aainstitutetext: Theoretical Physics Group, The Blackett Laboratory, Imperial
College London, Prince Consort Road London, SW7 2AZ, UKbbinstitutetext: Jesus
College, Oxford University, OX1 3DW, UKccinstitutetext: New College, Oxford
University, OX1 3BN, UK

# Orthosymplectic Implosions

Antoine Bourget b , Andrew Dancer a , Julius F. Grimminger a , Amihay Hanany c
, Frances Kirwan a and Zhenghao Zhong<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We propose quivers for Coulomb branch constructions of universal implosions
for orthogonal and symplectic groups, extending the work on special unitary
groups in DHK . The quivers are unitary-orthosymplectic as opposed to the
purely unitary quivers in the A-type case. Where possible we check our
proposals using Hilbert series techniques.

††preprint: Imperial/TP/21/AH/02

##  1 Introduction

We study examples of symplectic duality, that is, duality between the Higgs
and Coulomb branches of a 3-dimensional $N=4$ SUSY theory. Mathematically,
this is a duality between varieties with a complex-symplectic (or even
hyperKähler) structure admitting a rotating circle action.

We study quivers which are candidates for the symplectic duals of universal
hyperKähler implosions for various groups. We briefly recall some relevant
properties of these implosions here, referring the reader to
guillemin2002symplectic ; dancer2013implosion ; dancer2016symplectic  for a
more detailed description.

The universal hyperKähler implosion for a complex reductive group
$G=K_{\mathbb{C}}$ with maximal compact subgroup $K$, is supposed to be a
complex-symplectic variety, that is in fact hyperKähler in a suitable
stratified sense. It has a complex-symplectic action of $K_{\mathbb{C}}\times
T_{\mathbb{C}}$, where $T$ is the maximal torus of $K$. This action is
supposed to be the complexification of a hyperKähler action of the compact
group $K\times T$. Moreover, the dimension of the implosion is equal to the
dimension of $K_{\mathbb{C}}\times T_{\mathbb{C}}$.

HyperKähler reduction of the implosion by $T$, or equivalently complex-
symplectic reduction by the complex torus $T_{\mathbb{C}}$, will give the
Kostant varieties of the Lie algebra $\mathfrak{k}_{\mathbb{C}}$, that is, the
varieties we get by fixing the values of all Ad-invariant polynomials on this
Lie algebra. In particular, reduction at zero gives the nilpotent cone
$\mathcal{N}_{G}$. This is consistent with the dimension statements above, as
each Kostant variety has dimension equal to $\dim K_{\mathbb{C}}-{\rm
rank}\,K_{\mathbb{C}}$. The remaining $K_{\mathbb{C}}$ action on the implosion
then descends to the natural action on the Kostant varieties. We can thus
think of the implosion as a master space which yields the Kostant varieties on
reduction by the torus action at the appropriate level.

The implosion also has a description as a Geometric Invariant Theory quotient
by the maximal unipotent subgroup $N$ of $G=K_{\mathbb{C}}$. Explicitly, the
implosion is the GIT quotient

$(G\times{\mathfrak{n}}^{\circ})\mathbin{/\mkern-6.0mu/}N$ (1)

where $\mathfrak{n}^{\circ}$ denotes the annihilator of $\mathfrak{n}={\rm
Lie\;}N$ in the dual Lie algebra $\mathfrak{g}^{*}$. We can view this quotient
as the complex-symplectic quotient, in the GIT sense, of the cotangent bundle
$T^{*}G$ by $N$. The left action of $G$ on $T^{*}G$ survives, but the right
action is broken to a $T_{\mathbb{C}}$ action, giving the required action on
the implosion. We recall that Kronheimer has shown the existence of a complete
hyperKähler metric on $T^{*}G$, invariant under the action of $K\times K$.

On choosing an invariant inner product on $\mathfrak{g}$, we may identify
${\mathfrak{n}}^{\circ}$ with the Borel algebra $\mathfrak{b}$. Complex-
symplectic reduction by $T_{\mathbb{C}}=B/N$ then fixes the Cartan component
of $\mathfrak{b}$ to be zero, and replaces the $N$ quotient by a $B$ quotient,
yielding the GIT quotient $(G\times{\mathfrak{n}})\mathbin{/\mkern-6.0mu/}B$.
The fact that this gives the nilpotent cone is just the statement that the
Springer resolution $G\times_{B}{\mathfrak{n}}$ of the nilpotent cone is an
affinisation map. We can view the implosion in this way as occupying an
intermediate position between the cotangent bundle $T^{*}G$ and the nilpotent
cone–the implosion is the symplectic reduction in the GIT sense of $T^{*}G$ by
the maximal unipotent, and further reduction by the complex torus yields the
nilpotent cone.

In dancer2013implosion  two of the authors and Swann produced the implosion
for $K=\mathrm{SU}(n)$ as a hyperKähler quiver variety. As a complex-
symplectic variety the implosion is known to exist for general $K$ by work of
Ginzburg-Riche Ginzburg-Riche:2015  (we note that as $N$ is nonreductive it is
a nontrivial result that the quotient exists as an algebraic variety).

Implosion spaces should admit symplectic duals – in particular as mentioned
above the $\mathrm{SU}(n)$ implosion is a hyperKähler quotient of a linear
space at level zero, so has a $Sp(1)$ action rotating the complex structures.

In this paper we present candidates for the duals of the implosions for
$K=\mathrm{SO}(2n)$ and $\mathrm{SO}(2n+1)$, extending the discussion for
$\mathrm{SU}(n)$ in DHK . One test that we will perform is to check that the
dual spaces give the correct symmetry group of the original implosion. Another
check is on the dimension, using Nakajima’s observation that if the Higgs
branch is the hyperKähler quotient of a linear space $M$ by a compact group
$H$, then the Coulomb branch should be birational to the quotient by the Weyl
group of the cotangent bundle of the complexified dual maximal torus of $H$;
hence

$\dim_{\mathbb{R}}({\rm Coulomb\;branch})=4\;{\rm rank\;}H.$ (2)

We also do some checks that the hyperKähler quotient of the implosion by the
torus action gives the nilpotent cone, as expected.

##  2 Bouquet for Orthogonal groups

For this section, we will focus on quivers whose Coulomb branch is the
nilpotent cone of a complex semisimple group $G=K_{\mathbb{C}}$. We introduce
the process of explosion where a flavour node of rank $n$ is exploded into $n$
rank 1 gauge nodes. These unbalanced nodes generate the required Abelian
symmetries in the Coulomb branch, as required for the implosion. The number of
rank 1 nodes is chosen so as to preserve the balancing condition for the
remaining nodes in the original quiver, so that the $G$ symmetry in the
Coulomb branch is preserved as well, so the implosion has commuting symmetries
of $G$ and its maximal torus as required.

For unitary quivers this is well understood where we start with a
$T[\mathrm{SU}(n)]$ theory with $\mathrm{SU}(n)$ flavour node. We then explode
the $\mathrm{SU}(n)$ flavour node into $n$ $\mathrm{U}(1)$ gauge nodes (recall
that one has to ungauge an overall $U(1)$ in the resulting unframed quiver).

