UUITP-01/21

# Inozemtsev System as Seiberg-Witten Integrable System

Philip C. Argyres  Physics Department, University of Cincinnati, PO Box
210011, Cincinnati OH 45221, US Oleg Chalykh  School of Mathematics,
University of Leeds, Leeds, LS2 9JT, UK  Yongchao Lü  Department of Physics
and Astronomy, Uppsala university, Box 516, SE-75120 Uppsala, Sweden

###### Abstract

In this work we establish that the Inozemtsev system is the Seiberg-Witten
integrable system encoding the Coulomb branch physics of 4d $\mathcal{N}=2$
$\mathrm{USp}(2N)$ gauge theory with four fundamental and (for $N\geq 2$) one
antisymmetric tensor hypermultiplets. We describe the transformation from the
spectral curves and canonical one-forms of the Inozemtsev system in the $N=1$
and $N=2$ cases to the Seiberg-Witten curves and differentials explicitly,
along with the explicit matching of the modulus of the elliptic curve of
spectral parameters to the gauge coupling of the field theory, and of the
couplings of the Inozemtsev system to the field theory mass parameters. This
result is a particular instance of a more general correspondence between
crystallographic elliptic Calogero-Moser systems with Seiberg-Witten
integrable systems, which will be explored in future work.

###### Contents

  1. 1 Introduction and summary
  2. 2 Inozemtsev system
    1. 2.1 Hamiltonian description
    2. 2.2 Lax matrix
    3. 2.3 Spectral curve
    4. 2.4 Spectral curves for $N=1$ and $N=2$
    5. 2.5 Behaviour near marked points
    6. 2.6 Modular property
  3. 3 $\mathrm{USp}(2N)$ $N_{f}=4$ superconformal field theory
    1. 3.1 Field theory properties
    2. 3.2 Seiberg-Witten curve
  4. 4 Matching spectral curve to M5 brane curve
    1. 4.1 The $N=1$ case
    2. 4.2 The $N=2$ case
  5. A Appendix
    1. A.1 Elliptic functions and identities
    2. A.2 Calculating the $N=2$ spectral curve

##  1 Introduction and summary

Since the dawn of Seiberg-Witten era [1, 2], it has been recognized [3] that
there is close connection between 4d $\mathcal{N}=2$ systems and completely
integrable Hamiltonian systems. In particular, Donagi and Witten [4] explained
that for each 4d $\mathcal{N}=2$ supersymmetric field theory there exists a
complex integrable systems encoding its Coulomb branch physics. Following [5]
we will call such a complex integrable system a Seiberg-Witten integrable
system.

There are no known systematic ways to identify the Seiberg-Witten integrable
system for a given 4d $\mathcal{N}=2$ theory. Nevertheless, there have been
two main effective approaches in this regard. In the first approach, one tries
to match known many-body or spin chain integrable systems with particular 4d
$\mathcal{N}=2$ theories. There are several notable examples along this line.
For instance, 4d $\mathcal{N}=2$ pure YM theory with simple gauge algebra
$\mathrm{G}$ corresponds [6] to the twisted affine Toda chain of type
$(\widehat{\mathrm{G}}^{(1)})^{\vee}$, where
$(\widehat{\mathrm{G}}^{(1)})^{\vee}$ is the Langlands dual of the untwisted
affine Kac-Moody algebra $\widehat{\mathrm{G}}^{(1)}$. Another example [7, 8]
is the elliptic Calogero Moser system of $A_{N-1}$ type which describes the
Seiberg-Witten solution of 4d $\mathcal{N}=2^{\ast}$ theories with gauge group
$\mathrm{SU}(N)$ or $\mathrm{U}(N)$; this type of matching has been
generalized to arbitrary simple gauge groups (with $G_{2}$ as a potential
exception) [9]. It is also proposed [10, 11] that the inhomogeneous
$\mathfrak{sl}_{2}$ XXX spin chain provides solutions to 4d $\mathcal{N}=2$
$\mathrm{SU}(N_{c})$ gauge theories with $N_{f}\leq 2N_{c}$ fundamental
hypermultiplets. See the survey [12] for these and further connections.

A second approach identifies Seiberg-Witten integrable systems for a large
class of 4d $\mathcal{N}=2$ supersymmetric field theories as Hitchin systems
on Riemann surfaces with tame/wild ramified punctures. This class of 4d
$\mathcal{N}=2$ supersymmetric field theories are known as class-S theories
[13]. A precursor to this approach is the M-theory solution to certain 4d
$\mathcal{N}=2$ quiver gauge theories engineered with D4-NS5-D6 brane systems
[14].

These two approaches — matching to known integrable systems or to Hitchin
systems — have some overlap. For instance, it is known that the elliptic
Calogero Moser system of type $A_{N-1}$ can be interpreted as the
$\mathrm{SU}(N)$ Hitchin system on a torus with a puncture [15]. However, for
a majority of Hitchin systems there are no explicitly known many-body or spin
chain integrable systems.

In this and upcoming work [16], we will follow the line of the first approach
to identify the Seiberg-Witten systems for several series of 4d
$\mathcal{N}=2$ superconformal field theories which all admit F-theory
constructions. A common feature shared by those theories is that their Coulomb
branch chiral rings are given by the rings of symmetric polynomials with
respect to certain complex reflection groups [17].111We refer the reader to
the appendix in [18] for a nice account of complex reflection groups aimed at
physicists. On general grounds all the relevant complex reflection groups also
need to satisfy various physical constraints including Dirac quantization and
electric-magnetic duality which implies the relevant complex reflection groups
must be crystallographic — which means that there exists an invariant full-
rank lattice preserved by the complex reflection group. All such
crystallographic groups have been classified [19, 20].

Generalizations of elliptic Calogero-Moser systems — known as crystallographic
elliptic Calogero-Moser systems — have been constructed for all
crystallographic complex reflection groups [21]. Our proposal is that these
are candidates for Seiberg-Witten geometries. A nice feature of these
integrable systems is that their full set of parameters matches the mass
deformations of classes of $4d$ $\mathcal{N}=2$ quantum field theories. For
instance, we identify the elliptic Calogero-Moser systems attached to the
crystallographic complex reflection groups of type $G(m,1,N)$ with $m=2,3,4,6$
as Seiberg-Witten integrable systems for $4d$ $\mathcal{N}=2$ rank $N$ $D_{4}$
and $E_{6}$, $E_{7}$, $E_{8}$ theories [22, 23, 24]. Those theories belong to
the the category of class-S theories, therefore their Seiberg-Witten
integrable systems admit Hitchin system construction [25, 26, 27].

In this paper we will focus on the $G(2,1,N)$ case, which are also known as
the Inozemtsev system [28], which corresponds to $4d$ $\mathcal{N}=2$
$\mathrm{USp}(2N)$ gauge theory with one antisymmetric and four fundamental
hypermultiplets. Since $G(2,1,N)$ is the complexification of the Weyl group
$W(B_{N})\equiv W(C_{N})$ and depends on an elliptic modulus, it is natural to
guess that it describes the Coulomb branch of a superconformal gauge theory
with $\mathrm{USp}(2N)$ or $\mathrm{Spin}(2N{+}1)$ gauge group. What is
surprising is that, on the one hand, the Inozemtsev system has no direct Lie-
algebraic interpretation, and on the other hand the Inozemtsev systems has the
right pattern of couplings to match exactly with a single class of 4d
$\mathcal{N}=2$ gauge theories, namely, the $\mathrm{USp}(2N)$ superconformal
theories with one antisymmetric and $N_{f}=4$ fundamental hypermultiplets.

Since the $\mathrm{USp}(2N)$ $N_{f}=4$ theory admits class-S description, the
Inozemtsev system should be equivalent to an $\mathrm{SU}(2N)$ Hitchin system
on the orbicurve $T^{2}/\mathbb{Z}_{2}$, and we offer such an interpretation.
Furthermore, the Seiberg-Witten solutions for the particular
$\mathrm{USp}(2N)$ gauge theories are given in explicit form via an M5 brane
construction in [29]. The equivalence of the Seiberg-Witten solutions with the
Inozemtsev system is not at all obvious. In this work we check their
equivalence for the rank $N=1,2$ cases. We find that we need to modify some
choices made in [29] in the M5 brane construction of the Seiberg-Witten curve
in order to achieve an algebraically transparent matching to the integrable
system.

Our recognition of the Inozemtsev system as a Seiberg-Witten integrable system
has some independent interest. Specifically, one may be able to utilize the
gauge theory description to extract exactly solvable observables by various
powerful techniques including semi-classical methods, supersymmetric
localization, the gauge-Bethe correspondence, and the AGT correspondence, and
relate them to the Inozemtsev system.

This paper is organized as follows. In section 2 we discuss various aspects of
Inozemtsev system, and introduce the Lax representation following [30, 31].
Among other things, we give an interpretation of the Inozemtsev system as a
Hitchin system on the four-punctured sphere. In section 3, after recalling
some general properties of the series of $\mathrm{USp}(2N)$ $N_{f}=4$
theories, we describe the realization of their Coulomb branch physics in terms
of M5 brane curves. In section 4 we describe the transformation from the
spectral curves and canonical one-form of the Inozemtsev system in the $N=1$
and $N=2$ cases to the Seiberg-Witten curves and differentials explicitly,
along with the variable and parameter matching. We include an appendix which
summarizes some relevant elliptic functions and identities and outlines the
derivation of the $N=2$ spectral curve of the Inozemtsev system.

##  2 Inozemtsev system

###  2.1 Hamiltonian description

The Inozemtsev system, also known as the Calogero–Moser–Sutherland system of
$BC_{N}$-type, is described by the Hamiltonian [28]:

$h_{2}=\sum_{j=1}^{N}(p_{j}^{2}-u(q_{j}))-2g^{2}\sum_{j<k}^{N}\left(\wp(q_{j}-q_{k})+\wp(q_{j}+q_{k})\right)\,,$
(2.1)

where $\wp(q)$ is the Weierstrass $\wp$-function with periods $1,{\tau}$ and

$u(q)=\sum_{r=0}^{3}g_{r}^{2}\wp(q+{\omega}_{r})\,,\qquad({\omega}_{0},{\omega}_{1},{\omega}_{2},{\omega}_{3})=\left(0,\frac{1}{2},\frac{1+{\tau}}{2},\frac{{\tau}}{2}\right)\,.$
(2.2)

Here $(p_{i},q_{i})$, $i=1,\dots,N$ represent the momenta and positions of $N$
interacting particles on the line, subject to an external field with potential
$-u(q)$. Note that we have four coupling constants $g_{0,1,2,3}$ in the $N=1$
case and one additional coupling constant $g$ in the $N\geq 2$ cases. It is
customary to assume, in the repulsive regime, that the couplings $g^{2}$ and
$g_{r}^{2}$ are real negative. For our purposes, however, this is not
important, as we consider this system on the complexified phase space
$\mathbb{C}^{2N}$ with the standard (holomorphic) symplectic structure. As
such, it has the underlying symmetry associated with the complex
crystallographic group generated by the translations $q_{j}\mapsto q_{j}+1$,
$q_{j}\mapsto q_{j}+{\tau}$ together with the arbitrary permutations and sign
changes of $q_{j}$. This corresponds to the group $[G(2,1,N)]^{\tau}_{1}$ in
the classification [19].

