dx\left(\partial\phi\bar{\partial}\phi+|\bar{\partial}\gamma|^{2}e^{2\phi}\right),\qquad
h=\bigg{(}\begin{matrix}e^{-\phi}+|\gamma|^{2}e^{\phi}&\bar{\gamma}e^{\phi}\\\
{\gamma}e^{\phi}&e^{\phi}\end{matrix}\bigg{)},$ (6.2)

and the fields get represented by the functions $\Theta^{j}(h;x)$ defined as

$\Theta^{j}(h;x)=\frac{2j+1}{\pi}\bigg{(}(1,-x)\cdot
h\cdot\bigg{(}\begin{matrix}1\\\
-\bar{x}\end{matrix}\bigg{)}\\!\bigg{)}^{2j}.$ (6.3)

The matrices $h$ are hermitian, and have determinant equal to one,
representing elements of the three-dimensional hyperbolic space $H_{3}^{+}$.
This space is a symmetric space with symmetry group
$\mathrm{SL}(2)\equiv\mathrm{SL}(2,{\mathbb{C}})$ acting as $h\mapsto
g^{-1}\cdot h\cdot(g^{\dagger})^{-1}$. This symmetry gets enhanced to a
symmetry represented by a central extension of the loop group
$\widetilde{\mathrm{SL}}(2)$ in the $H_{3}^{+}$ WZNW model.

The notations used in (6.1) are adapted to the case of Riemann surfaces
$C=C_{0,n}$ of genus $0$ with $n$ punctures at the positions
$z_{1},\dots,z_{n}$. Equation (6.3) identifies the variables $x$ and $\bar{x}$
as auxilliary parameters, allowing us to represent the transformation law of
the generating functions $\Theta^{j}(h;x)$ under the $\mathrm{SL}(2)$-symmetry
in the form

$\Theta^{j}(h;x)\mapsto\Theta^{j}\big{(}g^{-1}h(g^{\dagger})^{-1};x\big{)}=|d+cx|^{4j}\,\Theta^{j}\Big{(}h;\frac{ax+b}{cx+d}\Big{)},\qquad
g=\bigg{(}\begin{matrix}a&b\\\ c&d\end{matrix}\bigg{)},$ (6.4)

characteristic for the spherical principal series ${\mathcal{P}}_{j}$ of the
group $\mathrm{SL}(2,{\mathbb{C}})$ if
$j\in-\frac{1}{2}+\mathrm{i}{\mathbb{R}}$.

The $H_{3}^{+}$ WZNW model has the parameter $k$. For many applications one is
interested in real values of $k\in(-\infty,-2)$, but let us anticipate that we
will later consider the analytic continuation to values where
$k+2\in\mathrm{i}{\mathbb{R}}$.

###  6.2 $H_{3}^{+}$ WZNW model on surfaces of higher genus

When the surface $C=C_{g,n}$ has genus $g>0$, it is natural to generalise the
definition of correlation functions by twisting with holomorphic
$G=\mathrm{SL}(2)$-bundles. Describing holomorphic bundles ${\mathcal{E}}$ in
terms of an atlas of local trivialisations, one assigns fields $h_{U}\in
H_{3}^{+}$ to open subsets $U\subset C$, related to each other as
$h_{U^{\prime}}=g_{U^{\prime}U}\cdot h_{U}\cdot g_{UU^{\prime}}^{\dagger}$,
with $g_{UU^{\prime}}$ being the transition function of the atlas defining
${\mathcal{E}}$ on the intersection of the open sets $U$ and $U^{\prime}$.

It is possible to construct families ${\mathcal{E}}_{\bm{x}}$ of holomorphic
$SL(2)$-bundles labelled by collections $\bm{x}=(x_{1},\dots,x_{3g-3+n})$ of
complex parameters. This can be done in such a way that $n$ out of the
$h:=3g-3+n$ parameters coincide with the variables $x_{1},\dots,x_{n}$
introduced with the help of (6.1) and (6.3). We will in the following use the
notation $\bm{x}$ for tuples of complex local coordinates on the moduli space
$\mathrm{Bun}_{G}$ of $G=SL(2)$-bundles generalising the variables
$x_{1},\dots,x_{n}$ explicitly introduced using (6.1) and (6.3).

The variables $z_{1},\dots,z_{n}$ used in (6.1) get generalised to $h=3g-3+n$
complex parameters $\bm{z}=(z_{1},\dots,z_{h})$ for families of complex
structures of $C$ if $C=C_{g,n}$ has $g>0$. The correlation functions defined
in (6.1) are generalised to functions ${\mathcal{G}}(\bm{x},\bm{z};\bm{j})$ of
$2h$ complex variables $\bm{x}$ and $\bm{z}$, parametrically depending on
$\bm{j}=(j_{1},\dots,j_{n})$. Away from certain singular loci, one expects a
dependence with respect to $\bm{x}=(x_{1},\dots,x_{h})$ and
$\bm{z}=(z_{1},\dots,z_{h})$ which is real-analytic, but not holomorphic.

The heuristics provided by (6.1) leads to the formulation of a precise
mathematical problem using the affine Lie algebra symmetry discussed next.

###  6.3 Affine Lie algebra symmetry

The $\mathrm{SL}(2,{\mathbb{C}})$-symmetry of the $H_{3}^{+}$-model is
accompanied by a symmetry under the Lie-algebra
$\mathfrak{sl}(2,{\mathbb{C}})$ which becomes enhanced to a symmetry under the
direct sum of affine Lie algebras
$\widehat{\mathfrak{sl}}_{2,k}\oplus\widehat{\mathfrak{sl}}_{2,k}$ extending
the complexification
$\mathfrak{sl}(2,{\mathbb{C}})_{{\mathbb{C}}}\simeq{\mathfrak{sl}}_{2}\oplus{\mathfrak{sl}}_{2}$
of $\mathfrak{sl}(2,{\mathbb{C}})$.

The affine Lie-algebra symmetry implies that the correlation functions locally
define modules of
$\widehat{\mathfrak{sl}}_{2,k}\oplus\widehat{\mathfrak{sl}}_{2,k}$,
represented by $\mathfrak{sl}_{2}$-valued differential operators
${\mathcal{J}}(z)$ and $\bar{{\mathcal{J}}}(\bar{z})$, with $z$ being a local
coordinate on $C$. The operator ${\mathcal{J}}(z)$ is a first order
differential operator in the variables $\bm{x}$, while
$\bar{{\mathcal{J}}}(\bar{z})$ is the complex conjugate of
${{\mathcal{J}}}({z})$. The action of ${\mathcal{J}}(z)$ and
$\bar{{\mathcal{J}}}(\bar{z})$ on ${\mathcal{G}}$ are interpreted as the
current expectation values.

The Sugawara construction allows one to derive differential equations of the
following form

$\displaystyle(k+2)\frac{\partial}{\partial
z_{r}}{\mathcal{G}}(\bm{x},\bm{z};\bm{j})={\mathsf{H}}_{\bm{x},r}^{\rm\scriptscriptstyle
KZB}\,{\mathcal{G}}(\bm{x},\bm{z};\bm{j}),$ (6.5)
$\displaystyle(k+2)\frac{\partial}{\partial\bar{z}_{r}}{\mathcal{G}}(\bm{x},\bm{z};\bm{j})=\bar{{\mathsf{H}}}_{\bar{\bm{x}},r}^{\rm\scriptscriptstyle
KZB}\,{\mathcal{G}}(\bm{x},\bm{z};\bm{j}),$

where ${\mathsf{H}}_{\bm{x},r}^{\rm\scriptscriptstyle KZB}$ are certain second
order differential operators with respect to the variables
$\bm{x}=(x_{1},\dots,x_{h})$, and the operators
$\bar{{\mathsf{H}}}_{\bar{\bm{x}},r}^{\rm\scriptscriptstyle KZB}$ are obtained
from ${\mathsf{H}}_{\bm{x},r}^{\rm\scriptscriptstyle KZB}$ by complex
conjugation.

The differential operators ${\mathsf{H}}_{\bm{x},r}^{\rm\scriptscriptstyle
KZB}$ reduce to the Hitchin Hamiltonians when $k=-2$ [66], and represent
quantisations of the isomonodromic deformation Hamiltonians [67, 68, 66].
These relations suggest that the quantum geometric Langlands correspondence is
a part of a larger picture discussed in [12] allowing two types of
quantisation and deformation parameters.

###  6.4 Holomorphic factorisation

Given that the correlation functions of the $H_{3}^{+}$ WZNW model represent
solutions to the KZB-equations (6.5), it is natural to expect that these
functions can be representated in a holomorphically factorised form,

${\mathcal{G}}(\bm{x},\bm{z};\bm{j})=\int_{{\mathbb{R}}^{h}}d\mu_{\rm\scriptscriptstyle
W}(\bm{p})\;\,{\mathcal{G}}_{\gamma}(\bm{x},\bm{z};\bm{p};\bm{j}){\mathcal{G}}_{\gamma}(\bar{\bm{x}},\bar{\bm{z}};\bm{p};\bm{j}),$
(6.6)

The functions ${\mathcal{G}}_{\gamma}$ in (6.6) are series solutions of the
KZB equations (6.5) having power-like singular behaviour parameterised by the
variables $\bm{p}$ near the boundary of the moduli space of Riemann surfaces
labelled by $\gamma$. The measure $d\mu_{\rm\scriptscriptstyle W}(j)$ can be
represented in terms of two explicitly known functions
$B_{\rm\scriptscriptstyle W}(j)$ and $C_{\rm\scriptscriptstyle
W}(j_{3},j_{2},j_{1})$ [65].

We shall illustrate the main idea underlying the definition of the functions
${\mathcal{G}}_{\gamma}$ in the case $C=C_{0,4}$, allowing us to replace the
variables $\bm{x}$ and $\bm{z}$ by the cross-ratios $x$ and $z$, formed out of
the variables $x_{1},\dots,x_{4}$ and $z_{1},\dots,z_{4}$ appearing in (6.1),
respectively. The KZB-equations (6.5) may then be represented explicitly in
the form

$(k+2)\frac{\partial}{\partial
z}\,{\mathcal{G}}_{\gamma}({x},{z};\bm{j})=\bigg{(}\frac{{\mathcal{P}}_{x}}{z}+\frac{\mathcal{Q}_{x}}{1-z}\bigg{)}{\mathcal{G}}({x},{z};\bm{j}),$
(6.7)

with ${\mathcal{P}}_{x}$ and ${\mathcal{D}}_{x}$ being second order
differential operators in $x$ independent of $z$. There exist power series
solutions ${\mathcal{G}}_{\gamma}({x},{z};p;\bm{j})$ of the form

${\mathcal{G}}({x},{z};p;\bm{j})=z^{\Delta_{j_{p}}-\Delta_{j_{1}}-\Delta_{j_{2}}}\sum_{k=0}^{\infty}z^{k}{\mathcal{G}}_{\gamma,k}(x;p;\bm{j}),$
(6.8)

with ${\mathcal{G}}_{\gamma,0}(x;p;\bm{j})$ being eigenfunctions of
${\mathcal{P}}_{x}$ with eigenvalue
$(k+2)(\Delta_{j_{p}}-\Delta_{j_{1}}-\Delta_{j_{2}})$, where

$\Delta_{j}=\frac{j(j+1)}{k+2},\qquad j_{p}=-\frac{1}{2}+\mathrm{i}p,$ (6.9)

and ${\mathcal{G}}_{\gamma,k}(x;p;\bm{j})$ for $k>0$ determined recursively in
terms of ${\mathcal{G}}_{\gamma,0}(x)$ by (6.7). The series are expected to be
absolutely convergent for $|z|<|x|<1$.

The papers [64, 65] presented evidence that there exist functions
$B_{\rm\scriptscriptstyle W}(j)$ and $C_{\rm\scriptscriptstyle
W}(j_{3},j_{2},j_{1})$ such that the linear combination
${\mathcal{G}}(\bm{z},\bm{x};\bm{j})$ defined in (6.6) admits a real-analytic
and single-valued continuation to the total space of the fibration with fibers
$\mathrm{Bun}_{G}\simeq{\mathbb{P}}^{1}\setminus\\{0,z,1,\infty\\}$ over the
moduli space ${\mathcal{M}}_{0,4}$ of four-punctured spheres parameterised by
the cross-ratio $z$.

This conjecture can be proven with the help of the relation to Liouville
theory [69] where the analogous problem has been completely solved, as will be
reviewed later in this paper.

###  6.5 Limits of the $H_{3}^{+}$ WZNW model

We will here review some predictions concerning the behaviour of the
$H_{3}^{+}$ WZNW model in the limits $k\to\infty$ and $k\to-2$ that can be
supported by fairly standard arguments from theoretical physics. These limits
will admit more explicit descriptions revealing relations of the WZNW model to
the harmonic analysis of $\mathrm{SL}(2,{\mathbb{C}})$, to the mathematical
theory of holomorphic $SL(2)$-bundles on Riemann surface, and to the analytic
Langlands correspondence. Relating these limits to the solutions of
mathematical problems which have been previously investigated, will offer
valuable insights into the mathematical nature of the $H_{3}^{+}$ WZNW model.

####  6.5.1 Mini-superspace limit

The limit $k\to\infty$ with representations labels $j_{r}$, $r=1,\dots,n$ kept
finite leads to the suppression of the derivatives of the field $h(z)$ with
respect to $z$ and $\bar{z}$. The consequence is a reduction to the quantum
mechanical system discussed in [70]. This limit reduces the correlation
functions defined in (6.1) to the finite-dimensional integrals

$G(\bm{x};\bm{j})=\int_{H_{3}^{+}}dh\;\prod_{r=1}^{n}\Theta^{j_{r}}(h;x_{k}),$
(6.10)

where $dh$ is the $\mathrm{SL}(2,{\mathbb{C}})$-invariant measure on
$H_{3}^{+}$.