$n-1$$n$1111$n$

Quiver for the nilpotent cone $\mathcal{N}_{\mathrm{SU}(n)}$

Explosion

Exploded quiver

1212$n-1$$n-2$

(3)

The resulting quiver has the global symmetry changed from $\mathrm{SU}(n)$ to
$\mathrm{SU}(n)\times\mathrm{U}(1)^{n-1}$. In DHK  various computational
checks for the duals of these examples were made, including calculating the
Hilbert series, verifying the dimension of the global symmetry group and
checking Nakajima’s equality, and checking that torus reduction yielded the
nilpotent cone.

We now extend this to orthosymplectic quivers following the extension in the
mathematical literature in dancer2016symplectic . We adopt the usual
convention in our diagrams that red nodes with label $m$ denote orthogonal
groups $\mathrm{SO}(m)$111The choice of $SO$ or $O$ is important here as the
Coulomb branch is sensitive to discrete factors in the gauge groups. For
orthosymplectic quivers that are closures of maximal nilpotent orbits of
$\mathrm{SO}(2n)$ and $\mathrm{SO}(2n+1)$, the Coulomb branch Hilbert series
are computed explicitly in Cabrera:2017ucb  and contains only $SO$ and $USp$
gauge groups.  and blue nodes with label $2k$ denote symplectic groups
$\mathrm{USp}(2k)=\mathrm{Sp}(k)$. We recall that a $\mathrm{USp}(2k)$ gauge
node is balanced if the neighboring $\mathrm{SO}(m_{j})$ nodes satisfy:

$4k=-2+\sum\limits_{j}m_{j}$ (4)

and an $\mathrm{SO}(m)$ node is balanced if the neighboring
$\mathrm{USp}(2k_{j})$ nodes satisfy:

$2m=2+\sum_{j}2k_{j}$ (5)

Note that for unframed unitary-orthosymplectic quivers, without $SO(m)$ nodes
with odd $m$, the gauge group $G$ is take to be the product of the groups
$G_{i}$ associated to the nodes $i$, modulo a diagonal $\mathbb{Z}_{2}$:

$G=\left(\prod_{i}G_{i}\right)/\mathbb{Z}_{2}.$ (6)

This is not a choice of convention, but rather a necessity. Taking the gauge
group to be $\prod_{i}G_{i}$ yields a different Coulomb branch which differs
by a $\mathbb{Z}_{2}$ quotient Bourget:2020xdz .

We start by looking at the Coulomb branch of quivers that are nilpotent cones
of $\mathrm{SO}(2n)$ and $\mathrm{SO}(2n+1)$. These are given explicitly in
GaiottoWitten . The unrefined Hilbert series of the nilpotent cone of $G$
takes a simple form:

$\mathrm{HS}_{\mathcal{N}_{G}}=\mathrm{PE}\left[\mathrm{dim}(G)t^{2}-\sum_{i=1}^{r}t^{2d_{i}}\right]$
(7)

where $d_{i}$ are the degrees of Casimir invariants of $G$ and $\mathrm{PE}$
is the plethystic exponential. For quivers whose Coulomb branch is the
nilpotent cone of $\mathrm{SO}(2n)$, the flavour node is $\mathrm{SO}(2n)$
which we explode into $n$ $\mathrm{U}(1)=\mathrm{SO}(2)$ gauge groups. The
$\mathrm{USp}(2n-2)$ node thus remains balanced, as do the nodes further down
the chain. The Coulomb branch Hilbert series for the exploded quivers take a
more complicated form and perturbative Hilbert series are presented in Table
LABEL:Dtypeimplosion.

Let us demonstrate this for $n=3$ which gives us the following explosion:

$n-1$$n$1111$n$

Quiver for the nilpotent cone $\mathcal{N}_{\mathrm{SU}(n)}$

Explosion

Exploded quiver

1212$n-1$$n-2$224462244111

Quiver for $\mathcal{N}_{\mathrm{SO}(6)}$

Explosion

Exploded quiver

(8)

Since the orthosymplectic quiver is unframed we ungauge a diagonal
$\mathbb{Z}_{2}$, as stated in (6). The Coulomb branch Hilbert series can be
readily computed and is found to be the same as the Coulomb branch Hilbert
series of the exploded unitary quiver in (3) for $n=4$. This is not surprising
due to the isomorphism $\mathfrak{so}(6)\cong\mathfrak{su}(4)$ but
nevertheless validates our approach in extending the explosion procedure to
orthosymplectic quivers.

For general $n$, we thus have:

$n-1$$n$1111$n$

Quiver for the nilpotent cone $\mathcal{N}_{\mathrm{SU}(n)}$

Explosion

Exploded quiver

1212$n-1$$n-2$224462244111

Quiver for $\mathcal{N}_{\mathrm{SO}(6)}$

Explosion

Exploded quiver

2244$2n-2$$2n$2244$2n-2$1111$n$

Quiver for $\mathcal{N}_{\mathrm{SO}(2n)}$

Explosion

Exploded quiver

(9)

The resulting global symmetry is the expected
$\mathrm{SO}(2n)\times\mathrm{U}(1)^{n}$. The Hilbert series and plethystic
logarithm for several members of this family are given in Table
LABEL:Dtypeimplosion. The coefficient of $t^{2}$ is

$2n^{2}=n+2n(2n-1)/2={\rm rank}\;\mathrm{SO}(2n)+\dim\mathrm{SO}(2n)\;,$ (10)

in agreement with the dimension of the global symmetry group. The group by
which we hyperKähler quotient has rank $n+2\sum_{i=1}^{n-1}i=n^{2}$, and the
implosion has complex dimension $2n^{2}$ so real dimension $4n^{2}$, so
Nakajima’s equality is satisfied.

For quivers whose Coulomb branch is the nilpotent cone of $\mathrm{SO}(2n+1)$,
the flavour node is an $\mathrm{USp}(2n)$ (the Langlands dual of
$\mathrm{SO}(2n+1)$). After explosion, the flavour node explodes into $n$
$\mathrm{U}(1)$ nodes. For purposes of the balancing condition at the
$\mathrm{SO}(2n)$ node, the $\mathrm{U}(1)$ nodes behave the same as
$C_{1}=\mathrm{USp}(2)$ (see the discussion in Bourget:2020xdz , for example)
hence the $\mathrm{SO}(2n)$ node remains balanced.

$n-1$$n$1111$n$

Quiver for the nilpotent cone $\mathcal{N}_{\mathrm{SU}(n)}$

Explosion

Exploded quiver

1212$n-1$$n-2$224462244111

Quiver for $\mathcal{N}_{\mathrm{SO}(6)}$

Explosion

Exploded quiver

2244$2n-2$$2n$2244$2n-2$1111$n$

Quiver for $\mathcal{N}_{\mathrm{SO}(2n)}$

Explosion

Exploded quiver

2244$2n$$2n$2244$2n$1111$n$

Quiver for $\mathcal{N}_{\mathrm{SO}(2n+1)}$

Explosion

Exploded quiver

(11)

The Hilbert series and plethystic logarithm for several members of this family
are given in Table LABEL:Btypeimplosion.

