCohomology^1(M)} \) with
\( \deRCohomology^2(M) \) and that \( \lambda_c^H = 0 \).
Then, the linear functional \( \deRCohomology^2(M) \to \R \) given by
integration against \( \sigma \wedge \omega^{n-2} \) with \( \sigma =
\frac{\mathrm{Av}_\omega(\lambda)}{2 (n-1)} \omega - \lambda \)
has to vanish, , \( \sigma \wedge \omega^{n-2} \) has to be exact.
But then \( 0 = \mathrm{Av}_\omega(\sigma) =
\mathrm{Av}_\omega(\lambda) \bigl(\frac{n}{n-1} - 1\bigr) \),
and thus \( \mathrm{Av}_\omega(\lambda)
= 0 \). Hence, \( \sigma = \lambda \) and
\( \lambda \wedge \omega^{n-2} \) has to be exact.

Applied to the non-equivariance cocycle of \( \SectionMapAbb{J} \), we
find the following.

The class of the non-equivariance cocycle of \( \SectionMapAbb{J} \) in the second
continuous Lie algebra cohomology of \( \VectorFieldSpace(M, \omega) \)
coincides with the pull-back along the natural map
\( \VectorFieldSpace(M, \omega) \to \deRCohomology^1(M) \) of the
antisymmetric bilinear form

\begin{equation}
\Sigma^H_{j_0}\bigl(\equivClass{\alpha}, \equivClass{\beta}\bigr) \defeq \int_M
\left((\mathrm{Ric}_{j_0})_{rs} - \frac{1}{2}\bar{S}_{j_0}\omega_{rs}
\right) \alpha^r \beta^s \, \mu_\omega
\end{equation}

on \( \deRCohomology^1(M) \), where the indices of \( \alpha \) and
\( \beta \) are raised using \( \omega \).
Moreover, the non-equivariance cocycle is trivial in Lie algebra
cohomology if \( c_1(M) \cup \equivClass{\omega}^{n-2} = 0 \), and
this condition is necessary when the cup product yields an
isomorphism \( \ExtBundle^2 {\deRCohomology^1(M)} = \deRCohomology^2(M) \).

Note that the class of the non-equivariance cocycle \( \Sigma^H_{j_0} \)
is independent of the reference complex structure \( j_0 \) and thus is
a well-defined invariant of the symplectic manifold \( (M, \omega) \).
Recall that a Kähler manifold with vanishing first real Chern class is
called a Calabi–Yau manifold. Thus, for Calabi–Yau manifolds, the
non-equivariance cocycle is trivial in Lie algebra cohomology.
To emphasize the close relationship, we say that a symplectic manifold
\( (M, \omega) \) is weakly Calabi–Yau if \( \Sigma^H_j = 0 \)
for some compatible almost complex structure \(  j \).

Let \( j_0 \in \SectionSpaceAbb{I} \) and let
\( X, Y \in \VectorFieldSpace(M, \omega) \).
By (<ref>), the non-equivariance 2-cocycle
\( \Sigma \) is given by \( \Sigma(X, Y) = -
\kappa\Bigl(\tangent_{j_0}\SectionMapAbb{J} \bigl(\difLie_X j_0\bigr),
Y\Bigr) \). On the other hand, by (<ref>), we find

\begin{equation}\begin{split}
\tangent_{j_0}J(j_0, \cdot) \bigl(\difLie_X j_0\bigr)
&= -\frac{1}{2} \tau^\nabla(j_0, \difLie_X j_0)
= - X \contr \mathrm{Ric}_{j_0} -\frac{1}{2} \dif \divergence(j_0X),
\end{split}\end{equation}

where the second equality follows from
[Theorem 2.7]Garcia-PradaSalamonTrautwein2018.
Thus, in summary,

\begin{equation}\begin{split}
\Sigma(X, Y)
&= \kappa\bigl((X \contr \mathrm{Ric}_{j_0})\wedge \omega^{n-1}, Y\bigr)
\\
&= \int_M (X \contr \mathrm{Ric}_{j_0}) \wedge \frac{\omega^{n-1}}{(n-1)!}
\wedge (Y \contr \omega)
\\
&= \int_M \mathrm{Ric}_{j_0}(X, Y) \, \mu_\omega.
\end{split}\end{equation}

Alternatively, this identity for the non-equivariance cocycle follows
directly from [Eq. (2.33)]Garcia-PradaSalamonTrautwein2018.
The claim now is a consequence of

From <ref> we know that the
non-equivariance cocycle of \( \SectionMapAbb{J} \) integrates to
a central extension of \( \DiffGroup(M, \omega) \).
In fact, the associated group \( 2 \)-cocycle \( c \) on
\( \DiffGroup(M, \omega) \) can be explicitly computed
using (<ref>), at least in principle.
The cocycle \( c \) also coincides with the cocycle given in
Reznikov1999 (where it appeared out of thin air).
Moreover, according to <ref>, the
cocycle \( c \) is bounded in the sense of Gromov.
This follows from the fact that \( \SectionSpaceAbb{I} \) is a
Domic-Toledo space, essentially because it is the space of sections
of a bundle whose typical fiber is a Domic-Toledo space; see
[Section 1.7]Shelukhin2014 for details.

Although this prescription yields a direct way to construct the
central extension of \( \DiffGroup(M, \omega) \), its geometric
interpretation still remains unclear.
This can be compared to the description of the momentum map above:
<ref> gives a concrete formula for the
momentum map \( \SectionMapAbb{J} \), but its geometric interpretation
in terms of the anti-canonical bundle is not obvious from this formula.
A first step towards a geometric interpretation of the central extension
is to better understand the prequantum bundle of \( \SectionSpaceAbb{I} \).
For example, realize it as a determinant line bundle of certain Dirac
operators or use the asymptotic prescription of FothUribe2007.

If \( \Sigma^H_{j} = 0  \) for some compatible almost complex structure
\( j \), , \( (M, \omega) \) is weakly Calabi–Yau, then the momentum
map \( \SectionMapAbb{J} \) is infinitesimally equivariant.
We can thus apply the general construction of Shelukhin2014
to obtain a quasimorphism on the universal covering of
\( \DiffGroup(M, \omega)_0 \).
By construction, the restriction of this quasimorphism to the subgroup
of Hamiltonian diffeomorphisms is the (non-trivial) quasimorphism
constructed in Shelukhin2014.
For completeness, let us record this observation.

If \( (M, \omega) \) is weakly Calabi–Yau, then the universal
covering of \( \DiffGroup(M, \omega)_0 \) admits a non-trivial
quasimorphism and hence has infinite commutator length.

Under the stronger assumption that the first Chern class vanishes,
Entov2004 constructed a quasimorphism on the universal
covering of \( \DiffGroup(M, \omega)_0 \) that coincides with the
Shelukhin quasimorphism on Hamiltonian diffeomorphisms.
Hence, it is natural to conjecture that the quasimorphism constructed
above is a natural generalization of the Entov quasimorphism; see
also [Point 3.2]Shelukhin2014.

§.§ Norm-squared momentum map

We identify the Lie algebra \( \HamVectorFields(M, \omega) \) of
Hamiltonian vector fields with the Lie algebra \( \sFunctionSpace_0(M) \)
of smooth functions with average zero by \( F \mapsto X_F \).
Using this identification, there is a natural inner product on
\( \HamVectorFields(M, \omega) \) defined by

\begin{equation}
\label{eq:kaehler:pairingOnHam}
\scalarProd{X_F}{X_G} = \frac{1}{2} \int_M F G \, \mu_\omega.
\end{equation}

To extend this inner product to \( \VectorFieldSpace(M, \omega) \),
consider the exact sequence

\begin{equation}
\label{eq:kaehler:symplecticVectorFieldsExactSequence}
0 \to \HamVectorFields(M, \omega) \to \VectorFieldSpace(M, \omega)
\to \deRCohomology^1(M) \to 0.
\end{equation}

We split this sequence by choosing a reference almost complex
structure \( j_0 \in \SectionMapAbb{I} \).
Let \( \mathfrak{har}_{j_0}(M) \isomorph \deRCohomology^1(M) \)
be the space of all vector fields \( \xi^h \) such that
\( \omega^\flat(\xi^h) \) is a \( g_{j_0} \)-harmonic
\( 1 \)-form. We call such infinitesimally
symplectic vector fields harmonic.
Then every \( \xi \in \VectorFieldSpace(M, \omega) \)
can be uniquely written as \( \xi = X_F + \xi^h \) for some
\( F \in \sFunctionSpace_0(M) \) and
\( \xi^h \in \mathfrak{har}_{j_0}(M) \). Define the following
inner product on \( \VectorFieldSpace(M, \omega) \):

\begin{equation}
\label{eq:kaehler:pairingOnLieAlgebra}
\scalarProd{\xi}{\eta}_{j_0} =
\int_M \frac{1}{2} F G \, \mu_\omega \, + \omega^\flat(\xi^h)
\wedge \hodgeStar_{g_{j_0}} \omega^\flat(\eta^h) =
\int_M \left(\frac{1}{2} F G +
g_{j_0}(\xi^h, \eta^h)\right) \, \mu_\omega
\end{equation}

for \( \xi = X_F + \xi^h \) and \( \eta = X_G + \eta^h \),
where \(\hodgeStar_{g_{j_0}}\) is the
Hodge star operator defined by the Riemannian metric \(g_{j_0}\).
From <ref>, we obtain
the momentum map relative to the inner
product \( \scalarProdDot_{j_0} \)
given in (<ref>).

For every \( j_0 \in \SectionSpaceAbb{I} \), the action
of \( \DiffGroup(M, \omega) \) on \( \SectionSpaceAbb{I} \)
has a momentum map
\( \SectionMapAbb{J}: \SectionSpaceAbb{I} \to
\VectorFieldSpace(M, \omega) \) relative to the inner
product \( \scalarProdDot_{j_0} \) given by assigning
to \( j \in \SectionSpaceAbb{I} \) the symplectic vector field

\begin{equation}
- X_{\bar{S}_j} + \omega^\sharp\bigl(j_0 J(j_0, j)\bigr)^h,
\end{equation}

where \( \bar{S}_j \) is the normalized Chern scalar curvature
given by (<ref>), the
\( 1 \)-form \( J(j_0, j) \) is defined
in (<ref>), and the
superscript \( h \) refers to taking the
\( g_{j_0} \)-harmonic part.

Moreover, \( \SectionMapAbb{J}_\HamDiffGroup:
\SectionSpaceAbb{I} \ni j \mapsto - X_{\bar{S}_j}
\in \HamVectorFields(M, \omega) \) is a momentum
map for the action of \( \HamDiffGroup(M, \omega) \),
relative to the inner product \( \scalarProdDot \)
defined in (<ref>).

With respect to the pairing \( \kappa \)
from (<ref>), we have for every
\( \xi = X_F + \xi^h \in \VectorFieldSpace(M, \omega) \)
and \( \alpha \in \DiffFormSpace^1(M) \):

\begin{equation}\begin{split}
\kappa\bigl(\equivClass{\alpha \wedge \omega^{n-1}}, \xi\bigr)
&= \frac{1}{(n-1)!} \int_M \alpha \wedge \omega^{n-1}
\wedge (\xi \contr \omega)
\\
&= \frac{1}{(n-1)!} \int_M F \, \dif \alpha \wedge \omega^{n-1}
+ \alpha \wedge (\xi^h \contr \omega) \wedge \omega^{n-1}
\\
&= \int_M \left(\frac{1}{2} F \tr_\omega(\dif \alpha)
+ \alpha(\xi^h)\right) \, \mu_\omega \,,
\end{split}\end{equation}

where the last equality follows
from (<ref>).
On the other hand, we have \( \alpha(\xi^h) =
g_{j_0}\bigl(\omega^\flat (\xi^h), j_0 \alpha\bigr) \)
with \( j_0 \alpha \defeq - \alpha(j_0 \cdot) \).
By assumption, \( \omega^\flat (\xi^h) \) is a
\( g_{j_0} \)-harmonic \( 1 \)-form.
Thus, \( \LFunctionSpace^2 \)-orthogonality of the
Hodge decomposition implies that only the harmonic
part \( (j_0\alpha)^h \) of \( j_0 \alpha \) contributes
to the integral, and we obtain

\begin{equation}\label{eq:kaehler:projectionDualSymp}\begin{split}
\kappa\bigl(\equivClass{\alpha \wedge \omega^{n-1}}, \xi\bigr)
&= \int_M \left(\frac{1}{2} F \tr_\omega(\dif \alpha) +
\xi^h\right)\right) \, \mu_\omega
\\
&= \scalarProd{\eta_\alpha}{\xi}_{j_0},
\end{split}\end{equation}

for \( \eta_\alpha = X_G + \omega^\sharp\bigl((j_0 \alpha)^h\bigr) \)
with \( G = \tr_\omega(\dif \alpha) -  \mathrm{Av}_\omega(\dif \alpha)\).
Note that if \( \alpha \) is exact, say \( \alpha = \dif f \), then the
Kähler identities, see [Proposition 1.14.1]Gauduchon2017,
imply \( j_0 \alpha = \diF (f \omega) \) so that \( (j_0 \alpha)^h = 0 \),
and hence \( \eta_\alpha = 0 \).
This verifies that \( \eta_\alpha \) depends only on the equivalence
class of \( \alpha \) modulo exact forms.

Finally, <ref> and (<ref>)
imply that \( \SectionMapAbb{J}(j) = X_F + \eta^h \) with

\begin{equation}
F = S_{j_0} - S_j \quad \text{and} \quad \omega^\flat (\eta^h)
= \bigl(j_0 J(j_0, j)\bigr)^h
\end{equation}

is a momentum map relative to the inner product
\( \scalarProdDot_{j_0} \).
Clearly, we can shift \( \SectionMapAbb{J} \) by a constant
and still obtain a momentum map.
The momentum map for the subgroup of Hamiltonian diffeomorphisms
can be calculated in a similar way.
This finishes the proof.

Assume that \( \mathrm{Ric}_{j_0} = \mathrm{Ric}_{j} \),
so that \( J(j_0, j) \) is closed by (<ref>).
In this situation, the harmonic form \( (j_0 J(j_0, j))^h \) in
<ref> has a nice
geometric interpretation.

To find it, choose an orthonormal basis
\( \set{\alpha_p} \) of \( g_{j_0} \)-harmonic \( 1 \)-forms,
and expand \( (j_0 J(j_0, j))^h = \sum_p c_p \alpha_p \).
The coefficients \(c_p\) are given by

\begin{equation}\begin{split}
c_p &= \dualPair{j_0 J(j_0, j)}{\alpha_p}_{j_0}
= \int_M g_{j_0}(j_0 J(j_0, j), \alpha_p) \, \mu_\omega
\\
&= \int_M \omega(J(j_0, j), \alpha_p) \, \mu_\omega
= \int_M J(j_0, j) \wedge \alpha_p \wedge \frac{\omega^{n-1}}{(n-1)!}\,,
\end{split}\end{equation}

where the last equality follows
from (<ref>).
Thus, in summary, \( c_p = \int_{\gamma_p} J(j_0, j) \)
for the Poincaré dual \( \gamma_p \) of the \( (2n-1) \)-form
\( \alpha_p \wedge \frac{\omega^{n-1}}{(n-1)!} \).
Now recall that \( J(j_0, j) \) is the difference of the Chern
connections of \( j_0 \) and \( j \) on \( \KBundle_{j_0} M \)
(relative to an identification of
\( \KBundle_{j_0} M \isomorph \KBundle_j M \)).
Thus, \( c_p \) is the difference of the holonomies of the
Chern connections of \( j_0 \) and \( j \) around the
loop \( \gamma_p \).

The norm-squared of the momentum map for the
action of Hamiltonian diffeomorphisms is  the
\( \LFunctionSpace^2 \)-norm of the (normalized) scalar curvature
that is, the Calabi energy functional on \( \SectionSpaceAbb{I} \):

\begin{equation}
\label{eq:kaehler:momentumMapHamSquared}
\norm{\SectionMapAbb{J}_\HamDiffGroup}^2(j) =
\frac{1}{2} \int_M {\bar{S}_j}^2 \mu_\omega \,.
\end{equation}

Critical points of \( \norm{\SectionMapAbb{J}_\HamDiffGroup}^2 \)
are called extremal almost-Kähler metrics; see
Calabi1985 in the Kähler setting and
Lejmi2010 without the integrability condition.
According to <ref>, these
are precisely the almost complex structures \( j \) for which
the Hamiltonian vector field \( X_{\bar{S}_j} \) is a real
holomorphic vector field, , \( \difLie_{X_{\bar{S}_j}} j = 0 \).
Constant scalar curvature metrics constitute an important special
case of extremal almost-Kähler metrics, and they correspond
to zeros of \( \SectionMapAbb{J}_\HamDiffGroup \).

