#  Symplectically Flat Connections and Their Functionals

Li-Sheng Tseng and Jiawei Zhou

(October 6, 2022)

###### Abstract

We continue our study of symplectically flat bundles. We broaden the notion of
symplectically flat connections on symplectic manifolds to $\zeta$-flat
connections on smooth manifolds. These connections on principal bundles can be
represented by maps from an extension of the base’s fundamental group to the
structure group. We introduce functionals with zeroes being symplectically
flat connections and study their critical points. Such functionals lead to
novel geometric flows. We also describe some characteristic classes of the
$\zeta$-flat bundles.

###### Contents

  1. 1 Introduction
  2. 2 Classification of $\zeta$-flat bundles
  3. 3 Curvature functionals whose zeroes are symplectically flat
    1. 3.1 Review of the twisted differentials of the TTY-algebra
    2. 3.2 Primitive Yang-Mills functional
    3. 3.3 Cone Yang-Mills functional
  4. 4 Discussion: flows and characteristic classes

##  1 Introduction

In our previous work [TZ], we introduced the notion of symplectically flat
connections on a symplectic manifold $(M^{2n},\omega)$. For a fiber bundle
$\pi:E\to M$ with covariant derivative $d_{A}$, and $A$ the corresponding
connection on $E$, symplectically flat connections have a curvature $F$ that
satisfies

$\displaystyle F$ $\displaystyle=\Phi\,\omega\,,\qquad d_{A}\Phi=0\,,$ (1.1)

where $\Phi\in\Omega^{0}(M,\mathrm{End}\,E)$. When the dimension $d=2n\geq 4$,
the second equation is implied by the first equation and the Bianchi identity.
When $d=2$, the first equation is trivial and $\Phi$ being covariantly
constant becomes the sole condition.

Let us recall two properties of symplectically flat connections pointed out in
[TZ]. First, with respect to a compatible metric, a symplectically flat
connection satisfies the Yang-Mills equations [TZ]*Example 3.3. (In this
paper, we will always assume that any Riemannian metric we consider on a
symplectic manifold is compatible to the symplectic structure.) In fact, in
dimension $d=2$, the notion of Yang-Mills connections with respect to a
compatible metric is equivalent to symplectically flat connections. This is
significant since Atiyah and Bott in a seminal work [Atiyah-Bott] gave the
classification of Yang-Mills bundles over Riemann surfaces. Their result thus
immediately provides us with the classification of the symplectically flat
bundles over closed, oriented two-manifolds.

In dimensions four and higher, it should be evident from (1.1) that the
symplectic flatness condition is a much stronger condition than the Yang-Mills
condition. Perhaps a better comparison for symplectic flatness in this case is
that of the standard flat connection condition, $F=0$. This relates to a
second property of symplectically flatness described in [TZ] that we now
recall. If the symplectic structure is integral class, i.e. $\omega\in
H^{2}(M,\mathbb{Z})$, then we can build what is called a prequantum circle
bundle $X$, i.e. $S^{1}\rightarrow X\rightarrow M$, which by definition has
Euler class given by $\omega$. Of interest, a symplectically flat connection
$A$ over $M$ satisfying (1.1) can be lifted to a flat $A-\theta\Phi$
connection over $X$, where $\theta$ is a global angular form of $X$
[TZ]*Corollary 3.9. And as is well-known, isomorphism classes of flat bundles
over any manifold is classified by the $G$-character variety of $\pi_{1}$,
i.e. the conjugacy classes of homomorphisms from the fundamental group of the
base space to the structure group $G$ of the fiber.

Together, we have for dimension $d=2$ a classification of symplectically flat
bundles from Atiyah-Bott, and for $d\geq 4$, we have in the integral $\omega$
case a suggestion of a possible classification related to flat bundles over a
space of one higher dimension. A natural question then is to ask whether we
can obtain a classification of bundles with a symplectically flat connection
in dimensions $d=2n>2$ over any symplectic manifold $(M^{2n},\omega)$, which
generalizes the classification of Atiyah-Bott’s in dimension two.

Indeed, we are able to obtain a classification of symplectically flat bundles
that holds for all dimensions. Our result is actually a corollary of a general
statement, that classifies principal bundles with curvature proportional to
any closed two-form. For this, it is useful to define a broader notion of
flatness.

######  Definition 1.1.

For a smooth manifold $M$, let $\zeta\in\Omega^{2}(M)$ be a closed 2-form on
$M$. We say a fiber bundle $E$ over $M$ with covariant derivative $d_{A}$,
where locally $d_{A}=d+A$ with $A$ being the connection 1-form on $E$, is
$\zeta$-flat if the curvature $F$ satisfies

$\displaystyle F$ $\displaystyle=\Phi\,\zeta\,,\qquad d_{A}\Phi=0\,.$ (1.2)

We shall call such a connection $A$ with curvature satisfying (1.2) a
$\zeta$-flat connection.

This definition does not require $\zeta$ to be a symplectic form and is also
applicable for bundles over odd dimensional base manifolds. For this general
$\zeta$-flat condition, we prove in Section 2 the following:

######  Theorem 1.2.

Let $M$ be a connected manifold, $\zeta\in\Omega^{2}(M)$ be a closed 2-form,
and $G$ be a Lie group. There exists a bijective correspondence between the
following sets:

$\displaystyle\left\\{\begin{matrix}\text{isomorphism classes of G-bundles
over M}\\\ \text{with a $\zeta$-flat
connection}\end{matrix}\right\\}\simeq\left\\{\begin{matrix}\text{conjugacy
classes of}\\\ \text{homomorphisms }\rho:\Gamma\to G\end{matrix}\right\\}.$

Here, $\Gamma$ is an $\mathbb{R}/\mkern 2.5mu\overline{\mkern-2.5muH\mkern
1.5mu}\mkern-1.5mu$-extension of $\pi_{1}(M)$, where $\mkern
2.5mu\overline{\mkern-2.5muH\mkern 1.5mu}\mkern-1.5mu\subset\mathbb{R}$ is the
closure of a group $H$ defined as

$H=\left\\{\int_{D}\zeta\ \big{|}\ D\text{ is a representative in
}\pi_{2}(M)\right\\}.$

When $\zeta=\omega$ is the symplectic structure, the above theorem gives a
classification of symplectically flat bundles. In $d=2$, this reproduces
exactly the classification of Yang-Mills bundles over Riemann surfaces of
Atiyah-Bott. In the case when $\mkern 2.5mu\overline{\mkern-2.5muH\mkern
1.5mu}\mkern-1.5mu=\mathbb{R}$, then $\Gamma=\pi_{1}(M)$ and the
classification reduces to exactly that for flat connections. And just like the
classification for flat connections, Theorem 1.2 holds for base manifolds of
even or odd dimensions.

That symplectically flat connections are Yang-Mills connections and also
related to flat connections make them rather special. Recall that flat
connections have a special role in Yang-Mills theory, which is described by
the functional

$\displaystyle S_{Y\\!M}(A)=\|F(A)\|^{2}=\int_{M}\mathrm{tr}~{}F\wedge*F\,.$
(1.3)

Flat connections have the distinguished role that they correspond to the
zeroes of the Yang-Mills functional. From this perspective, we ask in Section
3 whether there are functionals of connection 1-forms whose zeroes are exactly
symplectically-flat connections, or more generally $\zeta$-flat connections.
For this, we find two natural functionals.

######  Definition 1.3.

The primitive Yang-Mills functional is defined on a connection $A$ as

$\displaystyle
S_{pY\\!M}(A)=\|F_{p}(A)\|^{2}=\int_{M}\mathrm{tr}~{}F_{p}\wedge*F_{p}$

where under the Lefschetz decomposition of the curvature 2-form,
$F=F_{p}+\Phi\,\omega$, $F_{p}$ is the primitive component of $F$. We call the
critical points of $S_{pY\\!M}$ primitive Yang-Mills connections.

For the second functional, it can be easily motivated starting from the sum of
the norm-squared of the two conditions of symplectic flatness (1.1):
$\|F(A)-\omega\,\Phi\|^{2}+\|d_{A}\Phi\|^{2}$. If we let $\Phi$ to be
independent of $A$, or explicitly, replacing $\Phi$ with a general
endomorphism on the bundle denoted by $B$, this leads us to the following
functional.

######  Definition 1.4.

The cone Yang-Mills functional is defined on a connection $A$ and an
endomorphism $B$ on the bundle as

$\displaystyle S_{cY\\!M}(A,B)=\|F(A)+\omega
B\|^{2}+\|d_{A}B\|^{2}=\int_{M}\mathrm{tr}\left[(F+\omega B)\wedge*(F+\omega
B)+d_{A}B\wedge*d_{A}B\right]$

Clearly, the zeroes of both functionals satisfy (1.1) and correspond to
symplectically flat connections. But they interestingly have different systems
of equations for the critical points with solutions generally different from
the Yang-Mills connections. These functionals can also be modified so that the
zeroes are $\zeta$-flat instead.

It is worthwhile to point out that the second functional also has an
interpretation as the normed square of the cone curvature. The cone algebra of
interest here consists of elements
$\mathcal{C}^{*}(M):=\Omega^{*}(M)[\theta]=\\{\xi+\theta\eta\,|\,\xi,\eta\in\Omega^{*}(M)\\}$
with $d\theta=\omega$ (see [TT] for a discussion of this cone algebra in the
context of symplectic geometry). This algebra is quasi-isomorphic to the
algebra of differential forms of the prequantum circle bundle $X$,
$\Omega^{*}(X)$, when $\omega$ is integral class. But importantly,
$\mathcal{C}^{*}(M)$ is well-defined even when $\omega$ is not integral. As
discussed in [TZ], we can define a covariant derivative on the twisted
$\mathcal{C}^{*}(M,E)$ given by $D_{C}=d_{A}+\theta B$, with the cone
curvature
$D_{\mathcal{C}}^{2}=\left(F+\omega\,B\right)-\theta\left(d_{A}B\right)$. As
noted in [TZ]*Proposition 3.8, the cone curvature is flat, i.e.
$D_{\mathcal{C}}^{2}=0$, if and only if $A$ is a symplectically flat
connection. This justifies calling $S_{cYM}$ the cone Yang-Mills functional.

The equivalences between symplectically flat and cone flat, and also their
lift to flat connections on the prequantum circle bundle when $\omega$ is
integral do provide us with further insights. Though flat connections have
zero curvature, there are known characteristic classes associated to flat
bundles. (See for example [Morita]*Chapter 2.3 for an overview.) This led us
to consider characteristic classes of flat bundles over $X$ but with the
covariant derivative taken to be that of the cone. Specifically, using
$D_{C}=d_{A}-\theta\Phi$ with $F=\Phi\,\omega$, we are able to obtain
characteristic classes for the symplectically flat bundles. Moreover, the
Chern-Simons type functionals on $X$ also lead to novel functionals for
symplectically flat connections.

This paper is organized as follows. In Section 2, we give the proof of Theorem
1.2, the classification theorem of $\zeta$-flat bundles. In Section 3, we
introduce the primitive Yang-Mills and cone Yang-Mills functionals and
describe some of their properties. We provide explicit examples of connections
that are either Yang-Mills or primitive Yang-Mills, but not both. We also
write down functionals whose zeroes are more generally $\zeta$-flat. We
conclude in Section 4 with a brief discussion on the gradient flow of the
primitive Yang-Mills functional and some remarks concerning the characteristic
forms and classes for symplectically flat and $\zeta$-flat bundles.

Acknowledgements.  We would like to thank David Clausen, Si Li, Jianfeng Lin,
Xiang Tang, Jiaping Wang, Shing-Tung Yau and Xiangwen Zhang for helpful
discussions. We would like to acknowledge the support of the Simons
Collaboration Grant No. 636284 for the first author and the support of the
National Key Research and Development Program of China No. 2020YFA0713000 for
the second author.

