# Conformal Hypergeometry and Integrability

Volker Schomerus

To Jasper Stokman on the occasion of his 50th birthday Deutsches Elektronen
Synchroton DESY, Notkestrasse 85, 22607 Hamburg, Germany and Universität
Hamburg, Luruper Chausee 149, 22761 Hamburg, Germany<EMAIL_ADDRESS>
(Date: 4/30/2021)

###### Abstract.

Conformal field theories play a central role in modern theoretical physics
with many applications to the understanding of phase transitions, gauge
theories and even the quantum physics of gravity, through Maldacena’s
celebrated holographic duality. The key analytic tool in the field is the so-
called conformal partial wave expansion, i.e. a Fourier-like decomposition of
physical quantities into a basis of partial waves for the conformal group
SO$(1,d+1)$. This text provides an non-technical introduction to conformal
field theory and the conformal bootstrap program with some focus on the
mathematical aspects of conformal partial wave expansions. It emphasises
profound relations with modern hypergeometry, group theory and integrable
models of Gaudin and Calogero-Sutherland type.

###### 2010 Mathematics Subject Classification:

Primary: 20C35, 33C67, 33C90, 81R12, 81T40

##  1\. Introduction

During the last century physicists gradually learned to appreciate the
importance of conformal symmetry for our understanding of nature. By
definition, conformal symmetry is made up of angle preserving transformations.
The most commonly observed (conformal) symmetries are rotations and
translations. These are present in all homogeneous and isotropic systems.
Beyond that, dilations (scale transformations) are the most prominent examples
of conformal transformations. Scale invariance is an important phenomenon in
physics that refers to situations in which the features of a system do not
change upon zooming in or out. The first known occurrence in nature is the
critical point for the liquid-vapour transition in the phase diagram of water.
Many more of these points have been discovered in the meantime. Some of them
can be realized in spin systems such as the famous Ising model.

Particle physicists started to appreciate conformal field theories (CFTs)
through the influential work of Wilson on the renormalization group [74]. By
construction, scale invariance occurs at all fixed points of renormalization
group transformations. This gives CFTs a very special status within the space
of all theories, 111Throughout this introduction I will not distinguish
clearly between scale and conformal symmetry. I will briefly come back to the
issue in section 2. and makes them a good starting point to explore, in
particular, non-perturbative phenomena in quantum field theory. More recently,
CFTs became the main actors on another stage, Juan Maldacena’s celebrated
AdS/CFT duality [47]. Its most general formulation proposes an equivalence
between non-gravitational quantum systems in $d$ dimensions and gravitational
systems in one dimension higher. In this context, a $d$-dimensional CFT is
like a holographic screen for a gravitational theory in Anti-deSitter (AdS)
space. As we make progress in dealing with CFTs, the AdS/CFT duality can be
applied to provide novel views on the deep mysteries of a quantum theory of
gravity, such as the phenomenon of Hawking radiation and black hole
information.

The very brief history of scale/conformal invariance in physics that I
presented in the first two paragraphs may serve one main purpose: it explains
why a significant part of modern theoretical physics research is devoted to
CFTs and why the interest ranges over so many different communities, from
condensed matter theory and modern material science all the way to particle
physics and quantum gravity.

Invariance under conformal transformations turns out to be a surprisingly
powerful symmetry principle that strongly constrains physical observables such
as the correlations between different measurements. These constraints may be
exposed with the help of the most important analytical tool of CFT, the so-
called conformal partial wave expansions. One should think of these as an
extension of Fourier analysis, but for systems with conformal rather than mere
translational symmetry. Just as plane waves in Fourier analysis are uniquely
characterized by their behaviour under translations, conformal partial waves
are determined by conformal symmetry alone, i.e. they belong to the realm of
group and representation theory.

When conformal field theories were considered for the first time about 50
years ago, the relevant group theory was very far from being developed, which
halted initial progress. This changed only after physicists started to focus
on $d=2$ dimensions where conformal symmetry becomes infinite dimensional. For
that case a very beautiful theory was developed with many deep relations to
mathematics and integrable systems. But this is not our topic here. Instead,
what I shall describe below is relevant for CFTs in dimension $d>2$. At the
end of the last century, and in particular with the discovery of the AdS/CFT
duality, the case for studying CFTs in $d>2$ had become so strong that
physicists made another effort to develop at least some of its group theoretic
foundations. The influential work of Dolan and Osborn [16, 17] on conformal
partial waves for scalar 4-point functions and some subsequent work by other
authors, see in particular [35, 45, 44, 57], eventually provided a basis for
some new breakthrough developments in higher dimensional CFT, e.g. for the
3-dimensional critical Ising model.

At the time, physicists did not know that essentially all the required results
on conformal partial waves were already contained in the seminal papers of
Heckman and Opdam, see [31, 54, 30]. Somewhat surprisingly, before it was
uncovered in [37], the profound relation went unnoticed between the study of
integrable quantum mechanics of Calogero-Moser-Sutherland type on the one hand
and conformal partial waves on the other. The main intent of this text is to
help reduce the apparent and unfortunate divide between the physics
communities that work on conformal field theories on the one hand and
mathematicians with expertise in modern group theory, harmonic analysis,
hypergeometry and integrability on the other. The study of CFTs continues to
challenge profound advances of its group theoretic ingredients. So it is not
too late for the relevant communities to join forces.

After these introductory remarks, let me describe the plan of this short note.
The next section is meant as a smooth introduction to some basic notions of
critical systems and CFT. In the example of the critical Ising model I will
outline how statistical systems can become critical and possibly conformal. In
addition, I shall introduce the notions of (scalar and spinning) fields,
correlation functions, scaling weights, conformal partial waves etc. and I
will sketch the main idea of the so-called conformal bootstrap program.
Section 2 remains very descriptive, without precise definitions. Nevertheless,
I hope that it can build a first intuition into the basic concepts of the
area, in a spirit that is relatively close to the physicists’ approach. In
section 3, I then rephrase the basic setup in a more mathematical language.
There, the notions of fields, correlation functions and conformal partial
waves are embedded into group theory and harmonic analysis. Once this is
achieved, we can walk along well trodden paths into the world of hypergeometry
and integrable systems. We will see the first examples of the relation between
conformal partial waves and hypergeometry at the end of section 3. As I will
argue there, conformal partial waves for scalar 4-point functions can be
identified with the wave functions of hyperbolic Calogero-Moser-Sutherland
models for the root system $BC_{2}$, i.e. these conformal partial waves are
(close relatives of) $BC_{2}$ Heckman-Opdam hypergeometric functions. I
conclude section 3 with a lightning overview of extensions which are more or
less well understood, or at least have been actively researched already.

The final section is then devoted to a new important direction about which
very little seems to be known for now: multipoint correlation functions. In a
CFT, correlations of up to $N=3$ local measurements are essentially determined
by conformal symmetry. Hence, the first non-trivial correlations that can be
analysed with well studied conformal partial waves appears when $N=4$. The
territory we shall begin to explore in the final section lies just one step
beyond. For $N>4$, conformal partial waves were largely unexplored until very
recently. Section 4 embeds the profound mathematical challenges in gaining
control of these objects into the context of $N$-site Gaudin integrable models
for the conformal group [6]. As I will discuss in detail, the study of the
relevant conformal partial waves requires us to consider certain limits of the
Gaudin integrable models which extend the so-called caterpillar or bending
flow limits previously considered in [13, 14, 67]. In the final subsection I
highlight an intriguing new appearance of the lemniscatic elliptic Calogero-
Moser-Sutherland model of [20] in the context of $N$-point scalar conformal
blocks or, equivalently, in limits of $N$-site Gaudin integrable models.

While sections 2 and 3 of this note survey developments from several decades
and do not contain any new results, section 4 is based on some very recent
research in collaboration with I. Buric, S. Lacroix, J.A. Mann and L.
Quintavalle [5, 7].

Acknowledgments: I wish to thank my co-authors Ilija Buric, Misha Isachenkov,
Sylvain Lacroix, Pedro Liendo, Yanick Linke, Jeremy Mann, Lorenzo Quintavalle,
Evgeny Sobko for the inspiring collaboration on the material presented in this
note. At the same time I am very grateful to the organizers of the workshop
“Hypergeometry, Integrability and Lie Theory” at the Lorentz Center in Leiden
for giving me the opportunity to present our work and to Jasper for providing
the occasion. I have profited very much from many insightful questions,
comments and discussions with participants during and after the meeting. This
applies in particular to some of the new material included in the final
subsection which has been stimulated by discussions with Pavel Etingof.