There might be concerns about the nature of the bouquet nodes and why they are
$\mathrm{U}(1)$ rather than $\mathrm{SO}(2)$ or $\mathrm{USp}(2)$. For the
$\mathfrak{so}(2n)$ type quivers in (9), the $\mathrm{SO}(2n)$ flavour node
explodes into $n$ $\mathrm{SO}(2)$ nodes. Since
$\mathrm{SO}(2)\cong\mathrm{U}(1)$, we can use either one. If we take instead
the $\mathfrak{so}(2n+1)$ quivers in (11), the flavour group is
$\mathrm{USp}(2n)$ so in theory we can try to explode it into $n$
$\mathrm{USp}(2)$ nodes. This computation is done in Table LABEL:C1implosion.
Unfortunately, the global symmetry does not match the quivers we require for
the explosion quiver family (since there are no $\mathrm{U}(1)^{n}$ factors).
However, it does give interesting results such as for $n=4$, where there is a
symmetry enhancement from $\mathrm{SO}(9)$ to $F_{4}$.

Another argument for the explosion of the rank $n$ flavour nodes into $n$
$\mathrm{U}(1)$ gauge nodes follows from the property that taking the
hyperKähler quotient of the exploded quivers by $\mathrm{U}(1)^{n}$ returns
the quiver for the nilpotent cone, going from right to left in (9) and (11).
This is satisfied by all our exploded quivers, hence making them good
candidates for unitary-orthosymplectic quiver counterparts to the well known
cases with unitary quivers (3).

The $\mathrm{SO}(2n+1)$ universal implosion should have symmetry group
$\mathrm{SO}(2n+1)\times\mathrm{U}(1)^{n}$, whose complex dimension is
$2n(n+1)$. We see that this appears as the $t^{2}$ coefficient in the Hilbert
series for the displayed examples.

We can also check that this is consistent with Nakajima’s picture. The group
by which we perform the hyperKähler quotient has rank
$2\sum_{i=1}^{n}i=n(n+1)$, and the implosion has real dimension $4n(n+1)$, as
expected.

###  2.1 HyperKähler quotient

Let us look at $T[G]$, the nilpotent cone of
$G=\mathrm{SU}(n),\mathrm{SO}(2n),\mathrm{SO}(2n+1)$ and its explosion
$\widetilde{T}[G]$. As shown in DHK , one can recover $T[G]$ from the exploded
quiver through hyperKähler quotient.

$\mathcal{C}\left(T[G]\right)=\mathcal{C}\left(\widetilde{T}[G]\right)///U(1)^{\mathrm{rank}(G)}$
(12)

It is well known that orthosymplectic gauge groups lack the
$\mathrm{U}(1)_{J}$ topological symmetry that allows us to refine the Coulomb
branch Hilbert series. However, we can partially refine the Coulomb branch
Hilbert series by assigning fugacities $z_{i}$ with $i=1,\dots,n$ for the
bouquet of $n$ $\mathrm{U}(1)$ gauge nodes. The fugacities carry the
$\mathrm{U}(1)_{J}$ topological charge under each of the $\mathrm{U}(1)$ gauge
nodes. On the level of the Hilbert series the hyperKähler quotient with
respect to these charges takes the following form:

$\mathrm{HS}_{T[G]}(t)=(1-t^{2})^{n}\oint\prod\limits_{j=1}^{n}\frac{\mathrm{d}z_{j}}{2\pi
iz_{j}}\mathrm{HS}_{\widetilde{T}[G]}(z_{j};t)$ (13)

where $G=\mathrm{SU}(n),\mathrm{SO}(2n),\mathrm{SO}(2n+1)$.

The ability to do this further justifies our choice of using a bouquet of $n$
$\mathrm{U}(1)$ nodes for $\widetilde{T}[SO(2n+1)]$ rather than a bouquet of
$n$ $\mathrm{USp}(2)$ gauge nodes. This is because
$\mathrm{USp}(2)\cong\mathrm{SU}(2)$ does not have a $\mathrm{U}(1)_{J}$
topological symmetry with which we can refine our Hilbert series.

Let us demonstrate this with $\widetilde{T}[SO(5)]$:

$\begin{split}&(1-t^{2})^{2}\oint\frac{\mathrm{d}z_{1}\mathrm{d}z_{2}}{(2\pi
i)^{2}z_{1}z_{2}}\mathrm{HS}_{\widetilde{T}[SO(5)]}(z_{1},z_{2};t)=\\\
&(1-t^{2})^{2}\oint\frac{\mathrm{d}z_{1}\mathrm{d}z_{2}}{(2\pi
i)^{2}z_{1}z_{2}}\left[1+12t^{2}+\left(\frac{4(1+z_{1}^{2})(1+z_{2}^{2})}{z_{1}z_{2}}\right)t^{3}.\right.\\\
&\left.+\left(77+\frac{5}{z_{1}^{2}}+5z_{1}^{2}+\frac{5}{z_{2}^{2}}+5z_{2}^{2}\right)t^{4}+\dots\right]\\\
&=1+10t^{2}+54t^{4}+\dots=\mathrm{HS}_{T[\mathrm{SO}(5)]}(t)\end{split}$ (14)

An exact computation with the refined Hilbert series is time consuming so we
checked and matched the results to order $t^{40}$ using perturbative
computations.

#### Integer and half-integer magnetic lattice

In Bourget:2020xdz , it has been pointed out for unframed orthosymplectic
quivers made of $\mathrm{SO}(2n)$, $\mathrm{USp}(2k)$ and $\mathrm{U}(m)$
gauge groups, one can ungauge an overall diagonal $\mathbb{Z}_{2}$. The effect
of ungauging this $\mathbb{Z}_{2}$ translates to changing the magnetic lattice
of the gauge groups to include half-integer magnetic charges. Therefore, we
can write the Hilbert series as
$\mathrm{HS}=\mathrm{HS}_{\mathbb{Z}}+\mathrm{HS}_{\mathbb{Z}+\frac{1}{2}}$
where $\mathrm{HS}_{\mathbb{Z}}$ comes from integer magnetic charge
contributions and $\mathrm{HS}_{\mathbb{Z}+\frac{1}{2}}$ from half-integer
magnetic charge contributions. One thing to notice is that all the operators
from $\mathrm{HS}_{\mathbb{Z}+\frac{1}{2}}$ carry non-trivial
$\mathrm{U}(1)_{J}$ charges from _every_ $\mathrm{U}(1)$ node in the bouquet.
Therefore, the contribution from $\mathrm{HS}_{\mathbb{Z}+\frac{1}{2}}$ is
zero after taking a hyperKähler quotient. This result is expected because
taking a hyperKähler quotient over a $\mathrm{U}(1)$ gauge node is equivalent
to turning it into a $\mathrm{U}(1)$ flavor node. Therefore, even if we
quotient over one of the $\mathrm{U}(1)$ nodes in the bouquet, it changes the
exploded quiver from an unframed orthosymplectic quiver to a framed
orthosymplectic quiver. And for a framed orthosymplectic quiver, the magnetic
lattice only contains integer magnetic charges and the Hilbert series is
$\mathrm{HS}_{\mathbb{Z}}$.

