# Spin Calogero-Moser periodic chains and two dimensional Yang-Mills theory
with corners.

Nicolai Reshetikhin  N.R.: YMSC, Tsinghua University, Beijing, China,& BIMSA,
Beijing, China & Department of Mathematics, University of California,
Berkeley, CA 94720, USA<EMAIL_ADDRESS>
###### Abstract.

Quantum Calogero-Moser spin system is a superintegable system with the
spectrum of commuting Hamiltonians that can be described entirely in terms of
representation theory of corresponding simple Lie group. In this paper the
underlying Lie group $G$ is a compact connected, simply connected simple Lie
group. It has a natural generalization known as quantum Calogero-Moser spin
chain. In the first part of the paper we show that quantum Calogero-Moser spin
chain is a quantum superintegrable systems. Then we show that the Euclidean
multi-time propagator for this model can be written as a partition function of
a two-dimensional Yang-Mills theory on a cylinder. Then we argue that the two-
dimensional Yang-Mills theory with Wilson loops with ”outer ends” should be
regarded as the theory on space times with non-removable corners. Partition
functions of such theory satisfy non-stationary Calogero-Moser equations.

> Dedicated to the memory of Vaughan Jones.

## Introduction

In this paper we consider the two-dimensional Yang-Mills (2D YM) theory on
surfaces with open Wilson graphs. The partition function in this case is a
vector in the tensor product of finite dimensional representations of a simple
compact Lie group. We show that it is a solution to $N$-point spin Calogero-
Moser (CM) evolution with initial conditions determined by the structure of
the Wilson graph.

One can draw an analogy between these models and the Chern-Simons theory with
Wilson lines ending on the two-dimensional boundary of a 3-manifold.

Quantum spin Calogero-Moser systems have a long history. See for example
[7][14] and references therein. In this paper we will work with so called
$N$-spin version of CM systems. These models of many-particle quantum systems
with internal (spin) degrees of freedom are examples of quantum
superintegrable systems. For the superintegrability of quantum CM spin systems
and for the notion of quantum superintegrability see [14][18]. For the
superintegrability of classical $N$-point spin CM chain see [17]

Two dimensional model of Yang-Mills (YM) theory is a two dimensional
topological quantum field theory. A non-perturbative partition function for a
disc was introduced in [10]. The 2D YM theory was studied as a TQFT in papers
[22][23] where among other things a convincing argument was given that the
non-perturbative formula is a version of the Duijstermaat-Heckman
localization. For the latest study of perturbative aspects of the YM theory on
surfaces with boundary and for more references see [11]. To be more precise,
the YM theory on surfaces is not exactly topological. The partition function
depends on the total area of the surface (assuming it is finite), and,
possibly, on more parameters which are additive with respect to gluing, when
the rank of the gauge group is greater than one [23].

The partition function and expectation values of Wilson graphs (line
observables in this TQFT) can be interpreted as an analytic continuation of
the semiclassical limit of quantum Chern-Simons invariants [12][23]. If the
corresponding quantum group [13] has $q=\exp(i\frac{\pi}{r})$ with
$r=k+h^{\vee}$ where $k$ is the level of the quantum Chern-Simons theory [21],
$h^{\vee}$ is the dual Coxeter number, the semiclassical limit is when
$k\to\infty$. The quantum Chern-Simons invariant of of a circle bundle over
$\Sigma$ with the Chern class $m$ in the limit when $k\to\infty,m\to\infty$
and $a=\frac{m}{r}$ is finite becomes the partition function of the 2D YM with
the area $ia$ in the limit . When $m=0$ this was first observed in [23]. For
$a\neq 0$ see [12].

In this paper we show that the partition function of the 2D YM theory on
surfaces with open Wilson graphs are solutions to non-stationary Calogero-
Moser type differential equations. For example the partition function on a
cylinder with Wilson lines that are parallel to the cylinder is the multi-time
propagator for quantum $N$-spin Calogero-Moser system with Euclidean time
being areas of corresponding strips of the cylinder. We also argue that such
partition functions can be interpreted as a $2-1-0$ TQFT with non-removable
corners (points at the boundary where Wilson lines ”go out”).

In the first section we define quantum $N$-spin Calogero-Moser model for a
compact simple Lie group $G$. In the second section we explain the relation
between quantum $N$-spin Calogero-Moser system and the 2D YM with open Wilson
graphs.

###  0.1. Acknowledments

This paper was started as a joint project with Jasper Stokman. The author is
grateful to Jasper for many discussions and for the collaboration on this
paper. He also would like to thank S. Gukov, V. Kazakov, P. Mnev and G. Moore
for references, discussions and comments and to D. Solovyev for help with
figures. N.R. was partially supported by the Collaboration Grant ”Categorical
Symmetries” from the Simons Foundation, by the NSF grant DMS-1902226 and by
the RSF grant 21-11-00141. This paper grew out of the talk given by N.R. at
the ”Physics and Geometry” seminar at BIMSA in the Spring 2022. He is grateful
to participants of this seminar for interesting discussions.

##  1\. Quantum $N$-spin Calogero-Moser system and its superintegrability

###  1.1. Quantum superintegrable systems

Let $A_{h}$ be a family of associative algebras with trivial center. We think
of each algebra $A$ from this family as the algebra of quantum observables for
a quantum system.

A commutative algebra has rank $k$ means that it has maximum $k$ algebraically
independent elements, i.e. $k$ is the Krull dimension of $I$. We will say that
an associative algebra $A$ has rank $n$ if $n$ is the rank of maximal
commutative subalgebra in $A$.

######  Definition 1.

A commutative subalgebra $I\subset A$ of rank $k$ is a quantum superintegrable
system on $A$ if the rank of the centralizer of $I$ in $A$ is $n$ where $n$ is
the rank of $A$.

For a superintegrable quantum system we have embeddings of associative
algebras:

$I\subset Z(I,A)\subset A$

where $Z(I,A)$ is the centralizer of $I$ in $A$ and
$\mbox{rank}(Z(I,A))=\mbox{rank}(A)$.

An example of a superintegrable system is when $A$ is the algebra of
differential operators in variable $x_{1},\dots,x_{n}$ with polynomial
coefficients, $I$ is the algebra of differential operators in
$x_{1},\dots,x_{k}$ with constant coefficients. In this case $Z(I,A)$ is the
algebra of differential operators in $x_{1},\dots,x_{n}$ with polynomial
coefficients in $x_{k+1},\dots,x_{n}$.

###  1.2. Algebraic preliminaries

In this section we develop an algebraic set up for defining quantum $N$-spin
Calogero-Moser system.

#### 1.2.1.

Throughout this paper $G$ is a compact, simple, simply connected Lie group. We
will denote by $\mathcal{R}(G)$ the algebra of functions on $G$ generated by
matrix elements of finite dimensional representations. It is a span of matrix
elements of irreducible representations111 There is a natural isomorphism of
$G\times G$-modules
$\mathcal{R}(G)\simeq\bigoplus_{\lambda}V_{\lambda}\otimes V_{\lambda}^{*}$
where the sum is taken over all irreducible finite dimensional representation
of $G$ and $V_{\lambda}$ is the irreducible representation with the highest
weight $\lambda$. Note that space on the right consists of finite direct sums
of unbounded length..

The space $\mathcal{R}(G)$ is an open dense subspace in the Hilbert space
$L^{2}(G)$ of square integrable functions with respect the Haar measure on
$G$.

We denote by $\mathbb{D}(G)$ the algebra of differential operators on $G$
preserving $\mathcal{R}(G)$. Let $\mbox{Vect}(G)$ be the subalgebra in the Lie
algebra of all vector fields on $G$ which correspond to first order
differential operators in $\mathbb{D}(G)$.

The Lie group $G$ acts on itself by left and right translations:
$h\mathrel{\mathop{\ordinarycolon}}g\mapsto hg$ and
$h\mathrel{\mathop{\ordinarycolon}}g\mapsto gh^{-1}$. These actions give two
homomorphisms of Lie algebras $\mathfrak{g}\to\mbox{Vect}(G)$ and, as a
consequence, two algebra inclusions
$\widehat{\mu}_{\ell},\widehat{\mu}_{r}\mathrel{\mathop{\ordinarycolon}}U\mathfrak{g}\rightarrow\mathbb{D}(G)$
(quantum mement maps) as right and left $G$-invariant differential operators,
respectively.

Let $Z(\mathfrak{g})$ be the center of $U\mathfrak{g}$, and $S$ the antipode
for $U\mathfrak{g}$, i.e. the anti-involution of $U\mathfrak{g}$ that acts as
$x\to-x$ on elements $\mathfrak{g}\subset U\mathfrak{g}$.

#### 1.2.2.

Consider the group $G^{N}$ acting on itself by left translations :

$(h_{1},\dots,h_{N})\mathrel{\mathop{\ordinarycolon}}(g_{1},g_{2},\dots,g_{N})\mapsto(h_{1}g_{1},h_{2}g_{2},\dots,h_{N}g_{N})$

by right translations twisted by the cyclic permutation:

$(h_{1},\dots,h_{N})\mathrel{\mathop{\ordinarycolon}}(g_{1},g_{2},\dots,g_{N})\mapsto(g_{1}h_{2}^{-1},g_{2}h_{3}^{-1},\dots,g_{N}h_{1}^{-1})$

The combination of these two actions acts as:

(1)
$(h_{1},\dots,h_{N})\mathrel{\mathop{\ordinarycolon}}(g_{1},g_{2},\dots,g_{N})\mapsto(h_{1}g_{1}h_{2}^{-1},h_{2}g_{2}h_{3}^{-1},\dots,h_{N}g_{N}h_{1}^{-1})$

Note that the action by twisted conjugations has a natural interpretation as
the gauge action parallel transports for a connection in a trivial principal
$G$-bundle over a circle with with marked points. In this interpretation
$g_{i}$ is the holonomy along the interval connecting points $i$ and $i+1$,
and $h_{i}$ is the gauge transformation at the $i$-th point, see Fig. 2. This
is why we call (1) the gauge action.

Figure 1.  Holonomies along integravls and gauge transformations. Here $g_{i}$
stands for the holonomy between points $i$ and $i+1$. The gauge transformation
at points $i-1$ and $i$ acts on $g_{i}$ as $g_{i}\mapsto
h_{i}g_{i}h_{i+1}^{-1}$. Arrow show the orientation of the circle and of the
disc that it bounds.

The left, the right and the gauge action of $G^{N}$ on itself lift to actions
on $\mathbb{D}(G^{N})$. Quantum moment maps for these actions give injective
homomorphisms of associative algebras
$\widehat{\mu}_{L},\widehat{\mu}_{R},\widehat{\mu}_{ad}\mathrel{\mathop{\ordinarycolon}}U\mathfrak{g}^{\otimes
N}\subset\mathbb{D}(G)$.

This gives the following chain of inclusions of associative algebras:

$\mathbb{D}(G^{N})\stackrel{{\scriptstyle\widehat{\mu}_{L}\otimes\widehat{\mu}_{R}}}{{\hookleftarrow}}(U\mathfrak{g})^{\otimes
N}\otimes(U\mathfrak{g})^{\otimes N}\hookleftarrow(U\mathfrak{g})^{\otimes N}$

Here the right inclusion is simply mapping to the left factor in the tensor
product222 We could have taken the inclusion to the right factor. At the end
it does make a difference.