##  2\. Conformal Field Theory and Partial Waves

###  2.1. The critical Ising model

As we have discussed in the introduction, conformal field theories are used to
describe the critical behavior of many statistical systems. One of the most
well known is the Ising model. Let us use this simple model to illustrate some
of the key concepts. The Ising model can be defined on a $d$-dimensional
hypercubic lattice with lattice sites given by
$x_{n}=n^{\mu}\epsilon_{\mu}\in\mathbb{R}^{d}$. Here the summation over
$\mu=1,\dots,d$ is understood and $\epsilon_{\mu}=\epsilon e_{\mu}$ coincide
with the unit vectors $e_{\mu}$ of $\mathbb{R}^{d}$ up to a common scale
factor $\epsilon$. We refer to $\epsilon$ as the lattice spacing. The
coefficients $n^{\mu}$ are integers.

To each lattice site we now assign a spin $\sigma_{n}\in\\{\pm 1\\}$. Any such
choice of spin assignments on all lattice sites is referred to as a spin
configuration. The object $\sigma$ is (a discrete version) of what physicists
call a field. In order to define our theory we must specify a measure on the
on the space of configurations. For the Ising model, this measure is

(2.1) $P_{\zeta}(\\{\sigma\\})=\mathcal{N}e^{\zeta\sum_{\langle
nm\rangle}\sigma_{n}\sigma_{m}}\ .$

Here the sum extends over nearest neighbors $\langle nm\rangle$, i.e. over all
pairs of lattice sites that are a distance $|x_{n}-x_{m}|=\epsilon$ apart.

The nature of a typical configuration selected by the measure (2.1) depends
very much on the value of the parameter $\zeta$. If we set $\zeta=0$, for
example, neighboring spins do not interact, i.e. the value the measure assumes
is the same on all spin configurations. Hence, if we measure the spin at site
$n$, this won’t tell us anything about the value of the spin at another site
$m$, in other words there is no correlation between the value $\sigma_{n}$ of
the spin at site $n$ and the value of $\sigma_{m}$ at another sites $m\neq n$
of the lattice. In the opposite case with $\zeta=\infty$, configurations in
which spins assume different values are infinitely suppressed, i.e. the
measure only allows for the two constant configurations $\sigma_{n}=1$ or
$\sigma_{n}=-1$ for all sites $n$. In this case, the correlations between
spins are as strong as they can be. We only need to measure the spin
$\sigma_{n}$ at one site $n$ to know its value at all the other sites $m\neq
n$. Once we go away from this extreme case, spin flips are possible but still
costly. In other words, while the measure still assumes its maximum value on
the two constant configurations, it becomes less strongly peaked as we
decrease $\zeta$. Consequently, spin configurations with large ‘islands’ of
aligned spins will start to play a role. As we reduce the value of $\zeta$,
smaller islands contribute. It turns out that, for some very special value
$\zeta=\zeta_{c}$, islands of all sizes appear with equal probability. At this
very special point, the system appears to be self-similar on scales that are
large compared to the lattice spacing $\epsilon$ (and small compared to the
overall size $L$ of the lattice if we consider a finite lattice). It is this
‘critical’ value $\zeta=\zeta_{c}$ of the parameter $\zeta$ that we want to
focus on.

In order to probe the Ising model, physicists measure correlations between
spins. The simplest interesting example is the correlation between two spins
$\sigma_{n}$ and $\sigma_{m}$ at two different lattice sites $n,m$. The
quantity is defined as

(2.2)
$\langle\sigma_{n}\sigma_{m}\rangle_{\zeta}=\sum_{\\{\sigma\\}}P_{\zeta}(\\{\sigma\\})\,\sigma_{n}\sigma_{m}\
.$

To make this sum well defined, one can imagine that it is carried out on a
finite lattice with $L^{d}$ lattice sites, even though criticality requires
$L$ being sent to infinity at the end. As $\zeta$ approaches $\zeta_{c}$ the
system becomes self-similar in the sense described above and hence this
correlation function must take the form

(2.3)
$\langle\sigma_{n}\sigma_{m}\rangle_{\zeta_{c}}\sim\frac{1}{|x_{n}-x_{m}|^{2\Delta_{\sigma}}}\
.$

Here, $|x_{n}-x_{m}|$ denotes the Euclidean distance between the lattice
points $x_{n}$ and $x_{m}$ in $\mathbb{R}^{d}$. The $\sim$ means the function
on the right hand side approximates the correlation function on intermediate
scales or, equivalently, in the limit of an infinite lattice size $L$ and
vanishing lattice spacing $\epsilon$. The parameter $\Delta_{\sigma}$ that
describes how the correlation function falls off as we increase the distance
between the two lattice sites is referred to as the scaling weight of
$\sigma$. It is a very important observable of the critical system.

Determining the value of $\Delta_{\sigma}$ is one of the central challenges of
CFT. In $d=2$ dimensions, it is possible to give an exact value. As was
observed in [3], the scaling weight of the spin field in the critical
2-dimensional Ising model can be calculated using representation theory of the
Virasoro algebra with central charge $c=1/2$. This approach famously gives

(2.4) $\Delta^{\textrm{[2D]}}_{\sigma}=1/8\ .$

For the 3-dimensional critical Ising model, the exact value of the scaling
weight $\Delta_{\sigma}$ is not known, but there exist good numerical
simulations (NSs) of the correlation (2.2) near $\zeta=\zeta_{c}$ that give
the approximate result [29]

(2.5) $\Delta^{\textrm{[3D],NS}}_{\sigma}\sim 0.51813(5)\ .$

Let us point out right away that a more precise value of this exponent is
known by now. However, this precision determination is based not on numerical
simulation of the Ising model, but rather on the conformal bootstrap program
that we shall outline below.

Before we go there, let us briefly mention that the scaling weight
$\Delta_{\sigma}$ of the spin field $\sigma$ is not the only field in the
Ising model. Obviously, it is possible to construct many similar quantities.
To give just one additional example, let us introduce the object

(2.6)
$\varphi^{\mu}_{n}=\sigma(x_{n})\nabla^{\mu}\sigma(x_{n})=\sigma(x_{n})\sigma(x_{n}+\epsilon_{\mu})-1\
.$

Note that the definition of $\varphi^{\mu}$ depends on the choice of a
direction $\mu$ in the $d$-dimensional space. Physicists refer to
$\varphi^{\mu}$ as a vector field as opposed to the spin field $\sigma$ which
is scalar, i.e. does not single out any direction. One can go further and
consider fields that are sensitive to the value of the Ising spin at more than
two sites. The relevant sites may fall into a local cluster or be located
along lines, surfaces, etc. One calls the associated objects local, line,
surface operators, respectively. These operators can be inserted into
correlation functions of the form (2.2), just as we did for the spin field
$\sigma$. It turns out that the 2-point correlations of local operators also
fall off with some characteristic exponent $\Delta_{\varphi}$ whose value
depends on the field $\varphi$ that is inserted.

###  2.2. Conformal field theory

Critical systems are included in the wider class of homogeneous and isotropic
systems which are invariant under translations $x^{\mu}\mapsto
x^{\mu}+a^{\mu}$ and rotations $x^{\mu}\mapsto O^{\mu}_{\,\nu}x^{\nu}$ of
points $x=(x^{\mu})\in\mathbb{R}^{d}$. As we have described above, the
characteristic feature of critical system is self-similarity, i.e. at the
critical point the symmetry is enhanced to include scale transformations
$x^{\mu}\mapsto\lambda x^{\mu}$ with $\lambda\in\mathbb{R}^{+}$. It turns out,
however, that the symmetry of many critical systems, including the critical
Ising model, is further enhanced to the $d$-dimensional conformal group
$SO(1,d+1)$. In addition to translations, rotations and scale transformations,
the conformal group possesses $d$ generators of so-called special conformal
transformations. This enhancement has dramatic consequences as we shall
discuss now.

