[en-US]abbr=true,showyear=false,zone=atlantic

Proving the 6d ${a}$-theorem
with the double affine Grassmannian

part iii

Marco Fazzi,a,b,c Suvendu Giri,a,d,e and Paul Levyf

a Department of Physics and Astronomy, Uppsala University, SE-75120 Uppsala,
Sweden
b NORDITA, Hannes Alfvéns väg 12, SE-10691 Stockholm, Sweden
c School of Mathematics and Statistics, University of Sheffield,
Hounsfield Road, Sheffield, S3 7RH, UK
d Department of Physics, Princeton University, Princeton, New Jersey 08544,
USA
e Princeton Gravity Initiative, Princeton University, Princeton, New Jersey
08544, USA
f Department of Mathematics and Statistics, Fylde College,
Lancaster University, Lancaster, LA1 4YF, UK

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract

This paper contains two results of independent interest, the first being more
mathematical in nature whereas the second more physical. We first show that
the hierarchy of Higgs branch RG flows between the 6d $(1,0)$ SCFTs known as
A-type orbi-instantons is given by the Hasse diagram of certain strata and
transverse slices in the double affine Grassmannian of $E_{8}$. Secondly, we
leverage the partial order naturally defined on this Hasse diagram to prove
the $a$-theorem for orbi-instanton Higgs branch RG flows, thereby exhausting
the list of $c$-theorems in the even-dimensional supersymmetric setting.

MF would like to dedicate this paper to the late Luciano Girardello,
the first one to teach him anomalies and RG flows.

UUITP-39/23

###### Contents

  1. 1 Introduction and summary
    1. 1.1 Six-dimensional $a$-theorem
    2. 1.2 A-type orbi-instantons and hierarchies of RG flows
    3. 1.3 Higgs branch RG flows and slices in the Grassmannian
    4. 1.4 Three dimensions from six dimensions, and vice versa
    5. 1.5 Strategy of proof and results
    6. 1.6 Organization
  2. 2 A-type orbi-instantons and Kac diagrams
    1. 2.1 Kac diagrams and $\mathbb{Z}_{k}$-gradings of $E_{8}$
    2. 2.2 Orbi-instantons from M-theory
  3. 3 Affine and double affine Grassmannians
    1. 3.1 Affine Grassmannians
      1. 3.1.1 Coweights in type $E_{8}$
      2. 3.1.2 Orbits as weighted Dynkin diagrams
      3. 3.1.3 Geometry and slices in the affine Grassmannian
    2. 3.2 Double affine Grassmannians
      1. 3.2.1 Coroots and coweights
      2. 3.2.2 Partial order on coweights
      3. 3.2.3 Minimal degenerations for affine coweights
      4. 3.2.4 Relationship between $N$ and $n$
      5. 3.2.5 Geometry of the double affine Grassmannian
  4. 4 Hierarchy of flows as Hasse of dominant coweights
    1. 4.1 $k=1$
    2. 4.2 $k=2$
    3. 4.3 $k=3,\ldots,7$
  5. 5 Proving the $a$-theorem
    1. 5.1 The exact $a$ anomaly of A-type orbi-instantons
    2. 5.2 The proof
      1. 5.2.1 Considerations on sums of powers of roots
      2. 5.2.2 Finalizing the proof
  6. 6 Conclusions
  7. A Magnetic quivers
    1. A.1 $\operatorname{SU}(k)$ hyperkähler quotient of orbi-instanton magnetic quiver
  8. B RG flows for the decoupled system

##  1 Introduction and summary

Recently, there have been many fascinating developments in quantum field
theory (QFT). Yet, many simple but central dynamical questions remain open,
especially pertaining to the structure of renormalization group (RG) flows
along which QFTs sit, and which are “bounded” by fixed points, i.e. scale-
invariant field theories which (in the relativistic context) are typically
assumed to also be conformally invariant, i.e. to be full-fledged conformal
field theories (CFTs).111In two dimensions, scale and conformal invariance are
famously equivalent for unitary theories [1, 2]. In four dimensions, the
strong nonperturbative arguments of [3] were essentially proven by [4].
Subtleties can arise when nonlocal operators are included in the theory
however. See also the lecture notes [5].

In this paper we will answer one of these important questions, namely we will
prove the so-called $a$-theorem for an infinite class of six-dimensional
superconformal field theories (6d SCFTs) with minimal $(1,0)$ supersymmetry.
It is worth noting that this is the last case standing of _supersymmetric_
$c$-theorems in any even dimension, and moreover that there are no known
_non-_ supersymmetric interacting CFTs in six dimensions (or higher), so its
proof represents an important step in the study of (S)CFTs in general. The
proof reduces to an analysis of some properties of a mathematical object
called double affine Grassmannian (of $E_{8}$) and the (combinatorial)
construction of “transverse slices” between its “strata” (symplectic leaves).

The aim of the paper is thus to combine mathematics and physics to deliver a
powerful statement in QFT. Moreover (and perhaps most interestingly to the
string theory cognoscenti) it provides another glimpse into the physics of one
of the most mysterious extended objects in the theory: the Hořava–Witten
M9-wall.

###  1.1 Six-dimensional $a$-theorem

Concretely, by 6d supersymmetric $a$-theorem we mean:

  * _i)_

that there is a function defined along the flow which at the fixed points
equals the so-called $a$ conformal anomaly (or $a$ central charge) of the
SCFT, i.e. the coefficient of the 6d Euler density $E_{6}$ in the trace of its
stress-energy tensor on a curved background,

$\left\langle T_{\mu}^{\mu}\right\rangle=aE_{6}+\sum_{i=1}^{3}c_{i}I_{i}\ .$
(1.1)

Namely, the Ward identity $\left\langle T_{\mu}^{\mu}\right\rangle=0$ of the
CFT is violated by a c-number on a curved background, and for this reason
equation (1.1) is known as the trace, or Weyl, anomaly [6, 7, 8, 9, 10].222The
$I_{i}$ are Weyl invariants (or “cocycles”) of weight 6, and $c_{i}$ their
coefficients (which satisfy the relation $2c_{1}=c_{2}+c_{3}$ [11]); see [12,
13, 14, 15]. Equation (1.1) is sometimes written with a total derivative term
$\nabla_{i}J^{i}$ added to the RHS, which can be safely neglected for our
considerations (see e.g. [13]).

  * _ii)_

That this function decreases monotonically along any unitary RG flow
connecting ultraviolet (UV) and infrared (IR) SCFTs, implying that

$\Delta a:=a_{\text{UV}}-a_{\text{IR}}>0\ .$ (1.2)

It is known that $a>0$ for all unitary SCFTs [16], and this statement is
logically independent of $\Delta a>0$.333The statement $a\geq 0$ is also known
to be true for 4d CFTs [17], with the bound saturated only by a theory with no
local degrees of freedom. For a recent discussion about the role of $\Delta c$
in 4d see [18] and references therein.

It is a quantitative translation of the intuition that the number of degrees
of freedom should decrease along an RG flow as we integrate out high-energy
modes, and it also implies that the RG flow is irreversible. In other words,
we can only flow to the IR.

As stated, this 6d $a$-theorem should be thought of as another instance of the
$c$-theorem proven by Zamolodchikov in 2d [1]; and conjectured [19], proven
perturbatively [20, 21, 22], and finally non-perturbatively [23] also in
4d.444This hinges upon the findings of [24]. See also [3, 25] for further
discussions and [26] for an earlier investigation with a perturbative proof
for some theories. In both 2d and 4d the function that decreases along the
flow equals the $a$ conformal anomaly of the CFT at fixed points (historically
called $c$ in 2d, since it is equal to the central charge of the Virasoro
algebra – see e.g. [27]). In 5d or higher we do not have _conclusive_ evidence
in favor of the existence of non-supersymmetric interacting CFTs at the time
of writing.555See however [28, 29, 30, 31] for proposals in 5d, 6d, and even
8d from string constructions, even though their ultimate fate remains
uncertain (see e.g. [32, 33, 34, 35, 36, 37, 38] for a large-$N$ analysis in
5d and 6d from both a field theoretic and holographic perspective). Leaving
aside the odd-dimensional case (for which there is _no_ trace anomaly anyway
[13, Eq. (14)]), the strategy of [23] (i.e. the use of an effective action for
the “dilaton”, the Nambu–Goldstone boson of broken conformal invariance,
appearing as the conformal mode in a so-called local Riegert action [39, 40,
41]) does not generalize straightforwardly to 6d, and for this reason it does
not give rise to a general proof of the (non-supersymmetric) $a$-theorem [42,
43].666See also [44, 45, 46, 47], or [48] (and references therein), for an
alternative version of this statement (that uses the CFT entanglement entropy
in the latter case). See [26, 49, 50, 51] for early investigations on the
trace anomaly in arbitrary even dimensions (in particular 6d) in the context
of the $a$-theorem, and [52, 53, 54, 55, 56] for the use of the entanglement
entropy to prove various $c$-theorems (e.g. in 3d [54], where there is no
trace anomaly). See [57] for a proof that uses the averaged null energy
condition. Finally, see [58, 59] for a holographic proof of a $c$-theorem
which holds in any dimension. See [60] and references therein for a recent
reanalysis of this problem. Therefore we have to settle for supersymmetric
theories.

Luckily though, six is the largest dimension in which SCFTs can be defined
[61, 62] (see in particular [63, Sec. 5.1.4]), and a massive body of
literature has moreover shown that 6d SCFTs can be thought of as an
“organizing principle” for most lower-dimensional SCFTs, geometrizing their
construction, dualities, and interdependencies across dimensions. Therefore
studying the $a$-theorem for 6d SCFTs is a meaningful endeavor. With $(1,0)$
(or $(2,0)$) supersymmetry there are no relevant or marginal supersymmetry-
preserving deformations we can turn on [64, 65], so all RG flows are flows
onto the moduli space of supersymmetric vacua of the UV SCFT [66], obtained by
giving vacuum expectation values (VEVs) to some operators. This moduli space
is composed of two main branches: a tensor branch, parameterized by scalars in
the tensor multiplets taking VEVs; and a Higgs branch, where scalars in the
matter hypermultiplets take VEVs. Along flows onto either branch conformal
invariance is spontaneously broken,777See e.g. [66, Sec. 2.2.] for more
details. but in the latter case it is recovered in the deep IR, where a new
SCFT sits, with a global symmetry generically different from the UV
parent.888Likewise, the UV R-symmetry is broken by Higgs branch flows
generically since the matter hypermultiplets are charged under it, and a new
$\operatorname{SU}(2)$ R-symmetry emerges in the IR. There exist also mixed
branch flows, arising as a combination of the two aforementioned types of
flow; we will comment on them in the main text when appropriate. On the other
hand, for tensor branch flows the deep IR is a generalized quiver gauge theory
of massless vectors plus tensors,999This is an example of a scale-invariant
(but non-conformal) QFT, or SFT for short [3].  and the $a$-theorem was proven
in [66] (and in [67] in the $(2,0)$ case).101010See also [68] for earlier
investigations in the special $(2,0)$ case. The $a$-theorem is also valid for
flows from SCFT to supersymmetric SFT, see in particular [66, Sec. 6.2].

###  1.2 A-type orbi-instantons and hierarchies of RG flows

Exploration of the Higgs branch RG flows, on the other hand, has remained
elusive for longer. A proof of the $a$-theorem for a special but infinite
class of theories known as “T-brane theories” (descending from the “conformal
matter” of [69]) was finally provided in [70] (with prior evidence both in
field theory and holography given in [71, 72, 73, 74]). However there exists
another infinite class of 6d SCFTs which, together with a subclass of this
conformal matter, generates _all_ other known 6d SCFTs via “fission and
fusion” [75]. (This procedure involves subsequent ungaugings and gaugings of
symmetries, respectively.) The theories in this second class were dubbed “ADE-
type orbi-instantons” in [69], and are the data of: the number $N$ of
M5-branes simultaneously probing an M9-wall (i.e. acting as pointlike
instantons in the four codimensions) and the orbifold point of
$\mathbb{C}^{2}/\Gamma_{\text{ADE}}$ wrapped by the M9 (with
$\Gamma_{\text{ADE}}\subset\operatorname{SU}(2)$ finite); the order of said
orbifold; a boundary condition at the spatial infinity
$S^{3}/\Gamma_{\text{ADE}}$ of the orbifold, which is a representation
$\rho_{\infty}:\Gamma_{\text{ADE}}\to E_{8}$ (the $E_{8}$ gauge bundle
supported on the M9-wall acting as a flavor symmetry from the perspective of
the 6d worldvolume of the M5’s). The integer $N$ and the homomorphism
$\rho_{\infty}$ represent respectively the number of instantons and the
holonomy at infinity (of the flat connection of the gauge bundle) needed to
fully specify the instanton configuration on (the deformation/resolution of)
the orbifold (see e.g. [76, Sec. 2.1]).