This maps filter through algebra homomorphisms:

$\mathbb{D}(G^{N})\stackrel{{\scriptstyle\widehat{\mu}_{L}\otimes\widehat{\mu}_{R}}}{{\hookleftarrow}}(U\mathfrak{g})^{\otimes
N}\otimes_{Z(\mathfrak{g})^{\otimes N}}(U\mathfrak{g})^{\otimes
N}\hookleftarrow(U\mathfrak{g})^{\otimes N}$

Here $Z(\mathfrak{g})\subset U\mathfrak{g}$ is the center of the universal
enveloping algebra and $\otimes_{Z(\mathfrak{g})^{\otimes N}}$ is the quotient
of the algebraic tensor product. In this tensor product over
$Z(\mathfrak{g})^{\otimes N}$ we have:

$\displaystyle(a_{1}z_{1}\otimes a_{2}z_{2}\dots\otimes a_{N}z_{N})$
$\displaystyle\otimes_{Z(\mathfrak{g})^{\otimes N}}$
$\displaystyle(b_{1}\otimes b_{2}\dots\otimes b_{N})=$
$\displaystyle(a_{1}\otimes a_{2}\dots\otimes a_{N})$
$\displaystyle\otimes_{Z(\mathfrak{g})^{\otimes N}}$
$\displaystyle(S(z_{1})b_{1}\otimes S(z_{2})b_{2}\dots\otimes S(z_{N})b_{N}).$

Here $S(z)$ is the antipode of the central element $z$. The antipode in this
case is an algebra antiautomorphism of $U(\mathfrak{g})$ which acts as
$S(x)-x$ on linear elements $x\in\mathfrak{g}\subset U(\mathfrak{g})$.

The last embedding is $a\mapsto a\otimes_{Z(\mathfrak{g})^{\otimes N}}1$. Note
that we have a natural isomorphism

$(U\mathfrak{g})^{\otimes N}\otimes_{Z(\mathfrak{g})^{\otimes
N}}(U\mathfrak{g})^{\otimes
N}\simeq((U\mathfrak{g})\otimes_{Z(\mathfrak{g})}(U\mathfrak{g}))^{\otimes N}$

The action of the gauge group $G^{N}$ extends naturally to its action on
$\mathbb{D}(G)$. It acts by the adjoint action on each copy of
$U(\mathfrak{g})^{\otimes N}$ and it acts diagonally on
$(U\mathfrak{g})^{\otimes N}\otimes_{Z(\mathfrak{g})^{\otimes
N}}(U\mathfrak{g})^{\otimes N}$. The $G^{N}$-invariant part of these
embeddings is:

(2)
$\mathbb{D}(G^{N})^{G^{N}}\stackrel{{\scriptstyle\widehat{\mu}_{L}\widehat{\otimes}\widehat{\mu}_{R}}}{{\hookleftarrow}}(U\mathfrak{g}^{\otimes
N}\otimes_{Z(\mathfrak{g})^{\otimes N}}U\mathfrak{g}^{\otimes
N})^{G^{N}}\hookleftarrow Z(\mathfrak{g})^{\otimes N}$

Here $Z(\mathfrak{g})^{\otimes N}$ is the $G^{N}$-invariant part of
$U\mathfrak{g}^{\otimes N}$ with respect to the adjoint action.

###  1.3. Quantum $N$-spin Calogero-Moser system

####  1.3.1. The space of states

Let $V_{\mu_{1}},\dots,V_{\mu_{N}}$ be finite dimensional irreducible
representations of $G$ with highest weights $\mu_{i}$ and representation maps
$\pi^{\mu_{i}}$. We will use the same notation for associated irreducible
representations of $U\mathfrak{g}$,
$\pi^{\mu_{i}}\mathrel{\mathop{\ordinarycolon}}U\mathfrak{g}\rightarrow\textup{End}(V_{\mu_{i}})$.
Fix Hermitian scalar product $(\cdot,\cdot)_{\mu_{i}}$ on each $V_{\mu_{i}}$
turning it into a unitary representation. Denote $A^{\dagger}$ the Hermitian
conjugate to $A\in\textup{End}(V_{\mu_{i}})$ relative to
$(\cdot,\cdot)_{\mu_{i}}$. The unitarity of the representation $V_{\mu_{i}}$
can be written as

$\pi_{\mu_{i}}(g)^{\dagger}=\pi_{\mu_{i}}(g^{-1}).$

Now let us define the representation space for $\mathbb{D}(G^{N})^{G^{N}}$
corresponding to representations $\mu_{i}$,

Consider the following space of gauge group equivariant
$V_{\mu_{N}}\otimes\cdots\otimes V_{\mu_{1}}$-valued functions of $G^{N}$:

$\mathcal{R}_{\mu}=\mathcal{R}(G^{N}\to V_{\mu_{1}}\otimes\cdots\otimes
V_{\mu_{N}})\mathrel{\mathop{\ordinarycolon}}=\bigl{(}\mathcal{R}(G^{N})\otimes
V_{\mu_{1}}\otimes\cdots\otimes V_{\mu_{N}}\bigr{)}^{G^{N}}.$

were $\mathcal{R}(G^{N})$ is the span of matrix elements of all finite
dimensional representations of $G^{N}$. The $G^{N}$-equivariance condition for
$f\in\mathcal{R}_{\mu}$ is

(3)
$f(g^{h})=(\pi^{\mu_{1}}(h_{1})\otimes\cdots\otimes\pi^{\mu_{N}}(h_{N}))f(g_{1},\ldots,g_{N})$

where $g^{h}$ is the gauge action of $h\in G^{N}$ on $g\in G^{N}$ .

Clearly this is a module over $\mathbb{D}(G^{N})$ with natural action of
differential operators on functions.

This space has a Hermitian scalar product

$(f_{1},f_{2})=\int_{G^{N}}(f_{1}(g_{1},\dots,g_{N}),f_{2}(g_{1},\dots,g_{N}))_{\mu}dg_{1}\dots
dg_{N}.$

Here $(v_{1}\otimes\dots\otimes v_{N},u_{1}\otimes\dots\otimes
u_{N})_{\mu}=\prod_{i=1}^{N}(v_{i},u_{i})_{\mu_{i}}$ and $dg$ is the Haar
measure on $G$ normalized such that $\int_{G}dg=1$.

The completion $\mathcal{H}_{\mu}$ of $\mathcal{R}_{\mu}$ with respect to the
norm $||f||^{2}=(f,f)$ is the gauge equivariant subspace in
$V_{\mu_{1}}\otimes\cdots\otimes V_{\mu_{N}}$-valued square integrable
functions on $G^{N}$ satisfying (3). This is the space of states of quantum
$N$-spin Calogero-Moser system.

####  1.3.2. Quantum moment maps

Let $X\in\mathfrak{g}\subset U\mathfrak{g}$, define
$X_{j}=1^{\otimes(j-1)}\otimes X\otimes
1^{\otimes(N-j-1)}\in(U\mathfrak{g})^{\otimes N}$. The images of $X_{j}$ in
$\mathbb{D}(G^{N})$ with respect to the left and to the right quantum moment
map act on $\mathcal{R}(G^{N})\otimes V_{\mu_{N}}\otimes\cdots\otimes
V_{\mu_{1}}$ as follows

$\widehat{\mu}_{R}(X_{j})f(g_{1},\dots,g_{j},g_{j+1},\dots,g_{N})=\frac{d}{dt}f(g_{1},\dots,g_{j}e^{tX},\dots,g_{N})|_{t=0},$

where $X\mapsto e^{X}$ is the exponential map of $G$.

$\widehat{\mu}_{L}(X_{j})f(g_{1},\dots,g_{j},g_{j+1},\dots,g_{N})=\frac{d}{dt}f(g_{1},\dots,e^{-tX}g_{j},\dots,g_{N})|_{t=0}.$

The image of $X_{j}$ with respect to the quantum moment map for the gauge
action
$\widehat{\mu}(X_{j})=\widehat{\mu}_{L}(X_{j})+\widehat{\mu}_{R}(X_{j+1})$
acts as

$\widehat{\mu}(X_{j})f(g_{1},\dots,g_{j-1},g_{j},\dots,g_{N})=\frac{d}{dt}f(g_{1},\dots,g_{ij1}e^{-tX},e^{tX}g_{j},\dots,g_{N})|_{t=0}$

Gauge innvariance (3) implies that for each $f\in\mathcal{R}_{\mu}$, we have

(4)
$\widehat{\mu}(X_{j})f(g_{1},\ldots,g_{N})=\pi_{j}^{\mu_{j}}(X)f(g_{1},\ldots,g_{N})$

where
$\pi_{j}^{\mu_{j}}=\textup{id}_{V_{\mu_{N}}}\otimes\cdots\otimes\textup{id}_{V_{\mu_{j+1}}}\otimes\pi^{\mu_{j}}\otimes\textup{id}_{V_{\mu_{j-1}}}\otimes\cdots\otimes\textup{id}_{V_{\mu_{1}}}$.

####  1.3.3. Quantum Hamiltonians and quantum integrals

Recall that $I_{\mu}$ is the subalgebra of quantum Hamiltonians and $J_{\mu}$
is the subalgebra of quantum integrals.

Because $Z(\mathfrak{g})^{\otimes N}$,
$\bigl{(}(U\mathfrak{g}\otimes_{Z(\mathfrak{g})}\otimes
U\mathfrak{g})^{\otimes N}\bigr{)}^{G^{N}}$ are subalgebras of
$\mathbb{D}(G^{N})^{G^{N}}$, they act on the space $\mathcal{R}_{\mu}(G^{G})$.
Denote the images of the subalgebras $Z(\mathfrak{g})^{\otimes N}$,
$\bigl{(}(U\mathfrak{g}\otimes_{Z(\mathfrak{g})}\otimes
U\mathfrak{g})^{\otimes N}\bigr{)}^{G^{N}}$ and in
$\textup{End}(\mathcal{R}_{\mu})$ by $I_{\mu}$, $J_{\mu}$ and $A_{\mu}$. They
form a chain of subalgebras

(5) $A_{\mu}\hookleftarrow J_{\mu}\hookleftarrow I_{\mu}$

The center $Z(\mathfrak{g})$ as commutative algebra is a polynomial ring in
$r=\textup{rank}(\mathfrak{g})$ homogeneous generators
$c_{d_{1}},\ldots,c_{d_{r}}$ of degrees $d_{1}=2\leq d_{2}\leq\cdots\leq
d_{r}$ respectively333Numbers $d_{i}-1,1\leq i\leq r$ are called the exponents
of $\mathfrak{g}$.. Note that $c\mathrel{\mathop{\ordinarycolon}}=c_{2}\in
Z(\mathfrak{g})$ is the quadratic Casimir element determined by the Killing
form.

Denote by

$H^{\mu,(j)}_{d_{k}}\in I_{\mu},\qquad\quad 1\leq j\leq N,\,1\leq k\leq r$

the action of
$c^{(j)}_{k}\mathrel{\mathop{\ordinarycolon}}=1^{\otimes(j-1)}\otimes
c_{k}\otimes 1^{\otimes(N-j-1)}\in Z(\mathfrak{g})^{\otimes N}$ on
$\mathcal{R}_{\mu}$. The Hamiltonians $H^{\mu,(j)}_{d_{k}}$ commute and
generate the algebra $I_{\mu}$.

We will compute the radial components of the quadratic Hamiltonians
$H^{(j)}\mathrel{\mathop{\ordinarycolon}}=H^{(j)}_{2}$ ($1\leq j\leq N$) in
Subsection 1.4.

