

\magnification=1200
\documentstyle{amsppt}
\topmatter
\rightheadtext{Informal complexification}
\hskip 1.7in 
{\it  AMS Transl., Ser. 2, vol. 180 (1997), pp. 147--155}
\vskip .2in

\title Informal complexification and Poisson structures on moduli 
spaces\endtitle

\author Boris A. Khesin\endauthor

\hskip .5in 
{\it To Vladimir Igorevich Arnold on the occasion of his 60-th birthday}
\vskip 1in

\abstract
We show how the Cauchy (or, more generally, the Leray) residue
formula can be understood as an informal complex analog of the Stokes 
formula. It allows one to treat the  Poisson (and symplectic) 
structures on the moduli
spaces of   flat connections on real  manifolds and those structures
on the moduli spaces of holomorphic bundles on complex manifolds
in a parallel way.\endabstract



\endtopmatter
\document
\widestnumber\key{EFWS}
\baselineskip 18pt
\NoBlackBoxes


\head    Introduction\endhead

The formal  ``real-complex" correspondence extends to many 
mathematical notions
in a rather ``straightforward" way: for instance, the parallelism
 between the groups of 
orthogonal and unitary matrices, or, say, between the  Stiefel--Whitney   
and Chern classes of vector bundles.
In \cite{A1, A2} V.~Arnold presented many less formal ``dual pairs" in this  
$\Bbb R-\Bbb C$  correspondence of notions, such as, e.g.,
$$
\matrix
\text{Real version:} & \text{Complex version:}\\
~&~\\
\Bbb Z_2=\Bbb Z/2\Bbb Z & \Bbb Z\\
\pi_0 &\pi_1\\
\text{Morse theory} & \text{Picard-Lefschetz  theory}\\
\text{a manifold with boundary} & \text{a 2-ramified covering}\\
\text{manifold's orientation} & \text{homotopy class of Hermitian 
framings}\\
\endmatrix
$$
and posed a question about an informal analog of the DeRham 
cohomology theory.

In this paper we suggest an answer to this 
question, extending  the ``informal  complexification" table as follows:
$$
\matrix
\text{DeRham theory of smooth diff. forms} & \text{Leray theory of 
merom. forms}\\
\text{manifold's boundary}&\text{polar set of a form}\\
\text{restriction  to the boundary}&\text{taking residue}\\
\text{Stokes formula}&\text{Cauchy (Leray) formula.}
\endmatrix
$$
We develop here the idea from \cite{FK}, where this parallelism was 
used for 
an explicit realization  of the central extensions of 
current groups on Riemann surfaces as a ``complex  
analog" of affine groups. We show how  this 
heuristic principle can be applied to constructions of symplectic 
and Poisson structures on the 
moduli spaces of flat connections and of integrable (0,1)-connections 
(or holomorphic bundles) 
on real and complex two- and  three-dimensional manifolds. 
The real setting is classical, see \cite{AB, FR, Ad}. The complex 
counterpart   is
the result of our discussions with V.~Fock and A.~Rosly (cf. also \cite{K}). 
Our goal in this paper  is mostly  expository,
and we refer for all details, proofs,  and further applications to
\cite{FKR}, as well as to a relevant treatment in \cite{FKT}.

The statements are presented in ``real-complex" 
pairs  with the indexes {\bf R} or {\bf C} respectively.
We emphasize
that the complexification considered is not ``formal" in any way: 
we study 
{\it smooth} objects on complex manifolds, thus  doubling the 
domain dimension.
The  {\it holomorphic} objects arise as certain equivalence 
classes (orbits, symplectic leaves)
of these smooth ones (cf. \cite{AHS, EF,  FKR}). 