The Inozemtsev system is known to be completely integrable in Liouville’s
sense, which means that it admits $N$ independent Poisson-commuting
Hamiltonians $h_{2},h_{4},\dots,h_{2N}$. The higher Hamiltonians are of the
form $h_{4}=\sum_{i<j}p_{i}^{2}p_{j}^{2}+\ldots$,
$h_{6}=\sum_{i<j<k}p_{i}^{2}p_{j}^{2}p_{k}^{2}+\ldots$, etc., up to lower
degree terms. Explicit expressions for $h_{2k}$ are available for the quantum
case [32] from which the classical Hamiltonians are easily obtained. For
instance, in the $N=2$ case the quartic Hamiltonian can be taken as

$\displaystyle h_{4}$
$\displaystyle=\Big{(}p_{1}p_{2}+g^{2}\wp(q_{1}-q_{2})-g^{2}\wp(q_{1}+q_{2})\Big{)}^{2}$
$\displaystyle\quad\mbox{}-u(q_{1})p_{2}^{2}-u(q_{2})p_{1}^{2}+u(q_{1})u(q_{2})$
$\displaystyle\quad\mbox{}+\left(u(q_{1})+u(q_{2})\right)\left(g^{2}\wp(q_{1}-q_{2})+g^{2}\wp(q_{1}+q_{2})\right)$
$\displaystyle\quad\mbox{}-2g^{2}\sum^{3}_{i=0}g^{2}_{i}\wp(q_{1}+\omega_{i})\wp(q_{2}+\omega_{i})\,.$
(2.3)

###  2.2 Lax matrix

As another manifestation of the integrability of the model (2.1), it admits a
Lax representation, i.e., a pair of matrix-valued functions
$L,A\,:\,\mathbb{C}^{2N}\to\mathrm{Mat}(2N,\mathbb{C})$ such that the
Hamiltonian dynamics takes the form $\frac{d}{dt}L=[L,A]$. An immediate
corollary is that the quantities $\mathrm{tr}(L^{k})$, as well as the
eigenvalues of $L$, are constants of motion, which means that $L$ remains
isospectral for all $t$. Originally, Inozemtsev constructed in [28] a Lax pair
of size $3N\times 3N$ (see also [33]); other Lax pairs of smaller size have
since been found [9, 30]. We will use the Lax matrix of size $2N\times 2N$
from [30]. To write it down, we need the functions ${\sigma}_{\alpha}(x)$ and
$v_{\alpha}(x):=\sum_{r=0}^{3}g_{r}{\sigma}_{2\alpha}^{r}(x)$ whose definition
and basic properties are given in the Appendix. We have:

$\displaystyle L$
$\displaystyle=\sum_{i=1}^{N}\left(p_{i}E_{i,i}-p_{i}E_{i+N,i+N}+v_{\alpha}(q_{i})E_{i,i+N}+v_{\alpha}(-q_{i})E_{i+N,i}\right)$
(2.4) $\displaystyle+g\sum_{i\neq
j}^{N}\left(\sigma_{\alpha}(q_{ij})E_{i,j}+\sigma_{\alpha}(q_{ij}^{+})E_{i,j+N}+\sigma_{\alpha}(-q_{ij}^{+})E_{i+N,j}+\sigma_{\alpha}(-q_{ij})E_{i+N,j+N}\right)\,,$

where $E_{i,j}$ are the elementary matrices, and $q_{ij}$, $q_{ij}^{+}$ are
the shorthand notation for $q_{i}-q_{j}$ and $q_{i}+q_{j}$, respectively. This
Lax matrix $L$ contains an auxiliary parameter $\alpha$, usually referred to
as the spectral parameter, so we may write $L(\alpha)$ to emphasize this
dependence. We remark that the above expression for $L$ follows closely [31,
(5.15)]. It corresponds, in a different notation, to (3.37) and (3.39) in
[30].

As a function of $\alpha$, the Lax matrix $L$ has the following important
properties.

1. Periodicity:

$L(\alpha+1)=L(\alpha)\,,\quad L(\alpha+\tau)=CL(\alpha)C^{-1}\,,$ (2.5)

where $C=\begin{bmatrix}D&0\\\ 0&D^{-1}\end{bmatrix}$ with
$D=\mathrm{diag}(e^{2\pi iq_{1}},\dots,e^{2\pi iq_{N}})$.

2. Symmetry:

$L(-\alpha)=-ML(\alpha)M^{-1}\,,\quad\text{where}\
M=\begin{bmatrix}0&\mathrm{I}_{N}\\\ \mathrm{I}_{N}&0\end{bmatrix}.$ (2.6)

3. $L$ has simple poles at the half-periods: $L\sim L_{i}({\alpha}-{\omega}_{i})^{-1}+\mathrm{O}(1)$ near ${\alpha}={\omega}_{i}$. The residues $L_{i}$ are

$\displaystyle L_{i}$
$\displaystyle=-g_{i}^{\vee}\begin{bmatrix}0&\mathrm{I}_{N}\\\
\mathrm{I}_{N}&0\end{bmatrix}\quad(i=1,2,3)\,,$ (2.7) $\displaystyle L_{0}$
$\displaystyle=(g-g_{0}^{\vee})\begin{bmatrix}0&\mathrm{I}_{N}\\\
\mathrm{I}_{N}&0\end{bmatrix}-gT\,,$ (2.8)

where $T$ is the $2N\times 2N$ matrix with $0$’s along the main diagonal and
$1$’s elsewhere, and $g_{i}^{\vee}$ are the dual parameters,

$\begin{pmatrix}g^{\vee}_{0}\\\ g^{\vee}_{1}\\\ g^{\vee}_{2}\\\
g^{\vee}_{3}\\\
\end{pmatrix}=\frac{1}{2}\left({\begin{array}[]{rrrr}1&1&1&1\\\ 1&1&-1&-1\\\
1&-1&1&-1\\\ 1&-1&-1&1\\\ \end{array}}\right)\begin{pmatrix}g_{0}\\\ g_{1}\\\
g_{2}\\\ g_{3}\\\ \end{pmatrix}.$ (2.9)

Note that the residues $L_{i}$ are semi-simple (diagonalizable), with

$\displaystyle L_{i}$
$\displaystyle\sim\mathrm{diag}\Big{(}\underbrace{-g_{i}^{\vee},\dots,-g_{i}^{\vee}}_{\text{$N$
times}},\ \underbrace{g_{i}^{\vee},\dots,g_{i}^{\vee}}_{\text{$N$
times}}\Big{)}\qquad\text{for}\ i=1,2,3\,,$ (2.10) $\displaystyle L_{0}$
$\displaystyle\sim\mathrm{diag}\Big{(}-g_{0}^{\vee}-2(N-1)g,\
\underbrace{-g_{0}^{\vee}+2g,\dots,-g_{0}^{\vee}+2g}_{\text{$N{-}1$ times}},\
\underbrace{g_{0}^{\vee},\dots,g_{0}^{\vee}}_{\text{$N$ times}}\Big{)}\,.$
(2.11)

In [30], the Lax pair $L,A$ was constructed by an ad hoc method, and only for
the Hamiltonian flow corresponding to the quadratic Hamiltonian $h_{2}$. A
more general conceptual method for calculating $L,A$ was suggested in [31]. It
uses elliptic Dunkl operators [34, 21] and, apart from reproducing the above
$L$, it allows to construct a Lax partner $A$ for each of the commuting
Hamiltonian flows. This means that $L$ remains isospectral under each of the
flows governed by $h_{2},h_{4},\dots,h_{2N}$, cf. [31, Prop. 5.6]. As a
result, the quantities $f_{i}=\mathrm{tr}(L^{i})$ Poisson-commute with each of
$h_{2k}$, hence $f_{i}$ is a function of $h_{2},\dots,h_{2N}$. Taking into
account (2.5), we conclude that each of the functions
$f_{i}=\mathrm{tr}(L^{i})$ is a polynomial in $h_{2},\dots,h_{2N}$ whose
coefficients are elliptic functions of $\alpha$. Hence, the characteristic
polynomial of $L$ can be written as

$\det(L-k\mathrm{I})=k^{2N}+a_{1}k^{2N-1}+\dots+a_{2N}\,,$ (2.12)

where $a_{i}$ are polynomials in $h_{2},\dots,h_{2N}$, elliptic in the
spectral parameter.

###  2.3 Spectral curve

This puts us in the familiar setting of complex completely integrable systems.
Namely, the level sets of $N$ Poisson-commuting Hamiltonians
$h_{2},\dots,h_{2N}$ define a Lagrangian fibration
$\pi\,:\,\mathbb{C}^{2N}\to\mathbb{C}^{N}$. In addition to that, we have a
family of spectral curves

$f(k,\alpha):=\det(L(\alpha)-k\mathrm{I})=0\,.$ (2.13)

parametrized by the coordinates $h_{2},\dots,h_{2N}$ on the base of the
fibration $\pi$. Each spectral curve (2.13) is a $2N$-sheeted branched
covering of the base elliptic curve
$\Gamma=\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})$, with $(k,\alpha)$ viewed as
coordinates on the cotangent bundle $T^{*}\Gamma$. The curve (2.13) comes with
a meromorphic differential, obtained by restriction from the canonical
$1$-form $kd\alpha$ on $T^{*}\Gamma$, and a line bundle $\mathcal{L}$ (eigen-
bundle of $L$).

So far this looks parallel to the case of the usual Calogero–Moser system
[35]. Motivated by [36, 15, 37], one should think of the matrix-valued
$1$-form $\Phi:=L(\alpha)d\alpha$ as a Higgs field of some kind, so let us
sketch such an interpretation. First, instead of considering $\Phi$ over the
elliptic curve $\Gamma$, it is more natural to take into account the symmetry
(2.6). It implies that

$f(-k,-\alpha)=f(k,\alpha)$ (2.14)

and so the spectral curve can be viewed as a branched covering of the Riemann
sphere ${\Gamma}/\mathbb{Z}_{2}$, with the $\mathbb{Z}_{2}$ acting by
$\alpha\mapsto-\alpha$. Indeed, if we multiply $f(k,\alpha)$ by
$(\wp^{\prime}(\alpha))^{2N}$, we get

$\widetilde{f}:=(\wp^{\prime}(\alpha))^{2N}f(k,\alpha)=\det(\wp^{\prime}(\alpha)L(\alpha)-k\wp^{\prime}(\alpha)\mathrm{I})=\det(\widetilde{L}-y\mathrm{I})\,,$
(2.15)

where $\widetilde{L}=\wp^{\prime}(\alpha)L$ and $y=k\wp^{\prime}(\alpha)$. A
quick check confirms that $\widetilde{L}$ is regular at $\alpha=\omega_{r}$,
$r=1,2,3$, and that $\widetilde{L}(-\alpha)=M\widetilde{L}(\alpha)M^{-1}$.
Therefore, the expression (2.15) is a polynomial in $y$, whose coefficients
are even elliptic functions with the only singularity at $\alpha=0$. As a
result, the spectral curve (2.13) acquires polynomial form

$\widetilde{f}(x,y)=0\,,\quad\text{where}\ x=\wp(\alpha)\,,\
y=k\wp^{\prime}(\alpha)\,.$ (2.16)

Using $x=\wp(\alpha)$ as the coordinate on $\Gamma/\mathbb{Z}_{2}$, we also
obtain $\Phi=Ld\alpha=(\wp^{\prime}(\alpha))^{-1}Ldx$. The properties of $L$
tell us that such $\Phi$ should be viewed as a Higgs field on the Riemann
sphere with four marked points, more precisely, on an orbicurve
$\mathbb{CP}^{1}$ of type $(2,2,2,2)$. Recall [38] that Hitchin systems on
orbicurves can also be viewed as parabolic Hitchin systems, with (conjugacy
classes of) the residues of $\Phi$ at the marked points being associated with
the values of the moment map, cf. [37, 5]. Therefore, the formula (2.4) should
be interpreted as a parametrization, by $p_{i},q_{i}$, of the corresponding
$2N$-dimensional symplectic leaf of a parabolic $\mathrm{SL}(2N,\mathbb{C})$
Hitchin system on the Riemann sphere with four marked points
$e_{i}=\wp(\omega_{i})$, $i=0,1,2,3$. This provides an interpretation of the
Inozemtsev system as a Hitchin system. Note that this is different from the
approach of [39]. Note also that the pattern (2.10)–(2.11) of the residues of
$\Phi$ at the marked points is in good agreement with the SCFT picture (see
Sec. 3.2 below). Also, as is explained below in Sec. 2.5, the genus of the
spectral curve equals $N$, which is as expected from both the Hitchin-system
and the M5-brane perspectives.

Let us also recall that starting from a moduli space $\mathcal{M}$ of Higgs
bundles, the nonabelian Hodge correspondence and Riemann–Hilbert map associate
to $\mathcal{M}$ two other moduli spaces, of local systems and of monodromy
data (known as de Rahm and Betti models, see [40] for a nice overview). For
our case, these two other incarnations can be found in [41, 42], see also [43,
44, 33, 45, 46] for further links between the Inozemtsev system and
isomonodromic deformations.