##  2 Classification of $\zeta$-flat bundles

The definition of symplectically flat connection is given in (1.1). A natural
question to ask is how symplectically flat bundles can be classified. As we
have seen, when the base is two-dimensional, symplectically flat bundles are
Yang-Mills bundles, which are classified by Atiyah and Bott [Atiyah-Bott].
Hence, we seek a classification when the base space is of dimensions greater
than two. In this case, we find a classification of $\zeta$-flat bundles (see
Definition 1.1) that generalizes Atiyah-Bott’s result for Yang-Mills bundles
over Riemann surfaces. Our proof will be similar to that given by Morrison
[Morrison] for Yang-Mills bundles in the two-dimensional case.

######  Theorem 2.1.

Let $M$ be a connected manifold, $\zeta\in\Omega^{2}(M)$ be a closed 2-form,
and $G$ be a Lie group. There exists a bijective correspondence between the
following sets:

$\displaystyle\left\\{\begin{matrix}\text{isomorphism classes of G-bundles
over M}\\\ \text{with a $\zeta$-flat
connection}\end{matrix}\right\\}\simeq\left\\{\begin{matrix}\text{conjugacy
classes of}\\\ \text{homomorphisms }\rho:\Gamma\to G\end{matrix}\right\\}.$

Here, $\Gamma$ is an $\mathbb{R}/\mkern 2.5mu\overline{\mkern-2.5muH\mkern
1.5mu}\mkern-1.5mu$-extension of $\pi_{1}(M)$, where $\mkern
2.5mu\overline{\mkern-2.5muH\mkern 1.5mu}\mkern-1.5mu\subset\mathbb{R}$ is the
closure of a group $H$ defined as

$H=\left\\{\int_{D}\zeta\ \big{|}\ D\text{ is a representative in
}\pi_{2}(M)\right\\}.$

To give the exact definition of $\Gamma$, we use the following conventions.

Let $\Psi(M)$ be the path space defined as follows. It consists of equivalence
classes of closed, piecewise smooth paths on $M$. If we reparametrize a path
$\alpha_{1}:[0,1]\to M$ by choosing a different map $\alpha_{2}$ with the same
image and orientation, then $\alpha_{1}$ and $\alpha_{2}$ are equivalent in
$\Psi(M)$. In other words, $\alpha_{1}$ and $\alpha_{2}$ are equivalent if
there exists a piecewise smooth increasing function $\phi:[0,1]\to[0,1]$ with
$\phi(0)=0$, $\phi(1)=1$ and $\alpha_{2}=\alpha_{1}\circ\phi$.

We can define a multiplication in $\Psi(M)$ by the composition of paths. For
two paths $\alpha_{1},\alpha_{2}:[0,1]\to M$ with
$\alpha_{1}(1)=\alpha_{2}(0)$, set

$(\alpha_{2}\alpha_{1})(t)=\begin{cases}\alpha_{1}(2t),&t\in[0,\frac{1}{2}],\\\
\alpha_{2}(2t-1),&t\in[\frac{1}{2},1].\end{cases}$

Fix a base point $a\in M$. Let $\Psi^{a}_{a}(M)$ be the semigroup in $\Psi(M)$
that consists of classes of loops with base point $a$, $\Psi^{a}_{a,0}(M)$ be
the subsemigroup of $\Psi^{a}_{a}(M)$ that consists of classes of contractible
loops, and $\Psi^{a}_{a,\zeta}(M)$ be the subsemigroup of $\Psi^{a}_{a,0}(M)$
that consists of classes of loops which are the boundary of some disk
(contractible 2-chain) $D$ in $M$ satisfying $\int_{D}\zeta=0$. Then
$\Psi^{a}_{a}(M)/\Psi^{a}_{a,0}(M)$ and
$\Psi^{a}_{a,0}(M)/\Psi^{a}_{a,\zeta}(M)$ have group structures, where the
identity element is the equivalence class of the constant path at $a$, and the
inverse is reversing the orientation:

$\alpha^{-1}(t)=\alpha(1-t).$

Observe that $\Psi^{a}_{a}(M)/\Psi^{a}_{a,0}(M)=\pi_{1}(M)$, and
$\Psi^{a}_{a,0}(M)/\Psi^{a}_{a,\zeta}(M)$ is equivalent to a quotient group
$\mathbb{R}/H$ by the identification $\gamma\mapsto[\int_{D}\zeta]$, where
$H=\\{\int_{D}\zeta|D\text{ is a representative in }\pi_{2}(M)\\}$.

When $\mathbb{R}/H$ is a Lie group, we set
$\Gamma=\Psi^{a}_{a}(M)/\Psi^{a}_{a,\zeta}(M)$. When $\mathbb{R}/H$ is not a
Lie group, we set $\Gamma=\pi_{1}(M)$.

Explicitly, there are three possibilities for $H$. Let $H^{+}$ be the subset
of $H$ with positive numbers.

Case 1. When $H^{+}$ is empty, $H=0$ and $\mathbb{R}/H=\mathbb{R}$. In this
case $\Gamma=\Psi^{a}_{a}(M)/\Psi^{a}_{a,\zeta}(M)$ is an
$\mathbb{R}$-extension of $\pi_{1}(M)$. For Riemann surfaces, this occurs when
the genus $g\geq 1$.

Case 2. When $H^{+}$ has a minimal number, $H\simeq\mathbb{Z}$ and
$\mathbb{R}/H\simeq S^{1}$. In this case
$\Gamma=\Psi^{a}_{a}(M)/\Psi^{a}_{a,\zeta}(M)$ is an $S^{1}$-extension of
$\pi_{1}(M)$. For Riemann surfaces, this occurs when the genus $g=0$.

Case 3. When $H^{+}$ is non-empty and has no minimal number, $H$ is dense in
$\mathbb{R}$ and $\mathbb{R}/H$ is not a Lie group. In this case,
$\Gamma=\pi_{1}(M)$. This case is impossible when $c\,\zeta$ is an integral
cohomology class for some number $c\neq 0$.

In the remainder of this subsection, we prove the classification theorem.

Case 1 and 2, $\mathbb{R}/H=\mathbb{R}$ or $S^{1}$.

We first consider the case that $\mathbb{R}/H$ is a Lie group. By the
discussion above, $\Psi^{a}_{a}(M)/\Psi^{a}_{a,\zeta}(M)$ is either an
$\mathbb{R}$ or an $S^{1}$ extension of $\pi_{1}(M)$.

Step 1. Given a morphism $\rho:\Psi^{a}_{a}(M)/\Psi^{a}_{a,\zeta}(M)\to G$,
construct a corresponding principal $G$-bundle $P_{\rho}$ with a connection
$A_{\rho}$.

Let $\Psi^{a}(M)$ be the space of classes of paths starting at $a$ on $M$. Two
paths $\delta_{1},\delta_{2}$ are identified if $\delta_{1}(1)=\delta_{2}(1)$
and $\delta_{2}^{-1}\delta_{1}\in\Psi^{a}_{a,\zeta(M)}$. This $\Psi^{a}(M)$ is
a principal bundle over $M$. Its structure group is
$\Psi^{a}_{a}(M)/\Psi^{a}_{a,\zeta}(M)$, and the projection is
$\tau:\Psi^{a}(M)\to M,\delta\mapsto\delta(1)$. An element
$\gamma\in\Psi^{a}_{a}(M)/\Psi^{a}_{a,\zeta}(M)$ acts on
$\delta\in\Psi^{a}(M)$ on the right by $\delta\mapsto\delta\gamma$.

Let $P_{\rho}=\Psi^{a}(M)\times_{\rho}G$ be the associated $G$-bundle by
identifying $(\delta,g)$ and $\left(\delta\gamma,\rho(\gamma^{-1})g\right)$ in
$\Psi^{a}(M)\times G$ for each
$\gamma\in\Psi^{a}_{a}(M)/\Psi^{a}_{a,\zeta}(M)$. $G$ acts on $P_{\rho}$ by
$[\delta,g]h=[\delta,gh]$. To define a connection $A_{\rho}$ on $P$, we can
construct horizontal lifts for any path $\alpha$ on $M$ as follows. For
arbitrary $[\delta,g]\in(P_{\rho})_{\alpha(0)}$ with $\delta(1)=\alpha(0)$, we
define the horizontal lift of $\alpha$ starting from $[\delta,g]\in P_{\rho}$
by $\tilde{\alpha}(t)=[\alpha_{t}\delta,g]$. Here, $\alpha_{t}$ is part of
$\alpha$ starting at $\alpha(0)$ and ending at $\alpha(t)$. i.e.
$\alpha_{t}:[0,1]\to M,s\mapsto\alpha(st)$.

We verify that this connection is well defined. Let $\alpha$ and $\beta$ be
two paths on $M$ with $\alpha^{\prime}(0)=\beta^{\prime}(0)$. By definition,
we have $\tilde{\alpha}^{\prime}(0)=\tilde{\beta}^{\prime}(0)$. Thus, the
horizontal lifts do induce a distribution for horizontal vectors. Also,
suppose $\alpha$ is an arbitrary path on $M$. For
$[\delta_{(1)},g_{1}],[\delta_{(2)},g_{2}]\in(P_{\rho})_{\alpha(0)}$, there
exists an $h\in G$ such that $[\delta_{(1)},g_{1}]=[\delta_{(2)},g_{2}]h$. Let
$\tilde{\alpha}_{(1)}$ and $\tilde{\alpha}_{(2)}$ be the horizontal lifts of
$\alpha$ from $[\delta_{(1)},g_{1}]$ and $[\delta_{(2)},g_{2}]$, respectively.
By assumption, we have
$[\delta_{(1)},g_{1}]=\left[\delta_{(2)}\rho(g_{1}h^{-1}g_{2}^{-1}),g_{1}\right]$,
which implies
${\delta_{(1)}}^{-1}\delta_{(2)}\rho\left(g_{1}h^{-1}g_{2}^{-1}\right)\in\Psi^{a}_{a,\zeta}(M)$.
Hence,
$\big{(}\alpha_{t}\delta_{(1)}\big{)}^{-1}\alpha_{t}\delta_{(2)}\rho\left(g_{1}h^{-1}g_{2}^{-1}\right)\in\Psi^{a}_{a,\zeta}(M)$
for all $t\in[0,1]$. So the lifts satisfy

$\tilde{\alpha}_{(1)}(t)=[\alpha_{t}\delta_{(1)},g_{1}]=[\alpha_{t}\delta_{(2)},g_{2}]h=\tilde{\alpha}_{(2)}(t)h.$

This proves that the horizontal lift does not depend on the choice of
representative of $[\delta,g]$, and is equivariant along the fiber.

Step 2. Verify that the connection $A_{\rho}$ is $\zeta$-flat.

######  Lemma 2.2.

The connection $A_{\rho}$ constructed above is $\zeta$-flat.

###### Proof.

Let $p\in M$ be an arbitrary point and $v_{1},v_{2}\in T_{p}M$ be arbitrary
linearly independent vectors. Suppose $\zeta(v_{1},v_{2})=c$. We can find a
local coordinate $\\{x_{1},\cdots,x_{N}\\}$ such that
$v_{1}=\frac{\partial}{\partial x_{1}}$, $v_{2}=\frac{\partial}{\partial
x_{2}}$, and $\zeta=c\,dx_{1}\wedge dx_{2}+\bar{\zeta}$, where $\bar{\zeta}$
is generated by other $dx_{i}\wedge dx_{j}$ locally except for $dx_{1}\wedge
dx_{2}$. Let $D_{(t)}$ be the parallelogram spanned by $\sqrt{t}v_{1}$ and
$\sqrt{t}v_{2}$ in the local coordinate system, and $\gamma_{(t)}=\partial
D_{(t)}$ be its boundary. Then we have $\int_{D_{(t)}}\zeta=t\,c$.

At arbitrary $[\delta,g]\in P_{\rho}$ on the fiber of $p$, let $v_{1}^{H}$ and
$v_{2}^{H}$ be the horizontal lift of $v_{1}$ and $v_{2}$, respectively. Then
the curvature

$F\left(v_{1}^{H},v_{2}^{H}\right)=\frac{\partial}{\partial
t}hol\left(\gamma_{(t)}\right).$

Here, $hol\left(\gamma_{(t)}\right)\in G$ denotes the holonomy along
$\gamma_{(t)}$ at $[\delta,g]$.