##  3 Bouquet for Symplectic groups

Above, bouquets for $G=\mathrm{SU}(n),\mathrm{SO}(2n),\mathrm{SO}(2n+1)$ are
investigated. The reason we haven’t investigated $G=\mathrm{USp}(2n)$ is
because the orthosymplectic quivers are all ‘bad’. In particular, some of the
gauge nodes have negative balance. Therefore, the monopole formula diverges
and we cannot test any of our conjectures through explicit computation.
However, orthosymplectic quivers whose Coulomb branch are conjectured to be
closures of $G=\mathrm{USp}(2n)$ nilpotent orbits had been studied through
other means, in particular through brane configurations in Feng:2000eq ;
GaiottoWitten ; Cabrera:2017njm . With this, one can construct the quiver for
$T[\mathrm{USp}(2n)]$ where the $\mathrm{USp}(2n)$ global symmetry can be read
off following a new set of balancing conditions. Since the quivers are
obtained from brane configurations which are not sensitive to the difference
between $\mathrm{SO}$ and $\mathrm{O}$ gauge groups, we will stick with their
algebras $\mathfrak{so}$ in this section. For $\mathfrak{usp}(2k)$, we need
$\mathfrak{so}(m_{j})$ neighboring nodes such that $4k=\sum_{j}mj$. For
$\mathfrak{so}(2m+1)$, we need $\mathfrak{usp}(2k_{j})$ neighboring nodes such
that $4m+2=\sum_{j}2k_{j}$. For $2n$ nodes arranged linearly satisfying these
balancing conditions will give a $\mathrm{USp}(2n)$. For
$T[\mathrm{USp}(2n)]$, the flavor node is $\mathfrak{so}(2n+1)$ and the gauge
node it is connected to is $\mathfrak{usp}(2n)$. A very natural partition of
the bouquet is into $\mathfrak{so}(1)$ and $n$ $\mathfrak{u}(1)$s. Here, we
know that the global symmetry for $\mathrm{U}(1)$ with $n$ flavors where $n>2$
is $\mathrm{U}(1)$. Therefore, the global symmetry of the following explosion
based on the balance of the gauge nodes give the expected
$G_{\mathrm{global}}=\mathrm{USp}(2n)\times\mathrm{U}(1)^{n}$. Note that the
sum of the ranks of the groups at the gauge nodes is
$n+\sum_{i=1}^{n}i+\sum_{i=1}^{n-1}i=n(n+1)$, as in the $SO(2n+1)$ case
discussed in the previous section.

$n-1$$n$1111$n$

Quiver for the nilpotent cone $\mathcal{N}_{\mathrm{SU}(n)}$

Explosion

Exploded quiver

1212$n-1$$n-2$224462244111

Quiver for $\mathcal{N}_{\mathrm{SO}(6)}$

Explosion

Exploded quiver

2244$2n-2$$2n$2244$2n-2$1111$n$

Quiver for $\mathcal{N}_{\mathrm{SO}(2n)}$

Explosion

Exploded quiver

2244$2n$$2n$2244$2n$1111$n$

Quiver for $\mathcal{N}_{\mathrm{SO}(2n+1)}$

Explosion

Exploded quiver

1234$2n$$2n+1$1234$2n$1111$n$

Quiver for $\mathcal{N}_{\mathrm{USp}(2n)}$

Explosion

Exploded quiver

1 (15)

Another natural partition is into a $\mathfrak{so}(3)$ and $n-1$
$\mathfrak{u}(1)$s. The exploded quiver takes the following form:

$n-1$$n$1111$n$

Quiver for the nilpotent cone $\mathcal{N}_{\mathrm{SU}(n)}$

Explosion

Exploded quiver

1212$n-1$$n-2$224462244111

Quiver for $\mathcal{N}_{\mathrm{SO}(6)}$

Explosion

Exploded quiver

2244$2n-2$$2n$2244$2n-2$1111$n$

Quiver for $\mathcal{N}_{\mathrm{SO}(2n)}$

Explosion

Exploded quiver

2244$2n$$2n$2244$2n$1111$n$

Quiver for $\mathcal{N}_{\mathrm{SO}(2n+1)}$

Explosion

Exploded quiver

1234$2n$$2n+1$1234$2n$1111$n$

Quiver for $\mathcal{N}_{\mathrm{USp}(2n)}$

Explosion

Exploded quiver

11234$2n$$2n+1$1234$2n$1111$n-1$

Quiver for $\mathcal{N}_{\mathrm{USp}(2n)}$

Explosion

Exploded quiver

3 (16)

However, the Coulomb branch for a $\mathrm{SO}(3)$ gauge group with $n$
flavors has trivial global symmetry (no contribution at order $t^{2}$ in the
Hilbert series) for $n\geq 3$. Therefore, the global symmetry is
$\mathrm{USp}(2n)\times\mathrm{U}(1)^{n-1}$ which is not the expected result
for implosion. Based on the argument from global symmetry, we suggest the
correct explosion is as in (15).

#### Comment on bad quivers.

Since the quivers in this section are bad, and hence cannot be checked with
current Hilbert series techniques, we should address what we know exactly
about their moduli space. We will only consider the simplest example, the left
quiver for $n=1$ in (15). There is a choice to make, whether the leftmost
gauge node is an $\mathrm{SO}(1)$ or $\mathrm{O}(1)$. The difference in moduli
space is drastic. If we pick $\mathrm{SO}(1)$ then the theory is equivalent to
$\mathrm{USp}(2)$ with 4 fundamental half hypermultiplets. The classical Higgs
branch of this theory is the union of two cones,
$d_{2}=\mathbb{C}^{2}/\mathbb{Z}_{2}\cup\mathbb{C}^{2}/\mathbb{Z}_{2}$. It is
the nilpotent cone of $\mathrm{O}(4)$. There is a
$\mathbb{Z}_{2}=\mathrm{O}(1)$ symmetry exchanging the two cones. The Coulomb
branch of this theory is $D_{2}=(\mathbb{R}^{3}\times S^{1})/\mathbb{Z}_{2}$,
where the $\mathbb{Z}_{2}$ acts on both the $\mathbb{R}^{3}$ and the $S^{1}$.
The resulting space has two singular points, which are both of $A_{1}$ type
Seiberg:1996nz . The $\mathrm{O}(1)$ symmetry exchanges the two singularities.
The classical Higgs branch is distorted quantum mechanically. In the full
moduli space each $\mathbb{C}^{2}/\mathbb{Z}_{2}$ cone in the classical Higgs
branch emanates from a different singularity of the Coulomb branch.

We can now gauge the $\mathrm{O}(1)$ symmetry, leading to the second choice in
(15), the leftmost node now being O rather than SO. The resulting moduli space
is much simpler. The two cones in the Higgs branch are identified with each
other, as are the two singularities in the Coulomb branch. The Coulomb branch
of this quiver is expected to be $\mathbb{C}^{2}/\mathbb{Z}_{2}$ GaiottoWitten
. We can summarise this in Figure 1, depicting the moduli spaces by their
Hasse diagram Bourget:2019aer ; Grimminger:2020dmg , using red for Coulomb
branch and blue for Higgs branch directions.

