DESY 19-177

ROM2F/2019/06

The Geometry of SUSY Enhancement
Federico Carta,1 Simone Giacomelli,2 Hirotaka Hayashi,3 and Raffaele Savelli4

1 Deutches Electronen-Synchrotron, DESY, Notkestra$\beta$e 85,
22607 Hamburg, Germany.
2 Mathematical Institute, University of Oxford, Andrew Wiles Building
Radcliffe Observatory Quarter (550), Woodstock Road,
Oxford, OX2 6GG, United Kingdom
3 Department of Physics, School of Science, Tokai University,
4-1-1 Kitakaname, Hiratsuka-shi, Kanagawa 259-1292, Japan
4 Dipartimento di Fisica, Università di Roma “Tor Vergata” & INFN - Sezione di
Roma2
Via della Ricerca Scientifica, I-00133 Roma, Italy
Abstract

We provide a precise geometric picture that demystifies the phenomenon of
supersymmetry enhancement along certain RG flows of four-dimensional field
theories, recently discovered by Maruyoshi and Song. It applies to theories of
arbitrary rank and it is based on a hyperkähler-structure restoration on the
moduli space of solutions of (twisted) Hitchin systems, which underly the
class-$\mathcal{S}$ construction we use as an engineering tool. Along the way,
we formulate a necessary algebraic condition for supersymmetry enhancement,
and, when enhancement occurs, we are able to derive the Seiberg-Witten
geometry and all conformal dimensions of Coulomb-branch operators for the
infrared theory, without using a-maximization.

###### Contents

  1. 1 Introduction
  2. 2 SUSY enhancement and Hitchin systems
    1. 2.1 Preliminaries
    2. 2.2 RG flows and spectral curves
    3. 2.3 Twisted vs ordinary Hitchin systems
    4. 2.4 Comments about non-principal nilpotent vevs
  3. 3 Systematics of SUSY enhancement
    1. 3.1 $\mathcal{N}=1$ curves from branes
    2. 3.2 SQCD with $6$ flavors
    3. 3.3 Examples that do not enhance
  4. 4 Conclusions

##  1 Introduction

In the last few years the phenomenon of infrared supersymmetry enhancement in
quantum field theories, first observed by Maruyoshi and Song [1], has been a
subject of intense investigation. A remarkable outcome of these studies is the
discovery of four-dimensional (4d) $\mathcal{N}=1$ Lagrangian theories which
flow in the infrared (IR) to non-Lagrangian $\mathcal{N}=2$ theories, often of
generalized Argyres-Douglas type [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Such
renormalization group (RG) flows in 4d have been further generalized in [12,
13, 14, 15, 16, 17], and understood more deeply in [18, 19]. Apart from being
interesting in their own right, flows of this type have been used in order to
compute RG protected quantities of the IR Argyres-Douglas theories, such as
the superconformal index (see for example [1, 12, 13]).

In [20] three of the authors have initiated a study of the geometry underlying
supersymmetry enhancement. The aim was to make manifest the deep origin of
this phenomenon, which remained obscure at the field-theoretic level, and to
shed light on those features a theory needs to have in order to exhibit
enhancement. The main focus was on 4d rank-1 theories, which were engineered
by a D3-brane probe of singular geometries in F-theory [21]. In this case the
Seiberg-Witten (SW) curve [22, 23] of the field theory on the D3-brane may be
identified with the elliptic fibration of the F-theory geometry. While higher-
rank theories may also be realized in this context by simply adding more D3
probes, one inevitably looses the identification of the elliptic fiber of the
F-theory space with the SW curve of the field theory, which was a key aspect
of the construction proposed in [20]. The principal goal of the present paper
is to make instead use of the class-$\mathcal{S}$ realization of 4d
$\mathcal{N}=2$ field theories [24, 25, 8, 9, 10] in order to generalize the
geometric investigation of [20] to higher-rank theories.

Let us first briefly recall the Maruyoshi-Song procedure. One usually starts
with a (not necessarily Lagrangian) $\mathcal{N}=2$ theory in 4d, and deforms
it by adding the superpotential coupling

$\delta W={\rm Tr\,}(\mu M)\,,$ (1.1)

where $M$ is a chiral multiplet in the adjoint representation of the flavor
group we add by hand, $\mu$ is the moment map of the flavor symmetry, and the
trace is evaluated over flavor indices. The field $M$ is then given a
nilpotent vacuum expectation value (vev) and the deformation halves the amount
of preserved supersymmetry. Nevertheless, for specific choices of initial
theory and nilpotent vev, such a deformation triggers an RG flow which leads
(upon getting rid of a bunch of free fields) to a new (typically non-
Lagrangian) $\mathcal{N}=2$ theory in the IR.

One general conclusion that may be inferred from the analysis of [20] is that
the phenomenon of supersymmetry enhancement in the IR (at least the one
originating from the above-sketched procedure) seems to be intimately
correlated with the local structure of some auxiliary algebraic space $X_{n}$
around the origin. More precisely, if supersymmetry is to enhance, some non-
trivial factorization needs to take place, which turns said space locally into
a product of a lower-dimensional space times a trivial factor, i.e.
$X_{n}\simeq Y_{n-p}\times\mathbb{C}^{p}$.111In all interesting cases the
origin carries some singularity, because any space trivially factorizes in the
neighborhood of a smooth point. What the space $Y_{n-p}$ exactly is and what
the factorization of $X_{n}$ precisely means depends on the context. For
instance, in the case of field theories in three space-time dimensions,
$Y_{n-p}$ is identified with the Coulomb branch (CB) of the moduli space,
which gets geometrized in M-theory. The enhancement phenomenon is then
directly explained in purely geometric terms as a _holonomy reduction_ of
$X_{n}$, which is the space probed by M2-branes in M-theory.

In the 4d case, however, $X_{n}$ and $Y_{n-p}$ are rather auxiliary spaces,
whose interpretation changes according to the way one geometrically engineers
the field theory. For example, if one uses F-theory to realize rank-1 theories
(as was the case in [20]), $X_{n}$ is nothing but the elliptically-fibered
geometry probed by a D3-brane, and again supersymmetry enhances only when this
space locally exhibits a holonomy reduction (in this case down to $SU(2)$). In
contrast, if one considers theories of class-$\mathcal{S}$ of arbitrary rank
(as we are going to do here), the meaning of these spaces is more subtle.
$Y_{n-p}$ is closely related to the moduli space of solutions of the Hitchin
system underlying the class-$\mathcal{S}$ construction. More precisely, as is
well known, for a rank-$r$ theory such a moduli space has the structure of a
$T^{2r}$ fibered over the $r$-dimensional CB of the four-dimensional theory
(Hitchin fibration) [26, 25]. Associated to it there is another fibration (now
$r+1$-dimensional) over the same base, whose generic fiber is the spectral
curve of the Hitchin field: The genus of this curve turns out to be precisely
$r$, and this nicely encodes the SW geometry of the 4d theory. In analogy with
the F-theory case, we identify $Y_{n-p}$ with the latter fibration. A
necessary condition for supersymmetry to enhance is then that the space
$X_{n}$ (which also has the structure of a Riemann-surface fibration) locally
factorizes in such a way that the genus of the fiber of $Y_{n-p}$ coincides
with the dimension of its base. Only if this non-trivial factorization takes
place, can we associate to $Y_{n-p}$ a Hitchin fibration, which will determine
the low-energy dynamics of the ensuing $\mathcal{N}=2$ theory in the IR.

The above reasoning suggests that, for class-$\mathcal{S}$ theories, we can
geometrically explain the enhancement in terms of the _restoration_ of a
_hyperkähler structure_ on the moduli space of solutions of the corresponding
Hitchin system. As we will explain in more detail in Section 2, the starting
4d $\mathcal{N}=2$ theory will be associated to a Hitchin system on a two-
sphere with one regular and one irregular punctures, whose Hitchin field
$\Phi$ is a meromorphic section of $\mathcal{O}(-2)$ encoding the CB operators
in (some of) its Casimir invariants [27, 18]. The deformation (1.1) turns said
Hitchin system into a _generalized_ one [28, 29, 30, 31], consisting of two
Hitchin fields $\Phi_{1},\Phi_{2}$ which are now meromorphic sections of
$\mathcal{O}(-1)$, each being singular at just one of the two punctures. In
particular, in the neighborhood of the regular puncture where $\Phi_{2}$ has a
pole, $\Phi_{1}$ plays the role of the field $M$. In this context, a
deformation leading to supersymmetry enhancement corresponds to giving
$\Phi_{1}$ a nilpotent vev along the _principal_ orbit, which forces
$\Phi_{2}$ to become a holomorphic section of $\mathcal{O}(-1)$ on the two-
sphere, and thus to vanish identically. We are therefore left with a _twisted_
Hitchin system, whose solutions with given boundary conditions at the
irregular puncture222We restrict our attention to irregular punctures with the
property that boundary conditions for the Hitchin field can be univocally
inferred from its characteristic polynomial. Our approach is not refined
enough to treat irregular punctures where the Hitchin field has degenerate
eigenvalues, i.e. the so-called type III punctures [10]. We will briefly
comment about them in Section 4., as we will show, are in bijection with those
of an ordinary Hitchin system. This strongly suggests that a hyperkähler
structure can be restored on the corresponding moduli space, hence explaining
why supersymmetry enhances.

Armed with this understanding of the geometry underlying supersymmetry
enhancement, in Section 3 we will carry out a systematic analysis of the
Maruyoshi-Song flows. Given the starting theory and the nilpotent orbit, the
“interpolating” geometry $X_{n}$ remains the same at all energy scales, and
the IR behavior of the theory crucially depends on a possible factorization
$X_{n}\simeq Y_{n-p}\times\mathbb{C}^{p}$. Along the lines of [20], on the one
hand we will derive a simple algebraic criterion to rule out supersymmetry
enhancement. On the other hand, for the cases that exhibit supersymmetry
enhancement we will derive, using purely algebraic methods, the correct
scaling dimensions of CB operators as well as the explicit form of the
infrared SW geometry, including all masses and couplings.

The paper is organized as follows: In Section 2, after reviewing some material
about $\mathcal{N}=1$ class-$\mathcal{S}$ theories, we demonstrate how, for
flows exhibiting SUSY enhancement, the generalized Hitchin system turns into a
twisted Hitchin system in the IR. Its solutions are shown (Subsec. 2.3) to be
in bijection with those of an ordinary system. In Section 3 we derive in a
purely algebraic manner two necessary criteria for enhancement. Rather than
stating them abstractly, we present them in the context of two specific
Lagrangian models, in particular $\mathcal{N}=2$ SQCD with gauge group
$SU(3)$. Finally, we draw our conclusions in Section 4 and briefly comment on
open issues related to punctures of type III.

##  2 SUSY enhancement and Hitchin systems

In this section we explain how, in the context of $\mathcal{N}=1$
class-$\mathcal{S}$ field theories, supersymmetry enhancement originates from
the emergence of an ordinary Hitchin system out of a generalized Hitchin
system. The key intermediate step will be a bijection between solutions to the
ordinary Hitchin system and solutions to a suitably twisted one. Our method
uses deformations induced by principal nilpotent vevs only. Since there are
cases of non-principal deformations leading to enhancement too, in Subsection
2.4 we will explain how our approach allows us to recover those.

###  2.1 Preliminaries

Let us consider M-theory on the background $\mathbb{R}^{4}\times
X\times\mathbb{R}$, where $X$ is a Calabi-Yau threefold. A stack of $N$ M5
branes wrapping $\mathbb{R}^{4}\times\mathcal{C}$, where $\mathcal{C}$ is a
holomorphic two-cycle in $X$, describes an $\mathcal{N}=1$ theory on
$\mathbb{R}^{4}$. We consider backgrounds of the form

$X=L_{1}\oplus L_{2}\,,$

where $L_{1}$ and $L_{2}$ are holomorphic line bundles on $\mathcal{C}$ of
degree $p$ and $q$. Indeed the Calabi-Yau condition imposes the constraint
$p+q=2g-2$, where $g$ is the genus of the Riemann surface. In this paper we
will be concerned only with theories for which $\mathcal{C}$ is a sphere and
therefore, from now on, we will restrict to this case. The two line bundles
then satisfy the constraint $p+q=-2$, reflecting the Calabi-Yau condition

$L_{1}\otimes L_{2}=\mathcal{O}(-2)\,.$ (2.1)

As in the $\mathcal{N}=2$ case, a sphere with an arbitrary number of regular
punctures can be thought of as a collection of trinions (spheres with three
punctures) connected together, where connecting two trinions together is
physically interpreted as gauging the diagonal combination of their global
symmetry. The gauging can be either $\mathcal{N}=1$ or $\mathcal{N}=2$
depending on the details of the geometric construction [32, 33, 34, 35, 36,
37]. In order to describe the resulting four-dimensional theory, it therefore
suffices to understand what the trinions are.

