# Quantum Geometry, Integrability, and Opers

Peter Koroteev  Department of Mathematics, University of California, Berkeley,
CA 94720, USA and Department of Mathematics, State University of New York,
Buffalo, NY, 14068, USA

###### Abstract.

This review article discusses recent progress in understanding of various
families of integrable models in terms of algebraic geometry, representation
theory, and physics. In particular, we address the connections between soluble
many-body systems of Calogero-Ruijsenaars type, quantum spin chains, spaces of
opers, representations of double affine Hecke algebras, enumerative counts to
quiver varieties, to name just a few. We formulate several conjectures and
open problems. This is a contribution to the proceedings of the conference on
Elliptic Integrable Systems and Representation Theory, which was held in
August 2023 at University of Tokyo.

##  1\. Introduction

In this work, we review the current status of integrable models of the
Calogero-Ruijsenaars family in the context of recent developments in algebraic
geometry and representation theory. This study is strongly motivated by
progress in string theory and supersymmetric quantum field theories beginning
in the early 90s [Seiberg:1994rs, Seiberg:1994aj].

Out of plethora of SUSY theories with eight supercharges one particular theory
stands out – the $\mathcal{N}=2^{*}$ theory [Donagi:1995cf] where the star by
tradition denotes soft breaking of supersymmetry in half from the maximal
amount for a four-dimensional theory which does not include gravity. Unlike
many other theories $\mathcal{N}=2^{*}$ theory has very few parameters – gauge
coupling constant, gauge group type, and the SUSY breaking mass.

This relative simplicity of the parameter space of the model quickly attracted
experts in representation theory, algebraic geometry, and integrable systems.
Physics results have been expeditiously interpreted mathematically, fostering
a rich and fruitful synergy between these disciplines over the past couple of
decades.

### The Diamond

The content of the current review can be summarized in Fig. 2 by the $3\times
3$ diamond . Each of the nine ovals of the diamond contains the following data
which we exhibit in Fig. 1 and explain below.

Figure 1. Notations for the diamond in Fig. 2

  1. (1)

Phase Space. In the discussion of complex algebraic integrable systems with
$n$ degrees of freedom the phase space is Lagrangian fibration of complex
dimension $2n$, equipped with a holomorphic symplectic 2-form, over a smooth
base whose fibers are Abelian varieties. In proper Darboux coordinates
$(\textbf{p},\textbf{q})$ the symplectic form reads
$\Omega=\sum_{i}dp_{i}\wedge dq_{i}$ and $n$ Hamiltonians
$H_{1}(\textbf{p},\textbf{q}),\dots,H_{N}(\textbf{p},\textbf{q})$ Poisson
commute with each other. The Abelian nature of the Lagrangian fibers above
implies that each coordinate and momentum can take values in complex line
$\mathbb{C}$, in a complex line without origin $\mathbb{C}^{\times}$, or in an
elliptic curve $\mathcal{E}=\mathbb{C}^{\times}/\mathfrak{p}^{\mathbb{Z}}$,
where $\mathfrak{p}$ is modular parameter. Therefore each many-body integrable
model belongs either to rational, trigonometric or elliptic type depending on
where its coordinates and momenta take their values. For instance,
Hamiltonians of the tRS model in the middle of the diamond are periodic in
both coordinates and momenta hence the notation for its phase space –
$\mathbb{C}^{\times}_{p}\times\mathbb{C}^{\times}_{x}$.

  2. (2)

Type of Oper. The space of $G$-opers under certain conditions describes the
phase space of a classical many-body system according to [Koroteev:2022vg].
The space of opers thereby provides a bridge to establish the
quantum/classical duality between quantum spin chains and classical
$n$-particle systems. In the terminology of [Koroteev:2022vg] this description
corresponds to magnetic frame, i.e. the tCM model is dual to the XXX spin
chain in the magnetic frame. There is also an electric frame of the
quantum/classical duality under which the tCM is dual to tGaudin model and the
rRS is dual to the XXX chain. Here $e(G,q)$-opers stands for $(G,q)$-opers on
the elliptic curve (elliptic q-opers [GKSZ:2024Toappear]). Similarly,
$eG$-opers are elliptic opers from the same work in progress.

  3. (3)

Many-body system. The name of the corresponding $n$-particle model – Calogero-
Moser (CS), Ruijsenaars-Schneider (RS), double elliptic (DELL), and their
duals. Prefixes $r,t$, and $e$ stand for rational, trigonometric, and elliptic
respectively.

  4. (4)

Spin Chain. The quantum spin chain which is related to the many-body system
from the same oval via the quantum/classical duality in the magnetic frame.
The known models are XXX, XXZ, XYZ, and Gaudin spin systems. There are
rational, trigonometric, and elliptic Gaudin systems.

  5. (5)

Equation on Baxter functions. Baxter $Q$-functions play important roles in
representation theory and enumerative geometry. These functions depend on an
auxiliary (spectral) parameter and satisfy certain difference or differential
equations depending on the type of $G$-oper. For example, for the $(G,q)$-oper
they satisfy a difference $QQ$-relation.

  6. (6)

Algebra for $n$ particles. The phase space of a complex algebraic integrable
system is a holomorphic symplectic manifold $\mathcal{X}$. The ring of
holomorphic functions on this manifold admits deformation quantization
$\mathcal{O}^{q}(\mathcal{X})$ taken with respect to its holomorphic
symplectic form. In the case when $\mathcal{X}$ is the moduli space of
$SL(n)$-flat connections on the punctured torus the latter yields spherical
spherical subalgebra of the double affine Hecke algebra or DAHA for
$\mathfrak{gl}_{n}$ [oblomkov2004double, Cherednik-book]. This algebra admits
degenerations.

  7. (7)

Algebra for $n\longrightarrow\infty$. $\mathfrak{gl}_{n}$ DAHAs admit direct
$n\longrightarrow\infty$ limit which is known in the literature by various
names: Hall algebra of the elliptic curve, Ding-Iohara-Miki algebra, Drinfeld
double of shuffle algebra, quantum toroidal algebra
$U_{q,\hbar}\left(\widehat{\widehat{\mathfrak{gl}_{1}}}\right)$. This algebra
and its degenerations, i.e. affine Yangian
$Y_{\hbar,\epsilon}\left(\widehat{\mathfrak{gl}_{1}}\right)$ are interesting
at their own right, in particular, they act on equivariant K-theory
(cohomology) of Hilbert scheme of points on $\mathbb{C}^{2}$.

### Enumerative Geometry.

In addition, the middle and top northwest-southeast diagonals of Fig. 2 are
colored in yellow and green respectively which designates similarities of the
corresponding integrable models and representation theoretic and algebro-
geometric structures. For instance, the eigenfunctions of the tCM, tRS, and
dual eRS models up to a simple multiple coincide with quasimap vertex
functions of quantum equivariant cohomology, K-theory, and elliptic cohomology
of the cotangent bundle to the complete flags in $\mathbb{C}^{n}$.
Representation theoretically they describe equivariant Euler characteristics
of Laumon spaces. This will be addressed later in the paper.