When $\Gamma$ is of type A we can fix the order $k$ of the $\mathbb{Z}_{k}$
orbifold, allowing us to use the triple $(N,k,\rho_{\infty})$ to indicate an
orbi-instanton SCFT of this type. We will also say that $N$ is the number of
“full instantons” for reasons that will become clear later.111111Because the
four-dimensional space $\mathbb{C}^{2}/\Gamma_{\text{A}}$ contains an orbifold
singularity, the instanton number $\int F\wedge F$ on the
resolution/deformation of the singularity is _fractional_ and is given by
$N-\left\langle\overline{\lambda},\overline{\lambda}\right\rangle/(2k)$
($\overline{\lambda}$ is defined in (3.15) and the inner product in (5.1)), if
$\int F\wedge F=1$ on $\mathbb{C}^{2}$ for the smallest possible instanton
number. See also [77, p. 19] for the topological data of the instanton on the
(fully unresolved) singularity. Each flat connection $\rho_{\infty}$ is
conveniently specified by a choice of “Kac label” [76], that is an integer
partition $[k_{i}]$ of $k$ which uses only the Coxeter labels
$1,2,\ldots,6,4^{\prime},3^{\prime},2^{\prime}$ of the _affine_ $E_{8}$ Dynkin
diagram. In the mathematics literature, and in the following sections, such a
partition will be called a “Kac diagram $\lambda_{\text{Kac}}$ at level $k$”.
It is given by a weighted affine $E_{8}$ Dynkin diagram of the form

$\lambda_{\text{Kac}}:\quad
n_{1}-n_{2}-n_{3}-n_{4}-n_{5}-\overset{\overset{\displaystyle
n_{3^{\prime}}}{|}}{n_{6}}-n_{4^{\prime}}-n_{2^{\prime}}\ ,$ (1.3)

with

$k=\left(\sum_{i=1}^{6}in_{i}\right)+4n_{4^{\prime}}+3n_{3^{\prime}}+2n_{2^{\prime}}\
.$ (1.4)

(We briefly recap the theory of Kac diagrams in section 2.1.) The UV SCFT
whose Higgs branch we are exploring can be identified with the trivial choice
of boundary condition which preserves the full $E_{8}$ from the M9, that is
Kac label $\rho_{\infty}:k=[1^{k}]$ (which obviously exists for any $k$), or
$\lambda_{\text{Kac}}=\begin{smallmatrix}&&&&&0&&&\\\
k&0&0&0&0&0&0&0\end{smallmatrix}\\!\\!$. The IR SCFTs that can be reached by
(subsequent) Higgs branch RG flows are instead identified with other possible
choices of coefficients $n_{i},n_{i^{\prime}}$ (some of which may be zero),
namely

$\rho_{\infty}:k=[1^{n_{1}},2^{n_{2}},3^{n_{3}},4^{n_{4}},5^{n_{5}},6^{n_{6}},4^{n_{4^{\prime}}},2^{n_{2^{\prime}}},3^{n_{3^{\prime}}}]\
,$ (1.5)

(How to extract the flavor symmetry of the UV and IR SCFTs from the associated
Kac diagrams will be explained in section 2.2.) In what follows, we will use
the notations (1.3) and (1.5) interchangeably. There is another mathematical
structure naturally parameterized by pairs $(\lambda_{\rm Kac},n)$ of a Kac
diagram and an integer $n$: the set of dominant coweights for an affine
Kac–Moody Lie algebra. A major part of our work will be to make precise the
relationship between triples $(N,k,\rho_{\infty})$ and coweights
$(\lambda_{\rm Kac},n)$ for affine $E_{8}$.

In [78], exploiting the technology of “3d magnetic quivers” [79] and quiver
subtraction [79], the first two authors have constructed very intricate
hierarchies of allowed RG flows between UV and IR A-type orbi-instantons
defined by different Kac diagrams, and verified, by computing the $a$ anomaly
of UV and IR SCFTs connected by an RG flow, that each flow thus constructed
satisfies the $a$-theorem. (In [80], the same authors with Giacomelli have
extended this analysis to another infinite class of 6d SCFTs known as massive
E-string theories, which can be engineered from the orbi-instantons via the
above-mentioned fission procedure.) The three present authors have moreover
conjectured that, if we fix $k$ but allow $N$ to change, that is we consider
_mixed_ Higgs-tensor branch RG flows, these intricate hierarchies can be
understood as (semi-infinite periodic) Hasse diagrams of the so-called “double
affine Grassmannian of $E_{8}$”, which was introduced in the mathematics
literature [81, 82, 83] to generalize the better known affine Grassmannian
(see e.g. [84, Sec. 6.5]). Under the proposed identification, the SCFTs in the
hierarchy are certain symplectic leaves (or strata) of the Grassmannian, and
the Higgs branch RG flows that connect them are the transverse slices between
“neighboring” leaves (with the direction of flow increasing along the partial
order on strata). The larger the leaf (i.e. Higgs branch of the CFT), the
smaller its $a$ anomaly. Let us see how.

In finite type, the strata in the affine Grassmannian are parameterized by the
so-called dominant coweights (as we explain below in section 3.1). The closure
ordering on strata corresponds to the dominance ordering on coweights; the
“minimal degenerations” (adjacent pairs) are classified by an algorithm due to
Stembridge [85]. Slices in the affine Grassmannian are known [86] to be
isomorphic to Coulomb branches $\mathcal{M}_{\text{C}}$ of 3d $\mathcal{N}=4$
theories (in the sense of [77, 87]). The latter can also be realized in string
theory via Hanany–Witten brane setups (and the brane moves one can do within
them). See e.g. [84, 88, 89, 90] for many examples in type ABCD. It has
moreover been recently established that Coulomb branches are symplectic
singularities.121212Symplectic singularities have been introduced in [91] as
an analog to rational Gorenstein singularities in Calabi–Yau varieties in the
eight-supercharge setting, i.e. what physicists would call hyperkähler cones
[92].

The affine Grassmannian plays a crucial role in relation to the representation
theory of a finite-dimensional simple Lie algebra ${\mathfrak{g}}$ via the
geometric Satake correspondence [93]. The analogous object for the affine
counterpart ${\mathfrak{g}}_{\rm aff}$ is the “double affine” Grassmannian,
conjecturally defined by Braverman and Finkelberg in [81, 82, 83]. It is
conjectured that Braverman-Finkelberg’s double affine Grassmannian also gives
rise to Coulomb branches of 3d $\mathcal{N}=4$ theories – see e.g. the
proceedings [94] or [86, Sec. 3(viii) (b)].131313In affine type A, it is
proven that the Coulomb branch is given by a quiver variety of affine type
[95] known as Cherkis bow variety [96, 97, 98, 99]. The bow in the name comes
from a Hanany–Witten D3-D5-NS5 brane configuration on a circle. See e.g. the
video recordings of Cherkis’ 2018 mini course on _Instantons and monopoles_ at
ICTS, Bangalore [100]. In fact, because of the semi-infinite periodic
structure of its Hasse diagram (which will become apparent in later sections),
a definition of this object in toto is beyond current mathematical technology,
so that, practically speaking, it only makes sense to construct transverse
slices between strata. These transverse slices are finite-dimensional, hence
consist of finitely many strata ordered by the closure relation. It is known
that some (but not all) of the strata are parameterized by dominant affine
coweights $(\lambda_{\rm Kac},n)$; the minimal degenerations for affine
coweights have been classified by Roy’s generalization [101] of Stembridge’s
result in finite types. The basic objects of study are slices
${\mathcal{M}}_{\text{C}}(\mu,\lambda)$ between strata defined by dominant
coweights $\lambda<\mu$. Note that this requires $\lambda$ and $\mu$ to have
the same level $k$ (i.e. same order of the $\mathbb{Z}_{k}$ orbifold in
M-theory).

The symplectic leaves in ${\mathcal{M}}_{\text{C}}(\mu,\lambda)$ have been
classified in affine type A [95, Sec. 7.7]. In the following discussion we
will assume (as is expected) that an analogous classification holds in any
affine type. (We also assume $k>1$; the case $k=1$ is similar, but with fewer
strata.) Just as in the finite case, each coweight $\nu$ satisfying
$\mu\geq\nu\geq\lambda$ leads to a symplectic leaf in
${\mathcal{M}}_{\text{C}}(\mu,\lambda)$. However, in contrast with the finite
case, there are additional strata: if $\lambda\leq\nu\leq\mu-M\delta$ (where
$\delta$ is the minimal positive imaginary root in the affine root system),
then between $\nu$ and $\nu+M\delta$ there is also a sequence of strata
corresponding to
$\operatorname{Sym}^{M}({\mathbb{C}}^{2}/{\mathbb{Z}}_{k}\setminus\\{0\\})$
(and
$\operatorname{Sym}^{M-1}({\mathbb{C}}^{2}/{\mathbb{Z}}_{k}\setminus\\{0\\})$,
$\operatorname{Sym}^{M-2}({\mathbb{C}}^{2}/{\mathbb{Z}}_{k}\setminus\\{0\\})$,
$\ldots$). The strata in
$\operatorname{Sym}^{M}({\mathbb{C}}^{2}/{\mathbb{Z}}_{k}\setminus\\{0\\})$
are in one-to-one correspondence with the integer partitions $[m_{i}]$ of $M$,
and are (closure) ordered via refinement of partitions (in reverse, joining
parts). We can express this in terms of 3d Coulomb branches as follows:

$\displaystyle\text{strata:}\quad\mathcal{M}_{\text{C}}(\mu,\lambda)=\bigsqcup_{\nu,\,[m_{i}]}\mathcal{M}_{\text{C}}^{\text{smooth}}(\nu,\lambda)\times\operatorname{Sym}^{M}_{[m_{i}]}(\mathbb{C}^{2}/\mathbb{Z}_{k}\setminus\\{0\\})\
,$ (1.6)
$\displaystyle\text{slices:}\quad\mathcal{M}_{\text{C}}({\mu-M\delta,\nu})\times\prod_{i=1}^{m}\prescript{\text{c}}{}{\mathcal{U}_{[m_{i}]}}\
,$ (1.7)

where $\lambda\leq\nu\leq\mu-M\delta$ and $[m_{i}]$ is a partition of $M$, and
$\bigsqcup$ stands for disjoint union. Here (1.7) denotes the transverse slice
from a point of the stratum labeled by $(\nu,[m_{i}])$, and
$\prescript{\text{c}}{}{\mathcal{U}_{[m_{i}]}}$ is the (Uhlenbeck partial
compactification of) the centered moduli space of $m_{i}$ $E_{8}$-instantons
on ${\mathbb{C}}^{2}$. (See appendix A for more details.) Note that the smooth
locus of ${\mathcal{M}}_{\text{C}}(\nu,\lambda)$ is precisely the open
symplectic leaf (corresponding to $\nu$).

Before explaining how these 3d Coulomb branches come about in 6d, we remark
the following. A similar behavior (i.e. stratification) has been observed
[102] for “conformal matter” theories of type $(A,A)$ (i.e. bifundamentals of
$\operatorname{SU}(k)$) engineered by $N$ M5’s probing
$\mathbb{C}^{2}/\mathbb{Z}_{k}$, in the following sense. It is well-known that
if one flows onto the Higgs branch of the UV theory

$[\operatorname{SU}(k)]\,\underbrace{\overset{{\mathfrak{su}({k})}}{2}\cdots\overset{{\mathfrak{su}({k})}}{2}}_{N-1}\,[\operatorname{SU}(k)]$
(1.8)

by giving VEVs to matter hypermultiplets charged under the (nonabelian part of
the) flavor symmetry, i.e. $\operatorname{SU}(k)\times\operatorname{SU}(k)$,
one can reach new IR fixed points with a different flavor symmetry which is
specified by the choice of a (two) nilpotent orbit(s) in
$\operatorname{SU}(k)$ ($\operatorname{SU}(k)\times\operatorname{SU}(k)$);
that is, we reach a T-brane theory [69, 103].141414The abelian part of the
flavor symmetry is studied in detail in [104]. The AdS7 holographic duals to
T-brane theories were studied in [105, 71, 73, 74, 106]. We will call these
flows “flavor Higgsings”.

However, one can also explore other phases of the UV theory (still on its
Higgs branch) obtained by moving some (or all) of the M5’s off of the
singularity. This is done in [102], which constructs a Hasse diagram (via
quiver subtraction of 3d magnetic quivers, making use of an improved algorithm
based on “decorations” [107]) for the Higgs branch of the UV theory while
excluding the flavor Higgsings (which would lead to T-brane theories in the
IR). These phases correspond to decoupled products of lower-rank $(A,A)$
conformal matter (i.e. defined by some $M<N$) and a bunch of A-type $(2,0)$
theories given by stacks of M5’s (such that the total number of M5’s is $M$).
Some of the leaves in this Hasse diagram are symmetric products and correspond
to the Higgs branch of stacks of M5’s (away from the
singularity),151515Symmetric products also appear for the same reason in the
Higgs branch Hasse diagram of the rank-$N$ E-string, see [108, Fig. 3.6].
while the slices between adjacent symplectic leaves are either $A_{k-1}$
Kleinian singularities or certain non-normal singularities introduced in [109,
Sec. 1.8.4] (or unions of multiple copies thereof).

In this paper we will mostly be interested in flavor Higgsings, taking us from
UV SCFT to IR SCFT with (generically) different flavor symmetry (with an
exception for $k=1$, i.e. for E-string theories – see section 4.1).