Similarly, the norm-squared of the momentum map
for the full group of symplectomorphisms (see
yields the following
functional on \( \SectionSpaceAbb{I} \):

\begin{equation}
\label{eq:kaehler:momentumMapSquared}
\norm{\SectionMapAbb{J}}_{j_0}^2(j) =
\frac{1}{2} \int_M \left({\bar{S}_j}^2 +
2 \norm{\bigl(j_0 J(j_0, j)\bigr)^h}_{j_0} \right) \mu_\omega \,,
\end{equation}

where on the right-hand side the norm of the one-form
\( \bigl(j_0 J(j_0, j)\bigr)^h \) is taken with respect
to the metric \( g_{j_0} \).
The first summand is again the Calabi energy.
The second summand penalizes the difference between the
Chern connections of \( j \) and \( j_0 \).
In other words, \( \norm{\SectionMapAbb{J}}_{j_0}^2 \)
can be viewed as a localized Calabi energy.
Zeros of \( \norm{\SectionMapAbb{J}}_{j_0}^2 \) are, if they exist,
almost complex structures \( j \) with constant Chern scalar curvature
and \( j_0 J(j_0, j) \) having no \( g_{j_0} \)-harmonic component.
In analogy with Calabi's extremal metrics, we say that an almost
complex structure \( j \) is a \( j_0 \)-extremal metric
if it is a critical point of \( \norm{\SectionMapAbb{J}}_{j_0} \).
By <ref>, this is equivalent
to \( \SectionMapAbb{J}(j) \) being a real holomorphic vector field.

Our general results in
require a few technical properties. Let us check that these
are satisfied in the present situation.

The following holds:

* The group of symplectomorphisms \( \DiffGroup(M, \omega) \)
is an infinite-dimensional Fréchet Lie group and has a smooth
exponential map given by the flow.

* For every \( j \in \SectionSpaceAbb{I} \), the stabilizer
\( \DiffGroup(M, \omega)_j \) is a finite-dimensional Lie
subgroup of \( \DiffGroup(M, \omega) \) consisting of isometries
of \( g_j \).

* For every \( j \in \SectionSpaceAbb{I} \), the isotropy
representation of \( \DiffGroup(M, \omega)_j \) on
\( \TBundle_j \SectionSpaceAbb{J} \) is Hamiltonian with momentum
map given by

\begin{equation}
\label{eq:kaehler:momentumMapIsotropyRepresentation}
\widehat{\SectionMapAbb{J}_j}(A) = -\frac{1}{8} \pr_j
\left(X_{\tr_\omega(\dif \alpha)} +
\omega^\sharp \bigl((j_0 \alpha)^h\bigr)\right),
\end{equation}

where \( \alpha(Y) \defeq \tr(A j \nabla_Y A) -
2 \tr\bigl(\nabla (A j A) Y\bigr) \) for a torsion-free
connection \( \nabla \) preserving the volume form and
\( \pr_j \) is the orthogonal projection onto
\( \VectorFieldSpace(M, \omega)_j \).

For every \( X \in \VectorFieldSpace(M, \omega) \),
the adjoint of \( \difLie_X: \VectorFieldSpace(M, \omega) \to
\VectorFieldSpace(M, \omega) \) with respect to
\( \dualPairDot_{j_0} \) is

\begin{equation}
\label{eq:kaehler:adjointLieDerivative}
X_F + \xi^h \mapsto - \difLie_X X_F + \frac{1}{2} (F j_0 X)^h,
\end{equation}

for \( F \in \sFunctionSpace_0(M) \) and

\[ \xi^h \in \mathfrak{har}_{j_0}(M) \defeq \{\zeta \in
\mathfrak{X}(M, \omega) \mid
\omega^\flat(\zeta ) \text{ is a } g_{j_0}\text{-harmonic }
\]

\begin{equation}\begin{split}
\label{eq:kahler:adjointLieDerivative:pairing}
\dualPair*{\difLie_X (X_F + \xi^h)}{X_G + \eta^h}_{j_0}
&= - \dualPair*{X_F + \xi^h}{\difLie_X(X_G + \eta^h)}_{j_0}\\
&\qquad+ \frac{1}{2} \dualPair*{(Fj_0 X)^h}{\eta^h}_{j_0}\\
&\qquad+ \frac{1}{2} \dualPair*{\xi^h}{(Gj_0 X)^h}_{j_0}.
\end{split}\end{equation}

This shows that \( \dualPairDot_{j_0} \) is
not invariant under the Lie derivative by vector fields in
\(\VectorFieldSpace(M, \omega) \). However,
\( \dualPairDot_{j_0} \) is invariant under the Lie
derivative by elements of the stabilizer
\( \VectorFieldSpace(M, \omega)_{j_0} \).

* The almost complex structure \( \SectionMapAbb{j} \) is
equivariant with respect to the push-forward action of
\( \DiffGroup(M, \omega) \).

The group of symplectomorphisms is a Fréchet Lie group
by [Theorem 43.12]KrieglMichor1997 and the
automorphism group of an almost complex structure is a
finite-dimensional Lie group by
[Corollary I.4.2]Kobayashi1972.

Let \( \nabla \) be a torsion-free connection-preserving the
volume form. Using \( \difLie_X A =
\nabla_X A + \commutator{A}{\nabla X} \), we find

\begin{equation}
\tr(A j \difLie_X A)
= \tr(A j \nabla_X A) + 2 \tr(A j A \nabla X)
= \alpha(X) + 2 \tr\bigl(\nabla(A j A X)\bigr).
\end{equation}

On the other hand, \( \tr(\nabla Y) \) is the divergence of the
vector field \( Y \) so that upon integration over \( M \),
the last term vanishes and we obtain

\begin{equation}\begin{split}
\dualPair{\widehat{\SectionMapAbb{J}_j}(A)}{X}
\frac{1}{2} \Omega_j(A, -\difLie_X A) \\
&\, \, \, = - \frac{1}{8} \int_M \tr(A j \difLie_X A) \, \mu_\omega \\
&\, \, \, = - \frac{1}{8} \int_M \alpha(X) \, \mu_\omega \\
&\, \, \, = - \frac{1}{8} \int_M \alpha \wedge (X \contr \mu_\omega) \\
&\,\stackrel{\eqref{eq:kaehler:pairing}}{=} - \frac{1}{8}
\kappa\bigl(\equivClass{\alpha \wedge \omega^{n-1}}, X\bigr) \\
&\stackrel{\eqref{eq:kaehler:projectionDualSymp}}{=} -\frac{1}{8}
\dualPair*{X_{\tr_\omega(\dif \alpha)} +
\omega^\sharp \bigl((j_0 \alpha)^h\bigr)}{X}_{j_0}. \\
\end{split}\end{equation}

From this identity we directly read
off (<ref>).

Let \( X \in \VectorFieldSpace(M, \omega) \),
\( F, G \in \sFunctionSpace_0(M) \) and
\( \xi^h, \eta^h \in \mathfrak{har}_{j_0}(M) \).
Then we find

\begin{equation}\begin{split}
&\dualPair*{\difLie_X (X_F + \xi^h)}{X_G + \eta^h}_{j_0}\\
&\qquad= \dualPair*{X_{\omega(X, X_F + \xi^h)}}{X_G + \eta^h}_{j_0}
\\
&\qquad= \frac{1}{2} \int_M \omega(X, X_F + \xi^h) G \, \mu_\omega
\\
&\qquad= \frac{1}{2} \int_M \left(\dif F (X) G -
G \omega^\flat(\xi^h)(X)\right) \mu_\omega
\\
&\qquad= \frac{1}{2} \int_M - \dif G (X) F \mu_\omega -
G \omega^\flat(\xi^h) \wedge (X \contr \mu_\omega)
\\
&\qquad= - \dualPair*{X_F + \xi^h}{\difLie_X X_G}_{j_0} -
\frac{1}{2} \int_M G \omega^\flat(\xi^h)
\wedge *_{{g_j}_0} g^\flat_{j_0}(X)
\\
&\qquad= - \dualPair*{X_F + \xi^h}{\difLie_X X_G}_{j_0} +
\frac{1}{2} \int_M G \omega^\flat(\xi^h)
\wedge *_{{g_j}_0} \omega^\flat(j_0X)
\\
&\qquad= \dualPair*{X_F + \xi^h}{- \difLie_X X_G +
\frac{1}{2} (G j_0 X)^h}_{j_0}.
\end{split}\end{equation}

This verifies (<ref>).
Using this equation, we obtain

\begin{equation}\begin{split}
\dualPair*{\difLie_X (X_F + \xi^h)}{X_G + \eta^h}_{j_0}
&= -\dualPair*{X_F + \xi^h}{\difLie_X(X_G + \eta^h)}_{j_0}\\
&\qquad+ \frac{1}{2} \dualPair*{X_F + \xi^h}{\difLie_X \eta^h}_{j_0}\\
&\qquad+ \frac{1}{2} \dualPair*{X_F + \xi^h}{(Gj_0X)^h}_{j_0}.
\end{split}\end{equation}

Applying again (<ref>) on the
second summand, we get (<ref>).

Now, if \( X \) is Killing, then \( 0 = \difLie_X \alpha =
\dif (X \contr \alpha) \) for every harmonic
\( 1 \)-form \( \alpha \). Applied to
\( \alpha = \omega^\flat(\xi^h) \) in the above
chain of equalities at step \( 3 \), we conclude
that then \( \dualPair*{\difLie_X (X_F + \xi^h)}{X_G + \eta^h}_{j_0}
= \frac{1}{2} \int_M (\dif F(X) G) \, \mu_\omega =
-\dualPair*{X_F + \xi^h}{\difLie_X(X_G + \eta^h)}_{j_0} \),
which shows that \( \dualPairDot_{j_0} \) is invariant under
the adjoint action of Killing vector fields.

For every \( \phi \in \DiffGroup(M, \omega) \), we have

\begin{equation}
\phi_* \, \SectionMapAbb{j}_j(A) = \phi_* (A j) =
(\phi_* A) (\phi_* j) =
\SectionMapAbb{j}_{\phi_* j} (\phi_* A).
\end{equation}

Thus, \( \SectionMapAbb{j} \) is equivariant.

Let us discuss the analogs of the Lichnerowicz and Calabi operators.
Relative to the splitting of the exact
sequence (<ref>), every
operator \( T: \VectorFieldSpace(M, \omega) \to
\VectorFieldSpace(M, \omega) \) gives rise to operators
\( T^{SS}: \sFunctionSpace(M) \to \sFunctionSpace(M) \),
\( T^{HS}: \mathfrak{har}_{j_0}(M) \to \sFunctionSpace(M) \),
\( T^{SH}: \sFunctionSpace(M) \to \mathfrak{har}_{j_0}(M) \) and
\( T^{HH}: \mathfrak{har}_{j_0}(M) \to \mathfrak{har}_{j_0}(M) \).
The defining equations for these operators are

\begin{equation}
T(X_f) = X_{T^{SS}(f)} + T^{SH}(f),
\qquad T(X^h) = X_{T^{HS}\bigl(X^h\bigr)}
+ T^{HH} X^h
\end{equation}

with \( f \in \sFunctionSpace(M) \) and
\( X^h \in \mathfrak{har}_{j_0}(M) \), and where
we keep identifying \( \deRCohomology^1(M) \) with
\( g_{j_0} \)-harmonic \( 1 \)-forms.
Using this notation, we calculate the
Lichnerowicz operator introduced in the general setting in (<ref>).

For a \( j_0 \)-extremal almost Kähler metric
\( j \in \SectionSpaceAbb{I} \), the operators
\( L_j \xi = \tangent_j \SectionMapAbb{J} (j \, \difLie_\xi j) \)
and \( Z_j \xi = -\tangent_j \SectionMapAbb{J} (\difLie_\xi j) \)
on \( \VectorFieldSpace(M, \omega) \) are given by:

\begin{equation}\label{eq:kaehler:Lj}\begin{split}
L_j^{SS}(f) &= - \frac{1}{2}\tr_\omega
(\dif \tau^\nabla(j, j \difLie_{X_f} j))\\
L_j^{SH}(f) &= - \frac{1}{2} \omega^\sharp (
j_0 \tau^\nabla(j, j \difLie_{X_f} j))^h\\
L_j^{HS}(X^h) &= - \frac{1}{2}\tr_\omega
(\dif \tau^\nabla(j, j \difLie_{X^h} j)) \\
L_j^{HH}(X^h) &= - \frac{1}{2} \omega^\sharp (j_0 \tau^\nabla(j,
j \difLie_{X^h} j))^h,
\end{split}\end{equation}

\begin{equation}\label{eq:kaehler:Zj}\begin{split}
Z_j^{SS}(f) &= \poisson{f}{\bar{S}_j}\\
Z_j^{SH}(f) &= \omega^\sharp (j_0 (X_f \contr \mathrm{Ric}_j))^h\\
Z_j^{HS}(X^h) &= \dif \bar{S}_j(X^h)\\
Z_j^{HH}(X^h) &= \omega^\sharp (j_0 (X^h
\contr \mathrm{Ric}_j))^h
\end{split}\end{equation}

for \( f \in \sFunctionSpace(M) \) and every
\( X^h \in \mathfrak{har}_{j_0}(M) \).

These follow from direct computations.

*Calculation of \( Z_j \):
By <ref>, \( \tangent_j J(j_0, \cdot)(A) \)
equals \( - \frac{1}{2} \tau^\nabla(j, A) \) modulo an exact form.
On the other hand, from
[Theorem 2.7]Garcia-PradaSalamonTrautwein2018,
for every \( \xi \in \VectorFieldSpace(M, \omega) \), we have

\begin{equation}
\tau^\nabla(j, \difLie_\xi j) = 2 \xi \contr \mathrm{Ric}_j
+ \dif \divergence(j\xi).
\end{equation}

Hence, viewing \( \SectionMapAbb{J} \) as a map into
\( \DiffFormSpace^{2n-1} M \slash \dif \DiffFormSpace^{2n-2} M \),
we have

\begin{equation}
- \tangent_j \SectionMapAbb{J}(\difLie_\xi j) =
\frac{1}{2} \tau^\nabla(j, \difLie_\xi j) \wedge
\omega^{n-1} \textrm{ mod exact}
= (\xi \contr \mathrm{Ric}_j) \wedge \omega^{n-1}
\textrm{ mod exact}.
\end{equation}

Thus, composing with the projection (<ref>),
we obtain \( Z_j \xi = X_G + \omega^\sharp \beta^h \) for

\begin{equation}
G = \tr_\omega (\dif (\xi \contr \mathrm{Ric}_j))
= \tr_\omega (\difLie_\xi \mathrm{Ric}_j)
= \difLie_\xi S_j = \omega(\xi, X_{S_j})
\end{equation}

\begin{equation}
\beta^h = (j_0 (\xi \contr \mathrm{Ric}_j))^h.
\end{equation}

Using these equations for \( \xi = X_f \) or \( \xi = X^h \)
yields (<ref>).

*Calculation of \( L_j \):
Using a similar argument as above, we obtain
\( L_j \xi = X_G + \omega^\sharp \beta^h \) for

\begin{equation}
G = - \frac{1}{2}\tr_\omega (\dif \tau^\nabla(j, j \difLie_\xi j))
\end{equation}

\begin{equation}
\beta^h = - \frac{1}{2} (j_0 \tau^\nabla(j, j \difLie_\xi j))^h.
\end{equation}

Using these equations for \( \xi = X_f \) or \( \xi = X^h \)
yields (<ref>).