###  2.4 Spectral curves for $N=1$ and $N=2$

Here we present explicit equations for the spectral curves (2.13) in the cases
of $N=1$ and $N=2$. We write equations in terms of the variables $k,\alpha$.
They will be matched to M5 brane curves in Section 4.

####  2.4.1 $N=1$ curve

For $N=1$, the Lax matrix is (cf. [33])

$L=\begin{bmatrix}p&v_{\alpha}(q)\\\ v_{\alpha}(-q)&-p\end{bmatrix}\,.$ (2.17)

Using A.8, we find:

$\det L=-p^{2}-v_{\alpha}(q)v_{\alpha}(-q)=-p^{2}+u(q)-u^{\vee}(\alpha)\,,$
(2.18)

where $u^{\vee}(\alpha)$ is the dual version of (2.2), defined above in
(2.28). Hence, the spectral curve (2.13) takes the form

$f(k,z)=k^{2}-h_{2}-u^{\vee}(\alpha)=0\,,$ (2.19)

with $h_{2}=p^{2}-u(q)$ viewed as a complex parameter. Multiplying this by
$(\wp^{\prime}(\alpha))^{2}$ and using $x=\wp(\alpha)$,
$y=k\wp^{\prime}(\alpha)$ we obtain $y^{2}=\wp^{\prime
2}(\alpha)\,\left(h_{2}+u^{\vee}(\alpha)\right)$. Using (4.2) it is easy to
see that the right-hand side is a quartic polynomial in $x=\wp(\alpha)$ (it
reduces to a cubic if $g_{0}^{\vee}=0$). For generic $h_{2}$, the curves are
smooth of genus $1$.

The Lagrangian fibration $\pi\,:\,\mathbb{C}^{2}\to\mathbb{C}$ is by the level
sets $p^{2}-u(q)=h_{2}$. Singular fibers correspond to the stationary values
of the Hamiltonian, i.e. to the equilibria $(p,q)=(0,q_{0})$ with
$u^{\prime}(q_{0})=0$. Then we can find that for a number of $l\geq 1$ generic
couplings $g_{i}$, the number of stationary values of $h_{2}$ is $l+2$, in
agreement with the Seiberg-Witten geometry [2]. Indeed, the function
$u^{\prime}(q)=\sum_{i=0}^{3}g_{i}^{2}\wp^{\prime}(q+\omega_{i})$ is odd
elliptic of order $3l$, therefore it has $3l$ zeros; the genericity assumption
ensures that the multiplicity of each zero is always one. Then $4-l$ zeros are
given by the half-periods, for which the values of $h_{2}$ are distinct.
Furthermore, the other $4l-4$ zeros come in pairs $(q,-q)$ so give the same
stationary value of $h_{2}$. Thus, the number of singular fibers (or
stationary values of $h_{2}$) is $(4-l)+(4l-4)/2=l+2$, as claimed.

####  2.4.2 $N=2$ curve

For $N=2$, the Lax matrix is

$\displaystyle L$ $\displaystyle=$
$\displaystyle\begin{bmatrix}p_{1}&g\sigma_{\alpha}(q_{12})&v_{\alpha}(q_{1})&g\sigma_{\alpha}(q^{+}_{12})\\\
g\sigma_{\alpha}(-q_{12})&p_{2}&g\sigma_{\alpha}(q^{+}_{12})&v_{\alpha}(q_{2})\\\
v_{\alpha}(-q_{1})&g\sigma_{\alpha}(-q^{+}_{12})&-p_{1}&g\sigma_{\alpha}(-q_{12})\\\
g\sigma_{\alpha}(-q^{+}_{12})&v_{\alpha}(-q_{2})&g\sigma_{\alpha}(q_{12})&-p_{2}\end{bmatrix}$
$\displaystyle=$ $\displaystyle
P\begin{bmatrix}p_{1}&v_{\alpha}(q_{1})&g\sigma_{\alpha}(q_{12})&g\sigma_{\alpha}(q^{+}_{12})\\\
v_{\alpha}(-q_{1})&-p_{1}&g\sigma_{\alpha}(-q^{+}_{12})&g\sigma_{\alpha}(-q_{12})\\\
g\sigma_{\alpha}(-q_{12})&g\sigma_{\alpha}(q^{+}_{12})&p_{2}&v_{\alpha}(q_{2})\\\
g\sigma_{\alpha}(-q^{+}_{12})&g\sigma_{\alpha}(q_{12})&v_{\alpha}(-q_{2})&-p_{2}\end{bmatrix}P^{-1}\,,$

where

$\displaystyle P=\begin{bmatrix}1&0&0&0\\\ 0&0&1&0\\\ 0&1&0&0\\\
0&0&0&1\end{bmatrix}.$ (2.21)

The $N=2$ case is the first case with non-zero “antisymmetric mass” (related
to the coupling $g$). If we let $g=0$, we find that the Lax matrix reduces to
two $2\times 2$ blocks, each having the form of a $N=1$ Lax matrix. Similarly,
the general $2N\times 2N$ Lax matrix in the $g\to 0$ limit reduces to $N$
diagonal $2\times 2$ blocks. Subsequently, in this limit the spectral curve is
reducible, as it becomes a product of $N$ copies of the $N=1$ curve.

The $N=2$ spectral curve is given by

$\displaystyle 0=$
$\displaystyle(k^{2}-u^{\lor})^{2}-h_{2}(k^{2}-u^{\lor})+h_{4}$ (2.22)
$\displaystyle\quad-4g^{2}\Bigl{(}\wp(\alpha)(k^{2}-u^{\lor})+g^{\lor}_{0}\wp^{\prime}(\alpha)k+2(g^{\lor}_{0})^{2}\wp(\alpha)^{2}+\wp(\alpha)\sum_{r=1}^{3}(g^{\lor}_{r})^{2}\wp(\omega_{r})\Bigr{)}\,,$

where $u^{\vee}:=u^{\vee}(\alpha)$ and $h_{2},h_{4}$ represent the values of
two commuting Hamiltonians.

The derivation of (2.22) is outlined in appendix A.2.

###  2.5 Behaviour near marked points

In order to make a connection with the analysis of the Seiberg–Witten curve in
Sec. 3.2, it will be useful to look more closely at the singularities of the
Lax matrix (2.4). This will also allow us to confirm that the genus of the
spectral curves equals $N$, as expected.

Expanding $L$ at half-periods gives

$L=\sum_{j\geq-1}L_{i}^{(j)}(\alpha-\omega_{i})^{j}\,,\quad i=0,1,2,3\,,$
(2.23)

for some $L_{i}^{(j)}\in\mathrm{Mat}(2N,\mathbb{C})$ independent of $\alpha$,
with $L_{i}^{(-1)}$ being the residue matrices (2.7)–(2.8). The property (2.6)
implies that

$ML_{i}^{(j)}+(-1)^{j}L_{i}^{(j)}M=0\,,\qquad
M=\begin{bmatrix}0&\mathrm{I}_{N}\\\ \mathrm{I}_{N}&0\end{bmatrix}\,.$ (2.24)

Now consider the $2N$ sheets of the spectral curve $\det(L-k\mathrm{I})=0$
near one of the half-period $\alpha=\omega_{1,2,3}$. From (2.10), we know that
locally we can label these sheets so the roots $k_{1},\dots,k_{2N}$ near
$\alpha=\omega_{i}$ behave as follows:

$\displaystyle(k_{1},\dots,k_{2N})$
$\displaystyle\sim\frac{1}{{\alpha}-{\omega}_{i}}\Big{(}\underbrace{-g_{i}^{\vee},\dots,-g_{i}^{\vee}}_{\text{$N$
times}},\ \underbrace{g_{i}^{\vee},\dots,g_{i}^{\vee}}_{\text{$N$
times}}\Big{)}+\text{regular terms}\,.$ (2.25)

Series expansions for each $k_{r}(\alpha)$ can be worked out recursively, as a
perturbation series, together with the eigenvectors $v_{r}(\alpha)$ such that

$L(\alpha)v_{r}(\alpha)=k_{r}(\alpha)v_{r}(\alpha)\,,\qquad
v_{r}(\alpha)=\sum_{j\geq 0}v_{r}^{(j)}(\alpha-\omega_{i})^{j}\,,$ (2.26)

for a chosen “initial” eigenbasis $v_{r}^{(0)}$ of the residue matrix
$L_{i}^{(-1)}$. Since the residue matrix $L_{i}^{(-1)}$ commutes with $M$ for
all $i=0,1,2,3$ (for $i\neq 0$ it is simply proportional to $M$), the chosen
eigenvectors are also eigenvectors of $M$, and so half of them satisfy
$Mv_{r}^{(0)}=v_{r}^{(0)}$, with $Mv_{r}^{(0)}=-v_{r}^{(0)}$ for the other
half. The additional symmetry (2.24) of the Lax matrix imposes extra
constraints, which result in the following:

1. Near $\alpha=\omega_{i}$, each eigenvalue $k_{r}(\alpha)$ is odd, i.e. it changes sign under $\alpha\mapsto 2\omega_{i}-\alpha$.

2. The terms of the series for the eigenvector $v_{r}(\alpha)$ satisfy $Mv_{r}^{(j)}=\pm(-1)^{j}v_{r}^{(j)}$, with the sign $\pm$ determined by the initial eigenvector $v_{r}^{(0)}$.

An important corollary of the first property is that the regular terms in
(2.25) are in fact of order $O(\alpha-\omega_{i})$. Then by squaring the
spectral variable $k$ and shifting it appropriately, all the poles can be
cancelled. In particular,

$\displaystyle z$
$\displaystyle\sim\frac{1}{({\alpha}-{\omega}_{i})^{2}}\Big{(}\underbrace{0,\dots,0}_{2N\
\text{times}}\Big{)}+\text{regular terms}\quad(i=1,2,3)\,,$ (2.27)

where we have defined

$\displaystyle z$
$\displaystyle:=\frac{1}{4}\left(k^{2}-u^{\lor}+\text{constant}\right),$
$\displaystyle u^{\vee}$
$\displaystyle=\sum_{i=0}^{3}(g_{i}^{\vee})^{2}\wp({\alpha}+{\omega}_{i})\,.$
(2.28)

The factor of $1/4$ and the constant in (2.28) are for later convenience.

The same analysis for $\alpha\sim 0$ gives that

$\displaystyle(k_{1},\dots,k_{2N})$
$\displaystyle\sim\frac{1}{{\alpha}}\Big{(}-g_{0}^{\vee}-2(N-1)g,\
\underbrace{2g-g_{0}^{\vee},\dots,2g-g_{0}^{\vee}}_{\text{$N{-}1$ times}},\
\underbrace{g_{0}^{\vee},\dots,g_{0}^{\vee}}_{\text{$N$
times}}\Big{)}+{O}(\alpha)\,,$ (2.29)

and so by squaring and shifting it appropriately all but one of the $2N$ poles
there can be cancelled. In particular,

$\displaystyle\widetilde{z}:=z+\frac{g}{x}\left(y+\frac{1}{2}g^{\lor}_{0}x^{2}\right)$
$\displaystyle\sim\frac{1}{{\alpha}^{2}}\Big{(}Ng(g^{\lor}_{0}+(N-1)g),\
\underbrace{0,\dots,0}_{2N{-}1\ \text{times}}\Big{)}+\text{regular terms}\,,$
(2.30)

where we have defined

$\displaystyle x:=\wp({\alpha})\sim\frac{1}{{\alpha}^{2}}\,,\qquad
y:=\frac{1}{4}k\wp^{\prime}({\alpha}).$ (2.31)

(2.30) indicates that the coefficients of the spectral curve written in the
$(x,y,\widetilde{z})$ variables (as an $N$-fold cover of the sphere
parametrized by $x$) can only have simple poles at $x=\infty$, while (2.27)
indicates that if they are written in the $(x,y,z)$ variables they will be
regular away from $x=\infty$. In fact this observation will play an important
role in finding the change of variables needed to match the spectral curve to
the Seiberg-Witten curve, discussed in section 3.2.