This establishes a link with the harmonic analysis on the symmetric space
$H_{3}^{+}$. In order to describe this link more explicitly, one may expand

$\Theta^{j}(h;x)=\sum_{2l\in{\mathbb{Z}}_{\geq
0}}\sum_{m=-l}^{l}\Psi_{lm}^{p}(h)\Xi_{lm}^{p}(x),\qquad
j=-\frac{1}{2}+\mathrm{i}p,$ (6.11)

where the functions $\Xi_{lm}^{p}(x)$, $l\in\frac{1}{2}{\mathbb{Z}}_{\geq 0}$,
$m\in\\{-l,-l+1,\dots,l\\}$ generate a basis in ${\mathcal{P}}_{j}\simeq
L^{2}({\mathbb{C}})$ transforming as the irreducible representation with spin
$l$ under the compact subgroup of $\mathrm{SL}(2,{\mathbb{C}})$, and the
functions $\Psi_{lm}^{p}(h)$ represent distributions on $L^{2}(H_{3}^{+})$,
satisfying

$\int_{H_{3}^{+}}dh\;\big{(}\Psi_{l^{\prime}m^{\prime}}^{p^{\prime}}(h)\big{)}^{\ast}\Psi_{lm}^{p}(h)=\delta_{l^{\prime},l}\delta_{m,m^{\prime}}\delta(p-p^{\prime}).$

The collection of functions $\Psi_{lm}^{p}(h)$ forms a basis for the space of
square-integrable functions $L^{2}(H_{3}^{+})$. The transformation (6.11)
allows us to express the function $G(\bm{x};\bm{j})$ in terms of similar
integrals over products of the functions $\Psi_{lm}^{p}$.

The products $\Psi_{l_{2},m_{2}}^{p_{2}}(h)\Psi_{l_{1},m_{1}}^{p_{1}}(h)$ are
normalisable in $L^{2}(H_{3}^{+})$. Representing such products as integrals
over the functions $\Psi_{lm}^{p}(h)$ leads to expansions which become
equivalent to to (6.6) in the mini-superspace limit.

####  6.5.2 Classical limit

The limit $k\to\infty$ with representations labels $j_{r}$ diverging such that
$j_{r}/k$ stays finite for $r=1,\dots,n$ will be referred to as the classical
limit.

$\lim_{k\to\infty}\frac{1}{k}\log{\mathcal{G}}({z},{x};\bm{j})=S(\bm{x},\bm{z};\bm{j}),$
(6.12)

where

$S({z},{x};\bm{j})=S_{\rm\scriptscriptstyle
WZ}\big{[}h(.;\bm{x},\bm{z};\bm{j})\big{]},$ (6.13)

and $h(z)\equiv h(z;\bm{x},\bm{z};\bm{j})$ is the unique solution of the
following problem: For given families ${\mathcal{E}}_{\bm{x}}$ of holomorphic
bundles, the function $h$ is the single-valued solution to the equation

$\bar{\partial}_{\bar{z}}\big{[}(\partial_{z}h)h^{-1}\big{]}=0,$ (6.14)

which can be represented in the form

$h(z)=g(z)\cdot(g(z))^{\dagger},$ (6.15)

where $g(z)$ is a holomorphic multi-valued section of the stable parabolic
bundle ${\mathcal{E}}_{\bm{x}}$ having power-like growth at $z=z_{r}$,
$r=1,\dots,n$ determined by the parameters $\bm{j}$.272727The parameters
$\bm{j}$ are related to the data called parabolic weights in the mathematical
literature. See e.g. [71] for a thorough review of relevant background in a
related context. In order for $h_{\rm cl}(z)$ to be single-valued, it is
necessary that the monodromy of $g(z)$ is contained in $\mathrm{SU}(2)$. This
means that solutions to this problem correspond to the pairs
$({\mathcal{E}},\nabla)$ of holomorphic bundles ${\mathcal{E}}$ with
connection $\nabla$ having unitary holonomy.

This is precisely the classical form of the $B_{c}$ boundary condition we
introduced in complex Chern-Simons theory. Indeed, we propose that the WZNW
partition function at pure imaginary $k+2$ coincides with the wave-function of
the boundary state $|1\rangle$ created by $B_{c}$ in the polarisation
representing states by densities on $\mathrm{Bun}_{G}$.

Important results about this problem for $C=C_{0,n}$ can be found in [71] and
references therein.

####  6.5.3 Critical level limit

The limit $k\to-2$ of the KZB-equations has first been analysed in [72]. It is
elementary to verify that an ansatz of the form

$\displaystyle{\mathcal{G}}(\bm{x},\bm{z})\sim
e^{-b^{2}S(\bm{z})}\Psi(\bm{x},\bm{z})\big{(}1+{\mathcal{O}}(b^{-2})\big{)},$
(6.16)

yields solutions to (6.5) if

  * •

the functions $\Psi(\bm{x},\bm{z})$ are eigenfunctions $\Psi$ of the Hitchin
Hamiltonians,

${\mathcal{H}}_{x}^{(r)}\Psi(\bm{x},\bm{z})=E_{r}\Psi(\bm{x},\bm{z}),\qquad\overline{\mathcal{H}}_{\bar{x}}^{(r)}\Psi(\bm{x},\bm{z})=\bar{E}_{r}\Psi(\bm{x},\bm{z}),\qquad\text{and}$

  * •

the function $S$ is a generating function of the eigenvalues $E_{r}$ and
$\bar{E}_{r}$, $r=1,\dots,h$,

$\frac{\partial}{\partial
z_{r}}S(\bm{z})=E_{r}(\bm{z}),\qquad\frac{\partial}{\partial\bar{z}_{r}}S(\bm{z})=\bar{E}_{r}(\bm{z})\qquad
r=1,\dots,h.$ (6.17)

This observation suggests that the problem to construct single-valued
solutions to the KZB-equations represents a natural deformation of the
eigenvalue problem for the Hitchin Hamiltonians with deformation parameter
$k$.

###  6.6 Verlinde line operator insertions

We will now introduce a family of modification of the correlation functions of
the $H_{3}^{+}$-WZNW model labelled by half-integer measured laminations
$\Lambda$. For a given lamination $\Lambda$, let $\gamma$ be a boundary
component in the moduli space of curves $C$ corresponding to vanishing length
of the geodesics representing $\Lambda$. In the neighbourhood of such a
boundary one may start with (6.6), and simply modify it to

${\mathcal{G}}_{\Lambda}(\bm{x},\bm{z};\bm{j}):=2^{h}\\!\int_{{\mathbb{R}}^{h}}d\mu_{\rm\scriptscriptstyle
W}(\bm{p})\,\prod_{r=1}^{h}\cosh(4\pi
w_{r}p_{r})\;{\mathcal{G}}_{\gamma}(\bm{x},\bm{z};\bm{p};\bm{j}){\mathcal{G}}_{\gamma}(\bar{\bm{x}},\bar{\bm{z}};\bm{p};\bm{j}),$
(6.18)

where $w_{r}$, $r=1,\dots,h$, are the weights of $\Lambda$. The definition
(6.18) may look somewhat ad hoc at this point. We will explain later in this
paper why it is natural, relating the modification in (6.18) relative to (6.6)
to the Verlinde line operators.

It will in particular turn out that the correlation functions (6.18) admit a
continuation over the moduli space of curves and $\mathrm{Bun}_{G}$ which is
single valued and real-analytic over $\mathrm{Bun}_{G}$ away from the locus
represented by wobbly bundles. This continuation is uniquely determined by the
KZB-equations (6.5). A crucial result of this paper may be concisely
formulated as follows:

> The critical level limit $k\to-2$ of ${\mathcal{G}}_{\Lambda}$ takes a form
> similar to (6.16), with $\Psi(\bm{x},\bm{z})$ replaced by the eigenfunctions
> $\Psi_{\Lambda}(\bm{x},\bm{z})$ of the Hitchin-Hamiltonians associated by
> the analytic Langlands correspondence to the real oper obtained from the
> uniformizing oper by grafting along $\Lambda$.

##  7 A deformation of the Separation of Variables

This section will review the integral transformation relating correlation
functions of the $H_{3}^{+}$-WZNW model to Liouville correlation functions. We
will see that this transformation can be interpreted as a deformation of the
Separation of Variables transformation (3.5).

###  7.1 Liouville theory

The Liouville field theory can be defined on the classical level using the
action

$S_{\rm\scriptscriptstyle
Liou}=\frac{1}{4\pi}\int_{\mathbb{C}}dx\left(|\nabla\phi(x)|^{2}+4\pi\mu
e^{2b\phi(x)}\right)$ (7.1)

For Riemann surfaces of genus zero one may then define the correlation
functions by the path integral [73]

$\bigg{\langle}\;\prod_{k=1}^{n}V_{\alpha_{k}}(z_{k})\;\bigg{\rangle}{}\\!\\!\\!{\phantom{\bigg{|}}}_{\rm
L}:=\int\limits_{\phi:{\mathbb{C}}\to{\mathbb{R}}}{\mathcal{D}}[\phi]\;e^{-S_{\rm\scriptscriptstyle
Liou}[\phi]}\prod_{k=1}^{n}e^{2\alpha_{k}\phi(z_{k})}.$ (7.2)

The definition can be extended to surfaces $C$ of genus $g=1$ [74] and of
genus $g\geq 2$ [75].

It has been conjectured in [76] that there exists a range of values for
$\alpha_{1},\dots,\alpha_{n}$ for which these correlation functions have a
representation of the form,

$\bigg{\langle}\;\prod_{k=1}^{n}V_{\alpha_{k}}(z_{k})\;\bigg{\rangle}{}\\!\\!\\!{\phantom{\bigg{|}}}_{\rm
L}=\int_{{\mathbb{S}}^{h}}dC_{\mu}(\bm{\alpha})\;{\mathcal{F}}_{\mu}(\bm{\alpha};\bm{z}){\mathcal{F}}_{\mu}(\bm{\alpha};\bar{\bm{z}}),$
(7.3)

using the following notations:

  * •

${\mathbb{S}}_{\rm\scriptscriptstyle
L}=\frac{Q}{2}+\mathrm{i}{\mathbb{R}}_{+}$, $Q=b+b^{-1}$, and $h=3g-3+n$ if
$C$ has genus $g$ and $n$ punctures.

  * •

The integration is extended over the space
${\mathbb{S}}_{\rm\scriptscriptstyle L}^{h}$, the $h$-fold Cartesian product
of ${\mathbb{S}}_{\rm\scriptscriptstyle L}$, and the integration variable
$\alpha$ is an $h$-tuple $\alpha=(\alpha_{n+1},\dots,\alpha_{n+h})$ of
elements $\alpha_{r}\in{\mathbb{S}}$.

  * •

$\mu$ is a marking, a pair $\mu=(\Gamma,\Delta)$ consisting of a collection
$\Gamma=(\gamma_{1},\dots,\gamma_{h})$ of non-intersecting simple closed
curves defining a pants decomposition of $C$ and a trivalent graph $\Delta$ on
$C$ having exactly one vertex $v$ in each of the pairs of pants defined by
$\gamma$. Each edge of $\Delta$ is labelled by a number
$r\in\\{1,\dots,n+h\\}$ in such a way that $\alpha_{r}$ gets assigned to the
edge with number $r$.

  * •

For a given pants decomposition of $C$ one can represent the measure
$dC_{\mu}(\bm{\alpha})$ in terms of two known282828The explicit expressions
for the functions $R$ and $C$ have been conjectured in [77, 76]. Strong
arguments supporting this conjecture were later given in [78]. A rigorous
proof has recently been proposed in [79]. functions $R(\alpha)$ and
$C(\alpha_{3},\alpha_{2},\alpha_{1})$ as

$dC_{\mu}(\bm{\alpha})=\prod_{r=n+1}^{h+1}\frac{d\alpha_{r}}{R(\alpha_{r})}\prod_{v\in\Delta_{0}}C(\alpha_{r_{v,3}},\alpha_{r_{v,2}},\alpha_{r_{v,1}}),$
(7.4)

where $\Delta_{0}$ is the set of vertices of $\Delta$, and
$(r_{v,3},r_{v,2},r_{v,1})$ is the tuple of numbers of the edges incident to
the vertex $v\in\mu_{0}$.

  * •

The conformal blocks ${\mathcal{F}}_{\mu}$ are power series in $h$ gluing
parameters for a family of Riemann surfaces defined by gluing pairs of pants.
The coefficients of the power series ${\mathcal{F}}_{\mu}$ are recursively
defined by the representation theory of the Virasoro algebra (see e.g. [15]).

A rigorous proof of the holomorphic factorisation was recently given for the
case $g=0$ in [80].

###  7.2 Degenerate fields in Liouville theory

The fields $V_{\alpha_{n,m}}(z)$ with $\alpha_{n,m}=-nb/2-m/2b$,
$m,n\in\mathbb{Z}_{\geq 0}$, have special properties. Correlation functions
involving such fields satisfy partial differential equations first studied by
Belavin, Polyakov and Zamolodchikov (BPZ) in the context of minimal models.
Such fields are called degenerate fields. The differential equations can be
used to construct the fields $V_{\alpha_{r,s}}(z)$ recursively from the two
basic examples $V_{-b/2}(w)$ and $V_{-1/2b}(y)$. Of particular interest for us
will therefore be correlation functions of the form

$\bigg{\langle}\;\prod_{k=1}^{n}V_{\alpha_{k}}(z_{k})\prod_{j=1}^{m}V_{-b/2}(w_{j})\prod_{i=1}^{l}V_{-1/2b}(y_{i})\;\bigg{\rangle}{}\\!\\!\\!{\phantom{\bigg{|}}}_{\rm
L}.$ (7.5)

Such correlation functions satisfy a system of partial differential equations
of the form

$\displaystyle\left(\frac{1}{b^{2}}\frac{\partial^{2}}{\partial
w_{j}^{2}}+\mathbf{D}^{\rm\scriptscriptstyle
BPZ}_{j}\right)\bigg{\langle}\;\prod_{k=1}^{n}V_{\alpha_{k}}(z_{k})\prod_{j=1}^{m}V_{-b/2}(w_{j})\prod_{i=1}^{l}V_{-1/2b}(y_{i})\;\bigg{\rangle}{}\\!\\!\\!{\phantom{\bigg{|}}}_{\rm
L}=0,$ (7.6) $\displaystyle\left(b^{2}\frac{\partial^{2}}{\partial
y_{i}^{2}}+\tilde{\mathbf{D}}^{\rm\scriptscriptstyle
BPZ}_{i}\right)\bigg{\langle}\;\prod_{k=1}^{n}V_{\alpha_{k}}(z_{k})\prod_{j=1}^{m}V_{-b/2}(w_{j})\prod_{i=1}^{l}V_{-1/2b}(y_{i})\;\bigg{\rangle}{}\\!\\!\\!{\phantom{\bigg{|}}}_{\rm
L}=0,$

where $j=1,\dots,m$, $i=1,\dots,l$, and $\mathbf{D}^{\rm\scriptscriptstyle
BPZ}_{j}$, $\tilde{\mathbf{D}}^{\rm\scriptscriptstyle BPZ}_{i}$ are first
order differential operators.