### Supersymmetric Gauge Theories.

It was suggested long time ago [Gorsky:1995zq, Gorsky:2000px, Gorsky:1997mw],
and was confirmed in the subsequent publications by numerous authors, that the
wave functions of the quantum Hamiltonians of eCM, eRS, and DELL are the
supersymmetric partition functions in the presence of codimension-two defects
of the dual 4, 5, and 6-dimensional theories with adjoint matter
respectively111the 5d theory is compactified on a circle, while the 6d theory
is compatified on a torus. (see [Koroteev:2019gqi] for summary and
references). The corresponding eigenvalues are represented by vacuum
expectation values of local chiral observables in the respected dimensions.

One can immediately see that the diamond contains a plethora of question marks
which are placed instead of unknown models or algebraic structures. For
instance, to the best of the authors’ knowledge, there is no known oper
description of the phase space of the elliptic integrable systems on the green
diagonal, nor are there dual quantum lattice models for the eCM, eRS, and
DELL. We invite the reader to search for answers to these questions.

In the main part of the paper we describe various ovals of the diamond
together with interactions between its various components.

### Acknowledgements

The author is indebted to his colleagues and collaborators from whom he had
learned various lessons in mathematical physics over the years. In particular,
I would like to thank A. Zeitlin, E. Frenkel, and D. Sage for continuous
collaboration and encouragement. This conference paper precedes a larger
manuscript which will explain in greater detail beautiful mathematical
structures of the diamond in Fig. 2 which will appear in the near future. This
note will be continuously updated until then. The author will maintain
https://math.berkeley.edu/ pkoroteev/diamond33.pdf where more details and
references will be posted. I encourage my colleagues to submit materials and
missing references. Finally, I would like to thank H. Konno and J. Shiraishi
for inviting me to present this work at the conference on integrable systems
and representation theory in Tokyo, Japan in August 2023.

Figure 2. The Diamond

##  2\. $(G,q)$-Opers, tRS model, and XXZ Chain

The space of opers has recently become a natural framework for studying
integrable systems and the dualities between them [Frenkel:2020,
Koroteev:2022vg]. Below we review models of type $A$ leaving the study of the
other classical types for the upcoming work [KZSoSp:2024].

We begin our description in the middle oval of the diamond, viz. with
$(SL(N),q)$-opers (see [Koroteev:2022vg] for more details). Let
$X=\mathbb{P}^{1}\backslash\\{0,\infty\\}$. Consider the automorphism
$M_{q}:X\longrightarrow X$ sending $z\mapsto qz$ , where
$q\in\mathbb{C}^{\times}$.

######  Definition 2.1.

Let $U\subset X$ be a Zariski open dense subset and let $V=U\cap
M_{q}^{-1}(U)$. A meromorphic $(GL(r+1),q)$-oper on $\mathbb{P}^{1}$ is a
triple $(A,E,\mathcal{L}_{\bullet})$, where $E$ is a vector bundle of rank
$r+1$ on $\mathbb{P}^{1}$ and $\mathcal{L}_{\bullet}$ is the corresponding
complete flag of the vector bundles,

$\mathcal{L}_{r+1}\subset...\subset\mathcal{L}_{i+1}\subset\mathcal{L}_{i}\subset\mathcal{L}_{i-1}\subset...\subset\mathcal{L}_{1}=E,$

where $\mathcal{L}_{r+1}$ is a line bundle, so that the meromorphic
$(GL(r+1),q)$-connection $A\in\operatorname{Hom}_{\mathcal{O}_{U}}(E,E^{q})$,
where $E^{q}$ is the pullback of $E$ under $M_{q}$, satisfies the following
conditions:
i) $A\cdot\mathcal{L}_{i}\subset\mathcal{L}^{q}_{i-1}$,
ii) The restriction of
$A\in\text{Hom}(\mathcal{L}_{\bullet},\mathcal{L}^{q}_{\bullet})$ to $V$ is
invertible and satisfies the condition that the induced maps

$\bar{A}_{i}:\mathcal{L}_{i}/\mathcal{L}_{i+1}\longrightarrow\mathcal{L}^{q}_{i-1}/\mathcal{L}^{q}_{i},\qquad
i=2,\dots,r$

are isomorphisms on $V$.
An $(SL(r+1),q)$-$oper$ is a $(GL(r+1),q)$-oper with the condition that det
$A=1$ on $V$.

Changing the trivialization of $E$ via $g(z)\in SL(r+1)(z)$ modifies $A(z)$ by
the following $q$-gauge transformation

(2.1) $A(z)\mapsto g(qz)A(z)g(z)^{-1}.$

thereby endowing $A$ with the structure of $(SL(r+1),q)$-connection.

The above definition can be reformulated in a local form – given a section
$s(z)$ of $\mathcal{L}_{r+1}$, the oper condition is as follows. Consider the
following determinants

(2.2)
$W_{k}(s)(z)=\bigwedge_{i=0}^{k-1}\prod_{j=0}^{i-2}A(q^{i-2-j}z)\,s(q^{i-1}z)\Bigg{|}_{\Lambda^{k}\mathcal{L}_{r-k+2}^{q^{k-1}}}\,,\quad\quad
k=2,\dots,r+1\,.$

The oper conditions are equivalent to the fact that all the above determinants
are nonzero.

######  Definition 2.2.

We say that $(SL(r+1),q)$-oper has regular singularities defined by the
collection of polynomials $\\{\Lambda_{i}(z)\\}_{i=1,\dots,r}$ when
$\bar{A}_{i}$ is an isomorphism away from the zeros of $\Lambda_{i}(z)$ for
$i=1,\dots,r$.

In local terms, the above definition implies the following

(2.3) $\displaystyle W_{k}(s)(z)=P_{1}(z)\cdot P_{2}(qz)\cdots
P_{k}(q^{k-1}z),\qquad
P_{i}(z)=\Lambda_{r}(z)\Lambda_{r-1}(z)\cdots\Lambda_{r-i+1}(z)\,.$

######  Definition 2.3.

The $(SL(r+1),q)$-oper is called $Z-twisted$ if there exists $g(z)\in
SL(r+1)(z)$ such that

(2.4) $\displaystyle A(z)=g(qz)Zg(z)^{-1},$

where $Z$ is a diagonal element of $SL(r+1)$.

######  Definition 2.4.

A Miura $(SL(r+1),q)$-oper is a quadruple
$(A,E,\mathcal{L}_{\bullet},\hat{\mathcal{L}}_{\bullet})$, where
$(A,E,\mathcal{L}_{\bullet})$ triple is $(SL(r+1),q)$-oper and the complete
flag $\hat{\mathcal{L}}_{\bullet}$ of subbundles in $E$ is preserved by the
q-connection $A$.

One can immediately see that if $Z$ is regular semisimple element then there
are $(r+1)!$ Miura opers for a given $Z$-twisted $(SL(r+1),q)$-oper.