$n-1$$n$1111$n$

Quiver for the nilpotent cone $\mathcal{N}_{\mathrm{SU}(n)}$

Explosion

Exploded quiver

1212$n-1$$n-2$224462244111

Quiver for $\mathcal{N}_{\mathrm{SO}(6)}$

Explosion

Exploded quiver

2244$2n-2$$2n$2244$2n-2$1111$n$

Quiver for $\mathcal{N}_{\mathrm{SO}(2n)}$

Explosion

Exploded quiver

2244$2n$$2n$2244$2n$1111$n$

Quiver for $\mathcal{N}_{\mathrm{SO}(2n+1)}$

Explosion

Exploded quiver

1234$2n$$2n+1$1234$2n$1111$n$

Quiver for $\mathcal{N}_{\mathrm{USp}(2n)}$

Explosion

Exploded quiver

11234$2n$$2n+1$1234$2n$1111$n-1$

Quiver for $\mathcal{N}_{\mathrm{USp}(2n)}$

Explosion

Exploded quiver

3QuiverModuli Space Hasse
diagramSO$(1)$$2$$3$$=$$2$O$(4)$O$(1)$$A_{1}$$A_{1}$$A_{1}$$A_{1}$O$(1)$$2$SO$(3)$$A_{1}$$A_{1}$
Figure 1: Depiction of the difference in moduli space choosing O$(1)$ over
SO$(1)$ in the left quiver of (15) for $n=1$. In the Hasse diagram red lines
denote Coulomb branch directions, while blue lines denote Higgs branch
directions.

Since the Coulomb branch in the latter case is a hyper-Kähler cone, its
Hilbert Series is computable. However the monopole formula does not give the
desired result, but rather diverges. This is because something goes wrong with
the conformal dimension formula for ‘bad’ quivers. One could hope to adjust to
this formula for certain cases, where the quiver is bad, but the Coulomb
branch is a cone. We leave this for future work.

##  Appendix A A bouquet of $\mathrm{USp}(2)$ nodes

In this section, we list some interesting results for the explosion of
$T[SO(2n+1)]$ quivers with the choice of a bouquet of $\mathrm{USp}(2)$ gauge
nodes rather than $\mathrm{U}(1)$ gauge nodes that is discussed in Section 2.

For a $\mathrm{USp}(2)$ gauge group with $n$ flavors, we need $n\geq 3$,
otherwise the gauge node has negative imbalance and the monopole formula
diverges. The first such case is $T[\mathrm{SO}(7)]$, where a bouquet of
$\mathrm{USp}(2)$ gauge nodes takes the form:

$n-1$$n$1111$n$

Quiver for the nilpotent cone $\mathcal{N}_{\mathrm{SU}(n)}$

Explosion

Exploded quiver

1212$n-1$$n-2$224462244111

Quiver for $\mathcal{N}_{\mathrm{SO}(6)}$

Explosion

Exploded quiver

2244$2n-2$$2n$2244$2n-2$1111$n$

Quiver for $\mathcal{N}_{\mathrm{SO}(2n)}$

Explosion

Exploded quiver

2244$2n$$2n$2244$2n$1111$n$

Quiver for $\mathcal{N}_{\mathrm{SO}(2n+1)}$

Explosion

Exploded quiver

1234$2n$$2n+1$1234$2n$1111$n$

Quiver for $\mathcal{N}_{\mathrm{USp}(2n)}$

Explosion

Exploded quiver

11234$2n$$2n+1$1234$2n$1111$n-1$

Quiver for $\mathcal{N}_{\mathrm{USp}(2n)}$

Explosion

Exploded quiver

322446622446

Quiver for $\mathcal{N}_{\mathrm{SO}(7)}$

Explosion

Exploded quiver

222

(17)

We see here that all the gauge nodes are balanced, yet the Coulomb branch
Hilbert series diverges. This is not too surprising as a quiver can be bad
even if none of the individual gauge nodes are bad. This is often observed for
orthosymplectic quivers.

For $n\geq 4$, the Coulomb branch Hilbert series no longer diverges. Using the
fact that the Coulomb branch of $\mathrm{USp}(2)$ gauge group with $n$ flavors
and $n\geq 3$ has trivial global symmetry, we know that the global symmetry of
the exploded quiver will just be $\mathrm{SO}(2n+1)$. This is reported in
Table LABEL:C1implosion. However, for $n=4$, there is a contribution from
$\mathrm{HS}_{\mathbb{Z}+\frac{1}{2}}$ at order $t^{2}$, hence enhancing the
global symmetry. The enhancement from $\mathrm{HS}_{\mathbb{Z}+\frac{1}{2}}$
comes in the spinor representation of $\mathrm{SO}(9)$ which results in
$F_{4}$ of dimension 52.

## Acknowledgements

We would like to thank Rudolph Kalveks for helpful discussions. The work of
AB, JFG, AH and ZZ is supported by STFC grants ST/P000762/1 and ST/T000791/1.

## References

  * (1) A. Dancer, A. Hanany and F. Kirwan, _Symplectic duality and implosions_ , 2004.09620.
  * (2) V. Guillemin, L. Jeffrey and R. Sjamaar, _Symplectic implosion_ , _Trans. Group_ 7 (2002) 155–184.
  * (3) A. Dancer, F. Kirwan and A. Swann, _Implosion for hyperkähler manifolds_ , _Compositio Mathematica_ 149 (2013) 1592–1630.
  * (4) A. Dancer, B. Doran, F. Kirwan and A. Swann, _Symplectic and hyperkähler implosion_ , in _Arbeitstagung Bonn 2013_ , pp. 81–103.  Springer, 2016.
  * (5) V. Ginzburg and S. Riche, _Differential operators on G/U and the affine Grassmannian_ , _J. Math. Jussieu_ 14 (2015) 493–575.
  * (6) S. Cabrera, A. Hanany and Z. Zhong, _Nilpotent orbits and the Coulomb branch of $T^{\sigma}(G)$ theories: special orthogonal vs orthogonal gauge group factors_, _JHEP_ 11 (2017) 079, [1707.06941].
  * (7) A. Bourget, J. F. Grimminger, A. Hanany, R. Kalveks, M. Sperling and Z. Zhong, _Magnetic Lattices for Orthosymplectic Quivers_ , _JHEP_ 12 (2020) 092, [2007.04667].
  * (8) D. Gaiotto and E. Witten, _S-Duality of Boundary Conditions In N=4 Super Yang-Mills Theory_ , _Adv. Theor. Math. Phys._ 13 (2009) 721–896, [0807.3720].
  * (9) B. Feng and A. Hanany, _Mirror symmetry by O3 planes_ , _JHEP_ 11 (2000) 033, [hep-th/0004092].
  * (10) S. Cabrera and A. Hanany, _Branes and the Kraft-Procesi transition: classical case_ , _JHEP_ 04 (2018) 127, [1711.02378].
  * (11) N. Seiberg and E. Witten, _Gauge dynamics and compactification to three-dimensions_ , in _Conference on the Mathematical Beauty of Physics (In Memory of C. Itzykson)_ , pp. 333–366, 6, 1996.  hep-th/9607163.
  * (12) A. Bourget, S. Cabrera, J. F. Grimminger, A. Hanany, M. Sperling, A. Zajac et al., _The Higgs mechanism — Hasse diagrams for symplectic singularities_ , _JHEP_ 01 (2020) 157, [1908.04245].
  * (13) J. F. Grimminger and A. Hanany, _Hasse diagrams for 3d $\mathcal{N}$ = 4 quiver gauge theories — Inversion and the full moduli space_, _JHEP_ 09 (2020) 159, [2004.01675].