One may regard such correlation functions as objects associated to a surface
$C=C_{g,n}$ of genus $g$ with $n$ punctures at $z_{1},\dots,z_{n}$, on which
one has marked two further collections of distinguished points
$\bm{y}=(y_{1},\dots,y_{l})$ and $\bm{w}=(w_{1},\dots,w_{m})$. Families of
such surfaces can be constructed by the gluing construction from a collection
of pants $\\{S_{v};v\in\mu_{0}\\}$, each $S_{v}$ having two mutually disjoint
sets of punctures $\bm{w}_{v}=(w_{v,1},\dots,w_{v,m_{v}})$ and
$\bm{y}_{v}=(y_{v,1},\dots,y_{v,l_{v}})$.

####  7.2.1 Gluing construction

The conformal blocks appearing in Liouville correlation functions can be
constructed by the gluing construction. The geometric set-up underlying this
construction will be a refinement of the set-up used to define FN-type
coordinates in Section 4.4. It considers a decomposition of $C$ into punctured
annuli and pairs of pants, decorated with seams decomposing all annuli and
pairs of pants into simply-connected pieces.

Conformal blocks for $C$ can then be constructed from conformal blocks
associated to the building blocks by the gluing construction. One needs to
assign a representation of the Virasoro algebra to each cutting curve
appearing in this decomposition.

The assignment of representations to cutting curves must take into account the
so-called fusion rules. The space of conformal blocks assigned to an annulus
with degenerate representation $V_{-jb^{\pm 1}}$,
$j\in\frac{1}{2}{\mathbb{Z}}$, assigned to a single puncture is non-trivial
only if the representation labels $\alpha_{A}$, $\alpha_{A}^{\prime}$
associated to the two boundaries satisfy

$\alpha^{\prime}=\alpha-mb^{\pm 1},\qquad
m\in\big{\\{}-j,-j+1,\dots,j-1,j\big{\\}}.$ (7.7)

Similar selection rules for conformal blocks associated to annuli with any
number of degenerate punctures can be obtained by gluing annuli with lower
numbers of degenerate punctures.

The factorisation expansions of Liouville correlation functions in the
presence of degenerate insertions can then be represented in terms of
conformal blocks associated to intermediate representations from the set

$\hat{\mathbb{S}}:=\big{\\{}\alpha+jb+kb^{-1};\alpha\in\mathbb{S},j,k\in{\textstyle\frac{1}{2}}\mathbb{Z}\big{\\}},\qquad\mathbb{S}=\frac{Q}{2}+\mathrm{i}{\mathbb{R}}.$

We will consider decorations assigning elements of $\hat{\mathbb{S}}$ to each
cutting curve.

####  7.2.2 Factorised form of correlation functions with degenerate fields

Taking into account these differential equations one may then show that the
holomorphically factorised representation (7.3) has a generalization of the
following form

$\displaystyle\Bigg{\langle}\prod_{k=1}^{n}V_{\alpha_{k}}(z_{k})\prod_{j=1}^{m}V_{-b/2}(w_{j})\prod_{i=1}^{l}V_{-1/2b}(y_{i})\Bigg{\rangle}{}\\!\\!\\!{\phantom{\Bigg{|}}}_{\rm
L}\\!\\!=\sum_{\imath}\int_{{\mathbb{S}}_{\rm\scriptscriptstyle
L}^{h}}dC_{\mu,\imath}(\bm{\alpha})\;\big{|}{\mathcal{F}}_{\mu,\imath}(\bm{\alpha};\bm{z},\bm{w},\bm{y})\big{|}^{2},$
(7.8)

with conformal blocks now denoted as
${\mathcal{F}}_{\mu,\imath}(\bm{\alpha};\bm{z},\bm{w},\bm{y})$, and $\mu$
being a marking $\mu=(\Gamma,\Delta)$ of $C=C_{g,n}$ with numbered edges as
before. The functions
${\mathcal{F}}_{\mu,\imath}(\bm{\alpha};\bm{z},\bm{w},\bm{y})$ can be
characterised as solutions to the system (7.6) of PDE characterised by having
a particular asymptotic behaviour at the nodal degeneration of the complex
structure of $C$ described in terms of the gluing parameters. There will then
be unique solutions to (7.6) having leading behaviour in this nodal
degeneration which is proportional to the product of the conformal blocks
associated to the pants $S_{v}$, $v=1,\dots,2g-2+n$, from which $C$ was built.
The parameters $\alpha$ determine the labels associated to the three holes of
the pants by the same rule as used already in (7.4). The multi-indices
$\imath=(\imath_{1},\dots,\imath_{2g-2+n})$ collect the labels $\imath_{v}$
for elements of a basis for the space of solutions to the equations (7.6) for
each sphere $S_{v}$, $v=1,\dots,2g-2+n$.

Remarkably [81], the constraint that the solutions should have a good
degeneration limit with $w$’s and $y$’s placed in various ways with respect to
the degeneration locus turns out to be sufficiently strong to determine the
$z$ dependence of conformal blocks completely, without explicit use of the
representation theoretic definition employing a sum over Virasoro descendants
in intermediate channels.

###  7.3 Relation between Liouville theory and the $H_{3}^{+}$ WZNW model

It has been shown in [69] that the correlation functions of the $H_{3}^{+}$
WZNW model on Riemann surfaces with genus $g=0$ can be represented in terms of
certain correlation functions of Liouville theory. Correlation functions of
the $H_{3}^{+}$ WZNW model can be expressed in terms of an integral
transformation of the following form:

$\bigg{\langle}\prod_{k=1}^{n}\Phi^{j_{k}}(x_{k}|z_{k})\bigg{\rangle}{}\\!\\!\\!\\!{\phantom{\bigg{|}}}_{\rm\scriptscriptstyle
W}\\!\\!=\int d^{2}y_{1}\dots
d^{2}y_{h}\;{\mathcal{K}}_{\kappa}(\bm{x},\bm{y};\bm{z})\bigg{\langle}\prod_{k=1}^{n}V_{\alpha_{k}}(z_{k})\prod_{l=1}^{h}V_{-1/2b}(y_{l})\bigg{\rangle}{}\\!\\!\\!{\phantom{\bigg{|}}}_{\rm\scriptscriptstyle
L}\;,$ (7.9)

assuming the relations $h=n-2$, $b^{-2}=\kappa-2$, and
$2\alpha_{k}-Q=b(2j_{k}+1)$ for $k=1,\dots,n$. The relation (7.9) simplifies a
bit in the case where $z_{n}$ and $x_{n}$ are sent to infinity. In this case
one has $h=n-3$ in (7.9), and one finds the following formula for the kernel
${\mathcal{K}}_{\kappa}$ [38]:

$\displaystyle{\mathcal{K}}_{\rm\scriptscriptstyle
SOV}^{(n)}(\bm{x},\bm{y};\bm{j},\bm{z})=$ (7.10)
$\displaystyle\qquad=\left|\,\sum_{r=1}^{n-1}x_{r}\frac{\prod_{k=1}^{n-3}(z_{r}-y_{k})}{\prod_{s\neq
r}^{n-1}(z_{r}-z_{s})}\,\right|^{2J}\,\prod_{k<l}^{n-3}|y_{k}-y_{l}|^{-k}\left[\prod_{r=1}^{n-1}\frac{\prod_{s\neq
r}^{n-1}|z_{r}-z_{s}|^{2}}{\prod_{k=1}^{n-3}|z_{r}-y_{k}|^{2}}\right]^{\frac{\alpha_{r}}{b}},$

where $J=-j_{n}+\sum_{r=1}^{n-1}j_{r}$.

Similar relations can be derived for correlation functions on surfaces $C$ of
genus $g>0$ [40]. One can use relations (7.9) and their higher genus
generalizations to construct the correlation functions of the $H_{3}^{+}$ WZNW
model from Liouville theory.

If $h=3g-3+n$ one may solve the system of equations (7.6) for the derivatives
$\frac{\partial}{\partial z_{r}}$, leading to an equivalent representation of
the system of equations (7.6) in the form

$\left(\frac{1}{b^{2}}\frac{\partial}{\partial
y_{r}}-{{\mathsf{H}}}^{\rm\scriptscriptstyle
BPZ}_{\bm{y},r}\right)\\!\bigg{\langle}\prod_{k=1}^{n}V_{\alpha_{k}}(z_{k})\\!\prod_{l=1}^{3g-3+n}\\!\\!V_{-1/2b}(y_{l})\bigg{\rangle}{}\\!\\!\\!{\phantom{\bigg{|}}}_{\rm\scriptscriptstyle
L}=0,\quad r=1,\dots,h,$ (7.11)

where the second order differential operators
${{\mathsf{H}}}^{\rm\scriptscriptstyle BPZ}_{\bm{y},r}$ are quantisations of
the isomonodromic deformation Hamiltonians in Garnier form [12]. The integral
transformation (7.9) intertwines the action of
${{\mathsf{H}}}^{\rm\scriptscriptstyle KZB}_{\bm{x},r}$ with
${{\mathsf{H}}}^{\rm\scriptscriptstyle BPZ}_{\bm{y},r}$ [69, 40]. It enjoys
natural relations with the transformations for the nodal surfaces appearing at
the boundaries of the Deligne-Mumford compactification [69].

###  7.4 Critical level limit

We are now going to explain how the separation of variables transformation
(3.5) is obtained from the $H_{3}^{+}$-Liouville relation (7.9) in the
critical level limit.

Recall that the $b\to\infty$ limit is a classical limit for Liouville theory.
Accordingly, a Liouville theory correlation function behaves as

$\bigg{\langle}\;\prod_{k=1}^{n}V_{\alpha_{k}}(z_{k})\;\bigg{\rangle}{}\\!\\!\\!{\phantom{\bigg{|}}}_{\rm
L}\sim e^{-b^{2}S_{\rm\scriptscriptstyle Liou}[\phi_{\mathrm{cl}}]}$ (7.12)

for a classical solution $\phi_{\mathrm{cl}}$ of the Liouville equations of
motion, i.e. an uniformizing oper with a specific behaviour at the punctures.

The relation (7.12) can be made more precise by looking at the BPZ
differential equations satisfied by the “light” degenerate fields
$V_{-\frac{1}{2b}}$, which reduces to an oper differential equation for a
classical $t(z)$ which is the semiclassical limit of $b^{-2}T(z)$, a rescaled
stress tensor. Thus the semiclassical limit of a $V_{-\frac{1}{2b}}(z)$
insertion is the single-valued flat section $\chi$ of the uniformizing oper:

$\displaystyle F^{(n)}(\bm{y},\bm{z})\sim
e^{-b^{2}S(\bm{z})}Z^{L,n}_{\rm\scriptscriptstyle
1-loop}(\bm{z})\prod_{j=1}^{n-3}\chi(y_{j},\bar{y}_{j};\bm{z})\big{(}1+{\mathcal{O}}(b^{-2})\big{)},$
(7.13)

where

  * •

The functions $\chi(y,\bar{y})$ satisfy a pair of equations of the form

$\big{(}\partial_{y}^{2}+t(y)\big{)}\chi(y,\bar{y})=0,\qquad\big{(}\bar{\partial}_{\bar{y}}^{2}+\bar{t}(\bar{y})\big{)}\chi(y,\bar{y})=0,$
(7.14)

with $t(y)$ holomorphic on $C_{0,n}$. We refer to equations (7.14) as the oper
equations.

  * •

The function $S(\bm{z})$ is defined up to a constant by the equations (6.17),
where $E_{r}(\bm{z})$ are defined by the expansion

$t(y)=\sum_{r=1}^{n-1}\left(\frac{\delta_{r}}{(y-z_{r})^{2}}+\frac{E_{r}(\bm{z})}{y-z_{r}}\right),$
(7.15)

of the functions $t(y)$ defined by the oper equations (7.14).

The $b^{2}\to\infty$ limit is not a semi-classical limit in the WZW theory.
The relation to Liouville, though, still allows an asymptotic expansion,
leading to

$\displaystyle G^{(n)}(\bm{x},\bm{z})\sim e^{-b^{2}S_{\rm\scriptscriptstyle
Liou}[\phi_{\mathrm{cl}}](\bm{z})}\Psi(\bm{x},\bm{z})\big{(}1+{\mathcal{O}}(b^{-2})\big{)},$
(7.16)

where

  * •

The functions $\Psi(\bm{x},\bm{z})$ are eigenfunctions $\Psi$ of the Gaudin
Hamiltonians,

${\mathcal{H}}_{x}^{(r)}\Psi(\bm{x},\bm{z})=E_{r}\Psi(\bm{x},\bm{z}),\qquad\overline{\mathcal{H}}_{\bar{x}}^{(r)}\Psi(\bm{x},\bm{z})=\bar{E}_{r}\Psi(\bm{x},\bm{z}),$

  * •

The function $S$ is a generating function of the eigenvalues $E_{r}$ and
$\bar{E}_{r}$, $r=1,\dots,h$,

$\frac{\partial}{\partial
z_{r}}S(\bm{z})=E_{r}(\bm{z}),\qquad\frac{\partial}{\partial\bar{z}_{r}}S(\bm{z})=\bar{E}_{r}(\bm{z})\qquad
r=1,\dots,n-1.$ (7.17)

  * •

The wavefunction is of the SoV form

$\displaystyle\Psi(\bm{x},\bm{z})$ $\displaystyle=\int d^{2}y_{1}\dots
d^{2}y_{n-3}\;K_{\rm\scriptscriptstyle
SOV}^{(n)}(\bm{x},\bm{y};\bm{z})\prod_{j=1}^{n-3}\chi(y_{j};\bm{z}),$ (7.18)

Relation (7.9) therefore represents a natural deformation of (3.5).

In conclusion, the WZW partition function gives a single-valued solution of
KZB equations with $b^{2}\to\infty$ asymptotics controlled by the uniformizing
oper.

