Loops in SU(2), Riemann Surfaces, and Factorization, I

Loops in SU(2), Riemann Surfaces,
and Factorization, I\star\star$\star$This paper is a contribution to the
Special Issue on Asymptotics and Universality in Random Matrices, Random
Growth Processes, Integrable Systems and Statistical Physics in honor of Percy
Deift and Craig Tracy. The full collection is available at
http://www.emis.de/journals/SIGMA/Deift-Tracy.html

Estelle BASOR † and Doug PICKRELL ‡ † American Institute of Mathematics, 600
E. Brokaw Road, San Jose, CA 95112, USA<EMAIL_ADDRESS>
‡ Mathematics Department, University of Arizona, Tucson, AZ 85721, USA
<EMAIL_ADDRESS>E. Basor and D. Pickrell

Received October 24, 2015, in final form March 02, 2016; Published online
March 08, 2016

In previous work we showed that a loop $g\colon S^{1}\to{\rm SU}(2)$ has a
triangular factorization if and only if the loop $g$ has a root subgroup
factorization. In this paper we present generalizations in which the unit disk
and its double, the sphere, are replaced by a based compact Riemann surface
with boundary, and its double. One ingredient is the theory of generalized
Fourier–Laurent expansions developed by Krichever and Novikov. We show that a
${\rm SU}(2)$ valued multiloop having an analogue of a root subgroup
factorization satisfies the condition that the multiloop, viewed as a
transition function, defines a semistable holomorphic ${\rm SL}(2,\mathbb{C})$
bundle. Additionally, for such a multiloop, there is a corresponding
factorization for determinants associated to the spin Toeplitz operators
defined by the multiloop.

loop group; factorization; Toeplitz operator; determinant

22E67; 47A68; 47B35

This paper is dedicated to Percy Deift and Craig Tracy, both of whom have
contributed to the richness and beauty of mathematics and mathematical physics
in so many ways. Both have created techniques and theories that have enhanced
the understanding of fundamental problems and questions. And both with their
enthusiasm and fearlessness, have encouraged, inspired, and motivated others
to try to do the same.

The authors of this paper met in a summer meeting in 1984 in Laramie, Wyoming,
where Craig was one of the principal speakers. One of the topics discussed
informally at the meeting was determinants of Toeplitz operators, a subject
where both Percy and Craig have made considerable contributions, in particular
with the study of singular symbols and applications to statistical mechanics.
This paper concerns some generalized Toeplitz determinant calculations for
matrix-valued symbols, a topic that is a return to a long ago summer, yet one
which is still of current interest.

##  1 Introduction

Suppose that $\Sigma$ is a connected compact Riemann surface with nonempty
boundary $S$ (a disjoint union of circles). Let $\widehat{\Sigma}$ denote the
double,

$\displaystyle\widehat{\Sigma}=\Sigma^{*}\circ\Sigma,$

where $\Sigma^{*}$ is the adjoint of $\Sigma$, i.e., the surface $\Sigma$ with
the orientation reversed, and the composition is sewing along the common
boundary $S$. Let $R$ denote the antiholomorphic involution (or reflection)
fixing $S$.

The classical example is $\Sigma=D$, the closed unit disk. In this case
$S=S^{1}$, $\widehat{\Sigma}$ is isomorphic to the Riemann sphere, and (in
this realization) $R(z)=1/z^{*}$, where $z^{*}=\overline{z}$, the complex
conjugate. This example has the exceptional feature that there is a large
automorphism group, ${\rm PSU}(1,1)$, acting by linear fractional
transformations.

We now choose a basepoint, denoted by $(0)$, in the interior of $\Sigma$, and
we let $(\infty)$ denote the reflected basepoint for $\Sigma^{*}$. In the
classical case, without loss of generality because of the ${\rm PSU}(1,1)$
symmetry, we can assume the basepoint is $z=0$. Given the data $(\Sigma,(0))$,
following ideas of Krichever and Novikov, a reasonable function on $S$ has a
‘linear triangular factorization’

$\displaystyle f=f_{-}+f_{0}+f_{+},$ (1.1)

where $f_{\pm}$ is holomorphic in the interior of $\Sigma$ ($\Sigma^{*}$,
respectively), with appropriate boundary behavior, depending on the smoothness
of $f$, $f_{+}((0))=0$, $f_{-}((\infty))=0$, and $f_{0}$ is the restriction to
$S$ of a meromorphic function which belongs to a
$\text{genus}(\widehat{\Sigma})+1$-dimensional complementary subspace, which
we refer to as the vector space of zero modes (see Proposition 2.3). In the
classical case $f_{0}$ is the zero mode for the Fourier series of $f$.

A holomorphic map $\mathfrak{z}\colon\widehat{\Sigma}\to\widehat{D}$ is said
to be strictly equivariant if it satisfies

$\displaystyle\mathfrak{z}(R(q))=\frac{1}{\mathfrak{z}(q)^{*}}$

and maps $\Sigma$ to $D$ (and hence $\Sigma^{*}$ to $D^{*}$). When we refer to
the classical case ($\Sigma=D$), it will be understood that
$\mathfrak{z}(z)=z$. For a function $f\colon
U\subset\widehat{\Sigma}\to\mathcal{L}\big{(}\mathbb{C}^{N}\big{)}$, define
$f^{*}(q)=f(R(q))^{*}$, where $(\cdot)^{*}$ is the Hermitian adjoint. If $f\in
H^{0}(\Sigma)$ (i.e., a holomorphic function in some open neighborhood of
$\Sigma$), then $f^{*}\in H^{0}(\Sigma^{*})$. If $q\in S$, then
$f^{*}(q)=f(q)^{*}$, the ordinary complex conjugate of $f(q)$.

######  Theorem 1.1.

Suppose that $k_{1}\in C^{\infty}(S,{\rm SU}(2))$. Consider the following
three conditions:

  1. (I.1)

$k_{1}$ is of the form

$\displaystyle k_{1}(z)=\left(\begin{matrix}a(z)&b(z)\\\
-b^{*}(z)&a^{*}(z)\end{matrix}\right),\qquad z\in S,$

where $a$ and $b$ are boundary values of holomorphic function in $\Sigma$ with
$a((0))>0$, and $a$ and $b$ do not simultaneously vanish at a point in
$\Sigma$.

  2. (I.2)

$k_{1}$ has a “root subgroup factorization” of the form

$\displaystyle
k_{1}(z)=\lim_{n\to\infty}\mathbf{a}(\eta_{n})\left(\begin{matrix}1&-\overline{\eta}_{n}\mathfrak{z}^{n}\\\
\eta_{n}\mathfrak{z}^{-n}&1\end{matrix}\right)\cdots\mathbf{a}(\eta_{0})\left(\begin{matrix}1&-\overline{\eta}_{0}\\\
\eta_{0}&1\end{matrix}\right),$

for some rapidly decreasing sequence $\\{\eta_{0},\dots,\eta_{n},\dots\\}$ of
complex numbers, and for some strictly equivariant function $\mathfrak{z}$
with $\mathfrak{z}((0))=0$.

  3. (I.3)

$k_{1}$ has a (multiplicative triangular) factorization of the form

$\displaystyle\left(\begin{matrix}1&0\\\
y^{*}(z)+y_{0}(z)&1\end{matrix}\right)\left(\begin{matrix}a_{1}&0\\\
0&a_{1}^{-1}\end{matrix}\right)\left(\begin{matrix}\alpha_{1}(z)&\beta_{1}(z)\\\
\gamma_{1}(z)&\delta_{1}(z)\end{matrix}\right),$

where $a_{1}>0$, the third factor is a ${\rm SL}(2,\mathbb{C})$-valued
holomorphic function in $\Sigma$ which is unipotent upper triangular at $0$,
$y=y_{+}$ is holomorphic in $\Sigma$, and $y_{0}$ is a zero mode $($as in
(1.1)$)$.

Then (I.2) implies (I.1) and (I.3) $($with $y_{0}=0)$, and (I.1) and (I.3) are
equivalent.

Similarly, consider $k_{2}\in C^{\infty}(S,{\rm SU}(2))$ and the following
statements:

  1. (II.1)

$k_{2}$ is of the form

$\displaystyle k_{2}(z)=\left(\begin{matrix}d^{*}(z)&-c^{*}(z)\\\
c(z)&d(z)\end{matrix}\right),\qquad z\in S^{1},$

where $c$ and $d$ are boundary values of holomorphic functions in $\Sigma$,
$c((0))=0$, and $d((0))>0$, and $c$ and $d$ do not simultaneously vanish at a
point in $\Sigma$.

  2. (II.2)

$k_{2}$ has a “root subgroup factorization” of the form

$\displaystyle
k_{2}(z)=\lim_{n\to\infty}\mathbf{a}(\zeta_{n})\left(\begin{matrix}1&\zeta_{n}\mathfrak{z}^{-n}\\\
-\overline{\zeta}_{n}\mathfrak{z}^{n}&1\end{matrix}\right)\cdots\mathbf{a}(\zeta_{1})\left(\begin{matrix}1&\zeta_{1}\mathfrak{z}^{-1}\\\
-\overline{\zeta}_{1}\mathfrak{z}&1\end{matrix}\right),$

for some rapidly decreasing sequence $\\{\zeta_{1},\dots,\zeta_{n},\dots\\}$
of complex numbers, and for some strictly equivariant $\mathfrak{z}$ with
$\mathfrak{z}((0))=0$.

  3. (II.3)

$k_{2}$ has $($triangular$)$ factorization of the form

$\displaystyle\left(\begin{matrix}1&x^{*}(z)+x_{0}(z)\\\
0&1\end{matrix}\right)\left(\begin{matrix}a_{2}&0\\\
0&a_{2}^{-1}\end{matrix}\right)\left(\begin{matrix}\alpha_{2}(z)&\beta_{2}(z)\\\
\gamma_{2}(z)&\delta_{2}(z)\end{matrix}\right),$

where $a_{2}>0$, the third factor is a ${\rm SL}(2,\mathbb{C})$-valued
holomorphic function which is unipotent upper triangular at $(0)$, $x=x_{+}$
is holomorphic in $\Sigma$, and $x_{0}$ is a zero mode $($as in (1.1)$)$.

Then (II.2) implies (II.1) and (II.3) $($with $x_{0}=0)$, and (II.1) and
(II.3) are equivalent.

######  Remark 1.2.

  1. (a)

In the classical case, all three conditions are equivalent (with
$\mathfrak{z}=z$), and there are no constraints on $y^{*}$ or $x^{*}$ in (I.3)
and (II.3), respectively (see [15]).

  2. (b)

In nonclassical cases there does not exist an equivariant function
$\mathfrak{z}$ for which all three conditions are equivalent (because for
example (II.2) implies $x_{0}=0$ in (II.3)). In this paper we will primarily
work with the conditions (I.1) and (I.3), and (II.1) and (II.3).

  3. (c)

The second condition shows how to generate examples of loops with $y_{0}=0$
and $x_{0}=0$ in (I.3) and (II.3), respectively. We lack a method to generate
transparent examples with $y_{0}\neq 0$ or $x_{0}\neq 0$. At this point we do
not know whether there are constraints on the set of $x^{*}+x_{0}$ which occur
in (II.3).

  4. (d)

Below we will explain why it is essential to generalize this theorem by
allowing $a$ and $b$ ($c$ and $d$) to simultaneously vanish in $\Sigma$ (in a
controlled way) in (I.1) ((II.1), respectively), and to allow $a_{1}$
($a_{2}$) to be a function in (I.3) ((II.3), respectively). A tentative step
in this direction is discussed in Section 3.1.

In the classical case a loop $g\colon S^{1}\to{\rm SU}(2)$ has a root subgroup
factorization

$\displaystyle
g=k_{1}^{*}\operatorname{diag}\big{(}e^{\chi},e^{-\chi}\big{)}k_{2}$

if and only if $g$ has a triangular factorization, a generic condition, and
these factorizations are unique (see [15]). In our more general context we ask
similar questions: can we characterize the set of $g$ which have a
factorization $g=k_{1}^{*}\operatorname{diag}(e^{\chi},e^{-\chi})k_{2}$, is
the relevant condition generic, and is the factorization unique? An immediate
obstacle is that, to our knowledge, there does not currently exist an analogue
of the theory of triangular (or Birkhoff or Riemann–Hilbert) factorization for
matrix valued multiloops in the context of this paper (see [18] for a survey
of results in this direction, and [4] or [16, Chapter 8] for further
background and history). Partly for this reason, we express what we know about
these questions in terms of the holomorphic bundle defined by viewing a loop
as a transition function. For example in the classical case, as observed by
Grothendieck, a loop $g\colon S^{1}\to{\rm SL}(2,\mathbb{C})$ has a
Riemann–Hilbert factorization $g=g_{-}g_{0}g_{+}$, where $g_{-}\in
H^{0}(\Delta^{*},{\rm SL}(2,\mathbb{C}))$, $g_{0}\in{\rm SL}(2,\mathbb{C})$,
and $g_{+}\in H^{0}(\Delta,{\rm SL}(2,\mathbb{C}))$, if and only if the
corresponding holomorphic ${\rm SL}(2,\mathbb{C})$ bundle on $\mathbb{P}^{1}$
is trivial (see [7], or [16, Chapter 8]); in the case of the sphere, such a
bundle is trivial if and only if it is semistable (for the definition and
basic properties of semistability of principal bundles, see [17]; for an
alternative exposition, which inspires some of the goals of our project, see
[2]).

######  Theorem 1.3.

Suppose $g\in C^{\infty}(S,{\rm SU}(2))$. If $g$ has a factorization

$\displaystyle g(z)=k_{1}^{*}(z)\left(\begin{matrix}e^{\chi(z)}&0\\\
0&e^{-\chi(z)}\end{matrix}\right)k_{2}(z),$

where $\chi\in C^{\infty}(S,i\mathbb{R})$, and $k_{1}$ and $k_{2}$ are as in
(I.1) and (II.1), respectively, of Theorem 1.1, then $E(g)$, the holomorphic
${\rm SL}(2,\mathbb{C})$ bundle on $\widehat{\Sigma}$ defined by $g$ as a
transition function, is semistable, and the associated bundle for the defining
representation has a sub-line bundle with an antiholomorphic reflection
symmetry compatible with $R$.

######  Remark 1.4.

  1. (a)

For a multiloop $g$ as in the theorem (or more generally for a multiloop with
minimal smoothness), we explain how $g$ defines a holomorphic bundle in
Appendix A.

  2. (b)

Because of the residual symmetry condition on the sub-line bundle in the
theorem, the existence of a factorization as in the theorem is not a generic
condition, except in the classical case (this is explained further in Appendix
A.4).

  3. (c)

As an example of our ignorance, suppose that $g\colon S\to{\rm SU}(2)$ has a
factorization as in the theorem. Is the same true for the inverse, $g^{*}$?
This is true in the classical case, because in the classical case there exists
a root subgroup factorization if and only if there is a triangular
factorization, and for the latter condition, $g=lmau$ is a triangular
factorization if and only if $g^{-1}=u^{*}m^{*}al^{*}$ is a triangular
factorization. We suspect the answer is negative in general.