In the integrable case, the operator \( L_j^{SS}: \sFunctionSpace_0(M)
\to \sFunctionSpace_0(M) \) recovers the classical Lichnerowicz
operator, and a few different ways of writing it down are known
in the literature, see Gauduchon2017.
For non-integrable \( j \), similar expressions for \( L_j^{SS} \)
are obtained in Vernier2020,HeZheng2023.
In both cases, one concludes from these explicit expressions
that \( L_j^{SS} \) is a \( 4 \)-order elliptic differential
operator. In particular, \( L_j \) is a Fredholm operator.

Following the general procedure,
(<ref>), for
every \( j \in \SectionMapAbb{I} \), the Calabi operators
\( C^\pm_j: \VectorFieldSpace(M, \omega)_\C \to
\VectorFieldSpace(M, \omega)_\C \) are defined by

\begin{equation}
C^\pm_j = L_j \pm \I Z_j.
\end{equation}

Recall that a real vector field \( X \) is called holomorphic
if \( \difLie_X j = 0 \).
Using a similar argument as above, one sees that \( C^\pm_j \)
are Fredholm operators.
In particular, their kernels are finite-dimensional.

For every \( j \in \SectionMapAbb{I} \), the kernel of \( C^+_j \)
coincides with the stabilizer \( (\VectorFieldSpace(M, \omega)_\C)_j \)
under the complexified action. If \( j \) is integrable, then the
map \( \VectorFieldSpace(M, \omega)_\C \ni \xi +
\I \eta \mapsto \xi - j \eta \in \VectorFieldSpace(M) \)
restricts to a surjection from \( (\VectorFieldSpace(M, \omega)_\C)_j \)
onto the space of real holomorphic vector fields and it has kernel
\( \mathfrak{har}_{j}(M) \).

We do not know whether \( (\VectorFieldSpace(M, \omega)_\C)_j \)
is a Lie subalgebra of \( \VectorFieldSpace(M, \omega)_\C \).

The first statement follows directly from
For the second statement, we observe that \( \xi + \I \eta \)
is in the stabilizer \( (\VectorFieldSpace(M, \omega)_\C)_j \)
if and only if \( 0 = \xi \ldot j + \SectionMapAbb{j}_j (\eta \ldot j)
= - \difLie_\xi j  + j \difLie_\eta j =
- \difLie_\xi j + \difLie_{j \eta} j \),
where the last equality uses the integrability of \( j \),
see, , [Lemma 1.1.1]Gauduchon2017.
In other words, \( \xi - j \eta \) is a real holomorphic vector
field. Conversely, by [Lemma 2.1.1]Gauduchon2017,
every real holomorphic vector field \( X \) on a compact Kähler
manifold can be uniquely written as the sum \( X = j X^h + j X_f
+ X_g \) for \( X^h \in \mathfrak{har}_j(M) \) and
\( f,g \in \sFunctionSpace_0(M) \). This shows that the map
\( \xi + \I \eta \mapsto \xi - j \eta \) is surjective.
Finally, if \( \xi = j \eta \) with \( \xi, \eta \in
\VectorFieldSpace(M, \omega) \), then both \( \omega^\flat(\xi) \)
and \( \omega^\flat(j\xi) = j \omega^\flat(\xi) \) are closed.
By [Proposition 2.3.1]Gauduchon2017, this is equivalent
to \( \omega^\flat(\xi) \) being harmonic.

As a direct application of <ref>,
we obtain the following result.

Let \( (M, \omega) \) be a compact symplectic manifold and
\( j_0 \) a compatible almost complex structure. For every
\( j_0 \)-extremal almost complex structure \( j \) satisfying
\( \SectionSpaceAbb{J}(j) \in \VectorFieldSpace(M, \omega)_{j_0} \),
the following decomposition holds:

\begin{equation}
(\VectorFieldSpace(M, \omega)_\C)_j = \LieA{c} \oplus
\bigoplus_{\lambda \neq 0} \LieA{k}_\lambda,
\end{equation}

* \( \LieA{c} \) is the Lie subalgebra of
\( (\VectorFieldSpace(M, \omega)_\C)_j \) consisting of all
elements that commute with \( \SectionSpaceAbb{J}(j) \);

* \( \C \SectionSpaceAbb{J}(j) \subseteq \LieA{c} \);
\( \mathfrak{har}_j \subseteq \LieA{c} \);

* \( \LieA{k}_\lambda \) are eigenspaces of
\( 2 \I \difLie_{\SectionSpaceAbb{J}(j)} \) with eigenvalue
\( \lambda \in \R \) (with the convention that
\( \LieA{k}_\lambda = \set{0} \) if \( \lambda \) is not an
eigenvalue); in particular, $\mathfrak{c} = \mathfrak{k}_0$;

* \( \commutator{\LieA{k}_\lambda}{\LieA{k}_\mu}
\intersect (\VectorFieldSpace(M, \omega)_\C)_j \subseteq
\LieA{k}_{\lambda + \mu} \) if $\lambda + \mu$ is an eigenvalue
of \( 2 \I \difLie_{\SectionSpaceAbb{J}(j)} \); otherwise
\( \commutator{\LieA{k}_\lambda}{\LieA{k}_\mu} \intersect
(\VectorFieldSpace(M, \omega)_\C)_j = 0 \).

The only statement that does not follow directly from
<ref> is the
inclusion \( \mathfrak{har}_j \subseteq \LieA{c} \).
But this follows from the fact that the Lie derivative with
respect to a symplectic vector field commutes with the musical
isomorphism \( \omega^\flat \): for every
\( X^h \in \mathfrak{har}_j \), we have

\begin{equation}
\omega^\flat \difLie_{\SectionSpaceAbb{J}(j)} X^h
= \difLie_{\SectionSpaceAbb{J}(j)} (\omega^\flat X^h)
= 0,
\end{equation}

where the last equality follows from the fact that
\( \omega^\flat X^h \) is a \( g_j \)-harmonic form
and \( \SectionSpaceAbb{J}(j) \) is a \( g_j \)-Killing
vector field.

The assumption that \( \SectionSpaceAbb{J}(j) \in
\VectorFieldSpace(M, \omega)_{j_0} \) is not essential
and only serves to ensure that \( 2 \I \difLie_{\SectionSpaceAbb{J}(j)} \)
is symmetric and thus diagonalizable,
Without this assumption a similar statement
holds using generalized eigenspaces; see

Similarly, for extremal metrics, we obtain the following theorem
generalizing the classical result of Calabi1985 which holds
in the integrable case.

Let \( (M, \omega) \) be a compact symplectic manifold. For every
extremal almost complex structure \( j \in \SectionMapAbb{I} \),
the following decomposition holds:

\begin{equation}
\label{eq:kaehler:extremal:decompositionComplexStab}
(\VectorFieldSpace(M, \omega)_\C)_j = \LieA{c} \oplus
\bigoplus_{\lambda \neq 0} \LieA{k}_\lambda,
\end{equation}

* \( \LieA{c} \) is the subset of
\( (\VectorFieldSpace(M, \omega)_\C)_j \) consisting of
all elements that commute with \( X_{S_j} \);

* \( \C X_{S_j} \subseteq \LieA{c} \);
\( \mathfrak{har}_j \subseteq \LieA{c} \);

* \( \LieA{k}_\lambda \) are eigenspaces of \( 2 \I \difLie_{X_{S_j}} \)
with eigenvalue \( \lambda \in \R \) (with the convention that
\( \LieA{k}_\lambda = \set{0} \) if \( \lambda \) is not an
eigenvalue); in particular, $\mathfrak{c} = \mathfrak{k}_0$;

* \( \commutator{\LieA{k}_\lambda}{\LieA{k}_\mu} \subseteq
\LieA{k}_{\lambda + \mu} \intersect
(\VectorFieldSpace(M, \omega)_\C)_j \) if
\( \lambda + \mu \) is an eigenvalue
of \( 2 \I \difLie_{X_{S_j}} \); otherwise
\( \commutator{\LieA{k}_\lambda}{\LieA{k}_\mu}
\intersect (\VectorFieldSpace(M, \omega)_\C)_j = 0 \).

Moreover, if \( j \) is integrable, then the Lie algebra
\( \LieA{h}(M, j) \) of real holomorphic vector fields
admits the following decomposition:

\begin{equation}
\label{eq:kaehler:extremal:decompositionHolomorphic}
\LieA{h}(M, j) = \mathrlap{\overbrace{\phantom{\LieA{a}(M, g_j)
\oplus \LieA{k}_{\textnormal{ham}}(M,g_j)}}^{\LieA{k}(M,g_j)}}
\LieA{a}(M,j) \oplus \underbrace{\LieA{k}_{\textnormal{ham}}(M,g_j)
\oplus j \LieA{k}_{\textnormal{ham}}(M, g_j) \oplus
\bigoplus_{\lambda \neq 0} \LieA{h}_\lambda(M,j)}_{\LieA{h}_{
\textnormal{red}}(M, j)},
\end{equation}

where \( \LieA{a}(M, g_j) \) is the complex Abelian Lie
subalgebra of \( \LieA{h}(M, j) \) consisting of vector
fields that are parallel with respect to the Levi-Civita
connection of \( g_j \), \( \LieA{k}(M, g_j) \) is the Lie
algebra of Killing vector fields,
\( \LieA{k}_{\textnormal{ham}}(M, g_j) \) the subalgebra
of Hamiltonian Killing vector fields, and
\( \LieA{h}_{\textnormal{red}}(M, j) \) is the Lie
algebra of the reduced automorphism group
(see [Section 2.4]Gauduchon2017), and
\( \LieA{h}_\lambda(M,j) \) are
\( \lambda \)-eigenspaces of \( - 2 j \difLie_{X_{S_j}} \).

This statement does not directly follow from
<ref> applied to
the action of \( \HamDiffGroup(M, \omega) \), since this would
only yield a decomposition of \( (\HamVectorFields(M, \omega)_\C)_j \).
Instead, we use the fact that
\( \SectionMapAbb{J}_{\HamDiffGroup} (j) \) is an
element of the stabilizer \( \VectorFieldSpace(M, \omega)_j \)
as \( j \) is extremal. Then the first part of the statement follows
from <ref> relative to
the \( \DiffGroup(M, \omega) \) action, applied to the one-dimensional
subalgebra \( \LieA{t} \subseteq \VectorFieldSpace(M, \omega)_j \)
spanned by \( \SectionMapAbb{J}_{\HamDiffGroup} (j) = X_{S_j} \).

The image of the
decomposition (<ref>)
under the map
\( \VectorFieldSpace(M, \omega)_\C \ni \xi + \I \eta \mapsto
\xi - j \eta \in \VectorFieldSpace(M) \) yields the decomposition
\( \LieA{h}(M, j) = \bigoplus_{\lambda} \LieA{h}_\lambda(M,j) \),
<ref> (this uses
the fact that the kernel of this map is \( \mathfrak{har}_j \),
which is completely included in \( \LieA{c} \)).
A direct calculation shows that \( 2 \I \difLie_{X_{S_j}} \) under
this map takes the form \( -2 j \difLie_{X_{S_j}} \), which identifies
\( \LieA{h}_\lambda(M,j) \) as eigenspaces of \( -2 j \difLie_{X_{S_j}} \).
Finally, the further decomposition of the zero eigenspace and the
identification of \( \LieA{k}(M,g_j) \) and
\( \LieA{h}_{\textnormal{red}}(M, j) \) in this decomposition
are standard; see [Theorem 3.4.1]Gauduchon2017.
This finishes the proof of the
decomposition (<ref>).

Let \( (M, \omega) \) be a compact symplectic manifold.
The Hessian of the Calabi functional
\( \norm{\SectionMapAbb{J}_\HamDiffGroup}^2 \) at an
extremal \( j \in \SectionMapAbb{I} \) is given by

\begin{equation}
\frac{1}{2} \Hessian_j \norm{\SectionMapAbb{J}_\HamDiffGroup}^2
(\zeta \ldot j, \gamma \ldot j) =
\Re \, \dualPair{\zeta}{C^+_j C^-_j \smallMatrix{0 & 0\\0 & 1} \gamma}_{\C}
\end{equation}

for \( \zeta, \gamma \in \HamVectorFields(M, \omega)_\C \).
Moreover, the restriction of \( \Hessian_j
\norm{\SectionMapAbb{J}_\HamDiffGroup}^2 \) to
\( \VectorFieldSpace(M, \omega)_\C \ldot j \subseteq
\TBundle_j \SectionSpaceAbb{J} \) is positive semi-definite.

Similarly, for every \( j_0 \)-extremal \( j \in \SectionMapAbb{I} \)
(with respect to a given almost complex structure
\( j_0 \in \SectionMapAbb{I} \)), the Hessian of
\( \norm{\SectionMapAbb{J}}_{j_0}^2 \) at \( j \) is given by

\begin{equation}
\frac{1}{2} \Hessian_j \norm{\SectionMapAbb{J}}^2_{j_0}
(\zeta \ldot j, \gamma \ldot j) =
\Re  \, \dualPair{\zeta}{C^+_j R_j \gamma}_{j_0, \C},
\end{equation}

for \( \zeta, \gamma \in \VectorFieldSpace(M, \omega)_\C \) and

\begin{equation}
R_j = C^-_j \Matrix{0 & 0 \\ 0 & 1} +
\I \, \bigl(\difLie_{\SectionMapAbb{J}(j)} + Z_j\bigr)
\end{equation}

where \( Z_j \) is calculated in (<ref>).

The expression for the Hessian follows directly from
<ref> in both cases.
The fact that the restriction of
\( \Hessian_j \norm{\SectionMapAbb{J}_\HamDiffGroup}^2 \) to
\( \VectorFieldSpace(M, \omega)_\C \ldot j \subseteq
\TBundle_j \SectionSpaceAbb{J} \) is positive semi-definite
is a direct consequence of
In fact, the only missing assumption to verify is that the
Calabi operators \( C^\pm_j \) are essentially self-adjoint.
But this is clearly the case as these operators are elliptic
operators on \( \sFunctionSpace(M)_\C \isomorph
\sFunctionSpace(M, \C) \).

The first part concerning the Hessian of the Calabi energy
recovers the classical work of [Theorem 2]Calabi1985
in the integrable case (in which case, the vector space
\( \VectorFieldSpace(M, \omega)_\C \ldot j \) is identified with
the tangent
space to the space of Kähler forms in a given cohomology class
up to automorphisms; see
[p. 408f]Donaldson1997[Proposition 9.1.1]Gauduchon2017)
and the recent result of
[Theorem 1.1]HeZheng2023 in the non-integrable case.
The second part of the theorem is thus a natural generalization
of these insights to the case of \( j_0 \)-extremal metrics.

In Donaldson2015a, Donaldson introduced another symplectic
form on the space \( \SectionSpaceAbb{I} \) on a Fano manifold.
This new symplectic form is induced from the space of differential
\( n \)-forms with values in the prequantum bundle \( (L, \theta) \)
over \( M \), essentially via the Plücker embedding.
With respect to this symplectic form, the action of the group
\( \AutGroup(L, \theta) \) is Hamiltonian whose momentum map is
the logarithm of the Ricci potential. This momentum map is equivariant.
Zeros of the momentum map are precisely the Kähler–Einstein metrics.
The norm-squared of the momentum map yields the Ricci–Calabi
functional, whose critical points are generalized Kähler–Einstein
metrics, also known as Mabuchi solitons after Mabuchi2001.
As an application of our general results in
<ref>, we recover the
Matsushima type decomposition theorem for holomorphic vector fields
in the presence of generalized Kähler–Einstein metrics of
[Theorem 4.1]Mabuchi2001.
Moreover, from
<ref>, we recover the Hessian of the Ricci–Calabi
functional which has been calculated in
[Theorem 1.1]Nakamura2019a.

As an alternative to the norm-squared of the momentum map, one
may also consider the composition of the momentum map with a
certain convex function on the Lie algebra \( \sFunctionSpace(M) \)
of \( \AutGroup(L, \theta) \).
The resulting functional on \( \SectionSpaceAbb{I} \) is the
\( \SectionMapAbb{H} \)-functional introduced in He2016,
whose critical points are Kähler–Ricci solitons.
We expect that our results can be used to study the
\( \SectionMapAbb{H} \)-functional as well.
In fact, the results about the Hessian of the
\( \SectionMapAbb{H} \)-functional and the decomposition of
holomorphic vector fields in the presence of Kähler–Ricci
solitons obtained in Fong2016,Nakamura2019 and
[Theorem A]TianZhu2000, respectively, should follow
from an extension of our results to allow arbitrary convex functions
on the Lie algebra along the lines of the finite-dimensional/formal
picture of LeeSturmWang2022.