####  1.3.4. The spectrum of quantum Hamiltonians

Fix $N$ finite dimensional irreducible $G$-modules
$V_{\nu_{1}},\ldots,V_{\nu_{N}}$. Denote by $\mathcal{R}^{(\nu)}(G^{N})$ the
subspace of $\mathcal{R}(G^{N})$ spanned by matrix elements of the irreducible
$G^{N}$-representation $V_{\nu_{1}}\otimes\cdots\otimes V_{\nu_{N}}$444It does
not matter which basis we use to define matrix elements, the subspace will be
the same.. Then by Peter-Weyl theorem we have the decomposition with respect
to the left and the right action of $G^{N}$:

(6) $\mathcal{R}(G^{N})=\bigoplus_{\nu}\mathcal{R}^{(\nu)}(G^{N})$

Here $\nu$ runs over the set of $N$-tuples of dominant integral weights.

Subspaces $\mathcal{R}^{(\nu)}(G^{N})\subset\mathcal{R}(G^{N})$ are invariant
with respect to the gauge action of $G^{N}$, thus we can define subspaces

$\mathcal{R}^{(\nu)}_{\mu}\mathrel{\mathop{\ordinarycolon}}=(\mathcal{R}^{(\nu)}(G^{N})\otimes
V_{\mu_{1}}\otimes\cdots\otimes V_{\mu_{N}})^{G^{N}}\subset\mathcal{R}_{\mu},$

Note that this subspace is not empty only if $V_{\nu_{i}}\subset
V_{\nu_{i-1}}\otimes V_{\mu_{i}}$, cyclically for $i=1,\dots,N$.

Because $Z(\mathfrak{g})$ acts on an irreducible representation by the
multiplication on the corresponding central character, the subspace
$\mathcal{R}^{(\nu)}_{\mu}$ is an eigensubspace for $I_{\mu}$ with

$H_{d_{k}}^{(j)}f=c_{d_{k}}(\nu_{j})f$

for any $f\in\mathcal{R}^{(\nu)}_{\mu}$. Here $c_{d_{k}}(\eta)$ is the value
of $c_{d_{k}}\in Z(\mathfrak{g})$ on the irreducible finite dimensional module
$V_{\eta}$.

The Peter-Weyl theorem for $\mathcal{R}(G^{N})^{\otimes N}$ gives the
decomposition of $\mathcal{R}_{\mu}$ in simultaneous eigensubpaces for the
action of the Hamiltonians $H^{(j)}_{d_{k}}$:

(7) $\mathcal{R}_{\mu}=\bigoplus_{\nu}\mathcal{R}^{(\nu)}_{\mu}$

For intertwiners $a_{i}\in\textup{Hom}_{G}(V_{\nu_{i}},V_{\nu_{i-1}}\otimes
V_{\mu_{i}})$ (here $\nu_{o}=\nu_{N}$), define the trace function
$\Psi^{(\nu)}_{a,\mu}\in\mathcal{R}^{(\nu)}(G^{N})\otimes
V_{\mu_{1}}\otimes\cdots\otimes V_{\mu_{N}}$ on $G^{N}$ as

(8) $\begin{split}\Psi^{(\nu)}_{a,\mu}(g_{1},&\dots,g_{N})\\\
=&\textup{Tr}_{V_{\nu_{1}}}\bigl{(}(a_{1}\pi^{\nu_{1}}(g_{1})\otimes\textup{id}_{\textup{V}_{\mu_{2}}\otimes\cdots\otimes
V_{\mu_{N}}})\cdots(a_{N-1}\pi^{\nu_{N-1}}(g_{N-1})\otimes\textup{id}_{\textup{V}_{\mu_{N}}})a_{N}\pi^{\nu_{N}}(g_{N})\bigr{)}.\end{split}$

It is clear that $\Psi^{(\nu)}_{a,\mu}$ is gauge invariant and that
$\Psi^{(\nu)}_{a,\mu}\in\mathcal{R}^{(\nu)}_{\mu}$. The restriction of
$\Psi^{(\nu)}_{a,\mu}$ to $1^{\times(N-1)}\times H$ gives generalized trace
functions from [5].

######  Theorem 1.

Functions $\Psi^{(\nu)}_{a,\mu}$ form a linear basis in
$\mathcal{R}_{\mu}^{(\nu)}$ enumerated by a basis in
$\otimes_{i=1}^{N}\textup{Hom}_{G}(V_{\nu_{i}},V_{\nu_{i-1}}\otimes
V_{\mu_{i}})$.

######  Corollary 1.

We have a linear isomorphism

$\mathcal{R}_{\mu}^{(\nu)}(G^{N})\simeq\otimes_{i=1}^{N}\textup{Hom}_{G}(V_{\nu_{i}},V_{\nu_{i-1}}\otimes
V_{\mu_{i}})$

Let $Z(I_{\mu},A_{\mu})$ be the centralizer of the commutative subalgebra
$I_{\mu}$ in $A_{\mu}$. By the construction of $J_{\mu}\subset A_{\mu}$ it is
a subalgebra in the centralizer of $I_{\mu}$, i.e. $J_{\mu}\subset
Z(I_{\mu},A_{\mu})$. Since $\mathcal{R}^{(\nu)}_{\mu}\subset\mathcal{R}_{\mu}$
is an eigensubspace of $I_{\mu}$ the centralizer $Z(I_{\mu},A_{\mu})$ acts on
it. Thus, $\mathcal{R}^{(\nu)}_{\mu}$ is an $J_{\mu}$-module.

######  Theorem 2.

The decomposition (7) is the multiplicitly free decomposition of
$\mathcal{R}_{\mu}$ in irreducible $J_{\mu}$-modules.

The proof of this theorem and its generalization to a more general
representation theoretical context will be given in [19].

Denote by $J_{\mu}^{(\nu)}$ and $Z(I_{\mu},A_{\mu})^{(\nu)}$ images of
$J_{\mu}$ and $Z(I_{\mu},A_{\mu})$ in
$\textup{End}(\mathcal{R}_{\mu}^{(\nu)})$ respectively.

######  Corollary 2.

We have

$J_{\mu}^{(\nu)}=Z(I_{\mu},A_{\mu})^{(\nu)},$

Thus, the embeddings (5) define a simple quantum superintegrable system in the
sense of [18].

Using well known identities for integrals of matrix elements of irreducible
representations with respect to the Haar measure555All necessary identities
are summarized in the Appendix D it is easy to prove the following formula for
the scalar product of trace functions:

$\int_{G^{N}}(\Psi^{\\{\mu\\}}_{b,\lambda^{\prime}}(g_{1},\dots,g_{N}),\Psi^{\\{\mu\\}}_{a,\lambda}(g_{1},\dots,g_{N}))dg_{1}\dots
dg_{N}=\prod_{i=1}^{N}\delta_{\lambda_{i},\lambda_{i}^{\prime}}\prod_{i=1}^{N}(b_{i},a_{i})_{\lambda_{i}}$

Here $(b,a)_{\lambda_{i}}$ is a Hermitian scalar product on
$Hom_{\mathfrak{g}}(V_{\lambda_{i}},V_{\lambda_{i-1}}\otimes V_{\mu_{i}})$
defined as follows. Because all representations are equipped with Hermitian
structure for each $a\in
Hom_{\mathfrak{g}}(V_{\lambda_{i}},V_{\lambda_{i-1}}\otimes V_{\mu_{i}})$ we
have its Hermitian conjugate
$a^{\dagger}\mathrel{\mathop{\ordinarycolon}}V_{\lambda_{i-1}}\otimes
V_{\mu_{i}}\to V_{\lambda_{i}}$. For any such $a,b$, by Schur’s lemma the
composition $ab^{\dagger}$ is a multiple of the identity
$id_{V_{\lambda_{i}}}$. This defines the scalar product $(b,a)_{\lambda_{i}}$
as $ab^{\dagger}=(b,a)_{\lambda_{i}}id_{V_{\lambda_{i}}}$.

###  1.4. The gauge fixing

There $N$ natural ways to identify the coset $G^{N}/G^{N}$ for the gauge
action (1) with the coset $G/G$ with respect to the conjugation.

Indeed, fix $i\in 1,\dots,N$ and for $j\neq i+1$ set:

(9) $\displaystyle h_{j}=h_{i+1}g_{i+1}g_{i+2}\dots g_{j-2}g_{j-1}$

Here here the product is cyclic. Denote such $N$-tuple $(h_{1},\dots,h_{N})$
by $h_{g}$.

The gauge transformation by the element $h_{g}$ brings $g=(g_{1},\dots,g_{N})$
to
$g^{h_{g}}=(1,\dots,h_{i+1}(g_{i+1}\dots g_{N}g_{1}g_{2}\dots
g_{i-1}g_{i})h_{i+1}^{-1},\dots,1)$. This identifies the $G^{N}$ gauge orbit
through $(g_{1},\dots,g_{N})$ with the $G$ conjugation orbit through
$g_{1}\dots g_{N}$. Thus, we constructed the mapping

$G^{N}/G^{N}\simeq G/G$

which is easy to prove to be an isomorphism.

We can chose $h_{i+1}$ in such a way that $h_{i+1}(g_{i+1}\dots
g_{N}g_{1}g_{2}\dots g_{i-1}g_{i})h_{i+1}^{-1}\in H$ is an element of the
Cartan subgroup in $G$. This gives an isomorphism

$G^{N}/G^{N}\simeq G/G\simeq H/W$

for each $i=1,\dots,N$.

For each $i=1,\dots,N$, this gives an isomorphism of Hilbert spaces

(10)
$\phi_{i}\mathrel{\mathop{\ordinarycolon}}\mathcal{H}_{\mu}\simeq\mathcal{H}_{\mu}^{H}=L_{2}(H\to(V_{\mu_{1}}\otimes\dots\otimes
V_{\mu_{N}})[0])^{W}$

Here the Weyl group $W$ acts as $w(f)(h)=wf(w^{-1}(h))$. We took into account
that the Weyl group acts naturally on the zero weight subspace of any
$G$-module.

This isomorphism acts on functions as

$\phi_{i}(f)(h)=\pi_{\mu}(h_{g})f(g)=f(1,\dots,h,\dots,1),$

where $h_{g}$ is as above, $\pi_{\mu}(h)$ is as in (3) and
$h=h_{i}(g_{i+1}\dots g_{N}g_{1}g_{2}\dots g_{i-1}g_{i})h_{i}^{-1}$.

For the scalar product on $\mathcal{H}^{H}_{\mu}$ we have:

$(f,g)=\int_{H}(f(h),g(h))|\delta(h)|^{2}dh$

Here $H\subset G$ is the Cartan subgroup and $\delta(h)$ is the denominator in
the Weyl character formula.

####  1.4.1. Quantum Hamiltonians

I the theorem below we summarized computations of the action of quantum
Hamiltonians on the space $\mathcal{H}_{\mu}^{H}$ with the gauge fixing map
$\phi_{N}\mathrel{\mathop{\ordinarycolon}}\mathcal{H}_{\mu}\to\mathcal{H}_{\mu}^{H}$.
We show that operators $D_{k}=H^{(k+1)}-H^{(k)}$ are topological Knizhnik-
Zamilodchikov-Bernard operators and that $H^{(N)}$ is the spin Calogero-Moser
Hamiltonian.