A basic open question is whether a generic $g\colon S\to{\rm SU}(2)$ can be
factored as a product of ${\rm SU}(2)$-valued multiloops

$\displaystyle g(z)=\left(\begin{matrix}a^{*}(z)&-b(z)\\\
b^{*}(z)&a(z)\end{matrix}\right)\left(\begin{matrix}e^{\chi(z)}&0\\\
0&e^{-\chi(z)}\end{matrix}\right)\left(\begin{matrix}d^{*}(z)&-c^{*}(z)\\\
c(z)&d(z)\end{matrix}\right),$ (1.2)

where $a$, $b$, $c$, $d$ are appropriate boundary values (depending on the
smoothness of $g$) of holomorphic functions in $\Sigma$. Note that in Theorem
1.3, $a$ and $b$ ($c$ and $d$) are not allowed to simultaneously vanish in
$\Sigma$; the point is that this condition has to be relaxed. We will discuss
some results along these lines in Sections 3.1, 4.2 and 4.3.

###  1.1 Spin Toeplitz operators

Assume that $\Sigma$ has a spin structure. There is an induced spin structure
for $\widehat{\Sigma}$ which has an antiholomorphic reflection symmetry
compatible with $R$ (see [20, Chapter 7]). For simplicity we additionally
assume that the $\overline{\partial}$ operator for spinors on
$\widehat{\Sigma}$ is invertible. In this case there is a (pre-)Hilbert space
polarization for the space of ($\mathbb{C}^{2}$ valued) spinors along $S$,

$\displaystyle\Omega^{1/2}(S)\otimes\mathbb{C}^{2}=H^{1/2}(\Sigma)\otimes\mathbb{C}^{2}\oplus
H^{1/2}(\Sigma^{*})\otimes\mathbb{C}^{2},$

where $H^{1/2}(\Sigma)$ denotes the space of holomorphic spinors on $\Sigma$.
Given a (measurable) loop $g\colon S\to{\rm SU}(2)$, there is an associated
unitary multiplication operator $M_{g}$ on
$\Omega^{1/2}(S)\otimes\mathbb{C}^{2}$, and relative to the polarization

$\displaystyle M_{g}=\left(\begin{matrix}A(g)&B(g)\\\
C(g)&D(g)\end{matrix}\right).$

In the classical case $A(g)$ ($B(g)$) is the classical block Toeplitz operator
(Hankel operator, respectively), associated to the symbol $g$. In general the
‘spin Toeplitz (Hankel) operators’ $A(g)$ ($B(g)$, respectively) have many of
the same qualitative properties as in the classical case, because the
projection $\Omega^{1/2}\to H^{1/2}$ differs from the classical projection by
a smoothing operator.

######  Theorem 1.5.

Suppose that $g\colon S\to{\rm SU}(2)$ $($is smooth and$)$ has a factorization
as in Theorem 1.3. Then for any choice of spin structure for which
$\overline{\partial}$ is invertible,

$\displaystyle\det\big{(}A(g)A\big{(}g^{-1}\big{)}\big{)}=\det\big{(}A(k_{1})A\big{(}k_{1}^{-1}\big{)}\big{)}\det\big{(}\dot{A}(e^{\chi})\dot{A}(e^{-\chi})\big{)}^{2}\det\big{(}A(k_{2})A\big{(}k_{2}^{-1}\big{)}\big{)},$

where in the middle factor $\dot{A}(e^{\chi})$ is the compression to
$H^{1/2}(\Sigma)$ of $e^{\chi}$ as a $($scalar$)$ multiplication operator on
$\Omega^{1/2}(S)$ $($which accounts for the square of this factor$)$.

Even in the classical case, our proof of this is far more illuminating than
the one in [14]. For example we will prove the following much stronger
statement, at the level of operators:

######  Theorem 1.6.

Suppose that $k_{1}$, $k_{2}$ are measurable multiloops $S\to{\rm SU}(2))$ of
the form

$\displaystyle k_{1}=\left(\begin{matrix}a(z)&b(z)\\\
-b^{*}(z)&a^{*}(z)\end{matrix}\right)\qquad\text{and}\qquad
k_{2}=\left(\begin{matrix}d^{*}(z)&-c^{*}(z)\\\ c(z)&d(z)\end{matrix}\right),$

where $a$, $b$, $c$ and $d$ are boundary values of holomorphic function in
$\Sigma$. Then

$\displaystyle A(k_{1}^{*}k_{2})=A(k_{1}^{*})A(k_{2}).$

There are other senses in which the factors $k_{1}$, $\exp(\chi)$ and $k_{2}$
are expected to be “independent”. For example in the classical case we have
previously conjectured that these factors are independent random variables
with respect to the large temperature limit for Wiener measure on the loop
group, and that they Poisson commute with respect to the Evens–Lu homogeneous
Poisson structure on ${\rm LSU}(2)$ (see [14]). To properly formulate
nonclassical analogues of these conjectures, we need to prove the existence of
a factorization for a generic loop, as we discussed at the end of the previous
subsection.

######  Remark 1.7.

  1. (a)

One can replace the defining representation $\mathbb{C}^{2}$ by any
representation of ${\rm SU}(2)$ and there is a corresponding factorization.

  2. (b)

$A(g)$ is a Fredholm operator. As elegantly explained by Quillen and Segal,
$\det(A(g))$ can be defined as a section of a determinant line bundle, or
alternatively as a function on a central extension of maps $S\to{\rm SU}(2)$
(see [16]). In the classical case Theorem 1.5 can be refined to yield a
factorization for $\det(A(g))$, but this is open in general.

  3. (c)

$\det(A(g))$ is essentially the state corresponding to $\Sigma$ for the ${\rm
SU}(2)$ WZW model at level one. It is interesting to ask whether there might
be some factorization for the corresponding state at higher level. This is
true in the classical case, but for a trivial reason: the state at level $l$
is the $l$th power of the state at level one, i.e., one replaces the defining
representation by $(\mathbb{C}^{2})^{\otimes l}$.

###  1.2 Plan of the paper

In Section 2 we use ideas of Krichever and Novikov to obtain a linear
triangular factorization, as in (1.1), for a reasonable function $f\colon
S\to\mathbb{C}$ (in fact for any hyperfunction). In Section 3 we prove Theorem
1.1 and a technical refinement for Holder continuous loops. In Section 3.1 we
will present a generalization of Theorem 1.1 in which we relax the
simultaneously nonvanishing hypotheses in (I.1) and (II.1). In Sections 4 and
5, we prove Theorems 1.3 and 1.5, respectively. In Section 4.3 we further
discuss factorization and semistable bundles. In Section 5.2 we present some
calculations of determinants for spin Toeplitz operators in the scalar case
(the middle factor in Theorem 1.5); one suspects this can be substantially
improved. Finally there is Appendix A on hyperfunctions, which we use
sporadically in the text.

In part II of this paper [1], we will present a number of explicit
calculations, especially for elliptic and hyperelliptic surfaces. A motivating
question is whether it is possible to view spin Toeplitz operators in this
paper as deformations of classical Toeplitz operators by allowing the surface
to degenerate.

##  2 Linear triangular factorization

In the remainder of the paper, $\Sigma$ denotes a connected compact Riemann
surface with nonempty boundary $S$. Let $\widehat{\Sigma}$ denote the double,
i.e.,

$\displaystyle\widehat{\Sigma}=\Sigma^{*}\circ\Sigma,$

where $\Sigma^{*}$ is the adjoint of $\Sigma$, the surface $\Sigma$ with the
orientation reversed, and the composition is sewing along the common boundary
$S$. Let $R$ denote the antiholomorphic involution (or reflection) fixing $S$.

The singular cohomology group $H^{1}(\widehat{\Sigma},\mathbb{R})$ is a real
symplectic vector space with respect to wedge product. There is a positive
polarization of the complexification

$\displaystyle H^{1}\big{(}\widehat{\Sigma},\mathbb{C}\big{)}=H^{1,0}\oplus
H^{0,1},$

where we concretely identify $H^{1,0}$ with the complex
$\text{genus}(\widehat{\Sigma})$-dimensional vector space of holomorphic
differentials on $\widehat{\Sigma}$, and $H^{0,1}$ is the conjugate vector
space of antiholomorphic differentials. In this way we can think of $H^{1,0}$
as the dual of $H^{0,1}$ and vice versa. There is also a Dolbeault isomorphism

$\displaystyle H^{1}\big{(}\mathcal{O}\big{(}\widehat{\Sigma}\big{)}\big{)}\to
H^{0,1}\big{(}\widehat{\Sigma}\big{)},$

where $\mathcal{O}(\widehat{\Sigma})$ denotes the sheaf of holomorphic
functions (see [6, p. 45], but we will spell out what we need in (2.1)).

Let $H^{0}(S)$ denotes the space of analytic functions on $S$ (hence each such
function can be extended to a holomorphic function in an open neighborhood of
$S$ in $\widehat{\Sigma}$). Using Mayer–Vietoris for the sheaf of holomorphic
functions, corresponding to a slight thickening of the covering of
$\widehat{\Sigma}$ by $\Sigma$ and $\Sigma^{*}$, there is an exact sequence of
vector spaces

$\displaystyle 0\to H^{0}\big{(}\widehat{\Sigma}\big{)}\to H^{0}(\Sigma)\oplus
H^{0}(\Sigma^{*})\to H^{0}(S)\to H^{0,1}\big{(}\widehat{\Sigma}\big{)}\to 0.$

In terms of the Dolbeault isomorphism, the connecting map $H^{0}(S)\to
H^{0,1}(\widehat{\Sigma})$ is realized concretely as a map

$\displaystyle H^{0}(S)\to H^{0,1}\big{(}\widehat{\Sigma}\big{)}\colon\
f\to\begin{cases}\overline{\partial}f_{1}&\text{in}\ \Sigma,\\\
\overline{\partial}f_{2}&\text{in}\ \Sigma^{*},\end{cases}$ (2.1)

where in the $C^{\infty}$ category, $f=f_{1}-f_{2}$, $f_{1}$ is smooth in (a
slight open enlargement of) $\Sigma$, and $f_{2}$ is smooth in (a slight open
enlargement of) $\Sigma^{*}$.

In reference to the following statements, recall that the space of complex
hyperfunctions on an oriented analytic one-manifold is the dual of the space
of complex analytic one forms on the one-manifold, and is denoted by
$\operatorname{Hyp}(S)$; see Appendix A (and [19] for the original reference).
The introduction of hyperfunctions is possibly a distraction at this point of
the paper. The point is that eventually we will want to consider ‘transition
functions’ with minimal regularity; all such functions can be viewed as
hyperfunctions, and the natural domain for connecting maps, such as (2.1), is
a space of hyperfunctions.

######  Lemma 2.1.

The connecting map (2.1) is dual to the injective map

$\displaystyle C^{\omega}\Omega^{1}(S,\mathbb{C})\leftarrow
H^{1,0}\big{(}\widehat{\Sigma}\big{)}\equiv
H^{1}\big{(}\widehat{\Sigma},\mathbb{R}\big{)}\leftarrow 0$

given by restriction of a holomorphic differential to $S$. Consequently the
connecting map extends continuously to a surjective map

$\displaystyle\operatorname{Hyp}(S)\to
H^{0,1}\big{(}\widehat{\Sigma}\big{)}\equiv
H^{1}\big{(}\widehat{\Sigma},\mathbb{R}\big{)}\to 0.$

###### Proof.

We first show that the connecting map is dual to the restriction map. Suppose
that $f\in H^{0}(S)$ (i.e., functions which are holomorphic in a neighborhood
of $S$) and write $f=f_{1}-f_{2}$ as in the realization of the connecting map
(2.1), $f\to\overline{\partial}f_{j}$. Suppose also that $\omega\in
H^{1,0}(\widehat{\Sigma})$. To prove the formula for the transpose, we must
show that

$\displaystyle(\overline{\partial}f_{j},\omega)=(f,\omega|_{S}).$

The left-hand side equals (using $d\omega=0$ and Stokes’s theorem)

$\displaystyle\int_{\Sigma}\overline{\partial}f_{1}\wedge\omega+\int_{\Sigma^{*}}\overline{\partial}f_{2}\wedge\omega=\int_{S}f_{1}\wedge\omega-\int_{S}f_{2}\wedge\omega=\int_{S}f\wedge\omega$

and this equals the right-hand side.

The fact that the connecting map has a continuous extension to hyperfunctions
follows from the fact that it is the dual of the restriction map, which is
continuous. The extension is described more explicitly in Appendix A. The
basic idea (due to Sato) is that a hyperfunction $f$ on $S$ can be represented
as a pair, which we heuristically write as $f=f^{\prime}+f^{\prime\prime}$,
where $f^{\prime}$ is holomorphic in an annular region just outside $S$ and
$f^{\prime\prime}$ is holomorphic in an annular region just inside $S$. We
then compute the connecting map using three open sets, the interior of
$\Sigma$, a sufficiently small annular region containing $S$, and the interior
of $\Sigma^{*}$. One must check that this is independent of the choice of
representation for the hyperfunction: we can also represent
$f=(f^{\prime}+g)+(f^{\prime\prime}-g)$, where $g$ is holomorphic in a small
annular region containing $S$. ∎

Fix a basepoint $(0)\in\Sigma^{0}:=\text{interior}(\Sigma)$, and let
$(\infty)\in\Sigma^{*}$ denote the reflected basepoint. The starting point for
a series by papers by Krichever and Novikov (see for example [11], which
summarizes several of their papers) is the following

######  Lemma 2.2.

There exists a unique meromorphic differential $dk$ on $\widehat{\Sigma}$
which is holomorphic in $\widehat{\Sigma}{\setminus}\\{(0),(\infty)\\}$, has
simple poles at $(0),(\infty)$ with residues $\pm 1$, respectively, and
satisfies $\operatorname{Re}(\int_{\gamma}dk)$ $=0$, for all closed loops in
$\widehat{\Sigma}{\setminus}\\{(0),(\infty)\\}$.

###### Proof.

Let $\kappa$ denote the canonical line bundle, and $L$ the line bundle
corresponding to the divisor $(0)+(\infty)$. The holomorphic sections of the
line bundle $\kappa\otimes L$ are meromorphic differentials with at most
simple poles at $(0),(\infty)$. The Riemann–Roch theorem implies that

$\displaystyle\dim\big{(}H^{0}(\kappa\otimes
L)\big{)}-\dim\big{(}H^{1}(\kappa\otimes L)\big{)}=\deg(\kappa\otimes
L)-(g-1),$

where $g=\operatorname{genus}(\widehat{\Sigma})$. But

$\displaystyle\dim\big{(}H^{1}(\kappa\otimes
L)\big{)}=\dim\big{(}H^{0}\big{(}\kappa\otimes(\kappa\otimes
L)^{-1}\big{)}\big{)}=\dim\big{(}H^{0}\big{(}L^{-1}\big{)}\big{)}=0.$

Hence the dimension of the space of meromorphic differentials with at most
simple poles at $(0),(\infty)$ is $g+1$. A differential cannot have a single
pole. So any meromorphic differential in this space, which is not globally
holomorphic, will have simple poles at $(0)$ and $(\infty)$ with the sum of
the residues necessarily equal to zero. The normalizations in the statement of
the theorem uniquely determine $dk$. ∎

In the classical case $dk=dz/z$. Following Krichever and Novikov, set
$\tau=\operatorname{Re}(k)$, which is a single valued harmonic function on
$\widehat{\Sigma}{\setminus}\\{(0),(\infty)\\}$, which limits to $-\infty$ at
$(0)$ and $+\infty$ at $(\infty)$; this is thought of as a distinguished time
parameter.

The following summarizes a “linear triangular decomposition” for functions
which follows immediately from the existence of $dk$. The point is that the
existence of $dk$, which induces a metric on $S$, enables us to find a
complement to the sum of the two subspaces, holomorphic functions in the
interior of $\Sigma$, and holomorphic functions in the interior of
$\Sigma^{*}$.