In the special case of trinions with punctures which (locally) preserve 8
supercharges333These are the punctures appearing in the standard
$\mathcal{N}=2$ Class $\mathcal{S}$ construction and correspond to the $1/2$
BPS boundary conditions for $\mathcal{N}=4$ SYM [38]. In principle one could
consider more general punctures corresponding to $1/4$ BPS boundary conditions
[39, 40], but we will not need this in our paper. we can proceed as follows:
We decompose a trinion into a sphere with three holes (pair of pants) and
three caps with a puncture. For each of these building blocks we take the
canonical and the trivial line bundles. When we connect a cap to the pair of
pants, we also need to specify how the corresponding line bundles are glued
together: We can either glue the canonical bundles (and therefore the trivial
bundles) together, or we can glue the canonical bundle of one block to the
trivial bundle of the other. Once we have done that, we end up with our
trinion endowed with the two line bundles $L_{1,2}$.

We can encode these geometric data by attaching a sign to each puncture and to
the pair of pants. When the signs of the puncture and of the pair of pants
agree, it means that we are gluing the corresponding canonical bundles
together. Of course, if we change the sign of all the building blocks we are
simply interchanging $L_{1}$ and $L_{2}$ and we end up with the same theory.
We easily see that if all building blocks are of the same kind, one line
bundle gets identified with the trivial bundle on the sphere (and the other
with its canonical bundle) and the threefold is of the form
$T^{*}(S^{2})\times\mathbb{C}$. This special case corresponds to an
$\mathcal{N}=2$ class $\mathcal{S}$ trinion.

A nice feature of this construction is that the degrees of the two line
bundles $L_{1,2}$ can be computed straightforwardly: The first Chern class
receives a nontrivial contribution only from the canonical bundle on the
various building blocks and we simply need to sum the various contributions.
The canonical bundle of a cap contributes $-1$ whereas the sphere with three
holes contributes $+1$. In any case the constraint $p+q=-2$ is always
automatically satisfied.

Let us consider $T_{N}$ theory, which has three full punctures and all the
building blocks have the same sign. If we now modify the theory by changing
the sign of one puncture the two line bundles become $L_{1}\equiv
L_{2}=\mathcal{O}(-1)$. Physically, this is interpreted as follows: We start
from $T_{N}$ and we add by hand a chiral multiplet $M$ transforming in the
adjoint representation of the global symmetry carried by the puncture. We also
couple it to the corresponding moment map $\mu$ by adding the superpotential
term ${\rm Tr\,}(\mu M)$. Indeed we can generalize the construction by
including generic punctures, which are in one-to-one correspondence with
nilpotent orbits of the global symmetry. When the signs of the pair of pants
and the puncture agree and the puncture is not full, it means that we have
higgsed the theory with a full puncture by turning on a nilpotent vev for the
corresponding moment map. If instead the signs do not agree, it means that we
have turned on a nilpotent vev for the singlet $M$ rather than the moment map,
which is now set to zero in the chiral ring due to the F-term equation for
$M$. Combining these operations we can construct all of the $\mathcal{N}=1$
trinions starting from $T_{N}$ plus a collection of chiral multiplets.

The moduli space of these $\mathcal{N}=1$ theories (on $\mathbb{R}^{3}\times
S^{1}$) is described by the solutions of a generalized Hitchin system
involving two Hitchin fields ($\Phi_{1}$ and $\Phi_{2}$) which are sections of
the line bundles $L_{1}$ and $L_{2}$ respectively. The equations of the
generalized Hitchin system state that these fields are covariantly holomorphic
and commute ($[\Phi_{1},\Phi_{2}]=0$). Each field is singular at punctures of
a given sign only (for example $\Phi_{1}$ is singular only at punctures with
sign plus and analogously $\Phi_{2}$ is singular only at punctures with sign
minus). The singularity is the same as in the $\mathcal{N}=2$ case. Indeed, in
the $\mathcal{N}=2$ case, one field is a one-form and is singular at all the
punctures, whereas the other is a function without poles and is therefore
constant. Setting it to zero we recover the description of the Coulomb branch
of the $\mathcal{N}=2$ theory in terms of an ordinary Hitchin system.

In the rest of this paper we will be concerned with $D_{k}^{b}(J)$ theories,
which correspond to a sphere with two punctures, one is full and the other is
irregular [27, 18]. $J$ is an ADE group and labels the choice of the six-
dimensional $\mathcal{N}=(2,0)$ theory we compactify on the sphere. The
parameters $k$ and $b$ specify the choice of irregular puncture: If we take a
local coordinate $w$ on the sphere such that the irregular puncture is located
at $w=0$, the behavior of the Hitchin field near $w=0$ is

$\Phi\simeq\frac{T}{w^{1+k/b}}+\dots\,,$ (2.2)

where $T$ is a regular semi-simple element of the Lie algebra $J$ and the dots
stand for less singular terms.444Note that our notation slightly differs from
the one adopted in [27], whereby the parameter $k$ is shifted by one unit of
$b$ with respect to the $k$ appearing here. The parameter $k$ is an arbitrary
positive integer, whereas $b$ can take two or three different values depending
on the choice of $J$:

$\begin{array}[]{|c|c|}\hline\cr J&b\\\ \hline\cr A_{N-1}&N;\;N-1\\\ \hline\cr
D_{N}&2N-2;\;N\\\ \hline\cr E_{6}&12;\;9;\;8\\\ \hline\cr E_{7}&18;\;14\\\
\hline\cr E_{8}&30;\;24;\;20\\\ \hline\cr\end{array}$ (2.3)

Notice that the Coxeter number $h(J)$ is always an allowed value for $b$. In
the following we will drop the label $b$ whenever $b=h(J)$. A detailed
discussion about these theories can be found in [18].

The analysis with the two Hitchin fields briefly reviewed above does not
immediately apply to $D_{k}^{b}(J)$ models. However, by analogy with the case
of $\mathcal{N}=1$ theories labelled by a sphere with regular punctures only,
we propose that the $D_{k}^{b}(J)$ theory deformed by coupling an adjoint
chiral to the moment map associated with the symmetry carried by the full
puncture is described by a generalized Hitchin system in which both fields are
sections of $\mathcal{O}(-1)$. One field is singular at the irregular puncture
only (say $\Phi_{1}$) and the singularity is the same as in the parent
$\mathcal{N}=2$ theory, whereas the other field $\Phi_{2}$ is singular at the
regular puncture only. Moreover, giving a nilpotent vev to the adjoint chiral
(i.e. initiating a Maruyoshi-Song flow) can be implemented by changing the
boundary condition for $\Phi_{2}$. The nontrivial consistency checks we will
find below give strong evidence in favor of our claim.

###  2.2 RG flows and spectral curves

#### Extracting the SW curve of the IR theory

In this section we will use the results reviewed in the previous section about
the generalized Hitchin system to analyze the Maruyoshi-Song flow at the level
of the SW curve. As is well known, in the case of the ordinary Hitchin system
the SW curve for the underlying ${\cal N}=2$ theory is encoded in the spectral
equation for the Hitchin field $\Phi$ [25]:

$\det(\lambda-\Phi)=0\,,$ (2.4)

where $\lambda$ is the SW differential. If we now choose local coordinates for
the base and fiber of $T^{*}(\mathcal{C})$ and write $\lambda$ in terms of
those, (2.4) becomes the SW curve describing the theory. In the case of ${\cal
N}=1$ class $\mathcal{S}$ theories we have a similar result involving the
spectral equations of the generalized Hitchin fields $\Phi_{1,2}$ [28, 29, 30,
31]:

$\left\\{\begin{array}[]{l}\det(\lambda_{1}-\Phi_{1})=0\\\
\det(\lambda_{2}-\Phi_{2})=0\,,\\\ \end{array}\right.$ (2.5)

where $\lambda_{1,2}$ are sections of the corresponding line bundles. In
general the system (2.5) should then be supplemented by further equations
enforcing the commutativity constraint $[\Phi_{1},\Phi_{2}]=0$. As we will
explain later, this fact will not play any role in our discussion.

Now we use our guess that the $D_{k}^{b}(J)$ theory with a chiral multiplet in
the adjoint of $J$ coupled to the corresponding moment map is described by a
generalized Hitchin system, with both line bundles of degree $-1$. One field,
say $\Phi_{1}$, is singular at the irregular puncture only, whereas $\Phi_{2}$
is singular at the regular puncture. The boundary conditions at the two
punctures are the same as in the parent ${\cal N}=2$ theory. Upon giving a
principal nilpotent vev to the adjoint chiral we remove the regular puncture
completely. As a result, in the geometry describing the infrared fixed point,
the field $\Phi_{2}$ becomes a section of $\mathcal{O}(-1)$ on the sphere
without poles and therefore vanishes identically. We can therefore focus on
the spectral equation of $\Phi_{1}$ only.

Let us illustrate the procedure for the class of theories $D_{k}(SU(N))$ (i.e.
$J=SU(N)$ and $b=N$). The extension to other models with $J=SU(N)$ or
$J=SO(2N)$ is trivial. The SW curve and differential read

$x^{N}+z^{k}+...=0\,;\qquad\lambda=x\frac{dz}{z}\,,$

where the dots stand for subleading terms. We can rewrite it as in (2.4):

$\lambda^{N}+\sum_{\alpha=2}^{N}\lambda^{N-\alpha}P_{d_{\alpha}}(z)\left(\frac{dz}{z}\right)^{\alpha}=0\,.$
(2.6)

The polynomials $P_{d_{\alpha}}(z)$ have degree $d_{\alpha}$ equal to the
integer part of $k\alpha/N$ and $P_{d_{N}}(z)$ can be taken to be monic. The
degree $\alpha$ differentials have a pole of order $\alpha$ at $z=0$ (the
regular puncture) and a pole of order $\alpha+d_{\alpha}$ at $z=\infty$
(irregular puncture). In order to model the infrared fixed point of the
Maruyoshi-Song flow we now turn our attention to a twisted Hitchin field which
is a section of $\mathcal{O}(-1)$ and is singular only at infinity, where the
irregular puncture is located. The corresponding spectral equation then reads

$\lambda_{1}^{N}+\sum_{\alpha=2}^{N}\lambda_{1}^{N-\alpha}P_{d_{\alpha}}(z)(dz)^{\frac{\alpha}{2}}=0\,,$
(2.7)

where the various terms are chosen to reproduce the singular behaviour at the
irregular puncture555We denote with $(dz)^{\alpha/2}$ a section of
$\mathcal{O}(-\alpha)$ on the sphere without zeros and with a pole of order
$\alpha$ at infinity..

Our claim now is that the twisted Hitchin field $\Phi_{1}$ whose spectral
equation is given by (2.7) is equivalent to an ordinary Hitchin field
$\widetilde{\Phi}$ obtained by tensoring $\Phi_{1}$ with a reference section
of $\mathcal{O}(-1)$ having a simple pole at infinity. The corresponding
spectral equation is then obtained by tensoring (2.7) with $(dz)^{N/2}$:

$\widetilde{\lambda}^{N}+\sum_{\alpha=2}^{N}\widetilde{\lambda}^{N-\alpha}P_{d_{\alpha}}(z)(dz)^{\alpha}=0\,.$
(2.8)

This equation precisely encodes the SW data of the IR fixed point of the RG
flow, namely the theory $(A_{N-1},A_{k-1})$. In order to see this, we choose
local coordinates on $T^{*}(\mathbb{P}^{1})$ and set
$\widetilde{\lambda}=\tilde{x}dz$. Plugging this into (2.8) we find

$\tilde{x}^{N}+z^{k}+...=0\,;\quad\widetilde{\lambda}=\tilde{x}dz\,.$ (2.9)

We normalized the coordinates in such a way that $P_{d_{N}}(z)$ in (2.8) is
monic. We can also take advantage of the freedom to shift $z$ (which does not
change the SW differential up to exact terms) to remove all subleading terms
proportional to $z^{k-1}$. These are precisely the SW curve and differential
of the $(A_{N-1},A_{k-1})$ theory.