Relative position between flags $\mathcal{L}_{\bullet}$ and
$\hat{\mathcal{L}}_{\bullet}$ can be described using the following determinant

(2.5) $\mathcal{D}_{k}(s)=e_{1}\wedge\dots\wedge{e_{r+1-k}}\wedge s(z)\wedge
Z\,s(qz)\wedge\dots\wedge Z^{k-1}s(q^{k-1}z)\,$

for $k=2,\dots r+1$ and where $s(z)$ is vector of polynomials with components
$\\{s_{i}(z)\\}_{i=1,\dots,r+1}$ in the standard basis
$e_{1},e_{2},\dots,e_{r+1}$. The functions $\mathcal{D}_{k}(s)$ have a subset
of zeroes, which coincide with those of $W_{k}(s)(z)$. The remaining zeros of
$\mathcal{D}_{k}(s)$ are given by points at which the two flags fail to be in
a generic position. We have accordingly

(2.6)
$\underset{i,j}{\det}\left[\xi_{i}^{j-1}s^{(j-1)}_{r+1-k+i}(z)\right]=\beta_{k}W_{k}\mathcal{V}_{k}\,,$

where $i,j=1,\dots,k$ and $s_{l}^{(m)}=s_{l}(q^{m}z)$, $Z={\rm
diag}(\xi_{1},\dots,\xi_{r+1})$, and

(2.7) $\mathcal{V}_{k}(z)=\prod_{a=1}^{r_{k}}(z-v_{k,a})\,,$

while $\beta_{k}$ are some constants. The boundary conditions are
$\mathcal{V}_{0}=1$ and $\mathcal{V}_{r+1}=1$ since
$\mathcal{D}_{r+1}(s)=\mathcal{W}_{r+1}(s)$.

The remarkable property of polynomials (2.7) is that they solve the
$QQ$-system.

######  Theorem 2.5 ([KSZ]).

Polynomials $\\{\mathcal{V}_{k}(z)\\}_{k=1,\dots,r}$ provide the solution to
the $QQ$-system

(2.8)
$\xi_{i+1}Q^{+}_{i}(qz)Q^{-}_{i}(z)-\xi_{i}Q^{+}_{i}(z)Q^{-}_{i}(qz)=(\xi_{i+1}-\xi_{i})\Lambda_{i}(z)Q^{+}_{i-1}(qz)Q^{+}_{i+1}(z)\,,$

so that $Q^{+}_{j}(z)=\mathcal{V}_{j}(z)$. The polynomials
$Q^{+}_{j},Q^{-}_{j}$ for $j=1,\dots,r$ can be presented using minors

(2.9)
$Q^{+}_{j}(z)=\frac{1}{F_{i}(z)}\frac{\text{det}\Big{(}M_{1,\ldots,j}\Big{)}}{\text{det}\Big{(}V_{1,\ldots,j}\Big{)}}\,,\qquad
Q^{-}_{j}(z)=\frac{1}{F_{i}(z)}\frac{\text{det}\Big{(}M_{1,\ldots,j-1,j+1}\Big{)}}{\text{det}\Big{(}V_{1,\ldots,j-1,j+1}\Big{)}}\,,$

where $F_{i}(z)=W_{r-i}(q^{i-r}z)$,

(2.10)
$M_{i_{1},\ldots,i_{j}}=\begin{bmatrix}\,s_{i_{1}}&\xi_{i_{1}}s_{i_{1}}^{(1)}&\cdots&\xi_{i_{1}}^{j-1}s_{i_{1}}^{(j-1)}\\\
\vdots&\vdots&\ddots&\vdots\\\
s_{i_{j}}&\xi_{i_{j}}s_{i_{j}}^{(1)}&\cdots&\xi_{i_{j}}^{j-1}s_{i_{j}}^{(j-1)}\,\end{bmatrix}\,,\qquad
V_{i_{1},\ldots,i_{j}}=\begin{bmatrix}\,1&q\xi_{i_{1}}&\cdots&q^{j-1}\xi_{i_{1}}^{j-1}\\\
\vdots&\vdots&\ddots&\vdots\\\
1&q\xi_{i_{j}}&\cdots&q^{j-1}\xi_{i_{j}}^{j-1}\,\end{bmatrix}\,,$

are the quantum Wronskian and the Vandermonde matrix respectively.

We can now combine two theorems from [Frenkel:2020] and [Koroteev:2020mxs].

######  Theorem 2.6.

The following three spaces are pairwise isomorphic:

i) Space of solutions of the nondegenerate $SL(r+1)$ $QQ$-system (2.8)

ii) Space of solutions of the Bethe Ansatz equations for $\mathfrak{sl}(n+1)$
XXZ spin chain:

(2.11)
$\frac{Q^{+}_{i}(qs_{i,k})}{Q^{+}_{i}(q^{-1}s_{i,k})}\frac{\xi_{i}}{\xi_{i+1}}=-\frac{\Lambda_{i}(s_{i,k})Q^{+}_{i+1}(qs_{i,k})Q^{+}_{i-1}(s_{i,k})}{\Lambda_{i}(q^{-1}s_{i,k})Q^{+}_{i+1}(s_{i,k})Q^{+}_{i-1}(q^{-1}s_{i,k})},$

where $i=1,\ldots,r$ and $k=1,\ldots,r+1$.

iii) Space of $Z$-twisted Miura $(SL(r+1),q)$-opers with regular singularities
${\rm Fun}\big{(}q{\rm Op}_{Z}^{\Lambda}\big{)}$.

In i) above the nondegeneracy condition implies that the roots of polynomials
$Q_{i}^{+}$ are pairwise q-distinct.

###  2.1. The tRS model and $(SL(N),q)$-opers

In order to find connection with the many-body systems we need to consider a
special type of $q$-oper for which the polynomials
$\\{s_{i}(z)\\}_{i=1,\dots,r+1}$ describing the line bundle $\mathcal{L}_{r}$
are all of degree one [Koroteev:2022vg]. Provided that $Z$ is regular
semisimple and deg $\mathcal{D}_{k}$=k, we have
$\mathcal{D}_{r+1}(z)=\Lambda(z)$. Thus we have from Theorem 2.6

(2.12) $\displaystyle{\rm Fun}\big{(}q{\rm
Op}_{Z}^{\Lambda}\big{)}=\frac{\mathbb{C}(q,\xi_{i},p_{i},a_{i})}{(\mathcal{D}_{r+1}(z)=\Lambda(z))}=\frac{\mathbb{C}(q,\xi_{i},s_{k,l},a_{i})}{\rm
Bethe},$

where Bethe stands for the relations (2.11), specialized to $q{\rm
Op}_{Z}^{\Lambda}$.