We can now calculate the genus of the spectral curve (2.16). We follow the
same method as in [35]. First, consider the curve $\Gamma_{N}$ (2.13) and
denote its genus by $g$. Then $2g-2=\nu$,where $\nu$ is the number of the
branch points of $\Gamma_{N}$ viewed as a covering of the elliptic curve
$\Gamma$. This is the number of zeros of $\partial f/{\partial k}$ on
$\Gamma_{N}$; it also equals the number of poles of $\partial f/{\partial k}$.
The poles occur precisely at $2N$ points of $\Gamma_{N}$ above each of the
half-periods $\alpha=\omega_{i}$. Locally, we can factorize $f(k,\alpha)$ into
a product of factors $k-k_{r}(\alpha)$. For example, near
$\alpha=\omega_{1,2,3}$ we have

$f(k,\alpha)=\prod_{r=1}^{N}\left(k+\frac{g_{i}^{\vee}}{\alpha-\omega_{i}}+b_{r}({\alpha})\right)\prod_{r=N+1}^{2N}\left(k-\frac{g_{i}^{\vee}}{\alpha-\omega_{i}}+b_{r}({\alpha})\right)\,,$
(2.32)

where the $b_{r}({\alpha})$ are of order ${O}(\alpha-\omega_{i})$. By
differentiating this equation with respect to $k$, we find that $\partial
f/\partial k$ has a simple pole on each of the $2N$ sheets above $\omega_{i}$.
A similar analysis near $\alpha=0$ shows that $\partial f/\partial k$ has
there a pole of order $2N-1$ on one sheet, poles of order $3$ on $N-1$ sheets,
and simple poles on the remaining $N$ sheets. This gives $2g-2=3\times
2N+(2N-1)+3\times(N-1)+N=12N-4$, so $g=6N-1$.

The curve $\Gamma_{N}^{\prime}$ (2.16) is obtained from $\Gamma_{N}$ by taking
a quotient by the involution $(k,\alpha)\mapsto(-k,-\alpha)$. Thus,
$\Gamma_{N}$ can be viewed as a $2$-sheeted covering of $\Gamma_{N}^{\prime}$,
branched at the fixed points of the involution. These are precisely the points
above the half-periods, so there are $8N$ of them. Denoting by $g^{\prime}$
the genus of $\Gamma_{N}^{\prime}$, we get $12N-4=2g-2=2(2g^{\prime}-2)+8N$,
from which $g^{\prime}=N$, as claimed.

###  2.6 Modular property

The Lax matrix and the spectral curve exhibit a modular behaviour under
$\mathrm{SL}(2,\mathbb{Z})$-action. To state the result, recall that the Lax
matrix $L$ depends on the modular parameter $\tau$, the spectral parameter
$\alpha$, $2n$ variables $p_{i},q_{i}$, and the coupling constants $g$ and
$g_{0,1,2,3}$. Take ${\gamma}=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\mathrm{SL}(2,\mathbb{Z})$ and define
$L^{\prime}$ to be the Lax matrix with the variables changed to
$\tau^{\prime}$, $\alpha^{\prime}$, etc., in the following way:

$\displaystyle{\tau}^{\prime}$
$\displaystyle=\frac{a{\tau}+b}{c{\tau}+d}\,,\quad\alpha^{\prime}=(c{\tau}+d)^{-1}\alpha\,,$
(2.33) $\displaystyle p_{i}^{\prime}$ $\displaystyle=(c{\tau}+d)p_{i}\,,\quad
q_{i}^{\prime}=(c{\tau}+d)^{-1}q_{i}\,,$ (2.34) $\displaystyle g^{\prime}$
$\displaystyle=g\,,\quad g_{0}^{\prime}=g_{0}\,,\quad
g^{\prime}_{r}=g_{\pi_{{\gamma}}(r)}\quad\text{for}\ r=1,2,3\,.$ (2.35)

Here in the last formula we denote by $\pi_{\gamma}$ the permutation of
$\\{1,2,3\\}$ determined by the group homomorphism (A.10). With this notation,
we have:

$\displaystyle L^{\prime}=(c{\tau}+d)QLQ^{-1}\,,$ (2.36)

where $Q=\begin{bmatrix}R&0\\\ 0&R^{-1}\end{bmatrix}$ and
$R=\mathrm{diag}\left(\exp(-\frac{2\pi ic}{c{\tau}+d}\alpha
q_{1}),\dots,\exp(-\frac{2\pi ic}{c{\tau}+d}\alpha q_{N})\right)$.

The formula (2.36) is obtained in a straightforward way from the modular
properties of the functions $\sigma_{\alpha}(x)$ and $v_{\alpha}(x)$ given in
the Appendix. If we introduce $k^{\prime}=(c{\tau}+d)k$, then we also have

$\det(L^{\prime}-k^{\prime}\mathrm{I})=(c\tau+d)^{2N}\det(L-k\mathrm{I})\,.$
(2.37)

The physical interpretation of these properties on the QFT side is the
$\mathrm{SL}(2,\mathbb{Z})$ S-duality mixed with the Spin(8) triality (see
Sec. 3.1 below).

##  3 $\mathrm{USp}(2N)$ $N_{f}=4$ superconformal field theory

We consider the family of 4d $\mathcal{N}=2$ superconformal field theories
consisting of $\mathrm{USp}(2N)$ gauge theories with $N_{f}=4$ hypermultiplets
in the fundamental representation and (for $N\geq 2$) $N_{a}=1$
hypermultiplets in the traceless antisymmetric two-index tensor
representation.

###  3.1 Field theory properties

We list some long-established properties of these theories.

  * •

They are a family of interacting 4d $\mathcal{N}=2$ SCFTs labelled by a
positive integer $N$, which we call the rank of the $N_{f}=4$ theory. As
SCFTs, they are invariant under the 4d $\mathcal{N}=2$ superconformal group
$\mathrm{SU}(2,2|2)$.

  * •

The $N_{f}=4$ SCFTs have an exact $\mathrm{SL}(2,\mathbb{Z})$ S-duality. This
means that each theory has a one-complex-dimensional conformal manifold given
by the upper half complex plane modulo $\mathrm{SL}(2,\mathbb{Z})$ Möbius
transformations. Though the center of $\mathrm{SL}(2,\mathbb{Z})$ acts
trivially on the conformal manifold, it acts non-trivially as charge
conjugation in the field theory. Around a special point on the conformal
manifold the theory admits a weakly-coupled Lagrangian description in terms of
$\mathrm{USp}(2N)$ gauge theory with 4 fundamental and 1 antisymmetric
hypermultiplets. The weak coupling limit of the complex gauge coupling
constant ${\tau}$ parameterizing the conformal manifold is
$\mathrm{Im}({\tau})\to\infty$.

  * •

The internal global “flavor” symmetry is $\mathrm{Spin}(8)$ for $N=1$ and
$\mathrm{Spin}(8)\times\mathrm{SU}(2)$ for $N\geq 2$, under which the four
fundamental hypermultiplets (the same as eight fundamental half-
hypermultiplets) transform in the $(8_{v},1)$ representation, and the
antisymmetric hypermultiplet in the $(1,2)$ representation. Correspondingly,
there is a space of $\mathcal{N}=2$-preserving mass deformations given by the
complexified weight space of $\mathrm{Spin}(8)\times\mathrm{SU}(2)$. Introduce
mass (or deformation) parameters $(m_{1},\ldots,m_{4})$ for $\mathrm{Spin}(8)$
and $M$ for $\mathrm{SU}(2)$ as linear coordinates on this parameter space
such that $m_{i}$ is the complex mass of the $i$-th fundamental
hypermultiplet, and $M$ the mass of the antisymmetric hypermultiplet.222We use
an unconventional normalization for the mass such that our masses $m$ are
related to the canonically normalized masses $\widetilde{m}$ by
$\widetilde{m}=\sqrt{2}\,m$. The principal congruence subgroup
${\Gamma}(2)\subset\mathrm{SL}(2,\mathbb{Z})$ of the S-duality group acts
trivially on the $\mathrm{Spin}(8)$ masses, while the quotient
$\mathrm{SL}(2,\mathbb{Z})/{\Gamma}(2)\simeq S_{3}$ transforms the mass
parameters by the $\mathrm{Spin}(8)$ “triality” outer automorphism [2, 47].
The antisymmetric mass is invariant under S-duality transformations.

  * •

The operator content of an $N_{f}=4$ theory can be organized in terms of the
unitary representations of its global symmetry
$\mathrm{SU}(2,2|2)\times\mathrm{Spin}(8)\times\mathrm{SU}(2)$. In particular,
with respect to $\mathrm{SU}(2,2|2)$ there are various sectors of
supersymmetry-protected BPS operators, for instance, Coulomb branch operators
and Higgs branch operators. The condensate of the scalar components in the
$\mathcal{N}=2$ multiplets of BPS operators parameterize moduli spaces of
$\mathcal{N}=2$ invariant vacuum states.

  * •

The moduli space of vacua consists of various branches each of which is
locally a metric product of a Coulomb factor and a Higgs factor, with complex
dimension $n_{C}$ and quaternionic dimension $n_{H}$, respectively.
Conventionally, the branch with maximal $n_{C}$ is called the Coulomb branch
and the branch with maximal $n_{H}$ the Higgs branch. The rank $N$ $N_{f}=4$
theory has a Coulomb branch with $(n_{C},n_{H})=(N,N-1)$ and a Higgs branch
with $(n_{C},n_{H})=(0,6N-1)$. The $N-1$ quaternionic dimensional Higgs factor
of the Coulomb branch comes from the components of the antisymmetric
hypermultiplet carrying zero weight with respect to the $\mathrm{USp}(2N)$
gauge algebra.

  * •

The vector multiplet of the Lagrangian theory contains a scalar field $\Phi$
in the adjoint representation. The Coulomb branch coordinate ring is freely
generated by $u_{i}:=\mathrm{tr}(\wedge^{2i}\Phi)$ with $i=1,2,\ldots,N$,
corresponding to the primitive Casimir elements of $\mathrm{USp}(2N)$. The
Coulomb branch coordinate ring is graded by the scaling dimension, so the
weight of $u_{i}$ is $2i$. Since the Coulomb branch chiral operators are BPS
operators, this description of the Coulomb branch chiral ring is true at all
points of the conformal manifold, not just at the weak coupling point.

We are interested in the geometry of the Coulomb branch. The low energy
effective $\mathrm{U}(1)^{N}$ gauge theory on the Coulomb branch is encoded in
the special Kähler geometry [48] of the Coulomb branch. The $N-1$ massless
neutral hypermultiplets on the Coulomb branch decouple in the low energy
limit, so will be ignored.

On general grounds [4] a Coulomb branch special Kähler geometry is equivalent
to a classical complex completely integrable Hamiltonian system. In
particular, the Coulomb branch is the $N$-complex-dimensional manifold of the
action variables of the integrable system. The matrix of low energy
$\mathrm{U}(1)^{N}$ complex gauge couplings gives the period matrix of a
complex torus of dimension $N$, so the Coulomb branch parameterizes a family
of complex tori, giving the angle variables of the integrable system. The
complex tori are also endowed with principle polarization coming from the
Dirac pairing on the $\mathrm{U}(1)^{N}$ electric-magnetic charge lattice, and
hence are abelian varieties. The total space of this family of abelian
varieties is a complex symplectic variety, the complex phase space of the
integrable system, with holomorphic symplectic form ${\omega}$.

The next subsection describes the total space geometry by way of a holomorphic
family $\Sigma$ of genus-$N$ Riemann surfaces over the Coulomb branch, along
with a meromorphic one-form ${\lambda}$ on the fibers whose poles have
constant residues. $(\Sigma,{\lambda})$ are called the Seiberg-Witten curve
and one-form in the physics literature. The abelian variety fibers of the
integrable system are the Jacobian tori of the Riemann surfaces, and the
symplectic form is ${\omega}=d{\lambda}$. Thus we will match the field theory
Coulomb branch geometry to the Inozemtsev system by matching the Seiberg-
Witten curve and one-form to the spectral curve and canonical one-form of the
integrable system.

###  3.2 Seiberg-Witten curve

The $\mathrm{USp}(2N)$ $N_{f}=4$ SCFTs can be constructed as the low energy
effective theory of type IIA superstrings in the presence D4, NS5, D6, and O6-
branes generalizing the construction of [14]. The M-theory lift of the D6 and
O6- IIA brane configuration [49] is a specific choice of complex structure of
a $(T^{2}\times\mathbb{C})/\mathbb{Z}_{2}$ hyperkähler orbifold background.
The M-theory lift of the D4 and NS5 branes is a single M5 brane intersecting
the background except over points of $T^{2}$ corresponding to NS5 branes. This
intersection is the Seiberg-Witten curve, and the restriction of the
holomorphic hyperkahler form to the curve is the Seiberg-Witten one-form. This
is the spectral curve of a Hitchin system on the orbifolded torus with
punctures [49].