\medskip

Acknowledgements. 
I am sincerely grateful to V.~Fock, A.~Rosly, I.~Frenkel, 
 H.~Fan, R.~Lee, A.~Levin for many fruitful discussions. 
I am deeply obliged to V.I.~Arnold, who intrigued me with the question of 
informal complexification. 
This paper was conceived in the creative atmosphere of the MPI in Bonn,
and mostly completed in that of the FIM in Zurich. The research was 
partially
supported by NSF grant DMS-9627782.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bigskip   


\subhead \S 1. Complex analogs of the boundary operator and the Stokes-Leray 
formula\endsubhead


\subhead 1.1.  The Leray residue\endsubhead 
The Leray residue formula is a higher-dimensional generalization
of the Cauchy formula, which gives the value of a contour integral of 
a meromorphic 1-form via form's residue   at the pole.

Let $\omega$ be a closed meromorphic $k$-form on 
a  compact complex $n$-dimensional
manifold $M$ with poles  on a nonsingular complex 
hypersurface  $N\subset M$. All poles here and below are supposed to be 
of the first order.
Let $\psi$ be a function defining $N$ in a neighborhood of 
some point $p\in N$.
Then locally, in a certain neighborhood $U(p)$ the $k$-form $\omega$ can be 
decomposed into the sum 
$$
\omega = \frac{d\psi}{\psi} \wedge\alpha + \beta , 
$$
where $\alpha$ and $\beta$ are holomorphic in $U(p)$. One can show, that
the restriction $\alpha\big|_N$ is a well-defined (i.e. independent
 of   $\psi$) holomorphic $(k-1)$-form, see \cite{L}.



\definition{Definition 1.1}   The {\it form-residue}
$\text{res}~\omega$ of the  form $\omega$ is the holomorphic
$(k-1)$-form on $N$, such that in any neighborhood $U(p)$ of an
arbitrary point $p\in N$ it coincides with the form $\alpha\mid_N$ of the
decomposition above: $\text{res}~~\omega = \alpha\mid_N$.
The {\it coboundary Leray operator} $\delta$
 associates to every point $p\in N$ a topological circumference
$\delta p\subset M\setminus N$ satisfying certain conditions of 
continuity in $p$. Any $(k-1)$-cycle $\Gamma$ on $N$ gives rise to 
a $k$-cycle $\delta\Gamma$ in the complement $M\setminus N$.  
\enddefinition


\proclaim{Theorem 1.2 (\cite{L})}  Let $\Gamma\subset N$ be a
$(k-1)$-dimensional cycle in $N$ and let $\omega $ be a 
meromorphic $k$-form 
with a polar set $N$ of
order $1$.  Then
$$
\int_{\delta\Gamma} \omega = 2\pi i \int_{\Gamma} ~\text{res}~ \omega.
$$
\endproclaim

\bigskip

  The  theory above admits  natural generalizations to the case of closed 
$C^\infty$-smooth forms $\omega$, as well as to the case of polar sets 
  which consist of several  complex 
hypersurfaces in a general position in $M$.

\medskip

\subhead Remark 1.3\endsubhead According to the table in Introduction 
we will regard 
the poles of meromorphic forms as the ``complex analogs of the boundary",
while the boundary orientation (being $\Bbb Z_2$-valued) is replaced by 
an (integral valued) index of a contour with respect to the polar set. 
 
For several problems in algebraic geometry   V.~Arnold used  the 
following counterpart of the boundary operator
(see, e.g. \cite{A2, A3}). Let $f(x)$ be a function defined on the manifold 
with boundary, say $\{x~|~x\ge 0\}$. The latter is not an algebraic 
constraint, 
and hence it cannot be straightforwardly understood  over $\Bbb C$. 
To overcome this difficulty,   
introduce an additional variable $y$, and replace the condition
$x\ge 0$   by the algebraic condition $x=y^2$. The latter relation 
over $\Bbb C$ defines a two-fold covering of the $x$-axis  
ramified at $\{x=0\}$.
 Thus, the {\it passage to the ramified
covering} defines a would-be complexification of the 
{\it boundary operator}. 