#### Counting decoupled operators

In order to count decoupled operators we can make use of the one-to-one
correspondence between UV and IR CB operators discussed in [19], which we will
now review. We start by recalling that for $D_{k}(SU(N))$ theories the versal
deformations of the $A_{N-1}$ singularity are the mass Casimirs of the $SU(N)$
global symmetry. The vev of ultraviolet (UV) CB operators is instead described
by the $z$-dependent deformation terms. The $(A_{N-1},A_{k-1})$ theory is
described by the same curve but the CB operators correspond to all deformation
terms with coefficient of dimension larger than one. The one-to-one
correspondence between UV and IR CB operators is then described as follows:
given any UV CB operator $u$, divide the corresponding deformation term by
$z$. This operation maps the original term to another deformation and the
scaling dimension of the corresponding parameter $u^{\prime}$ is that of $u$
plus the dimension of $z$, which in the $D_{k}(SU(N))$ theory is equal to
$\frac{N}{k}$. Since by assumption $D(u)>1$, we conclude that

$D(u^{\prime})>1+\frac{N}{k}=\frac{k+N}{k}\,.$

Now we exploit the observation that the scaling dimension in the IR of
$u^{\prime}$ (provided it does not decouple) is $D(u^{\prime})$ times
$\frac{k}{k+N}$,666This can be seen e.g. by comparing the deformation term of
highest dimension in the UV and in the IR. See [18] for details. and due to
the above inequality, we clearly see that this quantity is larger than one. We
then conclude that the term $u^{\prime}$ always corresponds to a CB operator
of the IR theory $(A_{N-1},A_{k-1})$. Analogously, coupling constants of
dimension smaller than one in the UV are mapped to coupling constants in the
IR and mass parameters of dimension one are mapped to mass parameters.

Due to the fact that the curve describing UV and IR fixed points are the same,
the deformation parameters in the two cases are clearly equal in number. In
the UV there are $N-1$ parameters (the mass Casimirs of the $SU(N)$ global
symmetry, which in the $\mathcal{N}=1$ theory are rather interpreted as
expectation values for the singlets) on which the map described above is not
defined, and accordingly we expect to see $N-1$ parameters in the IR which do
not arise from our UV-IR map. These are easy to describe: For any integer
$n\leq N-2$ find the largest $j$ such that the monomial $x^{n}z^{j}$ appears
in the curve777For $n=0$ we take $j$ to be $k-1$.. Clearly all such terms (and
only those) cannot arise from our map and they are precisely $N-1$ in number.
Terms of the form $x^{n}z^{k-1}$ can actually be removed by shifting
$z$888This change of variables is allowed in the IR only because in the UV it
would change the location of the regular singularity. and do not arise in the
infrared theory. Their number can be easily determined to be the integer part
of $N/k$ plus one. All the other terms correspond to coupling constants in the
infrared theory. This is seen as follows: the dimension of the parameters
multiplying the monomials $x^{n}z^{j}$ has to be smaller than the dimension of
$z$, otherwise $x^{n}z^{j+1}$ would be an allowed deformation term. Combining
this with the fact that the dimension of $z$ in the IR is smaller than 1, we
reach the desired conclusion. So we conclude as expected that, out of the
singlets and UV CB operators, all except $N-1$ operators become CB operators
in the IR.

###  2.3 Twisted vs ordinary Hitchin systems

The analysis of the previous section relied on the equivalence of the moduli
space of solutions of two different Hitchin systems on a punctured Riemann
sphere: An ordinary one, with Hitchin field
$\widetilde{\Phi}\in\Gamma(\mathcal{O}(-2))$, and a twisted one, with Hitchin
field $\Phi_{1}\in\Gamma(\mathcal{O}(-1))$. Boundary conditions are such that
both these Hitchin fields are smooth sections everywhere on the sphere except
at one point (the same point for both), the irregular puncture, where they
develop a pole. We argue that there is a one-to-one correspondence between
solutions of the two systems with said boundary conditions. One way to see
this is to bijectively map one system of equations to the other.

The non-holomorphic equation of an ordinary Hitchin system reads

$\displaystyle F+[\widetilde{\Phi},\widetilde{\Phi}^{\dagger}]=0\,,$ (2.10)

where $F$ is the $(1,1)$-form gauge field strength and † simply denotes
complex conjugation and matrix transposition. This equation can also be
trivially written in terms of components and, in the local patch $U_{z}$ with
coordinate $z$, it takes the simple form:

$F_{z\bar{z}}+[\widetilde{\Phi}_{z},\widetilde{\Phi}^{\dagger}_{\bar{z}}]\,.$
(2.11)

On the contrary, the non-holomorphic equation of a twisted Hitchin system only
has a well defined expression in terms of components:

$h^{-1/2}_{z\bar{z}}F_{z\bar{z}}+[(\Phi_{1})_{z},(\Phi_{1})^{\dagger}_{\bar{z}}]\,,$
(2.12)

where $h$ is the hermitian metric on the tangent bundle of the sphere.

Let us now write

$\widetilde{\Phi}=\hat{s}\,\Phi_{1}\,,$ (2.13)

where $\hat{s}\equiv s/||s||$, and $s$ is a nowhere-vanishing reference
section of $\mathcal{O}(-1)$ that is smooth everywhere except at one point,
which we choose to be the same point where $\widetilde{\Phi}$ and $\Phi_{1}$
are singular. This condition on $s$ is needed in order not to change the
assigned boundary conditions of the two Hitchin fields by creating further
poles. There is only one such reference section (modulo rescaling by global
smooth functions), and it has obviously a pole of order $1$ at the irregular
puncture. For later convenience, we normalized this section by dividing it by
its norm, i.e. by the square root of the globally well-defined smooth function

$||s||^{2}=h^{-1/2}\bar{s}s\,.$ (2.14)

Placing the irregular puncture at the infinity of the patch $U_{z}$, the local
presentation of $\hat{s}$ in that patch is

$\hat{s}|_{U_{z}}=\frac{\sqrt{{\rm d}z}}{||\sqrt{{\rm
d}z}||}\,,\qquad\quad||\sqrt{{\rm d}z}||^{2}=\sqrt{\frac{{\rm d}z\,{\rm
d}\bar{z}}{1+|z|^{2}}}\,,$ (2.15)

where we have used the Fubini-Study metric on $\mathbb{P}^{1}$ to write down
the norm.

Plugging (2.13) into (2.10) and using (2.14) yields (2.12). Since $\hat{s}$ is
nowhere vanishing and unique, (2.13) is a bijective map between solutions to
the ordinary and the twisted Hitchin systems. This map respects the boundary
conditions of the Hitchin fields, but it changes their order of pole at the
irregular puncture. To see this, it is convenient to work in the local patch
$U_{w}$, where $w=1/z$, so that the irregular puncture is located at $w=0$.
Locally we can always switch to a gauge, the holomorphic gauge, where
$A^{0,1}=0$, and thus have the Hitchin field satisfy the equation

$\bar{\partial}_{\bar{w}}\widetilde{\Phi}_{w}=2\pi
i\sum_{i=0}^{p-1}a_{i}\partial_{w}^{i}\delta_{w}\,,$ (2.16)

where $\delta_{w}$ is the delta function on $w=0$ and $a_{i}$ are matrix-
valued coefficients determining the singular behavior of $\widetilde{\Phi}$
such that, around the irregular puncture

$\widetilde{\Phi}\sim{\rm
d}w\sum_{i=0}^{p-1}\frac{(-1)^{i}i!\,a_{i}}{w^{i+1}}\,.$ (2.17)

Using (2.15), it is immediate to see that our reference section around the
irregular puncture is

$\hat{s}|_{U_{w}}=\frac{1}{||\sqrt{{\rm d}w}||}\frac{\sqrt{{\rm d}w}}{w}\,,$
(2.18)

and therefore we have

$\Phi_{1}\sim\sqrt{{\rm
d}w}\sum_{i=0}^{p-2}\frac{(-1)^{i}i!\,b_{i}}{w^{i+1}}\,,$ (2.19)

where $b_{i}=||\sqrt{{\rm d}w}||a_{i+1}$. Hence the twisted Hitchin field has
a pole at the irregular puncture of order one unit less than the one of the
ordinary Hitchin field, i.e.

$\bar{\partial}_{\bar{w}}(\Phi_{1})_{w}=2\pi
i\sum_{i=0}^{p-2}b_{i}\partial_{w}^{i}\delta_{w}\,.$ (2.20)

###  2.4 Comments about non-principal nilpotent vevs

In [13] the authors found several examples of theories which exhibit
supersymmetry enhancement in the IR upon turning on a non-principal nilpotent
vev. At first sight these flows do not seem to fit in our discussion since
under a non-principal nilpotent vev the regular puncture is not removed
completely and the generalized Hitchin system does not reduce to a simpler
twisted Hitchin system. The scope of this section is to notice that, as was
already pointed out in [18], we are not actually missing any of the known
enhancing RG flows by focusing on our setup.

The argument is based on the simple observation that for any group we can get
several non-principal nilpotent orbits just by embedding the principal
nilpotent orbit of a subgroup. The point is the following: In our geometric
setup we are activating an expectation value for the moment map associated
with the full puncture only and since the global symmetry carried by the
regular puncture is in general only a subgroup of the actual global symmetry
of the theory, by considering the principal nilpotent vev for the
corresponding moment map we are actually considering (in general) a non-
principal nilpotent vev for the theory.

Our main observation is that whenever there are multiple choices of nilpotent
vev which lead to supersymmetry enhancement in the IR, there are also multiple
realizations of the theory in the $D_{k}^{b}(J)$ class (with different $J$)
and, by considering the principal nilpotent orbit for $J$ in the various
realizations, we always recover all the enhancing RG flows. We do not have an
a priori proof of this statement, but we will now check that we do recover all
the RG flows discussed in [13].

  * •

Let us start by the case of $SU(2)$ SQCD, which has three different
realizations in the $D_{k}^{b}(J)$ class: It is equivalent to
$D_{1}^{4}(SO(8))$, $D_{2}(SU(4))$ and $D_{2}^{2}(SU(3))$. We therefore
predict that the theory exhibits enhancement upon turning on a principal
nilpotent vev and also $SU(4)$ and $SU(3)$ induced nilpotent vevs
respectively. It is well-known that there is a unique way to embed $SU(3)$
inside $SO(8)$ up to conjugation and the corresponding induced nilpotent orbit
is labelled by the partition $[3^{2},1^{2}]$. Indeed it was found in [13] that
the corresponding vev does lead to enhancement in the IR. The remaining
enhancing orbits are the principal (as our construction correctly predicts),
the orbit $[5,1^{3}]$ and the two $[4,4]$ orbits. The last three all lead to
the same IR fixed point. This result is perfectly consistent with our
construction, which predicts enhancement in the case of an $SU(4)$ induced
nilpotent vev: There are three inequivalent embeddings of $SU(4)$ inside
$SO(8)$ and the corresponding nilpotent orbits are precisely the three listed
above.

  * •

Let us now discuss the other Lagrangian cases. The only relevant ones are
$SU(N)$ and $USp(2N)$ SQCD, since all other Lagrangian theories exhibit
enhancement upon turning on a principal nilpotent vev only (or do not exhibit
enhancement at all). In the case of $SU(N)$ SQCD ($N>2$) with $2N$ flavors
there are two choices of nilpotent vevs (principal and subregular), and
accordingly we have two different realizations of conformal $SU(N)$ SQCD in
our class: $D_{2}(SU(2N))$ and $D_{2}^{2N-2}(SU(2N-1))$. Analogously, the two
possible choices of nilpotent vev for $USp(2N)$ conformal SQCD, whose global
symmetry is $SO(4N+4)$, correspond to the two different realizations of this
theory in the $D_{k}^{b}(J)$ class: $D_{1}^{2N+2}(SO(4N+4))$ and
$D_{2}(SO(4N+2))$.