On the other hand,

$[\delta,g]hol\left(\gamma_{(t)}\right)=[\gamma_{(t)}\delta,g]=[\delta(\delta^{-1}\gamma_{(t)}\delta),g]=\left[\delta,g\left(g^{-1}\rho\left(\delta^{-1}\gamma_{(t)}\delta\right)g\right)\right].$

This implies

$hol\left(\gamma_{(t)}\right)=g^{-1}\rho\left(\delta^{-1}\gamma_{(t)}\delta\right)g.$

Note that $\delta^{-1}\gamma_{(t)}\delta$ is contractible and its base point
is $a$, i.e. $\delta^{-1}\gamma_{(t)}\delta\in\Psi^{a}_{a,0}(M)$. Since
$\Psi^{a}_{a,0}(M)/\Psi^{a}_{a,\zeta}(M)$ is equivalent to $\mathbb{R}/H$,
$\delta^{-1}\gamma_{(t)}\delta$ can be identified with
$\int_{D_{(t)}}\zeta=tc$ in $\mathbb{R}/H$. Let $\xi$ be an element of the Lie
algebra of $\Psi^{a}_{a}(M)/\Psi^{a}_{a,\zeta}(M)$ such that
$\mathrm{exp}(t\xi)$ generates the subgroup
$\Psi^{a}_{a,0}(M)/\Psi^{a}_{a,\zeta}(M)$ and $t$ can be identified with
$\mathbb{R}/H$. This $\xi$ is independent of the choice of $v_{1}$ and
$v_{2}$. Then we have

$F\left(v_{1}^{H},v_{2}^{H}\right)=\frac{\partial}{\partial
t}hol\left(\gamma_{(t)}\right)=c\mathrm{Ad}_{g^{-1}}\rho(\xi)=\zeta(v_{1},v_{2})\mathrm{Ad}_{g^{-1}}\rho(\xi).$

∎

Step 3. Show that the morphism $\rho\mapsto(P_{\rho},A_{\rho})$ is invariant
under conjugation.

######  Lemma 2.3.

Suppose $\rho$ and $\bar{\rho}$ are conjugate homomorphisms from
$\Psi^{a}_{a}(M)/\Psi^{a}_{a,\zeta}(M)$ to $G$. $(P_{\rho},A_{\rho})$ and
$(P_{\bar{\rho}},A_{\bar{\rho}})$ are principal bundles with $\zeta$-flat
connections constructed as above. Then $(P_{\rho},A_{\rho})$ and
$(P_{\bar{\rho}},A_{\bar{\rho}})$ are equivalent.

###### Proof.

Suppose $\bar{\rho}=g_{0}\rho g_{0}^{-1}$, we define an automorphism on
$\Psi^{a}(M)\times G$ by $(\delta,g)\mapsto(\delta,g_{0}g)$ and this
automorphism induces an isomorphism
$f:\Psi^{a}(M)\times_{\rho}G\to\Psi^{a}(M)\times_{\bar{\rho}}G$. For arbitrary
path $\alpha$ on $M$ and $[\delta,g]_{\rho}\in\Psi^{a}(M)$ such that
$\delta(1)=\alpha(0)$, the horizontal lift of $\alpha$ at $[\delta,g]_{\rho}$
is defined as $\tilde{\alpha}_{\rho}(t)=[\alpha_{t}\delta,g]_{\rho}$. On the
other hand, the horizontal lift of $\alpha$ at
$f\big{(}[\delta,g]_{\rho}\big{)}=[\delta,g_{0}g]_{\bar{\rho}}$ is defined as
$\tilde{\alpha}_{\bar{\rho}}(t)=[\alpha_{t}\delta,g_{0}g]_{\bar{\rho}}$. So we
have $\tilde{\alpha}_{\bar{\rho}}(t)=f\circ\tilde{\alpha}_{\rho}(t)$, which
implies $f_{*}$ sends horizontal vectors to horizontal vectors. Thus,
$f^{*}A_{\bar{\rho}}=A_{\rho}$. ∎

By this lemma, given any conjugacy classes $[\rho]$ of the morphisms from
$\Psi^{a}_{a}(M)/\Psi^{a}_{a,\zeta}(M)$ to $G$, there is a corresponding
$G$-bundle $P_{\rho}$ with a $\zeta$-flat connection $A_{\rho}$. It remains to
show that this correspondence is bijective.

Step 4. The morphism $[\rho]\mapsto(P_{\rho},A_{\rho})$ is injective.

######  Lemma 2.4.

Let $\rho,\bar{\rho}:\Psi^{a}_{a}(M)/\Psi^{a}_{a,\zeta}(M)\to G$. $\rho$ and
$\bar{\rho}$ are conjugate when $(P_{\rho},A_{\rho})$ and
$(P_{\bar{\rho}},A_{\bar{\rho}})$ are equivalent.

###### Proof.

Suppose $f:P_{\rho}\to P_{\bar{\rho}}$ is a $G$-bundle isomorphism and
$f^{*}A_{\bar{\rho}}=A_{\rho}$. Set $h:\Psi^{a}(M)\to G$ such that

$f\big{(}[\delta,e]_{\rho}\big{)}=[\delta,e]_{\bar{\rho}}h(\delta)$

where $\delta\in\Psi^{a}(M)$ and $e$ is the identity of $G$. For each
$\gamma\in\Psi^{a}_{a}(M)/\Psi^{a}_{a,\zeta}(M)$, we have

$\displaystyle[\delta,e]_{\bar{\rho}}h(\delta)\rho(\gamma)$
$\displaystyle=f\big{(}[\delta,e]_{\rho}\big{)}\rho(\gamma)=f\big{(}[\delta,e]_{\rho}\rho(\gamma)\big{)}$
$\displaystyle=f\big{(}[\delta\gamma,e]_{\rho}\big{)}=[\delta\gamma,e]_{\bar{\rho}}h(\delta\gamma)=[\delta,e]_{\bar{\rho}}\bar{\rho}(\gamma)h(\delta\gamma).$

This implies $\rho(\gamma)=h(\delta)^{-1}\bar{\rho}(\gamma)h(\delta\gamma)$.
We will prove that $h$ is a constant.

For arbitrary $\delta\in\Psi^{a}(M)$, let $\delta_{t}$ denote the part of the
path $\delta$ starting at $\delta(0)=a$ and ending at $\delta(t)$. i.e.
$\delta_{t}:[0,1]\to M,s\mapsto\delta(st)$. By the construction of the
connections, $[\delta_{t},e]_{\rho}$ and $[\delta_{t},e]_{\bar{\rho}}$ are
horizontal paths on $P_{\rho}$ and $P_{\bar{\rho}}\,$, respectively. The
projection of these two paths on $M$ are exactly $\delta$. By the definition
of $h$, we have

$f\big{(}[\delta_{t},e]_{\rho}\big{)}=[\delta_{t},e]_{\bar{\rho}}h(\delta_{t}).$

On the other hand, since $f^{*}A_{\bar{\rho}}=A_{\rho}$,
$f\big{(}[\delta_{t},e]_{\rho}\big{)}$ is a horizontal lift of $\delta$ on
$P_{\bar{\rho}}$ passing through $f\big{(}[\delta_{0},e]_{\rho}\big{)}$. As
horizontal curves are equivariant by right $G$-action,
$[\delta_{t},e]_{\bar{\rho}}h(\delta_{0})$ is also a horizontal lift of
$\delta$ on $P_{\bar{\rho}}$ passing through
$[\delta_{0},e]_{\bar{\rho}}h(\delta_{0})=f\big{(}[\delta_{0},e]_{\rho}\big{)}$.
Therefore, we have

$f\big{(}[\delta_{t},e]_{\rho}\big{)}=[\delta_{t},e]_{\bar{\rho}}h(\delta_{0}).$

Comparing the two equations, we have $h(\delta_{t})=h(\delta_{0})$ for all
$t\in[0,1]$. This implies $h(\delta)$ is equal to $h$ acting on the constant
path at $a$. So $h$ is a constant. Therefore, we have $\rho$ and $\bar{\rho}$
are conjugate. ∎

Step 5. The morphism $[\rho]\mapsto(P_{\rho},A_{\rho})$ is surjective.

The following lemma leads to surjectivity. It also holds when $\mathbb{R}/H$
is not a Lie group, which we will discuss later.

######  Lemma 2.5.

Suppose $\pi:P\to M$ be a principal $G$-bundle, and $A$ is a $\zeta$-flat
connection on $P$ with curvature $F=\Phi\zeta$. There exists
$\xi\in\mathfrak{g}$ such that for any contractible loop $\gamma$ starting at
$a$ and any oriented disk $D$ in $M$ with $\partial D=\gamma$ ($\partial D$
and $\gamma$ also have the same orientation), the holonomy along $\gamma$ is

$hol(\gamma)=-\mathrm{exp}\Big{(}\int_{D}\zeta\cdot\xi\Big{)}.$

###### Proof.

Take a point $u_{0}\in P$ on the fiber of $a$. Consider the holonomy bundle

$\hat{P}=\\{u\in P|\text{there exists horizontal path }\delta\text{ such that
}\delta(0)=u_{0},\delta(1)=u\\}.$

$\hat{P}$ is a principal bundle over $M$, and its structure group $\hat{G}$ is
the holonomy group of $P$ at $u_{0}$. Let $\hat{A}$ be the restriction of $A$
on $\hat{P}$. By the holonomy theorem of Ambrose and Singer [AS], the Lie
algebra of $\hat{G}$ is

$\hat{\mathfrak{g}}=\mathrm{span}\left\\{F\left(v_{1}^{H},v_{2}^{H}\right)\big{|}v_{1}^{H},v_{2}^{H}\text{
are horizontal vectors at }u\text{ for some }u\in\hat{P}\right\\}.$

By assumption, $F=\Phi\zeta$, or more precisely $F=\Phi(\pi^{*}\zeta)$. Since
$\Phi$ is covariantly constant, it is equal to some $\xi\in\hat{\mathfrak{g}}$
at any point in $\hat{P}$. Hence,
$F\left(v_{1}^{H},v_{2}^{H}\right)\in\mathbb{R}\xi$. So $\hat{\mathfrak{g}}$
is 1-dimensional and abelian. Then $\hat{A}=\xi\otimes\theta$ for some
$\theta\in\Omega^{1}(M,Ad\,\hat{P})$ and $d\theta=\hat{\pi}^{*}\zeta$, where
$\hat{\pi}:\hat{P}\to M$ is the projection.

For an arbitrary contractible loop $\gamma$ and a disk $D$ such that $\partial
D=\gamma$, there exists a contractible neighborhood $U\subset M$ of $D$. Then
$\hat{P}|_{U}=U\times\hat{G}$ is trivial. Let $\sigma:U\to\hat{P}|_{U}$ be a
local section and $\psi:\hat{P}|_{U}\to U\times\hat{G}$ be a trivialization
such that $\psi\circ\sigma(p)=(p,e)$ for $p\in M$. Then
$(\psi^{-1})^{*}\hat{A}=(\xi\otimes\sigma^{*}\theta,0)+(0,MC_{\hat{G}})$,
where $MC_{\hat{G}}:T\hat{G}\to\hat{\mathfrak{g}}$ is the Maurer-Cartan form
of $\hat{G}$ sending a vector to the corresponding invariant vector field.
Observe that $d(\sigma^{*}\theta)=\zeta$. The horizontal lift $\tilde{\gamma}$
of $\gamma$ with $\tilde{\gamma}(0)=\sigma(a)$ satisfies
$\psi\circ\tilde{\gamma}(t)=(\gamma(t),g(t))$ with $g(t)\in\hat{G}$ and
$g(0)=e$. Then

$\displaystyle
0=\hat{A}\big{(}\tilde{\gamma}^{\prime}(t)\big{)}\Big{|}_{\gamma(t_{0})}$
$\displaystyle=(\psi^{-1*}\hat{A})\big{(}\gamma^{\prime}(t),0\big{)}\Big{|}_{\big{(}\gamma(t_{0}),g(t_{0})\big{)}}+(\psi^{-1*}\hat{A})\big{(}0,g^{\prime}(t)\big{)}\Big{|}_{\big{(}\gamma(t_{0}),g(t_{0})\big{)}}$
$\displaystyle=(\sigma^{*}\theta)\big{(}\gamma^{\prime}(t)\big{)}\cdot\xi\Big{|}_{\big{(}\gamma(t_{0})\big{)}}+MC_{\hat{G}}\big{(}g^{\prime}(t)\big{)}\Big{|}_{\big{(}g(t_{0})\big{)}}.$

So we have

$g(t_{0})=-\mathrm{exp}\Big{(}\int_{0}^{t_{0}}(\sigma^{*}\theta)\big{(}\gamma^{\prime}(t)\big{)}dt\cdot\xi\Big{)},$

and the holonomy along $\gamma$ is

$hol(\gamma)=g(1)=-\mathrm{exp}\Big{(}\int_{\gamma}\sigma^{*}\theta\cdot\xi\Big{)}=-\mathrm{exp}\Big{(}\int_{D}\zeta\cdot\xi\Big{)}.$

∎

Now we prove the surjectivity.