Note, that this version of the boundary operator is consistent with
the complexification table above once we apply it to the set of  
{\it meromorphic} forms.  If we start with a meromorphic form $\omega(x)$ 
of the variable $x$ with a pole (of first order) at $x=0$ (say, $dx/x$), 
then after the change of variable $x=y^2$ we obtain the form 
$\tilde\omega(y)$ (respectively, equal to $2dy/y$) with the  pole of 
first order at $y=0$. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  
\bigskip

\subhead 1.2.  The   Stokes--Leray Theorem and its applications\endsubhead
The theorems of  Stokes and Leray admit the following ``joint" version.
Let $M$ be a complex compact manifold of complex dimension $n$, and $N$ a
complex hypersurface.

\proclaim{Theorem 1.4 (see e.g. \cite{GS, FK})} 
Assume that   a  $2n$-form $\mu=d\alpha\wedge\rho$   on $M\setminus N$   
is a wedge product of an exact smooth $n$-form $d\alpha$ with $\alpha\in
\Omega^{n-1}(M)$ and a meromorphic $n$-form $\rho$ on $M$ with 
a polar set $N$
of order $1$. Then the form $\mu$ 
is an   integrable form on $M$,
and
$$
\int_M  d\alpha\wedge\rho = - 2\pi i\int_N  \alpha\wedge~\text{res}~
\rho.\tag1 
$$
\endproclaim

The integrability of the form $\mu$ follows from the fact that $\mu$, 
as well as $\rho$ has a pole of first order on $N$, while the 
(real) codimension of $N$   equals 2. The equality is proved by 
successive application of the Stokes and Leray formulas
to the integral $\int_M\mu$, where $M$ is  split into the sum of 
a tubular neighborhood of $N\subset M$ and the neighborhood's complement.
  Theorem 1.4 remains true if $\rho$ is allowed to have poles of the 
first order along a union of hypersurfaces.


 Note, that the dimension of the manifold of integration in (1) 
drops by 2 at once.
Moreover, both the smooth objects and the meromorphic ones appear together
in the relation (1), and the meromorphic form $\rho$ can be thought of
as  ``measure-like", while smooth $\alpha$ as variable.
The formula (1), as well as that of  Cauchy,  will further be regarded as a
complexification of the Stokes formula, and it replaces the latter in 
complex versions of  proofs of the statements below, cf. \cite{FK}.


\medskip

\subhead Remark 1.5\endsubhead
We denote by $\Cal H^n(M\setminus N)$ the space of 
meromorphic $n$-forms on $M$ with poles on $N$ (of the first order).
For K\"ahler $M$ this is a finite-dimensional space, and it 
turns out to be isomorphic to the direct sum of the
space $\Cal H^{n-1}(N)$ of holomorphic $(n-1)$-forms
on $N$ and the space $\Cal H^{n}(M)$ of holomorphic $n$-forms
on $M$, provided that  $h^{n,1}(M,\Bbb C)=0$ (see e.g. \cite{Ch}).
In other words, under this assumption 
every meromorphic $n$-form  on $M$ is defined 
by its residue on $N$ uniquely up to a holomorphic $n$-form  on $M$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bigskip

\subhead \S 2. Spaces of flat connections on real and complex surfaces
\endsubhead

In this Section we recall several results on Poisson structures 
on the space 
of {\it smooth} connections on {\it real} 1-, 2-, and 3-dimensional 
manifolds. 
Their ``complexification" according to the table in Introduction allows 
one to describe the (holomorphic) 
Poisson structures on the spaces of {\it smooth} connections
on {\it complex} 1-, 2-, and 3-dimensional manifolds.
%We treat the real and complex settings in a parallel way, denoting
%the corresponding  theorems by the indexes {\bf R} and {\bf C}.

\medskip

We will use the following notations throughout the paper.