###  7.5 Analytic continuation

Discussions of the Liouville field theory often assume real values of the
parameter $b$. In order to identify the $H_{3}^{+}$ WZNW correlation functions
with a wavefunction in ${\mathcal{H}}_{s}$, though, we will need to consider
imaginary values of $b^{-2}=\mathrm{i}s$. Thanks to (7.9), we can analyse the
analytic continuation relating these two regimes using the Liouville
correlation functions. The conformal blocks in the factorised expression make
sense for all values of $b^{2}$. The measure and three-point functions have
poles at multiples of $b$ and $b^{-1}$. As we rotate the phase of $b^{2}$ by
$\pi/2$, these poles move away from the real axis to rays of slope $\pi/4$,
but remain well separated from the integration contour.

Another possible issue concerns the limits $b\to\infty$ of the relation
between the correlation functions of the $H_{3}^{+}$ WZNW model and Liouville
theory. The interplay of the limit $b\to\infty$ with analytic continuation
with respect to other parameters of the correlation functions might be
affected by Stokes phenomena. The line $\mathrm{i}s=b^{-2}$ is precisely the
phase at which a sub-dominant saddle may start competing with the dominant
one, if present. Based on the gauge-theory constructions we discussed, we
expect the standard Liouville correlation functions to reproduce a path
integral over a steepest descent contour associated to the uniformizing oper.
This is a stronger result than just claiming that the uniformizing oper
dominates asymptotics for real $b^{2}$: it ensures that no subdominant saddle
would appear at the first Stokes line.

##  8 Verlinde line operators

The deformation of the Separation of Variables transform discussed in Section
7 so far misses an important ingredient of the analytic Langlands
correspondence: The creation of generic eigenstates from a cyclic vector,
geometrically represented by the grafting of real projective structures. We
are here going to introduce the proposed counterparts of the grafting
operation away from the critical level, represented by the Verlinde line
operators. The following Section 9 will offer evidence for the claim that the
insertion of Verlinde line operators reproduces the grafting operation in the
critical level limit.

###  8.1 Summary

The main goal in this section is to define analogs of the Verlinde line
operators acting on $H_{3}^{+}$-correlation functions satisfying the following
crucial properties:

  * a)

Verlinde line operators are invariant under deformations of the contour used
as their label.

  * b)

Verlinde lines generate representations of the quantised algebra of functions
on $\mathcal{M}_{\rm flat}$, denoted ${\mathcal{O}}_{q}(\mathfrak{X}_{C})$. It
follows in particular that a linear basis can be labeled by laminations.

  * c)

Left- and right actions on correlation functions coincide.

  * d)

The mapping class group of the surface $C$ acts naturally.

This turns out to be somewhat subtle.

The usual definition of the Verlinde line operators reviewed below is based on
the holomorphic factorisation of correlation functions. The Verlinde line
operators can often be defined as operators acting on spaces of conformal
blocks. In our context we will find it useful to consider modifications of the
correlation functions in which holomorphic or anti-holomorphic conformal
blocks are replaced by the result of an action of a Verlinde line operator.
However, it turns out that the subtleties with the holomorphic factorisation
of $H_{3}^{+}$-correlation functions mentioned above prevent a straightforward
definition of Verlinde line operators in the $H_{3}^{+}$-WZNW-model.

We will therefore begin by describing the analogous construction in the case
of Liouville theory, where a fairly complete treatment is possible. We’ll then
note that a sub-algebra of the algebra of Verlinde line operators in Liouville
theory can be mapped to operators acting on $H_{3}^{+}$-correlation functions
by means of the map between the correlations functions of these two CFTs
discussed above. We will briefly mention a possible definition of Verlinde
line operators directly in the $H_{3}^{+}$-WZNW-model, leaving a more detailed
discussion for another occasion.

Regarding correlation functions as twisted-half densities on
$\mathrm{Bun}_{G}$, we may create a family of descendants by the natural
action of Verlinde lines on correlation functions. This action represents a
natural quantum deformation of the grafting action on Hitchin eigenfunctions.
One thereby gets a representation of ${\mathcal{O}}_{q}(\mathfrak{X})$ on a
natural space of densities on $\mathrm{Bun}_{G}$.

###  8.2 Definition of Verlinde line operators in Liouville theory

The definition of Verlinde loop operators goes back to [57] and has been
applied to Liouville field theory in [58, 59]. The following is a
reformulation which appears to be particularly well-suited for us. It depends
on a choice of pants decomposition as used in the gluing construction, and
will yield a direct relation with the Fenchel-Nielsen type coordinates
introduced in Section 4.4. We will later discuss in which sense this
construction defines objects that do not depend on the auxiliary choice of a
pants decomposition.

The definition of Verlinde line operators can be realised on spaces of
conformal blocks obtained by the gluing construction outlined in Section
7.2.1. For the definition of Verlinde line operators it turns out that we may
restrict attention to somewhat smaller spaces of conformal blocks described as
follows. We will restrict attention to conformal blocks associated to surfaces
decomposed into annuli and pairs of pants on which all punctures are located
inside of pairs of pants. We shall furthermore consider the cases where the
degenerate representations assigned to the punctures $P_{r}$ of $C$ all of the
type $V_{-j_{r}b}$, $j_{r}\in\frac{1}{2}{\mathbb{Z}}$. The data characterising
such punctures can then be collected in the divisors
$D_{C}=\sum_{r=1}^{d}2j_{r}P_{r}$.

Let $\bm{\alpha}$ be a map assigning an element $\alpha_{A}$ of ${\mathbb{S}}$
to each annulus $A$. For fixed $\bm{\alpha}$ we may consider decorations
$\Delta_{\bm{\alpha}}$ assigning discrete shifts
$\hat{\alpha}_{A}=\alpha_{A}+kb\in\alpha_{A}+\frac{b}{2}{\mathbb{Z}}$ of
$\alpha_{A}$ to each annulus $A$. A discrete infinite-dimensional space of
conformal blocks is defined for fixed $\bm{\alpha}$ by taking the direct sum
over all possible decorations $\Delta_{\bm{\alpha}}$ of the tensor products of
conformal blocks spaces assigned to the punctured annuli and pairs of pants,

$\mathsf{CB}(C,\bm{\alpha},{D}_{C})=\bigoplus_{\Delta_{\bm{\alpha}}}\bigotimes_{P}\mathsf{CB}(P,\Delta_{P,\bm{\alpha}},D_{P}),$
(8.1)

with $\Delta_{P,\bm{\alpha}}$ being the decoration of the boundary components
of a given pair of pants $P$ defined by the decoration $\Delta_{\alpha}$. One
should note that the spaces $\mathsf{CB}(P,\Delta_{P,\bm{\alpha}},D_{P})$
associated to punctured pairs of pants are all finite-dimensional.

####  8.2.1 Bubbling, de-bubbling and Verlinde line operators

Two operations are basic for the definition of Verlinde line operators, called
bubbling and fusion, intuitively describing “creation” and “annihilation” of
pairs of degenerate punctures.

In order to define the bubbling, let us consider two distributions $\bm{D}$
and $\bm{D}^{\prime}$ of $d$ and $d+2$ punctures, respectively, having the
property that $\bm{D}^{\prime}-\bm{D}=p_{1}+p_{2}$. The key observation
repeatedly used in the following is that the space of conformal blocks
$\mathrm{CB}(C^{\prime},\Delta^{\prime},\bm{D}^{\prime})$ contains a subspace
canonically isomorphic to $\mathrm{CB}(C,\Delta,\bm{D})$, as will now be
explained.

Given a surface $C$ with $d$ punctures we may construct a surface $C^{\prime}$
with $d+2$ punctures by gluing a twice-punctured disc $D$ into the open
surface obtained from $C$ by cutting out a disc containing no puncture.
Conformal blocks for $C^{\prime}$ can be constructed by taking conformal
blocks for $C$ and $D$, respectively, and applying the gluing construction. In
the case of degenerate fields with label $-b/2$ assigned to both $p_{1}$ and
$p_{2}$ there are only two possibilities for the representation assigned to
the boundary of $D$, the representations $V_{0}$ and $V_{-b}$. Let
$\delta_{0}$ be the decoration assigning $V_{0}$ to the boundary of $D$.

Fixing an element of the one-dimensional space
$\mathrm{CB}(D_{p_{1}p_{2}},\delta_{0},p_{1}+p_{2})$, we may then use the
gluing construction to assign to each element of
$\mathrm{CB}(C,\Delta,\bm{D})$ a unique element of
$\mathrm{CB}(C^{\prime},\Delta^{\prime},\bm{D}^{\prime})$, defining a map in
the following referred to as bubbling, and denoted by $\mathfrak{B}$.

Closely related is the map $\mathfrak{D}$ defined from the projection onto the
subspace of $\mathrm{CB}(C^{\prime},\bm{P},\bm{D}^{\prime})$ which is
isomorphic to $\mathrm{CB}(C,\bm{P},\bm{D})$. $\mathfrak{D}$ will be referred
to as de-bubbling.

Verlinde line operators may then be defined as the composition of bubbling,
parallel transport of one of the degenerate punctures defined by the BPZ-
equations, and de-bubbling. A concrete algorithm for constructing the parallel
transport will be described in Section 8.2.2 below. This composition defines a
deformed version of the trace of the holonomy of an oper connection.

####  8.2.2 Parallel transport of degenerate fields

In order to simplify the exposition we will restrict attention to the cases
where $C$ has at least one non-degenerate puncture $p$. We may then apply
bubbling to the pair of pants having a puncture of $C$ as one of its boundary
components. By cutting $C$ along a contour $\gamma_{p}$ one may get a
decomposition into two connected parts, one of which is a pair of pants
containing $p$ and one of the two punctures created by bubbling. It therefore
suffices to define the parallel transport for conformal blocks associated to
surfaces with a single degenerate puncture.

To simplify the exposition, we will furthermore restrict to the cases with a
single degenerate puncture associated to the representation $V_{-b/2}$ for the
moment. The puncture may be assumed to be located in any one of the pairs of
pants appearing in the refined pants decomposition described above. Natural
bases for the two-dimensional spaces of conformal blocks associated to a pair
of pants $P^{\prime}$ with one puncture can be defined by cutting $P^{\prime}$
along a curve $\delta$ into a punctured annulus $A^{\prime}$ glued to one of
the three boundaries of a pair of pants $P$ without punctures. Assigning
representations to the three boundaries of $P^{\prime}$ leaves two possible
choices for the representation assigned to the cutting curve $\delta$ on
$P^{\prime}$ according to (7.7).

The BPZ null vector decoupling equations define natural isomorphisms between
the spaces of conformal blocks associated to different locations of the
degenerate puncture. Motions of the puncture can be represented as
compositions of three types of elementary moves.

  * $T^{\scriptscriptstyle[ij]}_{P}$

(traversing a pair of pants) – Traversing a pair of pants $P$ along a path
connecting boundary components $i$ and $j$ without crossing any seams.

  * $T^{\scriptscriptstyle[i]}_{P}$

(half-monodromy) – Moving across the next seam along boundary $i$ of pair of
pants $P$.

  * $T_{A}$

(traversing an annulus $T_{A}$) – Move across the circle in the interior of
the annulus $A$.

These moves can be represented on spaces of conformal blocks of the type (8.1)
above by taking advantage of the fact that the representation of the
elementary moves on spaces of conformal blocks of the type (8.1) can be
induced from the representation on building blocks associated to sub-surfaces
of $C$.

For the sake of clarity let us stress that all these operations express the
analytic continuation of conformal blocks which can be defined with the help
of the gluing construction when the degenerate field is located in a specific
region of $C$ in terms of other conformal blocks of this type, defined by
gluing constructions that are suitable for locations of the degenerate field
in other regions of $C$ rather than the original one.

####  8.2.3 Realisation of elementary moves

We shall now describe these elementary moves more precisely.

$T^{\scriptscriptstyle[ij]}_{P}$-move: Let $P^{\prime}$ be the pair of pants
containing the puncture $p$, and let $\delta$ be a curve on $P^{\prime}$ such
that cutting along $\delta$ yields a punctured annulus $A^{\prime}$ and a pair
of pants $P$. If $\gamma_{i}$, $i\in\\{1,2,3\\}$, is the boundary of
$P^{\prime}$ which becomes a boundary of $A^{\prime}$ after cutting, and
$\alpha_{i}\equiv\hat{\alpha}_{\gamma_{i}}$ is associated to $\gamma_{i}$,
only the two assignments ${\alpha}_{i}\mp b/2$ of representation labels to
$\delta$ yield non-trivial conformal blocks. By fixing normalisations, one may
use the conformal blocks from the gluing construction to get a basis for
$\mathsf{CB}(P^{\prime},\Delta_{P^{\prime},\bm{\alpha}},p)$. There exist three
such bases, associated to the boundary components $\gamma_{i}$, $i=1,2,3$ of
$P^{\prime}$. The BPZ equations define a parallel transport pairwise relating
such bases, concretely represented by $2\times 2$ matrices
$\mathsf{T}^{\scriptscriptstyle[ij]}_{P}$ having matrix elements which are
functions of the three representation labels assigned to the boundaries of
$P^{\prime}$. By adopting suitable normalisations, one may get matrices
$\mathsf{T}^{\scriptscriptstyle[ij]}_{P}$ formed out of ratios of
trigonometric functions. Explicit formulae can be found e.g. in [59] or [82].

$L_{A}$-move: The monodromy of the bases introduced above, defined by analytic
continuation along a contour isotopic to $\gamma$, is represented diagonally,
multiplying the basis element corresponding to ${\alpha}_{i}\mp b/2$ by
$e^{2\pi\mathrm{i}b({Q}/{2}\pm({\alpha}_{i}-{Q}/{2}))}$. It is then natural to
introduce the half-monodromy operation, represented by the diagonal matrices
$\mathsf{L}_{A}$, defined as

$\mathsf{L}_{A}=e^{\frac{\pi\mathrm{i}}{2}(1+b^{2})}\Bigg{(}\begin{matrix}\mathsf{U}_{A}^{+\frac{1}{2}}&0\\\
0&\mathsf{U}_{A}^{-\frac{1}{2}}\end{matrix}\Bigg{)},\qquad\mathsf{U}_{A}^{\pm\frac{1}{2}}:=e^{\pm{\pi\mathrm{i}}b({\alpha}_{i}-{Q}/{2})}.$
(8.2)

This formula holds for a certain orientation of $\gamma$ which we will not
need explicitly.