######  Lemma 2.6.

Let $\pi:P\to M$ be a principal $G$-bundle with a $\zeta$-flat connection $A$.
There exists a morphism $\rho:\Psi^{a}_{a}(M)/\Psi^{a}_{a,\zeta}(M)\to G$ such
that $(P_{\rho},A_{\rho})$ and $(P,A)$ are equivalent.

###### Proof.

For each $\gamma\in\Psi^{a}_{a,\zeta}(M)$, there exists some disk $D$ such
that $\partial D=\gamma$, and $\int_{D}\zeta=0$. By Lemma 2.5, the holonomy
along $\gamma$ is
$hol(\gamma)=-\mathrm{exp}\Big{(}\int_{D}\zeta\cdot\xi\Big{)}=e$. So we can
define a morphism

$\rho:\Psi^{a}_{a}(M)/\Psi^{a}_{a,\zeta}(M)\to G,\quad\gamma\mapsto
hol(\gamma).$

We will show that $(P_{\rho},A_{\rho})$ and $(P,A)$ are equivalent.

Given $[\delta,g]\in P_{\rho}$, let $\tilde{\delta}$ be the horizontal lift of
$\delta$ in $P$ with $\tilde{\delta}(0)=u_{0}$. Set

$f:P_{\rho}\to P,\quad[\delta,g]\mapsto\tilde{\delta}(1)g.$

Suppose $[\delta_{(1)},g_{1}]$ and $[\delta_{(2)},g_{2}]$ represent the same
class in $P_{\rho}$, then there exists
$\gamma_{0}\in\Psi^{a}_{a}(M)/\Psi^{a}_{a,\zeta}(M)$ such that
$\delta_{(1)}=\delta_{(2)}\gamma_{0}$ in
$\Psi^{a}_{a}(M)/\Psi^{a}_{a,\zeta}(M)$ and
$g_{1}=\big{(}\rho(\gamma_{0})\big{)}^{-1}g_{2}$ in $G$. So
$\gamma_{0}^{-1}\delta_{(2)}^{-1}\delta_{(1)}\in\Psi^{a}_{a,\zeta}(M)$, and
its holonomy is trivial according to Lemma 2.5. Thus, the holonomy of
$\delta_{(2)}^{-1}\delta_{(1)}$ is $hol(\gamma_{0})=\rho(\gamma_{0})$. Let
$\tilde{\delta}_{(1)}$ and $\tilde{\delta}_{(2)}$ denote the horizontal lift
of $\delta_{(1)}$ and $\delta_{(2)}$, respectively, starting at $u_{0}$. Since
$\tilde{\delta}_{(1)}$ and the horizontal lift of
$\delta_{(2)}\gamma_{0}=\delta_{(2)}\delta_{(2)}^{-1}\delta_{(1)}$ starting at
$u_{0}$ have the same ending point, so do $\tilde{\delta}_{(1)}$ and the
horizontal lift of $\delta_{(2)}$ starting at $u_{0}\cdot hol(\gamma_{0})$.
Then the ending points of $\tilde{\delta}_{(1)}$ and $\tilde{\delta}_{(2)}$
satisfy $\tilde{\delta}_{(1)}(1)=\tilde{\delta}_{(2)}(1)\cdot
hol(\gamma_{0})$. Therefore, by definition,

$f\left([\delta_{(1)},g_{1}]\right)=\tilde{\delta}_{(1)}(1)g_{1}=\tilde{\delta}_{(2)}(1)\rho(\gamma_{0})g_{1}=\tilde{\delta}_{(2)}(1)g_{2}=f(\left[\delta_{(2)},g_{2}]\right).$

Hence, $f$ is well defined. It is straightforward to check that $f$ is a
$G$-bundle isomorphism.

Finally, we show that $f^{*}A=A_{\rho}$. For an arbitrary path $\alpha$ on $M$
and $[\delta,g]\in P_{\rho}$ such that $\delta(1)=\alpha(0)$, the horizontal
lift of $\alpha$ at $[\delta,g]$ is $\tilde{\alpha}(t)=[\alpha_{t}\delta,g]$.
Here, $\alpha_{t}$ is defined as before, which is the part of $\alpha$
starting at $\alpha(0)$ and ending at $\alpha(t)$. So
$f\left([\alpha_{t}\delta,g]\right)$ is the ending point of the path on $P$
which is the horizontal lift of $\alpha_{t}\delta$ starting at $u_{0}g$. In
other words, $f\circ\tilde{\alpha}(t)$ is on the horizontal lift of $\alpha$
starting at $f\left([\delta,g]\right)$ and on the fiber of $\alpha(t)$.
Therefore, $f\circ\tilde{\alpha}$ is a horizontal path on $P$. Hence, $f_{*}$
sends horizontal vectors to horizontal vectors and $f^{*}(A)=A_{\rho}$. ∎

Combining the lemmas above, we have proved that
$[\rho]\mapsto(P_{\rho},A_{\rho})$ is an isomorphism.

Case 3. $\mathbb{R}/H$ is not a Lie group

Now we turn to the case that $\mathbb{R}/H$ is not a Lie group. In this case
$H^{+}$ is non-empty and has no minimal number.

######  Lemma 2.7.

When $H^{+}$ is non-empty and has no minimal number, a $\zeta$-flat connection
is flat.

###### Proof.

By Theorem 2.5, there exists $\xi\in\mathfrak{g}$ such that for any
contractible loop $\gamma$ and disk $D$ with $\partial D=\gamma$, we have
$hol(\gamma)=-\mathrm{exp}\Big{(}\int_{D}\zeta\cdot\xi\Big{)}$. If $\xi\neq
0$, there exists some small enough $t_{0}$ such that $\mathrm{exp}(t\xi)\neq
e$, the identity element of $G$, for any $0<t<t_{0}$.

Since $H^{+}$ has no minimal number, there exists a closed sphere
$\Sigma\subset M$ such that $\int_{\Sigma}\zeta=t$ for some $0<t<t_{0}$. Let
$\gamma$ be a constant loop at some point $p\in\Sigma$, and
$D=\Sigma\setminus\\{p\\}$. Then $\partial D=\gamma$ so that
$hol(\gamma)=-\mathrm{exp}\Big{(}\int_{D}\zeta\cdot\xi\Big{)}$. But
$hol(\gamma)=e$ as $\gamma$ is the identity loop. On the other hand,
$\int_{D}\zeta=\int_{\Sigma}\zeta=t$. This implies $\mathrm{exp}(t\xi)=e$,
which is a contradiction.

Therefore, $\xi$ must be zero and the connection is flat. ∎

The classification of $G$-bundles with flat connections can be represented by
the conjugacy classes of morphisms from $\pi_{1}(M)\to G$ (c.f. [Morita]
Theorem 2.9). So we have proved the theorem in this case.

Below we will demonstrate the theorem for $M=T^{4}$ and $G=U(1)=S^{1}$.

######  Example 2.8.

We consider principal $U(1)$ bundles or circle bundles over $T^{4}$. We
describe $T^{4}$ as $\mathbb{R}^{4}/\sim$, with the identification
$(x_{1},x_{2},x_{3},x_{4})\sim(x_{1}+a,x_{2}+b,x_{3}+c,x_{4}+d),a,b,c,d\in\mathbb{Z}$,
and $S^{1}$ as $\\{z\in\mathbb{C}\mid|z|=1\\}$. Let $\zeta=c_{1}dx_{1}\wedge
dx_{2}+c_{2}dx_{3}\wedge dx_{4}$ be a closed 2-form with
$c_{1},c_{2}\in\mathbb{R}\setminus\\{0\\}$. Then the equivalent classes of
circle bundles with a $\zeta$-flat connection are the conjugacy classes of
morphisms $\rho:\Gamma\to S^{1}$, where
$\Gamma=\Psi^{a}_{a}(M)/\Psi^{a}_{a,\zeta}(M)$. Since $S^{1}$ is an abelian
group, this classification of $\zeta$-flat connections is just the morphisms
$\rho:\Gamma\to S^{1}$.

Since $\pi_{2}(T^{4})$ is trivial, $\Gamma$ is an $\mathbb{R}$-extension of
$\pi_{1}(T^{4})$. To describe its group structure explicitly, let $a_{i}$ be
the straight line path in $\mathbb{R}^{4}$ starting at the origin and ending
at $(0,\ldots,1,\ldots,0)$, where the $i$-th coordinate is 1 and the other
coordinates are 0. When projected to $T^{4}$, $a_{1},\ldots,a_{4}$ become the
generators of $\pi_{1}(T^{4})$. Although $\pi_{1}(T^{4})$ is abelian,
$a_{1},\ldots,a_{4}$ may no longer commute in its $\mathbb{R}$-extension
$\Gamma$. That is, $a_{j}^{-1}a_{i}^{-1}a_{j}a_{i}$ is some non-trivial class
of a contractible loop. For each contractible loop $b$ of $T^{4}$, there is an
oriented disk $D$ such that $\partial D=b$, and $\partial D$ has the same
orientation as $b$. Let $|b|=\int_{D}\zeta$, then $|b|$ is independent of the
choice of $D$. Moreover, contractible loops $b$ and $b^{\prime}$ represent the
same class in $\Gamma$ if and only if $|b|=|b^{\prime}|$. So a class of
contractible loops $b\in\Gamma$ can be represented by a real number $|b|$.
Therefore, $\Gamma$ is generated by $a_{1},\ldots,a_{4},b$ and their
multiplication are defined as

$\displaystyle|a_{2}^{-1}a_{1}^{-1}a_{2}a_{1}|=c_{1},$
$\displaystyle|a_{4}^{-1}a_{3}^{-1}a_{4}a_{3}|=c_{2},$
$\displaystyle|a_{j}^{-1}a_{i}^{-1}a_{j}a_{i}|=0,\text{ for other }i,j,$
$\displaystyle a_{i}b=ba_{i}$ $\displaystyle|b^{\prime}b|=|b|+|b^{\prime}|.$

The possible morphisms of $\rho:\Gamma\to S^{1}$ is dependent on whether
$\frac{c_{1}}{c_{2}}\in\mathbb{Q}$. Since $S^{1}$ is abelian,

$\rho(c_{1})=\rho(a_{2}^{-1})\rho(a_{1}^{-1})\rho(a_{2})\rho(a_{1})=\rho(a_{2})^{-1}\rho(a_{2})\rho(a_{1})^{-1}\rho(a_{1})=1.$

Similarly $\rho(c_{2})=1$, so $\rho(pc_{1}+qc_{2})=1$ for any
$p,q\in\mathbb{Z}$.

When $\frac{c_{1}}{c_{2}}\in\mathbb{Q}$,
$\\{\rho(pc_{1}+qc_{2})|p,q\in\mathbb{Z}\\}$ has a minimal positive number
$c_{0}$. Then $\rho(b)$ must have the form $e^{\frac{2\pi n|b|i}{c_{0}}}$ for
some $n\in\mathbb{Z}$. If $\rho(b)=e^{\frac{2\pi n|b|i}{c_{0}}}$, the Euler
class of the circle bundle is $\frac{n}{c_{0}}\zeta$, and the definition of
$\rho(a_{i})$ determines the connection chosen.