The deformations of this orbifold background and M5 brane curve corresponding
to turning on the $\mathrm{Spin}(8)$ fundamental masses and the
$\mathrm{SU}(2)$ antisymmetric mass was worked out in [29]. The connection to
a Hitchin system is no longer apparent in this description. We will describe
this solution for the $\mathrm{USp}(2N)$ $N_{f}=4$ Coulomb branch in more
detail shortly in preparation for showing its equivalence to the spectral
curve of the Inozemtsev system. But first, we make a few comments on two other
string constructions of the $\mathrm{USp}(2N)$ $N_{f}=4$ theories.

These theories naturally arise as the world volume theories on a stack of $N$
parallel D3 branes probing an F-theory singularity of $(I^{\ast}_{0},D_{4})$
type — i.e., an $\mathrm{O7}^{-}$ plane coinciding with four $\mathrm{D7}$
branes [50, 51, 52, 53]. But it is not known how to turn on the antisymmetric
mass $M$ deformation in the F-theory construction.

These theories also admit a class-S construction via a 6d $(2,0)$ $A_{2N-1}$
SCFT compactified on a sphere $C$ with four punctures all of type $[N,N]$.
This construction only makes manifest an $\mathrm{SU}(2)^{4}$ subgroup of the
$\mathrm{Spin}(8)$ flavor group, and does not make the antisymmetric
$\mathrm{SU}(2)$ flavor factor or its associated mass deformation apparent
[25]. $C$ is identified with $T^{2}/\mathbb{Z}_{2}$ with the four punctures
corresponding to the four $\mathbb{Z}_{2}$ orbifold fixed points. The
antisymmetric hypermultiplet appears upon taking an appropriate zero-area
limit of $C$ [54], and [27] showed that by modifying the type of one puncture
to be $[N,N-1,1]$, the theory manifests the antisymmetric $\mathrm{SU}(2)$
flavor symmetry. The class-S construction realizes the integrable system
underlying the Coulomb branch geometry as a Hitchin system [55].

The matching to the M5 brane curve, presented below, gives strong evidence
that the Hitchin system associated with the above class-S construction can be
identified with the Inozemtsev system.

In the rest of this section we review the M5 brane construction [29] of the SW
curve for the $\mathrm{USp}(2N)$ $N_{f}=4$ theory. The main ingredients in
this construction are:

  * •

The $\mathrm{USp}(2N)$ theory with the $\mathrm{Spin}(8)$ mass deformation is
realized by embedding one complex dimension of the M5 brane world volume in a
complex surface, $Q_{0}$. $Q_{0}$ carries a hyperkähler structure — from which
the SW 1-form is derived — and is a deformation of a
$(T^{2}\times\mathbb{C})/\mathbb{Z}_{2}$ orbifold. This surface can be thought
of (we will be more precise below) as fibered over $T^{2}/\mathbb{Z}_{2}$.

  * •

The intersection with the M5 brane then gives a curve which projects to an
$N$-fold cover of $T^{2}/\mathbb{Z}_{2}$ minus one of the orbifold points. At
the missing orbifold point the M5 brane is not transverse to $Q_{0}$; we will
call this point the “pole” of the M5 brane.

  * •

The $\mathrm{SU}(2)$ mass deformation, $M$, is realized by further deforming
the background surface to $Q_{M}$. Following the discussion of the analogous
deformation of the elliptic model in [14], describe $Q_{M}$ by two charts to
$Q_{0}$, one including the fibers above a neighborhood of a chosen point $p\in
T^{2}/\mathbb{Z}_{2}$, and the other encompassing the rest of the surface. The
two coordinate patches are isomorphic to the corresponding patches of $Q_{0}$,
and the $M$ deformation is realized by requiring that the transition map is a
shift of the fiber coordinate which has a pole with residue proportional to
$M$ at $p$. We call this transition map the “$M$ shift”. Changing $p$ and the
form of the transition map but keeping $M$ fixed does not change the complex
structure of $Q_{M}$.

  * •

The M5 brane curve for the mass-deformed $\mathrm{USp}(2N)$ $N_{f}=4$ SCFT is
then locally a degree-$N$ polynomial in the fiber coordinate on $Q_{M}$ whose
coefficients have at most a simple pole over a chosen orbifold point of
$T^{2}/\mathbb{Z}_{2}$.

The form of the SW curve for the $\mathrm{USp}(2N)$ $N_{f}=4$ (and many other
closely related) SCFTs found in [29] followed this procedure with the $M$
shift at a point $p$ not equal to one of the orbifold points of
$T^{2}/\mathbb{Z}_{2}$. Both the form of the spectral curve of the Inozemtsev
system as well as the above-mentioned S-class construction (where one of the
four punctures is modified to capture the $M$ deformation) suggest that they
will most easily match the form of the SW curve if the point $p$ of the $M$
shift should be taken to coincide with one of the orbifold points. This
involves a slight modification of the construction of [29] which we now
explain.

####  3.2.1 Background surface

We start with the orbifold $(T^{2}\times\mathbb{C})/\mathbb{Z}_{2}$. Think of
$T^{2}\times\mathbb{C}$ as an affine bundle over $T^{2}$ and let
$v\in\mathbb{C}$ be the fiber coordinate. Write the complex torus $T^{2}$ as a
curve $\eta^{2}=\prod_{i=1}^{4}(x-e_{i}w)$ in weighted projective space,
$[w:x:\eta]\in\mathbb{P}^{2}_{(1,1,2)}$. Note that $\mathrm{SL}(2,\mathbb{C})$
transformations of $(w,x)$ do not change the complex structure of $T^{2}$, but
change the $e_{i}$ by Möbius transformations. The $\mathbb{Z}_{2}$
identification on $\mathbb{C}\times T^{2}$ is
$(v,w,x,\eta)\simeq(-v,w,x,-\eta)$. Using the invariant coordinates on the
orbifold, $y=v\eta$, $z=v^{2}$ ($w$ and $x$ unchanged), the orbifolded
background space is given by the surface $y^{2}=z\prod_{i=1}^{4}(x-e_{i}w)$.

The $(T^{2}\times\mathbb{C})/\mathbb{Z}_{2}$ orbifold has a four-parameter
deformation into a complex surface $Q_{0}$ with the same asymptotic structure.
The mass-deformed orbifold surface $Q_{0}$ and SW 1-form are [29]

$\displaystyle{\lambda}$ $\displaystyle=\frac{y(wdx-xdw)}{P},$ $\displaystyle
P$ $\displaystyle:=\prod_{i}(x-e_{i}w),$ $\displaystyle y^{2}$
$\displaystyle=zP+Q,$ $\displaystyle Q$
$\displaystyle:=\sum_{j}\mu_{j}^{2}w\prod_{k\neq j}[(x-e_{k}w)(e_{j}-e_{k})],$
(3.1)

where $i,j,k\in\\{0,1,2,3\\}$. Note that we still have
$[w:x:y]\in\mathbb{P}^{2}_{(1,1,2)}$. The deformation parameters, ${\mu}_{i}$,
turn out to be related to the fundamental masses by [29]

${\mu}_{0}=\tfrac{1}{2}(m_{1}+m_{2}),\quad{\mu}_{1}=\tfrac{1}{2}(m_{1}-m_{2}),\quad{\mu}_{2}=\tfrac{1}{2}(m_{3}+m_{4}),\quad{\mu}_{3}=\tfrac{1}{2}(m_{3}-m_{4}).$
(3.2)

The topology of $Q_{0}$ can be pictured by noting that the $z=$ constant
“sections” are tori, and the $x={\xi}w$ (${\xi}=$ constant) “fibers” are
generically 2-sheeted covers of the $z$-plane branched over the point
$z=-Q/P$. But when $x=e_{i}w$ the fiber becomes two disconnected copies of the
$z$-plane, $S^{\pm}_{j}:=\bigl{\\{}\,x=e_{j}w,\
y=\pm{\mu}_{j}w^{2}{\textstyle\prod_{k\neq j}}(e_{j}-e_{k}),\ \forall
z\,\bigr{\\}}$. The existence of these “double fibers” over the Weierstrass
points in the deformed orbifold will play a central role in what follows. From
the point of view of the IIA string theory D4/NS5/O6- brane construction, the
generic $x={\xi}w$ fibers correspond to possible loci of (the M theory lift
of) an NS5 brane, while the $S^{\pm}_{j}$ curves correspond the possible loci
of “half” NS5 branes “stuck” at an O6- orientifold plane.

To get closer to the form of the integrable system spectral curve, we will
specialize (3.2.1) to Weierstrass form where the Weierstrass points are placed
at $e_{0}=\infty$ and $\sum_{j=1}^{3}e_{j}=0$. Then the $Q_{0}$ surface and
1-form become

$\displaystyle{\lambda}$ $\displaystyle=\frac{y(wdx-xdw)}{w\widetilde{P}},$
$\displaystyle\widetilde{P}$
$\displaystyle:=\prod_{i}(x-e_{i}w)=x^{3}+s_{2}w^{2}x-s_{3}w^{3},$
$\displaystyle y^{2}$
$\displaystyle=(zw+{\mu}_{0}^{2}x)\widetilde{P}+w^{2}\widetilde{Q},$
$\displaystyle\widetilde{Q}$
$\displaystyle:=\sum_{j}{\mu}_{j}^{2}{\epsilon}_{j}\prod_{k\neq j}(x-e_{k}w).$
(3.3)

where now indices only take the three values $i,j,k\in\\{1,2,3\\}$, and we
have defined the useful combinations

$\displaystyle s_{2}$ $\displaystyle:=\sum_{j<k}e_{j}e_{k},$ $\displaystyle
s_{3}$ $\displaystyle:=\prod_{j}e_{j},$ $\displaystyle{\epsilon}_{j}$
$\displaystyle:=\prod_{k\neq j}(e_{j}-e_{k}).$ (3.4)

Note that the equations for the disjoint fibers over the Weierstrass points
become

$\displaystyle S^{\pm}_{\infty}$
$\displaystyle:=\\{w=0,y=\pm\mu_{0}x^{2},\forall z\\},$ and $\displaystyle
S^{\pm}_{j}$
$\displaystyle:=\\{x{=}e_{j}w,\,y{=}\pm{\mu}_{j}{\epsilon}_{j}w^{2},\,\forall
z\\}.$ (3.5)

Now we discuss the $M$ deformation with the shift put at a branch point. To
motivate the construction, we first review, following [14], the corresponding
deformation of the unorbifolded $T^{2}\times\mathbb{C}$ background,
$\eta^{2}=P$. Put the $M$ shift at the Weierstrass point $w=0$ (which is
$x=\infty$ in the $w=1$ patch) by defining the transition map,

$\displaystyle\widetilde{v}=v+M\frac{\eta}{wx},$ (3.6)

where $\widetilde{v}$ is the fiber coordinate of a chart over a neighborhood
of the $w=0$ point of the $T^{2}$. This transition map has a pole with residue
$M$ over $w=0$, so describes a one-parameter complex deformation of
$T^{2}\times\mathbb{C}$ with parameter $M$. This is because the deformations
of the affine bundle $T^{2}\times\mathbb{C}$ are classified by
$H^{1}(T^{2},\mathcal{O}_{T^{2}})$ which is 1-dimensional, so there is just a
single deformation parameter, and furthermore this cohomology group vanishes
if a point is deleted from $T^{2}$.

In our case $Q_{0}$ is not an affine bundle, but is a deformation of a
$\mathbb{Z}_{2}$ orbifold of the this affine bundle. This leads to the
expectation (for which we do not have a rigorous justification) that there is
still only a single complex deformation preserving the asymptotic structure.
We can find a description of this deformation simply by orbifolding the $M$
shift given in (3.6), or more generally, by defining the transition map to be
any shift of the “fiber” ($z$) coordinate with a pole over the Weierstrass
point $w=0$ with residue proportional to $M$.