$K_{A}$-move: The representation of $K_{A}$ follows from the observation that
a basis defined in one half of the annulus $A$ trivially defines a basis in
the other half by analytic continuation. One needs to note, however, that the
representation label assigned to $A$ must change in this process, from
$\hat{\alpha}_{\gamma}$ to $\hat{\alpha}_{\gamma}\mp b/2$, corresponding to
the eigenvalues of $M_{A}$. We may therefore describe parallel transport
traversing $A$ by a matrix

$\mathsf{K}_{A}=\bigg{(}\begin{matrix}0&\mathsf{V}_{A}\\\
\mathsf{V}_{A}^{-1}&0\end{matrix}\bigg{)},$ (8.3)

with $\mathsf{V}_{A}$ being the shift operator mapping a conformal block with
intermediate representation label $\alpha_{A}$ to the one defined by replacing
$\alpha_{A}$ by $\alpha_{A}-b/2$, with other labels unchanged.

###  8.3 Generalisations, labelling and representation by difference operators

The action of a Verlinde line operator on a conformal block
${\mathcal{F}}_{\mu}(\bm{\alpha})$ can be represented as a linear combination
of the conformal blocks ${\mathcal{F}}_{\mu}(\bm{\beta})$ associated to
finitely many values $\bm{\beta}\in{\mathbb{C}}^{h}$ of the intermediate
dimension parameters.

We will find it convenient to replace the variables
$\bm{\alpha}=(\alpha_{1},\dots,\alpha_{h})$ by variables
$\bm{p}=(p_{1},\dots,p_{h})$, with $p_{r}$ related to $\alpha_{r}$ as
$p_{r}=\mathrm{i}(Q/2-\alpha_{r})$. With a slight abuse of notations we shall
identify ${\mathcal{F}}_{\mu}(\bm{p})$ with the function defined from
${\mathcal{F}}_{\mu}(\bm{\alpha})$ by this change of variables.

The representation is particularly simple if the contour defining the Verlinde
line operator is one of the curves $\gamma_{1},\dots,\gamma_{h}$ defining the
pants decomposition of $C$ underlying the definition of the conformal blocks
under consideration. In this case one finds [58, 59] that the Verlinde line
operators are represented simply by the operator of multiplication with the
function

$\xi(e^{2\pi bp_{r}}),\quad\text{where}\quad\xi(U):=U+U^{-1}.$ (8.4)

The definition of the Verlinde line operators leaves a freedom which amounts
to multiplying the function $\xi$ by a function of $b$. Formula (8.4) defines
a convention fixing this freedom.292929Our convention differs from the one
used in [59].

####  8.3.1 Generalisations

It is not hard to generalise the definition of Verlinde line operators by
using the degenerate representations $V_{-bj}$,
$j\in\frac{1}{2}{\mathbb{Z}}_{+}$, instead of $V_{-b/2}$. Appearance of the
vacuum representation in the fusion of two representations $V_{-bj}$ allows us
to generalise bubbling and de-bubbling. The parallel transport of an insertion
of $V_{-bj}$ is described by operator-valued $(2j+1)\times(2j+1)$-matrices.
Combined with bubbling and de-bubbling one gets a quantum trace of the
parallel transport defined by the BPZ-equations. Instead of (8.4) one finds
that the Verlinde line operators $W_{j}$ associated to $\gamma=\gamma_{r}$ are
represented as multiplication by

$\zeta_{j}(e^{2\pi
bp_{r}}),\quad\text{where}\quad\zeta_{j}(U)=\sum_{m=-j}^{j}U^{2m}.$ (8.5)

As another possibility one may consider parallel transport along contours
$2j\gamma$ wrapping $2j$ times a simple closed curve $\gamma$. If
$\gamma=\gamma_{r}$ the corresponding operator
${\mathcal{L}}_{2j\gamma}^{\mu}$ is represented by multiplication with

$\xi_{j}(e^{2\pi bp_{r}}),\quad\text{where}\quad\xi_{j}(U)=U^{2j}+U^{-2j}.$
(8.6)

A third possibility is to simply consider the product of $2j$ consecutive
applications of the same Verlinde line operator, in the case
$\gamma=\gamma_{r}$ represented by multiplication with $[\chi(U)]^{2j}$.

####  8.3.2 Labelling

In all three cases one finds Laurent polynomials in $U=e^{2\pi bp_{r}}$. The
three possibilities mentioned above correspond to three different bases in the
space of such Laurent polynomials, each having elements labelled by the half-
integer $j=0,1/2,1,\dots.$.

We will later see all algebraic relations among Verlinde line operators only
depend on the homotopy classes of the loops labelling them. By picking a
particular basis, one may therefore use half-integer measured laminations
$\mathcal{ML}_{C}({\textstyle\frac{1}{2}}{\mathbb{Z}})$ as labels for the
Verlinde line operators ${\mathcal{L}}_{\Lambda}^{\mu}$. To a lamination
$\Lambda$ represented by the collection of simple closed curves
$\lambda_{1},\dots,\lambda_{m}$ with $\lambda_{r}$ non-homotopic to
$\lambda_{s}$ for $r\neq s$, and weights
$j_{r}\in\frac{1}{2}{\mathbb{Z}}^{>0}$ associated to $\lambda_{r}$ for
$r=1,\dots,m$ we may associate the product
$\prod_{r}({\mathcal{L}}^{\mu}_{\lambda_{r}})^{2j_{r}}$, for example.

Notice that these three three labelling options lead to the same action of the
mapping class group on $\Lambda$. Also, the choice among the three does not
appear to affect the $b^{2}\to 0$ asymptotic analysis we use to relate
decorated partition functions to real opers. In the context of this work, we
see no compelling reason to prefer one labelling choice over the others.

####  8.3.3 Representation by difference operators

The explicit formulae for the Verlinde line operators are known in some
special cases [58, 59]. As an example we shall consider the case where one of
the curves $\gamma_{r}$ defining the pants decomposition $\mu$ is surrounded
by a sphere $C_{0,4}^{r}$ with four holes or punctures representing a
connected component of $C\setminus\bigcup_{s\neq r}\gamma_{s}$, and
$\Lambda=\delta_{r}$ is a simple closed curve in $C_{0,4}^{r}$ intersecting
$\gamma_{r}$ twice. The Verlinde loop operators
${\mathcal{L}}_{\delta_{r}}^{\mu}$ will then have a representation of the form
$({\mathcal{L}}_{\delta_{r}}^{\mu}{\mathcal{F}}_{\mu})(\bm{p})={\mathcal{D}}_{r}^{\mu}{\mathcal{F}}_{\mu}(\bm{p})$,
where ${\mathcal{D}}_{r}^{\mu}$ is a finite difference operator of the form

${\mathcal{D}}_{r}^{\mu}:={\mathsf{V}}_{r}\cdot
d_{+}({\mathsf{U}}_{r})\cdot{\mathsf{V}}_{r}+d_{0}({\mathsf{U}}_{r})+{\mathsf{V}}^{-1}_{r}\cdot
d_{-}({\mathsf{U}}_{r})\cdot{\mathsf{V}}_{r}^{-1}$ (8.7)

where ${\mathsf{U}}_{r}=e^{2\pi bp_{r}}$ and ${\mathsf{V}}_{r}$ is the shift
operator defined as
${\mathsf{V}}_{r}{\mathcal{F}}_{\mu}(\bm{p})={\mathcal{F}}_{\mu}(\bm{p}+\mathrm{i}\frac{b}{2}e_{r})$,
with $e_{r}\in{\mathbb{C}}^{h}$ being the vector with components
$(e_{r})_{s}=\delta_{rs}$. The coefficients $d_{\pm}$ in (8.7) depend on the
chosen normalisation of the conformal blocks. The normalisation adopted in
[59], for example, leads to

$\displaystyle d_{\pm}(U)=-\frac{\prod_{s=\pm}(1+UM_{1}^{s}M_{2}^{\mp
s})(1+UM_{3}^{s}M_{4}^{\mp s})}{U^{2}(U-U^{-1})(q^{\pm 1}U-q^{\mp 1}U^{-1})},$
(8.8)

where $M_{i}$, $i=1,\dots,4$ are related to the representation labels assigned
to the boundary components of $C_{0,4}^{r}$ as
$M_{i}=e^{-\pi\mathrm{i}b(2\alpha_{i}-Q)}$, assuming that $\delta_{r}$
separates the boundary components of $C_{0,4}^{r}$ with labels $2$ and $3$
from the other two boundary components.

General Verlinde line operators ${\mathcal{L}}_{\Lambda}^{\mu}$ can be
represented as
$({\mathcal{L}}_{\Lambda}^{\mu}{\mathcal{F}}_{\mu})(\bm{p})={\mathcal{D}}_{\Lambda}^{\mu}{\mathcal{F}}_{\mu}(\bm{p})$,
with finite difference operators ${\mathcal{D}}_{\Lambda}^{\mu}$ of the form

${\mathcal{D}}_{\Lambda}^{\mu}=\sum_{\ell\in{\mathbb{Z}}^{h}}d_{\Lambda,\ell}^{\mu}(\bm{U})\,{\mathsf{V}}^{\ell},$
(8.9)

by introducing shift operators ${\mathsf{V}}^{\ell}$,
$\ell\in{\mathbb{Z}}^{h}$ which act on the conformal blocks as

${\mathsf{V}}^{\ell}{\mathcal{F}}_{\mu}(\bm{p})={\mathcal{F}}_{\mu}(\bm{p}+\mathsf{i}{\textstyle\frac{b}{2}}\ell).$
(8.10)

We have $d_{l}(\mathsf{P})\neq 0$ for finitely many $\ell\in{\mathbb{Z}}^{h}$
only.

###  8.4 The algebra generated by the Verlinde line operators

It turns out that the algebra of Verlinde loop operators is isomorphic to a
non-commutative deformation ${\mathcal{O}}_{q}(\mathfrak{X}_{C})$ of the
algebra of functions on the character variety that is related to the skein
algebra on the one hand, and to the algebra of quantised geodesic length
functions in quantum Teichmüller theory on the other hand. This subsection
offers a brief review of the relevant results and conjectures.

####  8.4.1 Skein quantisation of the character variety

The ring of regular functions of the character variety
${\mathcal{O}}(\mathfrak{X}_{C})$ is generated by trace functions.
Quantisation of the canonical symplectic structure of
${\mathcal{O}}(\mathfrak{X}_{C})$ is known [54, 55, 56] to yield a non-
commutative algebra ${\mathcal{O}}_{q}(\mathfrak{X}_{C})$ also known as the
skein algebra $\mathrm{Sk}_{q}(C)$. This algebra has generators associated to
simple closed curves $\gamma$ on a surface $C$. The relations of
$\mathrm{Sk}_{q}(C)$ can be described in terms of relations between the curves
labelling the generators which follow from the simple local relations
pictorially represented as:

$\begin{aligned}
\includegraphics[width=64.01869pt]{cross}\end{aligned}=q^{\frac{1}{2}}\begin{aligned}
\includegraphics[width=64.01869pt]{resolu1}\end{aligned}+q^{-\frac{1}{2}}\begin{aligned}
\includegraphics[width=64.01869pt]{resolu2}\end{aligned}.$ (8.11)

The left hand side of (8.11) represents the product $L_{1}L_{2}$ of two
generators associated to curves $\gamma_{1}$ and $\gamma_{2}$ crossing each
other inside of a disc $D\subset C$, as indicated. From the union of
$\gamma_{1}$ and $\gamma_{2}$ one can define two new curves by a local surgery
replacing the region inside of the disc depicted on the left by one of the
regions depicted on the right side of (8.11). Relation (8.11) expresses the
product $L_{1}L_{2}$ as a linear combination of the generators associated to
the curves obtained by the surgeries which can be defined in this way.

It is clear that repeated use of the skein relations allows one to express any
product of generators in terms of the generators associated to non-
intersecting curves. By picking a basis associated to non-intersecting curves
one may construct a linear basis for the algebra
${\mathcal{O}}_{q}(\mathfrak{X}_{C})$.

####  8.4.2 Example $C=C_{0,4}$

In the case of the four-punctured sphere $C=C_{0,4}$ one may consider a set of
generators ${\mathcal{L}}_{i}$ associated to curves $\gamma_{i}$,
$i\in\\{1,\dots,4\\}$ encircling the four punctures, supplemented by
generators ${\mathcal{L}}_{ij}$ associated to the curves $\gamma_{ij}$, $i\neq
j$, $i,j\in\\{1,2,3\\}$, separating the pair of punctures labelled by $i$ and
$j$ from the two other punctures.

The algebra ${\mathcal{O}}_{q}(\mathfrak{X}_{C})$ can in this case be
explicitly characterised by two types of relations [83]: One has quadratic
relations of the form

$\displaystyle
e^{\pi\textup{i}b^{2}}{\mathcal{L}}_{12}{\mathcal{L}}_{23}-e^{-\pi\textup{i}b^{2}}{\mathcal{L}}_{23}{\mathcal{L}}_{12}=$
(8.12a) $\displaystyle\hskip
71.13188pt=(e^{2\pi\textup{i}b^{2}}-e^{-2\pi\textup{i}b^{2}}){\mathcal{L}}_{31}+(e^{\pi\textup{i}b^{2}}-e^{-\pi\textup{i}b^{2}})({\mathcal{L}}_{1}{\mathcal{L}}_{3}+{\mathcal{L}}_{2}{\mathcal{L}}_{4})\,,$
together with two similar relations for ${\mathcal{L}}_{12}{\mathcal{L}}_{31}$
and ${\mathcal{L}}_{31}{\mathcal{L}}_{23}$, respectively, and a cubic relation
$\displaystyle e^{\pi\textup{i}b^{2}}$
$\displaystyle{\mathcal{L}}_{12}{\mathcal{L}}_{23}{\mathcal{L}}_{31}+\big{(}2\cos\pi
b^{2})^{2}=\,e^{2\pi\textup{i}b^{2}}{\mathcal{L}}_{12}^{2}+e^{-2\pi\textup{i}b^{2}}{\mathcal{L}}_{23}^{2}+e^{2\pi\textup{i}b^{2}}{\mathcal{L}}_{31}^{2}$
(8.12b)
$\displaystyle+\,e^{\pi\textup{i}b^{2}}{\mathcal{L}}_{12}({\mathcal{L}}_{3}{\mathcal{L}}_{4}+{\mathcal{L}}_{1}{\mathcal{L}}_{2})+e^{-\pi\textup{i}b^{2}}{\mathcal{L}}_{23}({\mathcal{L}}_{2}{\mathcal{L}}_{3}+{\mathcal{L}}_{1}{\mathcal{L}}_{4})+e^{\pi\textup{i}b^{2}}{\mathcal{L}}_{31}({\mathcal{L}}_{1}{\mathcal{L}}_{3}+{\mathcal{L}}_{2}{\mathcal{L}}_{4})$
$\displaystyle+\,{\mathcal{L}}_{1}^{2}+{\mathcal{L}}_{2}^{2}+{\mathcal{L}}_{3}^{2}+{\mathcal{L}}_{4}^{2}+{\mathcal{L}}_{1}{\mathcal{L}}_{2}{\mathcal{L}}_{3}{\mathcal{L}}_{4}\,.$

These relations completely characterise the non-commutative algebra deforming
the ring of regular functions ${\mathcal{O}}(\mathfrak{X}_{C})$ on the
character variety $\mathfrak{X}_{C}$ for $C=C_{0,4}$ [83]. One can check
explicitly that the Verlinde line operators on Liouville conformal blocks
define a representation of this algebra [48, Appendix A].