###  1.3 Higgs branch RG flows and slices in the Grassmannian

We would now like to understand in 6d QFT what the different strata of the
double affine Grassmannian of $E_{8}$, and the slices between them, correspond
to. Under the identification of the hierarchy of RG flows with the Hasse
diagram of the strata already proposed in [78], the latter should correspond
to various (UV and IR) CFTs, whereas the slices to Higgs branch RG flows.

Generally speaking, there are two types of Higgsings (i.e. flows) that we can
realize among orbi-instantons. The first – flavor Higgsings, as we called them
above – correspond to giving a VEV to (scalar) operators in the matter
hypermultiplets charged under a flavor symmetry, in particular the left
symmetry factor $\mathfrak{f}\subseteq E_{8}$ of the orbi-instanton. (This
will also be denoted $[F]$ in the generic quiver on the tensor branch of the
SCFT – see [78] and (2.6) below for the notation.161616Of course, one could
also activate VEVs charged under the _right_ flavor symmetry factor, i.e.
$\mathfrak{g}_{\text{R}}=\mathfrak{su}(k)$ or $[\operatorname{SU}(k)]$, and
this has been considered e.g. in [80] for massive E-string theories and in
[76, Sec. 5.6] for orbi-instantons. We will not discuss such flows here, save
for a brief mention in the conclusions.) We then flow to a new IR SCFT on the
Higgs branch of the UV one, which is defined by a _different_ Kac diagram at
level $k$ (holonomy at infinity $\rho_{\infty}:\mathbb{Z}_{k}\to E_{8}$). This
type of flow is fairly easy to understand in QFT: the operators taking a VEV
satisfy a chiral ring relation which defines either a “minimal singularity”
$\mathfrak{a}_{i\leq 8},\mathfrak{d}_{i\leq
8},\mathfrak{e}_{6},\mathfrak{e}_{7},\mathfrak{e}_{8}$ (i.e. a singular
variety given by the closure of the minimal nilpotent orbit
$\overline{\text{min}_{\mathfrak{g}}}$ of that algebra $\mathfrak{g}$), or the
Kleinian singularity $A_{i-1}$ (i.e. $\mathbb{C}^{2}/\mathbb{Z}_{i}$). This
was described for any $k$ in [78] (with examples for some $k$ given before in
[110]), where moreover a connection with the so-called minimal degeneration
singularities of the $E_{8}[\\![z]\\!]$-orbits of the (singly) affine
Grassmannian of (non-affine) $E_{8}$ (classified by [111] thanks to work by
Stembridge [85]) was put forth. Notice that this type of flow lacks a “good”
description in terms of M-theory branes (or brane moves). It is the analog of
the T-brane VEVs for T-brane theories studied in [69, 103].171717It would be
interesting to study whether an analog to Nahm’s equations, probably involving
the $G_{4}$ flux, can be defined for M-theory flavor Higgsings. We thank
Alessandro Tomasiello for this suggestion.

The second type of flow has the opposite behavior, i.e. it has an easy
description in terms of branes but lacks an equally easy description in QFT,
at least for orbi-instantons.181818For a rank-2 E-string (which can be thought
of as a limiting case of an orbi-instanton with
$k=1,N=2,\rho_{\infty}:2=[1^{2}]$) a QFT description exists: the Higgsing is
induced by giving a VEV to an $\operatorname{SU}(2)$ moment map, where this
$\operatorname{SU}(2)$ is a factor of the flavor symmetry [112, 113]. We would
like to thank Simone Giacomelli for discussion on this point. It corresponds
to separating some of the M5’s from the stack at the singularity, while still
keeping them on the M9, and organizing them into substacks, potentially made
up of a single brane.191919It is the analog of the Higgs branch flows studied
in [102] for the $(A,A)$ theory of $N$ M5’s probing
$\mathbb{C}^{2}/\mathbb{Z}_{k}$. In [103] it is suggested that the
“separation” brane move should correspond to a diagonalizable VEV, i.e. a
semisimple element of the algebra $\mathfrak{f}={\mathfrak{su}({k})}$ of the
flavor factor $[\operatorname{SU}(k)]$; on the contrary, our flavor Higgsings
correspond in that context to nilpotent elements, i.e. T-brane VEV, as
reiterated above. We would like to thank Alessandro Tomasiello for discussion
on this point. At the orbifold point we are left with $N-M$ M5’s, i.e. a
“lower-rank” orbi-instanton. See figure 1 for clarity.

Figure 1: Top left: the generic point on the tensor branch of the UV orbi-
instanton. Top right: the origin of the tensor branch where the orbi-instanton
SCFT defined by the triple $(N,k,\rho_{\infty}^{\text{UV}})$ lives; all M5’s
are brought on top of each other and on top of the
$\mathbb{C}^{2}/\mathbb{Z}_{k}$ orbifold point, where the M9 sits (along
$x^{6}$). Bottom left: an IR orbi-instanton defined by a different Kac diagram
$(N,k,\rho_{\infty}^{\text{IR}})$. (The coweight corresponding to
$(N,k,\rho_{\infty}^{\text{IR}})$ lies _above_ the coweight corresponding to
$(N,k,\rho_{\infty}^{\text{UV}})$ in the partial order.) To reach it, we
activate a VEV for matter operators (in flavor hypermultiplets), and
generically break the UV flavor symmetry (a maximal subalgebra of $E_{8}$,
including the full $E_{8}$ by extension). Bottom right: a product of a “lower-
rank” orbi-instanton $(N-M,k,\rho_{\infty}^{\text{UV}})$ times $m$ substacks
of $m_{i}$ M5’s (with $\sum_{i=1}^{m}m_{i}=M$).

Each of the “separated” M5’s sees a copy of the singularity in its transverse
space (within the M9), so $M$ (indistinguishable) M5’s will see the symmetric
product
$\operatorname{Sym}^{M}(\mathbb{C}^{2}/\mathbb{Z}_{k}\setminus\\{0\\})$ (of
quaternionic dimension $\dim_{\mathbb{H}}=M$).202020The symmetric product is
known be the Coulomb branch (as a quiver variety) of the so-called “Jordan
quiver” [77]. The Higgs branch of the Jordan quiver is instead the (Uhlenbeck
partial compactification of the) moduli space of $k$ instantons of
$\operatorname{SU}(M)$ on $\mathbb{C}^{2}$ by ADHM [114]. Thanks to mirror
symmetry [115, 116] (boiling down to a “symplectic duality” [117, 118] in this
case), this Higgs branch is the same as the Coulomb branch of the mirror
(framed) necklace quiver, whereas its symmetric product Coulomb branch is the
same as the Higgs branch of the necklace. More precisely, we need to specify
an integer partition $[m_{i}]$ of $M$, say with $m$ parts, to describe $m$
substacks each containing $m_{i}$ branes. The symmetric product
$\operatorname{Sym}^{M}({\mathbb{C}}^{2}/{\mathbb{Z}}_{k}\setminus\\{0\\})$
has a non-trivial Hasse diagram: we can join together substacks, i.e. we can
glue together parts of the partition $[m_{i}]$. (As stated above, the slices
in this Hasse diagram are either the quotient
${\mathbb{C}}^{2}/{\mathbb{Z}}_{k}$ or a union of finitely many copies of
${\mathbb{C}}^{2}/{\mathbb{Z}}_{2}$ or the non-normal singularity $m$ studied
in [109].) Furthermore, the full symmetric product
$\operatorname{Sym}^{M}({\mathbb{C}}^{2}/{\mathbb{Z}}_{k})$ is the disjoint
union:

$\operatorname{Sym}^{M}({\mathbb{C}}^{2}/{\mathbb{Z}}_{k}\setminus\\{0\\})\bigsqcup\operatorname{Sym}^{M-1}({\mathbb{C}}^{2}/{\mathbb{Z}}_{k}\setminus\\{0\\})\bigsqcup\operatorname{Sym}^{M-2}({\mathbb{C}}^{2}/{\mathbb{Z}}_{k}\setminus\\{0\\})\bigsqcup\cdots\bigsqcup\,\\{0\\}\
,$ (1.9)

and hence the Hasse diagram extends to include all symmetric products
$\operatorname{Sym}^{M-i}$ for $i\leq M$; in this context one can also
“dissolve a stack of $i$ M5’s into flux” (within the M9), i.e. perform $i$
small $E_{8}$-instanton transitions transmuting $i$ tensors into $29i$
hypermultiplets [119], thus degenerating from $\operatorname{Sym}^{M}$ to
$\operatorname{Sym}^{M-i}$. Finally, we can remove any of the remaining M5’s
from the singularity (reducing the number $N-M$ of remaining M5’s, but in this
case degenerating from $\operatorname{Sym}^{M}$ to $\operatorname{Sym}^{M+i}$
for some $i$).

Notice that, because of the presence of the M9, each substack of separated
M5’s provides a decoupled copy of a rank-$m_{i}$ E-string theory,

$[E_{8}]\,\underbrace{12\cdots 2}_{m_{i}}\ ,$ (1.10)

itself a 6d $(1,0)$ interacting SCFT.

It is natural to identify orbi-instantons defined by triples
$(N,k,\rho_{\infty})$ (and without any decoupled E-string) with strata of the
double affine Grassmannian defined by a dominant coweight
$\lambda=(\lambda_{\text{Kac}},n)$ of affine $E_{8}$, since specifying
$\rho_{\infty}$ is the same as specifying a $\lambda_{\text{Kac}}$ at level
$k$. (We will elucidate the precise relationship between $N$ and $n$ at the
end of section 3.2.3). On the other hand, the $M$ separated M5’s organized in
$m$ substacks are identified with
$\operatorname{Sym}^{M}_{[m_{i}]}(\mathbb{C}^{2}/\mathbb{Z}_{k}\setminus\\{0\\})$.
The slices between UV and IR orbi-instantons $(N,k,\rho_{\infty}^{\text{UV}})$
and $(N,k,\rho_{\infty}^{\text{IR}})$ are identified with a degeneration
$\mu>\lambda$; the slices between symmetric products with the reduced moduli
spaces of $m_{i}$ $E_{8}$-instantons on $\mathbb{C}^{2}$. The quaternionic
dimension of the latter is
$\dim_{\mathbb{H}}\prescript{\text{c}}{}{\mathcal{U}_{[m_{i}]}}=30m_{i}-1$.
For $m_{i}=1$ (i.e. dissolving a single M5 from the separated stacks into
flux) this dimension is obviously 29, i.e. the dimension of
$\overline{\min_{E_{8}}}$ or the Higgs branch of a (rank-1) E-string. This is
precisely the number of hypermultiplets produced by a single small
$E_{8}$-instanton transition. Then the slice
$\prod_{i=1}^{m}\prescript{\text{c}}{}{\mathcal{U}_{[m_{i}]}}$ means we are
performing a total of $M=\sum_{i=1}^{m}m_{i}$ small $E_{8}$-instanton
transitions, reducing $N$ to $N-M$.212121For a rank-2 E-string, i.e. the orbi-
instanton $k=1,N=2,\rho_{\infty}:2=[1^{2}]$ according to footnote 18, the full
Hasse diagram of Higgs branch RG flows can be constructed [120, Fig. 45]. In
that figure, $\mathfrak{a}_{1}$ is the operator VEV charged under
$\operatorname{SU}(2)$ of [112, 113];
$\mathbb{H}=\mathbb{C}^{2}=\operatorname{Sym}^{1}(\mathbb{C}^{2})$;
$\mathfrak{g}=\mathfrak{e}_{8}$ indicates the $\mathfrak{e}_{8}$ minimal
degeneration, i.e. $\overline{\min_{E_{8}}}$, and corresponds either to a
slice of the type just described (going from a stack of 2 M5’s to 2 stacks of
1 M5 each, i.e. $m=2$ and $m_{i}=1$) or a small $E_{8}$-instanton transition,
when a rank-1 E-string (i.e. one tensor multiplet coupled to $E_{8}$ matter)
transmutes into $29=\dim_{\mathbb{H}}\overline{\min_{E_{8}}}$ hypermultiplets.
More generally, [108, Sec. 3.2] constructs in a schematic form the Higgs
branch Hasse diagram of the generic rank-$M$ E-string.

###  1.4 Three dimensions from six dimensions, and vice versa

The result
$\dim_{\mathbb{H}}\prescript{\text{c}}{}{\mathcal{U}_{[m_{i}]}}=30m_{i}-1$ was
recovered using the 3d magnetic quiver of the rank-$m_{i}$ E-string (which
will appear in section 4.1) in [121], and this is no coincidence, as we now
explain.

The Higgs branch of a supersymmetric theory is expected to be invariant under
torus compactification: compactifying on $T^{3}$ the F-theory “electric
quiver” (2.5) of a 6d orbi-instanton (i.e. its weak-coupling limit on the
tensor branch, obtained via the algorithm of [76] from a choice of
$(N,k,\rho_{\infty})$),222222Both 6d $(1,0)$ tensor and vector multiplets
reduce to 3d $\mathcal{N}=4$ vector multiplets. and applying mirror symmetry
to the 3d quiver gauge theory thus obtained, we land on a new “magnetic
quiver” [122]:232323By the results of [123], it is star-shaped with three
legs, given that it is mirror dual to the compactification on $S^{1}$ of the
class-S theory obtained by compactifying the orbi-instanton on $T^{2}$, itself
a fixture with three punctures (defined in [76, Eq. (4.1)]).