A different application of our results is to the coupled
Kähler–Einstein equations introduced by
Following DatarPingali2019, this setting fits into
our infinite-dimensional symplectic framework.
In this case, the Hessian and the Matsushima-type decomposition
recover the recent results of Nakamura2023.

Let \( f \) be a positive function on the symplectic
manifold \( (M, \omega) \), and denote its Hamiltonian vector field
by \( K = X_f \)We are skipping over some technical
points here, like the assumption that \( K \) has to lie in a
certain torus.. ApostolovMaschler2019 defined on the
space \( \SectionSpaceAbb{I}_K(M, \omega) \) of \( K \)-invariant
(almost) complex structures on \( M \) a \( f \)-deformed version
of the symplectic form (<ref>) as follows:

\begin{equation}
\Omega^f_j (A, B) = \frac{1}{4} \int_M \tr (A \, j \, B) \,
\frac{\mu_\omega}{f^{2n-1}}.
\end{equation}

They showed that the momentum map for the action of
\( \HamDiffGroup_K(M, \omega) \) on
\( \SectionSpaceAbb{I}_K(M, \omega) \) is given by
assigning to \( j \) the Hermitian scalar curvature
of \( f^{-2} g_j \).
Thus, zeros of the momentum map correspond to conformally
Kähler–Einstein metrics (cKEM) and the norm-squared
of the momentum map is the \( f \)-weighted Calabi functional, whose
critical points are called \( f \)-extremal Kähler metrics.
The Calabi program for \( f \)-extremal Kähler metrics has
been initiated by FutakiOno2017,Lahdili2019.
Naturally, this setting fits into our infinite-dimensional
symplectic framework and we can recover these results using
our general
Moreover, based on our discussion above, it would be interesting
to study the momentum map of all symplectic diffeomorphisms
preserving \( K \) (and not just the subgroup of Hamiltonian

In the odd-dimensional counterpart to Kähler geometry,
Sasakian metrics and their non-integrable pendant $K$-contact
structures are another important class of geometries that can
be studied using our results.
He2014,LejmiUpmeier2015 have shown that the space
of $K$-contact structures on a compact contact manifold is
an infinite-dimensional symplectic manifold and that the
action of the group of strict contactomorphisms is Hamiltonian
with momentum map given by the transverse Hermitian scalar
curvature. The critical points of the norm-squared of the
momentum map have been studied in BoyerGalickiSimanca2008
and are called extremal Sasakian metrics.
We expect that our results can be used to study the Hessian of
the norm-squared of the momentum map and the decomposition of
the complexified stabilizer of a $K$-contact structure.
In particular, the decomposition theorem
[Theorem 11.3.1]Boyer2008 of the space of transverse
holomorphic vector fields in the presence of an extremal
Sasakian metric should directly follow from
Moreover, it would be interesting to study the action of the
whole group of contactomorphisms (and not just the subgroup of strict
contactomorphisms) on the space of $K$-contact structures.
In parallel to our discussion of the Kähler case, one would
expect that the momentum map is then non-equivariant and one
obtains a natural central extension of the group of contactomorphisms.

§ APPLICATION: SYMPLECTIC CONNECTIONS

§.§ Momentum map for the action of
Diff(M, w)

First, we briefly review the necessary background on symplectic connections
summarizing definitions and conventions following Tondeur1961,
Hess1980, MarsdenRatiuEtAl1991, CahenGutt2005.
We make heavy use of the Penrose notation, which is reviewed in <ref>.
An affine connection \( \nabla \) on a symplectic manifold \( (M, \omega) \)
is a symplectic connection, if it is torsion-free and satisfies
\( \nabla \omega =0 \), , \( X[\omega(Y,Z)] =\omega(\nabla_X Y, Z) +
\omega(Y,\nabla_X Z) \), for all \( X, Y, Z \in  \VectorFieldSpace(M) \).
This condition is equivalent to the parallel transport operator being a
symplectic isomorphism between the tangent spaces to \( M \).
In contrast to the Levi-Civita connection on a Riemannian manifold, there does
not exist a unique symplectic connection on a given
symplectic manifold.
The non-uniqueness of torsion free symplectic connections cannot
be improved even if \( M = \CotBundle Q \), endowed with its standard exact
symplectic form and \(Q\) is Riemannian.
Then \( \TBundle Q\) has a naturally induced Riemannian metric.
Pull back this metric to \( \CotBundle Q\) using the given Riemannian metric
on \( Q \) to endow \( \CotBundle Q \) with a Riemannian metric.
So, we have now a Levi-Civita connection  on \( \CotBundle Q \).
It turns out that it is symplectic if and only if the given Riemannian
metric on \( Q \) is flat.
If \( \nabla^1 \) and \( \nabla^2 \) are symplectic connections on
\( (M,\omega) \), then

\begin{equation}
\label{two_nablas}
\nabla^2_i X^k = \nabla^1_i X^k + \tensor{A}{_{ij}^k} X^j
\end{equation}

for some tensor \( \tensor{A}{_{ij}^k} \) such that
\( \tensor{A}{_i_j_k} \defeq \tensor{A}{_{ij}^l}\omega_{lk}\)
is symmetric in all indices, see CahenGutt2005.
The Penrose index notation in (<ref>) stands for
the intrinsic formula \(\nabla^2_Y X = \nabla^1_Y X + A(Y, X, \cdot) \).
We abbreviate (<ref>) by writing
\( \nabla^2 = \nabla^1 + A \).
The important conclusion of these considerations is that the space
\(\ConnSpace_\omega(M) \) of symplectic connections on the symplectic
manifold \( (M, \omega) \) is an affine space whose linear model space is
isomorphic to the space \( \SymTensorFieldSpace_3(M) \) of symmetric
covariant \( 3 \)-tensor fields on \(M\).
In particular, \( \ConnSpace_\omega(M) \) is always non-empty.

In the following we assume \( M \) to be compact. We endow
\( \ConnSpace_\omega(M) \) with its natural \( \sFunctionSpace \)-Fréchet
topology. According to CahenGutt2005, the space
\( \ConnSpace_\omega(M) \) carries a natural affine (weak) symplectic
structure \( \Omega \) defined by

\begin{equation}
\label{symplectic_form_connections}
\Omega_\nabla(A, B) = \int_M \tensor{A}{_i_j_k} \tensor{B}{^i^j^k} \mu_\omega,
\end{equation}

where \( \nabla \in \ConnSpace_\omega(M) \), \( A, B \in
\SymTensorFieldSpace_3(M) \), and \( \mu_\omega = \frac{\omega^n}{n!} \)
is the Liouville volume form. Note that \( \Omega_\nabla \) does not depend
on \(\nabla \). The Fréchet Lie group \( \DiffGroup(M, \omega) \) of
symplectomorphisms of \( (M, \omega) \) acts on the left on
\( \ConnSpace_\omega(M) \) by push-forward according to

\begin{equation}
\label{eq:symplecticConnections:action}
(\phi \cdot \nabla)_X Y \defeq \phi_* \left(\nabla_{\phi^{-1}_* X} (\phi^{-1}_* Y)
\right)
\end{equation}

for \(\phi \in \DiffGroup(M, \omega)\) and \(\nabla \in \ConnSpace_\omega(M)\).
This action is clearly affine and the induced linear action is given by the
natural left action

\begin{equation}
(\phi \cdot A)(X, Y, Z) = A\bigl(\phi^{-1}_* X, \phi^{-1}_* Y, \phi^{-1}_*Z
\bigr)
\end{equation}

of \( \DiffGroup(M, \omega) \) on \( \SymTensorFieldSpace_3(M) \).
Using this expression for the linear action, it is straightforward to verify
that the \( \DiffGroup(M, \omega) \)-action on \( \ConnSpace_\omega(M) \)
preserves the symplectic form \( \Omega \). The infinitesimal action of a
symplectic vector field \( \xi \in \VectorFieldSpace(M, \omega) \)
on \( \ConnSpace_\omega(M) \) is given by

\begin{equation}
\label{eq:symplecticConnections:infAction}
\tensor{\bigl(\xi \ldot \nabla\bigr)}{_i_j^k}
= - \tensor{\bigl(\difLie_\xi \nabla\bigr)}{_{ij}^k}
= - \nabla_i \nabla_j \xi^k - \tensor{R}{_l_i_j^k} \xi^l.
\end{equation}

Similarly, the infinitesimal action of \( \VectorFieldSpace(M, \omega) \)
on \( \SymTensorFieldSpace_3(M) \) takes the form

\begin{equation}
\label{eq:symplecticConnections:infActionLinear}
\tensor{\bigl(\xi \ldot A\bigr)}{_i_j^k}
= - \tensor{\bigl(\difLie_\xi A\bigr)}{_{ij}^k}
= - \bigl(\xi^p \nabla_p \tensor{A}{_{ij}^k} + \tensor{A}{_{pj}^k}
\nabla_i \xi^p + \tensor{A}{_{ip}^k} \nabla_j \xi^p -
\tensor{A}{_i_j^q} \nabla_q \xi^k\bigr).
\end{equation}

As we are working in an infinite-dimensional setting, we have to pay
attention to functional analytic problems. We will be brief here and refer
the reader to DiezRatiuAutomorphisms,DiezThesis for background
information and further technical details.
For the construction of the momentum map, we need to clarify what we mean by
the dual space of \( \VectorFieldSpace(M, \omega) \). Note that the map
\( \xi \mapsto \xi \contr \omega \) identifies
\( \VectorFieldSpace(M, \omega) \) with the space of closed \( 1 \)-forms
on \( M \).
This suggests the choice \( {\VectorFieldSpace(M, \omega)}^* \defeq
\DiffFormSpace^{2n-1}(M) \slash \dif \DiffFormSpace^{2n-2}(M) \) for the dual
space of \( \VectorFieldSpace(M, \omega) \) relative to the pairing

\begin{equation}
\label{eq:symplecticConnections:pairing}
\kappa\bigl(\equivClass{\alpha},\xi\bigr)
= \frac{1}{(n-1)!}\int_M \alpha \wedge (\xi \contr\omega),
\end{equation}

where \( \equivClass{\alpha} \in \DiffFormSpace^{2n-1}(M) \slash \dif
\DiffFormSpace^{2n-2}(M) \) and \( \xi \in \VectorFieldSpace(M, \omega) \).
This is the same pairing as (<ref>)
used above in the Kähler setting.

The following proposition shows that the action has a momentum map.

For \( \nabla \in \ConnSpace_\omega(M) \) and \( A \in
\SymTensorFieldSpace_3(M) \), let \( J(\nabla + A) \in \DiffFormSpace^1(M) \)
be given by

\begin{equation}\begin{split}
\label{eq:symplecticConnections:momentumMapStrange:oneForm}
J(\nabla + A)_p
= - \nabla_j \nabla_i \tensor{A}{^i^j_p} + R_{pijk}A^{ijk} +
\frac{1}{2} (\nabla_p A_{ijk}) A^{ijk} -
\frac{3}{2} \nabla_i \bigl(A^{ijk}  A_{pjk} \bigr),
\end{split}\end{equation}

where \(R\) is the curvature operator of
\(\nabla \) and \( R_{pijk} = \tensor{R}{_{pij}^{s}} \omega_{sk} \).
For each \( \nabla \in \ConnSpace_\omega(M) \), the map

\begin{equation}
\label{eq:symplecticConnections:momentumMapStrange}
\SectionMapAbb{J}: \ConnSpace_\omega(M) \to \DiffFormSpace^{2n-1}(M) \slash
\dif \DiffFormSpace^{2n-2}(M),   \qquad
\nabla + A \mapsto \equivClass*{J(\nabla + A) \wedge
\omega^{n-1}}
\end{equation}

is the unique momentum map for the \( \DiffGroup(M, \omega) \)-action
on \( \ConnSpace_\omega(M) \) relative to the
pairing (<ref>) that vanishes
at \( \nabla \).

According to <ref>, the momentum map is
given by

\begin{equation}
\label{eq:symplecticConnections:momentumMapStrange:fromAffine}
\kappa\bigl(\SectionMapAbb{J}(\nabla + A), \xi\bigr) = \Omega(A, \xi \ldot \nabla)
+ \frac{1}{2} \Omega(A, \xi \ldot A).
\end{equation}

Let us start by evaluating the first summand on the right-hand side.
Using (<ref>)
and (<ref>), integrating by parts yields

\begin{equation*}\begin{split}
\Omega(A, \xi \ldot \nabla)
&= \int_M A_{ijk} \bigl(\xi \ldot \nabla\bigr)^{ijk} \mu_\omega \\
&= - \int_M A_{ijk} \bigl(\nabla^i \nabla^j \xi^k +
\tensor{R}{_p^i^j^k} \xi^p\bigr) \mu_\omega   \\
&= - \int_M \bigl( \nabla^j \nabla^i A_{ijp} +
A_{ijk}\tensor{R}{_p^i^j^k} \bigr) \xi^p \mu_\omega   \\
&= \int_M \bigl( - \nabla_j \nabla_i \tensor{A}{^{ij}_p} +
\tensor{R}{_p_i_j_k}A^{ijk} \bigr) \xi^p \mu_\omega .
\end{split}\end{equation*}

Using (<ref>) and the symmetry
of \( A_{ijk} \) we find for the second summand
in (<ref>):

\begin{equation*}\begin{split}
\Omega(A, \xi \ldot A)
&= \int_M A_{ijk} \bigl(\xi \ldot A\bigr)^{ijk} \mu_{\omega}  \\
&= - \int_M A_{ijk} \bigl(\xi^p \nabla_p \tensor{A}{^{ij}^k} +
\tensor{A}{_p^j^k} \nabla^i \xi^p + \tensor{A}{^i_p^k} \nabla^j \xi^p -
\tensor{A}{^i^j^q} \nabla_q \xi^k\bigr) \mu_{\omega}   \\
&= \int_M \bigl(- A_{ijk} \nabla_p \tensor{A}{^{ij}^k} +
\nabla^i (A_{ijk} \tensor{A}{_p^j^k}) +
\nabla^j(A_{ijk} \tensor{A}{^i_p^k}) -
\nabla_q (A_{ijp}\tensor{A}{^i^j^q})\bigr) \xi^p \mu_{\omega}    \\
&= \int_M \bigl(A^{ijk} \nabla_p \tensor{A}{_{ij}_k} -
3 \nabla_i (A^{ijk} \tensor{A}{_p_j_k})\bigr) \xi^p \mu_{\omega}.
\end{split}\end{equation*}

Thus, comparing
with (<ref>), we get

\begin{equation*}
\kappa\bigl(\SectionMapAbb{J}(\nabla + A),\xi\bigr)
= \Omega(A, \xi \ldot \nabla) + \frac{1}{2} \Omega(A, \xi \ldot A)
= \int_M J(\nabla + A)_p \xi^p \mu_\omega .
\end{equation*}

Finally, for every \( 1 \)-form \( \beta \), we have

\begin{equation*}
\int_M \beta_p \xi^p \mu_\omega
=  \frac{1}{(n-1)!} \int_M \beta \wedge\omega^{n-1} \wedge (\xi \contr \omega)
= \kappa\bigl(\beta \wedge \omega^{n-1} , \xi\bigr),
\end{equation*}

which yields the expression (<ref>)
for the momentum map \( \SectionMapAbb{J} \).