# ReLGAN: Generalization of Consistency for GAN with Disjoint Constraints and
Relative Learning of Generative Processes for Multiple Transformation Learning

Chiranjib Sur
Computer & Information Science & Engineering Department, University of
Florida.
Email<EMAIL_ADDRESS>
###### Abstract

Image to image transformation has gained popularity from different research
communities due to its enormous impact on different applications, including
medical. In this work, we have introduced a generalized scheme for consistency
for GAN architectures with two new concepts of Transformation Learning (TL)
and Relative Learning (ReL) for enhanced learning image transformations.
Consistency for GAN architectures suffered from inadequate constraints and
failed to learn multiple and multi-modal transformations, which is inevitable
for many medical applications. The main drawback is that it focused on
creating an intermediate and workable hybrid, which is not permissible for the
medical applications which focus on minute details. Another drawback is the
weak interrelation between the two learning phases and TL and ReL have
introduced improved coordination among them. We have demonstrated the
capability of the novel network framework on public datasets. We emphasized
that our novel architecture produced an improved neural image transformation
version for the image, which is more acceptable to the medical community.
Experiments and results demonstrated the effectiveness of our framework with
enhancement compared to the previous works.

###### Index Terms:

Relative Learning Transformation Learning Constraint Learning Image-to-Image.

##  I Introduction

Figure 1: A diagrammatic overview of ReLGAN with disjoint constraints are
illustrated through channels of (black and red) arrows. Figure 2: An
illustration of our architecture with Transformation Learning (TL) Contention
and Relative Learning (ReL) Contention for ReLGAN, compared with GAN
Contention.

What Claude Monet imagined was fancier than what we need for enabling the life
of a massive number of patients all over the world. Learning translation is
challenging. Previous works in image-2-image translations [3], [4] were
focused on mere feature translation. But what if the problem is more complex
like detection of organs and painting them with color, instead of using
segmentation. Or the texture of the image needs to be reconfigured, keeping
intact the feature composition and preventing any foreign object introduction.
[5], [6], [7] has provided some examples of such kinds of medical image
scenarios, which can make devices more sophisticated and smarter. There are
more significant challenges in these kinds of image-2-image translation
problems, and we have introduced a new architecture that can handle them.
Also, multiple translation learning can be challenging and requires extra
attention. We have introduced new learning strategies that can enhance
performance. We have adopted the baseline of CycleGAN [2], [8] architecture
for our experiments. Our model has the natural flow of learning the
translation in both ways, which can be very useful in many medical image
translations. We can define the problem as
$G_{AB}:\textbf{A}\rightarrow\textbf{B}$,
$G_{BA}:\textbf{B}\rightarrow\textbf{A}$ for $\textbf{A}\in S,\textbf{B}\in S$
for CycleGAN [2], [8]. Pairing training set generates additional constraint
and can be defined as $G_{AB}:\textbf{A}\rightarrow\textbf{B}$,
$G_{BA}:\textbf{B}\rightarrow\textbf{A}$ for $(\textbf{A},\textbf{B})\in S$.
We have generalized the constraints as Transformation Learning (TL) and is
defined as $G_{AB}:\textbf{A}\rightarrow\textbf{B}^{\prime}$,
$G_{BA}:\textbf{B}\rightarrow\textbf{A}^{\prime}$,
$G_{AB}^{\prime}:\textbf{A}^{\prime}\rightarrow\textbf{B}^{\prime\prime}$,
$G_{BA}^{\prime}:\textbf{B}^{\prime}\rightarrow\textbf{A}^{\prime\prime}$.
When the generators are separated individual, it solidifies the learning,
though a few of them will be useful. TL operates on the contention between
generation, discrimination, and multiple translations. But after some training
sessions, the learning gradient saturates because of the model limitations to
differentiate between background texture and the features. Hence, we introduce
TL for feature capturing and Relative Learning (ReL) for understanding the
difference between background texture. Figure 1 provided a diagrammatic
illustration of the new scheme.

ReL introduces these two new utilities that can enhance the learning process.
First is a new contention scheme for enhanced learning (without over-fitting)
and generalization of representations. Secondly, after the features are
translated, relative learning helps in learning the background. For most
image-2-image translation, the features are the main translated objects, but
ReL involves learning more about the background as well. This is the reason
why our novel architecture ReLGAN performs much better than the previous
architectures. We have shown through zebra-horse dataset that our architecture
is far better and outperformed CycleGAN. ReL also involves a smoothening
process in comparison to the TL, which introduces sharp gradients between the
generated images and the ground-truth images. In GAN [10] architectures,
multiple translations are independent learning strategies that do not
generalize well. But, ReL improves in coordination and cooperation among
multiple generators for enhanced generalization of the representations. While
TL helps in the determination of the critical feature transformation, ReL
helps in generalization through comparison of the relative generation. ReL is
learning from the relative knowledge of two generators. ReL converges only if
there is generalization among the generators. Compared to multiple loss based
learning models, which is dependent on TL, ReL is a space that can be defined
as a latent space contributed by all different translations and hence
generalization of consistencies for any GAN model.

The rest of the document is arranged with ReLGAN architectural details in
Section II, details of implementation and methodology, analysis of the
experiments and results in Section III, concluding remarks with future works
in Section IV, and two pages of Supplementary Material.

The main contributions of this work are: 1) ReLGAN with Relative Learning
strategy for enhanced image-to-image translation 2) The notion of disjoint
constraints in Transformation Learning and Relative Learning 3) applied to
public dataset, outperformed CycleGANThe main contributions of this work are:
1) ReLGAN with Relative Learning strategy for enhanced image-to-image
translation 2) The notion of disjoint constraints in Transformation Learning
and Relative Learning 3) applied to public dataset, outperformed CycleGAN.

The rest of the document is arranged with the details and scope of the problem
in Section LABEL:section:problem, the description and statistics of the data
in Section LABEL:section:data, Section LABEL:section:, Section LABEL:section:,
Section LABEL:section:, Section LABEL:section:, Section LABEL:section:, the
intricacies of our methodology in Section LABEL:section:methodology,
experiments, results and analysis in Section III, revisit of the existing
works in literature in Section LABEL:section:literature, concluding remarks
with future prospects in Section LABEL:section:discussion,

##  II ReLGAN Architecture

Relative Learning based Generative Adversarial Network (ReLGAN) works on the
principle of contention based learning like any Generative Adversarial Network
(GAN), but with enhanced contention strategies. GAN has learning limitations
due to the contention exhaustion of the discriminator and the generator. To
improve learning, we have introduced ReL based contention for the first time
in the literature. Relative because Relative Learning based gradient is much
smoother than the absolute difference based gradient. ReL participates as a
natural smoothening gradient for the model training.

Considering the generator for the transformation $A\rightarrow B$ as
$\text{G}_{{}_{A\rightarrow B}}(.)$ and for $B\rightarrow A$ as
$\text{G}_{{}_{B\rightarrow A}}(.)$, $\text{E}(.)$ as encoder, $\text{D}_{A}$,
$\text{D}_{B}$ is the cumulative discriminative losses for both A and B
respectively, we define the notions for defining the equations below. Compared
to previous works like CycleGAN [2], [8], Cycle Consistency Loss is the sole
constraint with the equation below.