The $\mathbb{Z}_{2}$ orbifold action identifies
$\widetilde{v}\leftrightarrow-\widetilde{v}$, so we define invariant
coordinates ${\widetilde{z}}={\widetilde{v}}^{2}$,
${\widetilde{y}}=\widetilde{v}\eta$. Then (3.6) gives the transition map

$\displaystyle{\widetilde{y}}$ $\displaystyle=y+M\frac{\widetilde{P}}{x},$
$\displaystyle{\widetilde{z}}$
$\displaystyle=z+2M\frac{y}{wx}+M^{2}\frac{\widetilde{P}}{wx^{2}},$ (3.7)

in a neighborhood of the $w=0$ fiber of $(\mathbb{C}\times
T^{2})/\mathbb{Z}_{2}$. Thus $y$ is shifted by a term regular at $w=0$ (in the
$x=1$ patch), while $z$ is shifted by a double pole at $w=0$ plus single pole
and regular terms. (Recall that in local coordinates around $w=0$ — i.e.,
$\sqrt{w}$ in the $x=1$ patch — $y$ has a simple zero and $w^{-1}$ a double
pole.)

So far this has all been in the undeformed orbifold. To go to the $Q_{0}$
surface where the orbifold is deformed by turning on the ${\mu}_{i}$ masses,
it was argued in [29] that (3.7) does not change, since one simply shifts
$z\to z+\frac{Q}{P}$ and the same for ${\widetilde{z}}$. In Weierstrass form
this applies without change; just rewrite
$\frac{Q}{P}={\mu}_{0}^{2}\frac{x}{w}+\frac{\widetilde{Q}}{w\widetilde{P}}$.

But (3.7) has a qualitatively different pole structure at $w=0$ in $Q_{0}$
than in the undeformed orbifold. In the undeformed orbifold $y\sim\sqrt{w}$
was the local coordinate vanishing at $w=0$, but in the deformed orbifold
$w=0$ is no longer a branch point for $y$; instead $y$ has two solutions,
giving two disjoint curves over $w=0$, denoted by $S^{\pm}_{\infty}$ in (3.5).
In the neighborhood of $S^{\pm}_{\infty}$ the transition map (3.7) has a pair
of distinct simple poles along $S^{\pm}_{\infty}$ rather than a single double
pole.

Although the form of the $M$ shift given in (3.7) is perfectly valid, the form
of the resulting M5 brane curves do not match to those of the Inozemtsev
system in an algebraically simple way. Confident that there is only a single
complex deformation $Q_{0}\to Q_{M}$, we can modify (3.7) to any other
convenient transition map which has a simple pole in ${\widetilde{z}}$ at
$w=0$.

The property (2.30) of the spectral curve indicates that ${\widetilde{z}}$
should be chosen to have only a single pole at $w=0$ ($x=\infty$). A simple
transition map which does this is

$\displaystyle{\widetilde{y}}$ $\displaystyle=y,$
$\displaystyle{\widetilde{z}}$
$\displaystyle=z+2M\frac{(y+\mu_{0}x^{2})}{wx}.$ (3.8)

since (3.8) behaves near $w=0$ as

$\displaystyle{\widetilde{z}}$
$\displaystyle=\begin{cases}(1+\tfrac{M}{{\mu}_{0}})z+2{\mu}_{0}M\frac{x}{w}&\text{at
$S^{+}_{\infty}$}\\\ (1-\tfrac{M}{{\mu}_{0}})z&\text{at
$S^{-}_{\infty}$},\end{cases}$ (3.9)

so has a simple pole only along the $S^{+}_{\infty}$ fiber over $w=0$, and is
regular along the $S^{-}_{\infty}$ fiber.

We will see below that this transition map gives an M5 brane curve which is
easily matched to the Inozemtsev system spectral curve. Indeed, comparing
(2.28), (2.30) and (2.31) to (3.8) already indicates how most of the variables
and parameters of the integrable system will have to be matched to those of
the SW curve.

####  3.2.2 M5 brane curve

We now have a choice of placing of a stuck NS5 brane at $w=0$ at either the
$S^{+}_{\infty}$ or the $S^{-}_{\infty}$ fiber. This choice gives two
different forms of the curve upon turning on the $M$ deformation since it
gives different regularity conditions in the shifted ${\widetilde{z}}$
coordinates depending on whether the stuck brane coincides with the shift pole
or not. However, once again the property (2.30) of the spectral curve
indicating that there should be only a single pole dictates that the stuck NS5
brane should be placed at the $S^{+}_{\infty}$ fiber to coincide with the
position of the $M$ shift pole.

Before turning on the $M$ deformation, the M5 brane curve of [29] in the
$Q_{0}$ background specialized to the case of the $\mathrm{USp}(2N)$ $N_{f}=4$
theory has the form $0=z^{N}+\mathcal{A}(w,x,y,z)$ where $\mathcal{A}$ is a
polynomial in $z$ of order $N-1$, homogeneous of weight 0 in $(w,x,y)$, and
can have a simple pole along either the $S^{+}_{\infty}$ or the
$S^{-}_{\infty}$ fiber over $w=0$. This comes from the IIA brane construction
where $N$ is the number of D4 branes (after orbifolding) corresponding to the
rank of the gauge group and the pole at $w=0$ is a single stuck NS5 brane. A
linear basis of functions of $(w,x,y)$ homogeneous of weight 0 with at most a
simple pole at $w=0$ is $\\{1,x/w\\}$. Thus $\mathcal{A}$ can be written more
explicitly as

$\displaystyle 0$ $\displaystyle=z^{N}+A_{0}(z)+\frac{x}{w}A_{1}(z)$ (3.10)

where $A_{0,1}$ are arbitrary polynomials of order $N-1$ in $z$. Since the
curve is allowed to have a pole only along either $S^{+}_{\infty}$ or
$S^{-}_{\infty}$, but not along both, and since $\frac{x}{w}$ has a pole along
both, we must, in fact, have that $A_{1}(z)\equiv 0$. Thus, when $M=0$ the
$\mathrm{USp}(2N)$ $N_{f}=4$ curve is generically $N$ disjoint sections of
$Q_{0}$ corresponding to the $N$ roots of the polynomial $z^{N}+A_{0}(z)$.
This reflect the well-known fact — reviewed at the beginning of the next
section — that when $M=0$ the Coulomb branch of the theory is the $N$-fold
symmetric product of the rank-1 Coulomb branch.

We now turn on the antisymmetric mass deformation parameter $M$ by using the
transition map (3.8). Concretely, the curve for the shifted model is like the
curve for the non-shifted model (3.10) except that we should now allow
singularities only $S^{+}_{\infty}$ in a coordinate patch covering $w=0$ with
coordinates $(w,x,y,{\widetilde{z}})$ related to $(w,x,y,z)$ by (3.8).

Since we are only adding poles at $w=0$, and the only functions of weight zero
in $(w,x,y)$ with poles only there are $(x/w)^{\alpha}$ and
$(y/w^{2})(x/w)^{\alpha}$ for non-negative ${\alpha}$, the general form of the
curve in the $z$ patch will be

$\displaystyle
0=F:=z^{N}+\sum_{a=0}^{\infty}\frac{w^{2}A_{a}+yE_{a}}{w^{2}}\Bigl{(}\frac{x}{w}\Bigr{)}^{a}$
(3.11)

where the $A_{a}$ and $E_{a}$ are arbitrary polynomials of order $N-1$ in $z$.

Though (3.11) is a correct general form for the curve, the infinite sum of
pole terms is intimidating. It is not too hard to bound the number of pole
terms that can contribute by using the condition that there is only at most a
first-order pole at $w=0$ in the shifted ${\widetilde{z}}$ variable. Under the
transition map (3.8), ${\widetilde{z}}=z+yP_{1}+P_{1}$, where $P_{a}$ refers
to a generic rational function of $w$ with poles of up to order $a$ at $w=0$
(work in the $x=1$ patch). Using the fact that $y^{2}\sim zP_{0}+P_{0}$, one
can recursively eliminate all higher powers of $y$ in
${\widetilde{z}}^{\ell}\sim z^{\ell}+\cdots$ to find that

$\displaystyle{\widetilde{z}}^{\ell}\lesssim
z^{\ell}+\sum_{a=1}^{\ell-1}z^{\ell-a}(P_{2a}+yP_{2a-1}).$ (3.12)

The $\lesssim$ sign means that we have pole orders bounded by the terms on the
right. In the ${\widetilde{z}}$ coordinate the curve is to have at most a
simple pole at $w=0$, so will have the form
${\widetilde{z}}^{N}+\sum_{\ell=0}^{N-1}{\widetilde{z}}^{\ell}P_{1}$.
Substituting (3.12) into this then shows that in the $z$ coordinate the
highest-order poles are of the form

$\displaystyle F$ $\displaystyle\ \lesssim\
z^{N}+\sum_{\ell=0}^{N-1}\sum_{a=0}^{\ell}z^{\ell-a}(P_{2a+1}+yP^{\prime}_{2a})\
\sim\ z^{N}+\sum_{\ell=1}^{N}z^{N-\ell}(P_{2\ell-1}+yP^{\prime}_{2\ell-2}),$
(3.13)

where by $P^{\prime}_{a}$ we mean the usual $a$th-order pole for $a\neq 0$,
but $P^{\prime}_{0}\equiv 0$. Comparing to (3.11) then implies that the curve
is

$\displaystyle
0=z^{N}+\sum_{\ell=1}^{N}z^{N-\ell}\Bigl{(}\sum_{a=0}^{2\ell-1}A_{a\ell}\frac{x^{a}}{w^{a}}+\sum_{a=0}^{2\ell-2}E_{a\ell}\frac{yx^{a}}{w^{a+2}}\Bigr{)}.$
(3.14)

Note that (3.12) and thus (3.14) does not give the optimal bound on the order
of the poles appearing in the curve, but instead just gives a reasonable upper
bound. This is not a big deal since any “extra” terms will be set to zero upon
demanding only a simple pole appear in the ${\widetilde{z}}$ patch.

The coefficients in (3.14) are determined by demanding the correct pole
behavior after shifting to the ${\widetilde{z}}$ variable. Concretely, make
the inverse change of coordinates (3.8) in the curve by substituting
$z\to{\widetilde{z}}-2M(y+{\mu}_{0}x^{2})/(wx)$ in (3.14). The 5 brane curve
(3.14) in the $x=1$ patch becomes in terms of the (3.8) shifted variables

$\displaystyle
0=\Bigl{(}{\widetilde{z}}-2M\frac{y+\mu_{0}}{w}\Bigr{)}^{N}+\sum_{\ell=1}^{N}\Bigl{(}{\widetilde{z}}-2M\frac{y+\mu_{0}}{w}\Bigr{)}^{N-\ell}\Bigl{(}\sum_{a=0}^{2\ell-1}\frac{A_{a\ell}}{w^{a}}+\sum_{a=0}^{2\ell-2}\frac{yE_{a\ell}}{w^{a+2}}\Bigr{)}.$
(3.15)

Expand this around $w=0$ keeping only pole terms
${\widetilde{z}}^{\ell}w^{-a}$ and ${\widetilde{z}}^{\ell}yw^{-a}$ for $a>0$.
We do this by using iteratively that
$y^{2}=({\widetilde{z}}w-2M(y+\mu_{0})+{\mu}_{0}^{2})\widetilde{P}+w^{2}\widetilde{Q}$,
with $\widetilde{P}=1+s_{2}w^{2}-s_{3}w^{3}$, and
$\widetilde{Q}=\sum_{j}{\mu}_{j}^{2}{\epsilon}_{j}\prod_{k\neq j}(1-e_{k}w)$
to reduce all terms to either ${\widetilde{z}}w^{-a}$ or
${\widetilde{z}}yw^{-a}$.

Motivated by the form of the spectral curve of the integrable system, as
discussed above, we choose the to put the stuck 5 brane at $S^{+}_{\infty}$.
This means that the $A_{a\ell}$ and $E_{a\ell}$ coefficients are determined by
requiring that all second- and higher-order poles along $S^{\pm}_{\infty}$ and
the simple poles along $S^{-}_{\infty}$ cancel in the ${\widetilde{z}}$
variables. Only a simple pole along $S^{+}_{\infty}$ is allowed, corresponding
to the stuck brane.