To begin exploring these consequences let us point out that in a conformal
field theory, correlation functions of three fields are fixed by symmetry, up
to some overall constant. This follows from the well known fact that conformal
transformation can be used to map any set of three points in $\mathbb{R}^{d}$
to the values $x_{1}=0,x_{2}=\vec{e}_{1},x_{3}=\infty$. Hence, a 3-point
function of scalar fields $\phi_{i}$ of scaling weight $\Delta_{i}$ must take
the form

(2.7)
$\langle\phi_{1}(x_{1})\phi_{2}(x_{2})\phi_{3}(x_{3})\rangle=\frac{\gamma_{123}}{|x_{12}|^{\Delta_{12,3}}|x_{23}|^{\Delta_{23,1}}|x_{13}|^{\Delta_{13,2}}}=\gamma_{123}\includegraphics[width=56.9055pt,valign={c}]{scalarvert.png}$

where $x_{ij}=x_{i}-x_{j}$ and
$\Delta_{12,3}=\Delta_{1}+\Delta_{2}-\Delta_{3}$ etc. In more mathematical
terms, the $x_{i}$-dependent function on the right hand side may be thought of
as Clebsch-Gordon coefficients for the conformal group and the prefactors
$\gamma_{123}$ provide a multiplication table for scalar fields of the theory.
This is the interpretation we have tried to stress in the symbolic
representation of the 3-point function on the right hand side.

What is so special about conformal field theories is indeed that their
‘multiplication table’ is really just a set of constants. To avoid confusion
we need to stress that the ‘multiplication’ does not close on scalar fields,
i.e. 3-point functions of two scalars and one vector field, for example, can
be non-vanishing. But this does not alter the basic conclusion that conformal
symmetry guarantees that there exists a ‘multiplication table’ of constants.
It was already observed in the early days of CFT that the multiplication table
of conformal field theories must obey stringent consistency conditions that
are reminiscent of associativity constraints on multiplication tables in
algebra. This observation is the main content of Polyakov’s conformal
bootstrap program [61]. The concrete implementation of this program, however,
is rather difficult and so far analytic solutions were only found for
2-dimensional conformal field theories. Nevertheless, the modern conformal
bootstrap has led to profound new insights on conformal field theories in
$d\geq 3$, see below.

###  2.3. Conformal partial waves and the bootstrap

The most basic associativity constraints arise from the study of 4-point
correlation functions. These are no longer determined by conformal symmetry
alone simply because one can form two independent conformally invariant cross
ratios from four points in $\mathbb{R}^{d}$. We shall parametrize these
through two variables $u_{1}$ and $u_{2}$. In order to analyse 4-point
functions it is useful to expand them in terms of a ‘basis’ of conformal
partial waves

(2.8)
$\langle\prod_{i=1}^{4}\phi_{i}(x_{i})\rangle\sum_{\Delta,l}\gamma_{12;\Delta,l}\gamma_{34;\Delta,l}G_{\Delta,l}(u_{1},u_{2})$

Here the sum extends over all primary (= non-derivative) fields of scaling
weight $\Delta$ and spin $l$, i.e. over fields that possess $l$ symmetric
vector indices $\mu_{1},\dots,\mu_{l}$. Note that the coefficients $\gamma$
are part of the multiplication table, though they also include multiplication
of scalar fields into fields that carry spin, unlike the coefficients we
introduced in eq. (2.7). Generalizing the graphical notation we introduced in
eq. (2.7) we can write such 3-point functions as

(2.9)
$\langle\phi_{1}(x_{1})\phi_{2}(x_{2})\Phi_{3}(x_{3})\rangle=\gamma_{12;\Delta,l}\includegraphics[width=65.44142pt,valign={c}]{SSSTTvert.png}\
.$

The Clebsch-Gordan coefficients for two scalar and one spinning field are
certainly known explicitly. But since we won’t need such detail, we shall
content ourselves with the graphical notation. The conformal partial waves
that appear in the expansion (2.8) of the scalar 4-point function can be
considered as 4-point Clebsch-Gordan maps of the form

(2.10)
$G_{\Delta,l}=\includegraphics[width=71.13188pt,valign={c}]{scalar4point.png}\
.$

The graphical notation in this equation emphases the construction of the
partial wave by contraction of two (3-point) Clebsch-Gordan coefficients. One
can actually translate this into a precise integral formula for $G$,

(2.11) $G_{\Delta,l}\sim\int
d^{d}x_{0}x_{10}^{l+a-\Delta}x_{20}^{l-a-\Delta}x_{30}^{l-b+\Delta-d}x_{40}^{l_{b}+\Delta-d}\left(|\mu||\tilde{\mu}|\right)^{l}Y^{d}_{l}\left(\frac{\mu\cdot\tilde{\mu}}{|\mu||\tilde{\mu}|}\right)$

where

(2.12)
$\mu=\frac{x_{10}}{x_{10}^{2}}-\frac{x_{20}}{x_{20}^{2}}\quad,\quad\tilde{\mu}=\frac{x_{40}}{x_{40}^{2}}-\frac{x_{30}}{x_{30}^{2}},$

the parameters $a,b$ are determined by the weights $\Delta_{i}$ as
$2a=\Delta_{2}-\Delta_{1}$, $2b=\Delta_{3}-\Delta_{4}$ and $Y^{d}_{l}$ denotes
zonal spherical functions. We have also used the notation $x^{t}=|x|^{t}$
whenever elements $x\in\mathbb{R}^{d}$ are raised to some power $t$. The
integral over $x_{0}\in\mathbb{R}^{d}$ takes care of the sum over intermediate
states when we construct a 4-point invariant from two 3-point invariants.
Since the relevant representations of the conformal group are infinite
dimensional (see next section), the summation is most easily expressed as an
integral over some insertion point $x_{0}$ of the intermediate field of weight
$\Delta$ and spin $l$.

Before we discuss which special functions the integrals (2.11) integrate to we
want to sketch one important application of conformal partial wave expansions
that should suffice to illustrate their importance. The attentive reader might
have noticed already that the expansion (2.8) breaks the symmetry of the
correlation functions with respect to the permutation of the four scalar
fields. The expansion we have displayed pairs the fields $\phi_{1}$ and
$\phi_{2}$ together. This arbitrary choice is referred to as the choice of a
channel. To match common convention, we shall use the term operator product
expansion (OPE) channel from now on, though it would be more appropriate to
call it a conformal partial wave (CPW) channel here. In any case, for $N=4$
scalar fields there are two more channels, in addition to the one we have
displayed, for which $\phi_{1}$ is paired with either $\phi_{3}$ or
$\phi_{4}$. Note that the conformal partial waves $G$ do depend on the choice
of the OPE channel. But after summing the partial waves with the coefficients
$\gamma$, the symmetry between the four fields should be restored. This
provides an important consistency condition on the coefficients $\gamma$ that
is known as crossing symmetry condition. The conformal bootstrap [61] is
designed to exploit these constraints in order to compute observables of the
theory, such as the values of the scaling weights $\Delta$ or the coefficients
$\gamma$. The program has had incredible success in $d=2$ dimensions after the
seminal paper of Belavin, Polyakov and Zamolodchikov [3]. During the last
decade, the conformal bootstrap has seen an impressive revival in higher
dimensions with new numerical as well as analytical incarnations, see [64, 63,
60, 18, 19] as well as [25, 42, 1, 2, 40, 41] for early contributions and [59,
70, 58] for reviews, though geared to physicists. This has produced many
stunning new insights including record precision computations of critical
exponents in the critical Ising model [46] through the numerical conformal
bootstrap (nCB),

(2.13) $\Delta^{\textrm{[3D],nCB}}_{\sigma}\sim 0.5181489(10)\ .$

Note that the bootstrap analysis succeeded to increase the precision to which
we know the scaling weight $\Delta_{\sigma}$ of the spin field of the Ising
model in $d=3$ dimensions by two orders of magnitude, as compared to all the
clever numerical simulations that had been developed over a long time. This
celebrated result has triggered a tremendous amount of new research that uses
the bootstrap program to compute observables in (super)conformal field
theories in higher dimensions.