######  Proposition 2.3.

A hyperfunction $\chi\colon\widehat{\Sigma}\to\mathbb{C}$ can be written
uniquely

$\displaystyle\chi=\chi_{-}+\chi_{0}+\chi_{+},$

where $\chi_{-}\in H^{0}(\Sigma^{0*},(\infty);\mathbb{C},0)$, $\chi_{+}\in
H^{0}(\Sigma^{0},(0);\mathbb{C},0)$, and $\chi_{0}dk$ is in the span of
$H^{1,0}(\widehat{\Sigma})$ and $dk$.

######  Remark 2.4.

  1. (a)

The zeros of $dk$ are the critical points for the time parameter $\tau$. For
$dk$ the number of zeros is $2\text{genus}(\widehat{\Sigma})$, because there
are two poles, and the degree of $\kappa$ is
$2\text{genus}(\widehat{\Sigma})-2$. The zeros, are located at the singular
points for the time slices of the surface.

  2. (b)

From (a) it follows that the set of $\chi_{0}$ can be characterized in the
following way: $\chi_{0}$ can have at most simple poles at the simple zeros of
$dk$, and $\chi_{0}(0)=\chi_{0}((\infty))$. (To explain the last condition,
note that $\chi_{0}dk=c_{0}dk+\text{holomorphic dif\/ferential}$, and hence
$\chi_{0}-c_{0}$ must vanish at $(0),(\infty)$.)

  3. (c)

Since the complementary functions $\chi_{0}$ can have poles at points other
than the basepoints $(0)$ and $(\infty)$, they do not necessarily belong to
the algebra $\mathcal{M}^{0}$ of meromorphic functions which are holomorphic
in $\widehat{\Sigma}{\setminus}\\{(0),(\infty)\\}$. Consequently our choice of
complementary subspace is not the same as the implicit choice made by
Krichiver and Novikov (this refers to the exceptional cases (1.5) after Lemma
1 in [11]). For emphasis, note our choice does not depend on the general
position hypothesis for $(0)$ in [11].

  4. (d)

Suppose that $\Sigma$ has genus zero. In this case there is an alternative
splitting, where $\chi_{0}$ is locally constant on $S$. This has the advantage
that there is no dependence on the choice of basepoint. We will consider this
elsewhere.

For smooth functions denote the decomposition in Proposition 2.3 by

$\displaystyle\Omega^{0}(S)=H_{+}\oplus H_{0}\oplus H_{-}.$

######  Lemma 2.5.

The action of multiplication by $F\in H^{0}(\Sigma)$ maps $H_{0}\oplus H_{+}$
into itself.

###### Proof.

First note it is obvious that multiplication by $F$ maps $H_{+}$ into itself.

Suppose that $\chi_{0}\in H_{0}$. Let $\chi_{0}^{(j)}$ denote a set of
functions in $H_{0}$ such that $\omega_{j}:=\chi_{0}^{(j)}dk$ is a basis for
the space of holomorphic differentials on $\widehat{\Sigma}$. We must show
that there exist constants $c_{1},\dots,c_{\text{genus}(\widehat{\Sigma})}$
such that (we write these as functions of a parameter $z$, to clarify the
notation)

$\displaystyle
F(z)\chi_{0}(z)-\sum_{j=1}^{\text{genus}(\widehat{\Sigma})}c_{j}\chi_{0}^{(j)}(z)\in
H^{0}(\Sigma),$

i.e., this function does not have any poles in $\Sigma$. Since the set of
poles of $F\chi_{0}$ (which are all simple) is a subset of the set of zeros of
$dk$, this is equivalent to showing the following. Let
$\\{z_{1}^{0},\dots,z^{0}_{\text{genus}(\widehat{\Sigma})}\\}$ denote the set
of zeros of $dk$ in $\Sigma$ (see (b) of Remark 2.4). Then there exist unique
constants $c_{1},\dots,c_{\text{genus}(\widehat{\Sigma})}$ such that

$\displaystyle
F(z)\chi_{0}(z)-\sum_{j=1}^{\text{genus}(\widehat{\Sigma})}c_{j}\chi_{0}^{(j)}(z)$

does not have any poles at the points
$z^{0}_{1},\dots,z^{0}_{\text{genus}(\widehat{\Sigma})}$.

To turn this into a system of scalar equations, for each $1\leq
i\leq\text{genus}(\widehat{\Sigma})$ choose a local coordinate $z_{i}$ with
$z_{i}(z_{i}^{0})=0$. Relative to this choice of coordinates, we can then
speak of residues. Our claim is then equivalent to the claim that the
$\text{genus}(\widehat{\Sigma})\times\text{genus}(\widehat{\Sigma})$ matrix
$\operatorname{res}^{j}_{i}:=\operatorname{Res}\big{(}\chi_{0}^{(j)},z^{0}_{i}\big{)}$
is nonsingular.

In the vicinity of $z^{0}_{i}$,

$\displaystyle\chi_{0}^{(j)}=\frac{\operatorname{res}^{j}_{i}}{z_{i}}+\text{holomorphic}(z_{i}),\qquad
dk=\big{(}f_{1}^{(i)}z_{i}+O\big{(}z_{i}^{2}\big{)}\big{)}dz_{i},$

where the first coefficient $f_{1}^{(i)}$ is not zero, and

$\displaystyle\chi_{0}^{(j)}dk=\big{(}\operatorname{res}_{i}^{j}f_{1}^{(i)}+O(z_{i})\big{)}dz_{i}.$

We can just as well show that the matrix
$\operatorname{res}_{i}^{j}f_{1}^{(i)}$ is nonsingular. In invariant terms we
must show that the evaluation map

$\displaystyle
H^{1,0}(\widehat{\Sigma})\to\bigoplus_{i=1}^{\text{genus}(\widehat{\Sigma})}T^{*}_{z_{i}}$

is an isomorphism of vector spaces. A holomorphic differential in the kernel
of this map would then have $2\text{genus}(\widehat{\Sigma})$ zeros and no
poles, which is impossible since the degree of $\kappa$ is
$2\text{genus}(\widehat{\Sigma})-2$. ∎

In the classical case it is a (difficult to prove but) well-known fact that if
$\chi$ is Holder continuous of order $s>0$, and $s$ is nonintegral, then
$\chi_{\pm}$ are Holder continuous of order $s$. This is equivalent to the
fact that the Hilbert transform $\mathcal{H}:=iP_{+}-iP_{-}\colon\chi\to
i\chi_{+}-i\chi_{-}$, which can be expressed as a principal value integral
relative to the Cauchy kernel

$\displaystyle\chi(z)\rightarrow\frac{1}{\pi}p.v.\int_{S^{1}}\frac{\chi(\zeta)}{\zeta-z}d\zeta$
(2.2)

is a bounded operator on $C^{s}(S^{1})$, the Banach space of Holder continuous
functions of order $s$ (the original reference in the classical case is [5,
Chapter III, Section 3]; this is cited for example in [4, p. 60]).

######  Theorem 2.6.

Suppose that $s>0$ and nonintegral. In reference to Proposition 2.3, if
$\chi\in C^{s}(S)$ $($i.e., Holder continuous of order $s)$, then
$\chi_{\pm}\in C^{s}(S)$.

This is equivalent to proving the continuity of an analogue of the Hilbert
transform on $C^{s}(S)$. The kernel is essentially the Szego kernel, which in
general is the classical Cauchy integral operator plus a smooth perturbation
(see [8], and [1] for examples). Thus Theorem 2.6 follows from the classical
case.

##  3 Proof of Theorem 1.1

In Theorem 1.1 it is obvious that (ii) implies (iii). This actually follows
from the classical example, because we can use $\mathfrak{z}$ to pull results
from the sphere back to $\widehat{\Sigma}$. It is also clear that (iii)
implies (i): This follows immediately by just multiplying the factors
together. In the course of completing the proof of Theorem 1.1, we will also
prove the following

######  Theorem 3.1.

Suppose that $k_{1}\in C^{s}(S,{\rm SU}(2))$, where $s>0$ and nonintegral. The
following are equivalent:

  1. (I.1)

$k_{1}$ is of the form

$\displaystyle k_{1}(z)=\left(\begin{matrix}a(z)&b(z)\\\
-b^{*}(z)&a^{*}(z)\end{matrix}\right),\qquad z\in S,$

where $a,b\in H^{0}(\Sigma)$ have $C^{s}$ boundary values, $a((0))>0$, and $a$
and $b$ do not simultaneously vanish at a point in $\Sigma$.

  2. (I.3)

$k_{1}$ has a factorization of the form

$\displaystyle\left(\begin{matrix}1&0\\\
y^{*}(z)+y_{0}(z)&1\end{matrix}\right)\left(\begin{matrix}a_{1}&0\\\
0&a_{1}^{-1}\end{matrix}\right)\left(\begin{matrix}\alpha_{1}(z)&\beta_{1}(z)\\\
\gamma_{1}(z)&\delta_{1}(z)\end{matrix}\right),$

where $y\in H^{0}(\Sigma^{0})$, $y_{0}$ is a zero mode, $a_{1}>0$, the last
factor is in $H^{0}(\Sigma^{0},{\rm SL}(2,\mathbb{C}))$ and is unipotent upper
triangular at the basepoint $(0)$, and the factors have $C^{s}$ boundary
values.

Similarly, the following are equivalent:

  1. (II.1)

$k_{2}$ is of the form

$\displaystyle k_{2}(z)=\left(\begin{matrix}d^{*}(z)&-c^{*}(z)\\\
c(z)&d(z)\end{matrix}\right),\qquad z\in S^{1},$

where $c,d\in H^{0}(\Sigma)$ have $C^{s}$ boundary values, $c((0))=0$,
$d((0))>0$, and $c$ and $d$ do not simultaneously vanish at a point in
$\Sigma$.

  2. (II.3)

$k_{2}$ has a factorization of the form

$\displaystyle\left(\begin{matrix}1&x^{*}(z)+x_{0}(z)\\\
0&1\end{matrix}\right)\left(\begin{matrix}a_{2}&0\\\
0&a_{2}^{-1}\end{matrix}\right)\left(\begin{matrix}\alpha_{2}(z)&\beta_{2}(z)\\\
\gamma_{2}(z)&\delta_{2}(z)\end{matrix}\right),$

where $x\in H^{0}(\Sigma^{0},(0);\mathbb{C},0)$, $x_{0}$ is a zero mode,
$x_{0}((0))=0$ $a_{2}>0$, the last factor is in $H^{0}(\Sigma^{0},{\rm
SL}(2,\mathbb{C}))$ and is upper triangular unipotent at the basepoint $(0)$,
and the factors have $C^{s}$ boundary values.

The triangular factorizations in (I.3) and (II.3) are uniquely determined.

######  Remark 3.2.

When $k_{2}$ is the restriction to $S$ of a function with entries in
$\mathcal{M}^{0}$, the determinant condition $c^{*}c+d^{*}d=1$ can be
interpreted as an equality of functions in $\mathcal{M}^{0}$. Together with
$d((0))>0$, this implies that $c$ and $d$ do not simultaneously vanish. Thus
the simultaneous vanishing hypotheses in (I.1) and (II.1) of Theorem 3.1 are
superfluous in that case.

###### Proof.

The two sets of conditions are proven in the same way. It is obvious that
(II.3) implies (II.1). The difficult task is to show that (II.1) implies
(II.3).

Consider the polarization

$\displaystyle\Omega^{0}\big{(}S,\mathbb{C}^{2}\big{)}=\mathcal{H}_{+}\otimes\mathbb{C}^{2}\oplus\mathcal{H}_{-}\otimes
C^{2},$

where $\mathcal{H}_{+}=H_{+}+\mathbb{C}$ (functions which are holomorphic in
$\Sigma$), and $\mathcal{H}_{-}$ is the sum of $H_{-}$ and the subspace of
zero modes $x_{0}\in H_{0}$ such that $x_{0}dk$ is a global holomorphic
differential (i.e., $x_{0}((0))=0$, as in the statement of (II.3)). We write
$P_{\pm}$ for the corresponding projections. We will view
$\Omega^{0}(S,\mathbb{C}^{2})$ as a preHilbert space, by using the measure
induced by the restriction of $(2\pi i)^{-1}dk$ to $S$. (Note that for a
differential $\omega$, if $\omega^{*}:=\overline{R^{*}\omega}$, then
$(dk)^{*}=-dk$, as follows by checking the asymptotics at $(0)$ and
$(\infty)$. Thus along $S$, $(2\pi i)^{-1}dk$ is real. It is also
nonvanishing; see (a) of Remark 2.4. The polarization is an orthogonal direct
sum.)

Relative to this polarization, we write the unitary multiplication operator
corresponding to a multiloop $g\colon S\to{\rm SU}(2)$ as

$\displaystyle M_{g}=\left(\begin{matrix}A&B\\\ C&D\end{matrix}\right).$

For a smooth loop the off diagonals (Hankel type operators) will be small and
$A$ and $D$ will be Fredholm.

We must show that the multiloop $k_{2}$ has a unique factorization as in
(II.3), i.e., we must solve for $a_{2}$, $x^{*}$, and so on, in

$\displaystyle\left(\begin{matrix}d^{*}&-c^{*}\\\
c&d\end{matrix}\right)=\left(\begin{matrix}1&x^{*}+x_{0}\\\
0&1\end{matrix}\right)\left(\begin{matrix}a_{2}&0\\\
0&a_{2}^{-1}\end{matrix}\right)\left(\begin{matrix}\alpha_{2}&\beta_{2}\\\
\gamma_{2}&\delta_{2}\end{matrix}\right).$

The second row implies

$\displaystyle(c,d)=a_{2}^{-1}(\gamma_{2},\delta_{2}).$

This determines $a_{2}^{-1}\gamma_{2}$ and $a_{2}^{-1}\delta_{2}$, and (using
the unipotence of the third factor in (II.3)), $a_{2}^{-1}=d((0))$.

The first row implies

$\displaystyle
d^{*}=a_{2}\alpha_{2}+(x^{*}+x_{0})c,\qquad-c^{*}=a_{2}\beta_{2}+(x^{*}+x_{0})d.$
(3.1)

The first term on the right-hand side of both of these equations belongs to
$\mathcal{H}_{+}$. To solve for $x^{*}+x_{0}$, we will apply a projection to
get rid of these $\mathcal{H}_{+}$ terms.

Consider the operator

$\displaystyle T\colon\
\mathcal{H}_{-}\to\mathcal{H}_{-}\oplus\mathcal{H}_{-}\colon\
x_{0}+x^{*}\to((c(x_{0}+x^{*}))_{-},(d(x_{0}+x^{*}))_{-}),$

where $(\cdot)_{-}$ is the projection to $\mathcal{H}_{-}$. To show that we
can uniquely solve for $x^{*}+x_{0}$, we will show $T$ is injective and that
$((d^{*})_{-},(-c^{*})_{-})$ is in the range of $T$.

The operator $T$ is the restriction of the Fredholm operator
$D(k_{2})^{*}=D(k_{2}^{*})$ to the subspace $\\{(x_{0}+x^{*},0)\\}$,
consequently the image of $T$ is closed. $T$ is also injective. For suppose
that both $((c(x_{0}+x^{*}))_{-}$ and $(d(x_{0}+x^{*}))_{-}$ vanish. Then

$\displaystyle c(x_{0}+x^{*})=g\qquad\text{and}\qquad d(x_{0}+x^{*})=h,$

where $g,h\in\mathcal{H}_{+}$. Since $c$, $d$ do not simultaneously vanish in
$\Sigma$, this implies that $x_{0}+x^{*}\in\mathcal{H}_{+}$. But this means
that $x_{0}+x^{*}$ must vanish. Thus $T$ is injective and has a closed image.