In order to see the connection with the Ruijsenaars model assume
$s_{i}(z)=z-p_{i}$ for some complex $p_{i}$. The latter will appear
momentarily as tRS momenta. Because of that

(2.13) $M_{1,\ldots,r+1}(z)=V_{1,\ldots,r+1}\cdot z+M_{1,\ldots,r+1}(0)\,.$

Now, using Theorem 2.5 for $j=r+1$ we get

(2.14)
$P(z)=\frac{\text{det}\Big{(}M_{1,\ldots,r+1}\Big{)}}{\text{det}\Big{(}V_{1,\ldots,r+1}\Big{)}}\,,$

or, after inverting the Vandermonde matrix $V_{1,\ldots,r+1}$, the formula
reads

(2.15) $P(z)=\text{det}\left(z-T\right)\,,$

where

(2.16) $T=-M_{1,\ldots,r+1}(0)\cdot\Big{(}V_{1,\ldots,r+1}\Big{)}^{-1}\,.$

As it was shown in [KSZ] the latter is the Lax matrix for the tRS model

(2.17) $\displaystyle T_{ij}=\frac{\prod\limits_{m\neq
j}^{r+1}\left(q^{-1}\xi_{i}\,-\xi_{m}\,\right)}{\prod\limits_{l\neq
j}^{r+1}(\xi_{j}-\xi_{l})}p_{i}\,,\qquad i,j=1,\dots,r+1\,$

whose characteristic polynomial (2.15) generates integrals of motion

(2.18)
$\textrm{det}\Big{(}z-T\Big{)}=\sum_{k}H^{tRS}_{k}(q,\\{\xi_{i}\\},\\{p_{i}\\})z^{k}\,,$

such that

(2.19)
$H^{tRS}_{k}=\sum_{\begin{subarray}{c}\mathcal{I}\subset\\{1,\dots,L\\}\\\
|\mathcal{I}|=k\end{subarray}}\prod_{\begin{subarray}{c}i\in\mathcal{I}\\\
j\notin\mathcal{I}\end{subarray}}\frac{q^{-1}\,\xi_{i}-\xi_{j}}{\xi_{i}-\xi_{j}}\prod\limits_{m\in\mathcal{I}}p_{m}\,.$

Assuming $\Lambda(z)=\prod_{i=1}^{r+1}(z-a_{i})$ we have

######  Theorem 2.7 ([KSZ]).

There is an isomorphism of algebras

(2.20) ${\rm Fun}\big{(}q{\rm
Op}_{Z}^{\Lambda}\big{)}\cong\frac{\mathbb{C}(q,\xi_{i},p_{i},a_{i})}{\left(H^{tRS}_{k}-e_{k}(a_{1},\dots,a_{r+1})\right)_{k=1,\dots,r+1}},$

where $e_{k}$ are elementary symmetric functions of their variables.

###  2.2. Quantum/Classical Duality – Magnetic Frame

Our results can be summarized via quantum/classical duality which was
geometerized in [Gaiotto_2013]. Consider the tRS phase space with symplectic
form $\Omega=\sum^{r+1}_{i=1}\frac{dp_{i}}{p_{i}}\wedge\frac{dx_{i}}{x_{i}}.$
The tRS Hamiltonians commute with each other with respect to the Poisson
bracket induced by $\Omega$.

######  Corollary 2.8.

We have the following isomorphisms:

(2.21) $\displaystyle\frac{\mathbb{C}(q,\xi_{i},s_{k,l},a_{i})}{\rm
Bethe}\cong{\rm Fun}\big{(}q{\rm
Op}_{Z}^{\Lambda}\big{)}\cong\frac{\mathbb{C}(q,\xi_{i},p_{i},a_{i})}{\left(\\{H^{tRS}_{k}-e_{k}(\\{a_{i}\\})\\}_{k=1,\dots,r+1}\right)}\,.$

The latter space is isomorphic to the space of functions on the intersection
of $\mathscr{L}_{1}\cap\mathscr{L}_{2}$ of two Lagrangian subvarieties with
respect to $\Omega$

(2.22)
$\displaystyle\mathscr{L}_{1}:\,\,\\{x_{i}=\xi_{i}\\},\quad\mathscr{L}_{2}:\,\,\\{H^{tRS}_{i}=e_{i}(\\{a_{i}\\})\\}\,,\quad
i=1,\dots,r+1\,.$

The above theorem establishes the quantum/classical duality in the magnetic
frame according to the terminology of [Koroteev:2023aa] since the tRS
coordinates $x_{i}$ are represented in terms of twist parameters $\xi_{i}$ of
the $q$-oper.

###  2.3. Quantum/Classical Duality – Electric Frame

Remarkably there is another way to see integrability in the oper setup – the
quantum/classical duality in the electric frame in which the tRS coordinates
are roots of polynomial $\Lambda(z)$ (see Section 4 of [Koroteev:2023aa]). The
corresponding tRS integrals of motion read

(2.23)
$H^{!,tRS}_{k}:=\sum_{\begin{subarray}{c}\mathcal{I}\subset\\{1,\dots,L\\}\\\
|\mathcal{I}|=k\end{subarray}}\prod_{\begin{subarray}{c}i\in\mathcal{I}\\\
j\notin\mathcal{I}\end{subarray}}\frac{q\,a_{i}-a_{j}}{a_{i}-a_{j}}\prod\limits_{m\in\mathcal{I}}p_{m}=e_{k}(\xi_{i})\,,\quad
q=1,\dots r+1\,.$

######  Theorem 2.9 ([Koroteev:2023aa]).

There is an isomorphism of algebras:

(2.24) $\displaystyle{\rm Fun}\big{(}q{\rm
Op}_{Z}^{\Lambda}\big{)}\cong\frac{\mathbb{C}(\\{\xi_{i}\\},\\{a_{i}\\},\hbar)(\\{p_{i}\\})}{\left((\ref{eq:tRSEqnsElectric})\right)}\,.$

The latter space is isomorphic to the space of functions on the intersection
of $\mathscr{L}_{1}\cap\mathscr{L}_{2}$ of two Lagrangian subvarieties with
respect to $\Omega$

(2.25)
$\displaystyle\mathscr{L}_{1}:\,\,\\{x_{i}=a_{i}\\},\quad\mathscr{L}_{2}:\,\,\\{H^{!\,tRS}_{i}=e_{i}(\\{\xi_{i}\\})\\}\,,\quad
i=1,\dots,r+1\,,$

where $H^{!\,tRS}_{i}$ are the left hand sides of (2.23).

###  2.4. 3d Mirror Symmetry

The electric and magnetic frames in the above description are related by the
3d mirror symmetry upon which the twist and singularity parameters
$\\{a_{i}\\}$ and $\\{\xi_{i}\\}$ are interchanged as well as $q$ is replaced
with $q^{-1}$.