When $\frac{c_{1}}{c_{2}}\notin\mathbb{Q}$,
$\\{\rho(pc_{1}+qc_{2})|p,q\in\mathbb{Z}\\}$ is dense in $\mathbb{R}$. So
$\rho(b)$ must be 1 for any $|b|\in\mathbb{R}$. Thus, the classification is
only dependent on $\rho(a_{i})$, and becomes equivalent to the classification
of flat connections. Actually, the Euler class in this case is $c\,\zeta$ for
some $c\in\mathbb{R}$ but it must be integral. So the only possible $c$ is 0,
i.e. every $\zeta$-flat connection is flat.

######  Remark 2.9.

The above example shows that different $\zeta$ can result in different
classifications of $\zeta$-flat principal bundles. In particular, for
symplectic manifolds, the classification of symplectically flat bundles can
vary with the symplectic structure.

##  3 Curvature functionals whose zeroes are symplectically flat

Let $P$ be a principal $G$-bundle over $M$. An inner product on the Lie
algebra of the structure group $G$ together with a Riemannian metric $g$ on
$M$ induces an inner product $\langle-,-\rangle$ on $\Omega^{*}(M,Ad\,P)$. The
squared norm of the curvature $F\in\Omega^{2}(M,Ad\,P)$ gives the Yang-Mills
functional

$\displaystyle\|F\|^{2}=\langle F,F\rangle=\int_{M}F\wedge*F\,.$ (3.1)

Flat connections, i.e. $F(A)=0$, are the zeroes of the Yang-Mills functional.
In this section, we will write down functionals whose zeroes correspond to
symplectically-flat and $\zeta$-flat connections and study some of their
properties. We will describe first functionals related to symplectically flat
connections. But before doing so, we will set our conventions and give a very
brief review of the differentials of the TTY algebra [TTY] of primitive forms
on symplectic manifolds and their twisting. Further details can be found in
[TZ]*Section 2.

###  3.1 Review of the twisted differentials of the TTY-algebra

Let $(M^{2n},\omega)$ be a symplectic manifold. Under the Lefschetz
decomposition, a differential form $\eta_{k}\in\Omega^{k}(M)$ can be expressed
as a polynomial in $\omega$:

$\displaystyle\eta_{k}=\beta_{k}+\omega\wedge\beta_{k-2}+\ldots+\omega^{p}\wedge\beta_{k-2p}+\ldots$
(3.2)

where $\\{\beta_{k},\beta_{k-2},\ldots,\beta_{k-2p},\ldots\\}$ are primitive
forms determined uniquely by $\eta_{k}$ and $\omega$. We use $P^{k}(M)$ to
denote the space of primitive $k$-forms, and $\Pi:\Omega^{k}(M)\to P^{k}(M)$
to denote the projection of forms into their primitive component.

Let

$\displaystyle L$ $\displaystyle:\Omega^{k}(M)\to~{}\Omega^{k+2}(M)$
$\displaystyle\qquad\alpha\quad~{}\mapsto\quad\omega\wedge\alpha$

be the Lefschetz operator. Given $g$ a Riemannian metric which is compatible
with $\omega$, we have the dual Lefschetz operator $\Lambda=L^{*}$. Actually,
$\Lambda$ is independent of the choice of the metric and can be defined
without it. The kernel of $\Lambda$ is exactly $P^{*}(M)$.

In [TY2], Tseng and Yau observed that
$d[\omega^{r}P^{k}(M)]\subset\omega^{r}P^{k+1}(M)\oplus\omega^{r+1}P^{k-1}(M)$.
This implies the decomposition of the exterior derivative
$d=\partial_{+}+\omega\wedge\partial_{-}$ where

$\partial_{+}:\omega^{r}P^{k}(M)\to\omega^{r}P^{k+1}(M),\qquad\partial_{-}:\omega^{r}P^{k}(M)\to\omega^{r}P^{k-1}(M)\,$

and ($\partial_{+}$, $\partial_{-}$) satisfy

$\partial_{+}^{2}=0\,,\quad\partial_{-}^{2}=0\,,\quad\omega\wedge(\partial_{+}\partial_{-}+\partial_{-}\partial_{+})=0\,.$

Based on this, they constructed an elliptic complex of primitive forms.

(3.7)

In [TZ]*Section 2.2, we showed that the covariant derivative $d_{A}$ when
acting on the space of twisted forms $\Omega^{*}(M,Ad\,P)$ also has a similar
decomposition $d_{A}={\partial_{+A}\,}+\omega\wedge{\partial_{-A}\,}$ where

${\partial_{+A}\,}:\omega^{r}P^{k}(M,Ad\,P)\to\omega^{r}P^{k+1}(M,Ad\,P),\qquad{\partial_{-A}\,}:\omega^{r}P^{k}(M,Ad\,P)\to\omega^{r}P^{k-1}(M,Ad\,P).$

But in general, $\partial_{+A}^{2}\,,\,\partial_{-A}^{2}$ and
$\omega\wedge({\partial_{+A}\,}{\partial_{-A}\,}+{\partial_{-A}\,}{\partial_{+A}\,})$
are non-trivial and dependent on the curvature $F=dA+A\wedge A$.

###  3.2 Primitive Yang-Mills functional

On a symplectic manifold $(M^{2n},\omega)$, the curvature 2-form $F$ can be
Lefschetz decomposed and expressed as $F=F_{p}+\Phi\,\omega$, where $F_{p}\in
P^{2}(M,Ad\,P)$ and $\Phi\in\Omega^{0}(M,Ad\,P)$. In this section, we will
assume that $\dim M=2n\geq 4$ so that $F_{p}$ is non-trivial. With the
Lefschetz decomposition of $F$, the Yang-Mills functional (3.1) decomposes
into the sum of two components

$\|F\|^{2}=\|F_{p}\|^{2}+\|\Phi\,\omega\|^{2}.$

As we shall see, a critical point of the Yang-Mills functional may not
necessarily be a critical point of both $\|F_{p}\|^{2}$ and
$\|\Phi\,\omega\|^{2}$. Hence, it is interesting to consider these two
functionals separately. The first component $\|F_{p}\|^{2}$ is the primitive
Yang-Mills functional introduced in Definition 1.3. It turns out to have many
properties similar to those of the Yang-Mills functional. Due to this
similarity, our description here will mirror that of Atiyah-Bott’s for Yang-
Mills connections in [Atiyah-Bott]*Section 4.

To begin, recall that a connection $A$ is a Yang-Mills connection if and only
if $d^{*}_{A}F=0$. The equation for the primitive Yang-Mills connections is
similar.

######  Proposition 3.1.

Let $P$ be a principal bundle over a symplectic manifold $(M,\omega)$, and $A$
is a connection over $P$. Then $A$ is a primitive Yang-Mills connection if and
only if $d_{A}^{*}F_{p}=-*d_{A}*F_{p}=0$.

###### Proof.

Let $\eta\in\Omega^{1}(M,Ad\,P)$ be an infinitesimal variation of $A$. Set
$A_{t}=A+t\eta$ and $F_{t}$ as the curvature of $A_{t}$. Then

$F_{t}=F(A)+td_{A}\eta+\frac{1}{2}t^{2}[\eta,\eta],$

and its primitive part

$F_{p}(A_{t})=F_{p}(A)+t\partial_{+A}\eta+\frac{1}{2}t^{2}\Pi[\eta,\eta].$

So we have

$\displaystyle\|F_{p}(A_{t})\|^{2}=\|F_{p}(A)\|^{2}+2t\langle\partial_{+A}\eta,F_{p}(A)\rangle+t^{2}\left(\|\partial_{+A}\eta\|^{2}+\langle
F_{p}(A),\Pi[\eta,\eta]\rangle\right)+o(t^{2})$ (3.8)

Thus, $A$ is a critical point of $\|F_{p}\|^{2}$ is equivalent to
$\langle\partial_{+A}\eta,F_{p}(A)\rangle=0$ for any $\eta$. Moreover, since
$\omega\wedge\partial_{-A}\eta$ is orthogonal to primitive forms, we have

$\displaystyle\langle\eta,d_{A}^{*}F_{p}(A)\rangle=\langle
d_{A}\eta,F_{p}(A)\rangle=\langle\partial_{+A}\eta,F_{p}(A)\rangle\,.$ (3.9)

This implies that $A$ is a primitive Yang-Mills connection if and only if
$d_{A}^{*}F_{p}(A)=0$.

Finally, on even dimensional manifolds, $d_{A}^{*}=-*d_{A}*$. This completes
the proof. ∎

######  Remark 3.2.

Observe that $\partial_{+A}^{*}F_{p}=d_{A}^{*}F_{p}$ according to (3.9). On
the other hand,

$\partial_{+A}F_{p}=\Pi\circ d_{A}F=0.$

Thus, $A$ is a primitive Yang-Mills connection if and only if $F_{p}$ is a
harmonic form of the operator $\partial_{+A}$. This is analogous to the
statement that $A$ is a Yang-Mills connection if and only if $F$ is a harmonic
form of $d_{A}$.

For the Yang-Mills functional, the Hessian of the functional at a critical
point gives the following quadratic form:

$Q(\eta,\eta)=\langle d_{A}^{*}d_{A}\eta+*[*F,\eta],\eta\rangle.$

The quadratic form for the primitive Yang-Mills functional has a similar form.

######  Proposition 3.3.

The quadratic form defined by the Hessian of the primitive Yang-Mills
functional at a critical point is given by

$Q_{p}(\eta,\eta)=\langle
d_{A}^{*}\partial_{+A}\eta+*[*F_{p},\eta],\eta\rangle.$

###### Proof.

By (3.8),

$Q_{p}(\eta,\eta)=\|\partial_{+A}\eta\|^{2}+\langle
F_{p}(A),\Pi[\eta,\eta]\rangle.$

As $\partial_{+A}\eta$ is a primitive 2-form, it is orthogonal to
$\omega\wedge\partial_{-A}\eta$. So

$\|\partial_{+A}\eta\|^{2}=\langle\partial_{+A}\eta,d_{A}\eta\rangle=\langle
d_{A}^{*}\partial_{+A}\eta,\eta\rangle.$

For the remaining term, we have

$\displaystyle\langle F_{p}(A),\Pi[\eta,\eta]\rangle$ $\displaystyle=\langle
F_{p}(A),[\eta,\eta]\rangle$ $\displaystyle=\int_{M}[\eta,\eta]\wedge*F_{p}$
$\displaystyle=\int_{M}\eta\wedge[\eta,*F_{p}]$
$\displaystyle=-\int_{M}\eta\wedge**^{-1}[*F_{p},\eta]$
$\displaystyle=\int_{M}\eta\wedge**[*F_{p},\eta]$
$\displaystyle=\langle*[*F_{p},\eta],\eta\rangle.$

The first line of equation holds because the primitive 2-form $F_{p}(A)$ is
orthogonal to $\omega$. The fifth line holds because $[*F_{p},\eta]$ is a
1-form and $*=-*^{-1}$ when acting on odd-degree forms. ∎

$Q(\eta,\eta)$ induces an operator $L_{A}=d_{A}^{*}d_{A}+*[*F,-]$ on
$\Omega^{*}(M,Ad\,P)$. $\eta\in\Omega^{1}(M,Ad\,P)$ is a tangent vector on the
space of Yang-Mills connections at $A$ if and only if $L_{A}\eta=0$. This also
works for the operator induced by $Q_{p}(\eta,\eta)$.

######  Theorem 3.4.

Let $(L_{p})_{A}=d_{A}^{*}\partial_{+A}+*[*F_{p},-]$ be the operator
$\Omega^{*}(M,Ad\,P)$ induced by $Q_{p}(\eta,\eta)$. Suppose $A$ is a
primitive Yang-Mills connection, then $\eta\in\Omega^{1}(M,Ad\,P)$ is a
tangent vector on the space of primitive Yang-Mills connections if and only if
$(L_{p})_{A}\eta=0$.