The adjoint of $T$ is given by

$\displaystyle T^{*}\colon\
\mathcal{H}_{-}\oplus\mathcal{H}_{-}\to\mathcal{H}_{-}\colon\
(f_{0}+f^{*},g_{0}+g^{*})\to(c^{*}(f_{0}+f^{*})+d^{*}(g_{0}+g^{*}))_{-}.$

If $(f_{0}+f^{*},g_{0}+g^{*})\in\operatorname{ker}(T^{*})$, then

$\displaystyle c^{*}f^{*}+d^{*}g^{*}+(c^{*}f_{0}+d^{*}g_{0})_{-}=0.$

By Lemma 2.5

$\displaystyle(c^{*}f_{0}+d^{*}g_{0})_{-}=c^{*}f_{0}+d^{*}g_{0}.$

Thus

$\displaystyle c^{*}(f_{0}+f^{*})+d^{*}(g_{0}+g^{*})=0$

viewed as a meromorphic function in $\Sigma^{*}$, vanishes in the closure of
$\Sigma^{*}$. Because $|c|^{2}+|d|^{2}=1$ around $S$,

$\displaystyle(f_{0}+f^{*},g_{0}+g^{*})=\lambda^{*}(d^{*},-c^{*}),$ (3.2)

where $\lambda^{*}$ is meromorphic in $\Sigma^{*}$ and vanishes at $(\infty)$
because $d^{*}((\infty))=d((0))>0$. Thus $\lambda$ is meromorphic in $\Sigma$
and vanishes at $(0)$.

We now claim that $((d^{*})_{-},-c^{*})\in\operatorname{ker}(T^{*})^{\perp}$.
To prove this, suppose that
$(f_{0}+f^{*},g_{0}+g^{*})\in\operatorname{ker}(T^{*})$, as in the previous
paragraph. Then

$\displaystyle\int_{S}((d^{*})_{-}(f_{0}^{*}+f)+(-c^{*})(g_{0}^{*}+g))dk=\int_{S}\lambda(d^{*}d+c^{*}c)dk=\int_{S}\lambda
dk.$

(Note that constants are orthogonal to $\mathcal{H}_{-}$, and hence we could
replace $(d^{*})_{-}$ by $d^{*}$.) We claim this integral vanishes. Since
$\lambda dk$ is a meromorphic differential in $\Sigma$ (by the previous
paragraph), this equals the sum of residues of $\lambda dk$ in $\Sigma$.
Because $f_{0}^{*}dk$ and $g_{0}^{*}dk$ are holomorphic differentials (the
defining characteristic for zero modes), and $c$ and $d$ do not simultaneously
vanish, the only point we need to worry about is $(0)$ (see (3.2)). Finally
the residue at $(0)$ is zero, because $\lambda((0))=0$ (by the previous
paragraph).

Because $T$ has closed image, there exists $x_{0}+x^{*}\in\mathcal{H}_{-}$
such that

$\displaystyle(d^{*})_{-}=((x_{0}+x^{*})c)_{-}\qquad\text{and}\qquad-c^{*}=((x_{0}+x^{*})d)_{-}.$

We can now solve for $a_{2}\alpha_{2}$ and $a_{2}\beta_{2}$ in (3.1). We
previously noted that $a_{2}^{-1}=d((0))$. Now that we have solved for the
factors, the form of the factorization immediately implies
$\alpha_{2}\delta_{2}-\gamma_{2}\beta_{2}=1$ on $S$; by holomorphicity of the
terms, this also holds in $\Sigma$.

Theorem 2.6 implies that the projections $P_{\pm}$ are continuous on
$C^{s}(S)$. Consequently the operator $T$ will be continuous and have the same
properties on the $C^{s}$ completions. Hence when $k_{2}\in C^{s}$, the
factors are $C^{s}$. This completes the proofs of Theorems 1.1 and 3.1. ∎

###  3.1 A generalization of Theorem 1.1

In the preceding proof a key step is to show that $\lambda dk$ is a
holomorphic differential in $\Sigma$. The proof is not sharp, in the sense
that $\lambda$ itself is holomorphic, whereas for the proof to go through, we
can allow $\lambda$ to have simple poles at the zeros of $dk$. By considering
(3.2), this suggests that there ought to be a generalization in which $c$ and
$d$ are allowed to have simultaneous zeros to first order, within the zero set
of $dk$. This in turn forces $a_{2}$ to be a function. It turns out to be
natural for $a_{1}$ and $a_{2}$ to be functions, for other reasons, as we will
explain in Section 4.3.

######  Theorem 3.3.

Suppose that $k_{1}\in C^{s}(S,{\rm SU}(2))$, where $s>0$ and nonintegral. The
following are equivalent:

  1. (I.1)

$k_{1}$ is of the form

$\displaystyle k_{1}(z)=\left(\begin{matrix}a(z)&b(z)\\\
-b^{*}(z)&a^{*}(z)\end{matrix}\right),\qquad z\in S,$

where $a,b\in H^{0}(\Sigma)$ have $C^{s}$ boundary values, $a((0))>0$, and $a$
and $b$ can simultaneously vanish only to first order, and this set of common
zeros is the set of poles in $\Sigma$ of a zero mode.

  2. (I.3)

$k_{1}$ has a factorization of the form

$\displaystyle\left(\begin{matrix}1&0\\\
y^{*}(z)+y_{0}(z)&1\end{matrix}\right)\left(\begin{matrix}a_{1}(z)&0\\\
0&a_{1}(z)^{-1}\end{matrix}\right)\left(\begin{matrix}\alpha_{1}(z)&\beta_{1}(z)\\\
\gamma_{1}(z)&\delta_{1}(z)\end{matrix}\right),$

where $a_{1}^{-1}$ is a nonvanishing sum of a zero mode and a holomorphic
function in $\Sigma$, $a_{1}((0))>0$, $y\in H^{0}(\Sigma^{0})$, the last
factor is in $H^{0}(\Sigma^{0},{\rm SL}(2,\mathbb{C}))$ and is unipotent upper
triangular at the basepoint $(0)$, and the factors have $C^{s}$ boundary
values.

Similarly, the following are equivalent:

  1. (II.1)

$k_{2}$ is of the form

$\displaystyle k_{2}(z)=\left(\begin{matrix}d^{*}(z)&-c^{*}(z)\\\
c(z)&d(z)\end{matrix}\right),\qquad z\in S^{1},$

where $c,d\in H^{0}(\Sigma)$ have $C^{s}$ boundary values, $c((0))=0$,
$d((0))>0$, and $c$ and $d$ can simultaneously vanish only to first order, and
this set of common zeros is the set of poles in $\Sigma$ of a zero mode.

  2. (II.3)

$k_{2}$ has a factorization of the form

$\displaystyle\left(\begin{matrix}1&x^{*}(z)+x_{0}(z)\\\
0&1\end{matrix}\right)\left(\begin{matrix}a_{2}(z)&0\\\
0&a_{2}(z)^{-1}\end{matrix}\right)\left(\begin{matrix}\alpha_{2}(z)&\beta_{2}(z)\\\
\gamma_{2}(z)&\delta_{2}(z)\end{matrix}\right),$

where $a_{2}$ is a nonvanishing sum of a zero mode and a holomorphic function
in $\Sigma$, $a_{2}((0)){>}0$, $x\in H^{0}(\Sigma^{0},(0);\mathbb{C},0)$,
$x_{0}$ is a zero mode, the last factor is in $H^{0}(\Sigma^{0},{\rm
SL}(2,\mathbb{C}))$ and is upper triangular unipotent at the basepoint $(0)$,
and the factors have $C^{s}$ boundary values.

######  Remark 3.4.

  1. (a)

In contrast to Theorem 1.1, we are not asserting that the triangular
factorizations in (I.3) and (II.3) are uniquely determined. In particular, as
the theorem is stated, $a_{1}$ and $a_{2}$ can be multiplied by nonvanishing
holomorphic functions in $\Sigma$.

  2. (b)

In the preceding theorem $a_{1}^{-1}$ and $a_{2}$ are not generally zero
modes. Nonconstant zero modes always vanish at some point in $\Sigma$, and
this would mean that for example $c$ or $d$ would have a pole. We want to
avoid poles, because for example the results in Section 5 are dependent on the
assumption that $a$, $b$, $c$, $d$ are holomorphic in $\Sigma$.

###### Proof.

The two sets of conditions are proven in the same way. It is obvious that
(II.3) implies (II.1). As before, the difficult task is to show that (II.1)
implies (II.3).

Assuming (II.1), we must show that the multiloop $k_{2}$ has a factorization
as in (II.3), i.e., we must prove existence of $a_{2}$, $x^{*}$, and so on, in

$\displaystyle\left(\begin{matrix}d^{*}&-c^{*}\\\
c&d\end{matrix}\right)=\left(\begin{matrix}1&x^{*}+x_{0}\\\
0&1\end{matrix}\right)\left(\begin{matrix}a_{2}&0\\\
0&a_{2}^{-1}\end{matrix}\right)\left(\begin{matrix}\alpha_{2}&\beta_{2}\\\
\gamma_{2}&\delta_{2}\end{matrix}\right).$

The second row implies

$\displaystyle(c,d)=a_{2}^{-1}(\gamma_{2},\delta_{2}).$

This determines $a_{2}^{-1}\gamma_{2}$ and $a_{2}^{-1}\delta_{2}$. In the
context of the previous subsection, i.e., when $c$ and $d$ do not
simultaneously vanish, $a_{2}$ is a constant, determined by the unipotence of
the third factor in (II.3), which (in general) implies
$a_{2}^{-1}((0))=d((0))$. In general, towards determining $a_{2}$, note that
$a_{2}$ must have simple poles at the common zeros for $c$ and $d$, and cannot
have zeros in $\Sigma$, so that $\gamma_{2}$ and $\delta_{2}$ are holomorphic
and do not simultaneously vanish in $\Sigma$. The hypotheses of (II.1)
guarantee that there is a zero mode with the appropriate (necessarily simple)
poles in $\Sigma$. Fix a slight open enlargement of $\Sigma$. Since this
surface is open, we can find a nonvanishing meromorphic function in this
enlargement with the same singular behavior as this zero mode; the difference
between this meromorphic function, which is our choice for $a_{2}$, and the
given zero mode is holomorphic in $\Sigma$. As we remarked above, this choice
is far from uniquely determined, and it is not clear how to pin down a
preferred choice.

The first row implies

$\displaystyle
d^{*}=a_{2}\alpha_{2}+(x^{*}+x_{0})c,\qquad-c^{*}=a_{2}\beta_{2}+(x^{*}+x_{0})d.$

The basic complication, compared to the proof of Theorem 1.1, is that
$a_{2}\alpha_{2}$ and $a_{2}\beta_{2}$ are no longer necessarily holomorphic
in $\Sigma$, because $a_{2}$ has poles. We will first solve (not necessarily
uniquely) for $x^{*}$, by using the same strategy as in the proof of Theorem
1.1, but using a different polarization.

Consider the polarization

$\displaystyle\Omega^{0}\big{(}S,\mathbb{C}^{2}\big{)}=\mathcal{H}_{+}\otimes
C^{2}\oplus\mathcal{H}_{-}\otimes C^{2},$

where $\mathcal{H}_{+}$ is now the sum of functions which are holomorphic in
$\Sigma$ and zero modes, and $\mathcal{H}_{-}=H_{-}$. We write $P_{+}$, and
$P_{-}$ for the projections onto the subspaces
$\mathcal{H}_{+}\otimes\mathbb{C}^{2}$ and $H_{-}\otimes C^{2}$, respectively.

Relative to this polarization, we write the multiplication operator defined by
a multiloop as

$\displaystyle M_{g}=\left(\begin{matrix}A&B\\\ C&D\end{matrix}\right).$

For a smooth loop the off diagonals (Hankel type operators) will be small and
$A$ and $D$ will be Fredholm.

Consider the operator

$\displaystyle T\colon\
\mathcal{H}_{-}\to\mathcal{H}_{-}\oplus\mathcal{H}_{-}\colon\
x^{*}\to(((cx^{*})_{-},(dx^{*})_{-}),$

where $(\cdot)_{-}$ is shorthand for the projection to $\mathcal{H}_{-}$. The
operator $T$ is the restriction of the Fredholm operator
$D(k_{2})^{*}=D(k_{2}^{*})$ to the subspace $\\{(x^{*},0)\\}$, consequently
the image of $T$ is closed. Unfortunately in general $T$ is not injective. For
suppose that both $((cx^{*})_{-}$ and $(dx^{*})_{-}$ vanish. Then

$\displaystyle cx^{*}=g\qquad\text{and}\qquad dx^{*}=h,$

where $g,h\in\mathcal{H}_{+}$. At a common simple zero for $c$ and $d$, it
could happen that $g$, $h$ also have a common pole, so $x^{*}$ does not
necessarily have simple poles.

The adjoint of $T$ is given by

$\displaystyle T^{*}\colon\
\mathcal{H}_{-}\oplus\mathcal{H}_{-}\to\mathcal{H}_{-}\colon\
(f^{*},g^{*})\to\left(c^{*}f^{*}+d^{*}g^{*}\right)_{-}.$

If $(f^{*},g^{*})\in\operatorname{ker}(T^{*})$, then

$\displaystyle c^{*}f^{*}+d^{*}g^{*}=0$

viewed as a meromorphic function in $\Sigma^{*}$. Because $|c|^{2}+|d|^{2}=1$
around $S$,

$\displaystyle(f^{*},g^{*})=\lambda^{*}(d^{*},-c^{*}),$

where $\lambda^{*}$ is meromorphic in $\Sigma^{*}$ and vanishes at $(\infty)$
because $d^{*}((\infty))=d((0))>0$. Thus $\lambda$ is meromorphic in $\Sigma$
and vanishes at $(0)$.

We now claim that $((d^{*})_{-},-c^{*})\in\operatorname{ker}(T^{*})^{\perp}$.
To prove this, suppose that $(f^{*},g^{*})\in\operatorname{ker}(T^{*})$, as in
the previous paragraph. Then

$\displaystyle\int_{S}((d^{*})_{-}(f)+(-c^{*})(g))dk=\int_{S}\lambda(d^{*}d+c^{*}c)dk=\int_{S}\lambda
dk.$

We claim this integral vanishes. Since $\lambda dk$ is a meromorphic
differential in $\Sigma$ (by the previous paragraph), this equals the sum of
residues of $\lambda dk$ in $\Sigma$. Because the poles of $\lambda$ occur at
common zeros of $c$, $d$, and hence only at zeros of $dk$, and because
$\lambda((0))=0$, $\lambda dk$ is holomorphic in $\Sigma$ (by the previous
paragraph). Thus the integral vanishes.

Because $T$ has closed image, there exists $x^{*}\in\mathcal{H}_{-}$ such that

$\displaystyle(d^{*})_{-}=(x^{*}c)_{-}\qquad\text{and}\qquad-c^{*}=(x^{*}d)_{-}.$

(Here and below we are using the fact that $a_{2}\in H_{0}+H_{+}$,
$\alpha_{2}\in H_{+}$, and Lemma 2.5, to conclude that $a_{2}\alpha_{2}\in
H_{0}+H_{+}$, so that it is killed by $(\cdot)_{-}$.)

Now consider the operator

$\displaystyle\mathcal{T}\colon\ (H_{+}+\mathbb{C})\oplus H_{0}\oplus
H_{+}\rightarrow(H_{+}+H_{0})\oplus(H_{+}+H_{0})\colon$
$\displaystyle\hphantom{\mathcal{T}\colon}\
(\alpha_{2},x_{0},\beta_{2})\to(a_{2}\alpha_{2}+x_{0}c,a_{2}\beta_{2}+x_{0}d).$

This is well-defined by Lemma 2.5.