######  Proposition 2.10 ([Koroteev:2022vg]).

The following algebras are isomorphic

(2.26)
$\displaystyle\frac{\mathbb{C}(q,\xi_{i},p_{i},a_{i})}{(H^{tRS}_{k}-e_{k}(a_{1},\dots,a_{r+1}))_{k=1,\dots,r+1}}\cong\frac{\mathbb{C}(q^{-1},a_{i},p^{\prime}_{i},\xi_{i})}{(H^{!,tRS}_{k}-e_{k}(\xi_{1},\dots,\xi_{r+1}))_{k=1,\dots,r+1}}$

######  Theorem 2.11.

The space of canonical trigonometrically-twisted q-opers on $\mathbb{P}^{1}$
is self-dual, namely:

(2.27) $\displaystyle{\rm Fun}\big{(}q{\rm
Op}_{Z^{\prime}}^{\Lambda^{\prime}}\big{)}\cong{\rm Fun}\big{(}q{\rm
Op}_{Z}^{\Lambda})\,.$

where $Z^{\prime}=\text{diag}(a_{1},\dots,a_{r+1})$ and
$\Lambda^{\prime}(z)=\prod_{i=1}^{r+1}(z-\xi_{i})$.

In the integrability world 3d mirror symmetry has been known as bispectral
duality (see [Koroteev:2023aa] for most recent review and references). In the
diamond Fig. 2 3d mirror symmetry is manifested by reflection about the
vertical line passing through diamonds 1, 5, and 9. Thus tCM is dual to rRS,
XXX is dual to tGaudin, and tRS, XXZ, rCM, and rGaudin are self-dual
[Feher:2009uw, Mukhin:2009uh, Feher:2010va, Feher:2012wc, Mukhin:wg]. Some
models in the diamond are defined as dual to known models, such as dual eCM
and dual eRS. The latter two can be also obtained via degenerate limits from
DELL which we shall address later.

###  2.5. Limits and Degenerations

The bottom four ovals (#5,7,8, and 9) of the diamond have been studied in
[Koroteev:2022vg]. In particular, the author together with A. Zeitlin studied
spaces of $(G,\epsilon)$-opers, trigonometrically twisted $G$-opers
(t$G$-opers), and rationally twisted $G$-opers (r$G$-opers). For the
$(G,\epsilon)$-opers, one pics $X=\mathbb{P}^{1}$ in the Definition 2.1 as
well as replaces $M_{q}$ with additive shift $M_{\epsilon}:z\mapsto
z+\epsilon$. As for the t$G$-opers and r$G$-opers one considers a vector
bundle $E$ together with a complete flag of subbundles $\mathcal{L}_{\bullet}$
over $\mathbb{P}_{1}$ with connection $\nabla:E\longrightarrow E\otimes K$ and
demands that the maps
$\mathcal{L}_{i}/\mathcal{L}_{i+1}\longrightarrow\mathcal{L}_{i-1}/\mathcal{L}_{i}\otimes
K$ are isomorphisms away, perhaps, from a set of isolated singularities. Then
the t$G$-oper connection can be brought to the form $\partial_{z}+Z$ meanwhile
the r$G$-oper connection can be set to $\partial_{z}+\frac{Z}{z}$ by a gauge
transformation $v(z)\in G(z)$ or $v(z)\in B_{+}(z)$ for Miura opers
[Brinson:2021aa].

Each type of opers leads to its own set of quantum/classical dual models in
the magnetic frame:

rRS/tGaudin tCM/XXX rCM/rGaudin

In the electric frame, the dualities go as follows

rRS/XXX tCM/rRS rCM/rGaudin

As was pointed out earlier the tRS model and the XXZ chain are self-dual under
3d mirror symmetry (bispectral duality).

The tRS, rRS, tCM, and rCM models can be obtained from the Calogero-Moser
space description by taking certain limits, see [Koroteev:2022vg] for review.

###  2.6. Elliptic Opers

The ongoing work [GKSZ:2024Toappear] is developing an oper description for
ovals #2 and #4 of the diamond. As for the e$(G,q)$-opers, we merely replace
the base of the oper bundle with the elliptic curve
$\mathcal{E}_{w}=\mathbb{C}^{\times}/w^{\mathbb{Z}}$ and provide a similar
definition to 2.1. In local trivialization, the components of sections of the
elliptic q-oper bundle are now theta functions $\theta_{1}(z/u_{i}|w)$. The
following statement holds

######  Theorem 2.12.

There is a one-to-one correspondence between the set of nondegenerate
solutions of the $sl_{n}$ XYZ Bethe equations and the set of elliptic
nondegenerate $Z$-twisted $(SL(n),q)$-opers with appropriate regular
singularities and weights.

Additionally, for special type of elliptic q-opers in which all components of
the section contain one theta-function, there exists and alternative
description in terms of the dual eRS model (the quantum/classical duality in
the magnetic frame)

######  Proposition 2.13.

(2.28) $\displaystyle{\rm Fun}\big{(}ell.q{\rm
Op}_{Z}^{\Lambda}\big{)}\cong\frac{\mathbb{C}((q,w,\xi_{i},p_{i},a_{i}))}{\left(\\{H^{dual\,eRS}_{k}\\}_{k=1,\dots,r+1}\right)}\,.$

The Hamiltonians will be specified below.

In the electric frame, it is expected that the eRS model will be dual to the
XYZ spin chain. To the best of our knowledge, this has not been shown yet.

##  3\. Enumerative Geometry and Quantum Many-Body Systems

The next step in our construction is to quantize each classical many-body
system from the diamond and find connections with enumerative geometry. We
begin with the quantum equivariant K-theory of quiver varieties and the tRS
system (oval #5). Below we shall introduce quantum K-theory vertex functions
and formulate difference equations that they satisfy. More details are
provided in [Koroteev:2018azn, Koroteev:2021aa].

For a given quiver variety $X$, the quantum multiplication in its localized
$T$-equivariant K-theory $K_{T}(X)[[z]]$ can be defined by studying moduli
spaces of quasimaps QM [Ciocan-Fontanine:2011tg] in all positive degrees from
a base curve into the variety.

$A\circledast B=A\otimes
B+\sum\limits_{\textbf{d}=1}\,z^{\textbf{d}}A\circledast_{\textbf{d}}B\,.$

This product enables us to construct a unital quantum K-theory ring. The
quasimap moduli spaces from $\mathcal{C}$ into $X$ have a natural action of
the maximal torus, lifting its action from $X$. When there are at most two
special points and the base curve $\mathcal{C}$ is ${\mathbb{P}}^{1}$, we
extend $T$ by additional torus $\mathbb{C}^{\times}_{q}$, which scales
${\mathbb{P}}^{1}$ such that the tangent space $T_{0}{\mathbb{P}}^{1}$ has
character denoted by $q$. We will include this action in the full torus by
$\mathsf{T}=T\times\mathbb{C}^{\times}_{q}$. We assume that the two fixed
points of $\mathbb{C}^{\times}_{q}$ are $0$ and $\infty$. As explained in
[Okounkov:2015aa], one can construct various enumerative invariants of $X$
using virtual structure sheaves $\mathscr{O}_{\text{vir}}$ for
$\textsf{QM}^{\textbf{d}}$. Using the above two marked points, one can define
a vertex function (analog of Givental’s J-function).