####  8.4.3 Relation between Verlinde line operator algebra and skein algebra

We claim that the algebra of Verlinde line operators is isomorphic to the
skein algebra $\mathrm{Sk}_{q}(C)$ for general Riemann surfaces $C$.

A proof of this claim can be based on the relation between Liouville conformal
blocks and quantum Teichmüller theory established in [48] and references
therein. This relation implies a direct relation between the Verlinde line
operators and the geodesic length operators of the quantum Teichmüller theory.
It is known [84, 85] that the geodesic length operators in quantum Teichmüller
theory generate a representation of the skein algebra $\mathrm{Sk}_{q}(C)$.

Another approach has been proposed in [86]. One may argue that the braid group
representation generated by Virasoro degenerate representations is conjugate
to the braid group representation from the quantum group
$\mathcal{U}_{q}(\mathfrak{sl}_{2})$. This relates the algebra of Verlinde
line operators to the Witten-Reshetikhin-Turaev representation of the Kauffman
bracket skein algebra.

####  8.4.4 Relation to Fenchel-Nielsen type coordinates

Recall that the Fenchel-Nielsen type coordinates for
${\mathcal{O}}(\mathfrak{X}_{C})$ introduced in Section 4.4 are Darboux
coordinates for the canonical symplectic structure on
${\mathcal{O}}(\mathfrak{X}_{C})$, as expressed in (4.6). One should note that
the description of the parallel transport of degenerate fields in the
definition of the Verlinde line operators exhibits a precise correspondence
with the set-up used to define the FN-type coordinates. This correspondence
suggests to view the parallel transport of degenerate fields on spaces of
conformal blocks as a quantised version of the holonomy of opers.

One may, in particular, compare the definition of the coordinates
$(\kappa_{A},\lambda_{A})$ using (4.5) with the building blocks of the
Verlinde line operators introduced in (8.2) and (8.3). This comparison
suggests the identifications

$\mathsf{U}_{A}=e^{\frac{1}{2}\hat{\lambda}_{A}},\qquad{\mathsf{V}}_{A}=e^{\frac{1}{2}\hat{\kappa}_{A}},\qquad\hat{\kappa}_{A}=4\pi\mathrm{i}\,b^{2}\frac{\partial}{\partial\lambda_{A}},$
(8.13)

identifying $(\hat{\lambda}_{A},\hat{\kappa}_{A})$ as quantised analogs of the
FN-type coordinates $(\lambda_{A},{\kappa}_{A})$.

###  8.5 Changes of pants decomposition and mapping class group

The mapping class group $\mathrm{MCG}(C)$ can be represented by
diffeomorphisms mapping the underlying smooth surface $S$ to itself. An
element $m\in\mathrm{MCG}(C)$ will map a curve $\gamma\subset S$ to
$m\gamma\subset S$, and a marking $\mu$ to a marking $m(\mu)$. The Verlinde
loop operators are, on the one hand, by definition invariant under the mapping
class group action in the sense that

${\mathcal{D}}_{m(\mu)}(m(\Lambda))={\mathcal{D}}_{\mu}(\Lambda).$ (8.14)

Indeed, all of the operations appearing in the definition of the Verlinde loop
operators only involve the local relations between $\Lambda$ and $\mu$.

The representation of the modular groupoid on the spaces of conformal blocks
[48] intertwines the different representations of the Verlinde loop operators
as

${\mathsf{F}}_{\mu^{\prime},\mu}\cdot{\mathcal{D}}_{\mu}(\Lambda)={\mathcal{D}}_{\mu^{\prime}}(\Lambda)\cdot{\mathsf{F}}_{\mu^{\prime},\mu}.$
(8.15)

It follows that one can map any given ${\mathcal{D}}^{\Lambda}_{\mu}$ to
diagonal form.

The representation of the modular groupoid on the spaces of conformal blocks
induces a representation of the mapping class group by taking advantage of the
fact that there is a canonical isomorphism
$\mathrm{CB}^{\mu}(C)\simeq\mathrm{CB}^{m(\mu)}(C)$, allowing one to define
${\mathsf{M}}_{\mu}(m):\mathrm{CB}^{\mu}(C)\to\mathrm{CB}^{\mu}(C)$ as
${\mathsf{M}}_{\mu}(m)={\mathsf{F}}_{m(\mu),\mu}$. It then follows from (8.15)
and (8.14) that

${\mathsf{M}}_{\mu}(m)\cdot{\mathcal{D}}_{\mu}({\Lambda})\cdot{\mathsf{M}}_{\mu}(m)^{-1}={\mathcal{D}}_{m(\mu)}(\Lambda)={\mathcal{D}}_{\mu}(m^{-1}(\Lambda)),$
(8.16)

expressing the equivariance of the Verlinde loop operators under the mapping
class group.

###  8.6 Representation on Liouville correlation functions

The definition of Verlinde loop operators can be extended to the level of
correlation functions. One may simply act on either the holomorphic or anti-
holomorphic functions appearing in the holomorphically factorised
representation with Verlinde loop operators. In the case $\Lambda=\gamma_{r}$,
for example, one finds that the holomorphically factorised representation
(7.3) of the Liouville correlation functions gets modified to

$\bigg{\langle}\;{\mathcal{L}}_{\Lambda}\prod_{k=1}^{n}V_{\alpha_{k}}(z_{k})\;\bigg{\rangle}{}\\!\\!\\!{\phantom{\bigg{|}}}_{\rm
L}=2\int_{{\mathbb{S}}^{h}}dC_{\mu}(\bm{\alpha})\;{\cos(\pi
b(2\alpha_{r}-Q))}{\mathcal{F}}_{\mu}(\bm{\alpha};\bm{z}){\mathcal{F}}_{\mu}(\bm{\alpha};\bar{\bm{z}}).$
(8.17)

More generally one finds expression of the form

$\bigg{\langle}\;{\mathcal{L}}_{\Lambda}\prod_{k=1}^{n}V_{\alpha_{k}}(z_{k})\;\bigg{\rangle}{}\\!\\!\\!{\phantom{\bigg{|}}}_{\rm
L}=\int_{{\mathbb{S}}^{h}}dC_{\mu}(\bm{\alpha})\;\big{(}{\mathcal{D}}_{\Lambda}^{\mu}{\mathcal{F}}_{\mu}(\bm{\alpha};\bm{z})\big{)}{\mathcal{F}}_{\mu}(\bm{\alpha};\bar{\bm{z}}),$
(8.18)

where ${\mathcal{D}}_{\Lambda}^{\mu}$ are the difference operators
representing the Verlinde line operators introduced in Section 8.3. The
expressions of the form (8.18) define the Verlinde line operator expectation
values in the region around the boundary component of the Deligne-Mumford
compactification associated to the marking $\mu$ within which there exist
convergent power series representations for the conformal block functions
${\mathcal{F}}_{\mu}(\bm{\alpha};\bm{z})$.

The representations of the form (8.18) admit analytic continuation over all of
Teichmüller space. This can be used to extend the definition of correlation
functions modified by the insertion of Verlinde line operators to all of
Teichmüller space. The resulting objects satisfy a generalised version of
modular and crossing invariance in the sense of being independent of the
choice of the marking $\mu$. While holomorphically factorised representations
of a given Verlinde line-operator expectation value certainly depend on the
choice of a marking $\mu$, one may relate the representations associated to
different pants decompositions with the help of (8.15).

An important feature, listed as point c) in Section 8.1 above, is the
coincidence of the action of Verlinde line operators on correlation functions
defined by the representations on holomorphic- and anti-holomorphic conformal
blocks, respectively. This is manifest in representations where the
representation of Verlinde line operators on conformal blocks is diagonal, as
e.g. in (8.17).

###  8.7 Weak degenerate fields in the presence of strong Verlinde loop
operators

The representation of correlation function in the $H_{3}^{+}$ WZNW model in
terms of Liouville correlation functions uses correlation functions with
insertion of the degenerate fields $V_{-1/2b}(y)$. When trying to define
Verlinde line operator expectation values with the help of this
correspondence, it will be important to understand if the correlation
functions depend on the choice of contour representing a lamination $\Lambda$
relative to the positions $y$ of the degenerate fields $V_{-1/2b}(y)$. We will
see that this dependence can be non-trivial for odd laminations, complicating
the definition of the corresponding Verlinde line operator expectation value
in the $H_{3}^{+}$ WZNW model in terms of Liouville correlation functions
somewhat.

In order to understand the origin of this subtlety, one may consider an
annulus $A$ with insertion of a Verlinde line operator of charge $j$ supported
on the non-contractible cycle $\gamma$ of $A$, and degenerate field
$V_{-1/2b}(y)$ inserted. Assigning representation label $\alpha$ to one of the
boundaries, here denoted $\gamma_{+}$ one needs to have representation labels
$\alpha-b^{-1}m$, $m=-j,\dots,j$ at the other boundary $\gamma_{-}$ according
to (7.7). If $\gamma$ is homotopic to $\gamma_{+}$, one finds that
$L_{(\gamma,j)}$ is represented by multiplication with

$\lambda_{j}(\alpha):=2{\cos(2\pi jb(2\alpha-Q))}.$

Otherwise one finds the factor $\lambda_{j}(\alpha-b^{-1}m)$ instead. We have
$\lambda_{j}(\alpha-b^{-1}m)=\lambda_{j}(\alpha)$ if $j=k/2$ with $k$ even,
and $\lambda_{j}(\alpha-b^{-1}m)=-\lambda_{j}(\alpha)$ if $j=k/2$ with $k$
odd.

It follows that Liouville correlation function with degenerate fields
$V_{-1/2b}(y)$ and odd Verlinde line operators has a sign discontinuity when a
degenerate field passes across an odd line. This discontinuity creates issues
in the definition of the integral transforms relating Liouville correlation
functions with Verlinde line operator insertions to solutions of the KZB
equations. The definition of the integrand would require additional choices in
this case, rendering the verification of some of the properties listed in
Section 8.1 difficult.

One may note, on the other hand, that this subtlety does not affect the cases
of even Verlinde line operators. For this class of Verlinde line operators one
may conveniently use the relation to Liouville theory in order to define
Verlinde line operators in the $H_{3}^{+}$-WZNW-model, and to study their
properties.

###  8.8 Verlinde lines in Liouville vs WZNW models

According to the discussion above, it is certainly possible to define the
Verlinde line operators with even weights through the relation with Liouville
theory. The integrand is independent of the choice of representatives of the
contour defining the Verlinde line operators on the surface $C_{g,n}$ obtained
by “filling the degenerate punctures”.

One may note, however, that this does not yield enough Verlinde line operators
to account for the expected spectrum of the quantised Hitchin Hamiltonians in
the critical level limit. There do exist eigen-functions associated to odd-
numbers of grafting lines, which one might expect to get from the limits of
Verlinde line operators with half-integer charges.

In order to account for this apparent mismatch one may recall from Sections 5
and 6.6 that it should be possible to define Verlinde line operators directly
in the $H_{3}^{+}$ WZNW model, taking advantage of the holomorphic
factorisation of its correlation functions. The holomorphic building blocks
appearing in this factorisation should allow one to define modifications
associated to insertions of degenerate representations with half-integer
weights. On the level of differential equations one may expect that the
strategy to define Verlinde line operators by fusing and parallel-transporting
degenerate representations has a close analog in the $H_{3}^{+}$ WZNW model.
More subtle appears to be mainly the interpretation of the results in terms of
summing over intermediate representations induced from principal series
representations of $\mathrm{SL}(2,{\mathbb{C}})$.

One may, however, expect that the outcome can be described in terms of finite
difference operators in a way that generalises the action of Verlinde line
operators with integer charges defined from Liouville theory in a natural way.
From the representation (6.18) it is clear, in particular, that the insertions
of holomorphic and anti-holomorphic Verlinde lines in the $H_{3}^{+}$ WZNW
partition function coincide for Verlinde line operators with both integer and
half-integer charges.

##  9 Grafting from Verlinde line insertions

We are now going to explain how the insertion of Verlinde operators gets
related to the grafting operation in the limit $b\to\infty$. The argument will
in particular establish the precise link between the lamination data
determining a basis for the algebra of Verlinde line operators, and the
grafting data determining real projective structures.

###  9.1 The main claim

We consider the limit $k\to-2$ of the relation (7.9), now represented in the
form

$\displaystyle G^{(n)}_{\Lambda}(\bm{x},\bm{z})$ $\displaystyle=\int
d^{2}y_{1}\dots d^{2}y_{n-3}\;{\mathcal{K}}_{\rm\scriptscriptstyle
SOV}^{(n)}(\bm{x},\bm{y};\bm{z})F^{(n)}_{\Lambda}(\bm{y},\bm{z}),$ (9.1)

where $F^{(n)}_{\Lambda}$ and $G^{(n)}_{\Lambda}$ are correlation functions in
Liouville theory and the $H_{3}^{+}$-WZNW model, respectively, modified by
insertions of Verlinde line operators labelled by a lamination $\Lambda$. The
field labels $\bm{\alpha}=(\alpha_{1},\dots,\alpha_{n})$ and
$\bm{j}=(j_{1},\dots,j_{n})$ are fixed and suppressed in the notations.