We must show that $(d^{*}-x_{0}c,-c^{*}-x_{0}d)$ (which is in the target of
$\mathcal{T}$ by the previous paragraph) is in the image of $\mathcal{T}$. The
operator $\mathcal{T}$ has a closed image. The (Hilbert space) adjoint of
$\mathcal{T}$ is given by

$\displaystyle(H_{+}+H_{0})\oplus(H_{+}+H_{0})\to(H_{+}+\mathbb{C})\oplus
H_{0}\oplus H_{+}\colon$
$\displaystyle(f,g)\to((a_{2}^{*}f)_{0+},(a_{2}^{*}g)_{0+},(c^{*}f+d^{*}g)_{0}).$
(3.3)

We need to show that $(d^{*}-x_{0}c,-c^{*}-x_{0}d)$ is orthogonal to the
kernel of $\mathcal{T}^{*}$. Suppose that
$(f,g)\in\operatorname{ker}(\mathcal{T}^{*})$ (so the three terms in (3.3)
vanish). Then

$\displaystyle(d^{*}-x_{0}c,f)+(-c^{*}-x_{0}d,g)=\text{const}\cdot\int_{S}(df^{*}-cg^{*}-x_{0}^{*}(c^{*}f+d^{*}g))dk.$

The last two terms immediately drop out. Thus, up to a constant, this equals

$\displaystyle\int_{S}(df^{*}-cg^{*})dk=\int_{S}\big{(}\big{(}da_{2}^{-1}\big{)}a_{2}f^{*}-\big{(}ca_{2}^{-1}\big{)}a_{2}g^{*}\big{)}dk.$

The vanishing of the first two terms in (3.3) imply that
$a_{2}f^{*},a_{2}g^{*}\in H_{+}$. Thus the integral vanishes since the
integrand is holomorphic in $\Sigma$.

Once we have solved for the factors, the form of the factorization immediately
implies $\alpha_{2}\delta_{2}-\gamma_{2}\beta_{2}=1$ on $S$; by holomorphicity
of the terms, this also holds in $\Sigma$. This completes the proof. ∎

##  4 Factorization and semistability

###  4.1 Proof of Theorem 1.3

We recall the statement of the theorem:

######  Theorem 4.1.

Suppose $g\in C^{\infty}(S,{\rm SU}(2))$. If $g$ has a factorization

$\displaystyle g(z)=k_{1}^{*}(z)\left(\begin{matrix}e^{\chi(z)}&0\\\
0&e^{-\chi(z)}\end{matrix}\right)k_{2}(z),$

where $\chi\in C^{\infty}(S,i\mathbb{R})$, and $k_{1}$ and $k_{2}$ are as in
(I.1) and (II.1), respectively, of Theorem 1.1, then $E(g)$, the holomorphic
$G$ bundle on $\widehat{\Sigma}$ defined by $g$ as a transition function, is
semistable, and the associated bundle for the defining representation has a
sub-line bundle with an antiholomorphic reflection symmetry compatible with
$R$.

######  Remark 4.2.

In the following proof, $a_{1}$ is a positive constant and hence
$a_{1}^{*}=a_{1}$. We will write $a_{1}^{*}$ at various points in the proof
because in the next subsection we will want to note that the calculations are
valid more generally when $a_{1}$ is a function.

###### Proof.

In the proof we will use the fact that, using a multiloop as a transition
function, there is a bijective correspondence between the double coset space

$\displaystyle
H^{0}\big{(}\Sigma^{0*},G\big{)}\backslash\operatorname{Hyp}(S,G)/H^{0}\big{(}\Sigma^{0},G\big{)}$

and the set of isomorphism classes of holomorphic $G$ bundles on
$\widehat{\Sigma}$, and that a smooth function defines a hyperfunction; see
Appendix A.3.

By assumption $k_{1}$ and $k_{2}$ have ‘triangular factorizations’ of the
following forms:

$\displaystyle k_{1}=\left(\begin{matrix}1&0\\\
y^{*}+y_{0}&1\end{matrix}\right)\left(\begin{matrix}a_{1}&0\\\
0&a_{1}^{-1}\end{matrix}\right)\left(\begin{matrix}\alpha_{1}&\beta_{1}\\\
\gamma_{1}&\delta_{1}\end{matrix}\right),$

and

$\displaystyle k_{2}=\left(\begin{matrix}1&x^{*}+x_{0}\\\
0&1\end{matrix}\right)\left(\begin{matrix}a_{2}&0\\\
0&a_{2}^{-1}\end{matrix}\right)\left(\begin{matrix}\alpha_{2}&\beta_{2}\\\
\gamma_{2}&\delta_{2}\end{matrix}\right).$

Given these ‘triangular factorizations’ for $k_{1}$ and $k_{2}$, we can derive
a ‘triangular factorization’ for $g$. (Note: we have not developed a general
theory of triangular factorization in the context of this paper, so we are
using this term somewhat loosely.) To simplify notation, let
$X=a_{2}^{-2}(x+x_{0}^{*})$, $Y=a_{1}^{2}(y+y_{0}^{*})$, and
$a=a_{1}^{*}a_{2}$. Then

$\displaystyle g=\left(\begin{matrix}\alpha_{1}&\beta_{1}\\\
\gamma_{1}&\delta_{1}\end{matrix}\right)^{*}\left(\begin{matrix}1&Y\\\
0&1\end{matrix}\right)\left(\begin{matrix}ae^{\chi_{-}+\chi_{0}+\chi_{+}}&0\\\
0&(ae^{\chi_{-}+\chi_{0}+\chi_{+}})^{-1}\end{matrix}\right)\left(\begin{matrix}1&X^{*}\\\
0&1\end{matrix}\right)\left(\begin{matrix}\alpha_{2}&\beta_{2}\\\
\gamma_{2}&\delta_{2}\end{matrix}\right)$
$\displaystyle\hphantom{g}{}=\left(\begin{matrix}\alpha_{1}^{*}&\gamma_{1}^{*}\\\
\beta_{1}^{*}&\delta_{1}^{*}\end{matrix}\right)\left(\begin{matrix}e^{\chi_{-}}&0\\\
0&e^{-\chi_{-}}\end{matrix}\right)\left(\begin{matrix}1&e^{-2\chi_{-}}Y\\\
0&1\end{matrix}\right)\left(\begin{matrix}ae^{\chi_{0}}&0\\\
0&(ae^{\chi_{0}})^{-1}\end{matrix}\right)\left(\begin{matrix}1&e^{2\chi_{+}}X^{*}\\\
0&1\end{matrix}\right)$
$\displaystyle\hphantom{g=}{}\times\left(\begin{matrix}e^{\chi_{+}}&0\\\
0&e^{-\chi_{+}}\end{matrix}\right)\left(\begin{matrix}\alpha_{2}&\beta_{2}\\\
\gamma_{2}&\delta_{2}\end{matrix}\right).$

The bundle $E(g)$ defined by $g$ as a transition function depends only on the
product of the middle three factors, because of the remark at the beginning of
the proof. The product of the middle three factors equals

$\displaystyle\left(\begin{matrix}ae^{\chi_{0}}&B\\\
0&(ae^{\chi_{0}})^{-1}\end{matrix}\right),$

where

$\displaystyle B=a^{-1}e^{-\chi_{0}-2\chi_{-}}Y+ae^{\chi_{0}+2\chi_{+}}X^{*}.$

We claim that the rank two bundle defined by this transition function is
semistable. Suppose otherwise. Then there exists a sub-line bundle which has
positive degree. The degree of this sub-line bundle is the negative of the
degree of a transition function $S\to{\rm GL}(1,\mathbb{C})$ (see our
conventions for transition functions in Remark A.1). This means that there
exists a factorization of the form

$\displaystyle\left(\begin{matrix}ae^{\chi_{0}}&B\\\
0&(ae^{\chi_{0}})^{-1}\end{matrix}\right)=g_{-}\left(\begin{matrix}\lambda&\beta\\\
0&\lambda^{-1}\end{matrix}\right)g_{+}^{-1},$

where $g_{\pm}=\left(\begin{matrix}\alpha_{\pm}&\beta_{\pm}\\\
\gamma_{\pm}&\delta_{\pm}\end{matrix}\right)$ are holomorphic ${\rm
SL}(2,\mathbb{C})$-valued functions in $\Sigma$ ($\Sigma^{*}$, respectively),
and the degree of $\lambda\colon S\to{\rm GL}(1,\mathbb{C})$ is negative. This
factorization implies

$\displaystyle ae^{\chi_{0}}\gamma_{-}=\lambda^{-1}\gamma_{+}$

and hence the line bundle defined by the transition function
$(ae^{-\chi_{0}}\lambda)^{-1}$ has a global holomorphic section. But this is
impossible, because the degree of this transition function is positive (i.e.,
the degree of the corresponding line bundle is negative). This implies that
$E(g)$ is semistable. (Note that in general, a principal bundle is semistable
if and only if the corresponding adjoint bundle is semistable; however for
${\rm SL}(2,\mathbb{C})$ it suffices to consider the defining representation,
see [17].)

Because $\chi_{0}=-\chi_{0}^{*}$, and because $a$ is a positive constant, the
line bundle defined by the transition function $ae^{\chi_{0}}$ has
antiholomorphic symmetry with respect to $R$. This completes the proof. ∎

###  4.2 A generalization of Theorem 1.3

######  Theorem 4.3.

Suppose $g\in C^{\infty}(S,{\rm SU}(2))$. If $g$ has a factorization

$\displaystyle g(z)=k_{1}^{*}(z)\left(\begin{matrix}e^{\chi(z)}&0\\\
0&e^{-\chi(z)}\end{matrix}\right)k_{2}(z),$

where $\chi\in C^{\infty}(S,i\mathbb{R})$, and $k_{1}$ and $k_{2}$ are as in
(I.1) and (II.1), respectively, of Theorem 3.3, then $E(g)$, the holomorphic
$G$ bundle on $\widehat{\Sigma}$ defined by $g$ as a transition function, is
semistable.

###### Proof.

For $a_{1}$ and $a_{2}$ as in Theorem 3.3 (viewed as transition functions)

$\displaystyle\deg(a_{1}^{*}a_{2})=\deg\big{(}a_{1}^{-1}\big{)}+\deg(a_{2}).$

Since $a_{1}^{-1}$ and $a_{2}$ do not vanish in $\Sigma$, this degree is the
negative of the sum of the number of common zeros of $(a,b)$ in $\Sigma$ and
the number of common zeros of $(c,d)$ in $\Sigma$. In particular the degree of
$a=a_{1}^{*}a_{2}$ is negative. With the exception of the last paragraph, we
can now repeat the preceding proof verbatim; the key point is that the degree
of the transition function $(ae^{-\chi_{0}}\lambda)^{-1}$, calculated in the
penultimate paragraph, remains positive. ∎

The main point of this theorem is that the antiholomorphic symmetry has been
broken. Consequently there is now a chance that the factorization of $g$, as
in the theorem, is generic.

###  4.3 Do there exist converses?

As of this writing, the questions of whether there are converses to Theorems
1.3 and 4.3 are open. We briefly discuss the issues involved. In this
subsection, for simplicity, we assume that $S$ has one connected component. We
first consider Theorem 1.3.

Suppose that $g\in C^{\omega}(S;{\rm SU}(2,\mathbb{C}))$ and $E(g)$ is
semistable. Any holomorphic vector bundle on a Riemann surface has a flag of
holomorphic subbundles. In particular for $E(g)$, there exists a holomorphic
sub-line bundle in the associated vector bundle for the defining
representation. This implies the existence of a factorization of the form

$\displaystyle g=\left(\begin{matrix}l_{11}&l_{12}\\\
l_{21}&l_{22}\end{matrix}\right)\left(\begin{matrix}\lambda&B^{\prime}\\\
0&\lambda^{-1}\end{matrix}\right)\left(\begin{matrix}u_{11}&u_{12}\\\
u_{21}&u_{22}\end{matrix}\right),$ (4.1)

where $l\in H^{0}(\Sigma^{*},{\rm SL}(2,\mathbb{C}))$, $u\in H^{0}(\Sigma,{\rm
SL}(2,\mathbb{C}))$, $\lambda\colon S\to{\rm GL}(1,\mathbb{C})$ has degree
zero (by semistability), and $B^{\prime}\colon S\to\mathbb{C}$. To proceed
rigorously, we would need to know more about this kind of factorization, but
in this heuristic discussion, we will put this aside. If we assume that the
line bundle has an antiholomorphic symmetry compatible with $R$, then the
degree of the line bundle is zero, and because we are assuming $S$ is
connected, we can assume that $\lambda=a^{\prime}\exp(\chi_{0})$, where
$\chi_{0}=-\chi_{0}^{*}$ and $a^{\prime}$ is a positive constant (we
eventually want $a^{\prime}=a_{1}a_{2}$). By comparing with the factorization
of $g$ in the previous subsection, we see that (putting questions about
uniqueness aside)

$\displaystyle l_{11}=\alpha_{1}^{*}\exp(\chi_{-}),\qquad
l_{21}=\beta_{1}^{*}\exp(\chi_{-}),\qquad
u_{21}=\gamma_{2}\exp(-\chi_{+}),\qquad u_{22}=\delta_{2}\exp(-\chi_{+}).$

Given this, one can possibly mimic the calculations in [15, Section 3] to
reconstruct the factorization. A key point here is that we need to know there
does exist a unitary transition function; this is discussed in Appendix A.4.

Theorem 4.3 asserts the existence of a diagram of the form,

$\displaystyle\begin{matrix}\\{g=k_{1}^{*}\operatorname{diag}(e^{\chi},e^{-\chi})k_{2}\\}&\subset&C^{\omega}(S;{\rm
SU}(2))&\subset&\operatorname{Hyp}(S;{\rm SL}(2))\\\
\downarrow&&\downarrow&&\downarrow\\\
\left\\{\begin{matrix}\text{semistable}\\\
\text{SL(2)-bundles}\end{matrix}\right\\}&\subset&\left\\{\begin{matrix}\text{holomorphic}\\\
\text{SL
(2)-bundles}\end{matrix}\right\\}&\equiv&\left\\{\begin{matrix}\text{holomorphic}\\\
\text{SL(2)-bundles}\end{matrix}\right\\}\end{matrix}$

where $k_{1}$ and $k_{2}$ satisfy the conditions in Theorem 3.3, and
isomorphic bundles are identified. The question is whether the first down
arrow is onto (for the second down arrow, see Appendix A.4). To understand
what is needed, suppose that $E(g)$ is semistable and there is a factorization
as in (4.1). To get started, we need a factorization of the form
$\lambda^{\prime}=a_{1}^{*}a_{2}$ (for some $\lambda^{\prime}$ equivalent to
$\lambda$, as a transition function), where $a_{1}$ and $a_{2}$ are functions
as in Section 3.1 and Theorem 4.3.

######  Remark 4.4.

To develop some intuition, we need to be able to solve the following problem:
Consider a decomposable bundle defined by a transition function
$\operatorname{diag}(a,a^{-1})$ where $a$ is positive function along $S$
(e.g., $a=\exp(\chi_{1})$, where $\chi_{1}$ is a zero mode with
$\chi_{1}=\chi_{1}^{*}$). Find a multiloop $g\colon S\to{\rm
SU}(2,\mathbb{C})$ which maps to this bundle.