  * •

Let us consider the theories called $(I_{N+1,1-N},F)$ in [13], whose global
symmetry is $SU(N+1)$. These models exhibit enhancement both for the principal
and subregular nilpotent orbits. When $N$ is odd the theory is equivalent to
$SU(n_{c})$ conformal SQCD with $n_{c}=\frac{N+1}{2}$ colors, whereas for $N$
even the model is not Lagrangian. We indeed recover this result! In our
notation these theories have the following realizations: $D_{2}(SU(N+1))$ and
$D_{2}^{N-1}(SU(N))$. The full global symmetry is manifestly visible in the
first realization only. When $N=2$ the theory coincides with $D_{4}$ Argyres-
Douglas theory (sometimes called $\mathcal{H}_{2}$), which has $SU(3)$ global
symmetry and flows to $\mathcal{N}=2$ SCFTs under both choices of nilpotent
vev (principal and minimal).

  * •

Finally, let us discuss Minahan-Nemeschansky theories. In the case of the
$E_{6}$ theory the authors of [13] found that there are three choices of
nilpotent vev which lead to supersymmetry enhancement in the IR. Accordingly,
it turns out that the $E_{6}$ Minahan-Nemeschansky theory appears three times
in the $D_{k}^{b}(J)$ class: It is equivalent to $D_{2}(SO(8))$,
$D_{1}^{5}(SO(10))$ and $D_{1}^{9}(E_{6})$. By activating a principal
nilpotent vev for the group $J$ we recover the three enhancing RG flows. We
find instead two different realizations of the $E_{7}$ Minahan-Nemeschansky
theory: $D_{1}^{14}(E_{7})$ and $D_{1}^{8}(E_{6})$, in agreement with the fact
that enhancement occurs only for two choices of nilpotent vev. Finally,
$E_{8}$ Minahan-Nemeschansky theory exhibits enhancement only in the case of a
principal nilpotent vev. As expected we find just one realization of this
model: $D_{1}^{24}(E_{8})$.

##  3 Systematics of SUSY enhancement

In this section we will derive a necessary algebraic criterion for
supersymmetry enhancement and, in case enhancement occurs, explain how to
systematically derive the SW curve and differential of the IR theory (as well
as the correct conformal dimensions of CB operators) without using any
maximization procedure. After discussing a few general facts about the
underlying geometries in Subsection 3.1, we will study in detail a specific
rank-$2$ Lagrangian case in Subsection 3.2, in order to illustrate the key
steps of our approach. We will then conclude by analyzing in Subsection 3.3 a
particular rank-$6$ case, whose peculiarities will lead us to an important
refinement of our algebraic criterion.

###  3.1 $\mathcal{N}=1$ curves from branes

For the analysis of this section it is crucial to understand how to implement
the SUSY breaking deformations at the level of the underlying SW geometry.
This will directly generalize the results of [20] for rank-1 theories, whereby
all Maruyoshi-Song flows were seen to originate from certain T-brane
deformations [41, 42] of the Weierstrass geometry in F-theory. To this end,
rather than aiming for a general treatment, we find it more convenient to work
with a simple class of SCFT’s. Extrapolating the rules of our approach to
treat more complicated theories (in particular any linear quiver) can be done
straightforwardly.

Figure 1: (a): Brane configuration for 4d $\mathcal{N}=2$ SQCD. The figure
shows the case with $N=3$. $\otimes$ represents an $\mathcal{N}=2$ D6-brane.
(b). Brane configuration obtained after rotating the $\mathcal{N}=2$ D6-brane
in Figure 1. The $\mathcal{N}=1$ D6-branes are depicted as red vertical lines.

Consider 4d $\mathcal{N}=2$ SQCD with $N$ colors and $2N$ flavors as the
starting theory. Its SW geometry can be easily derived from a standard Witten
cartoon involving D4, D6 and NS5-branes [43] (see Figure 1). The various
branes extend in ten-dimensional flat space as follows:

$\begin{array}[]{c|cccccccccc}\text{Witten cartoon}&0&1&2&3&4&5&6&7&8&9\\\
\hline\cr\text{D4}&\times&\times&\times&\times&&&\times&&\\\
\text{NS5}&\times&\times&\times&\times&\times&\times&&&&\\\
\text{D6${}_{\mathcal{N}=2}$}&\times&\times&\times&\times&&&&\times&\times&\times\\\
\hline\cr\text{D6${}_{\mathcal{N}=1}$}&\times&\times&\times&\times&\times&\times&&\times&&\end{array}$

where the subscripts N=2 and N=1 indicate the amount of supersymmetry
preserved by the orientation of the corresponding D6-branes. To engineer
$\mathcal{N}=2$ SQCD, $\mathcal{N}=2$ D6-branes can be placed in any place and
different configuraitons are related by the Hanany-Witten transitions [44].
Here we have chosen to work in the Hanany-Witten frame where all of the
$\mathcal{N}=2$ D6-branes are on one side of the two NS5-branes, which makes
the whole $U(2N)$ flavor symmetry manifest, and thus allows us to access all
of its nilpotent orbits when turning on the deformation. The SW geometry shows
up in the M-theory uplift as the internal world-volume of the M5-brane lifting
the above D4/NS5 configuration. This spans a holomorphic curve within the
Taub-NUT space lifting the D6N=2, and it has the following general form

$\displaystyle
z^{2}+c_{1}p_{N}(x)z+c_{2}\det\left(x\mathbb{1}_{2N}-M\right)=0\,,\qquad\lambda=x\frac{{\rm
d}z}{z}\,,$ (3.1)

where $z$ is a coordinate combining direction $6$ and M-theory circle, $x$
stands for directions $4,5$, $c_{1},c_{2}$ are constants depending on the
exactly marginal gauge coupling, $p_{N}(x)$ is a monic polynomial in $x$ of
degree $N$, encoding the $N-1$ CB parameters in its coefficients, $M$ is the
mass matrix for the $U(2N)$ flavors, and $\lambda$ indicates the SW
differential. From this perspective, the eigenvalues of $M$ are understood as
the relative position of the $N$ D6 branes in the directions $4$ and $5$ with
respect to the stack of $N$ D4-branes connecting the two NS5-branes.

The $\mathcal{N}=2\to\mathcal{N}=1$ coupling (1.1) can now be implemented
simply by rotating all of the D6-branes and taking them oriented like the
D6N=1 in the table above [45, 28]. Recalling that the meson $\mu$ is made of
fields originating from the strings stretching between the $N$ gauge D4-branes
and the $2N$ flavor D4-branes, this rotation has the effect of promoting the
mass matrix $M$ to a 4d dynamical chiral field, describing the now free motion
of the flavor D4-branes in directions $4,5$, which are now shared by NS5 and
D6-branes. The $\mathcal{N}=1$ brane configuration is depicted in Figure 1.
Therefore, in order to study all Maruyoshi-Song flows of SQCD, it will suffice
to insert in (3.1) the explicit form of the “flipping” field $M$

$M=\rho(\sigma^{+})+\sum_{j}M_{j,-j}\,,$ (3.2)

where $\rho$ indicates the nilpotent embedding and $M_{j,-j}$ the fluctuation
associated to the lowest component of the spin $j$ representation of the
embedded $SU(2)$. The sum extends over all spins appearing in the
decomposition of the adjoint representation of $SU(2N)$ (see [46, 47]).

The logic just described is completely general and can be applied to any
starting SCFT in 4d, even non-Lagrangian ones: The space (3.1), which we
dubbed $X_{n}$ in the introduction, has the general structure of a genus-$r$
Riemann-surface fibered over a base of dimension $n-r>r$, where $r$ is the
rank of the theory. Studying whether a given orbit leads to SUSY enhancement
is reduced to analyzing whether near the origin the fibration structure of
$X_{n}$ is non-trivial only on a $r$-dimensional base.

This picture nicely connects to the description of the enhancement via Hitchin
systems we discussed in Section 2. The Witten cartoon we have seen for SQCD
translates into a class-$\mathcal{S}$ configuration characterized by a two-
sphere with one regular maximal puncture, carrying $SU(2N)$ flavor symmetry,
and one irregular puncture, accounting for the two “unbalanced” NS5-branes of
Figure 1. The SW geometry, in turn, arises as a $2N$-branched cover of the
punctured sphere [25]

$\det\left(\lambda\mathbb{1}_{2N}-\Phi(z)\right)=0\,,$ (3.3)

where $z$ is the local coordinate on the sphere, $x$ is the local fiber
coordinate of its canonical bundle, and $\Phi$ is the Hitchin field, a
meromorphic section of $\mathcal{O}(-2)$ on the sphere, with poles at the
punctures. By reducing this configuration back to type IIA along a different
circle, say direction $3$, $\Phi$ acquires the interpretation of the field of
transverse deformations along directions $4,5$ of a stack of $2N$ D4-branes
wrapped on the punctured sphere:

$\begin{array}[]{c|cccccccccc}\text{class-$\mathcal{S}$}&0&1&2&4&5&6&7&8&9&10\\\
\hline\cr\text{D4}&\times&\times&\times&&&\times&&&&\times\\\
\text{$\Phi$}&&&&\times&\times&&&&&\\\
\text{$\Phi^{\prime}$}&&&&&&&&\times&\times&\end{array}$

In this different duality frame, the rôle of the mass matrix $M$ is played by
a second Hitchin field $\Phi^{\prime}$, constant over the sphere, representing
the transverse deformation of the D4 stack along directions $8,9$. This is
because $\Phi^{\prime}$ is identified with the complex scalar in the vector
multiplet of the three-dimensional $\mathcal{N}=4$ theory living on the stack,
and as such it couples to the matter $\mu$ localized at the regular puncture
as ${\rm Tr\,}(\mu\Phi^{\prime})$ [31]. As explained in Section 2, activating
the SUSY breaking deformation (1.1), therefore, translates in this context to
promoting $\Phi^{\prime}$ to a meromorphic section999In Section 2 this
meromorphic section was called $\Phi_{1}$. of $\mathcal{O}(-1)$, and to
viewing the $\mathcal{N}=1$ geometry $X_{n}$ (Eq. (3.1) with $M$ regarded as a
field) as the intersection of (3.3) with the second spectral equation

$\det\left(\lambda^{\prime}\mathbb{1}_{2N}-\Phi^{\prime}(z)\right)=0\,.$ (3.4)

Recall that $\Phi^{\prime}$ is taken completely smooth at the regular puncture
$z=0$. At this location, as is evident from Eq. (3.1), $\Phi^{\prime}$ has
exactly the same spectral data of $M$, thus elucidating the meaning of the
flipping field within the generalized Hitchin system of $\mathcal{N}=1$
class-$\mathcal{S}$ theories.

###  3.2 SQCD with $6$ flavors

With these geometric discussions in mind, we now analyze in detail a 4d
$\mathcal{N}=2$ SQCD with $N=3$, and the systematics of its Maruyoshi-Song
flows. We first focus on cases where the supersymmetry is enhanced to
$\mathcal{N}=2$ at IR. In order to see if the resulting curve describes a 4d
$\mathcal{N}=2$ superconformal field theory, we will make use of two necessary
conditions which are satisfied for an $\mathcal{N}=2$ superconformal field
theory. The first condition is that the genus of the curve should agree with
the number of CB operators. The second condition is that if there is a
parameter $a$ with $1<D(a)\leq 2$ in the curve, then there should be only one
parameter $b$ which satisfies $D(a)+D(b)=2$. Since these are necessary
conditions we cannot say exactly that the curve satisifying the two conditions
describes an $\mathcal{N}=2$ superconformal field theory. We can only say that
it is not inconsistent that it does. However the two conditions are more
powerful when we single out theories that do not lead to supersymmetry
enhancement. Indeed, in Subsection 3.3, we will see cases which do not satisfy
at least one of the two conditions, and hence the supersymmetry is not
enhanced for those cases.

We first start from the SW curve of 4d $\mathcal{N}=2$ $SU(3)$ gauge theory
with six flavors. The explicit form of the curve can be obtained from (3.1)
with $N=3$ and it is given by

$\displaystyle
z^{2}+\left(a_{1}x^{3}+a_{2}x+a_{3}\right)z+\prod_{i=1}^{6}\left(x-m_{i}\right)=0\,,$
(3.5)

where we chose

$\displaystyle M=\text{diag}(m_{1},m_{2},m_{3},m_{4},m_{5},m_{6})\,.$ (3.6)

Here $\text{diag}(a,b,c,\cdots)$ denotes a diagonal maxtrix with the entries
$a,b,c,\cdots$ and $m_{i},(i=1,\cdots,6)$ are mass parameters for the six
flavors. The SW diffrential is the same as the general form in (3.1), namely,

$\displaystyle\lambda=\frac{x}{z}dz\,.$ (3.7)

Note that, by a $z$-dependent shift of $x$, the SW differential changes only
by a total derivative term. We used a constant shift of $x$ to eliminate the
monomial $x^{2}z$ in (3.5). We have also rescaled $x$ to get rid of the
overall constant in front of the product term.