${\scriptstyle
1-2-\cdots-k-(r_{1}+\widetilde{M})-(r_{2}+2\widetilde{M})-(r_{3}+3\widetilde{M})-(r_{4}+4\widetilde{M})-(r_{5}+5\widetilde{M})-\overset{\overset{\scriptstyle
r_{3^{\prime}}+3\widetilde{M}}{|}}{(r_{6}+6\widetilde{M})}-(r_{4^{\prime}}+4\widetilde{M})-(r_{2^{\prime}}+2\widetilde{M})}\
.$ (1.11)

The latter is a 3d $\mathcal{N}=4$ unitary quiver gauge theory flowing to a
SCFT in the IR; the SCFT sits at the intersection between Higgs and Coulomb
branch. (We are selecting a vacuum of maximal breaking of the gauge symmetry,
i.e. the product gauge group of rank $r_{\text{V}}$ is Higgsed to maximal
torus $T^{r_{\text{V}}}$. $r_{\text{V}}$ is given by the sum of all gauge
ranks minus one.) The power of this construction lies in the fact that the 3d
Coulomb branch captures the Higgs branch of the starting 6d theory at weak
_and_ strong coupling [122] (i.e. at the generic point on the tensor branch
and at its origin, where the CFT sits).

The details of the above quiver (which can be found in appendix A, together
with the relevant notation) are not important for the present discussion. What
is important is that this quiver is neither of finite nor of affine type, so
the stratification result outlined in section 1.2 (a conjectural extension of
the results of [95] to affine type E) does _not_ apply. (Incidentally, it
would be interesting to develop the technology needed to construct its Coulomb
branch as a quiver variety [124]. We will briefly come back to this point in
the conclusions.)

As a first step toward understanding what the Coulomb branch
$\widehat{\mathcal{M}}_{\text{C}}$ of (1.11) is, it was proposed in [76, Sec.
4.3] that the hyperkähler quotient of the latter times the hyperkähler space
$\mathcal{O}_{\bm{\xi}}$ by $\operatorname{SU}(k)$, i.e.
$(\widehat{\mathcal{M}}_{\text{C}}\times\mathcal{O}_{\bm{\xi}})/\\!/\\!/\operatorname{SU}(k)$,
yields the Coulomb branch $\mathcal{M}_{\text{C}}^{\text{inst}}$ of a quiver
of affine type E, namely

${\scriptstyle\boxed{\scriptstyle
k}-(r_{1}+\widetilde{M})-(r_{2}+2\widetilde{M})-(r_{3}+3\widetilde{M})-(r_{4}+4\widetilde{M})-(r_{5}+5\widetilde{M})-\overset{\overset{\scriptstyle
r_{3^{\prime}}+3\widetilde{M}}{|}}{(r_{6}+6\widetilde{M})}-(r_{4^{\prime}}+4\widetilde{M})-(r_{2^{\prime}}+2\widetilde{M})}\
,$ (1.12)

for which the results of [86, 95] should apply. This fact is proven in
appendix A.1. Then, since $\mathcal{M}_{\text{C}}^{\text{inst}}$ is the
centered moduli space of $\widetilde{M}$ $E_{8}$-instantons on
$\mathbb{C}^{2}/\mathbb{Z}_{k}$, $\widehat{\mathcal{M}}_{\text{C}}$ should be
the instanton moduli space on the _deformation or resolution_ of
$\mathbb{C}^{2}/\mathbb{Z}_{k}$. In fact, $\mathcal{O}_{\bm{\xi}}$ can be
understood as a ($\operatorname{SU}(k)_{\mathbb{C}}$) “regular coadjoint
orbit” for
${\bm{\xi}}=(\xi_{\mathbb{C}},\xi_{\mathbb{R}})\in{\mathfrak{su}({k})}\otimes(\mathbb{C}\oplus\mathbb{R})$,
and has a hyperkähler metric specified by $\xi_{\mathbb{R}}$. The latter acts
as a resolution modulus for $\mathbb{C}^{2}/\mathbb{Z}_{k}$; on the other hand
a nonzero $\xi_{\mathbb{C}}$ deforms the singularity.

From a physics perspective, $\mathcal{O}_{\bm{\xi}}$ is nothing but the
Coulomb (or Higgs) branch of the $T(\operatorname{SU}(k))$ tail (which is
self-mirror dual [125]) on the left of (1.11), i.e.

$1-2-\cdots-(k-1)-\boxed{k}\ ,$ (1.13)

where now ${\bm{\xi}}$ can be understood as an $\operatorname{SU}(2)_{R}$
triplet of mass parameters of the $\operatorname{SU}(k)$ flavor symmetry
represented by $\boxed{k}$.

Because of this $T(\operatorname{SU}(k))$ tail, the stratification of the
Coulomb branch $\widehat{\mathcal{M}}_{\text{C}}$ will contain many more
strata than $\mathcal{M}_{\text{C}}^{\text{inst}}$: some strata will come from
the _base_ (which is parameterized by $\bm{\xi}$), while others from the
_fiber_ (i.e. the instanton moduli space for each possible value of
$\bm{\xi}$). However, we can stay in the vicinity of $\xi_{\mathbb{R}}=0$;
then, the stratification result (1.6) which is expected to hold for a quiver
of affine type E, i.e. for (1.12), should carry over to (1.11). In practice,
we are disregarding the strata given by symmetric products, as these are not
important for proving the $a$-theorem for flavor Higgsings.

We are finally ready to interpret strata and slices. We identify the Higgs
branch of the UV theory with $\mathfrak{f}=E_{8}$ flavor symmetry with the
Coulomb branch $\mathcal{M}_{\text{C}}(\mu,\lambda)$, where $\lambda,\mu$ are
dominant coweights of affine $E_{8}$ satisfying $\mu\geq\lambda$. We write
$\lambda=(\lambda_{\rm Kac},n)$ and similarly for $\mu$; then we must have
$\mu=\lambda+\sum_{i}v_{i}\alpha_{i}^{\vee}$ where $\alpha_{i}^{\vee}$ are the
simple coroots. In our case we will always set $\mu_{\rm Kac}$ to be the Kac
diagram $[1^{k}]$. Then $\mathcal{M}_{\text{C}}(\mu,\lambda)$ is the Coulomb
branch of the UV orbi-instanton magnetic quiver, _disregarding_ the
$T(\operatorname{SU}(k))$ tail (in the sense just explained). We now look at
the stratification of this Coulomb branch. There is a stratum associated with
each coweight $\nu$ satisfying $\lambda\leq\nu\leq\mu$, which is equal to the
open stratum (i.e. the smooth locus) in the Coulomb branch
${\mathcal{M}}_{\text{C}}(\nu,\lambda)$ of the magnetic quiver associated with
the interval $\nu\geq\lambda$. For each such coweight $\nu$ there are several
additional strata contained in ${\mathcal{M}}_{\text{C}}(\nu,\lambda)$ but not
in ${\mathcal{M}}_{\text{C}}(\xi,\lambda)$ for any $\xi<\nu$. These strata are
indexed by partitions $[m_{i}]$ of $M$ where $\nu-M\delta\geq\lambda$ and
“seen” by $M$ separated M5’s away from the singularity and organized in
substacks containing $m_{i}$ branes each. We will label the additional stratum
given by a partition $[m_{i}]$ of $M$ by
${\mathcal{M}}^{[m_{i}]}_{\text{C}}(\nu,\lambda)$. Note that
${\mathcal{M}}_{\text{C}}(\nu,\lambda)$ contains the closure of
${\mathcal{M}}^{[m_{i}]}_{\text{C}}(\nu,\lambda)$, which contains
${\mathcal{M}}_{\text{C}}(\nu-M\delta,\lambda)$ (but not
${\mathcal{M}}_{\text{C}}(\nu-(M-1)\delta,\lambda)$). The transverse slice to
${\mathcal{M}}^{\text{smooth}}_{\text{C}}(\nu,\lambda)$ is isomorphic to the
Coulomb branch ${\mathcal{M}}_{\text{C}}(\mu,\nu)$ (and can be understood by
quiver subtraction). The transverse slice to
${\mathcal{M}}^{(m_{i})}_{\text{C}}(\nu,\lambda)$ is isomorphic to the product
of ${\mathcal{M}}_{\text{C}}(\mu,\nu)$ and (the Uhlenbeck partial
compactifications of) the centered moduli spaces of $m_{i}$ $E_{8}$-instantons
on $\mathbb{C}^{2}$, i.e. the Higgs branch of a rank-$m_{i}$ E-string (given
by $m_{i}$ M5’s away from the singularity but lying on the M9), representing
small $E_{8}$-instanton transitions reducing $N$ to $N-M$.

###  1.5 Strategy of proof and results

Having spelled out the details of the various possible Higgsings, we will now
explain what we will do with them. First of all (and in analogy with the
T-brane case [103, 70]) we will entirely disregard flows from a UV orbi-
instanton to a lower-rank orbi-instanton times a collection of decoupled
E-strings in the IR.242424For an example of the full Higgs branch for
$N=1,k=2$ which includes the decoupled E-strings see [108, Fig. 3.1]. In that
Hasse diagram, the $a$ anomaly reduces from bottom to top. This is because the
$a$-theorem for the E-strings has already been proven (as we will review below
in section 4.1) both for tensor and Higgs branch flows, so we are reduced to
proving the $a$-theorem for flows from UV to IR orbi-instantons if we want to
complete the list of $c$-theorems in even dimension.

This can be done as follows. The 6d anomaly polynomial of the decoupled system
is given by the sum of the anomaly polynomials of the two ingredients, i.e.
lower-rank orbi-instanton and E-strings. The same is true for the total $a$
anomaly, being the sum of the $a$ anomalies of the two ingredients. Therefore
establishing an $a$-theorem for these flows boils down to separately proving
it for their “constituent” $a$ anomalies. It turns out that the $a$ anomaly of
an orbi-instanton defined by $(N,k,\rho_{\infty})$ is always higher than that
of the system $(N-M,k,\rho_{\infty})$ plus a collection of $n_{i}$
rank-$m_{i}$ E-strings (where $\left(n_{i},m_{i}\right)$ is an integer
partition of $M$ i.e. $\sum_{i}n_{i}m_{i}=M$) e.g. a single rank-$M$ E-string,
$M$ rank-$1$ E-strings etc. We prove this in appendix B. Therefore, these
flows are included within our proof of the $a$-theorem

Having identified orbi-instantons $(N,k,\rho_{\infty})$ with dominant
coweights $\lambda=(\lambda_{\text{Kac}},n)$ of affine $E_{8}$, and UV-IR
flavor Higgsings with degenerations $\mu>\lambda$, we will prove two facts.

  * _i_)

As we shall explain in section 3.2.1, affine coweights can be expressed as
triples $(k,\overline{\lambda},n)$, where $\overline{\lambda}$ is a weighting
of the finite $E_{8}$ Dynkin diagram and $n$ is an arbitrary integer. On the
basis of an explicit algorithm (due to Roy [101]) determining the Hasse
diagram of affine coweights, we will interpret $n$ in terms of the physical
number $N$ of M5-branes (in the process clarifying the relationship between
the numbers $N_{3}$, $N_{S}$ and $N_{6}$ introduced in [76]). This provides
the sought-after partial order on the space of homomorphisms
$\operatorname{Hom}(\mathbb{Z}_{k},E_{8})$ (a question asked e.g. in [75, 126,
127, 128]) and establishes, independently of 3d magnetic quivers and quiver
subtractions, a hierarchy between Higgs branch RG flows. As a result, the
hierarchies of [78] can be embedded as connected subdiagrams in the
aforementioned Hasse diagram. (In that paper, they were obtained by fixing the
sum $N+N_{\rho}$ for ease of illustration, where the number $N_{\rho}$ depends
solely on the choice of $\rho_{\infty}$ and is defined in (2.7) – see section
4 for an expanded explanation on this.)

  * _ii_)

Secondly, the $a$ anomaly of each SCFT can be written in terms of
combinatorial data of the associated stratum. (This is akin to what happens
for T-brane theories, where the hierarchy of nilpotent flavor Higgsings mimics
the Hasse diagram of nilpotent orbits of the flavor Lie algebra [103], and
writing $a$ in terms of the dimensions of these orbits ultimately allows to
prove that $\Delta a>0$ [70]. Here we see that this happens also in the case
of orbi-instantons, but with the dominant coweight strata of the double affine
Grassmannian.) The partial order on the dominant coweights induces a partial
order on the SCFTs (exactly the one obtained in [78, 110] via 3d magnetic
quiver subtraction and for $k=4$ via a 6d anomaly polynomial analysis in
[127]), and writing $a$ in terms of data of the strata allows us to prove
$\Delta a>0$ in full generality. There are finitely many cases to check, and
we have checked the required condition $\Delta a>0$ by computer in all cases.
However, we also sketch a conceptual proof which is valid for a large family
of cases (and which could in theory be adapted to other cases). This requires
a certain amount of analysis of the formula for the $a$ anomaly, especially
terms $\sum_{\alpha>0}\overline{\lambda}(\alpha)^{3}$ and
$\sum_{\alpha>0}\overline{\lambda}(\alpha)^{5}$ which appear in it.