####  3.2.3 The rank-1 SW curve

Specializing to rank $N=1$, there is no $M$ deformation, and the M5 brane
curve (3.10) becomes simply

$\displaystyle 0=z+A_{01}.$ (3.16)

We can use this to eliminate $z$ in the (3.2.1) to give the an elliptic curve
in Weierstrass form for the SW curve. We recall here for later convenience the
expressions for the $Q_{0}$ surface and 1-form written in the $w=1$ patch
coordinates,

$\displaystyle y^{2}$
$\displaystyle=(z+{\mu}_{0}^{2}x)\widetilde{P}+\widetilde{Q},$
$\displaystyle{\lambda}$ $\displaystyle=\frac{ydx}{\widetilde{P}},$ (3.17)

where

$\displaystyle\widetilde{P}$ $\displaystyle:=\prod_{i=1}^{3}(x-e_{i}),$
$\displaystyle\widetilde{Q}$
$\displaystyle:=\sum_{j=1}^{3}{\mu}_{j}^{2}{\epsilon}_{j}\prod_{k\neq
j}(x-e_{k}).$ (3.18)

####  3.2.4 The rank-2 SW curve

At rank $N=2$ the coefficients in the general M5 brane curve (3.14) are
determined by the procedure described below equation (3.15). For $N=2$ the
highest power of $y$ appearing in (3.15) is 2, and only a single iteration of
using the $Q_{0}$ surface equation to reduce the power of $y$ is needed. As a
result the constraints on the coefficients are not overly complicated, though
it is still useful to use a computer algebra system to solve the constraints.
The result is that the M5 brane curve is (written in the $w=1$ patch
coordinates)

$\displaystyle
0=z^{2}+A_{01}z+A_{20}-4M^{2}zx-8M^{2}{\mu}_{0}(y+{\mu}_{0}x^{2}).$ (3.19)

The intersection of (3.19) with the $Q_{0}$ surface (3.17) and the restriction
of the one-form to this intersection then give a genus-2 SW curve and
associated meromorphic 1-form.

##  4 Matching spectral curve to M5 brane curve

The Coulomb branch of the $\mathrm{USp}(2N)$ $N_{f}=4$ theory is isomorphic as
a complex space (though not as a metric space) to $\mathbb{C}^{N}$ with
coordinates given by the gauge invariant vacuum expectation values
$u_{i}:=\mathrm{tr}(\wedge^{2i}\Phi)$, $i=1,2,\ldots,N$ which have scaling
dimensions $2,4,\ldots,2N$ at the conformal point. The Coulomb branch of the
massless theory has the same complex structure as the classical moduli space.
At a generic point on the Coulomb branch of the massless theory, the adjoint
vev can be diagonalized,
$\Phi=\mathrm{diag}(\pm\phi_{1},\pm\phi_{2},\cdots,\pm\phi_{N})$, in which
case $u_{i}=e_{i}(\phi_{1}^{2},\phi_{2}^{2},\cdots,\phi_{N}^{2})$,
$i=1,2,\ldots,N$, where $e_{i}$ is the $i$-th elementary symmetric polynomial.
As long as the antisymmetric mass vanishes, the matrix of $\mathrm{U}(1)^{N}$
complex gauge couplings is diagonal,
${\tau}_{ij}={\delta}_{ij}{\tau}(\phi_{i}^{2})$.

In the case when all the masses vanish, ${\tau}(\phi_{i}^{2})={\tau}$, i.e.,
has the same constant value. We thus have the same abelian variety with period
matrix $\tau_{ij}=\delta_{ij}\tau$ at all points on the Coulomb branch except
the origin. The singular fiber above the origin is given by the orbifold
$T^{2N}/G(2,1,N)\simeq\mathbb{C}^{N}/([\mathbb{Z}+\tau\mathbb{Z})^{N}\rtimes
G(2,1,N)]$. Then the total space of Coulomb branch is identical to the phase
space of the Inozemtsev system with zero couplings.

Thus for vanishing masses the field theory Coulomb branch geometry is
correctly described by the Inozemtsev system. In the remainder of this section
we present parameter and variable identifications for the rank $N=1$ and $N=2$
cases, showing that the M5 brane SW curve and 1-form and the spectral curve
and 1-form of the Inozemtsev system coincide for non-vanishing masses
(deformation parameters). We stop at $N=2$ because the matching of parameters
becomes increasingly complicated for larger values of $N$.

###  4.1 The $N=1$ case

Recall that the $N=1$ spectral curve is given by (2.19), and the one-form by
${\lambda}=kd\alpha$. Introduce coordinates $(x,y)$ related to $(k,\alpha)$ by

$\displaystyle x$ $\displaystyle=\wp(\alpha),$ $\displaystyle y$
$\displaystyle=\frac{1}{4}\wp^{\prime}(\alpha)k,$ (4.1)

where the prime means derivative with respect to $\alpha$. These definitions
were motivated in (2.31) by the pole structure of the spectral curve. We then
find, using the Weierstrass $\wp$-function identities

$\displaystyle(\wp^{\prime}(\alpha))^{2}$
$\displaystyle=4\prod_{i=1}^{3}(\wp(\alpha)-e_{i}),$
$\displaystyle\wp(\alpha+{\omega}_{i})$
$\displaystyle=e_{i}+\frac{\prod_{j\neq
i}^{3}(e_{i}-e_{j})}{\wp(\alpha)-e_{i}},$ (4.2)

where

$\displaystyle e_{i}:=\wp({\omega}_{i}),\quad i=1,2,3,$ (4.3)

that the spectral curve and one-form become

$\displaystyle y^{2}$
$\displaystyle=\tfrac{1}{4}(h_{2}+{\gamma})\prod^{3}_{i=1}(x-e_{i})+\tfrac{1}{4}(g_{0}^{\vee})^{2}x\prod^{3}_{i=1}(x-e_{i})+\tfrac{1}{4}\sum^{3}_{i=1}(g_{i}^{\vee})^{2}\prod^{3}_{j\neq
i}(x-e_{i})(e_{i}-e_{j}),$ $\displaystyle kd\alpha$
$\displaystyle=\frac{ydx}{\prod^{3}_{i=1}(x-e_{i})},$ (4.4)

where ${\gamma}:=\sum^{3}_{i=1}(g_{i}^{\vee})^{2}e_{i}$. These are easily seen
to coincide with the SW curve and 1-form given in (3.16) and (3.17) with the
parameter identifications

$\displaystyle\mu_{i}^{2}$ $\displaystyle=\frac{1}{4}(g_{i}^{\vee})^{2},$
$\displaystyle A_{01}$ $\displaystyle=-\frac{1}{4}(h_{2}+{\gamma}).$ (4.5)

###  4.2 The $N=2$ case

Recall that the $BC_{2}$ spectral curve is given by (2.22). With the same
change of variables (4.1) as in the $BC_{1}$ case, which matched the 1-forms,
the $BC_{2}$ curve becomes

$\displaystyle(k^{2}-u^{\vee})^{2}-h_{2}(k^{2}-u^{\lor})+h_{4}-4g^{2}\left(x(k^{2}-u^{\vee})+4g^{\lor}_{0}y+2(g^{\vee}_{0})^{2}x^{2}+x{\gamma}\right)=0.$
(4.6)

Recall that
$u^{\vee}:=\sum_{r=0}^{3}(g_{r}^{\vee})^{2}\wp(\alpha+{\omega}_{r})$ and
${\gamma}:=\sum_{r=1}^{3}(g^{\lor}_{r})^{2}e_{r}$. Then with the parameter
identifications

$\displaystyle{\mu}_{0}$ $\displaystyle=\frac{1}{2}g_{0}^{\vee},$
$\displaystyle{\mu}_{i}^{2}$ $\displaystyle=\frac{1}{4}(g_{i}^{\vee})^{2}\
\text{ for }\ i\in\\{1,2,3\\},$ $\displaystyle M^{2}$
$\displaystyle=\frac{1}{4}g^{2},$ $\displaystyle A_{01}$
$\displaystyle=-\frac{1}{4}(h_{2}+2{\gamma}),$ $\displaystyle A_{02}$
$\displaystyle=\frac{1}{16}(h_{4}+{\gamma}h_{2}+{\gamma}^{2}),$ $\displaystyle
z$ $\displaystyle=\frac{1}{4}(k^{2}-u^{\vee}+{\gamma}),$ (4.7)

and using the Weierstrass identities (4.2), we find the spectral curve becomes
the pair of equations

$\displaystyle y^{2}$
$\displaystyle=(z+{\mu}_{0}^{2}x)\widetilde{P}+\widetilde{Q},$ $\displaystyle
0$
$\displaystyle=z^{2}+z\left(A_{01}-4M^{2}x\right)+\left(A_{02}-8M^{2}{\mu}_{0}^{2}x^{2}\right)-8M^{2}{\mu}_{0}y,$
(4.8)

which coincides with the M5 brane curve 3.19 and background surface 3.17. Note
that the definition of $z$ (up to a constant shift) was already motivated in
(2.28) by the pole structure of the spectral curve.

## Acknowledgement

We would like to thank Mario Martone, Joseph Minahan, Yiwen Pan, Oliver
Schlotterer for helpful discussions. YL is grateful to Yuji Tachikawa for
pointing out the paper [27] and for encouragement. The collaboration between
OC and YL started during the workshop “Elliptic integrable systems, special
functions, and quantum field theory” which took place in June, 2019 at
Nordita, Stockholm. OC is grateful to the organizers for the invitation, and
OC and YL would like to thank Nordita for the hospitality and stimulating
environment. The work of PCA is supported in part by DOE grant SC0011784. The
work of YL is supported by the Knut and Alice Wallenberg Foundation under
grant Dnr KAW 2015.0083.

##  Appendix A Appendix

###  A.1 Elliptic functions and identities

We use the following functions

$\sigma_{\alpha}^{r}(x)=\frac{\vartheta_{r+1}(x-\alpha)\vartheta_{1}^{\prime}(0)}{\vartheta_{r+1}(x)\vartheta_{1}(-\alpha)}\,,\quad
r=0,1,2,3\,,$ (A.1)

where $\vartheta_{1,2,3,4}(x|\tau)$ are the Jacobi theta functions. A summary
of their main properties can be found in [56]; in particular, we have

$\sigma_{\alpha}^{r}(x+\omega)=e^{2\pi
i\alpha\partial_{\tau}\omega}\sigma_{\alpha}^{r}(x)\quad\text{for}\
\omega\in\mathbb{Z}+\tau\mathbb{Z}\,.$ (A.2)

(Here we use the shorthand notation $\partial_{\tau}(a+b\tau)=b$.) We also
denote $\sigma_{\alpha}^{0}(x)$ simply by $\sigma_{\alpha}(x)$, that is,

$\sigma_{\alpha}(x)=\frac{\vartheta_{1}(x-\alpha)\vartheta_{1}^{\prime}(0)}{\vartheta_{1}(x)\vartheta_{1}(-\alpha)}\,.$
(A.3)

The functions (A.1) are related to each other by translations by the half-
periods
$(\omega_{0},\omega_{1},\omega_{2},\omega_{3})=(0,\frac{1}{2},\frac{1+\tau}{2},\frac{\tau}{2})$:

$\sigma_{\alpha}^{r}(x)=e^{2\pi
i\alpha\partial_{\tau}\omega_{r}}\sigma_{\alpha}(x-\omega_{r}).$ (A.4)

For given coupling parameters $g_{0,1,2,3}$, we further define

$v_{\alpha}(x)=v_{\alpha}(x;g_{0},g_{1},g_{2},g_{3})=\sum_{r=0}^{3}g_{r}\sigma_{2\alpha}^{r}(x).$
(A.5)

Note the properties

$\displaystyle\sigma_{-\alpha}(-x)$ $\displaystyle=-\sigma_{\alpha}(x)\,,\quad
v_{-\alpha}(-x)=-v_{\alpha}(x)\,,$ (A.6)

and the following identities:

$\displaystyle\sigma_{\alpha}(-x)\sigma_{\alpha}(x)$
$\displaystyle=\wp(\alpha)-\wp(x),$ (A.7) $\displaystyle
v_{\alpha}(-x)v_{\alpha}(x)$
$\displaystyle=\sum_{r=0}^{3}\Big{(}(g_{r}^{\vee})^{2}\wp(\alpha+\omega_{r})-(g_{r})^{2}\wp(x+\omega_{r})\Big{)}\,,$
(A.8)

where $g^{\vee}_{i}$ are the dual parameters (2.9). Using the notation (2.2),
(2.28), the last relation can be written as
$v_{\alpha}(-x)v_{\alpha}(x)=u^{\vee}(\alpha)-u(x)$.