We can determine the scaling dimension of the parameters $a_{1},a_{2},a_{3}$
in (3.5) from the fact that the scaling dimension of the SW differential (3.7)
is equal to one. This fixes the dimension of $x$ to be $1$. Then, from
homogeneity of the curve polynomial (3.5), the dimension of $z$ is $3$ and we
have

$\displaystyle D^{\rm UV}(a_{1})=0\,,\qquad D^{\rm UV}(a_{2})=2\,,\qquad
D^{\rm UV}(a_{3})=3\,,$ (3.8)

where the superscript UV reminds us that this is the UV theory. We can then
interpret $a_{1}$ as the coupling constant and $a_{2},a_{3}$ as CB operators.

We can see that the curve (3.5) satisfies the two necessary conditions for a
4d $\mathcal{N}=2$ superconformal field theory. The equation (3.5) describes a
genus-two curve at a generic point on the Coulomb-branch moduli space. Since
we have two CB operators $a_{2}$ and $a_{3}$, the genus of the curve indeed
agrees with the number of the CB operators. Regarding the second condition, we
have one parameter $a_{2}$ which satisfies $1<D^{\rm UV}(a_{2})\leq 2$. Then
we can see that the curve contains the associated parameter $a_{1}$ which
satisfies $D^{\rm UV}(a_{1})+D^{\rm UV}(a_{2})=2$.

As described in Subsection 3.1, when we turn on the $\mathcal{N}=1$ coupling
with an adjoint chiral multiplet, the equation of the curve is essentially the
same as (3.5) but the mass matrix $M$ is now promoted to a dynamical chiral
field. Namely, we consider the curve

$\displaystyle
z^{2}+\left(a_{1}x^{3}+a_{2}x+a_{3}\right)z+\det\left(x\mathbb{1}_{6}-M-\tilde{M}_{1}\mathbb{1}_{6}\right)=0\,,$
(3.9)

where $M$ is given by (3.2) and $\tilde{M}_{1}\mathbb{1}_{6}$ corresponds to
the trace component. Note that the SW differential is not necessarily the same
as (3.7), and needs to be determined for each example.

In this subsection we consider two examples, i.e. the nilpotent orbits $[6]$
and $[5,1]$. It is known from a-maximization that these cases lead to
supersymmetry enhancement [12, 13]. Here, instead, we carry out this analysis
in a purely algebraic manner, using the $\mathcal{N}=1$ curve (3.9).

#### Orbit $[6]$ of $SU(6)$

We first consider turning on a vev in the maximal nilpotent orbit of $SU(6)$,
labeled by $[6]$. For this nilpotent orbit, the raising operator of the
$\mathfrak{sl}(2)$ standard triple101010For a standard reference on building
standard triples see [48]. is canonically defined to be:

$\displaystyle\rho(\sigma^{+})=\left(\begin{array}[]{cccccc}0&1&0&0&0&0\\\
0&0&1&0&0&0\\\ 0&0&0&1&0&0\\\ 0&0&0&0&1&0\\\ 0&0&0&0&0&1\\\
0&0&0&0&0&0\end{array}\right).$ (3.16)

Under the background (3.16) the adjoint representation of the
$\mathfrak{su}(6)$ flavor algebra splits according to the branching rule

$\mbox{adj}\to V_{5}\oplus V_{4}\oplus V_{3}\oplus V_{2}\oplus V_{1}\,,$
(3.17)

where with $V_{j}$ we denote the $\mathfrak{sl}(2)$ irreducible representation
of spin $j$. The components that remain coupled are the lowest components of
each spin $j$ representation, namely $M_{j,-j}$ for $j=1,\cdots,5$. Hence the
$M$ in (3.9) is given by

$\displaystyle M=\left(\begin{array}[]{cccccc}0&1&0&0&0&0\\\
5M_{1,-1}&0&1&0&0&0\\\ 5M_{2,-2}&8M_{1,-1}&0&1&0&0\\\
5M_{3,-3}&9M_{2,-2}&9M_{1,-1}&0&1&0\\\
M_{4,-4}&8M_{3,-3}&9M_{2,-2}&8M_{1,-1}&0&1\\\
M_{5,-5}&M_{4,-4}&5M_{3,-3}&5M_{2,-2}&5M_{1,-1}&0\end{array}\right).$ (3.24)

Then the characteristic polynomial in the curve (3.9) becomes

$\displaystyle\det(x\mathbb{1}_{6}-M-\tilde{M}_{1}\mathbb{1}_{6})=$
$\displaystyle
x^{6}-6\tilde{M}_{1}x^{5}+\left(15\tilde{M}_{1}^{2}-35M_{1,-1}\right)x^{4}+$
(3.25)
$\displaystyle+\left(-20\tilde{M}_{1}^{3}+140\tilde{M}_{1}M_{1,-1}-28M_{2,-2}\right)x^{3}+$
$\displaystyle+\left(15\tilde{M}_{1}^{4}-210\tilde{M}_{1}^{2}M_{1,-1}+259M_{1,-1}^{2}+84\tilde{M}_{1}M_{2,-2}-18M_{3,-3}\right)x^{2}+$
$\displaystyle+\left(-6\tilde{M}_{1}^{5}+140\tilde{M}_{1}^{3}M_{1,-1}+-518\tilde{M}_{1}M_{1,-1}^{2}-84\tilde{M}_{1}^{2}M_{2,-2}+\right.$
$\displaystyle\left.+220M_{1,-1}M_{2,-2}+36\tilde{M}_{1}M_{3,-3}-2M_{4,-4}\right)x+$
$\displaystyle+\tilde{M}_{1}^{6}-35\tilde{M}_{1}^{4}M_{1,-1}+259\tilde{M}_{1}^{2}M_{1,-1}^{2}-225M_{1,-1}^{3}+28\tilde{M}_{1}^{3}M_{2,-2}+$
$\displaystyle+220\tilde{M}_{1}M_{1,-1}M_{2,-2}+25M_{2,-2}^{2}-18\tilde{M}_{1}^{2}M_{3,-3}+50M_{1,-1}M_{3,-3}+$
$\displaystyle+2\tilde{M}_{1}M_{4,-4}-M_{5,-5}\,.$

We can redefine $M_{j,-j},(j=1,\cdots,5)$ and $\tilde{M}_{1}$ to rewrite
(3.25) as

$\displaystyle\det(x\mathbb{1}_{6}-M-\tilde{M}_{1}\mathbb{1}_{6})=x^{6}+M_{1}x^{5}+M_{2}x^{4}+M_{3}x^{3}+M_{4}x^{2}+M_{5}x+M_{6}\,,$
(3.26)

where the $M_{i}$, $i=2,\cdots 6$ are now the Casimir invariants of $SU(6)$.
In the end we arrive at the equation

$\displaystyle
z^{2}+\left(a_{1}x^{3}+a_{2}x+a_{3}\right)z+x^{6}+M_{1}x^{5}+M_{2}x^{4}+M_{3}x^{3}+M_{4}x^{2}+M_{5}x+M_{6}=0\,,$
(3.27)

after turning on the $\mathcal{N}=1$ deformation (3.24).

We now interpret (3.27) as the IR curve after the RG flow. Since the curve
(3.27) is essentially the same equation as (3.5), it will have the same
holomorphic one-forms. On the other hand, the holomorphic one-forms can be
obtained by taking a derivative of the SW differential with respect to CB
operators. In order to determine the scaling dimension of the parameters in
(3.27), we use that relative scaling dimensions are RG-flow invariant, and
thus that the holomorphic one-form corresponding to the UV CB operator with
the maximal dimension, i.e. $a_{3}$, is the same as the holomorphic one-form
associated with the IR CB operator with the maximal scaling dimension, which
we postulate to be $M_{6}$. In other words, the new SW differential in the IR,
$\lambda_{[6]}$, must satisfy

$\displaystyle\frac{{\rm d}\lambda}{{\rm d}a_{3}}=\frac{{\rm
d}\lambda_{[6]}}{{\rm d}M_{6}}\,,$ (3.28)

where $\lambda$ is given by (3.7). Condition (3.28) leads to the following
relation between parameters and coordinates

$\displaystyle D^{\rm IR}(z)-5D^{\rm IR}(x)=1-D^{\rm IR}(M_{6})\,.$ (3.29)

Equation (3.27) also implies $D(z)=3D(x)$ and $2D(z)=D(M_{6})$111111The symbol
$D$ without subscript indicates the scaling dimension anywhere along the RG
flow.. Then, the scaling dimension of the parameter $M_{6}$ can be fixed as

$\displaystyle D^{\rm IR}(M_{6})=\frac{3}{2}\,.$ (3.30)

The scaling dimension of the other parameters can also be determined:

$\begin{split}&D^{\rm IR}(M_{5})=\frac{5}{4}\,,\quad D^{\rm
IR}(M_{4})=1\,,\quad D^{\rm IR}(M_{3})=\frac{3}{4}\,,\quad D^{\rm
IR}(M_{2})=\frac{1}{2}\,,\quad D^{\rm IR}(M_{1})=\frac{1}{4}\,,\quad\\\
&D^{\rm IR}(a_{1})=0\,,\quad D^{\rm IR}(a_{2})=\frac{1}{4}\,,\quad D^{\rm
IR}(a_{3})=\frac{1}{2}\,,\quad D^{\rm IR}(a_{4})=\frac{3}{4}\,.\end{split}$
(3.31)

Therefore the parameters $M_{6}$ and $M_{5}$ may be identified as the two CB
operators in the IR, being the only ones whose dimension is strictly above the
unitarity bound.

Let us see if the resulting curve (3.27) satisfies the two necessary
conditions for an $\mathcal{N}=2$ superconformal field theory. First, the
genus of the curve (3.27) is two and the number of the CB operators is also
two. Therefore the first condition is satisfied. For the second condition, we
need to see carefully if we can eliminate any parameters in (3.27) by a change
of coordinates which leaves the SW differential invariant up to a total
derivative. For that we need to determine the SW differential $\lambda_{[6]}$
explicitly for the IR theory.

Note that we can write the SW differential as
$\lambda_{[6]}=f(z,x(z,a))dz$,121212Here we are assuming that the SW
differential does not have an explicit dependence on $a$. In the next
subsection, we will analyze an example where we will need to relax this
hypothesis. where $a$ is any CB parameter and $x$ is regarded as a function of
$z$ from the curve equation (3.27), $F(z,x,a)=0$. Hence, the derivative of
$\lambda_{[6]}$ with respect to $a$ can be written as

$\displaystyle\frac{d\lambda}{da}=-\frac{\partial f(z,x)}{\partial
x}\frac{\partial F(z,x,a)}{\partial a}\left(\frac{\partial F(z,x,a)}{\partial
x}\right)^{-1}dz\,,$ (3.32)

Using (3.32), the relation (3.28) implies that

$\displaystyle\frac{d\lambda_{[6]}}{dx}=dz\,,$ (3.33)

which we can trivially solve, leading to

$\displaystyle\lambda_{[6]}=xdz\,,$ (3.34)

up to total-derivative terms.

As opposed to the SW differential in the UV, Eq. (3.7), the one in the IR, Eq.
(3.34), crucially allows for a new change of coordinates: an $x$-dependent
shift of $z$. It is also possible to shift $x$ by a constant. Therefore, Eq.
(3.27) can be further simplified by eliminating the term linear in $z$ and the
term proportional to $x^{5}$:

$\displaystyle z^{2}+x^{6}+M_{2}x^{4}+M_{3}x^{3}+M_{4}x^{2}+M_{5}x+M_{6}=0\,.$
(3.35)

It is now possible to see that the second condition is indeed satisfied.
Namely we have two pairs of coupling constant/CB operator, $(M_{2},M_{6})$ and
$(M_{3},M_{5})$, which satisfy

$\displaystyle D^{\rm IR}(M_{2})+D^{\rm IR}(M_{6})=2\,,\qquad D^{\rm
IR}(M_{3})+D^{\rm IR}(M_{5})=2\,.$ (3.36)

Finally, $M_{4}$ plays the rôle of mass parameter for the IR flavor symmetry.