###  1.6 Organization

The rest of the paper contains the technical proofs for the statements made so
far, and is organized as follows. In section 2 we define Kac diagrams as the
relevant objects needed to classify $\mathbb{Z}_{k}$-gradings of Lie algebras
($E_{8}$ in our case), i.e. homomorphisms
$\operatorname{Hom}(\mathbb{Z}_{k},E_{8})$, and we give a lightning review of
orbi-instantons. In section 3 we introduce the affine and double affine
Grassmannian of $E_{8}$, and give an explicit algorithm to construct strata of
the latter defined by dominant coweights. Using this algorithm, in section 4
we construct Hasse diagrams of dominant coweights for a few values of $k$, and
we verify that the hierarchies of RG flows (flavor Higgsings) obtained in [78,
110, 127] (via magnetic quiver subtraction or ’t Hooft anomaly matching) can
easily be recovered from these Hasse diagrams as connected subdiagrams. In
section 5 we prove the $a$-theorem, inducing the partial order on the $a$
anomalies of UV and IR orbi-instantons from the corresponding two strata
(dominant coweights) of the Grassmannian, such that $\Delta a>0$ for any pair
of SCFTs connected by RG flow (flavor Higgsing). We close in section 6 with a
few observations and an outlook. In appendix A we recall properties and
notation of 3d magnetic quivers associated with 6d orbi-instantons, and we
also prove that
$\mathcal{M}_{\text{C}}^{\text{inst}}=(\widehat{\mathcal{M}}_{\text{C}}\times\mathcal{O}_{\bm{\xi}})/\\!/\\!/\operatorname{SU}(k)$
as quiver varieties. In appendix B we prove the $a$-theorem for flows between
orbi-instantons and decoupled E-strings.

##  2 A-type orbi-instantons and Kac diagrams

We begin this section with a brief account of the theory of Kac diagrams, and
end it with a lightning review of orbi-instantons and how Kac diagrams enter
the physics discussion.

###  2.1 Kac diagrams and $\mathbb{Z}_{k}$-gradings of $E_{8}$

In mathematics, Kac diagrams were introduced in [129] to deal with gradings of
Lie algebras by finite cyclic groups (denoted $\mathbb{Z}_{k}$ here).
Therefore, in this section we briefly recall the relationship between “affine
Kac–Moody Lie algebras” and periodic automorphisms of (finite-dimensional)
simple Lie algebras, as explained in [129, Ch. 6-8]. For simplicity, we focus
on the untwisted case, which is the only one relevant to us.252525The reader
interested in the twisted case may consult [130] for a brief discussion on
outer automorphisms.

Let ${\mathfrak{g}}$ be a finite-dimensional simple Lie algebra and let
${\mathfrak{g}}_{\rm aff}$ be the corresponding (untwisted) affine Kac–Moody
Lie algebra. Fix once and for all a Cartan subalgebra ${\mathfrak{t}}$ of
${\mathfrak{g}}$ (contained in a Cartan subalgebra ${\mathfrak{t}}_{\rm aff}$
for ${\mathfrak{g}}_{\rm aff}$), and let
$\Phi=\Phi({\mathfrak{g}},{\mathfrak{t}})$ denote the root system of
${\mathfrak{g}}$ relative to ${\mathfrak{t}}$. Choose a positive system
$\Phi^{+}\subset\Phi$ (or equivalently, choose a Borel subalgebra
${\mathfrak{b}}$ containing ${\mathfrak{t}}$). Let
$\\{\alpha_{1},\ldots,\alpha_{r}\\}$ be a basis of simple roots in $\Phi^{+}$,
which we complete to a basis $\\{\alpha_{0},\ldots,\alpha_{r}\\}$ for the
affine root system $\Phi_{\rm aff}$. Recall that ${\mathfrak{g}}_{\rm aff}$ is
not simple: it is a semidirect product $[{\mathfrak{g}}_{\rm
aff},{\mathfrak{g}}_{\rm aff}]\oplus{\mathbb{C}}d$, where $d$ is any element
of the Cartan subalgebra not contained in the span of the coroots. (The
standard choice – see e.g. [129, Ch. 6.2] is $d$ such that $\alpha_{0}(d)=1$
and $\alpha_{i}(d)=0$ for $i>0$.) The derived subalgebra
${\mathfrak{g}}^{\prime}_{\rm aff}:=[{\mathfrak{g}}_{\rm
aff},{\mathfrak{g}}_{\rm aff}]$ is called the affine Lie algebra. Note that
${\mathfrak{g}}^{\prime}_{\rm aff}$ is not simple either: it has a one-
dimensional center ${\mathbb{C}}c$, where $c$ is the canonical central element
[129]. (The value of $c$ on a given representation is called, rather
evocatively, the “central charge” [27].262626See e.g. [131] for a brief
account on the Sugawara construction.) It is well known that the quotient
${\mathfrak{g}}^{\prime}_{\rm aff}/{\mathbb{C}}c$ is isomorphic to the loop
algebra
${\mathcal{L}}({\mathfrak{g}}):={\mathfrak{g}}\otimes{\mathbb{C}}[t,t^{-1}]$.

There is a more general loop algebra construction. Let $\theta$ be an
automorphism of order $k$ of the simple Lie algebra ${\mathfrak{g}}$ and let
$\zeta=e^{2\pi i/k}$. Since its minimal polynomial divides $t^{k}-1$, $\theta$
acts diagonalizably on ${\mathfrak{g}}$, hence there is a decomposition as a
direct sum of eigenspaces:

$\mathfrak{g}={\mathfrak{g}}(0)\oplus{\mathfrak{g}}(1)\oplus\ldots\oplus{\mathfrak{g}}(k-1)$
(2.1)

where ${\mathfrak{g}}(j)=\\{x\in{\mathfrak{g}}:\theta(x)=\zeta^{j}x\\}$. This
is a grading over $\mathbb{Z}_{k}$: for any integers $j,l$ modulo $k$ we have
$[{\mathfrak{g}}(j),{\mathfrak{g}}(l)]\subset{\mathfrak{g}}(j+l)$. We define
an infinite-dimensional, ${\mathbb{Z}}$-graded Lie algebra, called the
$\theta$-twisted loop algebra, as follows:

${\mathcal{L}}({\mathfrak{g}},\theta)=\bigoplus_{j\in{\mathbb{Z}}}{\mathfrak{g}}(j)\otimes
t^{j}\subset{\mathfrak{g}}\otimes{\mathbb{C}}[t,t^{-1}]\ .$ (2.2)

In the case $\theta=1$, we recover the loop algebra
${\mathcal{L}}({\mathfrak{g}})$ with the standard grading (i.e. with
${\mathfrak{g}}\otimes t^{j}$ in degree $j$).

Now let $\theta$ be an arbitrary inner automorphism. By an important theorem
of Kac [129, Thm. 8.5], ${\mathcal{L}}({\mathfrak{g}},\theta)$ is isomorphic
as an ungraded Lie algebra to ${\mathcal{L}}({\mathfrak{g}})$. This
isomorphism induces a non-standard grading on ${\mathcal{L}}({\mathfrak{g}})$,
and hence on ${\mathfrak{g}}_{\rm aff}$. After conjugating if necessary, any
such grading has ${\mathfrak{t}}_{\rm aff}\subset{\mathfrak{g}}_{\rm aff}(0)$
and ${\mathfrak{g}}_{{\rm aff},\alpha}\subset\sum_{i\geq 0}{\mathfrak{g}}_{\rm
aff}(i)$ for all simple roots $\alpha_{0},\ldots,\alpha_{r}$. The data of such
a grading therefore comes down to a non-negative integer weight for each
simple root, and can be represented by a Kac diagram, i.e. a copy of the
affine Dynkin diagram with the weights $(n_{i})_{0\leq i\leq r}$ attached (see
again (1.3) for $\mathfrak{g}=E_{8}$). We thus obtain a one-to-one
correspondence:

{periodic inner automorphisms of ${\mathfrak{g}}$}/conjugacy $\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{\text{1-to-1}}$ {Kac
diagrams}.

######  Remark 2.1.

  * _i)_

We here gloss over some subtleties concerning symmetries of the Dynkin
diagram. This will not be a problem for us since there are no such symmetries
in type $E_{8}$.

  * _ii)_

The periodic _outer_ automorphisms are classified via the same procedure,
leading to Kac diagrams on the _twisted_ affine root systems. We refer to
[129] for the details (which we do not need, since there are no outer
automorphisms in type $E_{8}$).

Each Kac diagram $\lambda_{\rm Kac}$ in (1.5) uniquely defines a homomorphism
${\mathbb{Z}}\Phi_{\rm aff}\to{\mathbb{Z}}$ which sends
$\sum_{i=0}^{r}m_{i}\alpha_{i}$ to $\sum_{i=0}^{r}m_{i}n_{i}$. We will use the
notation $\lambda_{\rm Kac}(\sum_{i}m_{i}\alpha_{i})$ for this weight. Recall
that a root $\alpha\in\Phi_{\rm aff}$ is _real_ if it is conjugate to a simple
root $\alpha_{i}$, and is _imaginary_ otherwise. In (untwisted) affine types,
there is a unique minimal positive imaginary root

$\delta:=\alpha_{0}+\sum_{i=1}^{r}m_{i}\alpha_{i}\ ,$ (2.3)

where $\widehat{\alpha}=\sum_{i=1}^{r}m_{i}\alpha_{i}$ is the (unique) highest
root in $\Phi^{+}$, and all imaginary roots are of the form $n\delta$ for
$n\in\mathbb{Z}\setminus\\{0\\}$. A Kac diagram $\lambda_{\rm Kac}$ determines
a $\mathbb{Z}$-grading of ${\mathfrak{g}}_{\rm aff}$ satisfying
${\mathfrak{g}}_{{\rm aff},\delta}\subset{\mathfrak{g}}_{\rm aff}(k)$, where

$k:=\lambda_{\rm Kac}(\delta)=a_{0}+\sum_{i=1}^{r}m_{i}n_{i}$ (2.4)

is the _level_ of $\lambda_{\text{Kac}}$ (again using terminology common from
2d CFTs). It follows that this grading induces a $\mathbb{Z}_{k}$-grading of
${\mathfrak{g}}$, hence an inner automorphism of ${\mathfrak{g}}$ of order
dividing $k$. If the $n_{i}$ have no common factor then this automorphism has
order $k$.

Note for instance that there are (up to conjugacy) three
$\mathbb{Z}_{2}$-gradings of a simple Lie algebra of type $E_{8}$, i.e. when
$k=2$ (see section 3.1.1 for our conventions on numbering):

  * _i)_

the trivial grading with ${\mathfrak{g}}={\mathfrak{g}}(0)$, given by
$n_{1}=2$ and all other $n_{i}=0$. This is Kac diagram
$\lambda_{\text{Kac}}=[1^{2}]$;

  * _ii)_

the grading with $n_{2}=1$ and all other $n_{i}=0$. This is Kac diagram
$\lambda_{\text{Kac}}=[2]$;

  * _iii)_

the grading with $n_{2^{\prime}}=1$ and all other $n_{i}=0$. This is Kac
diagram $\lambda_{\text{Kac}}=[2^{\prime}]$.

Kac diagrams incorporate a great deal of information about automorphisms. For
example, for a Kac diagram $\lambda_{\text{Kac}}$ with corresponding
automorphism $\theta$, the fixed point subalgebra
${\mathfrak{g}}^{\theta}={\mathfrak{g}}(0)$ can be read off as the reductive
subalgebra generated by ${\mathfrak{t}}$ and the simple root elements
$e_{\pm\alpha}$ such that $\lambda_{\text{Kac}}(\alpha)=0$; in this context we
identify the affine root elements $e_{\pm\alpha_{0}}$ with
$e_{\mp\widehat{\alpha}}$, where $\widehat{\alpha}$ is the highest root
element in $\Phi^{+}$. Such subalgebras are called _pseudo-Levi subalgebras_
in the mathematical literature. For the three Kac diagrams of order 2 listed
above, we have respectively ${\mathfrak{g}}(0)=\mathfrak{e}_{8}$,
$\mathfrak{e}_{7}\oplus{\mathfrak{su}({2})}$ and $\mathfrak{so}_{16}$. Notice
that this pseudo-Levi subalgebra is precisely what we called $\mathfrak{f}$
(i.e. the preserved flavor symmetry) in section 2.2.

###  2.2 Orbi-instantons from M-theory

Orbi-instantons are 6d $(1,0)$ SCFTs which are the datum of the number $N$ of
M5-branes probing an M9-wall and simultaneously the orbifold
$\mathbb{C}^{2}/\Gamma_{\text{ADE}}$ (with
$\Gamma_{\text{ADE}}\subset\operatorname{SU}(2)$ finite), the order of the
orbifold, and (the holonomy of) a flat connection at the spatial infinity
$S^{3}/\Gamma_{\text{ADE}}$ for the $E_{8}$ gauge bundle supported on the
M9-wall in eleven dimensions. Such flat connections are elements of
$\operatorname{Hom}(\pi_{1}(S^{3}/\Gamma_{\text{ADE}}),E_{8})\cong\operatorname{Hom}(\Gamma_{\text{ADE}},E_{8})$.
Hereafter we are going to focus exclusively on
$\Gamma_{\text{ADE}}=\mathbb{Z}_{k}$. Then $\rho_{\infty}:\mathbb{Z}_{k}\to
E_{8}\in\operatorname{Hom}(\mathbb{Z}_{k},E_{8})$ is a grading of $E_{8}$ by
$\mathbb{Z}_{k}$ of the type introduced in the previous section.