The determination of $\Delta_{\sigma}$ for the $d=3$ critical Ising model
required very good knowledge of the conformal partial waves which need to be
evaluated in a power series expansion around one special point
$u_{1}^{\ast},u_{2}^{*}$ in the space of cross ratios to very high orders.
When conformal partial wave expansions were first introduced in [23, 24] the
partial waves remained elusive objects. The necessary mathematical results
that made efficient evaluation of the partial waves possible were pretty much
out of reach back then. Physicists wondered for more than three decades what
kind of special functions conformal partial are in dimension $d>2$.222In $d=2$
dimensions it is easy to perform the integrals and to express them as a sum of
products of Gauss’ hypergeometric functions. They were interested to
understand how partial waves depend on the cross ratios $u_{i}$, the
parameters $\Delta,l$ of the ‘intermediate field’ and on the weights
$\Delta_{i}$ of the external fields. When higher dimensional conformal field
theories became a central research topic at the end of the last century, Dolan
and Osborn found a way to address these questions. Their idea was to
characterize the partial waves as eigenfunctions of Casimir differential
operators [16]. By analysing these operators they were able to uncover many
interesting properties and in particular to express conformal partial waves in
even dimensions through sums of products of Gauss’ hypergeometric functions
[16, 17]. After a number of relevant extensions [35, 45, 44, 57], this line of
research provided enough control over the conformal partial waves to implement
a numerical version of Polyakov’s bootstrap program.

The central observation of [37] implied later that all the results physicists
had obtained on conformal partial waves for scalar 4-point functions must be
contained in the seminal series of papers by Heckman and Opdam [33, 32, 53,
55], see also [30, 56]. From the vantage point of [38] it became clear that
the conformal partial waves (2.11) are sums of Harish-Chandra functions
associated with the root system $BC_{2}$ and very close relatives of the
corresponding Heckman-Opdam hypergeometric functions. At this point we will
leave the more physics oriented approach of the first section to explain the
relation between conformal partial waves and hypergeometry in a more
mathematical language that should be easier to follow than the original work
on the subject.

##  3\. Conformal Partial Waves and Hypergeometry

###  3.1. Group theoretic reformulation

In order to understand the natural home of conformal partial waves in
hypergeometry, we now want to start translating the physics ingredients we
described above into a more mathematical language, mostly in the context of
group theory. Recall that the symmetry group of a CFT is $G=SO(1,d+1)$ (or
some covering thereof). The fields we worked with in the previous sections
were inserted at points $x$ in the $d$-dimensional space. Since the action of
conformal transformations is transitive, it is clear that we can think of
space as a quotient of the conformal group by the stabilizer of a single
point. The latter is the parabolic subgroup $P$ that is generated by
dilations, rotations and special conformal transformations. Dilations form a
subgroup $SO(1,1)\subset G$ while rotations are elements of $SO(d)\subset G$.
These act on $\mathbb{R}^{d}$ in the usual way and hence they act on the
$d$-dimensional abelian subgroup $N\subset G$ that is generated by special
conformal transformations. Altogether, the stabilizer $P$ of a single point
takes the form $P=K\ltimes N$ where $K=SO(1,1)\times SO(d)$. Note that $K$ is
not compact. From now on we shall think of a point $x$ as an element $x\in
G/P$ in the quotient space of the conformal group $G$ and the parabolic
subgroup $P$.

The next ingredient we need to transfer to mathematics is the notion of a
field. Configurations of a scalar field $\phi$ of weight $\Delta$ are elements
in the space $\Gamma_{\Delta}$ of sections of a line bundle over the quotient
space $G/P$. More precisely, we define

(3.1)
$\Gamma_{\Delta}=\Gamma_{\Delta}(G/P)=\\{f:G\rightarrow\mathbb{C}\,|\,f(gp)=\chi_{\Delta}(p^{-1})f(g)\\}\
.$

The object $\chi_{\Delta}$ is a character of the parabolic subgroup $P\subset
G$ that is trivial on the subgroups $N$ and $R=SO(d)$. On the generator $D$ of
scale transformations, on the other hand, the character $\chi_{\Delta}$
assumes the value $\chi_{\Delta}(D)=\Delta$. Note that $D$ is an element of
the conformal algebra $\mathfrak{g}=\mathfrak{so}(1,d+1)$ and we did not
distinguish in notation between a character of the conformal group and its Lie
algebra. Let us also stress that the linear spaces $\Gamma_{\Delta}$ of
sections come equipped with the left regular action of the conformal group.
For generic values of $\Delta$ the representation of $G$ on $\Gamma_{\Delta}$
is irreducible and infinite dimensional.

After this preparation we can now state that an $N$-point correlation function
may be considered as a conformally invariant element in the tensor product of
$N$ spaces $\Gamma_{1},\dots,\Gamma_{N}$,

(3.2)
$\langle\varphi_{1}(x_{1})\cdots\varphi_{N}(x_{N})\rangle\in\left(\Gamma_{1}\otimes\cdots\otimes\Gamma_{N}\right)^{G}$

where the superscript $G$ is used to denote the subspace of $G$-invariant
elements. To be more precise, the correlation function itself is not
invariant, but there exists a map from the correlation function to the
associated invariant. The latter is mediated by multiplication with a certain
function $\Omega(x_{i};\Delta_{i})$ of the insertion points $x_{i}$. While
$\Omega$ is not unique, there exists a canonical choice [8].

###  3.2. Conformal partial waves as spherical functions

For the moment let us focus on the case with $N=4$ scalar field insertions.
The space of 4-point conformal invariants can be evaluated in two steps. First
we want to evaluate the tensor product of the tensor factors
$\Gamma_{1}=\Gamma_{\Delta_{1}}$ and $\Gamma_{2}=\Gamma_{\Delta_{2}}$. The
result we are going to spell out in a moment involves the following linear
space

(3.3)
$\Gamma_{\Delta}(G/K)\cong\\{f:G\rightarrow\mathbb{C}\,|\,f(gk)=\chi_{\Delta}(k^{-1})f(g)\\}\
.$

The definition is similar to the construction in eq. (3.1), except that the
right covariance condition only involves elements of the subgroup $K\subset
P\subset G$ of the conformal group. In an abuse of notation we have denoted
the character of this subgroup $K$ by the same symbol $\chi_{\Delta}$ as in
the case of the parabolic subgroup $P$ above. Note that the two sets of
characters are related by restriction since we assumed the character of $P$ to
be trivial on the abelian subgroup $N\subset P$. With this notation we can now
state [15]

(3.4)
$\Gamma_{i}(G/P)\otimes\Gamma_{j}(G/P)\cong\Gamma_{\Delta_{ij}}(G/K)\quad$

where $\Delta_{ij}=\Delta_{i}-\Delta_{j}$. We can put this formula to use in
the following analysis of conformal invariants in the 4-fold tensor product,

$\displaystyle\left(\Gamma_{\Delta_{1}}\otimes\Gamma_{\Delta_{2}}\otimes\Gamma_{\Delta_{3}}\otimes\Gamma_{\Delta_{4}}\right)^{G}$
$\displaystyle\cong$
$\displaystyle\left(\Gamma_{\Delta_{12}}(G/K)\otimes\Gamma_{\Delta_{34}}(G/K)\right)^{G}\noindent$
$\displaystyle\cong$
$\displaystyle\Gamma_{\Delta_{12},\Delta_{43}}(K\backslash G/K)\ .$

The final result is expressed in terms of the spaces

(3.5) $\Gamma_{\Delta,\Delta^{\prime}}(K\backslash
G/K)=\\{f:G\rightarrow\mathbb{C}\,|\,f(k_{l}gk_{r})=\chi_{\Delta_{21}}(k_{l})\chi_{\Delta_{34}}(k_{r}^{-1})f(g)\\}\
.$

These consist of sections of a line bundle over the double coset space
$G/\\!\\!/K$. It is not difficult to see that the left action of $K$ in $G/K$
is transitive with stabilizer subgroup $B=SO(d-2)$. It follows that the double
coset space has dimension

$\textit{dim}\,(G/\\!\\!/K)=\textit{dim}(G)-2\textit{dim}(K)+\textit{dim}(B)=2\
.$

The double quotient $G/\\!\\!/K$ coincides with the space of conformally
invariant cross ratios that can be build from $N=4$ points in $\mathbb{R}^{d}$
for $d\geq 2$.