##  5 Spin Toeplitz operators

A spin structure for a Riemann surface is the same thing as a choice of square
root for the canonical bundle, $\kappa$ (by way of explanation, holomorphic
sections of $\kappa$ are holomorphic differentials). We let $\kappa^{1/2}$
($\to\Sigma,\widehat{\Sigma},\Sigma^{*}$, depending on the surface) denote the
choice of square root (see [20, Chapter 7] for other points of view). There is
a natural (pre-) Hilbert space structure on
$\Omega^{0}\big{(}S,\kappa^{1/2}\big{)}\otimes\mathbb{C}^{2}$
($\mathbb{C}^{2}$ valued spinors along $S$) given by

$\displaystyle\langle\psi,\phi\rangle=\int_{S}\left(\psi,\overline{\phi}\right)_{\mathbb{C}^{2}},$

where $\left(v,\overline{w}\right)_{\mathbb{C}^{2}}$ denotes the standard
Hermitian inner product on $\mathbb{C}^{2}$, so that the integrand is a one
density on $S$.

For the sheaf of holomorphic sections of $\kappa^{1/2}$, and (a slight open
thickening of) the covering of $\widehat{\Sigma}$ by $\Sigma$ and
$\Sigma^{*}$, there is a Mayer–Vietoris long exact sequence,

$\displaystyle 0\to H^{0}\big{(}\widehat{\Sigma},\kappa^{1/2}\big{)}\to
H^{0}\big{(}\Sigma,\kappa^{1/2}\big{)}\oplus
H^{0}\big{(}\Sigma^{*},\kappa^{1/2}\big{)}$
$\displaystyle\hphantom{0}{}\to\Omega^{0}\big{(}S,\kappa^{1/2}\big{)}\to
H^{1}\big{(}\widehat{\Sigma},\kappa^{1/2}\big{)}\to 0.$

In terms of the $\overline{\partial}$ operator,
$H^{0}\big{(}\widehat{\Sigma},\kappa^{1/2}\big{)}=\operatorname{ker}(\overline{\partial})$
and
$H^{1}\big{(}\widehat{\Sigma},\kappa^{1/2}\big{)}=\operatorname{coker}(\overline{\partial})$.
The index of $\overline{\partial}$ is zero.

When $\overline{\partial}$ is invertible, a generic condition, by taking
Hilbert space completions, we obtain a Hilbert space polarization

$\displaystyle L^{2}\Omega^{1/2}(S)\otimes\mathbb{C}^{2}=H_{+}\oplus H_{-},$

where $L^{2}\Omega^{1/2}(S)$, $H_{+}$, and $H_{-}$ denote the completions of
$\Omega^{0}\big{(}S,\kappa^{1/2}\big{)}$,
$H^{0}\big{(}\Sigma,\kappa^{1/2}\big{)}\otimes\mathbb{C}^{2}$, and
$H^{0}\big{(}\Sigma^{*},\kappa^{1/2}\big{)}\otimes\mathbb{C}^{2}$,
respectively. When $\overline{\partial}$ is not invertible, there are two
distinct reasonable polarizations. The first possibility is that $H_{+}$ is
the completion of
$\Omega^{0}\big{(}S,\kappa^{1/2}\big{)}\otimes\mathbb{C}^{2}$, and $H_{-}$ is
the orthogonal complement. The second possibility is that $H_{-}$ is the
completion of
$H^{0}\big{(}\Sigma^{*},\kappa^{1/2}\big{)}\otimes\mathbb{C}^{2}$ and $H_{+}$
is the orthogonal complement.

Given an essentially bounded matrix valued function $g\colon
S\to\mathcal{L}(\mathbb{C}^{2})$, there is an associated bounded
multiplication operator $M_{g}$ on $\Omega^{1/2}(S)\otimes\mathbb{C}^{2}$, and
relative to a polarization as in the preceding paragraphs

$\displaystyle M_{g}=\left(\begin{matrix}A(g)&B(g)\\\
C(g)&D(g)\end{matrix}\right).$

If $\Sigma=D$, the unit disk, then $A(g)$ and $B(g)$ are the classical (block)
Toeplitz and Hankel operators associated to the symbol $g$ (see [3] and [12,
Chapter 3]). In general the basic qualitative properties of the operators
$A(g)$ and $B(g)$ (which we refer to as ‘spin Toeplitz and Hankel operators’,
respectively) are the same as in the classical case, because the projection to
$H_{+}$ differs from the classical projection by a smoothing operator. For
example (of most importance for our purposes), exactly as in [16, Chapter 5],
the map $g\to M_{g}$ defines an embedding of loops into the Hilbert–Schmidt
general linear group; the precise statement is that there is an embedding

$\displaystyle L^{\infty}\cap W^{1/2}(S;{\rm SL}(2,\mathbb{C}))\to{\rm
GL}(H_{+}\oplus H_{-})_{(2)}.$ (5.1)

The relevant Lie structure is described in [16, Chapters 6 and 7].

######  Remark 5.1.

The Krichever–Novikov theory, at least in principle, provides an explicit
means to calculate spin Hankel and Toeplitz operators. According to Krichever
and Novikov (following the notation in [11]), for certain spin structures and
for generic basepoints, there is an (orthonormal) basis for
$\mathcal{M}^{1/2}$ (meromorphic spinors which are regular in the complement
of the basepoints) of the form

$\displaystyle\dots,\phi_{3/2},\phi_{1/2},\phi_{-1/2}\phi_{-3/2},\dots$

(which in the classical case reduces to the usual basis

$\displaystyle\dots,z^{2}(dz)^{1/2},z(dz)^{1/2},z^{0}(dz)^{1/2},z^{-1}(dz)^{1/2},\dots)$

a basis for $\mathcal{M}^{0}$, $\\{f_{k}\colon k\in\frac{g}{2}+\mathbb{Z}\\}$
(which in the classical case reduces to the usual basis), and constants $Q$
such that

$\displaystyle f_{n}\phi_{m}=\sum_{|k|\leq g/2}Q^{k}_{n,m}\phi_{n+m-k}.$

Thus for each $n$, the matrices for the multiplication operator, and the spin
Toeplitz operator, defined by $f_{n}$ will have band width (the number of
nonzero diagonals that appear) equal to $g+1$,
$g=\operatorname{genus}(\hat{\Sigma})$ (in the classical case $f_{n}$ is a
shift operator, and the bandwidth is one). We will pursue this in part II of
this paper [1].

###  5.1 Determinants: proof of Theorem 1.5

######  Theorem 5.2.

Suppose that $g$ has a factorization as in Theorem 1.3. Then for any choice of
spin structure and polarization as in the preceding subsection,

$\displaystyle\det\big{(}A(g)A\big{(}g^{-1}\big{)}\big{)}=\det\big{(}A(k_{1})A\big{(}k_{1}^{-1}\big{)}\big{)}\det\big{(}A(e^{\chi})A(e^{-\chi})\big{)}\det\big{(}A(k_{2})A\big{(}k_{2}^{-1}\big{)}\big{)}.$

This theorem will follow from Theorem 5.3 and the lemmas below. To place the
calculations in some context, recall (see [16, pp. 88–89]) that for the
universal central extension (for the Hilbert–Schmidt general linear group)

$\displaystyle 0\to\mathbb{C}^{*}\to\widetilde{{\rm GL}}(H_{+}\oplus
H_{-})_{(2)}\to{\rm GL}(H_{+}\oplus H_{-})_{(2)}$

and the local cross-section (defined on the open dense set of $g$ such that
$A(g)$ is invertible)

$\displaystyle g\to[g,A(g)]$

the multiplication is given by

$\displaystyle\tilde{g}_{1}\tilde{g}_{2}=\mathbf{c}(g_{1},g_{2})\tilde{g}_{3},$

where the cocycle is given by

$\displaystyle\mathbf{c}(g,h)^{-1}=\det\big{(}A(gh)A(h)^{-1}A(g)^{-1}\big{)}=\det\big{(}A(g)^{-1}A(gh)A(h)^{-1}\big{)}$
$\displaystyle\hphantom{\mathbf{c}(g,h)^{-1}}{}=\det\big{(}1+A(g)^{-1}B(g)C(h)A(h)^{-1}\big{)}.$
(5.2)

The corresponding Lie algebra cocycle is given by

$\displaystyle\omega\colon\ \mathfrak{gl}(H_{+}\oplus
H_{-})_{(2)}\times\mathfrak{gl}(H_{+}\oplus H_{-})_{(2)}\to\mathbb{C},$
$\displaystyle\omega(X,Y)=\operatorname{tr}([A(X),A(Y)]-A([X,Y]))=\operatorname{tr}(B(Y)C(X)-B(X)C(Y)).$
(5.3)

We are interested in the restriction of the cocycle to $\Omega^{0}(S,{\rm
SL}(2,\mathbb{C}))$ via the injection (5.1). The basic fact about this
cocycle, which we will use repeatedly, is that if $g_{-}\in
H^{0}(\Sigma^{*},{\rm SL}(2,\mathbb{C}))$ and $g_{+}\in H^{0}(\Sigma,{\rm
SL}(2,\mathbb{C}))$, then

$\displaystyle\mathbf{c}(g_{-}g,hg_{+})=\mathbf{c}(g,h).$ (5.4)

This follows from $A(g_{-}g)=A(g_{-})A(g)$ and $A(hg_{+})=A(h)A(g_{+})$, which
in turn follow from $B(g_{-})=0$ and $C(g_{+})=0$, respectively. To see this,
suppose that $H_{+}$ is the completion of
$\Omega^{0}\big{(}\Sigma,\kappa^{1/2}\big{)}\otimes\mathbb{C}^{2}$. Given a
spinor $\phi_{+}\in H_{+}$, the product $g_{+}\phi_{+}$ is holomorphic in
$\Sigma$, hence $g_{+}\phi_{+}\in H_{+}$. This implies $C(g_{+})=0$. Now
consider a spinor $\phi\in H_{-}$, i.e., $\phi$ is orthogonal to $H_{+}$. We
claim that $g_{-}\phi$ is also orthogonal to $H_{+}$. If $\psi_{+}\in H_{+}$,
then the pointwise inner product (along $S$)

$\displaystyle\left(\psi,\overline{g_{-}\phi}\right)_{\mathbb{C}^{2}}=\left(g_{-}^{*}\psi_{+},\overline{\phi}\right)_{\mathbb{C}^{2}}.$

Since $g_{-}^{*}\in H^{0}(\Sigma,{\rm SL}(2,\mathbb{C}))$, it follows that
$\langle\psi_{+},g_{-}\phi\rangle=\langle g_{-}^{*}\psi_{+},\phi\rangle=0$. It
follows that $B(g_{-})=0$. A similar argument applies if $H_{+}$ is the
orthogonal complement of
$\Omega^{0}(\Sigma^{*},\kappa^{1/2})\otimes\mathbb{C}^{2}$.

To prove the theorem we need to show that $\mathbf{c}(g,h)=1$ when
$(g,h)=(k_{1}^{*},k_{2})$ and so on. In fact in some cases we can prove much
stronger statements.

######  Theorem 5.3.

$A(k_{1}^{*}k_{2})=A(k_{1}^{*})A(k_{2})$. In particular
$\mathbf{c}(k^{*}_{1},k_{2})=1$.

In the process of proving the first statement, we will prove Theorem 1.6,
i.e., we will only assume that $k_{1}$ and $k_{2}$ are measureable maps
$S\to{\rm SU}(2)$ of the appropriate form. We need to assume the loops are
$W^{1/2}$ for the second statement to be valid.

###### Proof.

Because

$\displaystyle A(k_{1}^{*}k_{2})=A(k_{1}^{*})A(k_{2})+B(k_{1}^{*})C(k_{2})$

this equivalent to showing that $B(k_{1}^{*})C(k_{2})=0$. We will prove this
by direct calculation. Suppose that $f:=\left(\begin{matrix}f_{1}\\\
f_{2}\end{matrix}\right)\in H_{+}=H^{1/2}(\Delta)$. Then

$\displaystyle
B(k_{1}^{*})C(k_{2})f=P_{+}\left(k_{1}^{*}P_{-}(k_{2}f)\right)=P_{+}\left(k_{1}^{*}\left(\begin{matrix}P_{-}(d^{*}f_{1}-c^{*}f_{2})\\\
0\end{matrix}\right)\right)$
$\displaystyle\hphantom{B(k_{1}^{*})C(k_{2})f}{}=P_{+}\left(\left(\begin{matrix}a^{*}P_{-}(d^{*}f_{1}-c^{*}f_{2})\\\
0\end{matrix}\right)\right)=0.$

Thus $B(k_{1}^{*})C(k_{2})=0$. By relation (5.2) (for sufficiently smooth
loops) this implies that $\mathbf{c}(k^{*}_{1},k_{2})=1$. ∎

######  Lemma 5.4.

$\mathbf{c}(k_{1}^{*},\operatorname{diag}(\exp(\chi),\exp(-\chi)))=\mathbf{c}(\operatorname{diag}(\exp(\chi),\exp(-\chi)),k_{2})=1$.

###### Proof.

There exists a generalized triangular factorization

$\displaystyle\left(\begin{matrix}a^{*}&-b\\\
b^{*}&a\end{matrix}\right)=\left(\begin{matrix}\alpha_{1}^{*}&\gamma_{1}^{*}\\\
\beta_{1}^{*}&\delta_{1}^{*}\end{matrix}\right)\left(\begin{matrix}a_{1}&0\\\
0&a_{1}^{-1}\end{matrix}\right)\left(\begin{matrix}1&y+y_{0}\\\
0&1\end{matrix}\right).$

Consequently, using (5.4),

$\displaystyle\mathbf{c}\left(\left(\begin{matrix}a^{*}&-b\\\
b^{*}&a\end{matrix}\right),\left(\begin{matrix}\exp(\chi)&0\\\
0&\exp(-\chi)\end{matrix}\right)\right)$
$\displaystyle=\mathbf{c}\left(\left(\begin{matrix}1&y+y_{0}\\\
0&1\end{matrix}\right),\left(\begin{matrix}e^{\chi}&0\\\
0&e^{-\chi}\end{matrix}\right)\right)$
$\displaystyle=\mathbf{c}\left(\exp\left(\left(\begin{matrix}0&y+y_{0}\\\
0&0\end{matrix}\right)\right),\exp\left(\left(\begin{matrix}\chi&0\\\
0&-\chi\end{matrix}\right)\right)\right).$

In the last line we are considering the cocycle pairing of an element of
$\Omega^{0}(S,N^{+})$, where $N^{+}$ is the group of unipotent upper
triangular matrices in ${\rm SL}(2,\mathbb{C})$, and an element of
$\Omega^{0}(S,H)$, where $H$ is the diagonal subgroup of ${\rm
SL}(2,\mathbb{C})$. We claim that the corresponding Lie algebra cocycle
pairing for $\Omega^{0}(S,\mathfrak{n}^{+})$ and $\Omega^{0}(S,\mathfrak{h})$
is zero. To check this suppose that $X:=\left(\begin{matrix}0&\beta\\\
0&0\end{matrix}\right)\in\Omega^{0}(S,\mathfrak{n}^{+})$ and
$Y:=\operatorname{diag}(\chi,-\chi)\in\Omega^{0}(S,\mathfrak{h})$. We must
calculate (5.3) for these two multiloops. Then

$\displaystyle B(X)C(Y)\left(\left(\begin{matrix}f_{1}\\\
f_{2}\end{matrix}\right)\right)=\left(\begin{matrix}-(Y(\chi
f_{2})_{-})_{+}\\\ 0\end{matrix}\right).$

From this it is clear that $\operatorname{tr}(B(X)C(Y))=0$. Similarly
$\operatorname{tr}(B(Y)C(X))=0$. Thus the cocycle pairing for
$\Omega^{0}(S,\mathfrak{n}^{+})$ and $\Omega^{0}(S,\mathfrak{h})$ is zero. It
follows that the cocycle pairing for $\Omega^{0}(S,N^{+})$ and
$\Omega^{0}(S,H)_{0}$ (the identity component) is trivial. This implies that

$\displaystyle\mathbf{c}(k_{1}^{*},\operatorname{diag}(\exp(\chi),\exp(-\chi)))=1.$

By the same argument
$\mathbf{c}(\operatorname{diag}(\exp(\chi),\exp(-\chi)),k_{2})=1$. ∎

######  Lemma 5.5.

$\mathbf{c}(k_{1}^{*}\operatorname{diag}(\exp(\chi),\exp(-\chi)),k_{2})=\mathbf{c}(k_{1}^{*},\operatorname{diag}(\exp(\chi),\exp(-\chi))k_{2})=1$.