###### Proof.

Let $A_{t}=A+t\eta+o(t)$. Since $\frac{\partial}{\partial
t}(d_{A_{t}})|_{t=0}=[\eta,-]$ and $\frac{\partial}{\partial
t}\big{(}F_{p}(A_{t})\big{)}|_{t=0}=\partial_{+A}\eta$, we have

$\frac{\partial}{\partial
t}\big{(}*d_{A_{t}}*F_{p}(A_{t})\big{)}|_{t=0}=*[\eta,*F_{p}(A)]+*d_{A}*(\partial_{+A}\eta)=-(L_{p})_{A}\eta.$

So $A_{t}$ is a path on the space of primitive Yang-Mills connections if and
only if $(L_{p})_{A}\eta=0$. ∎

As we mentioned in Remark 3.2, $\partial_{+A}^{*}=d_{A}^{*}$ when acting on
primitive 2-forms. So $(L_{p})_{A}$ can also be written as
$\partial_{+A}^{*}\partial_{+A}+*[*F_{p},-]$. On the other hand,
$A_{t}=A+t\eta+o(t)$ are all gauge equivalent to $A$ if and only if $\eta$ is
$d_{A}$-exact. As $d_{A}={\partial_{+A}\,}$ when acting on 0-forms, the
complement of $d_{A}$-exact forms can be chosen as $\partial_{A+}^{*}$-closed
forms. Thus, the tangent space of primitive Yang-Mills connections up to gauge
equivalence can be identified with the space of $\eta\in\Omega^{1}(M,Ad\,P)$
satisfying $(L_{p})_{A}\eta=0$ and $\partial_{A+}^{*}\eta=0$. Or equivalently,
$\partial_{A+}^{*}\eta=0$ and

$\partial_{+A}^{*}\partial_{+A}\eta+\partial_{+A}\partial_{A+}^{*}\eta+*[*F_{p},\eta]=0.$

The above operator acting on $\eta$ is elliptic because $\eqref{primcomplex}$
is an elliptic complex. Therefore, the solution space of primitive Yang-Mills
connections up to gauge equivalence is finite dimensional.

By the same arguments, we can prove similar properties for
$\|\Phi\,\omega\|^{2}$:

  * •

$A$ is a critical point of $\|\Phi\,\omega\|^{2}$ if and only if
$d_{A}^{*}(\Phi\,\omega)=0$. Moreover,

$d_{A}^{*}(\Phi\,\omega)=-*d_{A}*(\Phi\,\omega)=-\frac{1}{(n-1)!}*d_{A}(\Phi\,\omega^{n-1})=-\frac{1}{(n-1)!}*[(d_{A}\Phi)\omega^{n-1}].$

So these equivalent statements hold if and only if
$(d_{A}\Phi)\omega^{n-1}=0$. But since the map
$\omega^{n-1}\\!:\Omega^{1}\to\Omega^{2n-1}$ is an isomorphism, an equivalent
condition for a connection $A$ to be a critical point of
$\|\Phi\,\omega\|^{2}$ is $d_{A}\Phi=0\,$.

  * •

The quadratic form defined by the Hessian of $\|\Phi\,\omega\|^{2}$ at a
critical point is given by

$Q_{\Phi}(\eta,\eta)=\langle
d_{A}^{*}(\omega\partial_{-A}\eta)+*[*(\Phi\,\omega),\eta],\eta\rangle.$

  * •

Let $(L_{\Phi})_{A}=d_{A}^{*}(\omega\partial_{-A})+*[*(\Phi\,\omega),-]$ be
the operator acting on $\Omega^{*}(M,Ad\,P)$ induced by $Q_{\Phi}(\eta,\eta)$.
Suppose $A$ is a critical point of $\|\Phi\,\omega\|^{2}$, then $\eta$ is a
tangent vector on the space of the critical points of $\|\Phi\,\omega\|^{2}$
if and only if $(L_{\Phi})_{A}\eta=0$.

######  Remark 3.5.

However, the critical points of $\|\Phi\,\omega\|^{2}$ modulo gauge
equivalence can not be expressed as the kernel of some elliptic operator, as
$d_{A}\Phi=0$ is a weak condition. This makes the functional $\|F_{p}\|^{2}$
more appealing as compared to $\|\Phi\,\omega\|^{2}$.

To summarize, the conditions for the critical points of the three functionals
– $\|F\|^{2}$, $\|F_{p}\|^{2}$, and $\|\Phi\,\omega\|^{2}$ – are given in
Table 1. These conditions immediately give the following proposition.

Functional | Critical point $A$
---|---
$\|F\|^{2}$ (Yang-Mills) | $d_{A}^{*}F=0$
$\|F_{p}\|^{2}$ (primitive Yang-Mills) | $d_{A}^{*}F_{p}=0$
$\|\Phi\,\omega\|^{2}$ |  $d_{A}^{*}(\Phi\,\omega)=0$, or equivalently $d_{A}\Phi=0$.
Table 1: Three functionals and the conditions for their critical points.

######  Proposition 3.6.

If $A$ is a critical point of two of the functionals among
$\|F\|^{2},\|F_{p}\|^{2}$ and $\|\Phi\,\omega\|^{2}$, it is also a critical
point of the third one.

In the following, we will explore the relations between the critical points of
$\|F_{p}\|^{2}$ versus that of $\|F\|^{2}$.

######  Example 3.7.

When $A$ is a symplectically flat connection, it is Yang-Mills. Since
$F_{p}=0$ in this case, it is a primitive Yang-Mills connection, hence also a
critical point of $\|\Phi\,\omega\|^{2}$.

######  Example 3.8.

A Hermitian Yang-Mills connection $A$ is a Yang-Mills connection. Its
curvature can be written as $F_{p}^{1,1}+c\,I\omega$, where $F_{p}^{1,1}$ is a
primitive $(1,1)$-form, $c$ is a constant and $I$ is the identity map on
$Ad\,P$. So $\Phi=c\,I$ is $d_{A}$-closed, which implies $A$ is a critical
point of $\|\Phi\,\omega\|^{2}$. Therefore, it is also a primitive Yang-Mills
connection.

More generally, we have the following statement.

######  Lemma 3.9.

Suppose $P$ is a principal $G$-bundle over a closed manifold $M$ with a
connection $A$. Let $\Lambda$ be the dual Lefschetz operator and
$\Delta_{A}=d_{A}d_{A}^{*}+d_{A}^{*}d_{A}$. When the curvature $F$ satisfies
$[\Lambda,\Delta_{A}]$F=0, $A$ is Yang-Mills implies that it is a primitive
Yang-Mills connection. If in addition $G$ is an abelian group, $A$ is a
primitive Yang-Mills connection also implies that it is Yang-Mills.

###### Proof.

When $A$ is Yang-Mills, $\Delta_{A}F=0$. Write $F=F_{p}+\Phi\,\omega$ by the
Lefschetz decomposition. Then we have

$d_{A}^{*}d_{A}\Phi=\Delta_{A}\Phi=\Delta_{A}\Lambda F=\Lambda\Delta_{A}F=0.$

It follows that $d_{A}\Phi=0$. By Proposition 3.6, $A$ is also a primitive
Yang-Mills connection.

Conversely, when $A$ is a primitive Yang-Mills connection, $F_{p}$ is a
harmonic form of $\partial_{+A}$. If in addition $G$ is abelian, $F$ and
$F_{p}$ are invariant under each fiber. So they can be treated simply as
2-forms over $M$. In this case $d_{A}$ and $\partial_{+A}$ are just $d$ and
$\partial_{+}$, respectively. Then we can say that $F_{p}\in P^{2}(M)$ is a
harmonic form of $\partial_{+}$.

On the other hand, since $F\in\Omega^{2}(M)$ is $d$-closed, there exists some
$\xi\in\Omega^{1}(M)$ such that $F+d\xi$ is a harmonic form of $d$. By the
discussion of the previous part, we have $F_{p}+\partial_{+}\xi=\Pi(F+d\xi)$
is also $\partial_{+A}$-harmonic.

When $\dim M\geq 6$, by the ellipticity of (3.7), the $\partial_{+A}$-harmonic
forms, $F_{p}$ and $F_{p}+\partial_{+}\xi$, must be identical. This implies
that $d\xi=f\omega$ for some function $f\in\Omega^{0}(M)$. Then
$df\wedge\omega=d^{2}\xi=0$ and $f$ must be a constant. Since $[\omega]\in
H^{2}(M)$ is non-trivial for closed manifolds, $f=0$. Thus, $F$ is harmonic
and $A$ is Yang-Mills.

When $\dim M=4$, to use the ellipticity of 3.7, we need to show that $F_{p}$
and $F_{p}+\partial_{+}\xi$ are $\partial_{+}\partial_{-}$ and
$\partial_{+}^{*}$-closed, where the latter closeness has been proved. As
$dF=0$, $\omega(\partial_{-}F_{p}+\partial_{+}\Phi)=0$. So
$\partial_{+}\partial_{-}F_{p}=-\partial_{+}^{2}\Phi=0$. Similarly
$\partial_{+}\partial_{-}(F_{p}+\partial_{+}\xi)=0$. Then applying the same
arguments as above, we have $F_{p}=F_{p}+\partial_{+}\xi$ and it leads to that
$A$ is Yang-Mills. ∎

######  Corollary 3.10.

For a principal bundle over a Kähler manifold, we have the Hodge decomposition
$d_{A}=\partial_{A}+\bar{\partial}_{A}$. If the curvature of a Yang-Mills
connection satisfies
$\left[(\partial_{A}^{*})^{2}-(\bar{\partial}_{A}^{*})^{2}\right]F=0$, then
this connection is a primitive Yang-Mills connection.

###### Proof.

$[\Lambda,\Delta_{A}]=[\Lambda,[d_{A},d_{A}^{*}]]=[[\Lambda,d_{A}],d_{A}^{*}]+[d_{A},[\Lambda,d_{A}^{*}]].$

The second term is 0 because for any $\xi,\eta\in\Omega^{*}(M,Ad\,P)$, we have

$\langle[\Lambda,d_{A}^{*}]\xi,\eta\rangle=\langle\xi,[d_{A},L]\eta\rangle=0.$

For the first term, by the Kähler identities, we have

$[[\Lambda,d_{A}],d_{A}^{*}]=[i(\bar{\partial}_{A}^{*}-\partial_{A}^{*}),\partial_{A}^{*}+\bar{\partial}_{A}^{*}]=2i[(\partial_{A}^{*})^{2}-(\bar{\partial}_{A}^{*})^{2}].$

So by assumption it vanishes when acting on $F$. In this case, a Yang-Mills
connection is a primitive Yang-Mills connection. ∎

If in addition the structure group is abelian, $\partial_{A}^{*}$ and
$\bar{\partial}_{A}^{*}$ acting on $F\in\Omega^{2}(M,Ad\,P)$ can be viewed as
$\partial^{*}$ and $\bar{\partial}^{*}$ acting on $F\in\Omega^{2}(M)$
respectively. As $(\partial^{*})^{2}=(\bar{\partial}^{*})^{2}=0$, we have the
following statement.

######  Corollary 3.11.

If $P$ is a principal bundle over a Kähler manifold with an abelian structure
group, then the Yang-Mills connections and primitive Yang-Mills connections
are the same.

In the following, we give two examples that make clear that the critical
points of the Yang-Mills and primitive Yang-Mills functionals are in general
different. The first demonstrate this for abelian solutions on $T^{4}$. The
second shows that the BPST Yang-Mills instanton solutions on $\mathbb{R}^{4}$
[BPST] is not a primitive Yang-Mills solution with respect to the standard
symplectic structure.