######  Theorem 3.

Operators $H^{(N)}$ and $D_{j}=H^{(j+1)}-H^{(j)}$ for $j=1,\dots,N-1$ act as

$H^{(N)}_{2}=\Delta+2\sum_{\alpha>0}\frac{\pi^{V}(f_{\alpha}e_{\alpha})}{(1-h_{\alpha})(1-h_{-\alpha})}-||\rho||^{2}$

and

$D_{j}=(h^{(j)},\frac{\partial}{\partial\lambda})-\sum_{k=1}^{j-1}r_{kj}(\lambda)+\sum_{k=j+1}^{n}r_{jk}(\lambda)$

where
$\Delta=(\frac{\partial}{\partial\lambda},\frac{\partial}{\partial\lambda})$
is the Laplacian on $\mathfrak{h}\subset\mathfrak{g}$ determined by the
Killing form, $h_{\alpha}=e^{(\alpha,\lambda)}$ and $r(\lambda)$ is Felder’s
dynamical $r$-matrix [8]:

$r(\lambda)=-\frac{1}{2}\sum_{i=1}^{r}h_{i}\otimes
h_{i}-\sum_{\alpha}\frac{e_{-\alpha}\otimes e_{\alpha}}{1-h_{-\alpha}}$

Note that we consider compact simple $G$ which correspond to
$h_{\alpha}=e^{iq_{\alpha}}$ with $q_{\alpha}\in\mathbb{R}$.

As always explicit formulae for higher Hamiltonians $H_{k}^{(j)}$ are more
complicated.

###  1.5. The multi-time propagator

####  1.5.1. The propagator and the graphical calculus.

Figure 2.  The $N$-point trace function and the dual trace function.
Holonomies $g_{1},\dots,g_{N}$ are arguments of the trace function.

The multi-time propagator is simply an element of the semigroup generated by
all commuting Hamiltonians:

$U^{\\{\mu\\}}_{\\{A\\}}=\exp(-\sum_{k,j}H_{k}^{(j)}A_{k,j})$

It has $Nr$ independent times $A_{k,j}$ and satisfies natural composition
property

$U^{\\{\mu\\}}_{\\{A\\}}U^{\\{\mu\\}}_{\\{B\\}}=U^{\\{\mu\\}}_{\\{A+B\\}}$

where $(A+B)_{k,j}=A_{k,j}+B_{k,j}$.

The propagator is an integral operator acting on the space
$\mathcal{H}_{\mu}$. In the usual way, from the spectral decomposition we
derive its kernel:

(11)
$U^{\\{\mu\\}}_{\\{A\\}}(\\{g\\},\\{g^{\prime}\\})=\sum_{\\{\lambda\\},\\{a\\}}\exp(-\sum_{i=1}^{N}c_{2}(\lambda_{i})A_{i})\Psi^{\\{\mu\\}}_{\\{a\\},\\{\lambda\\}}(g_{1},\dots,g_{N})\Psi^{\\{\mu\\}}_{\\{\overline{a}\\},\\{\lambda\\}}(g_{1}^{\prime},\dots,g_{N}^{\prime})^{*}$

Here
$\Psi^{\\{\mu\\}}_{\\{\overline{a}\\},\\{\lambda\\}}(g_{1},\dots,g_{N})^{*}\in
V_{\mu_{1}}^{*}\otimes\dots\otimes V_{\mu_{N}}^{*}$ are dual trace functions
defined as

$\displaystyle\Psi^{\\{\mu\\}}_{\\{\overline{a}\\},\\{\lambda\\}}(g_{1},\dots,g_{N})^{*}=Tr_{V_{\lambda_{N}}^{*}}(\pi_{\lambda_{N}}^{*}(g_{N})b_{1}\dots(\pi_{\lambda_{1}}^{*}(g_{1})\otimes
id_{V_{\mu_{1}}})(b_{2}\otimes id_{V_{\mu_{2}}}\otimes id_{V_{\mu_{1}}})\dots$
(12) $\displaystyle(b_{N-1}\otimes id_{V_{\mu_{N-1}}}\otimes\dots\otimes
id_{V_{\mu_{1}}})(\pi_{\lambda_{1}}^{*}(g_{N-1})id_{V_{\mu_{1}}}\otimes\dots\otimes
id_{V_{\mu_{N-1}}}))$

For $a_{i}\mathrel{\mathop{\ordinarycolon}}V_{\lambda_{i}}\to
V_{\mu_{i}}\otimes V_{\lambda_{i-1}}$ the morphism
$\overline{a}_{i}\mathrel{\mathop{\ordinarycolon}}V_{\lambda_{i}}^{*}\otimes
V_{\mu_{i}}\to V_{\lambda_{i-1}}^{*}$ is a $\mathfrak{g}$ linear map obtained
by partial dualizing of
$a^{\dagger}\mathrel{\mathop{\ordinarycolon}}V_{\mu_{i}}\otimes
V_{\lambda_{i-1}}\to V_{\lambda_{i}}$.

Note that this kernel pointwise is a linear operator in
$End(V_{\mu_{N}}\otimes\dots\otimes
V_{\mu_{1}})\simeq(V_{\mu_{N}}\otimes\dots\otimes
V_{\mu_{1}})\otimes(V_{\mu_{1}}^{*}\otimes\dots\otimes V_{\mu_{N}}^{*})$.

From now on we will focus on multi-time propagators with non-zero times for
quadratic Hamiltonians only. We will also be using graphical calculus to
visualize algebraic operations. It is a version of Penrose’s graphical
calculus. For trace functions $\Psi$ we will write a circle marked points
corresponding to vertex operators $\\{a_{i}\\}$ andwith outgoing semi-open
intervals colored by representations $V_{m_{i}}$. For an edge colored by a
highest weight $\lambda$ and an element $g\in G$ we assign $\pi^{\lambda}(g)$.
To a vertex colored by a $G$-invariant linear map
$a\mathrel{\mathop{\ordinarycolon}}V_{\lambda}\to V_{\mu}\otimes V_{\nu}$
where $\lambda$ is the color of the incoming edge and $\mu$ and $\nu$ are
colors of outgoing edges, we assign the linear map $a$. Trace function is the
contraction of these maps, which correspond to assembling (gluing) the graph
on Fig. 3 from edges and vertices.

Figure 3. Here is the graphical representation of the propagator (11) for
$N=3$. We put the ”color” $\lambda_{i}$ on the $i$-th region. Both segments of
the boundary of this region which also belong to the boundary of the cylinder
inherit this color. Dashed lines are colored by representations $\mu_{i}$.
Colors $\lambda_{i}$ they satisfy the Clebsch-Gordan rules at each dashed
line, as it is shown below. There $V_{\lambda}\subset V_{\mu}\otimes
V_{\lambda^{\prime}}$ Here we use the orientation of the cylinder to
distinguish regions that are left and right to the dashed line.  Figure 4.
Coloring of faces and edges are related by Clebsch-Gordan rules and uses the
orientation of the surface.

####  1.5.2. The integral formula

######  Theorem 4.

The kernel (11) can be written as the following integral

(13) $\displaystyle
U^{\\{\mu\\}}_{\\{A\\}}(\\{g\\},\\{g^{\prime}\\})=\int_{G^{N}}Z_{A_{1}}({g_{1}^{\prime}}^{-1}h_{N}^{-1}g_{1}h_{1})Z_{A_{2}}({g_{2}^{\prime}}^{-1}h_{1}^{-1}g_{2}h_{2})\dots$
(14) $\displaystyle
Z_{A_{N}}({g_{N-1}^{\prime}}^{-1}h_{N-1}^{-1}g_{N}h_{N})\pi^{\mu_{N}}(h_{N})\otimes\dots\otimes\pi^{\mu_{1}}(h_{1})dh_{1}\dots
dh_{N}$

Here

$Z_{A}(g)=\sum_{\lambda}\chi_{\lambda}(g)dim(\lambda)\exp(-Ac_{2}(\lambda))$

is the partition function of the two-dimensional Yang-Mills theory with the
boundary holonomy $g\in G$ [10][22].

To prove this statement one should use standard integral identities from
Appendix D This integral formula also has a natural graphical representation,
see Fig. 5.

Figure 5. Here is the graphical representation of the propagator (11) in the
integral form (13) for $N=3$. Lines colored by representations $\mu_{i}$
should be regarded as open Wilson lines and each region which is homeomorphic
to a disc contributes as the corresponding factor in (13).  Figure 6. The
partition function for a disc.

The integral formula for the propagator reminds a lot the partition function
of pure 2-dimensional Yang-Mills theory [10][22]. An unusual feature of this
formula are ”open” Wilson lines which go to the outside of the cylinder as
outer edges. We build on this analogy in the next section where will define
2-dimensional Yang-Mills theory of surfaces with corners and embedded open
graphs.

##  2\. Quantum two dimensional Yang-Mills with corners

In this section we define two dimensional Yang-Mills theory on space time
manifolds with non-removable corners. We will not use the language of
2-categories, though it is natural in such setting.

###  2.1. Surfaces with open graphs.

Let $\Sigma$ be a topological oriented compact surface, possibly with a
boundary $\partial\Sigma$. Let $\Gamma$ be a graph, possibly with 1-valent
vertices. We assume that $\Gamma$ is partially embedded in $\Sigma$ in the
following sense: all vertices and all edges connecting vertices of valency
greater than one are embedded to $\Sigma$666In the most general setting we can
allow to have edges of $\Gamma$ which also belong to the boundary of
$\Sigma$.. Some of the vertices possibly to $\partial\Gamma$, we call them
boundary vertices. One valent vertices are connected only to boundary
vertices. Both, one valent vertices and edges connecting them to boundary
vertices do not belong to $\Sigma$, we will call them outer vertices and other
edges.

Now let us remove outer vertices. We will be left with $\Gamma$ partially
embedded to $\Sigma$ with outer edges being open ”away from $\Sigma$”. We will
call such pairs $(\Gamma,\Sigma)$ open graphs on $\Sigma$. An example of an
open graph on a surface is shown of Fig. 7

Note that each such graph defines a cell decomposition of $\Sigma$ with cells
that are not necessary contractible.

Figure 7. An example of an open graph on a surface, upper figure. The same
graph but enriched is shown on a lower figure. All edges of the graph are
oriented and all connected components of the boundary of the surface are
oriented by the orientation of the surface (”counterclockwise”)

Now for each open graph $\Gamma$ on $\Sigma$ we will construct enriched graph
$\widehat{\Gamma}$ as follows:

  * •

Add a circle to each connected component of $\Sigma\backslash\Gamma$ which
follows the boundary of this connected component.The orientation of this
circle is induced by the orientation of $\Sigma$.

  * •

Replace intervals of $\partial\Sigma$ between boundary vertices of $\Gamma$ by
segments of the corresponding inserted circle.

Figure 8. An illustration to how partition surfaces are glued along intervals
connecting non-removable corners. This may create multiple outer edges which
can still be regarded as a singe outer edge colored by the tensor product of
representations.

Finally, let us define the gluing of surfaces with enriched graphs.

A surface with an enriched graph in it can be glued to itself by identifying
two intervals at the boundary as it is shown on Fig. 8. The upper figure
corresponds to the case when the intervals do not shape a vertex. The lower
figure corresponds to the case when the intervals share a vertex. In the first
case they can belong to different connected components of the surface. Such
gluing of surfaces correspond to compositions of partition functions which is
also illustrated on Fig. 8 and will be discussed in the next section.

###  2.2. Quantum two-dimensional Yang-Mills theory on surfaces with open
Wilson graphs.

Figure 9. Enriched colored open graph on a surface. Here $\mu_{i}$ are highest
weights of representations assigned to edges and $a_{,}b,\dots$ are invariant
vectors assigned to vertices.

First fix a simple compact Lie group $G$ and define a $G$-coloring of a
surface with an open graph as follows:

  * •

We assign a finite dimensional representation of $G$ to each edge of the graph
$\Gamma$.