######  Definition 3.1.

The element

(3.1)
$V^{(\tau)}(z)=\sum\limits_{\textbf{d}=0}^{\infty}z^{\textbf{d}}{\rm{ev}}_{p_{2},*}\Big{(}\textsf{QM}^{\textbf{d}}_{{\rm{nonsing}}\,p_{2}},\widehat{{\mathscr{O}}}_{{\rm{vir}}}\tau(\left.\mathscr{V}_{i}\right|_{p_{1}})\Big{)}\in
K_{\mathsf{T}_{q}}(X)_{loc}[[z]]$

is called bare vertex with descendent $\tau\in K_{T}(X)$.

The vertex function is an element of localized quantum K-theory of $X$. Using
the localization theorem one can compute the above vertex functions as
equivariant integrals over the moduli space of quasimaps [Pushkar:2016qvw,
Koroteev:2021tl]. The ring structure of the K-theory arises in the saddle
point limit of the vertex function in the equivariant parameter $q$
corresponding to the $\mathbb{C}^{\times}$-action on the base curve.

######  Proposition 3.2.

For $X$ being a cotangent bundle to complete flag variety of type A the vertex
function coefficient for the identity class $\tau=1$ reads

(3.2) $V^{(1)}_{\textbf{p}}(z)=\sum\limits_{d_{i,j}\in
C}\prod_{i=1}^{n-1}\left(\frac{q}{\hbar}\frac{\xi_{i}}{\xi_{i+1}}\right)^{d_{i}}\prod\limits_{j,k=1}^{i}\frac{\left(q\frac{x_{i,j}}{x_{i,k}},q\right)_{d_{i,j}-d_{i,k}}}{\left(\hbar\frac{x_{i,j}}{x_{i,k}},q\right)_{d_{i,j}-d_{i,k}}}\cdot\prod_{j=1}^{i}\prod_{k=1}^{i+1}\frac{\left(\hbar\frac{x_{i+1,k}}{x_{i,j}},q\right)_{d_{i,j}-d_{i+1,k}}}{\left(q\frac{x_{i+1,k}}{x_{i,j}},q\right)_{d_{i,j}-d_{i+1,k}}}\,,$

where we define $x_{n,k}=a_{k}$ – equivariant parameters for the final node of
the quiver, and $z_{i}=\frac{\xi_{i}}{\xi_{i+1}}$ are the quantum (Kähler)
parameters.

In [Koroteev:2018azn, Koroteev:2021aa] it was shown that normalized quantum
K-theoretic vertex functions for cotangent bundles to flag varieties of type A
are the eigenfunctions of the tRS Hamiltonians.

In the magnetic frame

######  Theorem 3.3 ([Koroteev:2021aa]).

Consider $H_{r}^{tRS}(\xi)$ from (2.19) such that quantum parameters and
momenta satisfy the quantum torus relation
$p_{i}\xi_{j}=q^{\delta_{ij}}\xi_{j}p_{i}$. Then $\mathsf{V}_{\textbf{p}}$ are
eigenfunctions of $H_{r}^{tRS}(\xi)$ up to a simple multiplicative prefactor

(3.3)
$H_{r}^{tRS}(\xi)\mathsf{V}^{(1)}_{\textbf{p}}=e_{r}(\mathbf{a})\mathsf{V}^{(1)}_{\textbf{p}}\,,\qquad
r=1,\dots,n\,,$

where $e_{r}$ is elementary symmetric polynomial of degree $r$ of
$a_{1},\dots,a_{n}$ .

Similarly, in the electric frame

######  Theorem 3.4 ([Koroteev:2018azn]).

Consider $H_{r}^{!\,tRS}(a)$ from (2.23) such that
$p_{i}a_{j}=q^{\delta_{ij}}a_{j}p_{i}$. Then

(3.4)
$H_{r}^{!\,tRS}(a)\mathsf{V}^{(1)}_{\textbf{p}}=e_{r}(\xi)\mathsf{V}^{(1)}_{\textbf{p}}\,,\qquad
r=1,\dots,n\,,$

where $e_{r}$ is elementary symmetric polynomial of degree $r$ of
$\xi_{1},\dots,\xi_{n}$ .

The proof applied ideas of [2013arXiv1305.4759H, 2012arXiv1206.3787H,
2012JMP....53l3512H] to complex setup.

In other words, tRS wave functions coincide with K-theory vertex functions and
the information about the enumerative invariants can be obtained directly from
integrability. The semiclassical limit $q\longrightarrow 1$ will lead to
equations of motion of the tRS model which will serve as relations in the
ideal of quantum K-theory ring for the quiver variety in question.

######  Theorem 3.5.

For a given quiver variety $X$ of type $A_{r}$ the eigenvalues of the quantum
multiplication operator by quantum class $\widehat{\tau}(z)$ are given by the
character $\tau(s_{I})$ of the corresponding virtual representation of the
global symmetry evaluated on the solutions of Bethe ansatz equations for the
$\mathfrak{gl}_{r}$ XXZ spin chain.

The operator of quantum multiplication by the generating function of
tautological bundles in quantum K-theory of $X$ coincides with Baxter
$Q$-operator for the XXZ chain. One identifies each energy sector of the
Hilbert space of the spin chain with the space of equivariant localized
K-theory of $X$. The Hilbert space of the spin chain can be regarded as a
module of a quantum affine algebra for $\mathfrak{gl}_{r}$.

######  Theorem 3.6.

Quantum equivariant K-theory of $T^{*}\mathbb{F}l_{n}$ reads

(3.5) $QK_{T}(T^{*}\mathbb{F}l_{n})=\frac{\mathbb{C}[\zeta_{1}^{\pm
1},\dots,\zeta_{n}^{\pm 1};a_{1}^{\pm 1},\dots,a_{n}^{\pm 1},\hbar^{\pm
1};p_{1}^{\pm 1},\dots,p_{n}^{\pm
1}]}{(H^{tRS}_{r}(\zeta_{i},p_{i},\hbar)-e_{r}(a_{1},\dots,a_{n}))}\,.$

Note that geometrically tRS momenta $p_{i}$ are multiplication operators by
quantum tautological classes
$\hbar^{j-\frac{1}{2}}\widehat{\Lambda^{j}V_{j}}(z)\circledast\widehat{\Lambda^{j-1}{V^{*}}_{j-1}}(z)$
of $T^{*}\mathbb{F}l_{n}$.

###  3.1. Truncations to Macdonald Polynomials

Macdonald polynomials $P_{\lambda}(\xi;q,\hbar)$ can be obtained from infinite
series (3.2) by specifying equivariant parameters $a_{1},\dots,a_{n}$ on a
lattice spanned by $q$ and $\hbar$.