Let us rewrite the momentum map \( \SectionMapAbb{J} \) in such a way that
its geometric meaning becomes apparent. For this purpose, recall that the
Ricci curvature is defined by \( R_{ij} = \tensor{R}{_{kij}^k} \) and the
curvature \( 1 \)-form by

\begin{equation}
\label{eq:symplecticConnections:curvature1Form}
\rho_i = 2 \nabla^j \tensor{R}{_i_j} = - 2 \nabla_j \tensor{R}{_i^j} \, .
\end{equation}

We sometimes write \( \rho(\nabla) \) to emphasize the dependency on the
symplectic connection \( \nabla \).
Moreover, let \( \pontryaginClass \equiv \pontryaginClass(\nabla) \) be
the \( 4 \)-form

\begin{equation}
\label{eq:symplecticConnections:pontryaginForm}
\pontryaginClass_{ijkl} =
\frac{1}{4 \pi^2}
\bigl(
\tensor{R}{_i_j^p^q}\tensor{R}{_k_l_p_q}
+ \tensor{R}{_i_k^p^q}\tensor{R}{_l_j_p_q}
+ \tensor{R}{_i_l^p^q}\tensor{R}{_j_k_p_q}
\bigr)
\end{equation}

representing the first Pontryagin class of \( (M, \nabla) \).
Chern–Weil theory entails that the \( 4 \)-forms \( \pontryaginClass(\nabla
+ A) \) and \( \pontryaginClass(\nabla) \) associated with the connections
\( \nabla + A \) and \( \nabla \), respectively, are cohomologous.
Indeed, a straightforward (but lengthy) calculation shows that

\begin{equation}
\label{eq:symplecticConnections:pontryaginRelationTwoConnections}
\pontryaginClass(\nabla + A) = \pontryaginClass(\nabla) -
\frac{1}{4 \pi^2} \dif \sigma,
\end{equation}

where the \( 3 \)-form \( \sigma \equiv \sigma(\nabla, A) \) is defined by

\begin{equation}\label{eq:symplecticConnections:pontryaginPreForm}
\begin{split}
\sigma_{ijk}
&= \tensor{A}{_i_p^q}\tensor{R}{_j_k_q^p} +
\tensor{A}{_j_p^q}\tensor{R}{_k_i_q^p} +
\tensor{A}{_k_p^q}\tensor{R}{_i_j_q^p}
\\
&\quad + \frac{1}{2} \bigl(
\tensor{A}{_k_p^q}\nabla_i\tensor{A}{_j_q^p}
+ \tensor{A}{_i_p^q}\nabla_j\tensor{A}{_k_q^p}
+ \tensor{A}{_j_p^q}\nabla_k\tensor{A}{_i_q^p}
\\
- \tensor{A}{_j_p^q}\nabla_i\tensor{A}{_k_q^p}
- \tensor{A}{_k_p^q}\nabla_j\tensor{A}{_i_q^p}
- \tensor{A}{_i_p^q}\nabla_k\tensor{A}{_j_q^p}
\bigr)
\\
&\quad - \tensor{A}{_i_a^b} (\tensor{A}{_j_b^c}\tensor{A}{_k_c^a}
- \tensor{A}{_k_b^c}\tensor{A}{_j_c^a}).
\end{split}
\end{equation}

Using these notions, the momentum map \( \SectionMapAbb{J} \) has the
following expression.

The \( \DiffGroup(M, \omega) \)-action on \( (\ConnSpace_\omega(M), \Omega) \)
defined in (<ref>) is symplectic and has a
momentum map, relative to the pairing (<ref>),
given by

\begin{equation}
\label{eq:symplecticConnections:momentumMap}
\SectionMapAbb{J}: \ConnSpace_\omega(M) \to \DiffFormSpace^{2n-1}(M) \slash
\dif \DiffFormSpace^{2n-2}(M),
\quad
\nabla + A \mapsto \equivClass*{J(\nabla + A) \wedge
\omega^{n-1}},
\end{equation}

where \( J(\nabla + A) \in \DiffFormSpace^1(M) \) is defined by

\begin{equation}
J(\nabla + A)_i
= \frac{1}{2} \bigl( \rho(\nabla + A)_i - \rho(\nabla)_i -
\tensor{\sigma(\nabla, A)}{_i_j^j}\bigr) \,.
\qedhere
\end{equation}

The momentum map \( \SectionSpaceAbb{J} \) involves two ingredients
that have a different flavor. First, it contains the curvature
\( 1 \)-form \( \rho(\nabla) \) which has a clear geometric meaning.
Second, the correction term \( \sigma(\nabla, A) \) is closely related
to the Pontryagin class of \( M \) and thus has a more topological origin.
This is another manifestation of the general principle that momentum maps
for diffeomorphism groups involve geometric as well as topological data
(see DiezRatiuAutomorphisms for more examples).

Recall that the Ricci curvature \( \bar{R}_{ij} \) of the connection
\( \bar{\nabla} = \nabla + A \) is given by

\begin{equation}
\bar{R}_{ij} = R_{ij} + \nabla_k \tensor{A}{_i_j^k} -
\tensor{A}{_i_k^l} \tensor{A}{_j_l^k}
\end{equation}

and, for every tensor \( T^{ij} \), we have

\begin{equation}
\bar{\nabla}_k T^{ij} = \nabla_k T^{ij} + \tensor{A}{_k_l^i} T^{lj} +
\tensor{A}{_k_l^j} T^{il} \, .
\end{equation}

We hence obtain

\begin{equation}\begin{split}
\rho(\bar{\nabla})_i
&= - 2 \bar{\nabla}_j \tensor{\bar{R}}{_i^j}
\\
&= - 2 (\nabla_j \tensor{\bar{R}}{_i^j} +
\tensor{A}{_j_l_i} \tensor{\bar{R}}{^{lj}})
\\
&= - 2 (
\nabla_j \tensor{R}{_i^j}
+ \nabla_j \nabla_k \tensor{A}{_i^j^k}
- \tensor{A}{^j_l^k} \nabla_j \tensor{A}{_i_k^l} -
\tensor{A}{_i_k^l} \nabla_j \tensor{A}{^j_l^k}
\\
&\qquad+ \tensor{A}{_j_l_i} R^{lj}
+ \tensor{A}{_j_l_i} \nabla_k \tensor{A}{^l^j^k}
- \tensor{A}{_j_l_i} \tensor{A}{^l_k^p} \tensor{A}{^j_p^k} )
\\
&= \rho(\nabla)_i - 2 (\nabla_j \nabla_k \tensor{A}{_i^j^k} +
\tensor{A}{_j_k_i} R^{kj} )
\\
&\qquad+ 2 \tensor{A}{^j_l^k} \nabla_j \tensor{A}{_i_k^l} -
4 \tensor{A}{_j_l_i} \nabla_k \tensor{A}{^l^j^k}
+ 2 \tensor{A}{_j_l_i} \tensor{A}{^l_k^p} \tensor{A}{^j_p^k} \, .
\end{split}\end{equation}

On the other hand, we have

\begin{equation}\label{eq:symplecticConnections:pontryaginPreFormContracted}
\begin{split}
\tensor{\sigma}{_{ij}^j}
&= 2 \tensor{A}{_i_p^q}\tensor{R}{_q^p}
+ 2 \tensor{A}{_j_p^q}\tensor{R}{^j_i_q^p}
\\
+ \tensor{A}{^j_p^q}\nabla_i\tensor{A}{_j_q^p}
+ \tensor{A}{_i_p^q}\nabla_j\tensor{A}{^j_q^p}
- \tensor{A}{^j_p^q}\nabla_j\tensor{A}{_i_q^p}
\\
&\quad - 2 \tensor{A}{_i_a^b} \tensor{A}{_j_b^c}\tensor{A}{^j_c^a} \, .
\end{split}
\end{equation}

Comparing these identities
with (<ref>) shows that
the momentum map \( \SectionMapAbb{J} \) can be indeed written in the
form (<ref>).

If \( (M, \omega) \) is a two-dimensional symplectic manifold, then the
\( 3 \)-form \( \sigma \) necessarily vanishes and thus the momentum
map (<ref>) takes the simple form

\begin{equation}
\SectionMapAbb{J}: \ConnSpace_\omega(M) \to \DiffFormSpace^{1}(M)
\slash \dif \DiffFormSpace^{0}(M),
\qquad
\nabla + A \mapsto \frac{1}{2} \equivClass*{\bigl( \rho(\nabla + A) -
\rho(\nabla) \bigr)}.
\end{equation}

Thus, we recover the formula for the momentum map in this setting established
in [Theorem 1.2]Fox2014 (up to some constant).

Let us derive from the expression (<ref>)
the momentum map for the action of the subgroup \( \HamDiffGroup(M, \omega)
\subseteq \DiffGroup(M, \omega) \) of Hamiltonian diffeomorphisms.
As in <ref>, we identify the space
\( \HamVectorFields(M, \omega) \) of Hamiltonian vector fields
with \( \sFunctionSpace_0(M) \). Under this identification, the
space \( \dif \DiffFormSpace^{2n-1}(M) \) is dual to
\( \HamVectorFields(M, \omega) \) and the momentum map for the
action of \( \HamDiffGroup(M, \omega) \) is given by post-composition
of the momentum map for the action of \( \DiffGroup(M, \omega) \)
with the exterior differential.
Accordingly, the momentum map associated with the action of
\( \HamDiffGroup(M, \omega) \) on \( \ConnSpace_\omega(M) \) is given by

\begin{equation}
\label{eq:symplecticConnections:momentumMapHam}
\SectionMapAbb{J}_{\HamDiffGroup}: \ConnSpace_\omega(M) \to
\dif \DiffFormSpace^{2n-1}(M),
\qquad
\nabla + A \mapsto \dif J(\nabla + A) \wedge \frac{\omega^{n-1}}{(n-1)!} \, .
\end{equation}

Let \( \bar{\sigma}_i = \tensor{\sigma}{_i_j^j} \) be the contraction
of \( \sigma \). A straightforward calculation
using (<ref>)

\begin{equation}
\tensor{(\dif \bar{\sigma})}{_i^i}
= \frac{1}{2} \tensor{(\dif \sigma)}{_i^i_j^j}
= - 2 \pi^2 \tensor{\bigl(\pontryaginClass(\nabla + A) +
\pontryaginClass(\nabla)\bigr)}{_i^i_j^j} \, .
\end{equation}

Thus, using (<ref>), we can
rewrite the momentum map \( \SectionMapAbb{J}_{\HamDiffGroup} \)

\begin{equation}\begin{split}
\SectionMapAbb{J}_{\HamDiffGroup}(\nabla + A)
&= \dif J(\nabla + A) \wedge \frac{\omega^{n-1}}{(n-1)!}
\\
&= \frac{1}{2} \tensor{\bigl(\dif J(\nabla + A)\bigr)}{_i^i}
\mu_\omega
\\
&= \frac{1}{4} \tensor{\bigl(\dif \rho(\nabla + A) -
\dif \rho(\nabla)\bigr)}{_i^i} \mu_\omega
+ \frac{\pi^2}{2} \tensor{\bigl(\pontryaginClass(\nabla + A) -
\pontryaginClass(\nabla)\bigr)}{_i^i_j^j} \mu_\omega
\\
&\equiv \bigl(\SectionMapAbb{K}(\nabla + A) -
\SectionMapAbb{K}(\nabla)\bigr) \mu_\omega,
\end{split}\end{equation}

where, in the last line, we introduced the map

\begin{equation}
\label{eq:symplecticConnections:cahenGuttMomentumMap}
\SectionMapAbb{K}: \ConnSpace_\omega(M) \to \sFunctionSpace(M),
\qquad
\nabla \mapsto \frac{1}{2} \left(\nabla_i \rho(\nabla)^i +
\pi^2 \tensor{\pontryaginClass(\nabla)}{_i^i_j^j}\right) \, .
\end{equation}

It was shown in [Theorem 1.1]Fox2014 that \( \SectionMapAbb{K} \)
coincides with the Cahen–Gutt momentum map
[Proposition 1.1]CahenGutt2005Gutt2006 for the action of
the group of Hamiltonian diffeomorphisms on \( \ConnSpace_\omega(M) \).
In other words, \( \SectionMapAbb{J}_{\HamDiffGroup} \) recovers
the Cahen–Gutt momentum map (as a slight reformulation).
Let us record this observation.

The action of the subgroup of Hamiltonian diffeomorphisms on
\( \ConnSpace_\omega(M) \) has a momentum map \( \SectionMapAbb{J}_{\HamDiffGroup}:
\ConnSpace_\omega(M) \to \dif \DiffFormSpace^{2n-1}(M) \) given by

\begin{equation}
\SectionMapAbb{J}_{\HamDiffGroup}(\nabla + A)
= \dif J(\nabla + A) \wedge \frac{\omega^{n-1}}{(n-1)!}
= \bigl(\SectionMapAbb{K}(\nabla + A) - \SectionMapAbb{K}(\nabla)\bigr) \mu_\omega ,
\end{equation}

where \( \SectionMapAbb{K}: \ConnSpace_\omega(M) \to \sFunctionSpace(M) \) is the
Cahen–Gutt momentum map defined
in (<ref>).

Note that \( \SectionMapAbb{J}_{\HamDiffGroup} \) is
\( \HamDiffGroup(M, \omega) \)-equivariant.
Equivariance is, however, no longer the case for the
momentum map \( \SectionMapAbb{J} \)
for the full group of symplectomorphisms.

§.§ Central extension of
Diff(M, w)

As we have seen in <ref>, the momentum map
for an affine symplectic action is, in general, not equivariant.
For the action of \( \DiffGroup(M, \omega) \) on the space of symplectic
connections, we obtain the following.

The non-equivariance \( 2 \)-cocycle \( \Sigma: \VectorFieldSpace(M, \omega)
\times \VectorFieldSpace(M, \omega) \to \R \) associated with the momentum
map \( \SectionMapAbb{J} \) (see
<ref> or
<ref>) is given by

\begin{equation}
\label{eq:symplecticConnections:nonEquivariance:cocycle}
\Sigma(\xi, \eta) =
\frac{1}{2} \kappa\bigl(\rho(\nabla), \commutator{\xi}{\eta}\bigr) -
2 \pi^2 \int_M \tensor{\pontryaginClass}{_k^k_i_j}(\nabla) \xi^i \eta^j
\mu_\omega \, .
\qedhere
\end{equation}

The part of \( \Sigma \) not cohomologous to \( 0 \) is determined by the
contracted Pontryagin class \( \tensor{\pontryaginClass}{_k^k_i_j} \in
\DiffFormSpace^2(M) \) and thus carries topological information of the
symplectic manifold \( (M, \omega) \).

It is straightforward to verify that the curvature \( 1 \)-form \( \rho \)
transforms naturally under the action of \( \DiffGroup(M, \omega) \),
that is, we have

\begin{equation}
\rho(\phi \cdot \nabla) = (\phi^{-1})^* \rho(\nabla)
\end{equation}

for every \( \phi \in \DiffGroup(M, \omega) \) and \( \nabla \in
\ConnSpace_\omega(M) \).
Thus, according to (<ref>)
and (<ref>),
the non-equivariance \( 1 \)-cocycle \( \lambda: \DiffGroup(M, \omega) \to
\DiffFormSpace^{2n-1}(M) \slash \dif \DiffFormSpace^{2n-2}(M) \) is given by

\begin{equation}
\lambda(\phi)
= \SectionMapAbb{J}(\phi \cdot \nabla)
= \frac{1}{2} \equivClass*{\Bigl((\phi^{-1})^* \rho(\nabla) -
\rho(\nabla) - \bar{\sigma}(\nabla, \phi \cdot \nabla - \nabla)
\Bigr) \wedge \omega^{n-1}},
\end{equation}

where we recall that \( \bar{\sigma}_i =
\tensor{\sigma}{_i_j^j} \). Differentiating this relation with respect
to \( \phi \), we find for the non-equivariance Lie algebra \( 2 \)-cocycle:

\begin{equation}\begin{split}
\Sigma(\xi, \eta)
\kappa\bigl(\tangent_e \lambda (\xi), \eta \bigr)
\\
- \frac{1}{2} \dualPair*{\difLie_\xi \rho}{\eta}
+ \dualPair*{\tensor{(\difLie_\xi \nabla)}{_i_p^q}\tensor{R}{_q^p}
+ \tensor{(\difLie_\xi \nabla)}{_j_p^q}\tensor{R}{^j_i_q^p}}{\eta}
\\
\frac{1}{2} \dualPair*{\rho}{\commutator{\xi}{\eta}}
\\
&\qquad+ \dualPair*{\bigl(\nabla_i \nabla_p \xi^q +
\tensor{R}{_l_i_p^q} \xi^l\bigr)\tensor{R}{_q^p} +
\bigl(\nabla_j \nabla_p \xi^q +
\tensor{R}{_l_j_p^q} \xi^l\bigr)\tensor{R}{^j_i_q^p}}{\eta},
\end{split}\end{equation}

where we used the
expression (<ref>)
for \( \bar{\sigma} \) and (<ref>)
for the Lie derivative of \( \nabla \); the brackets \( \dualPairDot \)
in the last two lines denote the natural
pairing of \( 1 \)-forms with vector fields by integration against
\( \mu_\omega \). The terms on the right-hand side
involving two partial derivatives cancel, which can be seen using
integration by parts:

\begin{equation}\begin{split}
&\dualPair*{\bigl(\nabla_i \nabla_p \xi^q\bigr)\tensor{R}{_q^p} +
\bigl(\nabla_j \nabla_p \xi^q\bigr)\tensor{R}{^j_i_q^p}}{\eta}
\\
= \int_M \bigl(\nabla_i \nabla_p \xi^q\bigr)\tensor{R}{_q^p} \eta^i
+ \bigl(\nabla_j \nabla_p \xi^q\bigr)\tensor{R}{^j_i_q^p} \eta^i
\\
= - \int_M (\nabla_p \xi^q) (\nabla_i \tensor{R}{_q^p}) \eta^i +
(\nabla_p \xi^q) (\nabla_j \tensor{R}{^j_i_q^p}) \eta^i
\\
= 0,
\end{split}\end{equation}

where we used the fact that \( \nabla_i \eta_j \) is symmetric in \( (i, j) \)
and thus the terms involving the covariant derivatives of \( \eta^i \) vanish.
Indeed, \(0 = (\difLie_\eta \omega)_{ij} =
(\nabla_\eta \omega)_{ij} + \nabla_i \eta_j - \nabla_j \eta_i\) and
\(\nabla_\eta \omega =0\).
Thus, we get

\begin{equation}\label{eq:symplecticConnections:nonEquivariance:cocyclePrelim}
\begin{split}
\Sigma(\xi, \eta)
\frac{1}{2} \dualPair*{\rho}{\commutator{\xi}{\eta}}
+ \dualPair*{\bigl(\tensor{R}{_i_j_p^q} \tensor{R}{_q^p} +
\tensor{R}{_i_r_p^q} \tensor{R}{^r_j_q^p}\bigr) \xi^i}{\eta}.
\end{split}\end{equation}