$L_{C}=\mathop{\mathbb{E}}_{\begin{subarray}{c}\textbf{A}\sim\text{Pr}(\textbf{A})\\\
\textbf{B}\sim\text{Pr}(\textbf{B})\end{subarray}}[\parallel\textbf{A}-\text{G}_{{}_{B\rightarrow
A}}(\text{G}_{{}_{A\rightarrow B}}(\textbf{A}))\parallel]$ (1)

Augmented CycleGAN [1] was introduced with encoder based random variable
tensor. Here, encoder based distribution learning ($z\approx\text{E}(.)$) in
Consistency Loss is constrained as the following.

$L_{AC}=\mathop{\mathbb{E}}_{\begin{subarray}{c}\textbf{A}\sim\text{Pr}(\textbf{A})\\\
\textbf{B}\sim\text{Pr}(\textbf{B})\end{subarray}}[\parallel\textbf{A}-\text{G}_{{}_{B\rightarrow
A}}(\text{E}(\textbf{B}),\text{G}_{{}_{A\rightarrow
B}}(\text{E}(\textbf{A}),\textbf{A}))\parallel]$ (2)

But ReLGAN is about more generalized constraints and for multiple levels or
hierarchical and disjoint losses of the cycles. We define Transformer Learning
(TL) (same as Cycle Consistency Loss but with independent generators) for
learning of the transformation and Relative Learning (ReL) for improving the
quality of the transformation. Figure 1-2 explains the architecture. Apart
from learning the difference with the ground-truth, the key is bridging the
gap between different stages as constraints and thus can produce much better
feature capturing capability. These two kinds of constraints are very
important for medical image transformation as it helps capture minute details
and also important aspects that cannot be considered as fancy as an
intermediate hybrid. We define the equations for ReLGAN as the following,

$\begin{split}&L_{GC}=\mathop{\mathbb{E}}_{\begin{subarray}{c}\textbf{A}\sim\text{Pr}(\textbf{A})\\\
\textbf{B}\sim\text{Pr}(\textbf{B})\end{subarray}}[\text{ }\text{
}\parallel\textbf{A}-\text{G}_{{}_{B\rightarrow A}}(\text{G}_{{}_{A\rightarrow
B}}(\textbf{A}))\parallel\\\ &+\parallel\textbf{B}-\text{G}_{{}_{A\rightarrow
B}}(\textbf{A})\parallel+\parallel\text{G}_{{}_{A\rightarrow
B}}(\textbf{A})-\text{G}_{{}_{A\rightarrow B}}(\text{G}_{{}_{B\rightarrow
A}}(\textbf{B}))\parallel\text{ }]\end{split}$ (3)

Here, $\parallel\textbf{A}-\text{G}_{{}_{B\rightarrow
A}}(\text{G}_{{}_{A\rightarrow B}}(\textbf{A}))\parallel$ is the
Transformation Loss for Learning,
$\text{ReL}_{1}=\parallel\textbf{B}-\text{G}_{{}_{A\rightarrow
B}}(\textbf{A})\parallel$ is Relative Loss for Learning and
$\text{ReL}_{2}=\parallel\text{G}_{{}_{A\rightarrow
B}}(\textbf{A})-\text{G}_{{}_{A\rightarrow B}}(\text{G}_{{}_{B\rightarrow
A}}(\textbf{B}))\parallel$ is another Relative Loss for Learning. Initially,
$\text{ReL}_{1}$ is dominant until stagnant, when $\text{ReL}_{2}$ is
introduced for further contention. Later $\parallel\text{G}_{{}_{A\rightarrow
B}}(\textbf{A})-\text{G}_{{}_{A\rightarrow B}}(\text{G}_{{}_{B\rightarrow
A}}(\textbf{B}))\parallel$ is effective to prevent over-fitting and continues
the learning. At one point in GAN [10], the discriminator stops the learning
because of its enhanced ability to discriminate the generated samples from the
ground-truth. Additional constraints in ReL are
$\parallel\textbf{B}-\text{G}_{{}_{A\rightarrow B}}(\textbf{A})\parallel$ and
$\parallel\text{G}_{{}_{A\rightarrow
B}}(\textbf{A})-\text{G}_{{}_{A\rightarrow B}}(\text{G}_{{}_{B\rightarrow
A}}(\textbf{B}))\parallel$ as demonstrated in Figure 2. Some researchers argue
that $\parallel\textbf{B}-\text{G}_{{}_{A\rightarrow B}}(\textbf{A})\parallel$
is the same old MSE loss, but we argue that this loss $\text{ReL}_{1}$ helps
in learning the relative loss, while $\parallel\text{G}_{{}_{A\rightarrow
B}}(\textbf{A})-\text{G}_{{}_{A\rightarrow B}}(\text{G}_{{}_{B\rightarrow
A}}(\textbf{B}))\parallel$ drags the generated version of B to a better sub-
space, until generalization and identification of the necessaries and the
minute details. Here, $G_{XY}\equiv G_{X\rightarrow Y}$. Overall ReLGAN loss
function is denoted as,

$\begin{split}&\mathcal{L}(D_{B},D_{A},T_{L},R_{eL})=\mathop{\mathbb{E}}_{\textbf{B}\sim\text{Pr}(\textbf{B})}[\log
D_{B}(\textbf{B})]\\\
&+\mathop{\mathbb{E}}_{\textbf{A}\sim\text{Pr}(\textbf{A})}[\log(1-D_{B}(G_{AB}(\textbf{A})))]+\mathop{\mathbb{E}}_{\textbf{A}\sim\text{Pr}(\textbf{A})}[\log
D_{A}(\textbf{A})]\\\
&+\mathop{\mathbb{E}}_{\textbf{B}\sim\text{Pr}(\textbf{B})}[\log(1-D_{A}(G_{BA}(\textbf{B})))]\\\
&+\mathop{\mathbb{E}}_{\textbf{A}\sim\text{Pr}(\textbf{A})}[(\parallel\textbf{A}-G_{BA}^{\prime}(G_{AB}(\textbf{A}))\parallel)]\\\
&+\mathop{\mathbb{E}}_{\textbf{B}\sim\text{Pr}(\textbf{B})}[(\parallel\textbf{B}-G_{AB}^{\prime}(G_{BA}(\textbf{B}))\parallel)]\\\
&+\mathop{\mathbb{E}}_{\begin{subarray}{c}\textbf{A}\sim\text{Pr}(\textbf{A})\\\
\textbf{B}\sim\text{Pr}(\textbf{B})\end{subarray}}[(\parallel
G_{BA}^{\prime}(G_{AB}(\textbf{A}))-G(\textbf{B})\parallel)]\\\
&+\mathop{\mathbb{E}}_{\begin{subarray}{c}\textbf{A}\sim\text{Pr}(\textbf{A})\\\
\textbf{B}\sim\text{Pr}(\textbf{B})\end{subarray}}[(\parallel
G_{AB}^{\prime}(G_{BA}(\textbf{B}))-G_{AB}(\textbf{A})\parallel)]\end{split}$
(4)

###  II-A Translation Learning

Transformation Loss based learning works much better than mean square based
generative loss (like in Auto-Encoder, GAN) because of its steep gradient and
it cannot learn the latent transformed space for images. We are more
interested in the CycleGAN based transformations of both directions as it
captures hidden transformation truths that are beyond simple translation
problems like denoising [12], segmentation [10] and super-resolution [11].
Translation Learning can be defined mathematically as the following equation,

$G_{AB}:\textbf{A}\rightarrow\textbf{B}^{\prime}\text{ ,
}G_{BA}:\textbf{B}\rightarrow\textbf{A}^{\prime}\text{ ,
}G_{AB}^{\prime}:\textbf{A}^{\prime}\rightarrow\textbf{B}^{\prime\prime}\text{
, }G_{BA}^{\prime}:\textbf{B}^{\prime}\rightarrow\textbf{A}^{\prime\prime}$
(5)

This equation for TL is a generalized Cycle-Consistency Loss and imposes
disjoint constraints for learning representations.