######  Proposition 3.7 ([Koroteev:2021aa]).

Let $\lambda$ be a partition of $k$ elements of length $n$ and
$\lambda_{1}\geq\dots\geq\lambda_{n}$. Let

(3.6)
$\frac{a_{i+1}}{a_{i}}=q^{\ell_{i}}\hbar\,,\quad\ell_{i}=\lambda_{i+1}-\lambda_{i}\,,\quad
i=1,\dots,n-1\,.$

Then there exists a fixed point q of the maximal torus for which the
coefficient vertex function (3.2) truncates to

(3.7) $\mathsf{V}^{(1)}_{\textbf{q}}=P_{\lambda}(\xi;q,\hbar)\,.$

###  3.2. The DELL System

Together with Shakirov, we have introduced [Koroteev:2019gqi] the quantum
double elliptic or double periodic (DELL) integrable system whose Hamiltonians
enjoy double periodicity both in momenta and coordinates. This model sits in
oval # 1 and is at the top of the hierarchy of integrable many-body systems of
Calogero-Ruijsenaars type. Shortly after, the Hamiltonians were slightly
modified in [Grekov:2021zqq, Grekov:2020jpk] to be better fit the dualities
which are addressed here, in particular, a Lax operator has been produced.

The quantum DELL model for $N-1$ degrees of freedom (or $N$ degrees of freedom
with the removed center of mass) is described by the following Hamiltonians

(3.8) $\widehat{H}_{a}=\mathcal{O}_{0}^{-1}\mathcal{O}_{a}\,,\qquad
a=1,\ldots,N-1\,,$

where operators $\mathcal{O}_{0},\mathcal{O}_{1},\dots,\mathcal{O}_{N-1}$ are
Fourier modes of the following current

(3.9) $\mathcal{O}(z)\ =\ \sum\limits_{n\in{\mathbb{Z}}}\ \mathcal{O}_{n}\
z^{n}\ =\ \sum\limits_{n_{1},\ldots,n_{N}=-\infty}^{\infty}\ (-z)^{\sum
n_{i}}\ w^{\sum\frac{n_{i}(n_{i}-1)}{2}}\
\prod\limits_{i<j}\theta_{p}\big{(}\hbar^{n_{i}-n_{j}}x_{i}/x_{j}\big{)}\
p_{1}^{n_{1}}\ldots p_{N}^{n_{N}}\,.$

Here $\theta_{p}$ are certain modifications of $\theta$-functions while the
canonically conjugate position and momentum operators obeying canonical
q-commutation relation $x_{i}p_{j}=q^{\delta_{ij}}p_{j}x_{j}$. As one can see,
the DELL Hamiltonians are highly nonlocal as they involve formal infinite
series of functions of shift operators. As of this writing, it remains to be
proven that DELL Hamiltonians (3.8) commute with each other; the commutativity
has been checked up to several orders in expansion in elliptic parameters $p$
and $w$.

We have also found the formal spectrum of DELL Hamiltonians which can be
formulated using localization. The utilizes the elliptic version of affine
Laumon space [Negut_2009]

######  Conjecture 3.8.

Let $\mathcal{Z}^{DELL}_{\text{inst}}(p,x_{1},\dots,x_{N})$ be an equivariant
elliptic genus of the affine Laumon space. Then there exists a function
$\lambda(z,\textbf{a},w,p)$ such that

(3.10)
$\mathcal{O}(z)\mathcal{Z}^{DELL}_{\text{inst}}(p,x_{1},\dots,x_{N})=\lambda(z,\textbf{a},w,p)\
\mathcal{O}_{0}\mathcal{Z}^{DELL}_{\text{inst}}(p,x_{1},\dots,x_{N})\,.$

In particular, by expanding currents $\widehat{\mathcal{O}}(z)$ as in (3.9)
and $\lambda(z,\textbf{a},w,p)=\sum_{n}\lambda_{n}(\textbf{a},w,p)z^{n}$ in
$z$ we obtain similar relations for each operator
$\widehat{\mathcal{O}}_{n}(z)$, or, using (3.8) we obtain the eigenvalue
problem for DELL Hamiltonians

(3.11)
$\mathcal{H}_{n}\mathcal{Z}^{DELL}_{\text{inst}}(p,x_{1},\dots,x_{N})=\lambda_{n}(\textbf{a},w,p)\mathcal{Z}^{DELL}_{\text{inst}}(p,x_{1},\dots,x_{N})\,.$

At the moment there exists a plethora of computer computational evidence that
the above conjectures hold.

The classical DELL model was introduced in [Hollowood:2003cv, Braden:2001yc,
Braden:2003gv] using six dimensional gauge theories with eight supercharges
compactified on a torus. It was shown how to construct the spectral curve, or,
equivalently, the Seiberg-Witten curve of the corresponding 6d theory, from
M-theory. The two elliptic parameters of DELL, $p$ and $w$ are related to the
gauge coupling of the 6d theory and to the elliptic modulus of the
compactification torus respectively.

There has been parallel efforts to construct quantum DELL system based on
self-duality under 3d mirror symmetry. The summary of the up to date results
in that direction as well as references are given in [Mironov:2023ab].

###  3.3. The Elliptic RS Model

The eRS model from oval # 3 can be understood as either elliptic deformation
of the tRS model or as the limit $w\longrightarrow 0$ of the DELL system
above. Its Hamiltonians read

(3.12)
$\mathcal{H}^{eRS}_{r}=\sum_{\begin{subarray}{c}\mathcal{I}\subset\\{1,\dots,n\\}\\\
|\mathcal{I}|=r\end{subarray}}\prod_{\begin{subarray}{c}i\in\mathcal{I}\\\
j\notin\mathcal{I}\end{subarray}}\frac{\theta_{1}(\hbar\zeta_{i}/\zeta_{j}|\mathfrak{p})}{\theta_{1}(\hbar\zeta_{i}/\zeta_{j}|\mathfrak{p})}\prod\limits_{i\in\mathcal{I}}p_{k}\,,$

where $\mathfrak{p}\in\mathbb{C}^{\times}$ is the parameter of elliptic
deformation.

The current progress in finding the spectrum of the eRS model can summarized
as the following conjecture for the eigenfunctions and theorem of the
eigenvalues

######  Conjecture 3.9 ([Bullimore:2015fr]).

The eigenfunctions of the elliptic Ruijsenaars-Schneider Hamiltonians

(3.13)
$\mathcal{H}^{e\text{RS}}_{k}\mathcal{Z}^{{\text{RS}}}(\textbf{a},\textbf{x})=\lambda_{k}(\textbf{a})\mathcal{Z}^{{\text{RS}}}(\textbf{a},\textbf{x})\,,\qquad
k=1,\dots,N-1\,.$

are given by the K-theoretic holomorphic equivariant Euler characteristic of
the affine Laumon space $\mathcal{L}^{\text{aff}}_{\textbf{d}}$

(3.14)
$\mathcal{Z}^{{\text{RS}}}=\sum_{\textbf{d}}\mathfrak{q}^{\textbf{d}}\int\limits_{\mathcal{L}_{\textbf{d}}}1\,,$

where $\mathfrak{q}=(\mathfrak{q}_{1},\dots,\mathfrak{q}_{n})$ is a string of
$\mathbb{C}^{\times}$-valued coordinates on the maximal torus of
$\mathcal{L}^{\text{aff}}_{\textbf{d}}$.