  * •

For each vertex of $\Gamma$ fix a total ordering of adjacent edges which
agrees with the cyclic ordering induced by the orientation of $\Sigma$.
Enumerate adjacent edges $e_{1},\dots,e_{m}$ according to this total ordering.
Let $V_{e_{1}},\dots,V_{e_{m}}$ be colors of representations assigned to
adjacent edges. To the corresponding vertex we assign a $G$-invariant vector
in the tensor product

$a_{v}\in(V_{e_{1}}^{\epsilon_{1}}\otimes\dots\otimes
V_{e_{m}}^{\epsilon_{m}})^{G}$

Here $V^{+}=V$ and $V^{-}=V^{*}$ and $+$ is for edges oriented outward the
vertex and $-$ is for edges oriented inward.

We will call colored graphs Wilson graphs by analogy with Wilson loops
[23][12].

Now let us assign a variable to each connected component of
$\Sigma\backslash\Gamma$. We assume that this variable is additive with
respect to gluing. For example if a surface would be equipped with a volume
form, an area would be such additive variable. In the previous section when
the surfaces are cylinders, Euclidean times evolution for spin Calogero-Moser
model Hamiltonians are such variables.

Now let us describe the space of boundary states. Let
$\partial_{\alpha}\Sigma$ be connected components of the boundary of $\Sigma$,
$\partial\Sigma=\sqcup_{\alpha}\partial_{\alpha}\Sigma$. If a connected
component does not have boundary vertices of $\Gamma$, it is a circle. If it
does, it is a union of intervals to which boundary vertices of $\Gamma$
partition this connected component. We will call these intervals and circles
parts of $\partial\Sigma$.

To connected component $\partial_{\alpha}\Sigma$ we assign the following
spaces of boundary states:

  * •

If $\partial_{\alpha}\Sigma$ is a circle we assign to it

$\mathcal{H}(\partial_{\alpha}\Sigma)=L_{2}(G)^{G}$

It is a Hilbert space where characters of finite dimensional irreducible
representations of $G$ form an orthonormal basis.

  * •

If the connected component $\partial_{\alpha}\Sigma$ has boundary vertices of
$\Gamma$ on it, we assign the following space to this boundary component:

$\mathcal{H}(\partial_{\alpha}\Sigma)=L_{2}(G^{N_{\alpha}},\otimes_{i=1}^{N_{\alpha}}(V^{\epsilon_{1,i}}_{e_{1,i}}\otimes\dots\otimes
V^{\epsilon_{m_{i},i}}_{e_{m_{i},i}}))^{G^{N_{\alpha}}}$

Here $V_{e_{1,i}}^{\epsilon_{1,i}}\otimes\dots\otimes
V_{e_{m_{i},i}}^{\epsilon_{m_{i},i}}$ is the tensor product of colors of outer
edges adjacent to boundary vertex $i$, $G^{N_{\alpha}}$ acts by gauge
transformations on $G^{N_{\alpha}}$. On the tensor product it acts as

$(g_{1},\dots,g_{N_{\alpha}})(\otimes_{i=1}^{N_{\alpha}}(x_{e_{1,i}}\otimes\dots\otimes
x_{e_{m_{i},i}}))=\otimes_{i=1}^{N_{\alpha}}(g_{i}x_{e_{1,i}}\otimes\dots\otimes
g_{i}x_{e_{m_{i},i}})$

The partition function of the 2D Yang-Mills with for a surface with an open
Wilson graphs is defined as a vector the following vector in the space of
boundary states:

(15)
$Z_{\Gamma,A}(\\{g\\})=\int_{G^{E_{int}}}\prod_{D\in\Sigma\backslash\Gamma}Z_{D,A_{D}}(\\{g,h\\}_{\partial
D})W_{\Gamma}(\\{g,h\\})\prod_{e\in E_{int}}dh_{e}$

Here $D$ are parts of $\Sigma$ bounded by circles in the enriched open graph
$\Gamma$, $hol_{\partial D}$ is the product of group elements $h_{e}$ assigned
to internal edges in $\partial D$ and group elements $g_{e}$ corresponding to
boundary edges of $\partial D$ in the counter clock-wise order starting from
any edge.

(16) $Z_{D,A}(hol_{\partial D})=\sum_{\lambda}\chi_{\lambda}(hol_{\partial
D}){\dim(\lambda)}^{1-2g}e^{-c_{2}(\lambda)A}$

where $g$ is the genus of the surface obtained from $D$ by gluing a disc to
$\partial D$, i.e. by ”closing” the surface.

For example $hol_{\partial D}=1$ can be interpreted as if the boundary of $D$
does exists, i.e. shrinks to a point. Thus $Z_{D,A}(1)$ is the partition
function of 2D Yang-Mills on the ”closure” of $D$ [23]. If $D$ is a disk

(17)
$Z_{D,A}(g)=\sum_{\lambda}\chi_{\lambda}(g){\dim(\lambda)}e^{-c_{2}(\lambda)A}$

is the partition function of the 2D Yang-Mills on $D$ with boundary holonomy
$g$. It is easy to see that for a triangulated surface $D$ the partition
function (16) can be composed from elementary partition functions (17) [22].
This is the manifestation of locality of 2D Yang-Mills theory.

The function $W_{\Gamma}(\\{g,h\\})$ is the contraction of holonomies along
internal edges evaluated in the representation corresponding to this edge and
invariant vectors corresponding to vertices of $\Gamma$. Such function on
$G^{E_{int}}$ is know as Wilson graph evaluated on a graph connection see for
example [23](?)[12]. Wilson loop is a particular case of such a function.

Partition functions (15) satisfy the gluing property.

######  Theorem 5.

Let $\Sigma_{1}$ and $\Sigma_{2}$ be two surfaces with boundary edges
$e_{1}\in\partial\Sigma_{1}$ and $e_{2}\in\Sigma_{2}$ being identified as it
is shown of Fig. 8. Let $h$ be the holonomy along identified edges then

$\int_{G}Z(\Sigma_{1};h)\otimes\otimes\pi_{\mu}(h)Z(\Sigma_{2};h)dh=Z(\Sigma_{1}\cup_{e_{1}\simeq
e_{2}}\Sigma_{2})$

where in the partition function on the right there is an extra Wilson line
colored by $V_{\mu}$ ”sandwiching” between $\Sigma_{1}$ and $\Sigma_{2}$. And
a similar gluing identity we have for the partition functions in case we glue
two neighboring intervals on $\partial\Sigma$.

The proof follows immediately from integral identities in the Appendix.

###  2.3. Point observables

Wilson graphs can be regarded as line observables in 2D Yang-Mills. They has
been studies since many years ago, see for example [10][22][23][3][12].

Let $D\subset\Sigma$ be a disc. Let us assign to $D$ the following function on
$G/G$:

$\mathcal{O}_{A}^{F}=\sum_{\lambda}\chi_{\lambda}(g)F_{\lambda}\dim(\lambda)e^{-Ac_{2}(\lambda)}$

When $F_{\lambda}=1$ for all $\lambda$ this is the partition function of 2D
Yang-Mills on a disc. We will call such function the partition function of a
disc $D$ with topological point observables inserted in the disc. The position
of the point observable on $D$ is not given since it is a topological
observable. One should think of the function $F$ as ”sitting” at a point in
the interior of $D$, confined to $D$ by the Wilson loop at the boundary.

Naturally, if we glue a surface to $D$ along a connected component
$\partial_{\alpha}\Sigma$ of the boundary $\partial\Sigma$ of the surface
$\Sigma$ the resulting partition function is the partition function with the
point observable $F$ inserted in $\Sigma$.

The insertion of such observable in the partition function for a surface with
an open Wilson graph $\Gamma$ on it depends on into which region of
$\Sigma\backslash\Gamma$ it is inserted. But it does not depend at which point
of the region it is inserted.

If we put such point observable on a cylinder, the corresponding integral
operators

$U^{F}_{A}(g,h)=\sum_{\lambda}\chi_{\lambda}(g)\overline{\chi_{\lambda}(h)}F_{\lambda}\exp(-c_{2}(\lambda)A)$

form commutative algebra

$U_{A}^{F}\ast U_{B}^{G}=U^{FG}_{A+B}$

where $(FG)_{\lambda}=F_{\lambda}G_{\lambda}$.

An example of such observable was introduced in [23] as the insertion making
the partition function independent of the orientation. If
$\lambda^{*}=-w_{0}(\lambda)$777Here $w_{0}$ is the longest element of the
Weyl group. is the highest weight if the dual representation to $V_{\lambda}$,
such observable corresponds to $F_{\lambda}=\delta_{\lambda,\lambda^{*}}$.

##  3\. Conclusion

Here we will list some open problems and future directions.

  1. (1)

The study of large $n$ limit in the $SU_{n}$ 2D YM theory, is one of the most
interesting directions for a number of well known reasons. When $\Sigma$ is a
sphere there is a phase transition in this limit [4] at $A=\pi^{2}$. Similar
behavior for the cylinder and for a sphere with three holes was studied in
[6]. For the relation between the 2D YM in large $n$ limit and the string
theory see [9]. It would be interesting to extend these results to the case of
open Wilson graphs.

  2. (2)

In the follow up paper we will describe the 2D YM theory with mixed gauge
groups. Particular cases of such models are related to open spin Calogero-
Moser spin chains, see [20] and for its classical counterpart [17].

  3. (3)

There is a natural $q$-deformation of the $N$-spin CM systems described in
[2]. When $q$ is a root of unity this system is closely related to the quantum
Chern-Simons theory [21] for $U_{q}(\mathfrak{g})$ at roots of unity [13].
Corresponding deformations of 2D YM theory are invariants of twisted circle
bundles over surfaces. Quadratic Casimirs are special in this setting because
the ribbon structure in the corresponding modular tensor category is given by
the exponents of quadratic Casimirs. Classical counterparts of these
deformations are ”relativistic” versions of spin Calogero-Moser models on
moduli space of flat connections constructed in [1].

  4. (4)

As it was already mentioned partition functions of 2D YM theory can be
regarded as a semiclassical limit of invariants of twisted circle bundles over
surfaces corresponding to the quantum Chern-Simons theory with a compact Lie
group. This makes the study of a quantum 2D YM theory for a non-compact Lie
group even more interesting. It may be a toy model for quantum complex Chern-
Simons theory of a Chern-Simons theory for a non-compact real Lie group.
Quantum $N$-spin systems for noncompact real forms of simple complex Lie
groups were studied in [20].

##  Appendix A On quantum moment maps.

###  A.1. Quantum moment maps for $G$-actions

The left and the right quantum moment maps act as:

$\widehat{\mu}_{\ell}(X)f(g)=\frac{d}{dt}f(e^{-tX}g)|_{t=0},\ \
\widehat{\mu}_{r}(X)f(g)=\frac{d}{dt}f(ge^{tX})|_{t=0}$

For the left moment map acting on monomials in $U(\mathfrak{g})$ we have

$\widehat{\mu}_{\ell}(x)=\widehat{\mu}_{\ell}(X_{1})\dots\widehat{\mu}_{\ell}(X_{k})f(g)=\frac{\partial^{k}}{\partial
t_{1}\dots\partial t_{k}}f(e^{-t_{k}X_{k}}\dots e^{-t_{1}X_{1}}g)$

and a similar formula for the right action.