We are going to argue that (7.13) and (7.16) get modified to

$\displaystyle F^{(n)}_{\Lambda}(\bm{y},\bm{z})\sim
e^{-b^{2}S_{\Lambda}(\bm{z})}Z^{L(n)}_{\rm\scriptscriptstyle
1-loop}(\bm{z})\prod_{j=1}^{h}\chi_{\Lambda}(y_{j},\bar{y}_{j};\bm{z})\big{(}1+{\mathcal{O}}(b^{-2})\big{)},$
(9.2) $\displaystyle G^{(n)}_{\Lambda}(\bm{x},\bm{z})\sim
e^{-b^{2}S_{\Lambda}(\bm{z})}Z^{W(n)}_{\rm\scriptscriptstyle
1-loop}(\bm{z})\Psi_{\Lambda}(\bm{x},\bm{z})\big{(}1+{\mathcal{O}}(b^{-2})\big{)},$
(9.3)

where

  * •

The functions
$\chi_{\Lambda}(y,\bar{y})\equiv\chi_{\Lambda}(y,\bar{y};\bm{z})$ are related
to the metric $ds^{2}=e^{2\varphi_{\Lambda}(y,\bar{y})}dyd\bar{y}$ defined by
grafting the uniformising metric along $\Lambda$ as

$ds^{2}=\frac{dyd\bar{y}}{(\chi_{\Lambda}(y,\bar{y}))^{2}}.$

This implies in particular that the functions $\chi_{\Lambda}(y,\bar{y})$
satisfy a pair of equations of the form

$\big{(}\partial_{y}^{2}+t_{\Lambda}(y)\big{)}\chi_{\Lambda}(y,\bar{y})=0,\qquad\big{(}\bar{\partial}_{\bar{y}}^{2}+\bar{t}_{\Lambda}(\bar{y})\big{)}\chi_{\Lambda}(y,\bar{y})=0,$
(9.4)

with $t_{\Lambda}(y)\equiv t_{\Lambda}(y;\bm{z})$ holomorphic on $C$.

  * •

The function $S_{\Lambda}(\bm{z})$ is defined up to a constant by the
equations

$\frac{\partial}{\partial
z_{r}}S_{\Lambda}(\bm{z})=E_{r}^{\Lambda}(\bm{z}),\qquad r=1,\dots,h,$ (9.5)

where $E_{r}^{\Lambda}(\bm{z})$ are defined by the expansion of
$t_{\Lambda}-t_{0}$ into a basis for $H^{0}(C,K^{2})$. In the case of $g=0$,
for example, it can be explicitly represented as

$t_{\Lambda}(y;\bm{z})=\frac{\delta_{n}-\sum_{r=1}^{n-1}\delta_{r}}{y(y-1)}+\sum_{r=1}^{n-1}\frac{\delta_{r}}{(y-z_{r})^{2}}+\sum_{r=3}^{n-1}\frac{z_{r}(z_{r}-1)}{y(y-1)}\frac{E_{r}^{\Lambda}(\bm{z})}{y-z_{r}}.$
(9.6)

The dependence of $E_{r}^{\Lambda}(\bm{z})$ is real-analytic rather than
holomorphic.

  * •

The eigenfunctions $\Psi_{\Lambda}$ of the Gaudin Hamiltonians,

${\mathcal{H}}_{x}^{(r)}\Psi_{\Lambda}(\bm{x},\bm{z})=E_{r}^{\Lambda}(\bm{z})\Psi_{\Lambda}(\bm{x},\bm{z}),\qquad\overline{\mathcal{H}}_{\bar{x}}^{(r)}\Psi_{\Lambda}(\bm{x},\bm{z})=\bar{E}_{r}^{\Lambda}(\bm{z})\Psi_{\Lambda}(\bm{x},\bm{z}),$

are finally defined as

$\displaystyle\Psi_{\Lambda}(\bm{x},\bm{z})$ $\displaystyle=\int
d^{2}y_{1}\dots d^{2}y_{h}\;{\mathcal{K}}_{\rm\scriptscriptstyle
SOV}^{(n)}(\bm{x},\bm{y};\bm{z})\prod_{j=1}^{h}\chi_{\Lambda}(y_{j};\bm{z}).$
(9.7)

  * •

The explicit form of $Z^{L(n)}_{\rm\scriptscriptstyle 1-loop}(\bm{z})$ and
$Z^{W(n)}_{\rm\scriptscriptstyle 1-loop}(\bm{z})$ won’t be needed in the
following.

The functions characterising the leading order behaviour of the correlation
functions of the $H_{3}^{+}$-WZNW model are thereby expressed in terms of the
real projective structures obtained by grafting the uniformising metric.

###  9.2 Outline of the proof

To this aim we will start by considering laminations
$\Lambda=\Lambda(\gamma,\bm{m})$ specified by a cut system $\gamma$ with
multiplicities $\bm{m}$. The cut system $\gamma$ defines a component
$\partial_{\gamma}{\mathcal{M}}_{g,n}$ of the Deligne-Mumford moduli space
${\mathcal{M}}_{g,n}$. The systems of partial differential equations satisfied
by Liouville and $H_{3}^{+}$-WZNW correlation functions define real analytic
continuations from a neighbourhood of the boundary component
$\partial_{\gamma}{\mathcal{M}}_{g,n}$ to the Teichmüller space
${\mathcal{T}}_{g,n}$. We will therefore consider values of the arguments
$\bm{z}$ of the correlation functions associated to a sufficiently small
neighbourhood of $\partial_{\gamma}{\mathcal{M}}_{g,n}$. Considering the
extension to ${\mathcal{T}}_{g,n}$ one might expect phenomena of Stokes type
complicating the relation between Verlinde line operator labels and grafting
labels. As mentioned above, we suspect that such phenomena are absent in the
cases of interest for us.

We may first observe that the discussion in Section 7.4 reduces the claim to a
result about the semiclassical limit of certain Liouville correlation
functions. This result generalises and refines previous results from [87, 76,
88].

As example we will consider the case $g=0$, $n=4$, and the cut system defined
by the curve $\gamma$ separating the punctures at $0$ and $z$ from $1$ and
$\infty$, referring to the description used in Section 4.6.1. The
corresponding boundary component $\partial_{\gamma}{\mathcal{M}}_{0,4}$
corresponds to the limit $z\to 0$. As lamination $\Lambda$ we shall consider
$\Lambda=j\gamma$. This example is sufficiently typical.

Near the boundary $\partial_{\gamma}{\mathcal{M}}_{0,4}$ one may use the
holomorphically factorised representation (8.17), which in the case at hand
simplifies to the form

$F_{\Lambda}^{(4)}(z)=\int_{{\mathbb{R}}}dp\;\big{|}{\mathcal{F}}_{\gamma}\big{(}{\textstyle\frac{Q}{2}}+\mathrm{i}p,z\big{)}\big{|}^{2}\cosh(4\pi
bjp).$ (9.8)

We have absorbed the three-point functions into the normalisation for the
conformal blocks ${\mathcal{F}}_{\gamma}$.

####  9.2.1 Classical limit of conformal blocks

The first step is to verify that the conformal blocks in (8.17) behave in the
limit $b\to\infty$ as

$\log{\mathcal{F}}_{\gamma}({\alpha};{z})=-b^{2}{\mathcal{Y}}_{\gamma}({\lambda},{z})+{\mathcal{O}}(b^{0}),$
(9.9)

with $\alpha-\frac{Q}{2}=\mathrm{i}b\frac{\lambda}{4\pi}$, and
${\mathcal{Y}}_{\gamma}({\lambda},{z})$ being the generating function for the
sub-manifold of opers in the character variety defined in Section 4.5.

The verification of (9.9) can be decomposed into three steps.

  * i)

Exponentiation: The following limit exists:

${\mathcal{Y}}_{\gamma}({\lambda},{z}):=-\lim_{b\to\infty}\frac{1}{b^{2}}\log{\mathcal{F}}_{\gamma}({\alpha};{z}),\quad\text{with}\quad\lambda:=\frac{4\pi}{\mathrm{i}b}\Big{(}\alpha-\frac{Q}{2}\Big{)}$
(9.10)

kept finite for $b\to\infty$.

  * ii)

Relation with accessory parameters: Let the accessory parameter $E(\lambda,z)$
be defined as the solution to the inverse monodromy problem to find a function
$t(y)$ of the form

$t(y)=\frac{\delta_{1}}{y^{2}}+\frac{\delta_{2}}{(z-y)^{2}}+\frac{\delta_{3}}{(1-y)^{2}}+\frac{\delta_{n}-\sum_{r=1}^{3}\delta_{r}}{y(y-1)}+\frac{z(z-1)}{y(y-1)}\frac{E(\lambda,z)}{y-z}.$
(9.11)

such that the monodromy of $\partial_{y}^{2}+t(y)$ along a contour separating
$0$ and $z$ from $1$ and $\infty$ has trace equal to $2\cosh(\lambda/2)$. The
classical conformal blocks ${\mathcal{Y}}_{\gamma}(\bm{\lambda},\bm{z})$ are
then related to this accessory parameter as follows:

$\frac{\partial}{\partial
z}{\mathcal{Y}}_{\gamma}\big{(}{\lambda},{z}\big{)}=E({\lambda},{z}).$ (9.12)

  * iii)

Relation with FN-type coordinates:

$\frac{\partial}{\partial\lambda}{\mathcal{Y}}_{\gamma}({\lambda},{z})=\frac{1}{4\pi\mathrm{i}}{\kappa}({\lambda},{z}).$
(9.13)

Parts i) and ii) of the conjectures are due Al.B. Zamolodchikov [89]. More
recent related work in this direction includes [90, 91, 92, 93, 49, 47]. The
solution to the inverse monodromy problem in ii) can be represented as power
series in $z$ having coefficients which can be represented using continued
fractions [47].

In order to show that the conformal blocks have a limit of the required form
(9.10) one may start from the series expansions, use the AGT-conjectures, and
analyse the limit $b^{2}\to\infty$ using cluster expansions [10, 94].

One may next consider the conformal blocks
${\mathcal{F}}_{\gamma,\imath}({\alpha};y,y^{\prime},{z})$ modified by
insertion of the degenerate representations $V_{-1/2b}$ at a points $y$ and
$y^{\prime}$. Such conformal blocks satisfy the BPZ equations (7.6). To
leading order in $b^{-2}$ one can solve these equations in the form

${\mathcal{F}}_{\gamma,\imath}({\alpha};y,y^{\prime},{z})=\exp(-b^{2}{\mathcal{Y}}_{\gamma}({\lambda},{z}))Z^{L(4)}_{\rm\scriptscriptstyle
1-loop}({z})\,\chi_{i}(y)\chi_{i^{\prime}}(y^{\prime})\big{(}1+{\mathcal{O}}(b^{-2})\big{)},$
(9.14)

if $\imath=(i,i^{\prime})$, the functions $\chi_{i}(y)$, $i=1,2$ being
linearly independent solutions to the oper equations
$(\partial_{y}^{2}+t(y))\chi(y)=0$, provided that the function
${\mathcal{Y}}_{\gamma}({\lambda},{z})$ is related to $t(y)$ via (9.12), with
$t(y)$ being of the form (9.11).

One may, on the other hand apply the definition of Verlinde line operators
described in Section 8 to the conformal blocks
${\mathcal{F}}_{\gamma}({\alpha};{z})$. The limit $b\to\infty$ of conformal
blocks modified by Verlinde line operators can be computed in two ways. One
may, on the one hand, use the representations in terms of finite difference
operators discussed in Section 8, giving an expression containing the
derivative on the left side of equation (9.13). Another way to compute this
limit uses the representation of the Verlinde line operators in terms of the
modified conformal blocks
${\mathcal{F}}_{\gamma,\imath}({\alpha};y,y^{\prime},{z})$. The computation of
quantum monodromies underlying the definition of Verlinde line operators
reduces to the computation of classical monodromies of the oper
$\partial_{y}^{2}+t(y)$ in the limit $b\to\infty$. The comparison yields
(9.13), completing the argument.

####  9.2.2 Saddle point analysis of Liouville correlation functions

Recall that we may focus on the limit $z\to 0$. Taking into account the
asymptotic behaviour (4.15) of ${\mathcal{Y}}_{\gamma}$, one sees that the
asymptotic behaviour of the integrand in (8.17) has the form

$\exp\bigg{(}\\!-b^{2}\Big{[}\big{(}\delta(\lambda)-\delta(\lambda_{1})-\delta(\lambda_{2})\big{)}\log|z|^{2}+{\mathcal{Y}}_{0}(\lambda)+j\lambda+{\mathcal{O}}(z)\Big{]}\bigg{)},$
(9.15)

using the notations $\alpha-\frac{Q}{2}=\mathrm{i}b\frac{\lambda}{4\pi}$. The
integrals in (8.17) will therefore be dominated by a saddle point
$\lambda_{j}\equiv\lambda_{j}(z,\bar{z})$ determined by the equation

$2\frac{\partial}{\partial\lambda}\mathrm{Re}\big{[}{\mathcal{Y}}_{\gamma}({\lambda},{z})\big{]}\bigg{|}_{{\lambda}={\lambda}_{j}(z,\bar{z})}\\!\\!\\!=j\,.$
(9.16)

Recalling that
$\kappa(\lambda,z)=4\pi\mathrm{i}\frac{\partial}{\partial{\lambda}}{\mathcal{Y}}_{\gamma}(\lambda,z)$
we see that (9.16) is equivalent to $\mathrm{Im}(\kappa(\lambda_{j},z))=2\pi
j$. In Section 4.6.3 we have argued that there exists a unique solution of the
conditions (4.17) which can be identified with the conditions (9.16)
determining the saddle points of the integral (8.17) if we choose $n=0$ and
$m=2j$ in (4.17).