\definition{Definition 2.1R}
In the real case $G$ stands for a simple simply connected {\it compact} 
Lie group,
$\goth g=\text{Lie}(G)$ is its Lie algebra. Let $S$ (or $C,~D$)
be a real manifold 
(possibly with boundary), and $E$ be a principle $G$-bundle over $S$.
Let $\Cal A^S$ be the affine space of all {\it smooth} connections 
in $E$:
$\Cal A^S=\{ d+A~|~A\in \Omega^1(S,\goth g)\}$. 
For a nontrivial 
bundle $E$, $d$ stands for   the covariant differential $d_B$
for some flat ($d_B^2=0$) connection $B$ in $E$, and the identification 
of $\Cal A^S$ with $\Omega^1(S,\goth g)$, as well 
as the most structures below, depends on the choice of $B$.
\enddefinition


\definition{Definition 2.1C}
In the complex framework,   $G$ is a simple simply connected {\it complex} 
Lie group, $\goth g=\text{Lie}(G)$. All manifolds $N$ (or $\Sigma,~M$)
 are supposed to be  K\"ahler and, moreover,  equipped with 
some holomorphic or meromorphic form of highest degree.
Let  $E$ be a complex principle $G$-bundle over the (complex) 
manifold $N$.
Denote by $\Cal A_{\bar\partial}^N$ the space of all {\it smooth} 
$(0,1)$-connections  in $E$:
  $\Cal A_{\bar\partial}^N=\{ \bar\partial+A~|\break
A=A^{01}\in \Omega^{0,1}(N,\goth g)\}$. 
For a topologically nontrivial $E$ one replaces
${\bar\partial}$ by ${\bar\partial}_B$ for some integrable 
$(0,1)$-connection $B$ (i.e. ${\bar\partial}^2_B=0$). 
\enddefinition

For an (either real or complex) manifold $S$ and its 
submanifold $C\subset S$
denote by $G^S$ the gauge group of {\it smooth} currents 
$G^S=\{f\in C^\infty(S,G)\} $ with the pointwise product, 
and by $G^S_C$ the group of currents on $S$ ``based on $C$":
$G^S_C=\{f\in C^\infty(S,G)~|~f|_C=id\} $.  The 
corresponding Lie algebras
are denoted by $\goth g^S$ and $\goth g^S_C$.

\medskip

\subhead 2.1. Symplectic structures on the spaces of
 connections\endsubhead
Let $S$ be a  compact Riemann surface (with no complex structure fixed). 
  
\proclaim{Proposition 2.2R}
The space $\Cal A^S$ of all smooth $G$-connections in a bundle $E$ over 
$S$ is an affine  symplectic space with  the   symplectic structure 
$$
W(\delta A_1,\delta A_2):=\int_S tr(\delta A_1\wedge \delta A_2),
$$
where $\delta A_i\in \Omega^1(S,\goth g)$, and $tr(\delta A_1
\wedge \delta A_2)$ is 
the real-valued 2-form, being the wedge product of the 1-forms
$\delta A_1$ and $\delta A_2$ on the surface $S$
and the Killing form in the Lie algebra $\goth g$. 
The symplectic structure $W$ is invariant with respect 
to the gauge transformations
$A\mapsto g^{-1}Ag+g^{-1}dg$, where $g\in G^S$.
\endproclaim

\medskip

Now, let $N$ be a compact complex surface, and $\omega$ 
 a holomorphic  2-form on  $N$.
  
\proclaim{Proposition 2.2C}  Given a holomorphic 2-form $\omega$, 
the affine space $\Cal A_{\bar\partial}^N$ of all smooth 
$(0,1)$-connections in the bundle $E$ over  $N$ has a natural
holomorphic symplectic structure:
$$
W_{\Bbb C,\omega}(\delta A_1,\delta A_2):=
\int_N tr(\delta A_1\wedge \delta A_2)\wedge \omega. 
$$
The symplectic structure $W_{\Bbb C,\omega}$ is invariant with respect 
to the  gauge transformations
$A\mapsto g^{-1}Ag+g^{-1}\bar\partial g$, where $g\in G^N$.
\endproclaim