The tensor branch of each such orbi-instanton can be given an F-theory
description as follows [69] (to which we refer the reader for the relevant
notation):

$[E_{8}]\,\underbrace{\overset{{\mathfrak{su}({k})}}{1}\,\overset{{\mathfrak{su}({k})}}{2}\cdots\overset{{\mathfrak{su}({k})}}{2}}_{N}\,[\operatorname{SU}(k)]\
.$ (2.5)

However, this quiver represents only a _partial_ tensor branch. Depending on
the specific flat connection chosen (with this data being hidden in the
notation of (2.5)), we may have to introduce extra compact curves in the base
of F-theory. This is because the intersection
$[E_{8}]\,\overset{{\mathfrak{su}({k})}}{1}$ is too singular, and e.g. when
$k=[1^{k}]$ a further $k$ blowups in the base are required in the middle of
the two curves (each instance involves introducing a new $1$ curve, decorated
by ${\mathfrak{su}({k-i})}$, $i=1,...,k$, and blowing up the “old” $1$ into a
$2$). More generally, the full tensor branch of the orbi-instanton is given by
the following generalized F-theory quiver,

$[F]\underbrace{\overset{\mathfrak{g}}{1}\,\overset{\mathfrak{su}(m_{1})}{2}\,\overset{\mathfrak{su}(m_{2})}{2}\cdots\overset{\mathfrak{su}(m_{N_{\rho}-1})}{2}}_{\max\left(N_{\rho},1\right)}\underbrace{\overset{\mathfrak{su}(k)}{\underset{[N_{\text{f}}=k-m_{N_{\rho}-1}]}{2}}\overset{\mathfrak{su}(k)}{2}\cdots\overset{\mathfrak{su}(k)}{2}[\operatorname{SU}(k)]}_{N+1}\
,$ (2.6)

where $[F]$ is (the nonabelian part of) a maximal subalgebra $\mathfrak{f}$ of
$E_{8}$, and
$\mathfrak{g}\in\\{\emptyset,\mathfrak{usp}(m_{0}),\mathfrak{su}(m_{0})\\}$.
For $\mathfrak{g}=\mathfrak{su}(m_{0})$ we also have one hypermultiplet in the
two-index antisymmetric representation of $\mathfrak{su}(m_{0})$ for all
$m_{0}\neq 6$ (and a half hypermultiplet in the three-index antisymmetric
representation of $\mathfrak{su}(m_{0})$ for $m_{0}=6$). All ranks and matter
representations (i.e. hypermultiplets) are determined by the chosen
$\rho_{\infty}$ via a simple algorithm which can be found in [76]. The
subalgebra $\mathfrak{f}$ of $E_{8}$ that is unbroken by the Kac label is the
commutant of the image $\rho_{\infty}(\mathbb{Z}_{k})\subset E_{8}$, and its
Dynkin diagram is easily obtained by deleting the nodes of affine $E_{8}$
appearing in the partition (1.4) (i.e. the nodes of (1.3) whose $n_{i}$ are
nonzero), together with an Abelian subalgebra making the total rank 8 [129,
Sec. 8.6], i.e. a summand of the form
$\bigoplus_{i}\mathfrak{u}(1)_{i}$.272727The maximal subalgebras that do not
contain $\mathfrak{u}(1)$ summands are the semisimple regular ones; those that
do are non-semisimple regular. The former are preserved by Kac labels with a
single part, the latter by those with more than one part. See [78, Sec. 3.3]
for the $\mathfrak{u}(1)$ summands. We will also say that the orbi-instanton
has a plateau of length $N+1$, i.e. the region where the gauge algebras all
become ${\mathfrak{su}({k})}$, corresponding to having stacks of $k$ D6’s in
that region of the Type IIA reduction of M-theory.

What is $N_{\rho}$ in the above expression? The F-theory description, i.e. the
Type IIB background with varying axiodilaton and nonperturbative seven-branes,
is T-dual to a Type IIA setup with one O8--plane and 8 D8-branes. The
different ways in which the stack of D8’s split into substacks (one of which
may be on top of the O8-) encode the different subalgrebras $\mathfrak{f}$.
There are also $k$ D6-branes, which can end with different patterns on the
D8’s on the left of the configuration. (Each pattern is specified by a
different $\rho_{\infty}$.) Moreover the D6’s are suspended between
NS5-branes. (For details see e.g. [78].) For each M5-brane in the original
M-theory configuration we have one NS5-brane; however, because of the
orbifold, the M9 _fractionates_ [69], generating extra NS5’s once we reduce to
Type IIA. Then $N_{\rho}$ precisely captures this number. It depends on the
chosen $\rho_{\infty}$, and we will say that it gives the number of
“fractional instantons”. In F-theory, it signals that the intersection between
$[E_{8}]$ and $\overset{{\mathfrak{su}({k})}}{1}$ is too singular, and needs
to be blown up $N_{\rho}$ times to bring the model in Kodaira–Tate form.

$N_{\rho}$ can be determined in the following simple way in terms of
$\rho_{\infty}$ (i.e. (1.3)):

$N_{\rho}:=\sum_{i=1}^{6}n_{i}+p\ ,\quad
p:=\min\left(\left\lfloor\frac{n_{3^{\prime}}+n_{4^{\prime}}}{2}\right\rfloor,\left\lfloor\frac{n_{2^{\prime}}+n_{3^{\prime}}+2n_{4^{\prime}}}{3}\right\rfloor\right)\
.$ (2.7)

When the Kac diagram does not contain any primes, $N_{\rho}$ is identical to
the total number of unprimed parts.282828Notice that our $p$ also appears in
[132, Eq. (3.109)] with the same name, and in [76] denotes the difference
between $N_{S}$ and $N_{6}$. It may sometimes happen that $N_{\rho}=0$ (e.g.
picking $[4^{\prime}]$ for $k=4$); in this case the above electric quiver
reads

$[F]\,\overset{\mathfrak{g}}{1}\,\underbrace{\overset{\mathfrak{su}(m_{1})}{\underset{[N_{\text{f}_{1}}]}{2}}\,\overset{\mathfrak{su}(m_{2})}{\underset{[N_{\text{f}_{2}}]}{2}}\cdots\overset{\mathfrak{su}(k)}{\underset{[N_{\text{f}_{\text{plateau}}}]}{2}}\,\overset{\mathfrak{su}(k)}{2}\cdots\overset{\mathfrak{su}(k)}{2}[\operatorname{SU}(k)]}_{N+1}\
,$ (2.8)

where the ranks and matter representations depend nontrivially on the chosen
$k$ and $\mathfrak{g}$. (For concrete examples see [78].)

##  3 Affine and double affine Grassmannians

Having introduced orbi-instantons and Kac diagrams, we move on to the next
dramatis persona – _Grassmannians_. We begin the section by introducing affine
Grassmannians. We then move on to a discussion of double affine Grassmannians,
whose original (in some sense conjectural) definition is due to
Braverman–Finkelberg [81, 82, 83], and which have subsequently been studied in
the context of 3d Coulomb branches by the same authors in collaboration with
Nakajima [86, 95, 133, 134, 99, 135].

###  3.1 Affine Grassmannians

Let us define the affine Grassmannian of a simple algebraic group $G$ over
${\mathbb{C}}$. Let $\mathbb{C}[\\![z]\\!]$ denote the ring of formal complex
power series $f(z)=\sum_{i=0}^{\infty}a_{i}z^{i}$, and let
$\mathbb{C}(\\!(z)\\!)$ be its field of fractions, that is, all expressions
$f(z)/g(z)$ with $f(z),g(z)\in\mathbb{C}[\\![z]\\!]$ and $g(z)\neq 0$. An
element of $\mathbb{C}(\\!(z)\\!)$ can be written uniquely as a Laurent series
$h(z)=\sum_{i=N}^{\infty}a_{i}z^{i}$ with $N\in\mathbb{Z}$ and $a_{N}\neq 0$.
Let $G$ be an algebraic subgroup of $\text{GL}(n,\mathbb{C})$. By definition,
$G=\\{M\in\text{GL}(n,\mathbb{C})\ |\ P_{j}(M)=0\\}$ for some polynomials
$P_{j}$. (Assume $\\{P_{j}\\}$ is a complete set, i.e. generates the ideal of
all polynomials vanishing on $G$.) The exceptional groups, in particular
$E_{8}$, can be considered as algebraic subgroups in this way.292929Algebraic
varieties and hence algebraic groups are more properly defined
“intrinsically”, that is, in a way that is independent of any particular
embedding. The collection of polynomials $P_{j}$ allows us to define points of
$G$ in an arbitrary ring containing ${\mathbb{C}}$. Specifically, we have:
$G[\\![z]\\!]=\\{M\in\text{GL}(n,\mathbb{C}[\\![z]\\!])\ |\
P_{j}(M)=0\;\forall\,j\\}$ and
$G(\\!(z)\\!)=\\{M\in\text{GL}(n,\mathbb{C}(\\!(z)\\!))\ |\
P_{j}(M)=0\;\forall\,j\\}$. We consider $G[\\![z]\\!]$ as a subset of
$G(\\!(z)\\!)$ in the canonical way. (Note that $G(\\!(z)\\!)$ is sometimes
called the loop group; it can be considered as a completed version of the
group analog $G[z,z^{-1}]$ of the loop algebra considered in section 2.1.)

The affine Grassmannian of $G$ can be defined as the space of left cosets:

$\text{Gr}_{G}:=G(\\!(z)\\!)/G[\\![z]\\!]\ .$ (3.1)

Left multiplication defines an action of $G[\\![z]\\!]$ on $\text{Gr}_{G}$;
note that the set of orbits for this action is in one-to-one correspondence
with the set of double cosets
$G[\\![z]\\!]\\!\setminus\\!\\!\,G(\\!(z)\\!)/G[\\![z]\\!]$. There is a
natural topology on $\text{Gr}_{G}$, and the closure $\overline{\mathcal{O}}$
of a $G[\\![z]\\!]$-orbit in $\text{Gr}_{G}$ has a structure of (finite-
dimensional, generically singular) projective variety containing
${\mathcal{O}}$ as a Zariski-open subset. As we recount below, there are
countably many orbits, and any orbit closure $\overline{\mathcal{O}}$ is a
union of finitely many orbits, each a locally closed subset of
$\overline{\mathcal{O}}$. This gives $\text{Gr}_{G}$ a structure of infinite-
dimensional variety, or _ind-variety_. Given any $M\in G(\\!(z)\\!)$, denote
by $[M]$ the coset $M\cdot G[\\![z]\\!]\in\text{Gr}_{G}$, and by
$G[\\![z]\\!]\cdot[M]$ the corresponding $G[\\![z]\\!]$-orbit.

Recall that an algebraic group is simple if it is nonabelian and has no closed
connected normal subgroups; this includes $E_{8}$. Let $G$ be simple and let
${\mathfrak{g}}$ be the Lie algebra of $G$ (which is a simple Lie algebra). We
retain the notation (${\mathfrak{t}}$, $\Phi$, $\Phi^{+}$, $\alpha_{i}$)
introduced in section 2.1. Let $T$ be a maximal torus of $G$ with Lie algebra
${\mathfrak{t}}$, and identify $\Phi=\Phi({\mathfrak{g}},{\mathfrak{t}})$ with
the roots $\Phi(G,T)$ relative to $T$. Each $\alpha\in\Phi$ is therefore an
algebraic character $T\rightarrow{\mathbb{C}}^{\times}$, and hence the root
lattice ${\mathbb{Z}}\Phi$ embeds in the character group $X(T)$. Similarly,
for each $\alpha$ there is a corresponding coroot $\alpha^{\vee}$, which we
think of both as an element of ${\mathfrak{t}}$ and as a cocharacter, i.e. a
homomorphism ${\mathbb{C}}^{\times}\rightarrow T$. We denote the group of all
cocharacters by $Y(T)$. The coroot lattice ${\mathbb{Z}}\Phi^{\vee}$ therefore
embeds in $Y(T)$. In general, both inclusions are proper: the center $Z(G)$ is
isomorphic to $X(T)/{\mathbb{Z}}\Phi$, and the fundamental group $\pi_{1}(G)$
is isomorphic to $Y(T)/{\mathbb{Z}}\Phi^{\vee}$. Thus, $G$ is simply-connected
if $Y(T)={\mathbb{Z}}\Phi^{\vee}$. Obversely, $G$ is of _adjoint type_ if
$X(T)={\mathbb{Z}}\Phi$; for any simple $G$ the quotient $G/Z(G)$ is of
adjoint type.