In this way we are led to the conclusion that the leading behaviour of the
correlation functions (8.17) takes the form

$F^{(4)}_{\Lambda_{j}}({y},{z})\sim\exp\big{(}\\!-2b^{2}\mathrm{Re}\big{[}{\mathcal{Y}}_{\gamma}\big{(}{\lambda}_{j}(z,\bar{z}),{z}\big{)}\big{]}\big{)}\,\chi_{j}(y,\bar{y})\,Z^{L(n)}_{\rm\scriptscriptstyle
1-loop}({z})\big{(}1+{\mathcal{O}}(b^{-2})\big{)},$ (9.17)

where we are using the notation $\Lambda_{j}=j\gamma$, and
$\chi_{j}(y,\bar{y})\equiv\chi_{j}(y,\bar{y};z,\bar{z})$ satisfies the complex
conjugate pair of oper equations

$\big{(}\partial_{y}^{2}+t_{j}(y)\big{)}\chi_{j}(y,\bar{y})=0,\qquad\big{(}\bar{\partial}_{\bar{y}}^{2}+\bar{t}(\bar{y})\big{)}\chi_{j}(y,\bar{y})=0,$
(9.18)

with $t_{j}(y)\equiv t_{j}(y;z,\bar{z})$ of the form (4.11), and parameter
$K=zE_{j}(z,\bar{z})$ determined by

$E_{j}(z,\bar{z})=\frac{\partial}{\partial
z}{\mathcal{Y}}_{\gamma}\big{(}{\lambda},{z}\big{)}\bigg{|}_{\lambda=\lambda_{j}(z,\bar{z})}.$
(9.19)

It follows that the functions $\chi$ in (9.17) coincide with the functions
$\chi_{0,2j}$ introduced in (4.21).

##  10 Outlook

###  10.1 Higher rank generalisations

Although the analysis of this paper was specific to the quantum field theories
having moduli spaces of vacua represented by $SL(2)$ Higgs bundles, we believe
that many of the lessons we have learned can be generalized to wider classes
of quantum field theories and defects in various dimensions. Perhaps
confusingly, there are multiple constructions connecting complex integrable
systems to quantum field theories in various dimensions. Fortunately, the
moduli spaces ${\mathcal{M}}$ of Higgs bundles on a Riemann surface $C$ with
ADE gauge group $G$, possibly modified by various types of punctures303030The
only sharp characterization we know of $P$ is that the punctures should admit
a lift to half-BPS co-dimension two defects in the six-dimensional SCFTs which
are the ultimate source of the various gauge theory constructions [95, 96].,
are compatible with all of these constructions. In this Section outline some
of these conjectures, adopting a 2d CFT perspective.

The minimal QFT perspective involves a collection of non-chiral, non-compact
2d CFTs: the WZW models with target $G/G_{c}$. These 2d CFTs depend on a level
$\kappa$ and admit vertex operators we associate to punctures. They have
chiral and anti-chiral Kac-Moody currents at level $\kappa$. The partition
function $Z_{\kappa}$ is a single-valued twisted half-density on an
appropriate space $\mathrm{Bun}$ of $G$-bundles on $C$ with extra structure at
the punctures. The twist is controlled by the level $\kappa$.

The 2d CFTs are also endowed with a collection of “Verlinde” topological line
defects, labelled by irreducible representations of $G_{c}$. More precisely,
they are labelled by objects in the Kazhdan–Lusztig category
$\mathrm{KL}_{\kappa}[G]$. The Riemann surface $C$ can be decorated further by
a network $\ell$ of such lines, giving rise to a modified partition function
$Z_{\kappa;\ell}$. These are also single-valued twisted half-densities on
$\mathrm{Bun}$.

The 2d CFTs have a standard $\kappa\to\infty$ semi-classical limit. The
equations of motion of the model associate to a point in $\mathrm{Bun}$ the
corresponding unitary flat $G_{c}$ connection. The $\ell$ insertion simply
read off the monodromy data of that connection.

The critical level limit $\kappa\to\kappa_{c}$ is expected to have special
properties, in general. These properties should be made evident by relations
to the 2d Toda theories. Recall that the regular quantum Drinfeld-Sokolov
(qDS) reduction maps a Kac-Moody algebra to the corresponding W-algebra
$W_{\kappa}^{G}$. The Toda theories are non-chiral, non-compact 2d CFTs with
$W_{\kappa}^{G}$ algebra symmetries. One may in particular expect that there
should exist non-chiral versions of the qDS reduction, relating integral
transforms of the partition functions $Z_{\kappa;\ell}$ to the corresponding
partition functions $Z^{W}_{\kappa;\ell}$ of the relevant Toda theories,
possibly decorated by degenerate fields. A stronger conjecture is that a
quantum Separation of Variables transformation exists, reconstructing
$Z_{\kappa;\ell}$ as an integral transform of $Z^{W}_{\kappa;\ell}$ over a
space of positions of degenerate punctures.

A crucial property of $W_{\kappa}^{G}$ is the self-duality
$(\kappa-\kappa_{c})\to(\kappa-\kappa_{c})^{-1}$. This implies that the
critical level limit $\kappa\to\kappa_{c}$ is always related to a semi-
classical limit for the corresponding Toda CFTs. In that limit, the Verlinde
lines labelled by $\ell$ lines become “heavy” and strongly affect the
outcome/semi-classical saddle. The degenerate fields appearing in the qDS
reduction and possibly SoV, on the other hand, will become “light” and have a
finite limit to functions of the semi-classical saddle. Accordingly, the
conjectural quantum SoV transformation should have a finite limit of the form
$Z_{\kappa;\ell}\sim
e^{\frac{1}{\kappa-\kappa_{c}}S_{\ell}}\psi_{\ell}+\cdots,$ where the wave-
functions $\psi_{\ell}$ are half-densities on $\mathrm{Bun}$ produced by the
classical SoV transform. Both $\psi_{\ell}$ and $S_{\ell}$ depend on the
choice of $\ell$ in a way that might display a dependence on the given regime
in parameter space through the choice of saddle points dominating in different
regimes.

In the critical level limit, the Kac-Moody algebra develops a large center
[97], realized as the quantum Hitchin Hamiltonians acting on sections on
$\mathrm{Bun}$ [8]. This includes the Sugawara element, which is the limit of
$(\kappa-\kappa_{c})T$, where $T$ is the stress tensor. Although the stress
tensor is modified by the qDS reduction, the divergent parts of the stress
tensor and other similar generators matches the semi-classical limits of the
$W_{\kappa}^{G}$ generators.

As a consequence, the coefficient $\psi_{\ell}$ above must be an eigenfunction
of the quantum Hitchin Hamiltonians, with eigenvalues given by the values of
the $W_{\kappa}^{G}$ determined by the semiclassical saddle. We thus get a
family of eigenfunctions labelled by the networks $\ell$ of lines valued in
$\mathrm{KL}_{\kappa}[G]$, i.e. elements of the Skein algebra associated to
$C$, $G$ and the punctures.

Up to this point, our conjectures are based on reasonably well-understood
aspects of the WZW CFTs, together with the self-duality of Toda CFTs. These
are naturally defined for positive level $\kappa$ and admit conjectural
analytic continuation away from the $\kappa<\kappa_{c}$ negative real axis.
Next, we specialise to the unusual choice $\kappa\in\kappa_{c}+i\mathbb{R}$,
where the space of twisted half-densities on $\mathrm{Bun}$ is equipped with a
hermitian inner product and can be completed to an Hilbert space
$\mathcal{H}_{\kappa}$.

One basic conjecture is that $Z_{\kappa;\ell}$ is normalizable on
$\mathrm{Bun}$. Expected properties of inner products $\langle
Z_{\kappa;\ell}|Z_{\kappa;\ell^{\prime}}\rangle$ are discussed at length in a
companion paper [25]. From a 2d CFT perspective, a proof of this statement
requires a very careful analysis of the behaviour of the partition function
near loci in $\mathrm{Bun}$ where it becomes singular. Alternatively, one
could attempt to verify unitarity of a quantum SoV kernel and study the
normalizability of the wavefunction in the configuration space of degenerate
punctures. In [25] the authors invoke a chain of dualities mapping the inner
product to the Schur index of a theory of class ${\mathcal{S}}$.

In the critical level limit, we learn that $\psi_{\ell}$ itself is a
normalizable eigenfunction representing a part of the spectrum of the quantum
Hitchin Hamiltonians. A further conjecture is that $Z_{\kappa;\ell}$ and thus
$\psi_{\ell}$ give bases in the respective Hilbert spaces.

###  10.2 Real generalizations

Two-dimensional CFTs admit a rich collection of conformal boundary conditions.
The $G/G_{c}$ WZW model, in particular, admits boundary conditions which
relate the holomorphic and anti-holomorphic Kac-Moody currents by some anti-
linear map $\tau$. The theory of these boundary conditions [98, 99] and of
cross-caps [100] has been mostly developed for $G=SL(2)$. It generalizes the
rational WZW case [101] in a manner analogous to what happens in Liouville
theory [102, 103, 104]. The Separation of Variables approach is still
applicable [105].

Given a Riemann surface $C$ with boundaries decorated in such a manner, we can
associate to it a partition function $Z_{\kappa}$ which is a twisted half-
density on a space of “real” bundles $\mathrm{Bun}_{\mathbb{R}}$. Assuming we
also understand the interplay between boundaries and topological lines, we can
also produce a collection of partition functions $Z_{\kappa;\ell}$.

The logic of the previous analysis will apply verbatim to this situation,
leading to Toda theories with boundaries and to normalizable eigenfunctions of
quantum Hitchin Hamiltonians. The main extra piece of information needed for a
detailed analysis are the dictionaries between WZW and Toda boundaries and the
self-duality properties of boundary conditions.

##  Appendix A Real opers with more general types of singularities

###  A.1 Real opers with more general types of punctures

The discussion of grafting can be readily extended to the case of a surface
$C$ with punctures.

For a regular puncture at $z_{a}$, $t(z)$ has a double pole with a coefficient

$\frac{1}{4}-\frac{1}{4}(k_{a}+2it_{a})^{2}$ (A.1)

controlled by the parameters $t_{a}$ and $k_{a}$ of the principal series
representation there. Notice that $k_{a}$ can be half-integral depending on
the global form of the gauge group.

Solutions of the oper differential equations will behave as

$(z-z_{a})^{\frac{1}{2}\pm\frac{k}{2}\pm it}$ (A.2)

so that $\phi$ behaves as

$c(z-z_{a})^{k}|z-z_{a}|^{1-k+2it}+\mathrm{c.c.}$ (A.3)

for some constant $c$ and the local coordinate behaves as

$(z-z_{a})^{k_{a}+2it_{a}}$ (A.4)

If $k_{a}=0$, this is real along a semi-infinite collection of circles
converging to $z_{a}$ as a geometric series. The parameter $t_{a}$ controls
the density of the sequence in the radial direction. If $k_{a}\neq 0$, the
zero locus of $\phi$ includes $2k_{a}$ curves spiraling into $z_{a}$.

If all $k_{a}=0$, the zero locus minus the components wrapping around a single
puncture forms what is can be reasonably called an half-integer measured
lamination on the punctured Riemann surface. We still have a notion of
uniformizing oper: the real oper which only has the semi-infinite sequence of
circles around each regular puncture. This is a well-known construction of
hyperbolic metrics with geodesic boundaries of fixed length. Grafting along
closed geodesics then adds extra closed curves with a more general homotopy.

We can also connect the standard grafting operation to a form of gluing along
regular punctures. First, imagine cutting some $C$ along a geodesic, but
grafting in an infinite sequence of new lines of zeroes. The result will be a
real projective structure on a surface $C^{\prime}$ obtained from $C$ by
replacing an handle with two regular punctures, with a parameter $t$ which
encodes the length of the original geodesic, i.e. the monodromy eigenvalue for
the connection around the geodesic we cut.

If some $k_{a}\neq 0$, the story is somewhat different, as the zero locus now
includes open lines ending on punctures. We can define a notion of relative
half-integer measured lamination, including both closed curves and open
curves, with a specified number $k_{a}$ of curves ending at a given puncture.
We still lose some of the nice features we discussed above: there is no
natural notion of “minimal” real oper analogous to the uniformizing oper and
grafting only allows one to add closed lines. We thus are left with no proof
of the following natural conjecture: for every $t_{a}$ and $k_{a}$, each
relative lamination labels exactly one real oper.

###  A.2 Irregular singularities

One can also discuss irregular punctures. At an irregular puncture, we have a
behaviour

$e^{\pm(z-z_{a})^{-\mathrm{rk}_{a}}+\cdots}$ (A.5)

The natural generalization of the single-valuedness condition on $\phi$ is
that $\phi$ should not grow exponentially fast at infinity. This imposes a
reality condition on the Stokes data of the flat connection. Then $\phi$
behaves roughly as

$ce^{(z-z_{a})^{-\mathrm{rk}_{a}}-(\bar{z}-\bar{z}_{a})^{-\mathrm{rk}_{a}}\cdots}+\mathrm{c.c.}$
(A.6)

and the local coordinate as

$e^{2(z-z_{a})^{-\mathrm{rk}_{a}}+\cdots}$ (A.7)

which is real along $2\mathrm{rk}_{a}$ infinite collections of lines
approaching $z_{a}$ along the rays where $(z-z_{a})^{\mathrm{rk}_{a}}$ is
real, i.e. within specific Stokes sectors of the irregular puncture.

Semi-infinite collections of open lines will thus stretch between consecutive
Stokes sectors. Additionally, some finite collection of open lines may stretch
between non-consecutive sectors or between different regular punctures. We can
define irregular versions of half-integer measured lamination by including
such non-trivial open curves ending at Stokes sectors. Irregular punctures of
integral rank are labelled by “formal monodromy” parameters $k_{a}$ and
$t_{a}$. One can argue that $2k_{a}$ control the difference between non-
trivial curves ending at even or odd Stokes sectors.

The data of laminations in the presence of punctures may seem intricate.
Remarkably, in all the situations described above, the lamination data
coincides with the data one would employ to label holomorphic functions (or
sections of line bundles) on the corresponding character varieties. As we saw
in the main text, this is no coincidence and we expect it to remain remain
true even for general gauge group $G$.

A simple example of an irregular real oper, for $t(z)=-z$, is

$\mathrm{Ai}(z)\mathrm{Bi}(\bar{z})+\mathrm{Ai}(\bar{z})\mathrm{Bi}(z)$ (A.8)

leading to the pattern of zeroes in Figure 9.

Figure 9: The pattern of zeroes for the “Airy” real oper

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