For different 2-forms $\omega$ the spaces $\Cal A_{\bar\partial}^N$ are,
generally speaking, not symplectomorphic. One can join all these 
symplectic structures $W_{\Bbb C,\omega}$
into the ``symplectic structure" $W_{\Bbb C,*}$ on $\Cal A_{\bar\partial}^N$
with values in $(\Cal H^2(N))^*$: given a holomorphic 2-form
$\omega\in \Cal H^2(N)$ one obtains a complex valued symplectic form
$W_{\Bbb C,\omega}=:\langle ~W_{\Bbb C,*}~,~\omega~\rangle$.


\bigskip



The gauge group $G^S$ acts on $\Cal A^S$  in a Hamiltonian way (\cite{AB}).
Respectively, the complex group $G^N$  acts on
$\Cal A^N_{\bar\partial}$  in a Hamiltonian way, provided that
the holomorphic 2-form $\omega$ does not have zeroes on $N$ (i.e. for a
K3 surface or a two-dimensional complex torus $N$), \cite{K}.
The moment map of this action takes a connection into its
curvature: $A\mapsto F(A)=dA+\frac 12[A, A]$ 
(resp., $A\mapsto F_{\bar\partial}(A)\wedge\omega
:=(\bar{\partial} A+\frac 12[A, A])\wedge\omega$).
Preimage of zero value of the moment map consists of flat connections
(resp., integrable $(0,1)$-connections). 
The (nonsingular part of the) result of its Hamiltonian reduction 
carries a natural symplectic structure. Below we always mean the
 nonsingular parts of the moduli spaces when describing the  
symplectic (or Poisson) structures on them.

\proclaim{Theorem 2.3R (\cite{AB})} The  moduli space 
$\Cal M^S=\Cal A_{\operatorname{fl}}^S/G^S$
of flat  $G$-connections  carries a natural symplectic  structure
as the result of the Hamiltonian reduction.
\endproclaim


\proclaim{Theorem 2.3C (\cite{K, M})} 
The points of the moduli space $\Cal M^N=\Cal A^N_{\text{int}}/G^N$
of integrable $(0,1)$-connections are in 1-1 correspondence with the
equivalence classes of  holomorphic
$G$-bundles over $N$. If $N$ is a K3 surface or a two-dimensional 
complex torus then  
the moduli space carries a holomorphic symplectic 
structure (depending on the 2-form $\omega$)  as the result of the reduction.
\endproclaim

The first part of Theorem 2.3C is the classics of
four-dimensional Riemannian geometry, see e.g. \cite{AHS, K}.
The $(0,1)$-connections $\bar\partial+A$ can be regarded as 
the $(0,1)$-part of  Hermitian
connections in $E$, and the integrability condition $\bar\partial^2_A=0$ is
equivalent to the condition that the curvature of the Hermitian
connection is of $(1,1)$-type.
The latter implies the existence of the holomorphic structure in the
complex bundle $E$.
The existence of the (holomorphic) 
symplectic structure on the moduli space  $\Cal M^N$
is implied by the Hamiltonian reduction from the symplectic space 
$\Cal A^N_{\bar\partial}$.
 
\bigskip

\subhead 2.2. Poisson structures on the spaces of
connections on surfaces with boundaries\endsubhead
Let $S$ be a  Riemann surface with a boundary $C=\partial S$ 
consisting of several curves $C=\cup^k_1 C_j$. 
 
\proclaim{Theorem 2.4R (\cite{FR})} The  moduli space 
$\Cal M^S=\Cal A_{\operatorname{fl}}^S/G^S$
of flat  $G$-connections on $S$ with boundary admits 
a natural Poisson  structure. The symplectic leaves    
are determined by fixing the conjugacy classes of the connection  
holonomies  along the components  $C_j$ of the boundary of $S$.
\endproclaim

\medskip

A complex analog of a real surface with boundary is a complex surface $N$ 
equipped with a {\it meromorphic} 2-form $\omega$. Assume that 
$\omega$ does not have zeroes on $N$.
Let $\Sigma=\cup_1^k \Sigma_j$ be the polar set of $\omega$, consisting of
complex curves $\Sigma_j$. 