The lattices $X(T)$ and $Y(T)$ are dual, in the following sense. For $\xi\in
X(T)$ and $\lambda\in Y(T)$ we denote by $\lambda(\xi)$ the unique integer
such that $\xi(\lambda(t))=t^{\langle\xi,\lambda\rangle}$ for all
$t\in{\mathbb{C}}^{\times}$. This defines a _perfect pairing_ $X(T)\times
Y(T)\to{\mathbb{Z}}$, i.e. it identifies $Y(T)$ with ${\rm
Hom}(X(T),{\mathbb{Z}})$, and vice-versa. We can therefore identify elements
of $Y(T)$ with linear functions $X(T)\rightarrow{\mathbb{Z}}$. Hence we have
dual inclusions:

${\mathbb{Z}}\Phi\subset X(T)\subset{\rm
Hom}({\mathbb{Z}}\Phi^{\vee},{\mathbb{Z}})\ ,\quad{\rm
Hom}({\mathbb{Z}}\Phi,{\mathbb{Z}})\supset Y(T)\supset{\mathbb{Z}}\Phi^{\vee}\
,$ (3.2)

where the first (resp. second) inclusion in each pair is an equality if $G$ is
of adjoint type (resp. simply-connected). The homomorphisms
${\mathbb{Z}}\Phi^{\vee}\rightarrow{\mathbb{Z}}$ (resp.
${\mathbb{Z}}\Phi\rightarrow{\mathbb{Z}}$) are called weights (resp.
coweights). Since any coweight is determined by its values on the simple roots
$\alpha_{i}$, the lattice of coweights has a basis consisting of the
fundamental coweights $\varpi_{i}^{\vee}$ satisfying
$\varpi_{j}^{\vee}(\alpha_{i})=1$ if $i=j$ and $0$ otherwise.

A coweight (hence also any cocharacter) $\lambda$ is _dominant_ if
$\lambda(\alpha)\geq 0$ for all positive roots $\alpha$. Clearly, the dominant
coweights are those of the form $\sum_{i}n_{i}\varpi_{i}^{\vee}$ with
$n_{i}\in{\mathbb{Z}}_{\geq 0}$. To each dominant cocharacter $\lambda\in
Y(T)$ is naturally associated an element $\lambda_{z}\in
G({\mathbb{C}}(\\!(z)\\!))$ (which we can think of as “evaluating $\lambda$ at
$z$”), and hence a $G[\\![z]\\!]$-orbit:

$[\text{Gr}_{G}]^{\lambda}:=G[\\![z]\\!]\cdot[\lambda_{z}]\ ,$ (3.3)

also known as a Schubert cell. We have
$\dim_{\mathbb{C}}[\text{Gr}_{G}]^{\lambda}=2\lambda(\rho)$ where $\rho$ is
the Weyl vector, i.e. the half sum of the positive roots. (The Weyl vector can
also be defined by $\alpha_{i}^{\vee}(\rho)=1$ for all simple coroots
$\alpha_{i}^{\vee}$, that is, $\rho$ is the sum of the fundamental weights
$\varpi_{i}$.) It turns out that all $G[\\![z]\\!]$-orbits are of this form,
and the orbits of distinct dominant cocharacters are distinct. When $G$ is of
adjoint type, we therefore obtain the statement:

for $G$ adjoint, the $G[\\![z]\\!]$-orbits in $\text{Gr}_{G}$ are in
correspondence with dominant coweights.

There are finitely many Schubert cells of each dimension, hence there are
finitely many orbits in the closure of $[\text{Gr}_{G}]^{\lambda}$. We write
$\lambda\leq\mu$ when
$[\text{Gr}_{G}]^{\lambda}\subset\overline{[\text{Gr}_{G}]^{\mu}}$; this
defines a partial order on the dominant cocharacters. This partial order is
very well understood; we will return to it below.

It may be worth clarifying the relationship between coroots and coweights. For
a simple root $\alpha_{j}$ and a simple coroot $\alpha_{i}^{\vee}$, we have
$\alpha_{i}^{\vee}(\alpha_{j})=C_{ij}$ is the coefficient of the Cartan matrix
for ${\mathfrak{g}}$. Hence,
$\alpha_{i}^{\vee}=\sum_{j}C_{ij}\varpi_{j}^{\vee}$. (This shows that the
coroot lattice is of index $(\det C)$ in the coweight lattice.) We can
therefore express the fundamental coweights as (in general rational) linear
combinations of the simple coroots:
$\varpi_{i}^{\vee}=\sum_{j}A^{ij}\alpha_{j}^{\vee}$, where $(A^{ij})$ is the
inverse of the Cartan matrix.

####  3.1.1 Coweights in type $E_{8}$

We specialize to $G=E_{8}$ from now on. In this case $G$ is both simply-
connected and of adjoint type. (This follows from the fact that the Cartan
matrix has determinant $-1$. Note that $E_{8}$ is the only simply-laced root
system with this property.)

We use Coxeter labels
$i\in\\{2,3,4,5,6,4^{\prime},3^{\prime},2^{\prime}\\}=\Delta$ for the Dynkin
diagram of $E_{8}$:

$E_{8}:\quad\overset{2}{\underset{\alpha_{2}}{\circ}}-\overset{3}{\underset{\alpha_{3}}{\circ}}-\overset{4}{\underset{\alpha_{4}}{\circ}}-\overset{5}{\underset{\alpha_{5}}{\circ}}-\overset{6\overset{{\hbox
to0.0pt{\hss$\scriptstyle 3^{\prime}$}\displaystyle\circ{\hbox
to0.0pt{$\scriptstyle\alpha_{3^{\prime}}$\hss}}}}{\scriptstyle|}}{\underset{\alpha_{6}}{\circ}}-\overset{4^{\prime}}{\underset{\alpha_{4^{\prime}}}{\circ}}-\overset{2^{\prime}}{\underset{\alpha_{2^{\prime}}}{\circ}}\
.$ (3.4)

As above, denote by $\alpha_{i}$, $\alpha_{i}^{\vee}$ and $\varpi_{i}^{\vee}$
the simple root, simple coroot and fundamental coweight corresponding to $i$.
It follows from the simply-laced property that for any root
$\alpha=\sum_{i}m_{i}\alpha_{i}$, the corresponding coroot is
$\alpha^{\vee}=\sum_{i}m_{i}\alpha_{i}^{\vee}$. Denote by $\widehat{\alpha}$
the highest root in $\Phi$. Then we have
$\varpi_{2}^{\vee}=\widehat{\alpha}^{\vee}=\sum_{i=2}^{6}i\alpha_{i}^{\vee}+\sum_{i=2}^{4}i\alpha_{i^{\prime}}^{\vee}$,
which in particular is dominant. This holds more generally: if
$\widehat{\alpha}_{I}$ is the highest root element of any irreducible simply-
laced root system $\Phi_{I}$, then $\widehat{\alpha}_{I}^{\vee}$ is a dominant
coweight with respect to $\Phi_{I}$ (and is the only dominant coroot). We will
apply this in cases where $I$ is a connected subdiagram of the (affine) Dynkin
diagram of type $E_{8}$. We briefly note how $\widehat{\alpha}_{I}^{\vee}$ is
expressed in terms of fundamental coweights. In all cases, a node
$i\in\Delta\setminus I$ is joined via at most one edge to elements of $I$. We
let $\varpi_{I}^{-}=\sum_{\text{edge}\,I-j}\varpi_{j}^{\vee}$. Then
$\widehat{\alpha}_{I}^{\vee}=\varpi_{I}^{+}-\varpi_{I}^{-}$, where
$\varpi_{I}^{+}$ is the unique dominant coroot within the root subsystem
$\Phi_{I}$. We specify $\varpi_{I}^{+}$ in each case:

  * •

if $I$ is a singleton set then $\varpi_{I}^{+}=2\varpi_{i}^{\vee}$;

  * •

if $I=\\{i,\ldots,j\\}$ is of type $A_{m}$ ($m\geq 2$) then
$\varpi_{I}^{+}=\varpi_{i}^{\vee}+\varpi_{j}^{\vee}$;

  * •

if $I=\\{i,i+1,\ldots,j-2,j-1,j\\}$ is of type $D_{m}$ (with branch node
$(j-2)$ and “tail” $i-\cdots-(j-2)$) then
$\varpi_{I}^{+}=\varpi_{i+1}^{\vee}$;

  * •

if $I=\\{2^{\prime},3^{\prime},4^{\prime},6,5,4\\}$ is of type $E_{6}$ then
$\varpi_{I}^{+}=\varpi_{3^{\prime}}^{\vee}$;

  * •

if $I=\\{2^{\prime},3^{\prime},4^{\prime},6,5,4,3\\}$ is of type $E_{7}$ then
$\varpi_{I}^{+}=\varpi_{2^{\prime}}^{\vee}$.

####  3.1.2 Orbits as weighted Dynkin diagrams

As mentioned above, the $G[\\![z]\\!]$-orbits $[\text{Gr}_{E_{8}}]^{\mu}$ are
in one-to-one correspondence with the dominant coweights $\mu$, i.e. the non-
negative integer linear combinations

$\mu:=\sum_{i\in\Delta}n_{i}\varpi_{i}^{\vee}\ .$ (3.5)

The $n_{i}$ can be thought of as “multiplicities” of the Coxeter labels in the
$E_{8}$ Dynkin (3.4). Similarly to the Kac diagrams studied in section 2.1,
this allows us to represent orbits via “weighted Dynkin
diagrams”.303030However, this phrase often has a more restricted meaning: the
classification of nilpotent orbits via $\mathfrak{sl}_{2}$-triples
$\\{h,e,f\\}$ allows one to uniquely attach a coweight to each nilpotent
orbit; in the mathematical literature, the phrase “weighted Dynkin diagram”
usually refers only to these particular coweights.

The closure order on orbits induces a partial order on dominant coweights. In
fact we have
$[\text{Gr}_{E_{8}}]^{\lambda}\subset\overline{[\text{Gr}_{E_{8}}]^{\mu}}$ if
and only if $\mu-\lambda$ is a non-negative linear combination of simple
coroots. A pair $(\lambda,\mu)$ with $\lambda<\mu$ is called a _degeneration_
; it is a minimal degeneration if there is no $\nu$ with $\lambda<\nu<\mu$.
The partial order on coweights can be represented via a Hasse diagram, where
an edge connects adjacent coweights (i.e. minimal degenerations). According to
a result of Stembridge [85], there are (in the simply-laced case) exactly two
types of minimal degeneration $\lambda<\mu$:

  1. _i_)

pairs $(\lambda,\mu=\lambda+\widehat{\alpha}_{I}^{\vee})$, where $I$ is a
connected component of the subdiagram of zeros of $\mu$ (i.e. zero
multiplicities in the weighted Dynkin);

  2. _ii_)

pairs $(\lambda,\mu=\lambda+\alpha_{i}^{\vee})$, where $n_{j}>0$ for all $j$’s
connected via an edge to $i$.

Note that if we express $\lambda$ as sum $\sum_{i}n_{i}\varpi_{i}^{\vee}$ of
coweights and $\mu-\lambda=\sum_{i}l_{i}\alpha_{i}^{\vee}$ as a sum of
coroots, then in case _i_) we have $n_{i}l_{i}=0$ for all
$i\in\\{2^{\prime},3^{\prime},4^{\prime},6,5,4,3,2\\}$.

####  3.1.3 Geometry and slices in the affine Grassmannian

Each orbit $\text{Gr}_{\mu}$ in the affine Grassmannian is smooth, with
singular points lying on the boundary

$\overline{\text{Gr}_{\mu}}\setminus\text{Gr}_{\mu}=\bigcup_{\mu<\lambda}\text{Gr}_{\lambda}\
.$ (3.6)

The geometry of singular points can be understood using transverse slices.
Suppose $\lambda<\mu=\lambda+\sum l_{i}\alpha_{i}^{\vee}$ is a degeneration of
dominant coweights. The transverse slice ${\mathcal{S}}_{\lambda,\mu}$ to
$\overline{\text{Gr}_{\nu}}$ at $\lambda_{z}$ is a symplectic singularity. By
an earlier remark,
$\dim_{\mathbb{H}}({\mathcal{S}}_{\lambda,\mu})=(\mu-\lambda)(\rho)=\sum
l_{i}$, i.e. the _height_ of $\mu-\lambda$. In particular, in case _i)_ above
(with $\mu-\lambda=\widehat{\alpha}_{I}^{\vee}$) this dimension is $h_{I}-1$,
where $h_{I}$ is the Coxeter number of the root system spanned by $I$; in case
_ii_), $(\mu-\lambda)(\rho)=1$.

The main result of [111] is a classification of the singularities associated
to _minimal_ degenerations in affine Grassmannians (in which case the
singularity is isolated at $\lambda_{z}$). Specifically, the singularity in
case _i_) is the closure of the minimal nilpotent orbit of a simple Lie
algebra of type $I$ (a so-called _minimal singularity_); in case _ii_) it is
an $A_{n_{i}+1}$ type surface singularity, i.e.
${\mathbb{C}}^{2}/{\mathbb{Z}}_{n_{i}+2}$. These results are clearly
consistent with the dimension statement above. Note that the pairs in _i_) and
_ii_), and hence the partial order on dominant coweights, are naturally
expressed in purely combinatorial terms (i.e. without reference to the affine
Grassmannian).

###  3.2 Double affine Grassmannians

The affine Grassmannian has an important role in representation theory because
of the geometric Satake correspondence [93], which is a deep theorem relating
irreducible representations of ${\mathfrak{g}}$ to the geometry of
$\text{Gr}_{G}$. The double affine Grassmannian is a (somewhat conjectural)
object playing a similar role for the affine Lie algebra ${\mathfrak{g}}_{\rm
aff}$. To outline what is known about the double affine Grassmannian, we need
to clarify what are the coweights for the corresponding affine (i.e.
Kac–Moody) Lie algebra. We keep to type $E_{8}$ since this is the only case
that concerns us.