Finally, we find for the contraction of the Pontryagin form
(see (<ref>)):

\begin{equation}
4 \pi^2 \tensor{\pontryaginClass}{_k^k_i_j}
\tensor{R}{_k^k^p^q}\tensor{R}{_i_j_p_q}
+ \tensor{R}{_k_i^p^q}\tensor{R}{_j^k_p_q}
+ \tensor{R}{_k_j^p^q}\tensor{R}{^k_i_p_q}
= 2 \tensor{R}{^p^q}\tensor{R}{_i_j_p_q}
+ 2 \tensor{R}{_k_i^p^q}\tensor{R}{_j^k_p_q} \, .
\end{equation}

Inserting this relation into the
expression (<ref>)
for the non-equivariance cocycle \( \Sigma \)
yields (<ref>).

Recall from the discussion in <ref>, that
\( 2 \)-cocycles on \( \VectorFieldSpace(M, \omega) \) are sums of
extensions of certain \( 2 \)-cocycles on \( \HamVectorFields(M, \omega) \)
and pull-backs of elements of \( \ExtBundle^2 {\deRCohomology^1(M)}^* \).
Applied to the non-equivariance cocycle \( \Sigma \), we obtain the following.

The class of the non-equivariance cocycle \( \Sigma \) in the second
continuous Lie algebra cohomology of \( \VectorFieldSpace(M, \omega) \)
coincides with the pull-back along the natural map
\( \VectorFieldSpace(M, \omega) \to \deRCohomology^1(M) \) of the
antisymmetric bilinear form

\begin{equation}
\bigl(\equivClass{\alpha}, \equivClass{\beta}\bigr) \mapsto \pi^2 \int_M
\left(\SectionMapAbb{p} (\nabla) \, \varpi^{ij} -
2 \tensor{\pontryaginClass}{_k^k^i^j} (\nabla)\right) \alpha_i \beta_j \,
\mu_\omega
\end{equation}

on \( \deRCohomology^1(M) \), where

\begin{equation}
\SectionMapAbb{p}(\nabla) \defeq \tensor{\pontryaginClass}{_i^i_j^j}(\nabla)
- \frac{1}{\vol_{\mu_\omega}(M)} \int_M \tensor{\pontryaginClass}{_i^i_j^j}
(\nabla) \, \mu_\omega \, .
\qedhere
\end{equation}

It should not come as a big surprise that there is no contribution from the
second Lie algebra cohomology of \( \HamVectorFields(M, \omega) \), because
the momentum map \( \SectionMapAbb{K} \) for the action of the group of
Hamiltonian diffeomorphisms is equivariant.
We conclude that the momentum map for the action of the full group of
symplectomorphisms contains topological information of \( (M, \omega) \)
in terms of the Pontryagin form
while \( \HamVectorFields(M, \omega) \) is not sensitive to these
topological properties. A similar dichotomy has also been observed in
DiezRatiuAutomorphisms for different actions of
symplectomorphism groups.

The prequantum bundle construction in
<ref> shows that the \( 2 \)-cocycle
\( \Sigma \) integrates to a central Lie group extension of
\( \DiffGroup(M, \omega) \).

There exists a central Lie group \( \UGroup(1) \)-extension of the group
\( \DiffGroup(M, \omega) \) of symplectomorphisms whose corresponding
Lie algebra \( 2 \)-cocycle is cohomologous to the non-equivariance
\( 2 \)-cocycle \( \Sigma \).

Note that the central group extension in
<ref> has
been obtained by means of the action of \( \DiffGroup(M, \omega) \)
on the infinite-dimensional space of symplectic connections.
On the other hand, we have seen in
<ref> that
the non-equivariance cocycle \( \Sigma \) is essentially the pull-back
of a cocycle on the finite-dimensional space \( \deRCohomology^1(M) \).
One may thus hope for a finite-dimensional construction of the central
extension of \( \DiffGroup(M, \omega) \).
This is an issue for future research to explore.

§.§ Norm-squared momentum map

In this section, we apply our general results concerning the norm-squared
of the momentum map to the action of the symplectomorphism
group on the space
of symplectic connections.
As discussed in <ref>, the action of the
group \( \DiffGroup(M, \omega) \) of symplectomorphisms leaves
\( \Omega \) invariant and has a momentum map \( \SectionMapAbb{J}:
\ConnSpace_\omega(M) \to \Omega^{2n - 1}(M) \slash \dif \Omega^{2n-2}(M) \)
as calculated in <ref>.
Here, the target of \( \SectionMapAbb{J} \) is identified with the dual
of \( {\VectorFieldSpace(M, \omega)}^* \) by the
pairing (<ref>).
In order to fit this setting into the general framework of
<ref>, we need to realize \( \SectionMapAbb{J} \)
as a map into the space of symplectic vector fields.
For this purpose, let \( j \) be a complex structure
on \( M \) compatible with \( \omega \),
, \( \omega(j \,\cdot, j \,\cdot) =
\omega(\cdot, \cdot)  \) and \( \omega(X, j X) > 0 \) for all non-zero
\( X \in \TBundle M \).
Most results of this section hold with minor modification
also when \( j \) is not integrable.
Denote the associated Riemannian metric by \( g(\cdot, \cdot )
= \omega (\cdot, j \cdot ) \).
Using this data, consider the following non-degenerate
pairing on \( \VectorFieldSpace(M, \omega) \):

\begin{equation}
\label{eq:symplecticConnections:pairingOnLieAlgebra}
\kappa(\xi, \eta) = \int_M g(\xi, \eta) \, \mu_\omega \, .
\end{equation}

Relative to this pairing, the momentum
map (<ref>) takes the form

\begin{equation}
\SectionMapAbb{J}: \ConnSpace_\omega(M) \to \VectorFieldSpace(M, \omega),
\quad \nabla + A \mapsto -
\frac{1}{2} \tensor{j}{^i_k} \bigl( \rho(\nabla + A)^k -
\rho(\nabla)^k - \tensor{\sigma(\nabla, A)}{^k_j^j}\bigr) \,,
\end{equation}

where \( \rho_i \) and \( \sigma_{ijk} \) have been defined
in (<ref>)
and (<ref>), respectively.

In the following, it is often convenient to work on the complexified
tangent bundle and use an abstract index notation that is adapted to
the decomposition of \( \TBundle M \tensorProd \C =
\TBundle^{(1,0)} M \oplus \TBundle^{(0,1)} M \) into
\( \pm i \)-eigenspaces of \( j \).
For this purpose, we use capital Latin letters
\( \mathsf{A, B}, \ldots \)
to denote elements of \( \TBundle M \tensorProd \C \), Greek
letters \( \alpha, \beta, \ldots \) for elements of
\( \TBundle^{(1,0)} M \) and overlined Greek letters
\( \bar{\alpha}, \bar{\beta}, \ldots \) for elements of
\( \TBundle^{(0,1)} M \). For example, \( X^\mathsf{A}\) is a complex
vector field and \( Y^{\bar{\alpha}} \) is a \( (0,1) \)-vector field.
Moreover, we use only the symplectic form and not the metric to lower and raise

Using these conventions, the complex structure \( j \) on \( M \)
defines a constant almost complex structure \( \SectionMapAbb{j} \)
on \( \ConnSpace_\omega(M) \) by

$\mathsf{ABC}$     $(\SectionMapAbb{j} A)_\mathsf{ABC}$
$\alpha\beta\gamma$     $- \I A_{\alpha\beta\gamma}$

$\bar{\alpha}\beta\gamma$     $- \I A_{\bar{\alpha}\beta\gamma}$

$\alpha\bar{\beta}\bar{\gamma}$     $+ \I A_{\alpha\bar{\beta}\bar{\gamma}}$

$+ \I A_{\bar{\alpha}\bar{\beta}\bar{\gamma}}$

and symmetric extension, where $A$ is the symmetric covariant 3-tensor
defined in the text following (<ref>). Here, the
possible components are listed in the first
column, and the corresponding value of \( \SectionMapAbb{j} A \) is the entry
in the second column, the first row is equivalent to
\( (\SectionMapAbb{j} A)_{\alpha\beta\gamma} = - \I A_{\alpha\beta\gamma} \).
Note that this complex structure is not just precomposition of \( A \)
with \( j \), which would have a different sign in the second and third row.

A direct calculation yields \( \Omega(\SectionMapAbb{j} \cdot,
\SectionMapAbb{j} \cdot) = \Omega(\cdot, \cdot) \).
Moreover, in [Proposition 17 and Remark 20]Fuente-Gravy2015
(see also [Lemma 4.9]FutakiOno2018) it has been shown
that \( \Omega(\cdot, \SectionMapAbb{j} \cdot) \) is positive definite
on the complexified \( \DiffGroup(M, \omega) \)-orbit
if the Ricci curvature is non-negative.
In the general setting above, we only used the non-degeneracy of
\( \Omega(\cdot, \SectionMapAbb{j} \cdot) \) in
<ref> to determine
the kernel of the Calabi operators and, for this computation,
non-degeneracy along the
\( \VectorFieldSpace(M,\omega)_\C \)-orbit suffices.

The Levi-Civita connection \( \nabla^j \) associated with
the Riemannian metric defined by \( j \) and \( \omega \) is
a symplectic connection. We say that a compatible complex structure
\( j \) is a Cahen–Gutt critical if its Levi-Civita
connection \( \nabla^j \) is a critical point of the norm-squared of the momentum
map \( \norm{\SectionMapAbb{J}}_\kappa^2:
\ConnSpace_\omega(M) \to \R \).

We need the following generalization of
[Lemma 2.2 and 4.9]FutakiOno2018 from
Hamiltonian vector fields to symplectic vector fields.

Let \( (M, \omega, j) \) be a compact Kähler manifold with
Levi-Civita connection \( \nabla \). The following holds:

* For every \( X \in \VectorFieldSpace(M, \omega) \), \( \difLie_X \nabla = 0 \)
if and only if \( X \) is real holomorphic.

* For every \( X + \I Y \in \VectorFieldSpace(M, \omega)_\C \),
\( \difLie_X \nabla + \SectionMapAbb{j} \difLie_Y \nabla = 0 \)
if and only if \( (X + \I Y)^{(1,0)} \) is holomorphic.
Moreover, the map \( X + \I Y \mapsto X - j Y \) yields a surjection
from \( \VectorFieldSpace(M, \omega)_\C \) onto the space of
holomorphic vector fields.

Let \( Z^\mathsf{A} \in \VectorFieldSpace(M, \omega)_\C \). Since
\( \DiffGroup(M, \omega) \) acts on the space of
symplectic connections, we know that
\( (\difLie_Z \nabla)_\mathsf{ABC} \) is a symmetric tensor.
The only independent components are given as follows:

\begin{equation}
\label{eq:symplecticConnections:difLieNabla}
\begin{array}{ l c }
\mathsf{ABC} & (\difLie_Z \nabla)_\mathsf{ABC} \\ \midrule
\alpha\beta\gamma & \nabla_\alpha \nabla_\beta Z_\gamma \\
\bar{\alpha}\beta\gamma & \nabla_{\bar{\alpha}} \nabla_\beta Z_\gamma \\
\alpha\bar{\beta}\bar{\gamma} &
\nabla_\alpha \nabla_{\bar{\beta}} Z_{\bar{\gamma}} \\
\bar{\alpha}\bar{\beta}\bar{\gamma} &
\nabla_{\bar{\alpha}} \nabla_{\bar{\beta}} Z_{\bar{\gamma}} \\
\end{array}
\end{equation}

Here, we used (<ref>) and
the fact that the Riemann curvature of a Kähler metric
has additional symmetry properties, so that, for example,
\( R_{\mathsf{D}\alpha\beta\gamma} \) vanishes.	Thus,
\( \difLie_Z \nabla = 0 \) implies, using integration
by parts, that

\begin{equation}
\int_M g^{\alpha \bar{\gamma}} g^{\beta \bar{\delta}} \,
\bigl(\nabla_\alpha Z_\beta\bigr) \bigl(\nabla_{\bar{\gamma}}
\bar{Z}_{\bar{\delta}}\bigr) \, \mu_\omega = 0 \, ,
\end{equation}

hence \( \nabla_\alpha Z_\beta = 0 \). Similarly, we conclude
that \( \nabla_{\bar{\alpha}} Z_{\bar{\beta}} = 0 \).
Summarizing, \( \difLie_Z \nabla = 0 \) is equivalent to
\( \nabla_\alpha Z_\beta = 0 = \nabla_{\bar{\alpha}} Z_{\bar{\beta}} \).

On the other hand, by [Lemma 2.3]Futaki2006, we have

\begin{equation}
\tensor{(\difLie_Z j)}{_{\mathsf{A}}^{\mathsf{B}}} =
- 2 \I \tensor{\delta}{_{\bar{\beta}}^{\mathsf{B}}}
\tensor{\delta}{_{\mathsf{A}}^{\alpha}} \,
\nabla_\alpha Z^{\bar{\beta}}
+ 2 \I \tensor{\delta}{_\beta^{\mathsf{B}}}
\tensor{\delta}{_{\mathsf{A}}^{\bar{\alpha}}} \,
\nabla_{\bar{\alpha}} Z^{\beta} \, .
\end{equation}

Thus, upon lowering the last index, the only non-zero components
of \( \tensor{(\difLie_Z j)}{_{\mathsf{AB}}} \) are:

$\mathsf{AB}$     $(\difLie_Z j)_\mathsf{AB}$
$\alpha\beta$     $-2 \I \, \nabla_\alpha Z_\beta$

$\bar{\alpha}\bar{\beta}$     $2 \I \, \nabla_{\bar{\alpha}} Z_{\bar{\beta}}$

Thus, we see that \( \difLie_Z \nabla = 0 \) if and only if
\( \difLie_Z j = 0 \). In particular, this holds for \( Z = X \)
being a real vector field. This proves (i).