 
\proclaim{Theorem 2.4C (\cite{FKR})} 
Given a nonvanishing meromorphic 2-form $\omega$ on $N$,  
the moduli space $\Cal M^N=\Cal A_{\text{int}}^N/G^N$
of integrable $(0,1)$-connections  on $N $  admits a  
holomorphic Poisson  structure. The symplectic leaves are determined by
fixing the equivalence classes of the holomorphic $G$-bundles on all the 
components  $\Sigma_j$ of the polar set $\Sigma$ of $\omega$.
\endproclaim

 
Note that if all  curves $\Sigma_j$ are nonsingular and disjoint, then they 
are elliptic curves. Indeed, the residue of the
nonvanishing meromorphic 2-form $\omega$ is a holomorphic 1-form
on $\Sigma_j$ without zeroes. 

\flushpar {\bf Remark 2.5R.}  
To prove the real version one notices that
the Hamiltonian functions of the 
infinitesimal $G^S$-gauge action on $\Cal A^S$  
form the Lie algebra $\goth g^S$ extended by the 2-cocycle
$$
c(X,Y)=\int_S tr(dX\wedge dY)=\int_{C=\partial S}tr(XdY) 
\qquad\text{for }~~ X,Y\in \goth g^S, \tag2
$$ 
see e.g. \cite{Ad}.

The space of smooth $G$-connections 
$\{d+A~| ~A\in \Omega^1(C,\goth g)\}$ on every boundary curve $C$ 
can be regarded as the dual space to the  affine Lie algebra, i.e. 
the Lie algebra
$\goth g^C$ extended by the two-cocycle (2). This  
dual space carries the natural linear Lie-Poisson structure.
Coadjoint orbits of the affine group, or the symplectic leaves of 
this Lie--Poisson
structure,  consist of gauge-equivalent connections and differ by the 
holonomy around $C$, see  e.g. \cite{PS}.

\subhead Remark 2.5C\endsubhead
In the  complex case, the Hamiltonians of the infinitesimal 
$G^N$-action on $\Cal A^N$ form the Lie algebra $\goth g^N$ 
extended by the two-cocycle
$$
c_\omega(X,Y)=\int\limits_N tr(dX\wedge dY)\wedge \omega
=-2\pi i\int\limits_{\Sigma=\text{poles of } \omega}tr(XdY)
\wedge \text{res }\omega
$$
for $X,Y\in \goth g^N$ (according to the Stokes--Leray formula 
(1), see \cite{FK}). 

Denote by $\alpha=\text{res }\omega|_{\Sigma}$ the residue 1-form of 
the 2-form $\omega$ on the curve $\Sigma$. Let $\hat\goth g^{\Sigma}$
be the one-dimensional extension of the Lie algebra $\goth g^{\Sigma}$
by the two-cocycle 
$$
c_{\alpha}(X,Y)=\int_{\Sigma} tr(X{\bar\partial}Y)\wedge\alpha,
$$
where $X,Y\in \goth g^\Sigma$. The space of the connections 
 $\{\bar\partial+A(z,\bar z)
~|~A\in \Omega^{0,1}(\Sigma,\goth g)\}$ with complex Lie algebra $\goth g$ 
on a Riemann surface $\Sigma$ can be regarded as  (a hyperplane in) 
the dual 
space $\left(\hat{\goth g}^\Sigma\right)^*$, see \cite{EF}. 
%to the central extension of the current Lie algebra  on $\Sigma$, 

The symplectic leaves of the Lie--Poisson structure 
in this dual space  $\left(\hat{\goth g}^\Sigma\right)^*$
are enumerated by the equivalence classes of  
holomorphic $G$-bundles  on the curve $\Sigma$, \cite{EF}.
Thus, Theorems 2.4{\bf R-C} can be regarded as globalizations of  
considering
the coadjoint orbits in the affine Lie algebra, or in the current 
Lie algebra
$\hat{\goth g}^\Sigma$, respectively. 