######  Theorem 3.10 ([Gorsky:2022aa]).

The eigenvalues $\lambda_{k}(\textbf{a})$ are equivariant Chern characters of
bundles $\Lambda^{r}\mathscr{W}$, where $\mathscr{W}$ is the trivial bundle of
the corresponding ADHM space

(3.15)
$\lambda_{k}(\textbf{a})=\prod\limits_{n=0}^{k-1}\frac{\theta(\hbar^{N-n})}{\theta(\hbar^{n+1})}\cdot\frac{\mathcal{Z}^{{\text{RS}}}(\textbf{a},\hbar^{\rho}q^{\omega_{k}})}{\mathcal{Z}^{\text{RS}}(\textbf{a},\hbar^{\rho})}\,,\qquad
k=1,\dots,N-1\,,$

where $\mathcal{Z}^{{\text{RS}}}$ is defined in the previous theorem,
$\omega_{k}$ is the $k$-th fundamental weight of representation of $SU(N)$,
and $\rho$ is the $SU(N)$ Weyl vector.

###  3.4. The Calogero-Moser Systems

The Calogero-Moser systems have been shown to be related to quantum cohomology
of quiver varieties. It was also shown in [Negut_2009] that Euler
characteristic of the affine Laumon space is the eigenfunction of the elliptic
Calogero-Moser system following conjectures of [Braverman:aa].

More recently, the eCM eigenvalue problem was approached [Nekrasov:2015wsu,
Nekrasov:2017gzb] from the vantage point of $qq$-characters and the author
have found an implicit proof.

Representation-theoretically, the wavefunctions of the many-body models on the
yellow diagonal are described by Euler characteristic of the Laumon space
whereas those on the green diagonal are described by their affine analogs.

###  3.5. Spaces of Opers for Elliptic Many-Body Systems

A careful reader may find an obvious drawback in the diamond in Fig. 2. All
ovals on the yellow and white diagonals have spaces of opers assigned to the
respected integrable models. However, there are no known oper-like structures
for the elliptic (affine) models of the green diagonal. Presumably one needs
an additional geometric structure which reflects the elliptic/affine
deformation, i.e. another multiplicative/additive action on the base of the
oper bundle with respect to which one needs to work equivariantly. This
represents itself an intriguing problem. Perhaps certain ideas from the
original work of Cherednik [elldifdaha, Cherednik1995] could be geometrized.

As far as space of opers and dual integrable systems are concerned, another
interesting direction is to extend the setup to superalgebras. Some progress
in this direction has been made in [Zeitlin:2013aa, Huang:2018aa,
Zeitlin:2023ab].

###  3.6. DAHAs

The double affine Hecke algebra (DAHA) for Lie algebra
$\mathfrak{g}=\text{Lie}(G)$ can be described as follows. Consider an orbifold
fundamental group of the once-punctured torus $\pi_{1}(T^{2}\backslash{pt})/W$
where $W$ is the Weyl group of $G$. It is generated by A and B-cycle
monodromies $X_{i}$ and $Y_{i}$ as well as by braid generators $T_{i}$ modulo
a set of relations (see [Cherednik-book] and also [GKNS, Matsuo:2023aa]). Then
$\pi_{1}(T^{2}\backslash{pt})/W$ admits a central extension with parameter $q$
and is referred to as elliptic braid group. Finally, DAHA$(X,Y)_{q,\hbar}$ for
$\mathfrak{g}$, which in our classification lives in oval #5, is obtained by
taking the quotient of the latter by Hecke relations
$(T_{i}-\hbar)(T_{i}-\hbar^{-1})=0$.

The appearance of $T^{2}\backslash{pt}$ in the description of DAHA is not
surprising since the same Riemann surface describes the moduli space of vacua
of the $\mathcal{N}=2^{*}$ theory which gave rise to Calogero-Ruijsenaars
systems in the first place.

Next, we consider a spherical subalgebra of DAHA which can be obtained by
conjugating each element $\alpha\in$ DAHA$(X,Y)_{q,\hbar}$ with an idempotent
element $e$: $e\,\alpha\,e$. Remarkably, the spherical subalgebra can be
understood geometrically as the deformation quantization of the coordinate
ring of the moduli space of flat $G_{\mathbb{C}}$-connections on
$\mathcal{M}_{\text{flat}}(G_{\mathbb{C}},T^{2}\backslash{pt})$ with respect
to the Poisson bracket defined by complex structure $J$ (the latter space is
hyper Kähler) [Oblomkov:aa, oblomkov2004double].

The connection between DAHA and tRS model arises while studying functional and
polynomial representations of spherical DAHA. For $SL(N)$ DAHA $N-1$ of those
generators become tRS (or Macdonald) shift operators. Similar constructions
exist for trigonometric-elliptic DAHA [elldifdaha] in oval #3 and for rational
elliptic DAHA [Cherednik1995] in oval #6 where certain generators coincide
with elliptic RS and elliptic CM Hamiltonians respectively.

Spherical DAHA acts on the space of symmetric polynomials over
$\mathbb{Q}(q,\hbar)$ of $n$-variables. Let us specialize on the locus (3.6).
The q-hypergeometric functions coming from the spectrum of the tRS model (3.2)
truncate to Macdonald polynomials depending on the quantum parameters.
Macdonald (tRS) operators act on $P_{\lambda}$ by multiplication by symmetric
polynomial of parameters $s_{l}(a_{1},\dots,a_{n})$ evaluated at (3.6)
[Koroteev:2021aa]. The above vertex functions therefore can be treated as
highest weight vectors for DAHA modules over space of symmetric polynomials.
Unless we further specify the values of $q$ and $\hbar$ these modules are
infinite-dimensional and contain Macdonald polynomials parameterized by Young
tableaux $\lambda=\\{\lambda_{1},\dots,\lambda_{n}\\}$ with $n$ columns.
Proper combinations of DAHA generators acts on the states in the highest
weight module by adding to or removing boxes from tableau $\lambda$ thereby
producing a different Macdonald polynomials.

Representation theory of DAHA and its degenerations – trigonometric and
rational Cherednik algebras from ovals #7,8, and 9 – is extremely rich and has
deep connections with geometry of
$\mathcal{M}_{\text{flat}}(G_{\mathbb{C}},T^{2}\backslash{pt})$. Already when
$G=SL(2)$ the category of DAHA representations is highly sophisticated [GKNS].
The following has been concluded

######  Proposition 3.11.

There is a derived equivalence between the category of compact Lagrangian
branes on $\mathcal{M}_{\text{flat}}(G_{\mathbb{C}},T^{2}\backslash{pt})$ and
the category of finite-dimensional $SL(2)$ DAHA-modules.

This statement needs to be understood for higher rank algebras as well as for
other versions of DAHA from the diamond. Representation theory of elliptic
DAHAs is currently poorly understood. The double elliptic version of DAHA has
not yet been constructed and represents a formidable challenge.

## References