For $X\in\mathfrak{g}\subset U(\mathfrak{g})$ the left and the right quantum
moment maps are related as:

(18) $\widehat{\mu}_{r}(X)f(g)=-\widehat{\mu}_{\ell}(Ad_{g}(X))f(g)$

This relation extends to monomials $x=X_{1}\dots X_{k}$ as

$\widehat{\mu}_{r}(x)=\widehat{\mu}_{\ell}(Ad_{g}(S(x))$

Indeed

(19)
$\displaystyle\widehat{\mu}_{r}(x)f(g)=\widehat{\mu}_{r}(X_{1})\dots\widehat{\mu}_{r}(X_{k})f(g)=\frac{\partial^{k}}{\partial
t_{1}\dots\partial t_{k}}f(ge^{t_{1}X_{1}}\dots e^{t_{k}X_{k}})$ (20)
$\displaystyle=(-1)^{k}\widehat{\mu}_{\ell}(Ad_{g}(X_{k}))\dots\widehat{\mu}_{\ell}(Ad_{g}(X_{1}))f(g)=\widehat{\mu}_{\ell}(Ad_{g}(S(x)))f(g)$

Lie group $G$ acts on its Lie algebra $\mathfrak{g}$ via adjoint action
$g\mathrel{\mathop{\ordinarycolon}}X\mapsto Ad_{g}(X)$. This action lifts to
the action on $U(\mathfrak{g})$ which will also denote by $Ad_{g}$. The Lie
group $G$ also acts naturally on differential operators on $G$,
$h\mathrel{\mathop{\ordinarycolon}}D\mapsto D^{h}$

$D^{h}f(g)={h_{L}^{*}}^{-1}(Df(hg))$

Here ${h_{L}^{*}}^{-1}f(g)=f(h^{-1}g)$.

Quantum moment maps are $G$-equivariant:

$\widehat{\mu}_{\ell}^{h}(x)=\widehat{\mu}_{\ell}(Ad_{h}(x))$

Let $\\{e_{i}\\}$ be a basis in $\mathfrak{g}$ and
$z=\sum_{i_{1},\dots,i_{k}}f^{i_{1},\dots,i_{k}}e_{i_{1}}\dots e_{i_{k}}$It is
clear that differential operator $D=\widehat{\mu}_{\ell}(z)$ is $G$-invariant
iff tensor $f$ is $G$-invariant, i.e. iff $z\in Z(\mathfrak{g})$.

######  Lemma 1.

Let $z\in Z(\mathfrak{g})$, then

$\widehat{\mu}_{r}(z)=\widehat{\mu}_{\ell}(S(z)$

###### Proof.

Indeed, we have:

(21)
$\displaystyle\widehat{\mu}_{r}f(g)=\sum_{\\{i\\}}f^{\\{i\\}}\widehat{\mu}_{r}(e_{i_{1}})\dots\widehat{\mu}_{r}(e_{i_{k}})f(g)=\sum_{\\{i\\}}f^{\\{i\\}}\frac{\partial^{k}}{\partial
t_{1}\dots\partial t_{k}}f(ge^{t_{1}e_{i_{1}}}\dots e^{t_{k}e_{i_{k}}})$ (22)
$\displaystyle=\sum_{\\{i\\}}f^{\\{i\\}}\widehat{\mu}_{\ell}(e_{i_{k}})\dots\widehat{\mu}_{\ell}(e_{i_{1}})f(g)=\widehat{\mu}_{\ell}(S(z))f(g)$

∎

###  A.2. Quantum moment maps for $G^{N}$ actions

For the left and right action of $G^{N}$ on itself:

$\widehat{\mu}_{L}(X_{1},\dots,X_{N})=\widehat{\mu}^{(1)}_{\ell}(X_{1})+\dots+\widehat{\mu}^{(N)}_{\ell}(x_{N}),\
\
\widehat{\mu}_{R}(X_{1},\dots,x_{N})=\widehat{\mu}^{(1)}_{r}(X_{1})+\dots+\widehat{\mu}^{(N)}_{r}(X_{N})$

where $\widehat{\mu}^{(i)}_{\ell,r}$ are left and right moment maps for the
$i$-th factor.

For the gauge action of $G^{N}$ on $G^{N}$ by gauge transformations we have:

$\widehat{\mu}(X_{1},\dots,X_{N})=\widehat{\mu}_{L}(X_{1},X_{2},\dots,X_{N})+\widehat{\mu}_{R}(X_{2},X_{3}\dots,X_{N},X_{1})$

These Lie algebra homomorphisms extend uniquely to algebra homomorphisms
$\widehat{\mu},\widehat{\mu}_{L},\widehat{\mu}_{R}\mathrel{\mathop{\ordinarycolon}}U(\mathfrak{g})^{\otimes
N}\to\mathbb{D}(G^{N})$

Lemma 1 implies that

$\widehat{\mu}_{R}(z_{1}\otimes\dots\otimes
z_{N})=\widehat{\mu}_{L}(S(z_{1})\otimes\dots\otimes S(z_{N}))$

Because of this the natural algebra homomorphism

$\widehat{\mu}_{L}\otimes\widehat{\mu}_{R}\mathrel{\mathop{\ordinarycolon}}U(\mathfrak{g})^{\otimes
N}\otimes U(\mathfrak{g})^{\otimes N}\to\mathbb{D}(G^{N})$

filters through the algebra homomorphism

$\widehat{\mu}_{L}\otimes\widehat{\mu}_{R}\mathrel{\mathop{\ordinarycolon}}U(\mathfrak{g})^{\otimes
N}\widetilde{\otimes}_{Z(\mathfrak{g})^{\otimes N}}U(\mathfrak{g})^{\otimes
N}\to\mathbb{D}(G^{N})$

Here $U(\mathfrak{g})^{\otimes N}\widetilde{\otimes}_{Z(\mathfrak{g})^{\otimes
N}}U(\mathfrak{g})^{\otimes N}$ is the tensor product over
$Z(\mathfrak{g})^{\otimes N}$ where $(a_{1}z_{1}\otimes\dots\otimes
a_{N}z_{N})\otimes_{Z(\mathfrak{g})^{\otimes N}}(b_{1}\otimes\dots\otimes
b_{N})=(a_{1}\otimes\dots\otimes a_{N})\otimes_{Z(\mathfrak{g})^{\otimes
N}}(S(z_{1})b_{1}\otimes\dots\otimes S(z_{N})b_{N})$.

The mapping $\phi_{i}$ induces a linear isomorphism
$\textup{End}(\mathcal{H}_{\mu})\to\textup{End}(\mathcal{H}^{H}_{\mu})$ for
which we will use the same name $\phi_{i}$. Let us first compute how
$\widehat{\mu}_{L}(U(\mathfrak{g}))$ and $\widehat{\mu}_{R}(U(\mathfrak{g}))$
act on $\mathcal{H}^{H}_{\mu}$ and then the action of Hamiltonians
$H^{(j)}_{d_{k}}$.

###  A.3. Quantum moment maps after gauge fixing

Let $X\in\mathfrak{g}\subset U(\mathfrak{g})$ and
$X_{j}=1^{\otimes(j-1)}\otimes X\otimes 1^{\otimes(N-j)}$. The following is
the results of an elementary computation:

$\widehat{\mu_{R}}(X_{j})f(g)=-\widehat{\mu_{L}}(Ad_{g_{j}}X_{j})f(g),\ \ .$
$\widehat{\mu_{R}}(X_{j+1})+\widehat{\mu_{L}}(X_{j})=\pi_{j}^{\mu_{j}}(X)f(g)$

for functions in $\mathcal{R}_{\mu}$. Applying gauge fixing at $i$ to these
identities we obtain the left action of $X^{(j)}$ on $\mathcal{H}^{H}_{\mu}$:

(23)
$\phi_{i}(-\widehat{\mu_{L}}(X_{j+1})+\widehat{\mu_{L}}(X_{j}))f(h)=\pi_{j}^{\mu_{j}}(X)f(h),\
\ j\neq i$ (24)
$\phi_{i}(-\widehat{\mu_{L}}(Ad_{h}(X_{i}))+\widehat{\mu_{L}}(X_{i-1}))f(h)=\pi_{i-1}^{\mu_{i-1}}(X)f(h),$

where $f\in\mathcal{H}^{H}_{\mu})$.

Choose a basis $e_{\alpha}$ in each root subspace
$\mathfrak{g}_{\alpha}\subset\mathfrak{g}$. From the equations above we can
express the action of $e_{\alpha,j}$ in terms of $\pi_{j}^{\mu_{j}}(X)$and
$h\in H$:

$\phi_{i}(\widehat{\mu_{L}})(e_{\alpha,j})=\frac{1}{1-h_{\alpha}}S^{j}_{\alpha},$

where

(25) $S^{i}_{\alpha}=\sum_{j=1}^{N}\pi_{j}^{\mu_{j}}(e_{\alpha})$ (26)
$S^{j}_{\alpha}=h_{\alpha}(\pi_{i}^{\mu_{i}}(e_{\alpha})+\dots+\pi_{j-1}^{\mu_{j-1}}(e_{\alpha}))+\pi_{j}^{\mu_{j}}(e_{\alpha})+\dots+\pi_{i-1}^{\mu_{i-1}}(e_{\alpha}),\
\ j\neq i$

Here $h_{\alpha}$ is the coordinate function on $H$ corresponding to the root
$\alpha$, $he_{\alpha}h^{-1}=h_{\alpha}e_{\alpha}$ and the summation is
cyclic.

For the left action of the Cartan subalgebra we have

(27)
$\phi_{i}(\widehat{\mu}_{L}(H_{\alpha,j})-\widehat{\mu}_{L}(H_{\alpha,j+1}))f(g)=\pi_{j}(h)f(g)$

Here $j+1$ is defined cyclically $\mod N$.

After gauge fixing we have

$\phi_{i}(\widehat{\mu}_{L}(H_{\alpha,i})f(1,\dots,h,\dots,1)=-i\frac{\partial}{\partial
q_{\alpha}}f(1,\dots,h,\dots,1)$

The action of $\phi_{i}(\widehat{\mu}_{L}(H_{\alpha,j})$ with $j\neq i$ can be
computed from this formula and from equations (23)(24):

$\phi_{i}(\widehat{\mu}_{L}(H_{\alpha,j})=-\pi_{i}^{\mu_{i}}(H_{\alpha})-\pi_{i+1}^{\mu_{i+1}}(H_{\alpha})-\dots-\pi_{j}^{\mu_{j}}(H_{\alpha})-i\frac{\partial}{\partial
q_{\alpha}}$

Here the sum is cyclic.