###  3.3. Hypergeometry of conformal partial waves

It is obvious that the Laplace operator on the conformal group $G$ descends to
the space $\Gamma(K\backslash G/K)$. The associated radial Laplacian has been
constructed and studied extensively in the mathematical literature [28, 34].
In particular, it is well known [51, 52] to be intimately related the
trigonometric/hyperbolic Calogero-Moser-Sutherland Hamiltonian [11, 72, 49]

(3.6) $\displaystyle H_{\textrm{CS}}$ $\displaystyle=$
$\displaystyle-\sum_{i=1}^{2}\frac{\partial^{2}}{\partial
u_{i}^{2}}+\frac{k_{m}(k_{m}-1)}{2}\left[\sinh^{-2}\left(\frac{u_{1}+u_{2}}{2}\right)+\sinh^{-2}\left(\frac{u_{1}-u_{2}}{2}\right)\right]$
$\displaystyle\hskip
28.45274pt+\sum_{i=1}^{2}\left[k_{l}(k_{l}-1)\sinh^{-2}(u_{i})+\frac{k_{s}(k_{s}+2k_{l}-1)}{4}\sinh^{-2}\left(\frac{u_{i}}{2}\right)\right]$

associated to the roots system $BC_{2}$ with multiplicity parameters given by

(3.7) $k_{s}=\Delta_{43}\ ,\quad k_{m}=\frac{1}{2}(\Delta_{21}-\Delta_{43}+1)\
,\quad k_{l}=\frac{1}{2}(d-2)$

for the short, middle and long roots, respectively. These Hamiltonians have
also been discussed in the talks of Margit Rössler and Eric Opdam at the
meeting. The line of reasoning we have outlined here now allows to draw an
important conclusion: In the case of scalar 4-point functions, the sought
after conformal partial waves are eigenfunctions of a Calogero-Moser-
Sutherland Hamiltonian and hence they are closely related with Heckman-Opdam
hypergeometric functions for the $BC_{2}$ root system. To be quite precise,
the boundary conditions one has to impose in the case of conformal partial
waves differ from those imposed in the work of Heckman and Opdam. The correct
conditions are described in [38]. It turns out that conformal partial waves
are a linear combination of four of the eight Harish-Chandra functions in the
problem.

In spite of this minor difference, the relation with Heckman-Opdam
hypergeometry can be used to deduce many important properties in conformal
partial waves, see [38]. The position of poles and their residues can be
obtained easily from those of the corresponding Harish-Chandra functions, see
e.g. [30, 56]. To obtain series expansions we realized the partial waves as
the $q\mapsto 1^{-}$ limit of virtual Koornwinder polynomials [62] and
employed the binomial expansions of Koornwinder polynomials from [50] to
perform the degeneration with the help of [71, 43], see appendix A of [38] for
details. Interesting shift equations for the various parameters of partial
waves, finally, arise from the bispectral duality [65] between the
trigonometric Calogero-Moser-Sutherland model with the rational Ruijsenaars-
Schneider model for the same root system [66]. In this way, all properties of
conformal partial waves physicists needed to prepare for applications in the
modern conformal bootstrap can be recovered with relatively little effort from
results mathematicians had obtained at the end of the last century.

The relation between conformal partial waves and Calogero-Moser-Sutherland
models was first observed in [37] based on a direct analysis of the Casimir
operators in the work of Dolan and Osborn. The direct group theoretical
explanation I have outlined in the previous section has been described in [69,
68], with inspiration from earlier work in particular by Feher [21].

This group theoretic approach to conformal partial waves admits many relevant
extensions. Among them is the extension to correlation functions of fields
with spin, i.e. fields that transform non-trivially under the action of the
rotation group $SO(d)\subset G$. In the general setup we described, this
extension can be implemented by replacing the characters $\chi_{\Delta}$ of
the parabolic subgroup $P$ or its subgroup $K=SO(1,1)\times SO(d)$ in eqs.
(3.1) and (3.3), respectively, by some finite dimensional representations. The
radial Laplacians on the associated vector bundles over the double coset
$G/\\!\\!/K$ have been worked out very explicitly [69, 68] for a few
representations of the rotation group $SO(d)$ in $d=3,4$. A general theory of
the Hamiltonians and their eigenfunctions remains to be developed, even though
for applications in physics sufficient control has been achieved, see [12, 39]
and further references therein.

Another very important extension is required to study superconformal field
theories, i.e. quantum field theories in which the conformal Lie algebra
$\mathfrak{g}=\mathfrak{so}(1,d+1)$ gets extended to a superconformal one.
This is particularly important since most known conformal quantum field
theories are actually supersymmetric, especially in dimension $d>3$. For
superconformal algebras of type I a general theory is obtained in [9, 10]. It
reduces the theory of superconformal partial waves to that of ordinary
conformal partial waves for fields with spin, see previous paragraph. The CFT
literature contains few results in superconformal partial waves, except for
very exceptional choices of field insertions, see e.g. [9] for references to
the physics literature.

There is one more extension we want to briefly mention before we conclude this
section. So far we have mostly discussed correlation functions of fields
$\varphi(x)$ that are inserted at a point $x$ in the $d$-dimensional Euclidean
space $x\in\mathbb{R}^{d}$ or its conformal compactification. But physics
applications require to also consider operators that are localized along
higher dimensional geometries such as lines, surfaces etc. One such example
are the famous Wilson line operators in conformal gauge theories. Such non-
local operators (fields) can be treated within the same framework, except that
the stabilizer group $P$ of a point $x$ gets replaces by the stabilizer group
$P_{p}=SO(1,p+1)\times SO(d-p)\ltimes\mathbb{R}^{d}$ of some $p$-dimensional
submanifold. In this context, 2-point correlation functions of non-local
fields can be treated pretty much in the same way as the 4-point function of
local fields.333A non-local field that is inserted along a submanifold of
dimension $p=0$ may be considered as a ‘compound’ of two local fields. The
theory of conformal partial waves for 2-point correlation functions of non-
local fields in CFT has been developed systematically in [36]. Not
surprisingly, it can be reduced to the study of the Laplace operator on double
quotients of the form $K_{p}\backslash G/K_{p^{\prime}}$ with
$K_{p}=SO(1,p+1)\times SO(d-p)$. Therefore it is related to Heckman-Opdam
hypergeometric functions for root systems $BC_{n}$ with the rank $n$ given by

(3.8) $n=\textit{min}\left(p+2,p^{\prime}+2,d-p,d-p^{\prime}\right)\ .$

Note that the rank is $n=2$ in case at least one of the non-local fields is
inserted along a surface of dimension $p=0$ and $d\geq 2$. Higher ranks appear
e.g. for correlation functions of two line operators (i.e. $p=p^{\prime}=1$)
in $4$-dimensional conformal field theories etc. The theory of the associated
conformal partial waves is relatively well developed, but this time only due
to its relations with $BC_{n}$ Heckman-Opdam theory [36].

##  4\. Integrability of Multipoint Conformal Partial Waves

Now that have described the link between conformal partial waves and
hypergeometry at the example of scalar 4-point functions and have sketched a
few extensions at the end of the previous section, I want to dedicate the
final section to one challenging new direction that is only beginning develop.
In the study of concrete conformal field theories physicists often compute
correlation functions of more than $N=4$ fields. And even though the conformal
bootstrap in its original formulation is focused on 4-point functions, one
could very well imagine an alternative program that is based multipoint
partial waves. Note that associativity of an algebra can be studied either
within the set of all products that one can form from four elements of some
linear basis or, alternatively, by studying associativity for higher products
of a set of generating elements in the algebra. Some first attempt to carry
the conformal bootstrap into this direction was recently made in [4], though
in a somewhat restricted setting.

From these short comments we can conclude that it is of significant interest
to develop a theory of multipoint conformal partial waves for $N>4$ insertion
points. The following exposition of the subject in the context of
integrability is based on several recent publications and work in progress [6,
5, 7]. There have also been other activities in physics to study multipoint
partial waves, in particular directly through their integral representations.
The most recent list of references to this research can be found in [7].