###  II-B Disjoint Constraints

Differentiating $G_{AB}$ with $G_{AB}^{\prime}$ and $G_{BA}$ with
$G_{BA}^{\prime}$ have provided significant scope for improvement and were not
present in CycleGAN. The main reason is the disjoint constraints like
$G_{AB}^{\prime}(G_{BA}(\textbf{B}))\rightarrow\textbf{B}$ and
$G_{BA}^{\prime}(G_{AB}(\textbf{A}))\rightarrow\textbf{A}$ instead of
$G_{AB}(G_{BA}(\textbf{B}))\rightarrow\textbf{B}$ and
$G_{BA}(G_{AB}(\textbf{A}))\rightarrow\textbf{A}$ in CycleGAN. ReL is also
part of the notion of disjoint constraints as more constraints are imposed for
generalization, as shown Equation 3, instead of relying on TL. ReL helps in
information exchange between the cycle and also in making sure that the
learning is reciprocating among different parts of the network. TL and ReL are
disjoint constraints because they manage the learning phases of two different
criteria. In Transformation Learning, data $A$ is considered, while in
Relative Learning, data $B$ is considered and vice-versa for the other cycle.

###  II-C Relative Learning

Relative Learning Generative Adversarial Network (ReLGAN) initially uses
Transformation Learning (TL) and later fine-tunes with Relative Learning
(ReL), which has a relative finer gradient due to the relative knowledge
comparison between two generators. For an enhanced transformation, it is very
important to impose constraints. Apart from learning the TL loss for
transformation, we introduced the novel idea of ReL to impose more constraints
on the model, make the different transformation interdependent, and extract
more generalization information. Relative Learning can be defined
mathematically as the following equation,

$\mathcal{L}_{AB}^{ReL}:G_{AB}^{\prime}\text{ ,
}G_{AB}\equiv\text{Pr}(B^{\prime})\approx\text{Pr}(B^{\prime\prime})\text{ and
}\\\ \mathcal{L}_{BA}^{ReL}:G_{BA}^{\prime}\text{ ,
}G_{BA}\equiv\text{Pr}(A^{\prime})\approx\text{Pr}(A^{\prime\prime})$ (6)

ReL will be enhance different translation learning as medical applications are
getting more delicate and models need to learn very minute details. Medical
applications are sensitive to generative processes and require more control
over the features. ReL will provide such improved controls.

##  III Results & Discussion

Image translation problems like denoising [12], segmentation [10] and super-
resolution [11] are spatially proportional/similar translation and much easier
to solve like pseudo-conditional probability
$\text{Pr}(\textbf{A}+\delta(\textbf{A})\mid\textbf{A})$. We are trying to
solve problems like removing haziness [5] in images, or perform some non-
conventional transformations like 3T to 7T MRI [7] or locating a specific
organ and provide a clearer and detailed view or true conditional probability
$\text{Pr}(\textbf{B}\mid\textbf{A})$.

###  III-A Dataset Description

To demonstrate the effectiveness of our strategy we have chosen zebra-horse
translation dataset and performed a comparison with CycleGAN [2], [8].
Learning the latent distribution and intermediates, unsupervised, can be
challenging. But, to learn properties like locating the horse and not disturb
the non-horse brown structures, the task is relatively tricky. Our proposed
joint learning method contributed to this conceptual framework and produced
promising results. While two transformations can be easier to train but are
lengthy, and the inference is fast. Experiments were done in Tensorflow in K80
Tesla GPU with 32 GB RAM with image dimension $256\times 256$. We used the
training-testing split of the dataset as it is.

###  III-B Experiments & Results

The model is trained end-to-end for different horse-zebra combination subset
for several epochs. Later, the test set is generated, and some instances are
provided in Figure 3 and more in Supplementary Material. In the absence of any
particular ground-truth, other numerical metrics like SSIM cannot be reported,
but a visual inspection will surely conclude that ReLGAN outperformed
CycleGAN. ReLGAN also performed much better than CycleGAN for many internal
datasets for an organization.

There are severe drawbacks in CycleGAN, mainly when the generators try to
learn both the transformation, where one is inverse of the other. However,
these latent representations are more driven towards the inverse generation
consistency, leading to hybridization, and not complete learning of the usable
intermediate spaces. CycleGAN’s inverse constraint strategy fails for cross-
domain translations. [1] proposed a transformed random variable for style
encoding, which is difficult to define for medical images. In [1] situations,
learning both way translations are beneficial as paired data converges to a
definite random variable based latent space. In fact, many aesthetic
applications can tolerate distortions or artifacts. However, in many
applications related to health-care and sensitive technologies, the
translations and its parity with the ground-truths are more important. ReLGAN
is driven to incorporate different aspects of the transformation through
learning the transformations and yet preventing the dual-direction
transformation from getting rid of the chances that can move in either way,
which is not desirable.

Figure 3: Generated Instances with Zebra, For ReLGAN. Supplementary.

###  III-C Implications of TL and ReL

Learning joint transformations can be challenging and requires extra sets of
information. For cross-domain translation problems, it is very difficult to
incorporate all possibilities and complications of a problem. In such a
scenario, ReL provides an additional edge over previous works. While TL is an
unsupervised fashion of learning and yet to gain full effectiveness in
applications related to medical devices, ReL is a smoothening gradient. ReL
introduces a supervised way of learning with comparative knowledge like
between two trained generators. This kind of joint learning of image-to-image
transformations for better contrast and feature prominence can help in many
medical applications, and in this work, we have provided a model that can help
to learn multiples effectively and is part of the strategy to use it
extensively in many applications, where it is needed. Capturing unique
characteristics of image collections and configuring these characteristics to
be translated into another is the main objective of ReLGAN. In the absence of
any paired training samples, this is even more challenging. Some of the
highlights of our results are marked with black color ovals in Figure 3 and
Supplementary materials. It is to be noted that the generated images have
horse(s) pointed out, transformed to zebra, and the non-horse parts are left
out. CycleGAN distorted some non-horse parts, mainly when they are brown in
color. CycleGAN has limitations generating original looking images, which can
be a serious issue, mainly in the medical image domain.

##  IV Conclusion

In this work, we introduced ReLGAN, a novel architecture for image-to-image
transformation, mainly for cross-domain and non-conventional translations.
ReLGAN is a contention based strategy to learn delicate characteristics of
images. With the horse-zebra dataset, the experiments demonstrated that ReLGAN
outperformed CycleGAN. ReLGAN is more generalized and gathers minute details
that previous models could not. ReLGAN is based on a series of constraints
defined as TL and ReL, and these kinds of disjoint constraints are the first
of its kind in the literature.

## References

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