##  Appendix B Quadratic Hamiltonian for $N=1$

For$N=1$ the gauge invariance of functions from the space $\mathcal{R}_{\mu}$
is $\psi(hgh^{-1})=\pi(h)\psi(g)$. The transformation property of functions
from $\mathcal{R}_{\mu}$ imply:

$(\widehat{\mu}_{\ell}(X)+\widehat{\mu}_{r}(X))f(g)=\pi(X)f(g)$

Combining this with (18) , for functions $f$ from $\mathcal{R}_{\mu}$ we have:

$(\widehat{\mu}_{\ell}(X)-\widehat{\mu}_{r}(Ad_{g}(X)))f(g)=\pi(g)f(g)$

Now let us compute $\widehat{\mu}_{\ell}(Y)\widehat{\mu}_{\ell}(X)f(g)$. From
the definition (19) we have:

$\widehat{\mu}_{\ell}(Y)(\widehat{\mu}_{\ell}(X)-\widehat{\mu}_{\ell}(Ad_{g}(X)))f(g)=\pi(X)\widehat{\mu}_{\ell}(Y)f(g)$

similarly

$-\widehat{\mu}_{\ell}(Ad_{g}(Y))(\widehat{\mu}_{\ell}(X)-\widehat{\mu}_{\ell}(Ad_{g}(X)))f(g)=-\pi(X)\widehat{\mu}_{\ell}(Ad_{g}(Y)f(g)$

Combining these identities we have:

######  Theorem 6.

(28)
$\displaystyle\pi(X)\pi(Y)f(g)=(\widehat{\mu}_{\ell}(Y)-\widehat{\mu}_{\ell}(\widetilde{Z}))(\widehat{\mu}_{\ell}(X)-\widehat{\mu}_{\ell}(Z))\psi|_{Z=Ad_{g}(X),\widetilde{Z}=Ad_{g}(Y)}+$
(29)
$\displaystyle\widehat{\mu}_{\ell}([Y,Ad_{g}(X)])\psi-\widehat{\mu}_{\ell}(Ad_{g}([Y,X]))f(g)$

From here, setting $X=e_{\alpha},Y=e_{-\alpha}$ and specializing $g$ to $h\in
H$ we obtain:

$\pi(e_{\alpha})\pi(e_{-\alpha})f(h)=(1-h_{\alpha})(1-h_{-\alpha})\widehat{\mu}_{\ell}(e_{-\alpha})\widehat{\mu}_{\ell}(e_{\alpha})f(h)+(1-h_{\alpha})\widehat{\mu}_{\ell}(H_{\alpha})f(h)$

or

$\widehat{\mu}_{\ell}(e_{-\alpha})\widehat{\mu}_{\ell}(e_{\alpha})f(h)=\frac{\pi(e_{\alpha})\pi(e_{-\alpha})}{(1-h_{\alpha})(1-h_{-\alpha})}f(h)+\frac{\widehat{\mu}_{\ell}(H_{\alpha})}{(1-h_{\alpha})}f(h)$

Setting $h=e^{\lambda}$ we can write
$\widehat{\mu}_{\ell}(H_{\alpha})=(\alpha,\frac{\partial}{\partial\lambda})$
and for the second Casimir we obtain

$\widehat{\mu}_{\ell}(c_{2})f(h)=(\Delta+2\sum_{\alpha>0}\frac{\pi(e_{\alpha})\pi(e_{-\alpha})}{(1-h_{\alpha})(1-h_{-\alpha})}-D)f(h)$

where
$D=\sum_{\alpha>0}\frac{1+h_{\alpha}}{1-h_{\alpha}}(\alpha,\frac{\partial}{\partial\lambda})$.

##  Appendix C Quantum quadratic Hamiltonians for $N>1$.

Here we will compute quadratic Hamiltonians from the action of
$U(\mathfrak{g})^{\otimes N}$ on trace functions by left and right quantum
moment map. Since trace functions form a basis in the space
$\mathcal{R}_{\mu}$ this is equivalent to computing quantum Hamiltonians
directly from the action on functions from $\mathcal{R}_{\mu}$ as it was done
above for $N=1$, but it is easier.

###  C.1. Dynamical $r$-matrices

For $\lambda\in\mathfrak{h}$ define
$\xi_{\alpha}=e^{\frac{(\lambda,\alpha)}{2}}$, $h_{\alpha}=\xi_{\alpha}^{2}$.
The dynamical $r$-matrix [8] is the following
$\mathfrak{g}\otimes\mathfrak{g}$-valued function on $\mathfrak{h}$

$r(\lambda)=-\frac{1}{2}\sum_{i=1}^{r}h_{i}\otimes
h_{i}-\sum_{\alpha}\frac{e_{-\alpha}\otimes e_{\alpha}}{1-h_{-\alpha}}$

or

$r(\lambda)=-\frac{1}{2}\sum_{i=1}^{r}h_{i}\otimes
h_{i}+\sum_{\alpha>0}(\frac{e_{\alpha}\otimes
f_{\alpha}}{h_{\alpha}-1}-\frac{h_{\alpha}f_{\alpha}\otimes
e_{\alpha}}{h_{\alpha}-1}$

Here are some basic properties of $r(\lambda)$:

  * •

$r(-\lambda)=r^{21}(\lambda)$,

  * •

$r(\lambda)+r(-\lambda)=-\Omega$,

  * •

$(e^{-\lambda})_{2}r_{12}(\lambda)(e^{\lambda})_{2}=-\sum_{i=1}^{r}h_{i}\otimes
h_{i}-r_{21}(\lambda)$,

  * •

$D_{1}D_{2}r(\lambda)D_{1}^{-1}D_{2}^{-1}=r(\lambda)$

  * •

The dynamical Yang-Baxter equation.

###  C.2. Quantum topological Knizhnik-Zamolodchikov-Bernard equation

All the computations from this sections can be done from the definition of the
space $\mathcal{R}_{\mu}$. However, it is more convenient to do them using the
basis of trace functions after the gauge fixing for $i=N$.

Choose a basis $e_{\alpha}$ is root subspaces of $\mathfrak{g}$ and
orthonormal (with respect to the Killing form) basis in $\mathfrak{h}$. Let
$c_{2}$ be the quadratic Casimir for $\mathfrak{g}$:

$c_{2}=\sum_{i=1}^{r}h_{i}^{2}+\sum_{\alpha>0}(e_{\alpha}f_{\alpha}+f_{\alpha}e_{\alpha})$

where $f_{\alpha}=e_{-\alpha}$. If $h_{\alpha_{i}}$ is a basis in
$\mathfrak{h}$ corresponding to simple roots,
$[e_{\alpha},f_{\alpha}]=H_{\alpha}$, the Cartan part of the Casimir element
can be written as

$\sum_{i=1}^{r}h_{i}^{2}=\sum_{i,j=1}^{r}H_{\alpha_{i}}(B^{-1})_{ij}H_{\alpha_{j}}$

where $B$ is the symmetrized Cartan matrix.

Denote by $\Omega$ the ”mixed Casimir”:

$\Omega=\sum_{i=1}^{r}h_{i}\otimes h_{i}+\sum_{\alpha>0}(e_{\alpha}\otimes
f_{\alpha}+f_{\alpha}\otimes e_{\alpha})$

If $\Delta$ is the compultiplication for $U(\mathfrak{g})$, we have

$\Delta c_{2}=c_{2}\otimes 1+1\otimes c_{2}+2\Omega$

For $b\in Hom_{\mathfrak{g}}(M_{\nu},M_{\mu}\otimes V_{\eta})$ where
$M_{\mu},M_{\nu}$ are Verma modules and $V_{\lambda}$ is an irreducible finite
dimensional module with the highest weight $\eta$. We have

$bc_{2}(\nu)=(c_{2}(\mu)+c_{2}(\eta)+2\Omega)b$

This identity can be written in terms of dynamical $r$-matrices as

(30)
$\frac{c_{2}(\nu)-c_{2}(\mu)-c_{2}(\eta)}{2}b=(r_{12}(\lambda)+r_{21}(\lambda))b$

Here $\lambda\in\mathfrak{h}$.

For $b_{i}\in\textup{Hom}_{G}(V_{\nu_{i}},V_{\nu_{i-1}}\otimes V_{\mu_{i}})$,
with $0=N$, consider the trace function (8) evaluated at
$g=(1,\dots,1,e^{\lambda})$, i.e. for the gauge fixing with $i=N$:

(31) $\Psi^{(\nu)}_{b,\mu}(\lambda)=Tr_{a}(b_{1}\dots b_{N}(e^{\lambda})_{a})$

Here we denote representation $V_{\mu_{N}}$ by index $a$ (auxiliary space) and
take trace over this space.

Denote

$d(\lambda)=\frac{1}{2}\sum_{\alpha>0}\frac{\xi_{\alpha}+\xi_{-\alpha}}{\xi_{\alpha}-\xi_{-\alpha}}H_{\alpha}$

######  Theorem 7.

Trace function $F$ satisfies differential equations

(32)
$((h^{(i)},\frac{\partial}{\partial\lambda})-\sum_{k=1}^{i-1}r_{ki}(\lambda)+\sum_{k=i+1}^{n}r_{ik}(\lambda)-d(\lambda)_{i})\Psi^{(\nu)}_{b,\mu}(\lambda)=\frac{c(\nu_{i+1})-c(\nu_{i})}{2}\Psi^{(\nu)}_{b,\mu}(\lambda)$

###### Proof.

Let us apply the identity (30) to the i-th factor in the definition of the
trace function. We have:

$\frac{c_{2}(\nu_{i})+c_{2}(\mu_{i})-c_{2}(\nu_{i+1})}{2}\Psi^{(\nu)}_{b,\mu}(\lambda)=Tr_{a}(b_{1}\dots
b_{i-1}(r_{ia}(\lambda)+r_{ai}(\lambda))b_{i}\dots b_{N}(e^{\lambda})_{a})$

From the intertwining property of $b_{i}$ we have:

$b_{1}\dots
b_{i-1}r_{ai}(\lambda)=(r_{ai}(\lambda)+r_{1i}(\lambda)+\dots+r_{i-1,i}(\lambda)b_{1}\dots
b_{i-1}$

Now let us use the cyclicity of the trace:

(33) $\displaystyle Tr_{a}(r_{ai}(\lambda)b_{1}\dots
b_{N}(e^{\lambda})_{a})=Tr_{a}(b_{1}\dots
b_{N}(e^{\lambda})_{a}r_{ai}(\lambda)(e^{-\lambda})_{a}(e^{\lambda})_{a})=$
$\displaystyle Tr_{a}(b_{1}\dots
b_{N}(-\sum_{k}h^{(i)}_{k}h^{(a)}_{k}-r_{ia}(\lambda))(e^{\lambda})_{a})=$
$\displaystyle-(h^{(i)},\frac{\partial}{\partial\lambda})Tr_{a}(b_{1}\dots
b_{N}(e^{\lambda})_{a})-Tr_{a}(b_{1}\dots
b_{N}r_{ia}(\lambda)(e^{\lambda})_{a}))$

Here we used properties of the dynamical $r$-matrix listed in section C.1.

Now let us use the intertwining properties of $b_{j}$’s:

$b_{i+1}\dots
b_{N}r_{ia}(\lambda)=(r_{i,i+1}(\lambda)+\dots+r_{iN}(\lambda)+r_{ia}(\lambda))b_{i+1}\dots
b_{N}$

and

$b_{i}r_{ia}(\lambda)=(r_{ia}(\lambda)+m(r(\lambda))_{i})b_{i}$

Here $m(a\otimes b)=ab$ for $a,b\in U(\mathfrak{g})$.

$m(r(\lambda))=-\frac{1}{2}\sum_{i}h_{i}^{2}-\sum_{\alpha}\frac{e_{-\alpha}e_{\alpha}}{1-h_{-\alpha}}=-\frac{1}{2}c_{2}+d(\lambda)$

where $d(\lambda)$ is as above.

Thus,

$b_{i}b_{i+1}\dots
b_{N}r_{ia}(\lambda)=(r_{ia}(\lambda)-\frac{c_{2}(\mu_{i})}{2}+d(\lambda)_{i}+r_{i,i+1}(\lambda)+\dots+r_{iN}(\lambda))b_{i}\dots
b_{N}$

Combining all identities we obtain (32).