To set the stage, let us state the well-known fact that the number of
independent conformal invariants one can build from $N$ points on
$\mathbb{R}^{d}$ is given by

(4.1) $n_{\textit{cr}}(N,d)=\left\\{\begin{array}[]{ll}\frac{1}{2}N(N-3)&N\leq
d+2\\\\[8.53581pt] Nd-\frac{1}{2}(d+2)(d+1)&N>d+2\end{array}\right.$

Following the approach we have described above, a complete characterization of
the associated conformal partial waves requires the same number of independent
commuting differential operators. As in the case of the 4-point function, the
set of differential operators depends on the choice of an OPE channel, see
below for a precise definition.

###  4.1. Gaudin integrable models

The main idea for how to construct a complete set of commuting differential
operators for each OPE channel of an N-point function exploits an interesting
relation with Gaudin integrable systems. In order to make precise statements
we need a bit of background on Gaudin models [26, 27]. Let us begin with a
central object, the so-called Lax matrix,

(4.2)
$\mathcal{L}(w)=\sum_{i=1}^{N}\frac{\mathcal{T}^{(i)}_{\alpha}T^{\alpha}}{w-w_{i}}=\mathcal{L}_{\alpha}(w)T^{\alpha}\
.$

Here, $w_{i}$ are a set of complex numbers, $T_{\alpha}$ denotes a basis of
generators of the conformal Lie algebra in $d$ dimensions and $T^{\alpha}$ its
dual basis with respect to an invariant bilinear form. The object
$\mathcal{T}^{(i)}_{\alpha}$ denotes the set of right invariant vector fields
on the space $\Gamma^{\Delta_{i}}_{G/P}$.444Our discussion is tailored to the
study of N-point functions of scalar fields. To accomodate tensor fields one
needs to extend to more general flag manifolds.

Given some conformally invariant symmetric tensor $\kappa_{p}$ of degree $p$
one can construct a family $\mathcal{H}_{p}(w)$ of commuting operators as [22,
73, 48]

(4.3)
$\mathcal{H}_{p}(w)=\kappa^{\alpha_{1}\cdots\alpha_{p}}_{p}\mathcal{L}_{\alpha_{1}}(w)\cdots\mathcal{L}_{\alpha_{p}}(w)+\dots\
,$

where the dots represent correction terms expressible as lower degree
combinations of the Lax matrix components $\mathcal{L}_{\alpha}(w)$ and their
derivatives with respect to $w$. For $p=2$ such correction terms are absent.
The correction terms are necessary to ensure that the families commute,

(4.4) $[\,\mathcal{H}_{p}(w)\,,\,\mathcal{H}_{q}(w^{\prime})\,]=0\ ,$

for all $p,q$ and all $w,w^{\prime}\in\mathbb{C}$. It is a well-known fact
that these families commute with the diagonal action of the conformal algebra,
i.e.

(4.5) $[\,\mathcal{T}_{\alpha}\,,\,\mathcal{H}_{p}(w)\,]=0\quad\textit{ where
}\ \mathcal{T}_{\alpha}=\sum_{i=1}^{N}\mathcal{T}_{\alpha}^{(i)}\ .$

Hence the commuting families $\mathcal{H}_{p}(w)$ of operators descend to
differential operators on functions of the $n_{\textit{cr}}(N,d)$ conformally
invariant cross ratios.

###  4.2. OPE limits of the Gaudin model

At this point, the commuting families $\mathcal{H}_{p}(w)$ still contain $N$
continuous parameters $w_{i},i=1,\dots,N$. Without loss of generality we can
set three of these complex numbers to some specific values, e.g.
$w_{1}=0,w_{N-1}=1,w_{N}=\infty$ so that we remain with $N-3$ complex
parameters our Gaudin Hamiltonians depend on. The set of differential
equations which characterize conformal partial waves, on the other hand,
should only depend on the discrete choice of an OPE channel. As we shall see
now, any OPE channel determines a certain limit of the parameters $w_{i}$. In
other words, the parameters $w_{i}$ are entirely fixed by the channel.

In order to make a precise statement, we need a bit of preparation. Let us
pick some OPE channel $\mathcal{C}$ along with an (arbitrary) external edge in
the diagram. The latter serves as a reference point and we shall assign it the
label $N$. As this edge is external, it is attached to a unique vertex which
we will denote by $\rho_{\ast}$. Such a choice of reference vertex defines a
so-called rooted tree representation of the channel, see Figure 1 for a
concrete example with $N=8$. We draw the OPE diagram on a plane, with the
vertex $\rho_{\ast}$ situated at the top and with each vertex having two
downward edges attached. The representation on a plane forces us to make a
choice of which edges are pointing towards the left and which edges are
pointing towards the right: this choice is arbitrary, and gives rise to what
is called a plane (or ordered) representation of the underlying rooted tree.

Figure 1. Plane rooted tree representation of an OPE diagram with 8 external
fields.

Obviously, each vertex $\rho\in V$ from the set $V$ of $N-2$ vertices of the
OPE diagram defines a partition
$\underline{N}=I_{\rho,1}\mathbin{\mathchoice{\leavevmode\vtop{\halign{\hfil$\m@th\displaystyle#$\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\textstyle#$\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptstyle#$\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptscriptstyle#$\hfil\cr\cup\cr\cdot\crcr}}}}I_{\rho,2}\mathbin{\mathchoice{\leavevmode\vtop{\halign{\hfil$\m@th\displaystyle#$\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\textstyle#$\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptstyle#$\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptscriptstyle#$\hfil\cr\cup\cr\cdot\crcr}}}}I_{\rho,3}$,
with the sets $I_{\rho,j}$ formed by the labels of the external fields
attached to the three branches of the vertex. We shall fix the labelling of
the three branches using the plane rooted tree representation of the channel:
choose the branch $I_{\rho,1}$ to be the one pointing to the bottom left while
the branch $I_{\rho,2}$ is pointing the bottom right. By construction, the
last branch $I_{\rho,3}$ then always points to the top and contains the
reference point $N$. Let us also note that the vertices $\rho\neq\rho^{*}$ are
associated with a unique internal edge, namely the one that points upward.
Thereby the plane rooted tree representation allows us to identify the set $E$
of internal edges with $E\equiv V\setminus\\{\rho^{*}\\}$.

So far, vertices were just enumerated by an element $\rho\in V$. Once we have
drawn the diagram in a plane, we can assign a sequence
$s^{\rho}=(s^{\rho}_{1},s^{\rho}_{2},\dots,s^{\rho}_{\delta_{\rho}})$ of
elements $s^{\rho}_{a}\in\\{1,2\\}$ to each vertex $\rho$. This sequence
$s^{\rho}$ encodes the path from $\rho_{\ast}$ to $\rho$. It tells us whether
we have to move to the left (for $s_{a}^{\rho}=1$) or right (for
$s_{a}^{\rho}=2$) every time we reach a new vertex until we arrive at $\rho$
after $\delta_{\rho}$ steps. We shall also refer to the length $\delta_{\rho}$
of the sequence as the depth of the vertex and to $s^{\rho}$ as the binary
sequence of $\rho$. Note that the top vertex $\rho_{\ast}$ has depth
$\delta_{\rho_{\ast}}=0$.

In order to construct the limit of the Gaudin model associated with the OPE
channel $\mathcal{C}$, we need now construct a polynomial $g_{\rho}(\varpi)$
for each vertex $\rho$. If $s^{\rho}$ is the binary sequence associated with
the vertex $\rho$, the polynomial $g_{\rho}$ is defined as

(4.6)
$g_{\rho}(\varpi)=\sum_{a=1}^{\delta_{\rho}}\varpi^{a-1}\delta_{s_{a}^{\rho},2}\
.$

Obviously the top vertex $\rho_{\ast}$ is assigned to
$g_{\rho_{\ast}}(\varpi)=0$. The vertices of depth $\delta_{\rho}=1$ are
associated with $g_{\rho_{1}}(\varpi)=0$ or $g_{\rho_{2}}(\varpi)=1$,
depending on whether they are reached from $\rho_{\ast}$ by going down to the
left ($\rho_{1}$) or to the right ($\rho_{2}$) etc.