Finally, let \( X, Y \in \VectorFieldSpace(M, \omega) \) be
such that \( \difLie_X \nabla + \SectionMapAbb{j} \difLie_Y \nabla = 0 \).
The definition of \( \SectionMapAbb{j} \)
and (<ref>) imply that
this is equivalent to

\begin{equation}\begin{split}
\nabla_\alpha \nabla_\beta (X_\gamma - \I Y_\gamma)
&= 0 = \nabla_{\bar{\alpha}} \nabla_\beta (X_\gamma - \I Y_\gamma), \\
\nabla_\alpha \nabla_{\bar{\beta}} (X_{\bar{\gamma}} + \I Y_{\bar{\gamma}})
&= 0 = \nabla_{\bar{\alpha}} \nabla_{\bar{\beta}} (X_{\bar{\gamma}}
+ \I Y_{\bar{\gamma}}).
\end{split}\end{equation}

Using integration by parts as above, we see that these equations
themselves are equivalent to \( \nabla_{\bar{\beta}} (X_{\bar{\gamma}}
+ \I Y_{\bar{\gamma}}) = 0 \), that is \( \nabla_{\bar{\beta}} (X^{\gamma}
+ \I Y^{\gamma}) = 0 \) upon lifting the second index, which
proves the first part of (ii).
The second part follows as in <ref>:

\begin{equation}
(\difLie_X j - \difLie_{jY} j)_{{\bar{\alpha}} {\bar{\beta}}}
= (\difLie_X j - j \difLie_Y j)_{{\bar{\alpha}} {\bar{\beta}}}
= 2 \I \nabla_{\bar{\alpha}} X_{\bar{\beta}} -
2 \nabla_{\bar{\alpha}} Y_{\bar{\beta}}
= 2 \I \nabla_{\bar{\alpha}}(X_{\bar{\beta}} +
\I Y_{\bar{\beta}}) = 0.
\qedhere
\end{equation}

This shows that every vector field \( X \) in the stabilizer
\( \VectorFieldSpace(M, \omega)_\nabla \) of the Levi-Civita
connection \( \nabla \) is real holomorphic, and thus Killing.
Hence, \( \kappa \) and \( \SectionMapAbb{j} \) are invariant
under \( \VectorFieldSpace(M, \omega)_\nabla \).
Moreover, the stabilizer of \( \nabla \) under the
\( \VectorFieldSpace(M,\omega)_\C \)-action
projects onto the Lie algebra of
holomorphic vector fields; in particular,
it is finite dimensional, too.
<Ref> implies that a compatible complex
structure \( j \) is Cahen–Gutt critical if and only if
\( \SectionMapAbb{J}(\nabla^j) \in \VectorFieldSpace(M, \omega) \) is
real holomorphic.
Note that our notion of extremality is hence slightly different
from FutakiOno2018,Fox2014.

As a consequence of <ref>
we obtain the following.

Let \( (M, \omega) \) be a compact symplectic manifold and let \( j \)
be a compatible Cahen–Gutt critical
complex structure on \( M \). Assume that the Ricci curvature of
the Levi-Civita connection \( \nabla \) associated with \( g_j \) is non-negative.
Then the Lie algebra of real holomorphic vector fields admits the
following decomposition:

\begin{equation}
\label{eq:symplecticConnections:decompositionAut}
\LieA{h}(M, j) = \LieA{c} \oplus
\bigoplus_{\lambda \neq 0} \LieA{k}_\lambda,
\end{equation}

* \( \LieA{c} \) is the Lie subalgebra of
\( \LieA{h}(M, j) \) consisting of
all elements that commute with \( \SectionMapAbb{J}(\nabla) \);

* \( \C \SectionMapAbb{J}(\nabla) \subseteq \LieA{c} \);

* \( \LieA{k}_\lambda \) are eigenspaces of
\( -2 j \difLie_{\SectionMapAbb{J}(\nabla)} \)
with eigenvalue \( \lambda \in \R \) (with the convention that
\( \LieA{k}_\lambda = \set{0} \) if \( \lambda \) is not an
eigenvalue); in particular, $\mathfrak{c} = \mathfrak{k}_0$;

* \( \commutator{\LieA{k}_\lambda}{\LieA{k}_\mu} \subseteq
\LieA{k}_{\lambda + \mu} \) if
\( \lambda + \mu \) is an eigenvalue
of \( -2 j \difLie_{\SectionMapAbb{J}(\nabla)} \); otherwise
\( \commutator{\LieA{k}_\lambda}{\LieA{k}_\mu} = 0 \).

Moreover, the Hessian of \( \norm{\SectionMapAbb{J}}^2 \) at
\( \nabla \) is given by

\begin{equation}
\frac{1}{2} \Hessian_\nabla \norm{\SectionMapAbb{J}}^2
(\zeta \ldot \nabla,
\gamma \ldot \nabla) =
\Re  \, \dualPair{\zeta}{C^+_\nabla R_\nabla \gamma}_{\C},
\end{equation}

for \( \zeta, \gamma \in \VectorFieldSpace(M, \omega)_\C \) and

\begin{equation}
C_\nabla^\pm = L_\nabla \pm \I Z_\nabla, \qquad
R_\nabla = C^-_\nabla \Matrix{0 & 0 \\ 0 & 1} + \I \,
\bigl(\difLie_{\SectionMapAbb{J}(\nabla)} + Z_\nabla\bigr)
\end{equation}

where \( L_\nabla = - \tangent_\nabla \SectionMapAbb{J} (\SectionMapAbb{j} \,
\difLie_\xi \nabla) \) and \( Z_\nabla =
- \tangent_\nabla \SectionMapAbb{J} (\difLie_\xi \nabla) \).

We can apply <ref>
to obtain a decomposition of the stabilizer
\( \VectorFieldSpace(M, \omega) \ldot \nabla \), where \( \nabla \)
is the Levi-Civita connection associated with \( g_j \).
The claims concerning the
decomposition (<ref>)
follow then directly under the map \( \VectorFieldSpace(M, \omega) \ldot
\nabla \ni X + \I Y \mapsto X - j Y \in \LieA{h}(M, j) \),
<ref> and the
proof of <ref>.
The expression for the Hessian follows directly from

* If one proceeds in an analogous way for the action of the
subgroup of Hamiltonian diffeomorphism, then one recovers
[Theorem 4.7 and 4.11]FutakiOno2018 as a direct application of

* A similar theorem holds for connections that are critical points
of the momentum map squared (seen as a functional on
\( \ConnSpace_\omega(M) \))
without being necessarily the Levi-Civita connection of some compatible
Riemannian metric. However, in this case, we do not know of a result
similar to <ref> that
allows us to identify the stabilizers.

* Instead of using the almost complex structure \( \SectionMapAbb{j} \)
on \( \ConnSpace_\omega(M) \) defined above, one could also work with
the almost complex structure that sends
\( A \in \SymTensorFieldSpace_3(M) \)
to \( A(j \cdot, j \cdot, j \cdot) \).
In this case, the stabilizer of the Levi-Civita connection under the
complexified action is a proper subalgebra of \( \LieA{h}(M, j) \).

* In FutakiOno2018,Fuente-Gravy2016 a slightly different
viewpoint is used: instead of working on the symplectic manifold
\( \ConnSpace_\omega(M) \) of symplectic connections as we do above,
the pull-back of the symplectic form \( \Omega \) along the Levi-Civita
map \(\SectionMapAbb{lc}: j \mapsto \nabla^j\) to the space
\( \SectionSpaceAbb{I}(M, \omega) \) of integrable complex structures
on \( M \) compatible with \( \omega \) is used.
In this setting, the condition of non-negative Ricci curvature is necessary
to guarantee the non-degeneracy of \( \SectionMapAbb{lc}^* \Omega \); see
[Proposition 17]Fuente-Gravy2016.

§ APPLICATION: YANG–MILLS

Let \( G \) be a compact connected Lie group and let \( P \to M \) be a
principal \( G \)-bundle over a closed connected Riemann surface \( M \).
Fix an \( \AdAction \)-invariant pairing on the Lie algebra \( \LieA{g} \)
of \( G \). The space \( \ConnSpace(P) \) of connections on \( P \) is an
affine space modeled on the tame Fréchet space
\( \DiffFormSpace^1(M, \AdBundle P) \)
of \( 1 \)-forms on \( M \) with values in the adjoint bundle
\( \AdBundle P \).
The \( 2 \)-form \( \omega \) on \( \ConnSpace(P) \) defined by the
integration pairing

\begin{equation}
\label{eq:yangMillsSurface:symplecticForm}
\omega_A (\alpha, \beta) = \int_M \wedgeDual{\alpha}{\beta}
\end{equation}

for \( \alpha, \beta \in \DiffFormSpace^1(M, \AdBundle P) \) is a
symplectic form, where \( \wedgeDualDot \) denotes the wedge
product relative to the
\( \AdAction \)-invariant pairing on \( \LieA{g} \).
For \( \alpha, \beta \in \DiffFormSpace^1(M, \AdBundle P) \)
and \( X, Y \in \VectorFieldSpace(M) \), we have
\( \wedgeDual{\alpha}{\beta}(X,Y) = \dualPair{\alpha(X)}{\beta(Y)} -
\dualPair{\alpha(Y)}{\beta(X)} \).
The natural action on \( \ConnSpace(P) \) of the group \( \GauGroup(P) \)
of gauge transformations of \( P \) is smooth and preserves the symplectic
structure \( \omega \). The \( \AdAction \)-invariant pairing on
\( \LieA{g} \) induces a natural pairing

\begin{equation}
\kappa: \sSectionSpace(\AdBundle P) \times
\sSectionSpace(\AdBundle P) \to \R, \qquad (\phi, \varrho) \mapsto
\int_M \dualPair{\phi}{\varrho} \, \vol_g \, .
\end{equation}

A straightforward calculation verifies that the map

\begin{equation}
\SectionMapAbb{J}: \ConnSpace(P) \to \sSectionSpace(\AdBundle P),
\qquad A \mapsto - \hodgeStar F_A
\end{equation}

is an equivariant momentum map for the \( \GauGroup(P) \)-action on
\( \ConnSpace(P) \), see AtiyahBott1983.
The norm-squared of the momentum map
\( \norm{\SectionMapAbb{J}}_\kappa^2(A)
= \int_{M} F_A \wedge \hodgeStar F_A \) is the Yang–Mills action, whose
critical points are, according to <ref>,
precisely the Yang–Mills connections, , connections \( A \)
satisfying \( \dif \hodgeStar F_A = 0 \). This observation goes back
at least to [Proposition 4.6]AtiyahBott1983. Moreover,
the Hodge star operator squares to minus the identity on \( 1 \)-forms
and so yields an almost complex structure
\( \hodgeStar: \DiffFormSpace^1(M, \AdBundle P ) \to
\DiffFormSpace^1(M, \AdBundle P ) \) that is compatible
with \( \omega \). Upon complexification, we obtain a decomposition

\begin{equation}
\DiffFormSpace^1(M, \AdBundle P \tensorProd \C)
= \DiffFormSpace^{1,0}(M,
\AdBundle P) \oplus \DiffFormSpace^{0,1}(M, \AdBundle P)
\end{equation}

in eigenspaces of \( \hodgeStar \) with eigenvalues \( -i \) and \( i \),
For a connection \( A \), the associated
exterior derivative \( \dif_A:
\DiffFormSpace^0(M, \AdBundle P \tensorProd \C) \to \DiffFormSpace^1(M,
\AdBundle P \tensorProd \C) \) decomposes accordingly into
\( \dif_A = \difp_A + \difpBar_A \). Under the
\( \hodgeStar \)-\( i \)-complex
linear identification \( \DiffFormSpace^1(M, \AdBundle P) \ni
\alpha \mapsto \I \alpha + \hodgeStar \alpha \in
\DiffFormSpace^{0,1}(M, \AdBundle P) \), the
complexified action on the Lie algebra level is the operator
\( - 2 \I \difpBar_A: \GauAlgebra(P)_\C \to
\DiffFormSpace^{0,1}(M, \AdBundle P) \).
Thus, the stabilizer \( \bigl(\GauAlgebra(P)_\C\bigr)_A \)
is identified with
the space \( \holSectionSpace_A(\AdBundle P \tensorProd \C) \) of
holomorphic sections of \( \AdBundle P \tensorProd \C \).
Moreover, in [p. 556]AtiyahBott1983, it was shown that
the eigenvalues of the endomorphism
\( - 2 \I \, \commutator{\hodgeStar F_A}{\cdot} \) on
\( \AdBundle P \) are locally constant, and that one thus
obtains an eigenspace decomposition

\begin{equation}
\label{eq:yangMillsSurface:decompositionAdBundle}
\AdBundle P \tensorProd \C = \bigoplus_{\lambda} \AdBundle_\lambda P \, ,
\end{equation}

where \( \AdBundle_\lambda P \) is the eigenspace of \( - 2 \I \,
\commutator{\hodgeStar F_A}{\cdot} \) corresponding to the eigenvalue
\( \lambda \). Since \( \bigl(\GauAlgebra(P)_\C\bigr)_A \) is
finite-dimensional, we can apply
<ref> to obtain the

Let \( G \) be a compact connected Lie group and let \( P \to M \)
be a principal \( G \)-bundle over a closed connected Riemann
surface \( M \). For every Yang–Mills connection \( A \) on \( P \),
the following decomposition of
the complex Lie algebra of holomorphic sections of
\( \AdBundle P \tensorProd \C \)

\begin{equation}
\holSectionSpace_A\bigl(\AdBundle P \tensorProd \C\bigr) =
\bigl(\GauAlgebra(P)_A\bigr)_\C \oplus \bigoplus_{\lambda < 0}
\holSectionSpace_A\bigl(\AdBundle_\lambda P\bigr)
\end{equation}

such that \( \hodgeStar F_A \) lies in the center of
\( \GauAlgebra(P)_A \) and

\begin{equation}
\commutator*{\holSectionSpace_A\bigl(\AdBundle_\lambda P\bigr)}
{\holSectionSpace_A\bigl(\AdBundle_\mu P\bigr)} \subseteq
\holSectionSpace_A\bigl(\AdBundle_{\lambda + \mu} P\bigr),
\end{equation}

with the convention that \(\holSectionSpace_A\bigl(
\AdBundle_{\lambda+\mu} P\bigr)\) is trivial if \( \lambda + \mu \)
is not an eigenvalue of \( - 2 \I \, \commutator{\hodgeStar F_A}{\cdot} \).

This follows from <ref> but for
completeness we give a sketch of a direct proof.

The decomposition (<ref>) of
\( \AdBundle P \tensorProd \C \) induces decompositions on the level of
differential forms:

\begin{equation}
\DiffFormSpace^k(M, \AdBundle P \tensorProd \C) =
\bigoplus_{\lambda} \DiffFormSpace^k(M, \AdBundle_\lambda P).
\end{equation}

As a consequence of the Yang–Mills equation, the operators \( \difpBar_A \)
and \( \commutator{\hodgeStar F_A}{\cdot} \) commute.
Hence, \( \difpBar_A \) decomposes into the sum of operators
\( \difpBar_{A, \lambda}: \DiffFormSpace^0(M, \AdBundle_\lambda P) \to
\DiffFormSpace^1(M, \AdBundle_\lambda P) \) and so

\begin{equation}
\holSectionSpace_A\bigl(\AdBundle P \tensorProd \C\bigr) =
\bigoplus_{\lambda} \holSectionSpace_A\bigl(\AdBundle_\lambda P\bigr) \, .
\end{equation}

By considering appropriate Laplacian operators, one can show that
\( \holSectionSpace_A\bigl(\AdBundle_\lambda P\bigr) \) is isomorphic
to \( \bigl(\GauAlgebra(P)_A\bigr)_\C \) for \( \lambda = 0 \) and
is trivial for \( \lambda > 0 \); see
[Lemma 5.9 (iii) and p. 559]AtiyahBott1983.

If \( P \) is a reduction of a \( G^\C \)-principal bundle
\( P^\C \) to \( G \subseteq G^\C \), the space
\( \holSectionSpace_A\bigl(\AdBundle P \tensorProd \C\bigr) \) is
naturally identified with the space of sections of \( \AdBundle P^\C \)
that are holomorphic with respect to the holomorphic structure
\( \difpBar_A \) on \( P^\C \) induced by the connection \( A \).
Hence, \( \holSectionSpace_A\bigl(\AdBundle P \tensorProd \C\bigr) \)
can be viewed as the stabilizer algebra of \( \difpBar_A \) under
the action of \( \GauGroup(P^\C) \) on the space of holomorphic
structures on \( P^\C \).