∎

###  C.3. The quantum spin Calogero-Moser Hamiltonian

For $b\in Hom_{\mathfrak{g}}(M_{\nu},M_{\nu}\otimes V)$, where $V$ is a finite
dimensional representation of $\mathfrak{g}$ and $M_{\nu}$ is the Verma module
with the highest weight $\nu$, we define the trace function

$\Phi_{b,V}^{(\nu)}(\lambda)=Tr_{a}(b(e^{\lambda})_{a})$

Here the trace is takes over the space $M_{\nu}$ which we denote by index $a$
(auxiliary). For our purposes it is enough to consider this trace function as
a Laurent power series in $e^{\lambda}$.

Let $\Delta$ be the Laplacian on $\mathfrak{h}$:

$\Delta=(\frac{\partial}{\partial\lambda},\frac{\partial}{\partial\lambda})$

where $(.,.)$ is the bilinear form on $\mathfrak{h}^{*}$ which is dual to the
Killing from on $\mathfrak{h}$. We have:

$\sum_{i}h^{2}_{i}e^{\lambda}=\Delta e^{\lambda}$

We halso have the identity

$c_{2}(\nu)\Psi_{b,V}^{(\nu)}(\lambda)=Tr_{a}(b(c_{2})_{a}(e^{\lambda})_{a})$

######  Lemma 2.

Following identities hold:

(34)
$Tr_{M_{\nu}}(be_{\alpha}e^{\lambda})=\frac{\pi^{V}(e_{\alpha})}{1-h_{\alpha}}\Psi_{b,V}^{(\nu)}(\lambda)$

(35)
$Tr_{M_{\nu}}(bf_{\alpha}e^{\lambda})=\frac{\pi^{V}(f_{\alpha})}{1-h_{-\alpha}}\Psi_{b,V}^{(\nu)}(\lambda)$

Indeed, the first identity follows from the intertwining property of $b$ and
from the cyclicity of the trace:

$Tr(be_{\alpha}e^{\lambda})=\pi^{V}(e_{\alpha})\Psi_{b,V}^{(\nu)}(\lambda)+Tr(be^{\lambda}e_{\alpha})=\pi^{V}(e_{\alpha})\Psi_{b,V}^{(\nu)}(\lambda)+h_{\alpha}Tr(be_{\alpha}e^{\lambda})$

The proof of (35) is similar.

######  Proposition 1.

(36)
$Tr_{M_{\nu}}(be_{\alpha}f_{\alpha}e^{\lambda})=\frac{\pi^{V}(e_{\alpha}f_{\alpha})}{(1-h_{\alpha})(1-h_{-\alpha})}\Psi_{b,V}^{(\nu)}(\lambda)-\frac{h_{\alpha}}{1-h_{\alpha}}(\alpha,\frac{\partial}{\partial\lambda}\Psi_{b,V}^{(\nu)}(\lambda)$
(37)
$Tr_{M_{\nu}}(bf_{\alpha}e_{\alpha}e^{\lambda})=\frac{\pi^{V}(f_{\alpha}e_{\alpha})}{(1-h_{\alpha})(1-h_{-\alpha})}\Psi_{b,V}^{(\nu)}(\lambda)-\frac{1}{1-h_{\alpha}}(\alpha,\frac{\partial}{\partial\lambda}\Psi_{b,V}^{(\nu)}(\lambda)$

###### Proof.

(38) $\displaystyle
Tr_{M_{\nu}}(be_{\alpha}f_{\alpha}e^{\lambda})=\pi^{V}(e_{\alpha})Tr_{M_{\nu}}(bf_{\alpha}e^{\lambda})+Tr_{M_{\nu}}(bf_{\alpha}e^{\lambda}e_{\alpha})=\frac{\pi^{V}(f_{\alpha})}{1-h_{-\alpha}}\Psi_{b,V}^{(\nu)}(\lambda)+$
$\displaystyle
h_{\alpha}Tr_{M_{\nu}}(bf_{\alpha}e_{\alpha}e^{\lambda})=\frac{\pi^{V}(f_{\alpha})}{1-h_{-\alpha}}\Psi_{b,V}^{(\mu)}(\lambda)+h_{\alpha}Tr_{M_{\nu}}(be_{\alpha}f_{\alpha}e^{\lambda})-h_{\alpha}(\alpha,\frac{\partial}{\partial\lambda}\Psi_{b,V}^{(\mu)}(\lambda)$

The proof of (37) is similar.

∎

######  Corollary 3.

$Tr_{M_{\nu}}(b(e_{\alpha}f_{\alpha}+f_{\alpha}e_{\alpha})e^{\lambda})=\frac{2\pi^{V}(f_{\alpha}e_{\alpha})}{(1-h_{\alpha})(1-h_{-\alpha})}\Psi_{b,V}^{(\nu)}(\lambda)-\frac{1+h_{\alpha}}{1-h_{\alpha}}(\alpha,\frac{\partial}{\partial\lambda}\Psi_{b,V}^{(\nu)}(\lambda)$

Here we use that $\Psi_{b,V}^{(\mu)}(\lambda)$ is an element of the zero
weight subspace in $V$.

Thus, we proved the following theorem.

######  Theorem 8.

(39)
$\Delta\Psi_{b,V}^{(\nu)}(\lambda)+2\sum_{\alpha>0}\frac{\pi^{V}(f_{\alpha}e_{\alpha})}{(1-h_{\alpha})(1-h_{-\alpha})}\Psi_{b,V}^{(\nu)}(\lambda)-\frac{1+h_{\alpha}}{1-h_{\alpha}}(\alpha,\frac{\partial}{\partial\lambda})\Psi_{b,V}^{(\nu)}=c_{2}(\mu)\Psi_{b,V}^{(\mu)}$

It is easy to see that if we replace Verma module $M_{\nu}$ by an irreducible
representation $V_{\nu}$ with the highest weight $\mu$, the theorem and its
proof hold. Note that when $V=V_{\mu_{1}}\otimes\dots\otimes V_{\mu_{N}}$ with
the diagonal action of $\mathfrak{g}$, and $b=b_{1}\dots b_{N}$, where $b_{i}$
are the same intertwines as in (31), the trace function $\Phi_{b,V}^{(\nu)}$
becomes the trace function $\Psi_{a,\mu}^{(\nu)}$ (31) with $\nu=\nu_{N}$.

This proves that trace functions (31) satisfy (39).

###  C.4. Normalization

######  Lemma 3.

(40) $\Delta(\delta^{-1})=-(\rho,\rho)-D(\delta^{-1}$

where
$D=\sum_{\alpha>0}\frac{\xi_{\alpha}+\xi_{-\alpha}}{\xi_{\alpha}-\xi_{-\alpha}}(\alpha,\frac{\partial}{\partial\lambda})$
and $\delta$ is the denominator in the Weyl character formula:

$\delta=\prod_{\alpha>0}(\xi_{\alpha}-\xi_{-\alpha})$

###### Proof.

Consider the trace function for the trivial representation $V$ with
$b=id\mathrel{\mathop{\ordinarycolon}}M_{\nu}\to M_{n}u$. In this case
$\Psi_{b,V}^{(\nu)}=\frac{e^{(\nu+\rho,\lambda)}}{\delta}$ is the character of
$M_{\nu}$. In this case the identity (39) immediately implies (40). ∎

######  Proposition 2.

$\delta\circ(\Delta+D)\circ\delta^{-1}=\Delta-||\rho||^{2}$
$\delta\circ(h,\frac{\partial}{\partial\lambda})\circ\delta^{-1}=-\sum_{\alpha>0}\frac{\xi_{\alpha}+\xi_{-\alpha}}{\xi_{\alpha}-\xi_{-\alpha}}H_{\alpha}$

Define the normalized trace function as

$F_{b,V}^{(\nu)}(\lambda)=\delta^{-1}\Psi_{b,V}^{(\nu)}$

We proved:

######  Theorem 9.

Normalized trace functions satisfy differential equations

$H^{(N)}_{2}F_{b,V}^{(\nu)}(\lambda)=c_{2}(\nu)F_{b,V}^{(\nu_{N})}(\lambda),\
\
D_{i}F_{b,V}^{(\nu)}(\lambda)=\frac{c_{2}(\nu_{i+1})-c_{2}(\nu_{i})}{2}F_{b,V}^{(\nu)}(\lambda)$

where $i=1,\dots,N$ and

$H^{(N)}_{2}=\Delta+2\sum_{\alpha>0}\frac{\pi^{V}(f_{\alpha}e_{\alpha})}{(1-h_{\alpha})(1-h_{-\alpha})}-||\rho||^{2}$

and

$D_{i}=(h^{(i)},\frac{\partial}{\partial\lambda})-\sum_{k=1}^{i-1}r_{ki}(\lambda)+\sum_{k=i+1}^{n}r_{ik}(\lambda)$

##  Appendix D Useful identities with the Haar measure

Figure 10. Graphical representation of the identity (41).  Figure 11.
Graphical representation of the identity (42).

Choose a basis $\\{e_{i}\\}$ in the representation space $V_{\lambda}$ and
denote by $\pi^{\lambda}(g)^{i}_{j}$ matrix elements of linear operator
$\pi_{\lambda}(g)$ acting on this space
$\pi_{\lambda}(g)e_{j}=\sum_{i}\pi^{\lambda}(g)^{i}_{j}e_{i}$. Let $dh$ be the
Haar measure on $G$, then

(41) $\int_{G^{\times
2}}\pi^{\lambda}(h^{-1})_{j}^{i}\pi^{\mu}(h)^{k}_{l}dh=\frac{1}{\dim(\lambda)}\delta_{\lambda,\mu}\delta_{l}^{i}\delta_{j}^{k}$

Note that if $\\{e^{i}\\}$ is the dual basis then
$\pi_{\lambda}(g)^{*}(e^{i})=\sum_{j}\pi^{\lambda}(g)^{i}_{j}e^{j}$. In the
basis free form this identity can be written as

$\int_{G^{\times
2}}(\pi^{\lambda})^{*}(h^{-1})\otimes\pi^{\mu}(h)dh=\frac{1}{\dim(\lambda)}\delta_{\lambda,\mu}\iota\circ
ev$

where $ev\mathrel{\mathop{\ordinarycolon}}V_{\lambda}^{*}\otimes
V_{\lambda}\to\mathbb{C}$ is the evaluation map and
$\iota\mathrel{\mathop{\ordinarycolon}}\mathbb{C}\to V_{\lambda}^{*}\otimes
V_{\lambda}$ is the injection map.

(42) $\int_{G^{\times
3}}\pi^{\mu_{1}}(h^{-1})^{i}_{j}\pi^{\mu_{2}}(h)^{k}_{l}\pi^{\mu_{3}}(h)^{s}_{t}dh=\sum_{a\in(V_{\mu_{1}}^{*}\otimes
V_{\mu_{2}}\otimes V_{\mu_{3}})^{G}}a^{iks}\overline{a}_{jlt}$

where $\\{a\\}$ is a basis in subspace of $G$-invariant vectors in the triple
tensor product $(V_{\mu_{1}}^{*}\otimes V_{\mu_{2}}\otimes V_{\mu_{3}})^{G}$
and $\\{\overline{a}\\}$ is the dual basis in the dual vector space.

These and other similar identities can be written in a basis free form as

$\int_{G^{\times
m}}\pi^{\mu_{1}}(h)\otimes\pi^{\mu_{2}}(h)\otimes\dots\otimes\pi^{\mu_{m}}(h)dh=P_{0}$

where $P_{0}$ is the orthogonal projector onto the subspace
$(V_{\mu_{1}}\otimes V_{\mu_{2}}\otimes\dots\otimes V_{\mu_{m}})^{G}\subset
V_{\mu_{1}}\otimes V_{\mu_{2}}\otimes\dots\otimes V_{\mu_{m}}$ of
$G$-invariant vectors.

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