Similarly, we can assign polynomials $f_{i}(\varpi)$ to each external edge
$1\leq i<N$ at the bottom of the plane rooted tree. Once again, we can encode
the path from $\rho_{\ast}$ down to the edge $i$ by a binary sequence
$s^{i}=(s^{i}_{1},s^{i}_{2},\dots,s_{\delta_{i}}^{i})$. The length
$\delta_{i}$ of the sequence $s^{i}$ is also referred to as the depth of the
edge $i$. Now we introduce

(4.7)
$f_{i}(\varpi)=\sum_{a=1}^{\delta_{i}}\varpi^{a-1}\delta_{s^{i}_{a},2}+\varpi^{\delta_{i}}\delta_{s^{i}_{\delta_{i}},1}\
.$

and set

(4.8) $f_{N}(\varpi)=\varpi^{-1}$

for the external edge of the reference field at the top of the plane rooted
tree. We have thus set up all the necessary notation that is needed to
construct the relevant scaling limits of the $N$-site Gaudin model that remove
the dependence on the continuous parameters $w_{i}$

The main idea in defining the channel dependent limit is to set
$w_{i}=f_{i}(\varpi)$ and to send $\varpi$ to zero. When we perform this limit
of the objects $\mathcal{H}^{(p)}(w;w_{i})$ of the $N$-site Gaudin model we
must adjust the spectral parameter $w$ and multiply with some appropriate
powers of $\varpi$ in order to obtain sufficiently many independent commuting
operator. More precisely, for given a vertex $\rho$ we introduce

(4.9) $\mathcal{H}_{\rho}^{(p)}(w)=\lim_{\varpi\to
0}\varpi^{pn_{\rho}}\mathcal{H}^{(p)}\bigl{(}\varpi^{n_{\rho}}w+g_{\rho}(\varpi),w_{i}=f_{i}(\varpi)\bigr{)}\
.$

It can be shown that the set of these objects with $\rho\in E$ indeed contain
$n_{\textit{cr}}(N,d)$ independent operators on the space of conformal
invariants. These include operators of the form

(4.10)
$\mathcal{C}^{(p)}_{\rho}=\lim_{w\to\infty}w^{p}\mathcal{H}_{\rho}^{(p)}(w)\quad\mathit{for}\quad\rho\neq\rho^{*}\
.$

These measure the weight and spin of the field which is exchanged along the
internal edge $\rho\in E$ above the vertex $\rho\neq\rho^{*}$. We refer to the
operators $\mathcal{C}^{(p)}_{\rho}$ as Casimir operators of the edge $\rho$.
For a given edge $\rho\neq\rho^{*}$ there can be relations between operators
with different values of $p$. Taking these into account, the internal edge
$\rho\neq\rho^{*}$ provides us with

(4.11)
$n_{\textit{Cas}}(\rho,d):=\textrm{min}\left(|I_{\rho,3}|,N-|I_{\rho,3}|,\textrm{rank}_{d}\right)=:\mathfrak{d}_{\rho}$

independent operators. Here $|I|$ denotes the order of the subset
$I\subset\underline{N}$ and $\mathrm{rank}_{d}$ is the rank of the
$d$-dimensional conformal algebra. It is easy to see that for $N>4$, the
integers $n_{\textit{Cas}}$ do not add up to $n_{\textit{cr}}$. This means
that the Casimir operators do not suffice to characterize multipoint conformal
partial waves. The missing operators can all be constructed from the objects
(4.9). It turns out that given a vertex $\rho$ and an index $p$ one can
prepare

(4.12)
$n_{\textit{vert}}^{(p)}(\rho,d)=\textit{max}\left[\left((p-2)-\sum_{e\in\overline{E}_{\rho}}^{3}\Theta_{0}(p-2\mathfrak{d}_{e})(p-2\mathfrak{d}_{e}-1)\right),0\right]\,,$

additional independent vertex differential operators. Here $\Theta_{0}$ is the
Heaviside step function with $\Theta_{0}(0)=0$ and $\overline{E}_{\rho}$
consists of all external and internal edges that have one endpoint at the
vertex $\rho$. For internal edges $e=\rho^{\prime}$, the quantity
$\mathfrak{d}_{e}$ was introduced as a shorthand in eq. (4.11) and for
external edges we set $\mathfrak{d}_{e}=1$. The formula applies for all $p$ as
long as $d$ is odd. It requires some adjustment in even $d$, see [7] for
details and a derivation.

Given this counting of independent commuting differential operators it is now
possible to check that

(4.13)
$n_{\textit{cr}}(N,d)=\sum_{\rho\neq\rho^{*}}n_{\textit{Cas}}(\rho,d)+\sum_{\rho}\sum_{p}n_{\textit{vert}}^{(p)}(\rho,d)\
,$

i.e. that the set of independent differential operators suffices to completely
characterize the multipoint conformal partial waves.

The OPE limits we described here are generalizations of the so-called
caterpillar or bending flow limits that have been considered in the
mathematical literature, see e.g. [13, 14, 67]. The caterpillar limit
corresponds to the special case of an OPE limit for which the OPE diagram has
the shape of a comb, see Figure 2. Following our conventions in [6], we
enumerate the vertices $\rho=[r]$ by an integer $r=1,\dots,N-2$ that is
related to its depth $\delta_{\rho}=\delta_{[r]}$ as $\delta_{[r]}=N-2-r$.

Figure 2. Choice of plane rooted tree representation of the comb channel OPE
diagram with $N$ points.

One can compute the limit of the Gaudin model associated with this tree using
the construction outlined in the previous subsection. For the polynomials
$f_{i}$ that determine the parameters $w_{i}$ of the Gaudin model, one finds
from eq. (4.7) that

(4.14)
$w_{i}=f_{i}(\varpi)=\varpi^{N-1-i},\qquad\forall\,i\in\\{1,\cdots,N\\}\ .$

while all the vertex polynomials vanish, i.e. $g_{[r]}=0$ for $r=1,\dots,N-2$.
Our general prescription (4.9) thus becomes

(4.15) $\mathcal{H}_{[r]}^{(p)}(w)=\lim_{\varpi\to
0}\varpi^{p(N-2-r)}\mathcal{H}^{(p)}\bigl{(}\varpi^{N-2-r}w;w_{i}=\varpi^{N-1-i}\bigr{)}\
.$

In order to illustrate our results and also prepare for the discussion in the
next subsection we want to take a closer look at the case of $N=6$ points for
$d>4$. According to equation (4.11), the three internal edges $[r],r=1,2,3,$
in the comb channel diagram contribute

(4.16) $n_{\textit{Cas}}([1],d)=2=n_{\textit{Cas}}([3],d)\quad,\quad
n_{\textit{Cas}}([2],d)=3$

and hence a total number of $7$ Casimir differential operators. Our result
(4.12), on the other hand shows that the four vertices $[r],r=1,\dots,4,$
contribute

(4.17)
$n_{\textit{vert}}^{(p)}([1],d)=0=n_{\textit{vert}}^{(p)}([4],d)\quad,\quad
n_{\textit{vert}}^{(p)}([2],d)=1=n_{\textit{vert}}^{(p)}([3],d)$

and hence a total of two vertex differential operators which are both of order
four. When combined, the number of differential operators matches the number
of $n_{\textit{cr}}=9$ cross ratios.

###  4.3. Relation with lemniscatic elliptic CMS model

The two inner vertices of the $N=6$ point partial wave are among the simplest
vertices that carry degrees of freedom, i.e. for which $n_{\textit{vert}}\neq
0$. As one can infer from our general formula (4.12), it is in fact the unique
vertex in $d>4$ that gives rise to a single parameter. Let us focus on the
vertex $\rho=[2]$ of the $N=6$ point function, see our discussion at the end
of the previous subsection, which connects the internal edges $[1]$ and $[2]$
with the external line $i=3$. According to our list (4.16), the edge $[2]$
admits two independent Casimir operators whose eigenvalues we parametrize
through $\Delta_{1}$ and $l_{1}$. The internal edge $[3]$, on the other side,
admits three independent Casimir operators. Their eigenvalues can be
parametrized by three parameters $\Delta_{2},l_{2},\ell_{2}$. The external
scalar edge comes with one parameter, namely the scaling weight $\Delta_{3}$
of the external scalar field.555This is different from the weight on the third
internal edge which we also denoted by $\Delta_{3}$ in the previous
subsection. Let us note that for $d=4$, the vertex which we consider here is
the special case of a single variable vertex with one additional parameter
$\ell_{1}\neq 0$. On the other hand, if we specialize the non-trivial vertex
of the $N=6$ point function to $\ell_{2}=0$ it can be considered down to
$d=3$.