It is possible to extend the above results to the case when the
base \( M \) is a compact symplectic manifold of arbitrary dimension;
see [Section 4]Donaldson1985 for the setup of the
infinite-dimensional symplectic framework.
Then global minima and critical points of the norm-squared of the
momentum map correspond to Kähler–Einstein connections and
Hermitian Yang–Mills connections, respectively.
This extension is especially fruitful when coupled to other geometric
structures on the base, such as one of the special Kähler metrics
discussed in <ref>.
For example, we expect that our general results directly yield the
reductiveness obstruction of solutions of the
Kähler–Yang–Mills–Higgs equations [Theorem 3.6]
AlvarezConsulGarciaFernandezGarciaPrada2019 (note, however, that
the assumption of vanishing first Betti number in that theorem calls
for a careful treatment, so one might expect to again encounter
central extensions of the symplectomorphism group).

§ NOTATION AND CONVENTIONS

Penrose Notation.
In <ref>, we shall make extensive use of
Penrose's abstract index notation. In this notation, indices are
used as labels indicating the type of a tensor and do not
denote the components of a tensor with respect to a local frame.
For example, a vector field is denoted by \( X^i \).
The superscript \( i \) in \( X^i \) does not refer to a particular
component in local coordinates but serves as a label telling us
that \( X \) is a vector field. Similarly, a \( 1 \)-form is
written as \( \alpha_j \). Contraction is indicated by labeling one
covariant index and one contravariant
index with the same letter, , \( \alpha(X) \equiv \alpha_i X^i \).
Thus, the resulting calculus resembles the usual coordinate
expressions but has the important advantage of being completely intrinsic
and coordinate-free.

Indices are raised and lowered using the symplectic form
\( \tensor{\omega}{_i_j} \) as follows:

\begin{align}
\label{flat}
\omega^\flat: X^i &\mapsto X_i \equiv \omega_{ji} X^j \, ,   \\
\label{sharp}
\omega^\sharp: \alpha_j &\mapsto \alpha^j \equiv \varpi^{ji} \alpha_i \, ,
\end{align}

where \( \varpi^{ij} \) is the Poisson tensor associated with
\( \omega_{ij} \) according to \( \varpi^{ik}\omega_{kj} =
- \tensor{\delta}{^i_j} \).
For a symplectic form \( \omega \) with associated Poisson
tensor \( \varpi \), the Poisson bracket is given by \( \poisson{f}{g} =
\omega(X_f, X_g) = \varpi(\dif f, \dif g) \), where \( X_f \) is the
Hamiltonian vector field satisfying \( X_f \contr \omega = - \dif f \).
We thus have

\begin{equation*}
\poisson{f}{g}
= \varpi^{ij} (\dif f)_i (\dif g)_j
= \varpi^{ij} \omega_{ki} (X_f)^k \omega_{lj} (X_g)^l
= \omega_{kl} (X_f)^k (X_g)^l.
\end{equation*}

In other words, \( \varpi^{ij} \omega_{ki} \omega_{lj} = \omega_{kl} \),
which is equivalent to \( \varpi^{ik}\omega_{kj} =
- \tensor{\delta}{^i_j} \).

Note that \( \omega^\flat \) and \( \omega^\sharp \) are inverses of
each other. The minus sign in the definition of the Poisson tensor
is a consequence of the skew-symmetry of \( \omega_{ij} \) and leads
to some subtle consequences for the index calculus that are different
from the Riemannian context. In particular, the position of the
indices is important even if they are summed-over. For example,
we have \( A_{ij} = \tensor{A}{_i^l} \omega_{lj}\)
and \( B^{jk} = \varpi^{jp} \tensor{B}{_p^k} \) so that

\begin{equation}
A_{ij} \tensor{B}{^{jk}}
= \tensor{A}{_i^l} \omega_{lj} \varpi^{jp} \tensor{B}{_p^k}
= - \tensor{A}{_i^p}\tensor{B}{_p^k} \, .
\end{equation}

Moreover, lowering the index of the identity map \( \tensor{\delta}{_i^j}:
\TBundle M \to \TBundle M \) yields the skew-symmetric map
\( \delta_{ij} = \omega_{ij}: \TBundle M \times_M \TBundle M \to \R \).

Let \( \TensorFieldSpace^r_s(M) \) be the space of \( r \)-times
contravariant and \( s \)-times covariant tensor fields.
An affine connection on \( M \) is a linear map

\begin{equation}
\nabla: \VectorFieldSpace(M) \to \TensorFieldSpace^1_1(M),
\qquad
X^j \mapsto \nabla_i X^j,
\end{equation}

which satisfies the Leibniz rule \( \nabla_i (fX^j) =
{(\dif f)}_i X^j + f \, \nabla_i X^j \).
The covariant derivative extends uniquely to all tensor fields
by requiring \( \nabla _i \) to preserve the type of the tensor
and to be a \( \R \)-linear tensor derivation, ,
\( \nabla_i (t \otimes s) = (\nabla_i t) \otimes s +
t \otimes (\nabla_i s) \) for any \( t, s \in \TensorFieldSpace(M) \),
and to commute with contractions. In abstract Penrose index notation,
the covariant derivative of a tensor field
\( t^{j_1 \dotso j_p}_{k_1 \dotso k_q} \) is
denoted by \( \nabla_i t^{j_1 \dotso j_p}_{k_1 \dotso k_q} \).

The Lie derivative of a connection is defined by the requirement
that it behaves like a derivation on all symbols, , for each given
\( X \in \VectorFieldSpace(M) \), the formula

\begin{equation}
\difLie_X (\nabla_Y Z) = (\difLie_X \nabla)_Y Z + \nabla_{\difLie_X Y} Z
+ \nabla_Y \difLie_X Z,
\end{equation}

for all \( Y,Z \in \VectorFieldSpace(M) \), defines a new covariant
derivative \( (\difLie_X \nabla)_Y \) along the vector field \( Y \).

The torsion of \( \nabla \) is the \( 1 \)-contravariant,
\( 2 \)-covariant tensor field \( \tensor{T}{_i_j^k} \) defined by

\begin{equation}
\label{torsion_definition}
\tensor{T}{_i_j^k} X^i Y^j =
X^i \nabla_i Y^k - Y^j \nabla_j X^k - [X,Y]^k
\quad
\text{for all}\quad X^i,Y^j \in \VectorFieldSpace(M),
\end{equation}

The curvature \( \tensor{R}{_i_j_k^l} \) of the connection
\( \nabla \) is defined by

\begin{equation}
\label{curvature_definition}
\tensor{R}{_i_j_k^l} Z^k = \nabla_i \nabla_j Z^l - \nabla_j \nabla_i Z^l
+ \tensor{T}{_i_j^k} \, \nabla_k Z^l \, .
\end{equation}

Since we will rely heavily on the Penrose notation, this is a good place
to compare it with the standard coordinate free notation. Remember,
the indices are not coordinate components of tensors.
For example, in (<ref>), $X^i \nabla_i Y^k$ actually
means $\nabla_X Y$. So, formula (<ref>), even though it
looks like the coordinate expression of the torsion tensor, it really
means $T(X,Y)= \nabla_X Y - \nabla_Y X - [X,Y]$, the standard coordinate
free definition of the torsion. This brings us to the interpretation
of (<ref>), which would be a standard formula
had the sub- and superscripts been indices in a coordinate system. Note
that (<ref>) does not state that
$R(X,Y)Z=\nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z +
\nabla_{T(X,Y)}Z$, which is false, even though one is tempted
to interpret it in this manner. To
see how one can recover the standard definition of the curvature from
the Penrose index formula (<ref>), we multiply
both sides by $X^i Y^j$ and get (again, the indices and their position
only reflect what kind of tensor is considered, so the index $l$ in
the computation below is not a “free index” in Penrose notation;
it only tells us that the result is a vector field and, similarly,
we need to interpret $\nabla_X Y^j$ as $(\nabla_X Y)^j$ since the
upper index only indicates that the expression is a vector field):

\begin{align*}
R(X,Y)Z&= X^i Y^j\tensor{R}{_i_j_k^l} Z^k = X^i Y^j\nabla_i \nabla_j Z^l -
X^i Y^j\nabla_j \nabla_i Z^l +  X^i Y^j\tensor{T}{_i_j^k} \, \nabla_k Z^l\\
&= Y^j \nabla_X \nabla_j Z^l - X^i \nabla_Y \nabla_i Z^l +
T(X,Y)^k \nabla_k Z^l \\
&= \nabla_X(Y^j \nabla_j Z^l) - (\nabla_X Y)^j \nabla_j Z^l -
\nabla_Y(X^i \nabla_i Z^l) + (\nabla_Y X)^i \nabla_i Z^l+
\nabla_{T(X,Y)}Z \\
&= \nabla_X \nabla_Y Z - \nabla_{\nabla_X Y} Z -\nabla_Y \nabla_X Z +
\nabla_{\nabla_Y X} Z + \nabla_{T(X,Y)}Z \\
&= \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z +
\nabla_{T(X,Y)-\nabla_X Y + \nabla_Y X}Z  \\
&= \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z -
\nabla_{[X,Y]}Z\,,
\end{align*}

which is the definition of the curvature tensor. This simple computation
illustrates the power of the Penrose notation: (<ref>)
looks like the correct local formula in coordinates for the curvature
tensor, whereas, in reality, it gives an intrinsic expression of the
curvature tensor and one can recover the classical definition after a simple
computation. It is in this spirit that all the formulas that appear later
on should be interpreted; they have Penrose indices, which means that they
are intrinsic, and the index free expressions can be easily obtained
after a computation analogous to the one above.

Lie Group and Lie Algebra Actions. The
left (right) action of of a Lie group $G$ on a manifold $M$ is denoted
by $(g,m) \mapsto g.m$ ($m.g$) for $g \in G$ and $m \in M$. The induced
left (right) Lie algebra action of $\mathfrak{g}$, the Lie algebra of $G$,
on $M$ is denoted by $(\xi,m) \mapsto \xi.m$ ($m.\xi$) for $\xi \in
\mathfrak{g}$ and $m \in M$, where

\[
\xi.m \defeq \xi^*(m) \defeq
\left.\frac{d}{dt}\right|_{t=0} \exp(t \xi).m
\]

is the value of the fundamental vector field (or infinitesimal
generator) \( \xi^* \) defined by $\xi$ at $m$; analogous notation
for a right action. Recall
that for left (right) Lie algebra actions we have $[\xi^*, \eta^*] =
- [\xi, \eta]^*$ ($[\xi^*, \eta^*] =[\xi, \eta]^*$).

Throughout the paper we think of $\operatorname{U}(1)
= S^1$ as $\mathbb{R}/\mathbb{Z}$ and write hence the group multiplication

Conventions in Symplectic Geometry.
Since the sign conventions in symplectic geometry are not uniform,
we specify them at the outset. The canonical one-form on the cotangent
bundle is in canonical local cotangent bundle coordinates
$(q^i, p_i)$ equal to $\theta =p_i {\rm d}q^i$ and the symplectic
form is $\omega = {\rm d}\theta = {\rm d} p_i \wedge {\rm d}q^i$.
The Hamiltonian vector field $X_h$ of a function $h$ on a general
symplectic manifold $(M,\omega)$ is defined hence by
${\rm d}h = - X_h \contr \omega$ and Hamilton's equations
in Poisson bracket form are $\dot{f}=\{h,f\}$ for any smooth function,
which, in local Darboux coordinates $(q^i, p_i)$ on $M$
(, $\omega = {\rm d} p_i \wedge {\rm d}q^i$) are the standard
Hamilton equations $\frac{dq^i}{dt}  = \frac{\partial h}{\partial p_i}$,
$\frac{dp_i}{dt} = -\frac{\partial h}{\partial q^i}$.
We have $[X_f, X_g] = X_{\{f,g\}}$ for any $f,g \in C^{\infty} (M)$.

The trace of a \( 2 \)-form \( \alpha \) with respect to \( \omega \)
is defined by first raising the second index of \( \alpha \)
with \( \omega \) and then taking the ordinary trace of the
resulting endomorphism of \( \TBundle M \), that is,
\( \tr_\omega (\alpha) \defeq \tensor{\alpha}{_i^i} \).
The following formula

\begin{equation}
\label{eq:symplectic:formWedgeOmegaNMinus1}
\alpha \wedge \frac{\omega^{n-1}}{(n-1)!} =
\frac{1}{2} \tr_\omega (\alpha)\, \mu_\omega \, ,
\end{equation}

where \( \mu_\omega = \omega^n/n! \) is the volume
form on the \(2n\)-dimensional manifold \( M \) induced by \( \omega \),
will often be used in <ref>;
it is checked using a canonical basis in each tangent space.
For all \( 1 \)-forms \( \sigma, \tau \), we have

\begin{equation}
\label{eq:symplectic:symplecticHodge1Form}
\omega(\sigma, \tau) \, \mu_\omega = \sigma \wedge \tau \wedge
\frac{\omega^{n-1}}{(n-1)!} \, .
\end{equation}

A Lie group action on the symplectic manifold $(M, \omega)$ is called
symplectic or canonical if the diffeomorphism on $M$ defined by each
$g \in G$ preserves the symplectic form $\omega$ on $M$. This implies
$\dif (\xi^*\contr \omega) = \difLie_{\xi^*} \omega = 0$, where $\difLie$
denotes the Lie derivative; this condition
is equivalent to the action being symplectic if the Lie group $G$ is
We use a weakly nondegenerate pairing $\kappa: \mathfrak{g}^\ast \times
\mathfrak{g} \rightarrow  \mathbb{R}$ and think of $\mathfrak{g}^\ast$
as the “dual” of $\mathfrak{g}$ (even though it is not the functional
analytic dual in infinite dimensions); nondegenerate always means weakly
nondegenerate. The momentum map $J:M \rightarrow
\mathfrak{g}^\ast$ is defined by the requirement $\xi^\ast = X_{J_\xi}$
for any $\xi \in  \mathfrak{g}$, where $J_\xi (m):= \kappa (J(m), \xi)$
for any $m \in  M$. Thus, $J$ is infinitesimally equivariant if and only
if it is an anti-Poisson map, , $\{ J_\xi , J_\eta \} +
J_{[ \xi , \eta ]}  = 0$ for any $\xi, \eta \in  \mathfrak{g}$.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

groupsSymplectic connection
journaltitlePure Appl. Math. Q.
titleOn the Kähler-Yang-Mills-Higgs equations

groupsSymplectic connection
journaltitleJournal of the European Mathematical Society
titleConformally Kähler, Einstein–Maxwell geometry

journaltitlePhilos. Trans. Roy. Soc. London Ser. A
titleThe Yang-Mills equations over Riemann surfaces

Birkhäuser Boston

booktitleArithmetic and Geometry
titleConvexity and Loop Groups

editionSoftcover reprint of the original 2nd ed. 2004
groupsMomentum polytopes
titleTorus Actions on Symplectic Manifolds (Progress in Mathematics)

seriesLecture Notes in Mathematics
titleDifferential Characters

journaltitleThe Annals of Mathematics
titleOn unitary ray representations of continuous groups

Basel: Springer

booktitleRecent trends in dynamical systems. Proceedings of the international conference, Munich, Germany, January 11–13, 2012, in honor of Jürgen Scheurle on the occasion of his 60th birthday
titleGradient flows in the normal and Kähler metrics and triple bracket generated metriplectic systems

Fondation L'Enseignement Mathématique

journaltitleEnseign. Math. (2)
titleOn the characteristic classes of groups of diffeomorphisms

Oxford University Press

groupsSymplectic connection
noteIncludes bibliographical references and index. - Formerly CIP
seriesOxford Math. Monogr.
titleSasakian Geometry

groupsSymplectic connection
journaltitleCommun. Math. Phys.
titleCanonical Sasakian metrics
Scalar Curvature,Einstein Metrics,Constant Scalar Curvature,Sasakian Manifold,Reeb Vector,paper

Academia Press

booktitleLiber Amicorum Richard Delanghe: Een Veelzijdig Wiskundige
groupsSymplectic connection
titleMoment map for the space of symplectic connections

Princeton Univ. Press, Princeton, N.J.

booktitleSeminar on Differential Geometry
seriesAnn. of Math. Stud.
titleExtremal Kähler metrics
53C55 (58E30)

Springer, Berlin

booktitleDifferential Geometry and Complex Analysis
groupsSymplectic connection
titleExtremal Kähler metrics II
53C55 (58E30)

American Mathematical Society (AMS)