Since the curve (3.35) with the SW differential (3.34) satisfies the two
conditions, the theory described by the curve is compatible with an
$\mathcal{N}=2$ superconformal field theory. Indeed in this case we know that
the curve is nothing but the SW curve of the $(A_{1},A_{5})$ generalized
Argyres-Douglas theory, which has $U(1)$ flavor symmetry. This is consistent
with the result of [12].

#### Orbit $[5,1]$ of $SU(6)$

Let us now consider turning on a vev for the subregular nilpotent orbit of
$SU(6)$. Such an orbit is labeled by the partition $[5,1]$ of the number
$N_{f}=6$. For this nilpotent orbit, the raising operator of the
$\mathfrak{sl}(2)$ standard triple is canonically defined to be

$\rho(\sigma^{+})=\left(\begin{array}[]{cccccc}$0$&$1$&$0$&$0$&$0$&$0$\\\
$0$&$0$&$1$&$0$&$0$&$0$\\\ $0$&$0$&$0$&$1$&$0$&$0$\\\
$0$&$0$&$0$&$0$&$1$&$0$\\\ $0$&$0$&$0$&$0$&$0$&$0$\\\
$0$&$0$&$0$&$0$&$0$&$0$\end{array}\right).$ (3.37)

Under the background (3.37) the adjoint representation of the
$\mathfrak{su}(6)$ flavor algebra splits according to the branching rule

$\mbox{adj}\to V_{4}\oplus V_{3}\oplus 3V_{2}\oplus V_{1}\oplus V_{0}\,,$
(3.38)

where with $V_{j}$ we denote the $\mathfrak{sl}(2)$ irreducible representation
of spin $j$. As usual, the components of the field $M$ that remain coupled
after turning on the vev (3.37) are given by the lowest component of each
$\mathfrak{sl}(2)$ spin $j$ representation appearing in (3.38), namely
$M_{j,-j}$ for $j=1,\cdots,5$. Hence the $M$ in (3.9) is given in this case by

$\displaystyle M=\left(\begin{array}[]{cccccc}M_{0,0}&1&0&0&0&0\\\
2M_{1,-1}&M_{0,0}&1&0&0&0\\\
2\sum_{i=1}^{3}M_{2,-2}^{(i)}&3M_{1,-1}&M_{0,0}&1&0&0\\\
M_{3,-3}&3\sum_{i=1}^{3}M_{2,-2}^{(i)}&3M_{1,-1}&M_{0,0}&1&0\\\
M_{4,-4}&M_{3,-3}&2\sum_{i=1}^{3}M_{2,-2}^{(i)}&2M_{1,-1}&M_{0,0}&2M^{(2)}_{2,-2}\\\
2M^{(3)}_{2,-2}&0&0&0&0&-5M_{0,0}\end{array}\right).$ (3.45)

where we have denoted with $M^{(i)}_{2,-2}$, $i=1,2,3$ the lowest spin
component of the three different $V_{2}$ representations appearing in (3.38).

Now we need to compute the characteristic polynomial of the matrix
$M-\mathbb{1}_{6}\tilde{M}_{1}$, and perform an analysis analog to that of
section (3.2). However, for ease of presentation of the result, let us just
write the characteristic polynomial of $M$, and re-install the trace part at a
later stage. Such characteristic polynomial can be computed as

$\displaystyle\det\left(\mathbb{1}_{6}x-M\right)$
$\displaystyle=x^{6}-x^{4}\left(15M_{0,0}^{2}+10M_{1,-1}\right)+x^{3}\left(40M_{0,0}^{3}-20M_{0,0}M_{1,-1}-7\sum_{i=1}^{3}M_{2,-2}^{(i)}\right)+$
(3.46)
$\displaystyle+x^{2}\left(-45M_{0,0}^{4}+120M_{0,0}^{2}M_{1,-1}-21M_{0,0}\sum_{i=1}^{3}M_{2,-2}^{(i)}+16M_{1,-1}^{2}-2M_{3,-3}\right)+$
$\displaystyle
x\left(24M_{0,0}^{5}-140M_{0,0}^{3}M_{1,-1}+63M_{0,0}^{2}\sum_{i=1}^{3}M_{2,-2}^{(i)}+\right.$
$\displaystyle+\left.64M_{0,0}M_{1,-1}^{2}-8M_{0,0}M_{3,-3}+8M_{1,-1}\sum_{i=1}^{3}M_{2,-2}^{(i)}-M_{4,-4}\right)+$
$\displaystyle+40M_{0,0}M_{1,-1}\sum_{i=1}^{3}M_{2,-2}^{(i)}-5M_{0,0}M_{4,-4}-4M_{(2,-2)}^{(2)}M_{(2,-2)}^{(3)}+$
$\displaystyle-5M_{0,0}^{6}+50M_{0,0}^{4}M_{1,-1}-35M_{0,0}^{3}\sum_{i=1}^{3}M_{2,-2}^{(i)}-80M_{0,0}^{2}M_{1,-1}^{2}+10M_{0,0}^{2}M_{3,-3}\,.$

We can now redefine the singlets as follows

$\displaystyle M_{2}$ $\displaystyle:=15M_{0,0}^{2}+10M_{1,-1}\,,$ (3.47)
$\displaystyle M_{3}$
$\displaystyle:=40M_{0,0}^{3}-20M_{0,0}M_{1,-1}-7\sum_{i=1}^{3}M_{2,-2}^{(i)}\,,$
$\displaystyle M_{4}$
$\displaystyle:=-45M_{0,0}^{4}+120M_{0,0}^{2}M_{1,-1}-21M_{0,0}\sum_{i=1}^{3}M_{2,-2}^{(i)}+16M_{1,-1}^{2}-2M_{3,-3}\,,$
$\displaystyle M_{5}$
$\displaystyle:=24M_{0,0}^{5}-140M_{0,0}^{3}M_{1,-1}+63M_{0,0}^{2}\sum_{i=1}^{3}M_{2,-2}^{(i)}+$
$\displaystyle+64M_{0,0}M_{1,-1}^{2}-8M_{0,0}M_{3,-3}+8M_{1,-1}\sum_{i=1}^{3}M_{2,-2}^{(i)}-M_{4,-4}\,.$

The characteristic polynomial in terms of these new variables can be written
in a much more compact form, namely

$\det\left(\mathbb{1}_{6}x-M-\mathbb{1}_{6}\tilde{M}_{1}\right)=x^{6}+M_{1}x^{5}+M_{2}x^{4}+M_{3}x^{3}+M_{4}x^{2}+M_{5}x+M_{6}\,,$
(3.48)

where we have re-installed the trace part, and we have defined

$\left.M_{6}\right|_{\tilde{M}_{1}=0}:=-15625M_{0,0}^{6}+625M_{0,0}^{4}M_{2}+125M_{0,0}^{3}M_{3}-25M_{0,0}^{2}M_{4}+5M_{0,0}M_{5}-4M_{2,2}^{(2)}M_{2,2}^{(3)}\,.$
(3.49)

We stress that while in equation (3.48) the quantities $M_{2}$, $\cdots$,
$M_{5}$ have to be considered independent variables, $M_{6}$ is instead
explicitly dependent on all of the $M_{i}$’s and also on $M_{0,0}$,
$M_{2,-2}^{(2)}$, $M_{2,-2}^{(3)}$ and $\tilde{M}_{1}$.

The $\mathcal{N}=1$ deformed curve is therefore given by

$\displaystyle
z^{2}+\left(a_{1}x^{3}+a_{2}x+a_{3}\right)z+x^{6}+M_{1}x^{5}+M_{2}x^{4}+M_{3}x^{3}+M_{4}x^{2}+M_{5}x+M_{6}=0\,,$
(3.50)

As in Subsection (3.2), our strategy to compute the dimension of all the
operators and couplings entering the SW curve of the infrared theory consists
in making an Ansatz for the parameter playing the rôle of the IR CB operator
with the highest dimension. Since relative dimensions are RG-flow invariant,
we are led to identify the highest spin $M_{5}$ as such operator. Then
equating the holomorphic one-forms associated to the UV and IR CB operators of
highest dimension, we get

$\dfrac{{\rm d}\lambda}{{\rm d}a_{3}}=\dfrac{{\rm d}\lambda_{[5,1]}}{{\rm
d}M_{5}}\,,$ (3.51)

where we denoted by $\lambda_{[5,1]}$ the IR SW differential. At the level of
dimensions, this equation implies

$D^{\rm IR}(z)-5D^{\rm IR}(x)=1-D^{\rm IR}(M_{5})\,.$ (3.52)

Now crucially equation (3.52), together with the homogeneity of the curve
(3.50) fixes the dimensions of the coordinates $x$ and $z$ and the operator
$M_{5}$ as

$D^{\rm IR}(M_{5})=\frac{5}{3}\,,\quad D^{\rm IR}(x)=\dfrac{1}{3}\,,\quad
D^{\rm IR}(z)=1\,.$ (3.53)

This in turn fixes the dimensions of all the other parameters as

$\displaystyle D^{\rm IR}(M_{4})$ $\displaystyle=\dfrac{4}{3}\,,\quad D^{\rm
IR}(M_{3})=1\,,\quad D^{\rm IR}(M_{2})=\dfrac{2}{3}\,,\quad D^{\rm
IR}(M_{0,0})=\dfrac{1}{3}\,,$ (3.54) $\displaystyle D^{\rm IR}(a_{1})$
$\displaystyle=0\,,\quad D^{\rm IR}(a_{2})=\dfrac{1}{3}\,,\quad D^{\rm
IR}(a_{3})=\dfrac{2}{3}\,,\quad D^{\rm IR}(a_{4})=1\,.$

The parameters $M_{2,-2}^{(2)}$ and $M_{2,-2}^{(3)}$ only appear in the
deformed curve (3.50) through their product, so it seems that we could only
infer

$D^{\rm IR}(M_{2,-2}^{(2)})+D^{\rm IR}(M_{2,-2}^{(3)})=2\,.$ (3.55)

However, we also know they must have the same dimension as they have the same
spin under the Jacobson-Morozov $\mathfrak{sl}(2)$, as shown in (3.38). This
is enough to conclude that

$D^{\rm IR}(M_{2,-2}^{(2)})=D^{\rm IR}(M_{2,-2}^{(3)})=1\,.$ (3.56)

Nevertheless, these two as well as $M_{0,0}$ are not to be considered as
independent parameters, since they only enter the low-energy effective theory
through the combination $M_{6}$ (3.49).

We can therefore see that the parameters $M_{5}$ and $M_{4}$ can be identified
as CB operators in the IR, as they are the only ones of dimension stricly
greater than one.

Let us now check if the chosen Ansatz leads to a SW geometry that satisfies
the two necessary conditions for an $\mathcal{N}=2$ superconformal field
theory. We see that the genus of the curve (3.50) is two, and also the number
of CB operators is two, so the first condition is satisfied. In order to check
for the second condition, we need to eliminate any reduntant parameter in
(3.50) by a coordinate trasformation which leaves the SW differential fixed
(up to a total derivative).

In order to do this, we need first of all to solve for the SW differential of
the IR theory. By using equation (3.32) we find

$\lambda_{[5,1]}=\frac{z}{x}\ dx=z\ d\log(x)\,,$ (3.57)

up to total-derivative terms. This form of the SW differential allows us to
freely shift $z$ by a generic polynomial in $x$. We will use such a shift in
order to reabsorb all the terms of the form $a_{i}x^{4-i}z$ for $i=1,\cdots 4$
in the curve (3.50).

In particular, the equation (3.50) can be thus simplified to

$\displaystyle
z^{2}+x^{6}+M_{1}x^{5}+M_{2}x^{4}+M_{3}x^{3}+M_{4}x^{2}+M_{5}x+M_{6}=0\,,$
(3.58)

where $D^{\rm IR}(M_{6})=2$ and $D^{\rm IR}(M_{1})=\frac{1}{3}$.