Another useful property of $v_{\alpha}(x)$ is the following duality:

$v_{\alpha}(x;g_{0},g_{1},g_{2},g_{3})=v_{-x}(-\alpha;g^{\vee}_{0},g^{\vee}_{1},g^{\vee}_{2},g^{\vee}_{3})=-v_{x}(\alpha;g^{\vee}_{0},g^{\vee}_{1},g^{\vee}_{2},g^{\vee}_{3})\,.$
(A.9)

This can be checked by comparing translation properties and residues in the
$x$-variable.

Finally, let us state how $\sigma_{\alpha}(x)$ and $v_{\alpha}(x)$ behave
under action of $\gamma\in\mathrm{SL}(2,\mathbb{Z})$. We will use the group
homomorphism $\pi$ from $\mathrm{SL}(2,\mathbb{C})$ to the permutation group
$S_{3}$ defined on the generators as follows:

$\pi\,:\,\mathrm{SL}(2,\mathbb{C})\to
S_{3}\,,\quad\gamma\mapsto\pi_{\gamma}\,,\quad\left(\begin{smallmatrix}1&1\\\
0&1\end{smallmatrix}\right)\mapsto
s_{23}\,,\quad\left(\begin{smallmatrix}0&1\\\
-1&0\end{smallmatrix}\right)\mapsto s_{13}\,.$ (A.10)

Note that the kernel of $\pi$ is the principal congruence subgroup
${\Gamma}(2)\subset\mathrm{SL}(2,\mathbb{C})$.

Take ${\gamma}=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\mathrm{SL}(2,\mathbb{Z})$ and define
$\tau^{\prime}$, $\alpha^{\prime}$, $x^{\prime}$, $g^{\prime}_{i}$ in the
following way:

$\displaystyle{\tau}^{\prime}$
$\displaystyle=\frac{a{\tau}+b}{c{\tau}+d}\,,\quad\alpha^{\prime}=(c{\tau}+d)^{-1}\alpha\,,\quad
x^{\prime}=(c{\tau}+d)^{-1}x\,,$ (A.11) $\displaystyle g_{0}^{\prime}$
$\displaystyle=g_{0}\,,\quad
g^{\prime}_{r}=g_{\pi_{{\gamma}}(r)}\quad\text{for}\ r=1,2,3\,.$ (A.12)

With this notation, we have:

$\displaystyle{\sigma}_{\alpha^{\prime}}(x^{\prime}|{\tau}^{\prime})$
$\displaystyle=(c{\tau}+d)\exp\left(-\frac{2\pi ic}{c{\tau}+d}\alpha
x\right){\sigma}_{\alpha}(x|{\tau})\,,$ (A.13)
$\displaystyle{\sigma}^{\pi_{\gamma}(r)}_{\alpha^{\prime}}(x^{\prime}|{\tau}^{\prime})$
$\displaystyle=(c{\tau}+d)\exp\left(-\frac{2\pi ic}{c{\tau}+d}\alpha
x\right){\sigma}^{r}_{\alpha}(x|{\tau})\,,\quad r=1,2,3\,.$ (A.14)

These transformations can be deduced easily using the modular transformations
of Jacobi theta functions. As a corollary,

$\displaystyle
v_{\alpha^{\prime}}(x^{\prime};g_{0}^{\prime},g_{1}^{\prime},g_{2}^{\prime},g_{3}^{\prime}|{\tau}^{\prime})=(c{\tau}+d)\exp\left(-\frac{4\pi
ic}{c{\tau}+d}\alpha x\right)v_{\alpha}(x;g_{0},g_{1},g_{2},g_{3}|{\tau})\,.$
(A.15)

###  A.2 Calculating the $N=2$ spectral curve

The $N=2$ spectral curve is defined by the characteristic polynomial

$\det(L-k\mathrm{Id})=k^{4}+a_{1}k^{3}+a_{2}k^{2}+a_{3}k+a_{4}$ (A.16)

of the Lax matrix (2.4.2). By direct calculation,

$\displaystyle a_{1}$ $\displaystyle=0$ $\displaystyle a_{2}$
$\displaystyle=-\Bigl{(}p_{1}^{2}-p_{2}^{2}+2g^{2}\bigl{(}\sigma_{\alpha}(-q_{12})\sigma_{\alpha}(q_{12})+\sigma_{\alpha}(-q^{+}_{12})\sigma_{\alpha}(q^{+}_{12})\bigr{)}$
$\displaystyle\quad+v_{\alpha}(-q_{1})v_{\alpha}(q_{1})+v_{\alpha}(-q_{2})v_{\alpha}(q_{2})\Bigr{)}$
$\displaystyle a_{3}$
$\displaystyle=-2g^{2}\Big{(}v_{\alpha}(q_{2})\sigma_{\alpha}(q_{12})\sigma_{\alpha}(-q^{+}_{12})+v_{\alpha}(q_{1})\sigma_{\alpha}(-q_{12})\sigma_{\alpha}(-q^{+}_{12})$
$\displaystyle\quad+v_{\alpha}(-q_{1})\sigma_{\alpha}(q_{12})\sigma_{\alpha}(q^{+}_{12})+v_{\alpha}(-q_{2})\sigma_{\alpha}(-q_{12})\sigma_{\alpha}(q^{+}_{12})\Big{)}$
$\displaystyle a_{4}$
$\displaystyle=p_{1}^{2}p_{2}^{2}+v_{\alpha}(-q_{2})v_{\alpha}(q_{2})p_{1}^{2}+v_{\alpha}(-q_{1})v_{\alpha}(q_{1})p_{2}^{2}$
$\displaystyle\quad+2g^{2}\Big{(}\sigma_{\alpha}(-q^{+}_{12})\sigma_{\alpha}(q^{+}_{12})-\sigma_{\alpha}(-q_{12})\sigma_{\alpha}(q_{12})\Big{)}p_{1}p_{2}$
$\displaystyle\quad+v_{\alpha}(-q_{1})v_{\alpha}(q_{1})v_{\alpha}(-q_{2})v_{\alpha}(q_{2})$
$\displaystyle\quad-g^{2}\Big{(}v_{\alpha}(q_{1})v_{\alpha}(q_{2})\sigma_{\alpha}(-q^{+}_{12})^{2}+v_{\alpha}(-q_{1})v_{\alpha}(q_{2})\sigma_{\alpha}(q_{12})^{2}$
$\displaystyle\quad+v_{\alpha}(q_{1})v_{\alpha}(-q_{2})\sigma_{\alpha}(-q_{12})^{2}+v_{\alpha}(-q_{1})v_{\alpha}(-q_{2})\sigma_{\alpha}(q^{+}_{12})^{2}\Big{)}$
$\displaystyle\quad+g^{4}\Big{(}\sigma_{\alpha}(-q_{12})\sigma_{\alpha}(q_{12})-\sigma_{\alpha}(-q^{+}_{12})\sigma_{\alpha}(q^{+}_{12})\Big{)}^{2}$
(A.17)

where we have used the abbreviations $q_{ij}=q_{i}-q_{j}$ and
$q_{ij}^{+}=q_{i}+q_{j}$.

Using (A.7) and (A.8), we easily find that

$\displaystyle a_{2}$
$\displaystyle=-\Big{(}h_{2}+4g^{2}\wp(\alpha)+2\sum_{r=0}^{3}(g_{r}^{\vee})^{2}\wp(\alpha+\omega_{r})\Big{)}=-(h_{2}+4g^{2}\wp(\alpha)+2u^{\vee}(\alpha))\,,$
(A.18)

where

$h_{2}=p_{1}^{2}+p_{2}^{2}-u(q_{1})-u(q_{2})-2g^{2}\left(\wp(q_{12})+\wp(q_{12}^{+})\right)\,.$
(A.19)

To calculate $a_{3}$, we first note that it is elliptic in $q_{1,2}$ with
possible first order poles along the mirrors $q_{i}=0$ for $i=1,2$ and
$q_{1}\pm q_{2}=0$. However, it is symmetric under interchanging $q_{1},q_{2}$
and changing their signs arbitrarily. Hence, $a_{3}$ cannot have a first order
pole along any mirror, thus it is regular elliptic, i.e. constant independent
of $q_{1}$, $q_{2}$. After than we can evaluate $a_{2}$ at convenient values
of $q_{1},q_{2}$. The result is

$\displaystyle a_{3}$
$\displaystyle=-2g^{2}\Big{(}\sum_{i=0}^{3}g_{i}\Big{)}\wp^{\prime}(\alpha)=-4g^{2}g_{0}^{\vee}\wp^{\prime}(\alpha).$
(A.20)

It remains to deal with $a_{4}$. By using (A.7) and (A.8) repeatedly, we
rearrange it into

$\displaystyle a_{4}$
$\displaystyle=\left(\sum_{r=0}^{3}(g_{r}^{\vee})^{2}\wp(\alpha+\omega_{r})\right)h_{2}+\left(\sum_{r=0}^{3}(g_{r}^{\vee})^{2}\wp(\alpha+\omega_{r})\right)^{2}$
$\displaystyle\qquad+\left(p_{1}p_{2}+g^{2}\wp(q_{12})-g^{2}\wp(q_{12}^{+})\right)^{2}$
$\displaystyle\qquad\mbox{}-u(q_{1})p_{2}^{2}-u(q_{2})p_{1}^{2}+u(q_{1})u(q_{2})-g^{2}b$
(A.21)

where we have introduced

$\displaystyle b$
$\displaystyle=v_{\alpha}(q_{1})v_{\alpha}(q_{2})\sigma_{\alpha}(-q^{+}_{12})^{2}+v_{\alpha}(-q_{1})v_{\alpha}(q_{2})\sigma_{\alpha}(q_{12})^{2}$
$\displaystyle\ \
\qquad+v_{\alpha}(q_{1})v_{\alpha}(-q_{2})\sigma_{\alpha}(-q_{12})^{2}+v_{\alpha}(-q_{1})v_{\alpha}(-q_{2})\sigma_{\alpha}(q^{+}_{12})^{2}\,.$
(A.22)

Calculating $b$ is more involved, so we just give a sketch. As the first step,
we analyse the $2$nd order poles in $q_{1},q_{2}$ and find that the following
expression agrees with $b$ up to an extra term $c$ having first order poles
only:

$\displaystyle b$
$\displaystyle=(2u^{\vee}(\alpha)-u(q_{1})-u(q_{2}))(\wp(q_{12})+\wp(q_{12}^{+}))+\sum_{r=1}^{3}2g_{r}^{2}\wp(q_{1}+\omega_{r})\wp(q_{2}+\omega_{r})+c\,.$
(A.23)

Using the symmetry arguments once more, we conclude that $c$ must be regular,
i.e. it is just a function of $\alpha$. In addition, we know that
$c=c(\alpha)$ is even elliptic. It is also easy to check that $c(\alpha)$ has
a 4th order pole at $\alpha=0$ and 2nd order poles at $\alpha=\omega_{1,2,3}$.
To determine $c(\alpha)$ from that, we analyse the Laurent expansion of $b$ in
$\alpha$ near $\alpha=0$ and $\alpha=\omega_{1,2,3}$. We skip the details and
just give the answer:

$\displaystyle c$
$\displaystyle=2\left[4(g_{0}^{\vee})^{2}\wp(\alpha)^{2}-2\wp(\alpha)\left(u^{\vee}(\alpha)-\sum_{i=1}^{3}(g^{\vee}_{r})^{2}e_{r}\right)\right]+d\,,$
(A.24)

up to a possible constant $d$ which may depend on $g_{i}$ and
$e_{i}=\wp(\omega_{i})$, but not on $\alpha$.

Backward substitution of (A.24) and (A.23) into (A.2) gives the answer for
$a_{4}$, after which all that remains is to rearrange the terms based on the
form of the quartic Hamiltonian $h_{4}$ (2.1). The constant $d$ in (A.24) can
always be absorbed into $h_{4}$, so can be ignored.

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