######  Example 3.12.

Let $T^{4}=\mathbb{R}^{4}/2\pi\mathbb{Z}^{4}$ be a 4-torus, and
$\omega=dx_{1}\wedge dx_{2}+dx_{3}\wedge dx_{4}$ be its symplectic form. Take
the Riemannian metric to be

$g=dx_{1}^{2}+dx_{2}^{2}+\frac{1}{f}dx_{3}^{2}+fdx_{4}^{2}$

where $f=\dfrac{3+2\sin 2x_{2}\cos x_{3}}{1-\frac{1}{2}\sin 2x_{2}\cos
x_{3}}$. Construct a circle bundle $X=\mathbb{R}^{5}/\sim$ over $T^{4}$ by
identifying

$\displaystyle x_{i}\sim x_{i}+2\pi n_{i}\,,\qquad y\sim y+2\pi
n_{5}-n_{2}x_{3}\,$

for $i=1,\ldots,4,$ and $n_{i},n_{5}\in\mathbb{Z}$, Set the connection as

$A=dy+\frac{1}{2\pi}x_{2}dx_{3}+\frac{1}{4\pi}\sin 2x_{2}\sin x_{3}dx_{1}.$

Then the curvature

$F=-\frac{1}{2\pi}\cos 2x_{2}\sin
x_{3}dx_{1}dx_{2}+\frac{1}{2\pi}(1-\frac{1}{2}\sin 2x_{2}\cos
x_{3})dx_{1}dx_{3}$

can be treated as a 2-form over $M$. As

$\displaystyle d*F$ $\displaystyle=\frac{1}{2\pi}d[-\cos 2x_{2}\sin
x_{3}dx_{3}dx_{4}-(1-\frac{1}{2}\sin 2x_{2}\cos x_{3})fdx_{2}dx_{4}]$
$\displaystyle=\frac{1}{2\pi}d[-\cos 2x_{2}\sin x_{3}dx_{3}dx_{4}-(3+2\sin
2x_{2}\cos x_{3})dx_{2}dx_{4}]$ $\displaystyle=0,$

$F$ is $d$-harmonic so that $A$ is Yang-Mills. However, under the Lefschetz
decomposition $F=F_{p}+\Phi\,\omega$, we have $\Phi=-\frac{1}{4\pi}\cos
2x_{2}\sin x_{3}$, which is not $d$-closed. This implies that $A$ is not a
primitive Yang-Mills connection.

On the other hand, by the discussion in the last paragraph of the proof of
Lemma 3.9, $F_{p}\in P^{2}(M)$ is $\partial_{+}\partial_{-}$-closed. By the
ellipticity of (3.7), there exists some $\xi\in P^{1}(M)$ such that
$F_{p}+\partial_{+}\xi$ is $\partial_{+}^{*}$-closed. Consider the connection
$A^{\prime}=A+\xi$. Its curvature is $F^{\prime}=F+d\xi$ and the primitive
part is exactly $F^{\prime}_{p}=F_{p}+\partial_{+}\xi$. Hence, $A^{\prime}$ is
a primitive Yang-Mills connection. But by $\partial_{+}^{*}F_{p}\neq 0$,
$\partial_{+}\xi$ can not be zero. Then $d\xi\neq 0$ and $F^{\prime}=F+d\xi$
is not $d$-harmonic. Thus, $A^{\prime}$ is not Yang-Mills.

######  Example 3.13.

Let $P$ be an $SU(2)$-bundle over $M=\mathbb{R}^{4}$ and $\omega=dx^{1}\wedge
dx^{2}+dx^{3}\wedge dx^{4}$. Set

$A=\frac{\eta^{a}_{\mu\nu}x^{\nu}}{|x|^{2}+1}T_{a}dx^{\mu}$

be a BPST instanton [BPST] (see also the review article [BVV]). Here
$\\{T_{1},T_{2},T_{3}\\}$ is a basis of $\mathfrak{su}(2)$ satisfying
$[T_{a},T_{b}]=\epsilon^{abc}T_{c}$. For example, we can set
$T_{a}=\frac{1}{2i}\sigma_{a}$ for the Pauli matrices
$\sigma_{1},\sigma_{2},\sigma_{3}$. $\eta^{a}_{\mu\nu}$ is the ’t Hooft
symbol.

$\eta^{a}_{\mu\nu}=\begin{cases}\epsilon^{a\mu\nu},&\mu,\nu=1,2,3\\\
-\delta^{a\nu},&\mu=4\\\ \delta^{a\mu},&\nu=4\\\ 0,&\mu=\nu=4\end{cases}.$

The curvature is

$F=\frac{4}{(|x|^{2}+1)^{2}}[-T_{1}(dx^{1}\wedge dx^{4}+dx^{2}\wedge
dx^{3})+T_{2}(dx^{1}\wedge dx^{3}+dx^{2}\wedge dx^{4})-T_{3}(dx^{1}\wedge
dx^{2}+dx^{3}\wedge dx^{4})].$

$F$ is self-adjoint so $A$ is a Yang-Mills connection. On the other hand, we
have

$\Phi=-\frac{4}{(|x|^{2}+1)^{2}}T_{3}.$

Since $\frac{4}{(|x|^{2}+1)^{2}}$ is not $d$-closed, $d\Phi$ has a non-trivial
$T_{3}$-component. But $[T_{a},T_{3}]$ has no $T_{3}$-component for any $a$,
neither is $[A,\Phi]$. Therefore, $d_{A}\Phi$ has a non-trivial
$T_{3}$-component and cannot be 0. $A$ is not a critical point of
$\|\Phi\,\omega\|^{2}$, so not a primitive Yang-Mills connection, either.

######  Remark 3.14.

The construction of BPST instanton makes use of the homotopy of
$S^{3}\rightarrow S^{3}$, mapping from the boundary of
$\mathbb{R}^{4}\rightarrow SU(2)=S^{3}$ [BVV]. However, the presence of
$\omega$ in the primitive Yang-Mills conditions breaks the symmetry of
$\mathbb{R}^{4}$ coordinates by separating them into two groupings of
$\\{x^{1},x^{2}\\}$ and $\\{x^{3},x^{4}\\}$.

###  3.3 Cone Yang-Mills functional

Suppose $(M^{2n},\omega)$ is a symplectic manifold and $[\omega]\in
H^{2}(M,\mathbb{Z})$. We can construct a circle bundle $\pi:X\to M$ whose
Euler class is $\omega$. Let $P$ be a principal bundle over $M$ with a
symplectically flat connection $A$. Then $A-\theta\,\Phi$ is a flat connection
on the pullback bundle $\pi^{*}P$ [TZ]*Corollary 3.9. Given an inner product
structure over the associated vector bundle $Ad\,P$ compatible with $\omega$,
we can extend it to an inner product over $\pi^{*}Ad\,P$ and make the global
angular form $\theta\in\Omega^{1}(X)$ orthogonal to $\pi^{*}\Omega^{*}(M)$.
These two inner products induce Yang-Mills functionals on $Ad\,P$ and
$\pi^{*}Ad\,P$ respectively, and they have the following relationship.

$S_{Y\\!M}(A-\theta\Phi)=\|F(A)-\omega\Phi-\theta
d_{A}\Phi\|^{2}_{\pi^{*}Ad\,P}=\|F(A)-\omega\Phi\|^{2}_{Ad\,P}+\|\theta
d_{A}\Phi\|^{2}_{Ad\,P}.$

Observe that the right hand side is the cone Yang-Mills functional defined in
Definition 1.4 if we replace $-\Phi$ by a function $B\in\Omega^{0}(M,Ad\,P)$,
and is well defined even if $[\omega]$ is not an integral class. This suggests
the consideration of the mapping cone of differential forms
$\Omega^{*}(M)[\theta]=\Omega^{*}(M)\oplus\,\theta\,\Omega^{*}(M)$, which is
quasi-isomorphic to both $\Omega^{*}(X)$ and the space of primitive forms
[TT]. Here, $\theta$ has degree 1 and satisfies $d\theta=\omega$, and further,
$[\omega]$ is allowed to be non-integral. For a connection $A$ on a principal
bundle $P$ over $M$, its covariant derivative $d_{A}$ induces an operator
$d_{A}-\theta\Phi$ over $\Omega^{*}(M,Ad\,P)[\theta]$. Then
$(d_{A}-\theta\Phi)^{2}=0$ if and only if $A$ is a symplectically flat
connection [TZ]*Proposition 3.8.

We can also extend the inner product $\langle-,-\rangle$ on
$\Omega^{*}(M,Ad\,P)$ to $\langle-,-\rangle_{\mathcal{C}}$ on
$\Omega^{*}(M,Ad\,P)[\theta]$, by setting

$\langle\xi_{1}+\theta\eta_{1},\xi_{2}+\theta\eta_{2}\rangle_{\mathcal{C}}=\langle\xi_{1},\xi_{2}\rangle+\langle\eta_{1},\eta_{2}\rangle.$

It induces an operator

$\displaystyle*_{\mathcal{C}}:\Omega^{*}(M,Ad\,P)[\theta]~{}$
$\displaystyle\to~{}~{}\Omega^{*}(M,Ad\,P)[\theta]$
$\displaystyle\xi+\theta\eta~{}\qquad$
$\displaystyle\mapsto~{}~{}(-1)^{|\xi|}\theta*\xi+*\eta$

satisfying

$\langle\xi_{1}+\theta\eta_{1},\xi_{2}+\theta\eta_{2}\rangle_{\mathcal{C}}=\int_{M}\frac{\partial}{\partial\theta}[(\xi_{1}+\theta\eta_{1})\wedge*_{\mathcal{C}}(\xi_{2}+\theta\eta_{2})]$

where $\frac{\partial}{\partial\theta}(\theta\alpha)=\alpha$ for any
$\alpha\in\Omega^{*}(M,Ad\,P)$.

Given a connection $A$ and a function $B\in\Omega^{0}(M,Ad\,P)$, the curvature
of the operator $d_{A}+\theta B$ is $\tilde{F}=(d_{A}+\theta
B)^{2}=F(A)+\omega B-\theta d_{A}B$. It suggests the following cone Yang-Mills
functional for this curvature

$\|\tilde{F}\|_{\mathcal{C}}^{2}=\|F(A)+\omega B\|^{2}+\|d_{A}B\|^{2}.$

The zero points of the functional satisfies

$F=-\omega B,\quad d_{A}B=0,$

which identifies $A$ as a symplectically flat connection.

A straightforward calculation shows that $A+\theta B$ is a critical point of
$\|\tilde{F}\|_{\mathcal{C}}^{2}$ if and only if

$\begin{cases}d_{A}^{*}(F_{A}+\omega B)-[d_{A}B,B]=0,\\\ n(\Lambda
F_{A}+B)+d_{A}^{*}d_{A}B=0.\end{cases}$

We may ask for what connection $A$ there is some $B$ such that $A+\theta B$ is
a critical point of the cone Yang-Mills functional. For a connection which is
both Yang-Mills and primitive Yang-Mills, $B=-\Phi=-\Lambda F_{A}$ is an
obvious solution. However, if only one of the condition above holds, there may
not be solutions.

######  Proposition 3.15.

Suppose the structure group is abelian. If a connection $A$ is Yang-Mills but
not primitive Yang-Mills, or is primitive Yang-Mills but not Yang-Mills, then
there is no $B$ such that $A+\theta B$ is a critical point of the cone Yang-
Mills functional.

###### Proof.

As the structure group is abelian, the condition that $A+\theta B$ is a
critical point of the cone Yang-Mills functional
$\|\tilde{F}\|_{\mathcal{C}}^{2}$ becomes

$\displaystyle\begin{cases}d^{*}(F_{A}+\omega B)=0,\\\ n(\Lambda
F_{A}+B)+d^{*}dB=0.\end{cases}$ (3.10)

Write $F_{A}=F_{p}+\Phi\omega$ by the Lefschetz decomposition. In both cases
$d\Phi\neq 0$ because of Proposition 3.6.

When $A$ is Yang-Mills, the first equation of (3.10) is equivalent to
$d*(\omega B)=\frac{\omega^{n-1}}{(n-1)!}dB=0$. So $dB=0$ and $B$ is a
constant. Then the second equation implies that $\Phi=\Lambda F_{A}$ is a
constant, which is contradicted to $d\Phi\neq 0$. Thus, such $B$ does not
exist.