It is now possible to check that the second condition for the enhancement is
indeed satisfied. Namely we have two pairs of coupling constant/CB operator,
$(M_{1},M_{5})$ and $(M_{2},M_{4})$, satisfying

$D^{\rm IR}(M_{5})+D^{\rm IR}(M_{1})=2\,,\qquad D^{\rm IR}(M_{4})+D^{\rm
IR}(M_{2})=2\,.$ (3.59)

Finally $M_{3}$ and $M_{6}$ play the rôle of mass terms for the IR curve.

In conclusion we claim that, considering the orbit $[5,1]$, our method leads
to a geometry specified by the curve (3.58) and the differential (3.57). Such
a pair passes both our criteria of SUSY enhancement, so we expect that the
$\mathcal{N}=1$ geometry has enhanced to $\mathcal{N}=2$. Our expectation is
confirmed by the $a$-maximization analysis of [13]. This flow is believed to
land on the $(A_{1},D_{6})$ generalized Argyres-Douglas theory, which has
$SU(2)\times U(1)$ flavor symmetry [10]. Indeed, as can be seen in (3.58 and
(3.57), our method naturally and explicitly reproduced the SW curve and
differential of such a theory.

###  3.3 Examples that do not enhance

In Subsection 3.2, we have considered the deformations corresponding to the
orbits $[6]$ and $[5,1]$ of $SU(6)$. Both cases satisfy the two conditions and
they showed supersymmetry enhancement in the IR. In this section, we turn to
cases that do not exhibit supersymmetry enhancement. One case is considering a
different deformation in the 4d $SU(3)$ gauge theory with six flavors. In the
other case we will use an SO-Sp quiver theory with the deformation
corresponding to the maximal nilpotent orbit of the symplectic flavor
symmetry, which was discussed in [19]. It is known that neither case leads to
supersymmetry enhancement and we are going to confirm this claim using our
algebraic criteria.

#### Orbit $[4,2]$ of $SU(6)$

For the first case we consider a deformation with a vev corresponding to the
nilpotent orbit labeled by $[4,2]$. Namely the vev is given by

$\displaystyle\rho(\sigma^{+})=\left(\begin{array}[]{cccccc}0&1&0&0&0&0\\\
0&0&1&0&0&0\\\ 0&0&0&1&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&1\\\
0&0&0&0&0&0\end{array}\right).$ (3.66)

The fluctuations around the background that remain coupled are as usual the
lowest components of the spin $j$ representations and the matrix form of the
deformation becomes

$\displaystyle M=\left(\begin{array}[]{cccccc}M_{0}&1&0&0&0&0\\\
-3M_{1,-1}&M_{0}&1&0&0&0\\\ M_{2,-2}&-4M_{1,-1}&M_{0}&1&M^{(1)}_{1,-1}&0\\\
-M_{3,-3}&M_{2,-2}&-3M_{1,-1}&M_{0}&-M^{(1)}_{2,-2}&3M^{(1)}_{1,-1}\\\
3M^{(2)}_{1,-1}&0&0&0&-2M_{0}&1\\\
-M^{(2)}_{2,-2}&M^{(2)}_{1,-1}&0&0&-M^{(3)}_{1,-1}&-2M_{0}\end{array}\right).$
(3.73)

Inserting (3.73) into (3.9) yields the $\mathcal{N}=1$ curve.

Let us then determine the scaling dimension of the parameters appearing in the
curve equation. Our Ansatz is that the CB operator with the highest dimension
in the IR is $M_{3,-3}$. Denoting by $\lambda_{[4,2]}$ the IR SW differential,
this yields the relation

$\displaystyle\frac{{\rm d}\lambda}{{\rm d}a_{3}}=\frac{{\rm
d}\lambda_{[4,2]}}{{\rm d}M_{3,-3}}\,,$ (3.74)

which leads to

$\displaystyle D^{\rm IR}(z)-5D^{\rm IR}(x)=1-D^{\rm IR}(M_{3,-3})\,.$ (3.75)

Then the relation (3.75) together with the curve equation fixes the scaling
dimension of some of the parameters as

$\displaystyle D^{\rm IR}(M_{3,-3})=2\,,\quad D^{\rm
IR}(M_{2,-2})=\frac{3}{2}\,,\quad D^{\rm IR}(M^{(3)}_{1,-1})=1\,,\quad D^{\rm
IR}(M_{0})=\frac{1}{2}\,.$ (3.76)

Moreover, using that relative scalings are RG-flow invariant, we also find

$\displaystyle D^{\rm IR}(M^{(1)}_{1,-1})=D^{\rm
IR}(M^{(2)}_{1,-1})=1\,,\qquad D^{\rm IR}(M^{(1)}_{2,-2})=D^{\rm
IR}(M^{(2)}_{2,-2})$ $\displaystyle=\frac{3}{2}\,.$ (3.77)

Hence $M_{3,-3}$ and $M_{2,-2}$ may serve as CB operators. But the relation
(3.77) implies that also $M^{(1)}_{2,-2}$ and $M^{(2)}_{2,-2}$ remain coupled
in the IR. On the other hand, we know that the SW curve remains a genus-two
curve throughout the flow. Therefore we now encounter a situation where the
number of the CB operators does not agree with the genus of the curve,
implying that the curve cannot describe an $\mathcal{N}=2$ superconformal
field theory. This is consistent with the result of [12].

#### An orthosymplectic quiver

Here we want to apply a similar analysis to a Lagrangian theory with different
gauge and flavor groups. The aim is to show an example that is known not to
exhibit supersymmetry enhancement, but that nevertheless satisfies the
criterion that the genus of the IR curve matches the dimension of the base
over which it is fibered. Consider the quiver of Figure 2 where four flavors
are attached to the $SO(8)$ gauge node.

$USp(2)$$SO(8)$$USp(2)$$USp(8)$ Figure 2: The orthosymplectic quiver.

This theory was found in [19] to give no enhancement, because it violates an
intricate relation imposed by ’t Hooft anomaly matching, while preserving the
rank. Using the underlying SW geometry, we would like to argue that the
absence of enhancement originates from a mismatch between CB operators of
dimension between $1$ and $2$ and coupling constants. This suggests an elegant
geometric counterpart to the third criterion for enhancement discussed in
Subsection 3.2 of [19]131313The first two criteria in [19] are incorporated in
our geometric condition that, if the theory is to preserve $\mathcal{N}=2$ in
the IR, the genus of the SW curve must be equal to the dimension of the base
of the fibration..

The SW curve of the theory can be obtained from a brane configuration
realizing the quiver theory. This involves an O4-plane and the schematic
picture is depicted in Figure 3.

Figure 3: Brane picture realizing the orthosymplectic quiver of Figure 2.

The four D6-branes in the upper half-plane give four flavors to the $SO(8)$
gauge node. In order to read off the SW curve, it is useful to use a
configuration which does not have any D6-branes. For that we move the
D6-branes in between the middle NS5-branes in the right direction for example.
A D4-brane is created when a D6-brane crosses an NS5-brane and the final
configuration is given in Figure 3.

To write the SW curve we follow the procedure developed in [49]. Due to the
orientifold, the curve is invariant under $x\to-x$. Also, the charge of the
orientifold affects the asymptotic behavior of the NS5-branes, which changes
the powers of $v$ compared to the cases without an orientifold. In the end,
the SW curve for the quiver in Figure 2 is given by

$\begin{split}z^{4}+(a_{2}x^{4}+a_{1}x^{2}+a_{0})z^{3}&+(b_{4}x^{8}+b_{3}x^{6}+b_{2}x^{4}+b_{1}x^{2}+b_{0})z^{2}\\\
&+\prod_{i=1}^{4}(x^{2}-m_{i}^{2})(c_{2}x^{4}+c_{1}x^{2}+c_{0})z+\prod_{i=1}^{4}(x^{2}-m_{i}^{2})^{2}=0\,,\end{split}$
(3.78)

where $a_{0}$ and $c_{0}$ are fixed by the constraint

$\displaystyle
z^{4}+a_{0}z^{3}+b_{0}z^{2}+\prod_{i=1}^{4}m_{i}^{2}c_{0}z+\prod_{i=1}^{4}m_{i}^{4}=(z-\alpha)^{2}(z-\beta)^{2}\,.$
(3.79)

The SW differential is still given by (3.7). Since its dimension is one, we
have

$D^{\rm UV}(z)=4\,,\qquad D^{\rm UV}(x)=1\,.$ (3.80)

Hence the dimension of the various parameters are

$\displaystyle D^{\rm UV}(a_{2})$ $\displaystyle=0\,,\quad D^{\rm
UV}(a_{1})=2\,,\quad D^{\rm UV}(b_{4})=0\,,$ $\displaystyle D^{\rm UV}(b_{i})$
$\displaystyle=8-2i\;\;(i=0,1,2,3)\,,\quad D^{\rm UV}(c_{2})=0\,,\quad D^{\rm
UV}(c_{1})=2\,.$

$a_{1}$ is the CB operator of the first $USp(2)$, $b_{i},(i=0,\cdots,3)$ are
the CB operators of the $SO(8)$ and $c_{1}$ is the CB operator of the second
$USp(2)$. Note that the highest Casimir of $SO(8)$ is reducible,
$b_{0}=\tilde{b}_{0}^{2}$ and $m_{i},(i=1,\cdots,4)$ are the mass parameters
for the four flavors141414The field that is charged under both $USp(2)$ and
$SO(8)$ is a half-hypermultiplet in the bifundamental representation and it
has no mass term.. Finally, $a_{2},b_{4},c_{2}$ are the gauge coupling
constants associated to the three CB operators of dimension $2$. Recall that,
in an $\mathcal{N}=2$ theory, each CB operator with scaling dimension $1<D\leq
2$ has a corresponding coupling constant with scaling dimension $2-D$.

Let us now deform the above theory as usual by a coupling of the form (1.1),
and let us consider turning on a vev for $M$ corresponding to the maximal
nilpotent orbit of $USp(8)$, i.e.

$\displaystyle\rho(\sigma^{+})=\left(\begin{array}[]{cccccccc}0&1&0&0&0&0&0&0\\\
0&0&1&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0\\\ 0&0&0&0&0&0&0&1\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&-1&0&0&0\\\ 0&0&0&0&0&-1&0&0\\\ 0&0&0&0&0&0&-1&0\end{array}\right)\,.$
(3.89)

The decomposition of the adjoint representation of $USp(8)$ under the embedded
$SU(2)$ is

$\text{adj}=V_{1}\oplus V_{3}\oplus V_{5}\oplus V_{7}\,,$ (3.90)

and the fluctuation of $M$ will depend on the fields corresponding to the
lowest components of the spin $1,3,5,7$ representations. This means that we
have:

$\displaystyle M=\left(\begin{array}[]{cccccccc}0&1&0&0&0&0&0&0\\\
-7M_{1,-1}&0&1&0&0&0&0&0\\\ 0&-12M_{1,-1}&0&1&0&0&0&0\\\
-7M_{3,-3}&0&-15M_{1,-1}&0&0&0&0&1\\\
-M_{7,-7}&0&-7M_{5,-5}&0&0&7M_{1,-1}&0&7M_{3,-3}\\\
0&12M_{5,-5}&0&-16M_{3,-3}&-1&0&12M_{1,-1}&0\\\
-7M_{5,-5}&0&20M_{3,-3}&0&0&-1&0&15M_{1,-1}\\\
0&-16M_{3,-3}&0&-16M_{1,-1}&0&0&-1&0\end{array}\right).$ (3.99)

The characteristic polynomial of the above matrix is:

$\begin{split}P(x)=&\det\left(x\mathbb{1}_{6}-M\right)\\\
=&\;x^{8}+84M_{1,-1}x^{6}+(66M_{3,-3}+1974M_{1,-1}^{2})x^{4}+\\\
&+(-26M_{5,-5}+1364M_{3,-3}M_{1,-1}+12916M_{1,-1}^{3})x^{2}+\\\
&+(-M_{7,-7}+49M_{3,-3}^{2}-98M_{5,-5}M_{1,-1}+2450M_{3,-3}M_{1,-1}^{2}+11015M_{1,-1}^{4})\,,\end{split}$
(3.100)

and therefore, the SW curve (3.78) is deformed as:

$\begin{split}z^{4}+(a_{2}x^{4}+a_{1}x^{2}+a_{0})z^{3}+(b_{4}x^{8}+b_{3}x^{6}+b_{2}x^{4}+b_{1}x^{2}+b_{0})z^{2}+\\\
+P(x)(c_{2}x^{4}+c_{1}x^{2}+c_{0})z+P(x)^{2}=0\,.\end{split}$ (3.101)