Figure 3. The two nontrivial vertices in the 6-point function are
characterized by six quantum numbers each.

Our goal now is to identify the fourth order vertex differential operator for
the single variable vertices. The easiest way to describe the findings of [7]
involves a new algebra that is generated by operators $A,A^{\dagger}$ and $N$
which obey the following relations

(4.18) $\displaystyle[N,A^{\dagger}]=A^{\dagger}\quad$ ,
$\displaystyle\quad[N,A]=-A\,,$ (4.19) $\displaystyle
AA^{\dagger}=R_{\alpha}(N+1)\quad$ , $\displaystyle\quad
A^{\dagger}A=R_{\alpha}(N)$

where

$R_{\alpha}(N)=\frac{N(N+2\alpha-1)}{(N+\alpha-1)(N+\alpha)}(N+\nu_{1}+2\alpha)(N-\nu_{1}-1)(N+\nu_{2}+2\alpha)(N-\nu_{2}-1)\,.$

The algebra depends parametrically on the three spin parameters as
$\alpha=\ell_{2}+(d-3)/2$ and $\nu_{k}=l_{k}-\ell_{2}$, but is independent of
the conformal weights $\Delta_{i}$. The family of algebras comes equipped with
an involutive anti-automorphism $\ast$ defined by $N^{\ast}=N$ and
$A^{\ast}=A^{\dagger}$, provided that the parameters obey

(4.20)
$\bar{\alpha}=\alpha\,,\qquad\bar{\nu}_{1}=-(2\alpha+1+\nu_{1})\,,\qquad\bar{\nu}_{2}=-(2\alpha+1+\nu_{2})\,.$

The vertex operator $H$ can now be written as a rational combination of the
three generators as,

(4.21) $\displaystyle H^{(d;\Delta_{i};l_{i};\ell_{i})}=$ $\displaystyle
B^{\dagger}B-\Gamma(N+\alpha)^{2}+\frac{\alpha(\alpha-1)\,K}{(N+\alpha)^{2}-1}+E^{(d;\Delta_{i};l_{k};\ell_{2})}.$

Here, we defined the operators

(4.22) $\displaystyle B^{\dagger}:=$
$\displaystyle\frac{A-A^{\dagger}}{2\mathrm{i}}-\mathrm{i}(2\gamma_{1}\gamma_{2}+\mathrm{i}\gamma_{3})(N+\alpha)\,,$
(4.23) $\displaystyle B:=$
$\displaystyle\frac{A-A^{\dagger}}{2\mathrm{i}}+\mathrm{i}(2\gamma_{1}\gamma_{2}-\mathrm{i}\gamma_{3})(N+\alpha)\,,$

and the two parameters

(4.24) $\displaystyle\Gamma:=$
$\displaystyle\frac{1}{4}(1+4\gamma_{1}^{2})(1+4\gamma_{2}^{2})\,,$ (4.25)
$\displaystyle K:=$
$\displaystyle(\nu_{1}+\alpha)(\nu_{2}+\alpha)(\nu_{1}+\alpha+1)(\nu_{2}+\alpha+1)\,.$

An expression for the constant term $E$ in our formula for the vertex operator
(4.21) can be found in subsection 5.2 of [5]. The three parameters
$\gamma_{i}$ that appear in the last few formulas are related to the conformal
weights $\Delta_{i}$ of the three fields at the vertex as

$\Delta_{i}=\frac{d}{2}+\mathrm{i}\gamma_{i}\ .$

The operator $H$ is self-adjoint provided that the parameters $\gamma_{i}$ are
real.

As one can easily check, the algebra generated by $A,A^{\dagger}$ and $N$
admits a representation in a basis $C_{n}^{(\alpha)}$ of the form

(4.26) $\displaystyle NC_{n}^{(\alpha)}$ $\displaystyle:=nC_{n}^{(\alpha)},$
(4.27) $\displaystyle AC_{n}^{(\alpha)}$
$\displaystyle:=(n+\nu_{1}+2\alpha)(n+\nu_{2}+2\alpha)\frac{n+2\alpha-1}{n+\alpha}C_{n-1}^{(\alpha)}\,,$
(4.28) $\displaystyle A^{\dagger}C_{n}^{(\alpha)}$
$\displaystyle:=(n-\nu_{1})(n-\nu_{2})\frac{n+1}{n+\alpha}C_{n+1}^{(\alpha)},$

where eq. (4.27) applies to all integers $n>0$, and $AC_{0}^{(\alpha)}=0$ when
$n=0$, i.e. the state $C_{0}^{(\alpha)}$ is annihilated by the lowering
operator $A$. Similarly, the action of the raising operator $A^{\dagger}$
vanishes if $n=\nu_{1}$ or $n=\nu_{2}$. Consequently, one can restrict the
action of $A,A^{\dagger}$ and $N$ to the finite dimensional subspace that is
spanned by $C_{n}$ for $n=0,\dots,\mathrm{min}(\nu_{1},\nu_{2})$. When the
basis is realized as Gegenbauer polynomials
$C^{(\alpha)}(s)=C^{(\alpha)}(1-2\mathcal{X})$, the operators $A,A^{\dagger}$
describe the action of second order differential operators in the variable
$\mathcal{X}$. We then showed that the resulting fourth order differential
operator $H=H_{\mathcal{X}}$ can be mapped to the Hamiltonian $L$ of a
crystallographic elliptic Calogero-Moser-Sutherland model that was discovered
originally by Etingof, Felder, Ma and Veselov [20]. The latter acts on
functions in a variable $z$ on the lemniscatic elliptic curve. We found the
variables $\mathcal{X}$ to be related to $z$ as

(4.29) $\mathcal{X}=\frac{\wp(1/2)^{2}}{\wp(1/2)^{2}-\wp(z)^{2}}\ ,$

where $\wp$ is the Weierstrass elliptic function with lemniscatic periodicity
conditions $\wp(z)=\wp(z+1)=\wp(z+i)$. Given this change of variables, our
vertex Hamiltonian $H=H_{\mathcal{X}}$ and the elliptic CMS Hamiltonian
$L=L_{z}$ of [20] are related as

(4.30) $\Theta^{-1}H_{\mathcal{X}}\Theta\sim L_{z}$

where

(4.31)
$\Theta\sim\,\mathcal{X}^{(l_{1}+l_{2}-2\ell_{2}+\Delta_{3}+(1-d)/2)/4}(1-\mathcal{X})^{(l_{1}+l_{2}-2\ell_{2}-\Delta_{3}+(1+d)/2)/4}\
.$

The lemniscatic Calogero-Moser-Sutherland Hamiltonian $L$ involves seven
parameters (multiplicities) whose values are determined by the three conformal
weights $\Delta_{i}$, the spin parameters $l_{1},l_{2},\ell_{2}$ and the
dimension $d$. Precise formulas can be found in [7]. A similar relation for a
3-point vertex with one scalar and two generic spinning legs of the $d=4$
dimensional conformal group also holds. In that case, the dimension $d$ is
fixed and the seven multiplicities of the lemniscatic model $L$ are obtained
from the three conformal weights and four spin variables
$l_{1},\ell_{1},l_{2},\ell_{2}$, see [7] for precise formulas.

##  5\. Concluding comments

This brings us to the end of this short note on higher dimensional CFT and its
profound relation to multivariate hypergeometry. In the past, the interaction
between these two fields has not lived up to its true potential. I hope that
this text can make a modest contribution to developing a culture of mutual
interest and understanding. It would clearly be very beneficial to join forces
on overcoming some of the challenges that arise within CFT.

In my outline of concrete open problems I have focused on the study of
multipoint partial waves, mostly because it is particularly timely and has
seen some recent advances that are partly due to the very fruitful
interactions I had during and after the workshop. Let me stress, however, that
there are many other interesting problems that could profit from fresh
mathematical input. One of these concerns the study of hypergeometric
functions associated with (conformal) supergroups. I have only scratched this
topic very briefly at the end of section 3. Given that most of the CFTs we
know about are supersymmetric, progress in the theory of superconformal
partial waves is guaranteed to have many applications. While physicists have
made some advances over the last years, very much remains to be done.

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