###### Proof.

Using (5.4)

$\displaystyle\mathbf{c}(k_{1}^{*}\operatorname{diag}(\exp(\chi),\exp(-\chi)),k_{2})=\mathbf{c}\left(\left(\begin{matrix}e^{\chi}&y+y_{0}\\\
0&e^{-\chi}\end{matrix}\right),\left(\begin{matrix}1&x^{*}+x_{0}\\\
0&1\end{matrix}\right)\right).$

As in the proof of the preceding lemma, this is a cocycle pairing between
elements from two (connected) groups. The corresponding pairing of Lie algebra
elements is trivial. So this cocycle pairing is trivial. ∎

These lemmas imply Theorem 5.2.

###  5.2 Calculating the determinant: scalar case

Our goal is to calculate the second determinant appearing on the right-hand
side of the statement in Theorem 5.2. Recall that in that statement we think
of $e^{\chi}$ as a multiplication operator on scalar valued functions. We
first consider the simplest possibility, which can be handled in the same way
as done by Widom in the classical case (see [21]).

######  Proposition 5.6.

Suppose that $\chi=\chi_{+}+\chi_{-}$, where $\chi_{+}$ is holomorphic in
$\Sigma$ and $\chi_{-}$ is holomorphic in $\Sigma^{*}$. Then

  1. $(a)$

$\det(A(e^{\chi})A(e^{-\chi}))=\exp(\operatorname{tr}(B(\chi_{+})C(\chi_{-})))$,

  2. $(b)$

if $\chi$ is $i\mathbb{R}$ valued, then this equals

$\displaystyle\exp(-\operatorname{tr}(B(\chi_{+})B(\chi_{+})^{*}))=\exp\big{(}{-}|B(\chi_{+})|^{2}_{\mathcal{L}_{2}}\big{)}.$

In the classical case $\Sigma=D$,

$\displaystyle\operatorname{tr}(B(\chi_{+})B(\chi_{+})^{*})=\sum_{n=1}^{\infty}n|\chi_{n}|^{2}.$

###### Proof.

Because $A(e^{\chi})=A(e^{\chi_{-}})A(e^{\chi_{+}})$,
$\det(A(e^{\chi})A(e^{-\chi}))$ equals

$\displaystyle\det\big{(}A(e^{\chi_{-}})A(e^{\chi_{+}})A(e^{-\chi_{-}})A(e^{-\chi_{+}})\big{)}=\det\big{(}e^{A(\chi_{-})}e^{A(\chi_{+})}e^{A(-\chi_{-})}e^{A(-\chi_{+})}\big{)}.$

The Helton–Howe formula (see [10]) implies that this equals
$\exp(\operatorname{tr}([A(\chi_{-}),A(\chi_{+})]))$. Now observe that

$\displaystyle\begin{split}[A(\chi_{-}),A(\chi_{+})]&=A(\chi_{-})A(\chi_{+})-A(\chi_{+})A(\chi_{-})\\\
&=A(\chi_{-})A(\chi_{+})-\left(A(\chi_{+}\chi_{-})-B(\chi_{+})C(\chi_{-})\right)=B(\chi_{+})C(\chi_{-}).\end{split}$

This implies part (a).

For part (b) the only thing we need to comment on is the last equality. This
follows from writing out the matrix for $B$, relative to the standard basis.
Up to a multiple, it also follows abstractly by the ${\rm SU}(1,1)$ symmetry
of $D$ and Schur’s lemma. ∎

######  Remark 5.7.

It is obvious that the formula in (a) depends on the complex structure of
$\Sigma$. Does it depend on the spin structure?

The shortcoming of the preceding proposition is that when
$\text{genus}(\widehat{\Sigma})>0$, we have not accounted for zero modes.

######  Proposition 5.8.

$\displaystyle\det\big{(}A(e^{\chi})A(e^{-\chi})\big{)}=\mathbf{c}(e^{\chi_{-}},e^{\chi_{0}})\mathbf{c}(e^{\chi_{0}},e^{\chi_{+}})\det\big{(}\big{\\{}e^{-A(\chi_{+})},e^{A(\chi_{-})}\big{\\}}\big{)}\det\big{(}A(e^{\chi_{0}})A(e^{-\chi_{0}})\big{)}.$

In the case that $\chi$ has values in $i\mathbb{R}$, this equals

$\displaystyle\det\big{(}\big{|}\big{\\{}A(e^{-\chi_{0}}),e^{-A(\chi_{+})}\big{\\}}\big{|}^{2}\big{)}\det\big{(}A(e^{\chi_{0}})A(e^{\chi_{0}})^{*}\big{)}\exp\big{(}{-}\operatorname{tr}(B(\chi_{+})B(\chi_{+})^{*})\big{)},$

where $\\{A,B\\}=ABA^{-1}B^{-1}$ denotes the group commutator.

###### Proof.

$\displaystyle
A(e^{\chi})A(e^{-\chi})=e^{A(\chi_{-})}A(e^{\chi_{0}})e^{A(\chi_{+})}e^{-A(\chi_{-})}A(e^{-\chi_{0}})e^{-A(\chi_{+})}.$

The right-hand side is of the form $xyzx^{-1}Yz^{-1}$. The conjugate of this
by $z^{-1}$ equals

$\displaystyle
z^{-1}xyzx^{-1}Y=\big{\\{}z^{-1}x,y\big{\\}}y\big{\\{}z^{-1},x\big{\\}}y^{-1}yY,$

where we have temporarily assumed that $y$ is invertible. This implies

$\displaystyle\det\big{(}A(e^{\chi})A(e^{-\chi})\big{)}=\det\big{(}\big{\\{}e^{-A(\chi_{+})}e^{A(\chi_{-})},A(e^{\chi_{0}})\big{\\}}\big{)}$
$\displaystyle\hphantom{\det\big{(}A(e^{\chi})A(e^{-\chi})\big{)}=}{}\times\det\big{(}\big{\\{}e^{-A(\chi_{+})},e^{A(\chi_{-})}\big{\\}}\big{)}\det\big{(}A(e^{\chi_{0}})A(e^{-\chi_{0}})\big{)}$

(each of these determinants is the determinant of an operator of the form
identity + trace class). Using the Helton–Howe formula, this equals

$\displaystyle\det\big{(}\big{\\{}e^{-A(\chi_{+})}e^{A(\chi_{-})},A(e^{\chi_{0}})\big{\\}}\big{)}\exp\big{(}{-}\operatorname{tr}(B(\chi_{+})C(\chi_{-}))\big{)}\det\big{(}A(e^{\chi_{0}})A(e^{-\chi_{0}})\big{)}.$

Now we use the identity

$\displaystyle\big{\\{}z^{-1}x,y\big{\\}}=z^{-1}xyx^{-1}y^{-1}zz^{-1}yzy^{-1}.$

This implies that the first determinant in the expression above satisfies

$\displaystyle\det\big{(}\big{\\{}e^{-A(\chi_{+})}e^{A(\chi_{-})},A(e^{\chi_{0}})\big{\\}}\big{)}=\det\big{(}\big{\\{}e^{A(\chi_{-})},A(e^{\chi_{0}})\big{\\}}\big{)}\det\big{(}\big{\\{}e^{A(\chi_{+})},A(e^{\chi_{0}})\big{\\}}\big{)}.$

In terms of the group cocycle notation, this equals

$\displaystyle\mathbf{c}(e^{\chi_{-}},e^{\chi_{0}})\mathbf{c}(e^{\chi_{0}},e^{\chi_{+}}).$

This completes the proof. ∎

######  Remark 5.9.

In the classical case this determinant is nonvanishing. It would seem unlikely
that this is true in general. But the expression we have produced does not
seem to help in deciding this question. It should be possible to calculate the
determinant for $\exp(\chi_{0})$ exactly.

##  Appendix A Hyperfunctions and holomorphic bundles

###  A.1 Hyperfunctions on a circle

Suppose that $X$ is an oriented analytic compact $d$-manifold. Let
$C^{\omega}\Omega^{d}(X)$ denote the vector space of real analytic forms of
order $d$ on $X$. There is a standard topology on this space. By definition
the space of (real) hyperfunctions on $X$, denoted
$\operatorname{Hyp}(X;\mathbb{R})$, is the dual of this topological vector
space; the space of complex hyperfunctions is the complexification, denoted
$\operatorname{Hyp}(X;\mathbb{C})$, or more simply $\operatorname{Hyp}(X)$
(see [9, p. 35]). In the case of $X=S^{1}$, any complex hyperfunction has a
unique representation of the form

$\displaystyle f=f_{-}+f_{0}+f_{+},$

where $f_{-}\in H^{0}(\Delta^{*},(\infty);\mathbb{C},0)$,
$f_{0}\in\mathbb{C}$, and $f_{+}\in H^{0}(\Delta,0;\mathbb{C},0)$ (this is how
hyperfunctions were first introduced, see [19]). The corresponding functional
is given by

$\displaystyle C^{\omega}\Omega^{1}\big{(}S^{1}\big{)}\to\mathbb{C}\colon\
\omega\to\int_{|z|=1-\epsilon}(f_{0}+f_{+})\omega+\int_{|z|=1+\epsilon}f_{-}\omega,$

where $\omega=gdz$ and $g$ is complex analytic in a collar of $S^{1}$, for
sufficiently small $\epsilon$.

An integrable function $f\colon S^{1}\to\mathbb{C}$ naturally defines a
hyperfunction by integrating an analytic form against $f$ around $S^{1}$. Note
that for the linear triangular decomposition for the Fourier series of $f$,
where $f_{\pm}=\sum\limits_{\pm n>0}f_{n}z^{n}$, $f_{\pm}$ are not necessarily
integrable on the circle, but they do define holomorphic functions in $\Delta$
($\Delta^{*}$, respectively).

###  A.2 Nonabelian hyperfunctions on a circle

Suppose that $G$ is a simply connected complex Lie group, e.g., $G={\rm
SL}(2,\mathbb{C})$.

In [16] it is observed that the group of analytic loops,
$C^{\omega}(S^{1};G)$, is a complex Lie group. A neighborhood of the identity
consists of those loops which have a unique Riemann–Hilbert factorization

$\displaystyle g=g_{-}\cdot g_{0}\cdot g_{+},$ (A.1)

where $g_{-}\in H^{0}(D^{*},(\infty);G,1)$, $g_{0}\in G$, $g_{+}\in
H^{0}(D,0;G,1)$. A model for this neighborhood is

$\displaystyle H^{1}(D^{*},\mathfrak{g})\times G\times H^{1}(D,\mathfrak{g}),$

where the linear coordinates are determined by $\theta_{+}=g_{+}^{-1}(\partial
g_{+})$, $\theta_{-}=(\partial g_{-})g_{-}^{-1}$. The (left or right)
translates of this neighborhood by elements of $L_{\rm f\/in}K$ cover
$H^{0}(S^{1},G)$; a key point is that the transition functions are functions
of a finite number of variables, in an appropriate sense (see [13, Chapter 2,
Part III]).

The hyperfunction completion, $\operatorname{Hyp}(S^{1},G)$, is modeled on the
space

$\displaystyle H^{1}(\Delta^{*},\mathfrak{g})\times G\times
H^{1}(\Delta,\mathfrak{g})$

and the transition functions are obtained by continuously extending the
transition functions for the analytic loop space of the preceding paragraph.
The global definition is

$\displaystyle\operatorname{Hyp}\big{(}S^{1},G\big{)}=\lim_{r\downarrow
1}H^{0}\big{(}\\{1<|z|<r\\},G\big{)}\times_{H^{0}(S^{1},G)}\lim_{r\uparrow
1}H^{0}\big{(}\\{r<|z|<1\\},G\big{)}.$ (A.2)

From this point of view, a hyperfunction is an equivalence class $[g,h]$,
where $g$ ($h$) is a $G$-valued holomorphic map in an annulus to the left
(right, respectively) of $S^{1}$). The correct way to topologize this space
does not seem clear from this point of view, and we will put this aside. From
the global definition it is clear that the group $H^{0}(S^{1},G)$ acts
naturally from both the left and right of $\operatorname{Hyp}(S^{1},G)$, e.g.,
for the left action, $g_{l}\colon[g,h]\to[g_{l}g,h]$. There is a generalized
Birkhoff decomposition

$\displaystyle\operatorname{Hyp}\big{(}S^{1},G\big{)}=\bigsqcup_{\lambda\in\operatorname{Hom}(S^{1},T)}\Sigma^{\rm
hyp}_{\lambda},\qquad\Sigma_{\lambda}^{\rm
hyp}=H^{0}(\Delta^{*},G)\cdot\lambda\cdot H^{0}(\Delta,G).$

The top stratum (the piece with $\lambda=1$ above) is open and dense, and for
each point $g$ in the top stratum, there is a unique factorization as in
(A.1), where $g_{\pm}$ are $G$-valued holomorphic functions in the open disks
$\Delta$ and $\Delta^{*}$, respectively. We will refer to $g_{-},g_{0},g_{+}$
($\theta_{-},g_{0},\theta_{+}$, respectively) as the Riemann–Hilbert
coordinates (linear Riemann–Hilbert coordinates, respectively) of $g$ (see
[13, Chapter 2, Part III]).

Given a continuous function $g\colon S^{1}\to G$ there exists a generalized
Riemann–Hilbert factorization, where the factors $g_{\pm}$ are not necessarily
continuous (see [4, Theorem 1.1 of Chapter VIII] for a precise statement). In
this way any reasonable loop in $G$ can be regarded as an element of
$\operatorname{Hyp}(S^{1},G)$.

The space $\operatorname{Hyp}(S^{1},G)$ depends only on the orientation and
real analytic structure of $S^{1}$, i.e., there is a natural action of real
analytic homeomorphisms on hyperfunctions: in terms of the global definition
(A.2)

$\displaystyle\sigma\colon\ [g,h]\to\big{[}\sigma^{-*}g,\sigma^{-*}h\big{]}.$

Consequently if $S$ is an oriented real analytic one-manifold, then
$\operatorname{Hyp}(S,G)$ is well-defined.

###  A.3 Hyperfunctions and holomorphic $\boldsymbol{G}$ bundles

Suppose that $\widehat{\Sigma}$ is a closed Riemann surface. In this
subsection it is not necessary to assume that $\widehat{\Sigma}$ is a double.
We also suppose that $c\colon S^{1}\to\widehat{\Sigma}$ is a real analytic
embedding, i.e., a simple analytic loop. We could more generally suppose that
$c$ is an embedding of several disjoint analytic loops, but we will focus on
one to simplify notation. A basic (nongeneric) example is the case when
$\widehat{\Sigma}$ is a double and $c$ is a parameterization of $S$, the fixed
point set of the involution $R$.