As usual, along the RG flow the functional form of the SW differential will
change, but those of the six holomorphic ($1,0$)-forms of the curve will
remain the same. The CB operator with the largest scaling dimension in the UV
is $b_{1}$, and, given that relative scalings are RG-flow invariant, the
candidate field to play the rôle of $b_{1}$ in the IR is the singlet with the
largest spin, i.e. $M_{7,-7}$. Hence, we are led to impose

$\displaystyle\frac{{\rm d}\lambda}{{\rm d}b_{1}}=\frac{{\rm
d}\lambda^{\prime}}{{\rm d}M_{7,-7}}\,,$ (3.102)

where $\lambda^{\prime}$ is the SW differential in the IR. Since the SW
differential in the UV (3.7) does not have an explicit dependence on $b_{1}$,
but depends on it only through $v$, we can write the l.h.s. of (3.102) as

$\displaystyle\frac{{\rm d}\lambda}{{\rm
d}b_{1}}=-\frac{x^{2}zdz}{16x^{15}+\cdots}\,.$ (3.103)

Since the scaling dimension of $\lambda^{\prime}$ is $1$, we obtain

$\displaystyle 1-D^{\rm IR}(M_{7,-7})=2D^{\rm IR}(z)-13D^{\rm IR}(x)\,.$
(3.104)

The explicit form of the curve (3.101) implies

$\displaystyle D(z)=4D(x)\,,\qquad 8D(x)=D(M_{7,-7})\,.$ (3.105)

Combining (3.104) with (3.105) yields

$\displaystyle D^{\rm IR}(z)=\frac{4}{3}\,,\qquad D^{\rm
IR}(x)=\frac{1}{3}\,.$ (3.106)

Then the scaling dimension of the various parameters in the SW curve (3.101)
is given by

$\displaystyle D^{\rm IR}(a_{2})=0\,,\quad D^{\rm IR}(a_{1})=\frac{2}{3},$
$\displaystyle D^{\rm IR}(b_{4})=0\,,\quad D^{\rm
IR}(b_{3})=\frac{2}{3}\,,\quad D^{\rm IR}(b_{2})=\frac{4}{3}\,,\quad D^{\rm
IR}(b_{1})=2\,,\quad D^{\rm IR}(\tilde{b}_{0})=\frac{4}{3}\,,$ $\displaystyle
D^{\rm IR}(c_{2})=0\,,\quad D^{\rm IR}(c_{1})=\frac{2}{3}\,,$ $\displaystyle
D^{\rm IR}(M_{1,-1})=\frac{2}{3}\,,\quad D^{\rm
IR}(M_{3,-3})=\frac{4}{3}\,,\quad D^{\rm IR}(M_{5,-5})=2\,,\quad D^{\rm
IR}(M_{7,-7})=\frac{8}{3}\,.$

From this we conclude that there are still $6$ operators above the unitarity
bound, playing the rôle of the would-be CB operators in the IR, and hence also
in the IR the genus of the SW curve matches the dimension of the base over
which it is fibered. However, as one can see from the above scaling
dimensions, there is naïvely no matching between CB operators with $1<d\leq 2$
and coupling constants of dimension $2-d$: The are $2$ CB operators of
dimension $2$, but $3$ coupling constants of dimension $0$, and also $3$ CB
operators of dimension $4/3$, but $4$ coupling constants of dimension $2/3$.
To confirm that this expectation is correct, we should make sure that there
exists no change of variables leaving the IR SW differential invariant (up to
total derivatives), which eliminates from (3.78) the two extra coupling
constants preventing the match. Unfortunately, this is very hard here, because
we do not know the explicit form of the IR SW differential. Nevertheless, we
can give some evidence in this direction. First, as opposed to the previously-
discussed examples, here we cannot exclude an explicit dependence of
$\lambda^{\prime}$ from the new would-be CB operators. Thus, focusing solely
on the operator of largest dimension, and writing with no loss of generality

$\lambda^{\prime}=f[x(M_{7},,z),z,M_{7}]\,{\rm d}z\,,$ (3.108)

Eq. (3.102) reads151515Note that $f$ must satisfy five more partial
differential equations, which originate from the other holomorphic
($1,0$)-forms. Given the degeneration in dimension of the other would-be CB
operators, we do not know the explicit expression of these extra equations.

$\frac{\partial f}{\partial
x}\left(2P(x)+(c_{2}x^{4}+c_{1}x^{2}+c_{0})z\right)-\frac{\partial f}{\partial
M_{7}}(16x^{15}+\cdots)=x^{2}z\,,$ (3.109)

where we have renamed $M_{7}:=M_{7,-7}$. Consider changing $x\to
x+g(z,M_{7})$, leaving everything else invariant. This change of variable
induces a modification of (3.108) which amounts to a total derivative if and
only if $f$ depends linearly on $x$. But such a dependence can never satisfy
Eq. (3.109) for generic values of the parameters. A similar argument can be
drawn swapping $x$ and $z$. However, one can think of a more general change of
variables, such as

$\begin{split}x\to x+g_{x}(M_{7},z)\,,\\\ z\to z+g_{z}(M_{7},x)\,.\end{split}$
(3.110)

The change of the IR SW differential then reads

$\Delta\lambda^{\prime}_{SW}=\left[f(x+g_{x},z+g_{z},M_{7})-f(x,z,M_{7}))\right]{\rm
d}z+f(x+g_{x},z+g_{z},M_{7})\frac{\partial g_{z}(x,M_{7})}{\partial x}\,{\rm
d}x\,.$ (3.111)

One condition for the above to be a total derivative is that the change of
variables (3.110) must be such that
$f(x+g_{x}(z,M_{7}),z+g_{z}(x,M_{7}),M_{7})$ looses any explicit dependence on
$z$. Though we lack a proof of this, we argue that this cannot happen
compatibly with the six differential equations that the function $f$ must
satisfy.

Consequently, the mismatch between CB operators and couplings that we found in
the IR would explain why this theory does not exhibit supersymmetry
enhancement, despite we found the right CB dimension.

The matching condition refines our necessary criterion for enhancement, and
seems to give a geometric meaning to the condition (3.12) of [19], which every
theory displaying supersymmetry enhancement should meet.

##  4 Conclusions

In this paper we have extended our geometric understanding of the phenomenon
of SUSY enhancement to 4d field theories of rank higher than $1$. In [20] the
origin of the enhancement for rank-$1$ theories was traced in the holonomy
reduction of the F-theory internal space used to engineer the field theory.
Here, instead, we have used class-$\mathcal{S}$ constructions to track the
enhancement down to a hyperkähler-structure restoration on the moduli space of
solutions of the underlying Hitchin system. As in [20], we have formulated a
simple necessary algebraic criterion for enhancement in terms of an auxiliary
geometry given by a Riemann-surface fibration: If SUSY enhancement occurs in
the IR, this geometry needs to factorize in such a way that the dimension of
the base of the fibration reduces and becomes equal to the genus of the fiber.
We have refined this criterion, supplementing it by a matching condition
between CB operators of dimension $1<D\leq 2$ and coupling constants of
dimension $2-D$161616Such a condition is trivially satisfied for all theories
of rank $1$ which exhibit enhancement.. For theories exhibiting enhancement,
we have been able to write down the complete SW geometry (including masses and
couplings) of the IR theory, and compute all conformal dimensions of CB
operators by purely algebraic techniques, i.e. without relying on any
maximization procedure.

An important remark is in order. The above-mentioned factorization implies
that some of the fields hitting the unitarity bound disappear from the IR
theory. Our technique is able in a purely geometric manner to distinguish them
from those becoming instead masses and coupling constants of the IR theory:
The functional form of the SW differential gets modified by the flow in such a
way that new changes of coordinates become available in the IR and this allows
us to get rid precisely of those monomials containing the decoupled fields. It
would be very interesting to further investigate the deeper geometric meaning
of these specific RG-flow-induced modifications of the SW differential. We
hope to come back to this matter in a future publication.

As already mentioned in the introduction, our treatment of class-$\mathcal{S}$
theories in this paper does not cover irregular punctures featuring a
nontrivial degeneracy among the eigenvalues of the Hitchin field (Type III
irregular punctures [10]). Let us briefly illustrate here what the issue is.
Consider a Hitchin field on the sphere (parametrized by the coordinate $z$)
with a puncture of Type III at $z=\infty$ (we consider the $SU(N)$ case for
simplicity). Locally around the puncture the field can be diagonalized and
expanded in powers of $z$ as follows:

$\Phi=M_{n+1}z^{n}dz+\dots+M_{0}\frac{dz}{z}+M^{\prime}\frac{dz}{z^{2}}+\dots\,,$
(4.1)

where $n>0$, and the $M_{i}$’s and $M^{\prime}$ are diagonal $N\times N$
traceless matrices. The matrices $M_{0},\dots M_{n+1}$ encode the data
defining the boundary condition at the irregular puncture. The matrix
$M^{\prime}$ and subsequent terms are determined instead by solving the
differential equation of the Hitchin system; the corresponding terms are not
singular at infinity. In the case of type III punctures, the matrices
$M_{0},\dots M_{n+1}$ have degenerate eigenvalues and the degeneracy for the
matrix $M_{i}$ is _not_ arbitrary, but instead it is constrained by the
degeneracies of $M_{i+1}$.

Assuming the UV theory has a puncture of Type III, upon activating the
Maruyoshi-Song RG flow, we are left with a twisted Hitchin field $\Phi_{1}$
with the same boundary condition as in (4.1) at $z=\infty$:

$\Phi_{1}=M_{n+1}z^{n+1}\sqrt{dz}+\dots+M_{0}\sqrt{dz}+\frac{M^{\prime}}{z}\sqrt{dz}+\dots\,.$
(4.2)

Again, the term proportional to $M^{\prime}$ is not singular at infinity and
the matrix $M^{\prime}$ is not part of the data defining the boundary
condition, it is determined by solving the differential equation of the
twisted Hitchin system.

On the one hand, according to our prescription, the new Hitchin field
$\widetilde{\Phi}$ in the infrared should read

$\widetilde{\Phi}=M_{n+1}z^{n+1}dz+\dots+M_{0}dz+M^{\prime}\frac{dz}{z}+\dots\,.$
(4.3)

On the other hand, the term proportional to $M^{\prime}$ is now singular at
infinity and is therefore part of the data defining the boundary condition.
This in particular means that the eigenvalue degeneracy for $M^{\prime}$
cannot be arbitrary and is actually constrained by the form of the matrix
$M_{0}$. For the twisted and ordinary Hitchin systems to be equivalent, it
must be the case that the form of $M^{\prime}$ in (4.2) as determined by the
differential equation of the twisted Hitchin system is automatically
consistent with the constraint imposed by $M_{0}$. This is not necessarily
true. We conclude that in the case of Type III punctures the twisted Hitchin
system and the ordinary one are generically inequivalent, thus preventing any
enhancement. However, we are unable to decide whether accidental equivalences
may occur, leading to SUSY enhancements for theories with type III punctures.
We plan to come back to this issue in the near future.

Acknowledgments

We would like to thank A. Collinucci for initial collaboration and many useful
discussions. We also thank P. Agarwal, I. Bah, F. Bonetti, H-C. Kim, K.
Maruyoshi, J. Song, W. Yan, and M. Zimet for discussions.

HH and RS are grateful to the Aspen Center for Physics (which is supported by
National Science Foundation grant PHY-1607611), for hospitality during the
initial stage of this work. We are pleased to acknowledge the 2019 Pollica
summer workshop where some of this work was performed, and we are grateful to
its supporting organizations: the Simons Foundation (Simons Collaboration on
the Nonperturbative Bootstrap) and the INFN. The work of FC is supported by
the ERC Consolidator Grant STRINGFLATION under the HORIZON 2020 grant
agreement no. 647995. The work of SG is supported by the ERC Consolidator
Grant 682608 “Higgs bundles: Supersymmetric Gauge Theories and Geometry
(HIGGSBNDL)”. The work of HH is supported in part by JSPS KAKENHI Grant Number
JP18K13543. The work of RS is supported by the program “Rita Levi Montalcini”
for young researchers (D.M. n. 975, 29/12/2014).

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