When $A$ is primitive Yang-Mills, $d^{*}$ acting on $F_{p}$ is zero, the first
equation of (3.10) is equivalent to $d*(\omega\,\Phi+\omega\
B)=\frac{\omega^{n-1}}{(n-1)!}d(\Phi+B)=0$. Thus, $\Phi+B=\Lambda F_{A}+B$
must be a constant then so is $d^{*}dB$. Since the only $d^{*}$-exact constant
is zero, $d^{*}dB$ must vanish. It follows that $B=-\Phi$ and $dB=0$. This
contradicts $d\Phi\neq 0$, so no such $B$ exists either. ∎

On the other hand, it is easy to show that $B$ is unique for abelian structure
group.

######  Proposition 3.16.

Suppose the structure group is abelian. For each connection $A$ there is at
most one $B$ such that $A+\theta B$ is a critical point of the cone Yang-Mills
functional.

###### Proof.

Suppose both $A+\theta B_{1}$ and $A+\theta B_{2}$ are critical points of
$\|\tilde{F}\|_{\mathcal{C}}^{2}$. Then $(A,B_{1})$ and $(A,B_{2})$ both need
to satisfy (3.10). By the first equation of (3.10) we have

$\frac{\omega^{n-1}}{(n-1)!}dB_{1}=d*(\omega B_{1})=d*(\omega
B_{2})=\frac{\omega^{n-1}}{(n-1)!}dB_{2}.$

It follows that $dB_{1}=dB_{2}$. Then the second equation implies

$n(B_{1}-B_{2})=d^{*}d(B_{2}-B_{1})=0.$

Hence, $B_{1}=B_{2}$. ∎

######  Remark 3.17.

We comment here on the generalization of the above functionals to those with
zeroes being $\zeta$-flat connections. For any closed 2-form
$\zeta\in\Omega^{2}(M)$, an inner product structure on $\Omega^{*}(M,Ad\,P)$
induces the decomposition of the curvature

$F=F_{\perp}+\Phi\zeta,$

where $F_{\perp}$ is orthogonal to $\zeta$. Thus, the Yang-Mills functional
can also be decomposed as

$\displaystyle\|F\|^{2}=\|F_{\perp}\|^{2}+\|\Phi\,\zeta\|^{2},$ (3.11)

and $\zeta$-flat connections are exactly the zero points of
$\|F_{\perp}\|^{2}$ when $\dim M\geq 4$.

Similarly, if we set $d\theta=\zeta$ for the cone $\Omega^{*}(M)[\theta]$, the
functional $\|\tilde{F}\|_{\mathcal{C}}^{2}$ will become

$\displaystyle\|F_{A}+\zeta B\|^{2}+\|d_{A}B\|^{2}.$ (3.12)

Its zero points $A+\theta B$ are then identifies with the $\zeta$-flat
connections $A$.

However, unlike for a compatible symplectic structure $\omega$,
$*\,\zeta\neq\frac{\zeta^{n-1}}{(n-1)!}$ generally. Hence, the properties
described above do not generalize straightforwardly to the $\zeta$-flat case.

##  4 Discussion: flows and characteristic classes

In previous section, we introduced new Yang-Mills type functionals and
explicitly showed that their Euler-Lagrange equations which specify the
critical points can have different solutions. It would be interesting to study
the moduli space of solutions modulo gauge equivalence and identify any
structures.

It is also natural to consider how the critical points can be connected by
gradient flows of the functionals. Analogous to the Yang-Mills gradient flow,
first studied in two dimension by Atiyah and Bott [Atiyah-Bott], we can write
down a gradient flow for each functional that we have introduced. As an
example, take the primitive Yang-Mills functional of Definition 1.3. It gives
what we can call the primitive Yang-Mills flow:

$\displaystyle\dfrac{\partial A}{\partial t}=-d_{A}^{*}F_{p}\,.$ (4.1)

We expect this flow to share similar properties of the Yang-Mills flow. The
flow can be used to link critical points of the primitive Yang-Mills
functional. But additionally, we can also consider the flow of Yang-Mills
solutions that are not primitive Yang-Mills. The primitive Yang-Mills flow
(4.1) could therefore provide a link between critical points of the Yang-Mills
functional and that of the primitive Yang-Mills.

Our motivation in writing down the new functionals in Section 3 was that the
zeroes of the functionals correspond to symplectically-flat or more generally
$\zeta$-flat connections. If we remove this criterion, there are other
functionals that are natural to write down. A concrete example comes from the
Chern-Simons characteristic forms. We will describe how we can obtain such a
functional for symplectically flat connections.

Recall that a symplectically flat connection $A$ can be associated to a cone-
flat connection $A+\theta\,B$ where $d\theta=\omega$ and $F(A)=-B\,\omega$.
When $\omega$ is integral class, cone-flat connections are just flat
connections on the prequantum circle bundle whose Euler class is $\omega$ and
$\theta$ is a global angular form. We consider the Chern-Simons form of the
Chern character, which we will denote by $Q_{2k+1}$. Using the cone-flat
connection, we decompose $Q_{2k+1}(A+\theta B)$ as
$Q^{\prime}_{2k+1}(A,B)+\theta Q^{\prime\prime}_{2k}(A,B)$ with both
$Q^{\prime}_{2k+1}(A,B),Q^{\prime\prime}_{2k}(A,B)\in\Omega^{*}(M,Ad\,P)$. But
note that if $A+\theta B$ is a cone-flat connection, then $d\,Q_{2k+1}=0$ and
also $d\,Q^{\prime\prime}_{2k}(A,B)=0$. Hence, $Q^{\prime\prime}_{2k}(A,B)$ is
a characteristic form on $M$. Furthermore, with $\dim M=2n$,
$P_{2n}=\int_{M}Q^{\prime\prime}_{2n}(A,B)$ gives us a functional. We will
write down explicitly two cases, $\dim M=2,4$.

For $\dim M=2$, we find

$\displaystyle P_{2}(A,B)$
$\displaystyle=\frac{-1}{8\pi^{2}}\int_{M}\text{tr}\left[\omega\,B^{2}+2\,B\,F-d(A\,B)\right],$
(4.2)

and for $\dim M=4$, we find

$\displaystyle
P_{4}(A,\Phi)=\frac{-i}{48}\int_{M}\text{tr}\left[3\,B\,F^{2}+3\,\omega\,B^{2}\,F+\omega^{2}\,B^{3}-d(BdA\,A+\omega\,B\,A\,B+\frac{3}{2}BA^{3}-BAdA)\right].$
(4.3)

We will assume $M$ is a closed manifold. Then the critical points of
$P_{2}(A,B)$ are exactly symplectically flat connections, i.e. satisfying
$F(A)=-B\,\omega$ and $d_{A}B=0$. For $P_{4}(A,B)$, the symplectically flat
connections are a subset of the critical points.

Furthermore, if we substitute $F=\Phi\,\omega$ and $B=-\Phi$ into (4.2) and
(4.3), we obtain the Chern-Simons invariants for symplectically flat
connections

$\displaystyle P_{2}(A,-\Phi)$
$\displaystyle=\frac{1}{8\pi^{2}}\int_{M^{2}}\text{tr}\,\Phi^{2}\,\omega$
$\displaystyle P_{4}(A,-\Phi)$
$\displaystyle=\frac{i}{48}\int_{M^{4}}\text{tr}\,\Phi^{3}\,\omega^{2}$

which are just proportional to the integral of $\mathrm{tr}\,\Phi^{n+1}$ and
generally are non-zero.

Finally, let us describe some characteristic classes of symplectically flat
$G$-bundles, or more generally $\zeta$-flat $G$-bundles, when the Lie group
$G$ is non-compact. In essence, we can extend the theory of characteristic
classes of flat bundles to $\zeta$-flat bundles by means of cone-flat
connections. (Note that a cone-flat connection is equivalent to a $\zeta$-flat
connection if $d\theta=\zeta$, and to a symplectically flat connection in the
special case where $d\theta=\omega$.) Our description will follow Morita’s
exposition of characteristic classes in [Morita]*Chapter 2.3

Let $G$ be a Lie group and $\pi:P\to M$ be a $\zeta$-flat $G$-bundle. Suppose
$A\in\Omega^{1}(P,\mathfrak{g})$ is the corresponding connection form and its
curvature is $\Phi\,\zeta$. Then $A$ induces a linear map

$\mathcal{A}:\mathfrak{g}^{*}\to\Omega^{1}(P)\oplus\theta\,\Omega^{0}(P),\qquad\alpha\mapsto\alpha\circ
A-\theta\,\alpha(\Phi).$

$\mathcal{A}$ can be extended to a morphism from $\Lambda^{*}\mathfrak{g}^{*}$
to the cone $\Omega^{*}(P)[\theta]$ naturally. Explicitly, it sends
$\alpha\in\Lambda^{k}\mathfrak{g}^{*}$ to $\alpha\circ A^{\otimes
k}-\theta(\iota_{\Phi}\alpha\circ
A^{\otimes(k-1)})\in\Omega^{k}(P)\oplus\theta\,\Omega^{k-1}(P)$. It is
straightforward to verify that this morphism is a DGA morphism, where the
differential on $\Lambda^{*}\mathfrak{g}^{*}$ is the exterior differentiation,
and $d\theta=\zeta$ on $\Omega^{*}(P)[\theta]$. So it induces a morphism
$H^{*}(\mathfrak{g})=H^{*}(\Lambda^{*}\mathfrak{g}^{*})\to
H^{*}(\Omega^{*}(P)[\theta])$ between cohomologies.

Take a maximal compact subgroup $K$ of $G$. Then the projection $\pi:P\to M$
can be decomposed as two maps

$P\to P/K\to M.$

Set $C^{*}(\mathfrak{g})=\Lambda^{*}\mathfrak{g}^{*}$ and
$C^{*}(\mathfrak{g},K)$ be the space of $K$-invariant elements in
$C^{*}(\mathfrak{g})$. That is,

$\displaystyle C^{k}(\mathfrak{g},K)$ $\displaystyle=\\{\alpha\in
C^{k}(\mathfrak{g})|\iota_{X}\alpha=0,\alpha(Ad_{g}X_{1},\ldots,Ad_{g}X_{k})=\alpha(X_{1},\ldots,X_{k})$
$\displaystyle\quad\text{ for any }X,X_{1},\ldots,X_{k}\in\mathfrak{k},g\in
G\\}$

where $\mathfrak{k}$ is the Lie algebra of $K$. The image of $\mathcal{A}$
acting on $C^{*}(\mathfrak{g},K)$ is in the pullback of the projection from
$\Omega^{*}(P/K)[\theta]$. So it induces the following commutative diagram

$\textstyle{C^{*}(\mathfrak{g})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{*}(P)[\theta]}$$\textstyle{C^{*}(\mathfrak{g},K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{*}(P/K)[\theta]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

So we obtain a morphism from $H^{*}(\mathfrak{g},K)$ to
$H^{*}(\Omega^{*}(P/K)[\theta])$.

On the other hand, $P/K$ is a $G/K$ bundle over $M$ and it is well know that
$G/K$ is diffeomorphic to a Euclidean space and in particular is contractible.
Therefore, $P/K$ is homotopy equivalent to $M$ and we have
$H^{*}(\Omega^{*}(P/K)[\theta])\simeq H^{*}(\Omega^{*}(M)[\theta])$.

The homomorphism $\mathcal{A}:H^{*}(\mathfrak{g},K)\to
H^{*}(\Omega^{*}(M)[\theta])$ is the characteristic homomorphism, and for any
$[\alpha]\in H^{*}(\mathfrak{g},K)$, the cohomology class
$\mathcal{A}[\alpha]\in H^{*}(\Omega^{*}(M)[\theta])$ gives us the
characteristic class of the $\zeta$-flat $G$-bundle corresponding to
$[\alpha]$. Notice that if $G$ is compact, then $K=G$ and the characteristic
class becomes trivial since $H^{*}(\mathfrak{g},K)=0$.

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Department of Mathematics, University of California, Irvine, CA 92697, USA
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Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China
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