Let $\mathcal{O}$ denote the structure sheaf of $\widehat{\Sigma}$. There is
an associated mapping

$\displaystyle E=E_{c,\mathcal{O}}\colon\
\operatorname{Hyp}\big{(}S^{1},G\big{)}\to H^{1}(\mathcal{O}_{G})\colon\
[g,h]\to E_{c}([g,h]),$

where $H^{1}(\mathcal{O}_{G})$ denotes the set of isomorphism classes of
holomorphic $G$-bundles on $\widehat{\Sigma}$. For an ordinary analytic
$G$-valued loop $g$, $E$ maps the loop to the isomorphism class of the
holomorphic $G$-bundle defined by using the loop as a holomorphic transition
function in a tubular neighborhood of the image of $c$.

######  Remark A.1.

Our convention for transition functions is the following. A (holomorphic)
section of $E(g)$ (which one should think of as a frame) is a (holomorphic)
function $s\colon\widehat{\Sigma}{\setminus}c\to G$ such that
$s_{+}g^{-1}=s_{-}$, where $s_{+}(s_{-})$ is the restriction of $s$ to a
sufficiently small annulus to the left (respectively, the right) of the
oriented loop $c$. Consequently for an associated bundle $E(g)\times_{G}V$ a
(holomorphic) section is represented by a (holomorphic) function
$v\colon\widehat{\Sigma}{\setminus}c\to V$ such that $gv_{+}=v_{-}$.

With this convention, for scalar transition functions, the degree of a line
bundle is the negative of the degree of a corresponding transition function,
e.g., for differentials on $\mathbb{P}^{1}$, $dw=dz(-z^{-2})$ ($s_{-}=(dw)$
and $s_{+}=(dz)$); the degree of the canonical bundle is $2$, and the degree
of the transition function $-z^{2}$ is $-2$.

The point is that $E$ can be extended naturally to hyperfunctions in the
following way. The map $c$ extends uniquely to a holomorphic embedding
$c\colon\\{1-\epsilon<|z|<1+\epsilon\\}\to\widehat{\Sigma}$ for some
$\epsilon>0$. Given the pair $(g,h)$, we obtain a holomorphic bundle on
$\widehat{\Sigma}$ by using $g$ as a transition function on an
$\epsilon^{\prime}$-collar to the left of $c$ and $h$ as a transition function
on an $\epsilon^{\prime}$-collar to the right of $c$, for some
$\epsilon^{\prime}<\epsilon$, depending upon the pair $(g,h)$. The isomorphism
class of this bundle is independent of the choice of $\epsilon^{\prime}$, and
depends only upon $[g,h]\in\operatorname{Hyp}(S^{1},G)$.

The basic properties of the mapping $E_{c}$ are summarized as follows.

######  Proposition A.2.

  1. $(a)$

If $\phi\in C^{\omega}\operatorname{Hom}(S^{1})$, then the induced map

$\displaystyle\operatorname{Hyp}\big{(}S^{1},G\big{)}\stackrel{{\scriptstyle\phi}}{{\rightarrow}}\operatorname{Hyp}\big{(}S^{1},G\big{)}\stackrel{{\scriptstyle
E_{c}}}{{\rightarrow}}H^{1}(\mathcal{O}_{G})$

equals $E_{c\circ\phi^{-1}}$.

  2. $(b)$

There is a holomorphic action

$\displaystyle
H^{0}\big{(}\widehat{\Sigma}{\setminus}c\big{(}S^{1}\big{)}\big{)}\times\operatorname{Hyp}\big{(}S^{1},G\big{)}\to\operatorname{Hyp}\big{(}S^{1},G\big{)}\colon\
f,[g,h]\to\big{[}f|_{S^{1}_{-}}g,hf|_{S^{1}_{+}}^{-1}\big{]}.$

  3. $(c)$

Inclusion and the mapping $E_{c}$ induce isomorphisms of sets

$\displaystyle
H^{0}\big{(}S^{1},G\big{)}/H^{0}\big{(}\overline{\widehat{\Sigma}{\setminus}c},G\big{)}\to\operatorname{Hyp}\big{(}S^{1},G\big{)}/H^{0}\big{(}\widehat{\Sigma}{\setminus}c\big{)}\to
H^{1}(\mathcal{O}_{G}),$

where $\overline{\widehat{\Sigma}{\setminus}c}$ denotes the closed Riemann
surface obtained by cutting along $c$ and adding two boundary components
$($see [16, Section 8.11]).

An alternate way to think about the projection $E_{c,\mathcal{O}}$ is as
follows. Suppose that $(g,h)$ represents
$[g,h]\in\operatorname{Hyp}(S^{1},G)$. In the $C^{\infty}$ category the
principal bundle defined using the transition functions $(g,h)$ is trivial.
Therefore we can find smooth functions

$\displaystyle s\colon\ \widehat{\Sigma}{\setminus}c\big{(}S^{1}\big{)}\to
G\qquad\text{and}\qquad s_{0}\colon\ S^{1\epsilon}\to G$

(where $S^{1\epsilon}$ is an $\epsilon$-collar neighborhood of $S^{1}$ in
$\widehat{\Sigma}$, and the $\epsilon$-collar will depend upon $(g,h)$) such
that

$\displaystyle s_{-}=gs_{0}\qquad\text{and}\qquad s_{0}=hs_{+}$

on an $\epsilon$ collar to the left (respectively, the right) of $S^{1}$. If
$s$ is replaced by $s^{\prime}$, then there exists $f\in
C^{\infty}(\widehat{\Sigma},G)$ such that $s^{\prime}=sf$, i.e.,
$s_{-}^{\prime}=s_{-}f$ and so on with $+$ and $0$ in place of $-$.

Define $a=s^{-1}\overline{\partial}s$. Then the gauge equivalence class of $a$
depends only upon $[g,h]$. For if we change $g$, $h$ to $g_{-}gg_{0}$,
$g_{0}^{-1}hg_{+}$, where $g_{\pm}$ is holomorphic in
$\widehat{\Sigma}^{\pm}$, and $g_{0}$ is holomorphic in $S^{1\epsilon}$, then
the choice of $s$ is modified in an obvious way, and this does not change $a$,
and when $s\to s^{\prime}=sf$, then $a\to f^{-1}af+f^{-1}\overline{\partial}f$
(a gauge transformation). Thus we obtain a well-defined map

$\displaystyle\operatorname{Hyp}\big{(}S^{1},G\big{)}\to\Omega^{0,1}\big{(}\widehat{\Sigma},\mathfrak{g}\big{)}/C^{\infty}(\widehat{\Sigma},G)=H^{1}(\mathcal{O}_{G}).$

###  A.4 The image of $\boldsymbol{LK}$

Given a simple loop $c$ on a closed Riemann surface $\widehat{\Sigma}$,
$H^{0}(S^{1},G)$ maps onto the set of all isomorphism classes of holomorphic
$G$ bundles on $\widehat{\Sigma}$. The main point of this subsection is to
show that, assuming $K$ is simply connected, the subgroup of $K$ valued loops
also maps onto the set of all isomorphism classes of holomorphic bundles.

We first explain why this is true generically, but with critical exceptions,
for non-simply connected $K$.

######  Proposition A.3.

Suppose $K=\mathbb{T}$. The map from circle valued loops $($of degree zero$)$
to isomorphism classes of holomorphic line bundles $($of degree zero$)$,

$\displaystyle E_{c}\colon\
(L\mathbb{T})_{0}\to\operatorname{Jac}\big{(}\widehat{\Sigma}\big{)}$ (A.3)

is not surjective if and only if $c$ is a straight line with respect to the
flat geometry defined by some holomorphic differential.

###### Proof.

We must determine when the connecting map, as in (2.1),

$\displaystyle C^{\omega}(\operatorname{Im}(c);\mathbb{R})\to
H^{0,1}\big{(}\hat{\Sigma}\big{)}\colon\ f\to\bar{\partial}F$

is surjective, where in the $C^{\infty}$ category, $f=F_{+}-F_{-}$, $F$ is a
smooth function on the complement of $c$, with limits $F_{\pm}$ from the left
(right, respectively) of $c$. This map fails to be surjective if and only if
there exists a nonzero holomorphic differential $\omega$ such that
$\operatorname{Re}(\int_{\hat{\Sigma}}\bar{\partial}F\wedge\omega)=0$ for all
$f$, i.e., there is a nonzero real harmonic one form
$\eta=\omega+\overline{\omega}$ such that $\int_{c}f\eta=0$ for all real
functions on $c$, which is equivalent to saying that the pullback of $\eta$ to
$c$ vanishes. Locally $\omega=wdz$ and the corresponding flat metric (with
conical singularities at the zeros of $\omega$) is $|dW|^{2}=|w|^{2}|dz|^{2}$.
The vanishing of the pullback of $\operatorname{Re}(dW)$ along $c$ is
equivalent to saying that $c$ is a straight line. ∎

Suppose that $\widehat{\Sigma}$ is a double of $\Sigma$, a compact Riemann
surface with boundary, as in the text, and $\operatorname{Im}(c)=S$. The
Jacobian of $\widehat{\Sigma}$ fits into a short exact sequence

$\displaystyle 0\to H^{1}\big{(}\widehat{\Sigma},2\pi\mathbb{Z}\big{)}\to
H^{1}\big{(}\widehat{\Sigma},\mathbb{R}\big{)}\to\operatorname{Jac}\big{(}\widehat{\Sigma}\big{)}\to
0.$

$R$ acts equivariantly on this sequence, using pullback. $R$ also acts
equivariantly on the projection
$(LT)_{0}\to\operatorname{Jac}(\widehat{\Sigma})$, where the action on
transition functions along $S$ is given by

$\displaystyle g\to g^{*}:=(R^{*}g)^{-1}.$

In this case the image consists of line bundles having an antiholomorphic
reflection symmetry compatible with $R$, and these correspond to cohomology
classes which are fixed by $R$. This is precisely the kind of (sub-line
bundle) degeneracy that we encountered in Theorem 1.3.

Suppose now that $K$ is simply connected. Given $R$ and the involution $\tau$
defining $K$ as a real form of $G$, there is an involution of
$\Omega^{0}(S,G)$ which fixes $\Omega^{0}(S,K)\colon g\to\tau(g\circ R)$. This
does not have an extension to $\operatorname{Hyp}(S,G)$ (in sharp contrast to
the abelian case). For in the nonabelian case this involution is not
compatible with the action in part (b) of Proposition A.2. Consequently there
does not exist a real form for isomorphism classes of $G$ bundles on
$\widehat{\Sigma}$ which would confine the image of $\Omega^{0}(S,K)$, as in
the abelian case.

In the nonabelian case we conjecture that the ordered product map

$\displaystyle H^{0}(\Sigma^{*},G)\times\Omega^{0}(S,K)\times
H^{0}(\Sigma,G)\to\Omega^{0}(S,G)$

is surjective. Our objective is to explain why this is plausible.

For a group $H$ use left translation to identify $H\times\mathfrak{h}$ with
$TH$. With this convention at the point $(g_{-},k,g_{+})$, the derivative of
the ordered product mapping is given by the formula

$\displaystyle
H^{0}(\Sigma^{*},\mathfrak{g})\times\Omega^{0}(S,\mathfrak{k})\times
H^{0}(\Sigma,\mathfrak{g})\to\Omega^{0}(S,\mathfrak{g}),$
$\displaystyle(X_{-},x,X_{+})\to\frac{d}{dt}\Big{|}_{t=0}\big{(}g_{+}^{-1}k^{-1}g_{-}^{-1}g_{-}e^{tX_{-}}ke^{tx}g_{+}e^{tX_{+}}\big{)}=g_{+}^{-1}k^{-1}X_{-}kg_{+}+g_{+}^{-1}xg_{+}+X_{+}.$

######  Lemma A.4.

The derivative of the ordered product map at $(g_{-},k,g_{+})$ is critical
$($i.e., not surjective$)$ if $k$ is degenerate in the sense that

$\displaystyle\cap_{z\in
S^{1}}\operatorname{ker}(\operatorname{Ad}(k(z))\neq\\{0\\}.$

######  Remark A.5.

  1. (a)

Conjecturally the converse to the lemma holds.

  2. (b)

In a heuristic way, this is telling us when the map

$\displaystyle\Omega^{0}(S,K)\to
H^{0}(\Sigma^{*},G)\backslash\Omega^{0}(S,G)/H^{0}(\Sigma,G)$

is regular, i.e., has a surjective derivative.

###### Proof.

The derivative of the ordered product map at $(g_{-},k,g_{+})$ is surjective
if and only if the the map

$\displaystyle
H^{0}(\Sigma^{*},\mathfrak{g})\times\Omega^{0}(S,\mathfrak{k})\times
H^{0}(\Sigma,\mathfrak{g})\to\Omega^{0}(S,\mathfrak{g}),$
$\displaystyle(X_{-},x,X_{+})\to X_{-}+x+kX_{+}k^{-1}$

is surjective if and only if the dual map is one to one. To compute the dual
map, we identify the dual of $\Omega^{0}(S,\mathfrak{g})$ with itself using
the invariant form. The dual map is then given by

$\displaystyle\Omega^{0}(S,\mathfrak{g})\to
H^{0}(\Sigma^{*},\mathfrak{g})\times\Omega^{0}(S,\mathfrak{k})\times
H^{0}(\Sigma,\mathfrak{g}),$
$\displaystyle\phi\to\big{(}\phi_{-},\phi_{L\mathfrak{k}},\big{(}k^{-1}\phi
k\big{)}_{+}\big{)}.$

This map is one to one if and only if the map (with finite-dimensional domain)

$\displaystyle\\{\phi_{0}\colon\phi_{0}^{*}=\phi_{0}\\}\to
H^{0}(\Sigma,\mathfrak{g})\colon\phi_{0}\to\big{(}k^{-1}(\phi_{0}+\phi_{0}^{*})k\big{)}_{+}$
(A.4)

is one to one (here $\\{\phi_{0}\\}$ is the vector space of zero modes with
values in $\mathfrak{g}$). If $k$ is degenerate as in the statement of the
lemma, then given $X\in\operatorname{ker}(\operatorname{Ad}(k(z))$ for all
$z\in S^{1}$, and any zero scalar zero mode $\chi_{0}$ satisfying
$\chi_{0}=\chi_{0}^{*}$, $\phi_{0}=\chi_{0}X$ is in the kernel of (A.4).

Conversely suppose $\phi_{0}$ is in the kernel of (A.4). Then

$\displaystyle\psi:=k^{-1}\phi_{0}k=\psi_{0}+\psi_{-}=\psi^{*}=\psi_{0}^{*}+\psi_{+}.$

Consequently $k^{-1}\phi_{0}k$ is another zero mode satisfying
$\psi_{0}=\psi_{0}^{*}$. This would seem to force $k$ to be degenerate, but
this is not entirely clear. ∎

At the point $(1,1,1)$, the derivative is the sum mapping

$\displaystyle
H^{0}(\Sigma^{*},\mathfrak{g})\times\Omega^{0}(S,\mathfrak{k})\times
H^{0}(\Sigma,\mathfrak{g})\to\Omega^{0}(S,\mathfrak{g}).$

The image of this sum is proper and equal to

$\displaystyle\Omega^{0}(S,\mathfrak{k})+H^{0}(\Sigma,\mathfrak{g}).$

However at a generic point, $(g_{-},k,g_{+})$, because of noncommutativity,
the derivative is surjective. At these points the map is locally open.

This derivative calculation strongly supports the conjecture, but it is not
clear how to parlay this into a proof.

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