\medskip

\subhead Question 2.6\endsubhead
For a given graph on a surface $S$,
 the Poisson manifold $\Cal M^S$ admits a 
 combinatorial description 
as the quotient of the space of graph connections by the graph gauge 
group, see \cite{FR}.
 What would be an analogous (complex) combinatorial 
description of the Poisson manifold $\Cal M^N$?

\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subhead \S 3. Complex Chern--Simons functional 
\endsubhead

\subhead 3.1. CS functional and flat connections\endsubhead
First, let D be a real compact three-dimensional manifold, E a 
trivial vector 
$G$-bundle on $D$, and $S=\partial D$ the boundary of $D$. 
Define the {\it Chern--Simons functional} 
$\operatorname{CS}: \Cal A^D\to S^1$
on the space of connections $\Cal A^D$ by the formula
$$
\operatorname{CS}(A)=\frac{1}{4\pi}\int_D tr(A\wedge dA+\frac 23 
A\wedge A\wedge A) \qquad\mod 2\pi.
$$

\proclaim{Theorem 3.1R (see e.g. \cite{W1})} 
A) The Chern--Simons functional $\operatorname{CS}: 
\Cal A^D\to \Bbb R/2\pi\Bbb Z$
is gauge invariant with respect to the group $G^D_S$ 
of currents based on the boundary $S=\partial D$.
In particular, for a closed manifold $D$ ($\partial D=\emptyset$) 
the $\operatorname{CS}$ functional is gauge invariant with respect 
to $G^D$.

B) If $S=\partial D$ is nonempty, the variation of the 
$\operatorname{CS}$ functional is a 1-form 
on the space $\Cal A^S$ of  boundary values  of the connections.
The differential of this 1-form is proportional to
the symplectic 2-form of Proposition 2.2R
on the space $\Cal A^S$ of smooth connections on the 
two-dimensional surface $S$.
\endproclaim

\medskip

Furthermore, as we discussed above, the moduli space 
$\Cal M^S=\Cal A_{\operatorname{fl}}^S/G^S$ of flat connections on $S$
possesses a natural symplectic structure. 

\proclaim{Theorem 3.2R (see e.g. \cite{At, CLM})} 
If a surface $S$ is the boundary of a three-fold $D$
then  the (equivalence classes of all) those 
flat connections on $S$ that can be
extended to   flat connections on $D$ form an isotropic variety in 
the symplectic orbifold $\Cal M^S$. 
\endproclaim

In particular, a  splitting of a three-fold $D=D_1\bigcup_S D_2$
along a surface $S$ into two handle bodies 
defines two  Lagrangian varieties in the symplectic orbifold
$\Cal M^S$. For a rational homology 3-sphere $D$
the intersection number of these varieties can be understood 
as the  {\it Casson invariant} of  $D$, see \cite{CLM}. 

\medskip

\subhead 3.2. Complex CS setting\endsubhead
Now we consider a complex three-manifold $M$, a complex 
hypersurface $N\subset M$, and the space $\Cal H^3(M\setminus N)$ of all
meromorphic 3-forms on $M$ with poles  on $N$.
To a  3-form $\eta\in \Cal H^3(M\setminus N)$ one associates  
the {\it complex-valued Chern--Simons functional} 
$\operatorname{CS}_{\Bbb C,\eta}$
 on the space $\Cal A^M_{\bar\partial}$ by the formula
$$
\operatorname{CS}_{\Bbb C,\eta}(A)
:= \frac{1}{8\pi^2}\int_M tr(A\wedge {\bar\partial} A+
\frac 23 A\wedge A\wedge A)\wedge \eta.
$$
(This integral converges, since the singularities are of the 
first order and
confine to the set $N$ of real
codimension 2.) 
Thus the {\it complex} $\operatorname{CS}_{\Bbb C,*}$
 {\it functional} can be regarded as the 
$(\Cal H^3(M\setminus N))^*$-valued mapping $\langle 
\operatorname{CS}_{\Bbb C,*}~,~\eta\rangle:=
\operatorname{CS}_{\Bbb C,\eta}$.

Note, that the homology group $H_3(M\setminus N,\Bbb Z)$ forms a lattice in 
$(\Cal H^3(M\setminus N))^*$: one can integrate the meromorphic form $\eta$
over integral 3-cycles lying off $N$ in $M$. 
 In the case of Calabi--Yau manifolds $M$ the form $\eta$
is chosen to be the nonvanishing holomorphic 3-form, see  
 \cite{W2}, cf.\cite{FKT}.


\proclaim{Theorem 3.1C (\cite{FKR})} A) The complex 
$\operatorname{CS}_{\Bbb C,*}$  functional defines the
mapping  from $\Cal A^M_{\bar\partial}$ to  the quotient
$(\Cal H^3(M\setminus N))^*/ H_3(M\setminus N,\Bbb Z)$ 
gauge invariant with respect to the 
group $G^M_N$  of currents on $M$ based on the polar set $N$.

B)  Given meromorphic 3-form $\eta\in \Cal H^3(M\setminus N)$, 
the differential of the  variation 1-form of
the complex $\operatorname{CS}_{\Bbb C,*}$ functional defines a 
holomorphic symplectic structure 
on the affine space 
$\Cal A^N_{\bar\partial}$ of smooth connections on the polar set $N$. 
This symplectic structure
coincides with the structure of Proposition 2.2C, where the 2-form 
$\omega$ on $N$
is the residue of the 3-form $\eta$: $\operatorname{res} \eta|_N=\omega$.
\endproclaim


\medskip

Let $N$ be a K3 surface or complex torus of dimension 2 (i.e. $N$
admits a nonvanishing holomorphic 2-form).
Similarly to the real case, the moduli space $\Cal M^N$
of holomorphic $G$-bundles (or the moduli space of the 
integrable $(0,1)$-connections) on the complex surface $N$ possesses a 
 (holomorphic) symplectic structure. 

\proclaim{Theorem 3.2C} Suppose that the surface $N$ is the polar set 
of a meromorphic 3-form in a complex 
three-dimensional manifold $M$, i.e. $N$ is a complex hypersurface in $M$
and the space $\Cal H^3(M\setminus N)$ is nontrivial.
 Then those holomorphic $G$-bundles
on the surface $N$ that can be extended to holomorphic $G$-bundles on the 
three-fold $M$ form an isotropic variety in the moduli space $\Cal M^N$.
\endproclaim

In other words, the natural restriction  $\Cal M^M\to \Cal M^N$ 
has an isotropic image (see \cite{FKR} for more detail).
If the meromorphic 3-form on $M$ has no zeroes, then this restriction map
      is Lagrangian (cf. \cite{T}).

A transverse intersection of two such three-dimensional complex manifolds 
$M_1$ and $M_2$ along the complex surface $N= M_1\bigcap M_2$ 
defines a pair 
of Lagrangian varieties in the holomorphic symplectic orbifold $\Cal M^N$.

\proclaim{Question 3.3} What  analog of the Casson invariant  is
associated with the intersection of these two  Lagrangian varieties 
in $\Cal M^N$?
\endproclaim


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\vskip .5cm

Department of Mathematics, Yale University,
New Haven, CT  06520, USA  
and 
\newline

Department of Mathematics, University of Toronto,
Toronto, ON M5S 3G3, Canada 
\newline

e-mail:~~khesin\@math.yale.edu

\endRefs

\end
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