####  3.2.1 Coroots and coweights

Extend the Dynkin diagram for $E_{8}$ by adding an affine node, labeled 1; set
$\Delta_{\text{aff}}=\Delta\cup\\{1\\}$:

$E_{8}^{(1)}:\quad\overset{1}{\underset{\alpha_{1}}{\circ}}-\overset{2}{\underset{\alpha_{2}}{\circ}}-\overset{3}{\underset{\alpha_{3}}{\circ}}-\overset{4}{\underset{\alpha_{4}}{\circ}}-\overset{5}{\underset{\alpha_{5}}{\circ}}-\overset{6\overset{{\hbox
to0.0pt{\hss$\scriptstyle 3^{\prime}$}\displaystyle\circ{\hbox
to0.0pt{$\scriptstyle\alpha_{3^{\prime}}$\hss}}}}{\scriptstyle|}}{\underset{\alpha_{6}}{\circ}}-\overset{4^{\prime}}{\underset{\alpha_{4^{\prime}}}{\circ}}-\overset{2^{\prime}}{\underset{\alpha_{2^{\prime}}}{\circ}}\
.$ (3.7)

Correspondingly, we introduce an affine root $\alpha_{1}$ and coroot
$\alpha_{1}^{\vee}$, generating (along with $\Phi$ and $\Phi^{\vee}$) the
affine root lattice $\mathbb{Z}\Phi_{\text{aff}}$ and coroot lattice
$\mathbb{Z}\Phi_{\text{aff}}^{\vee}$. These lattices are _not_ dual via the
affine Cartan matrix, because of the imaginary roots in $\Phi_{\text{aff}}$
and the central elements in $\Phi_{\text{aff}}^{\vee}$. Thus we need to extend
further (in the language of Kac [129], to a _realization_ of the generalized
Cartan matrix, allowing for a perfect pairing between weights and coweights).
Following the notation in [129, Ch. 6], we therefore introduce an additional
weight $\Lambda_{0}$ and an additional coweight $d$. Then the weight lattice
(of the affine Kac–Moody group) is
$\widehat{X}_{\text{aff}}:=\mathbb{Z}\Phi_{\text{aff}}\oplus\mathbb{Z}\Lambda_{0}$
and the coweight lattice is
$\widehat{Y}_{\text{aff}}:=\mathbb{Z}\Phi_{\text{aff}}^{\vee}\oplus\mathbb{Z}d$.

The elements $d$ and $\Lambda_{0}$ are defined such that ${\rm
Hom}(\widehat{X}_{\text{aff}},{\mathbb{Z}})=\widehat{Y}_{\text{aff}}$, and
vice versa. (We can and should think of $\widehat{Y}_{\text{aff}}$ as the set
of cocharacters in a 10-dimensional torus with character lattice
$\widehat{X}_{\text{aff}}$.) To this end we need to describe the pairing
$\widehat{X}_{\text{aff}}\times\widehat{Y}_{\text{aff}}\to\mathbb{Z}$.

We use the generalized Cartan matrix to define the pairing between roots and
coroots:

$\alpha_{j}^{\vee}(\alpha_{i})=\begin{cases}2&\text{if $i=j$,}\\\ -1&\text{if
$i$ and $j$ are connected by an edge,}\\\ 0&\text{otherwise.}\end{cases}$
(3.8)

(Note that this pairing extends the pairing on simple roots and coroots for
the finite diagram.) We can extend this by $\alpha_{i}^{\vee}(\Lambda_{0})=0$
for $i\neq 1$ and $\alpha_{1}^{\vee}(\Lambda_{0})=1$; similarly,
$d(\alpha_{i})=0$ for $i\neq 1$ and $d(\alpha_{1})=1$; finally
$d(\Lambda_{0})=0$. This gives the perfect pairing: with respect to the bases
$\\{\alpha_{1},\ldots,\alpha_{6},\alpha_{4^{\prime}},\alpha_{3^{\prime}},\alpha_{2^{\prime}},\Lambda_{0}\\}$
and
$\\{\alpha_{1}^{\vee},\ldots,\alpha_{6}^{\vee},\alpha_{4^{\prime}}^{\vee},\alpha_{3^{\prime}}^{\vee},\alpha_{2^{\prime}}^{\vee},d\\}$,
it is given by the matrix

$\begin{pmatrix}2&-1&0&0&0&0&0&0&0&1\\\ -1&2&-1&0&0&0&0&0&0&0\\\
0&-1&2&-1&0&0&0&0&0&0\\\ 0&0&-1&2&-1&0&0&0&0&0\\\ 0&0&0&-1&2&-1&0&0&0&0\\\
0&0&0&0&-1&2&-1&-1&0&0\\\ 0&0&0&0&0&-1&2&0&-1&0\\\ 0&0&0&0&0&-1&0&2&0&0\\\
0&0&0&0&0&0&-1&0&2&0\\\ 1&0&0&0&0&0&0&0&0&0\end{pmatrix}\ .$ (3.9)

Note in particular that all entries in the matrix are integers (so the pairing
is well-defined) and the determinant equals $-1$ (so the pairing is perfect,
i.e. it identifies $\widehat{Y}_{\rm aff}$ with the linear functions
$\widehat{X}_{\rm aff}\rightarrow{\mathbb{Z}}$). The analog here of the
fundamental coweights in $\mathbb{Z}\Phi^{\vee}$ is the basis for
$\widehat{Y}_{\text{aff}}$ which is dual to the basis
$\\{\alpha_{1},\ldots,\Lambda_{0}\\}$ for $\widehat{X}_{\text{aff}}$. Let

$\xi_{2}=\varpi_{2}^{\vee}+2d\ ,\quad\xi_{3}=\varpi_{3}^{\vee}+3d\ ,\ \ldots\
,\quad\xi_{2^{\prime}}=\varpi_{2^{\prime}}^{\vee}+2d\ .$ (3.10)

It is easy to check that the basis for $\widehat{Y}_{\text{aff}}$ which is
dual to $\\{\alpha_{1},\ldots,\Lambda_{0}\\}$ is
$\\{d,\xi_{2},\ldots,\xi_{2^{\prime}},c\\}$ where $c$ is the canonical central
element (or central charge)

$c=\left(\sum_{i=1}^{6}i\alpha_{i}^{\vee}\right)+4\alpha_{4^{\prime}}^{\vee}+3\alpha_{3^{\prime}}^{\vee}+2\alpha_{2^{\prime}}^{\vee}\
.$ (3.11)

Hence we can denote $d$ by $\xi_{1}$. (Note that the fundamental coweights
$\varpi_{i}^{\vee}$ in ${\mathbb{Z}}\Phi^{\vee}$ are no longer fundamental
coweights for the affine root system; this role is now played by the
$\xi_{i}$.)

In the affine case, a _dominant coweight_ is an element of
$\widehat{Y}_{\text{aff}}$ of the form:

$\lambda:=\left(\sum_{i=1}^{6}n_{i}\xi_{i}\right)+n_{4^{\prime}}\xi_{4^{\prime}}+n_{3^{\prime}}\xi_{3^{\prime}}+n_{2^{\prime}}\xi_{2^{\prime}}+nc$
(3.12)

where all the $n_{i}$ are non-negative integers. (To connect to well known
physics, $n$ can be an arbitrary integer, and would correspond to the energy
of a 2d WZW model. Moreover there is a natural $\mathbb{Z}$-shift acting on
$n$ which will have an interpretation in terms of tensor branch flows shifting
$N$.) We say that $\lambda$ has _level_

$k:=\left(\sum_{i=1}^{6}in_{i}\right)+4n_{4^{\prime}}+3n_{3^{\prime}}+2n_{2^{\prime}}\
.$ (3.13)

This gives another way of understanding the Kac diagrams mentioned in section
2.1. Moreover,

$k=\lambda(\delta)\,,$ (3.14)

where $\delta$ is the smallest positive imaginary root. Hence the dominant
coweights of level $k$ can be identified with the pairs
$(\lambda_{\text{Kac}},n)$ of a Kac diagram of level $k$ _and_ an arbitrary
integer $n$. In the notation of [82], one writes

$\lambda_{\text{Kac}}=(k,\overline{\lambda})\
,\quad\overline{\lambda}=\sum_{i\neq
1}n_{i}\varpi_{i}^{\vee}=\lambda_{\text{Kac}}-kd\ .$ (3.15)

Note that $\lambda(\alpha)=\lambda_{\rm Kac}(\alpha)$ for any
$\alpha\in\Phi_{\rm aff}$ and $\lambda(\alpha)=\overline{\lambda}(\alpha)$ for
any $\alpha\in\Phi$. The point of the above discussion is to highlight the
three different ways of writing dominant coweights for affine $E_{8}$:

  1. _i_)

for [82], these are triples $(k,\overline{\lambda},n)$ where
$\overline{\lambda}$ is a dominant coweight for the _finite_ diagram, of level
(i.e. $\overline{\lambda}(\widehat{\alpha})$) less than or equal to $k$. In
our notation, this form for $\lambda$ arises from the direct sum
$\widehat{Y}_{\text{aff}}=\mathbb{Z}d\oplus\mathbb{Z}\Phi^{\vee}\oplus\mathbb{Z}c$.
In particular, $\overline{\lambda}$ is a $\mathbb{Z}$-linear combination of
$\alpha_{2}^{\vee},\ldots,\alpha_{2^{\prime}}^{\vee}$, hence also of the
fundamental coweights $\varpi_{2}^{\vee},\ldots,\varpi_{2^{\prime}}^{\vee}$
introduced in the previous subsection.

  2. _ii_)

Alternatively, we can write $\lambda$ in (3.12) as $(\lambda_{\text{Kac}},n)$
where $\lambda_{\text{Kac}}$ is a Kac diagram. Here $\lambda_{\text{Kac}}$ is
a linear combination of $\xi_{i}$ for $i\in\Delta_{\text{aff}}$, and this
decomposition arises from _i_) by considering
$\mathbb{Z}d\oplus\mathbb{Z}\Phi^{\vee}$ as the span of the $\xi_{i}$. (This
form helps to understand the connection with Kac diagrams.) In particular, a
given Kac diagram gives rise to infinitely many dominant coweights in
$\widehat{Y}_{\rm aff}$.

  3. _iii_)

Finally, we can express $\lambda$ as a pair $(k,\lambda_{\text{comb}})$, where
$\lambda_{\text{comb}}$ is a $\mathbb{Z}$-linear combination of the simple
coroots $\alpha_{i}^{\vee}$ for $i\in\Delta_{\text{aff}}$, subject to the
positivity conditions $\langle\alpha_{i},\lambda_{\text{comb}}\rangle\geq 0$
for $i\neq 1$ and $\langle\alpha_{1},\lambda_{\text{comb}}\rangle\geq-k$. This
arises from the decomposition
$\widehat{Y}_{\text{aff}}=\mathbb{Z}d\oplus\mathbb{Z}\Phi^{\vee}_{\text{aff}}$.
(In the notation of _i_), we have
$\mathbb{Z}\Phi^{\vee}\oplus\mathbb{Z}c=\mathbb{Z}\Phi^{\vee}_{\text{aff}}$.)

####  3.2.2 Partial order on coweights

Recall from the previous subsection that the closure order on orbits in the
affine Grassmannian induces a partial order on the dominant coweights in
${\mathfrak{g}}$, with respect to which the adjacent pairs $(\lambda,\mu)$
were classified by Stembridge [85]. We now explore the analogous partial order
for dominant coweights for the _affine_ Lie algebra. Similarly to the affine
Grassmannian, we write $\lambda\leq\mu$ if $\mu-\lambda$ is a non-negative
integer linear combination of simple roots $\alpha_{i}^{\vee}$, where
$i\in\Delta_{\text{aff}}$. We explore the statement $\mu\geq\lambda$ in each
of the settings _i_)-_iii_) above for the dominant coweights.

  * _iii_)

If $(k,\lambda_{\text{comb}})\leq(l,\mu_{\text{comb}})$ then clearly $k=l$.
Thus, there are infinitely many connected components of
$\widehat{Y}_{\text{aff}}$ with respect to the partial order (at least one for
each $k$); the partial order arises solely from its restriction to
$\mathbb{Z}\Phi_{\text{aff}}^{\vee}$. On the other hand, the meaning of
$\lambda_{\text{comb}}\leq\lambda_{\text{comb}}$ is not immediately clear.

  * _ii_)

If $(\lambda_{\text{Kac}},n)\leq(\mu_{\text{Kac}},n^{\prime})$ then it follows
from _i_) that $\lambda_{\text{Kac}}$ and $\mu_{\text{Kac}}$ are Kac diagrams
of the same order; further, we clearly have $n\leq n^{\prime}$. A
(mathematically, but not physically) trivial case is
$(\lambda_{\text{Kac}},n)<(\lambda_{\text{Kac}},n+1)$, by addition of $c$. (We
will see in section 4.1 that this minimal degeneration corresponds to
performing a tensor branch flow from a rank-$\left(n+1\right)$ E-string to a
rank-$n$ E-string.)

  * _i_)

If $\overline{\lambda}$, $\overline{\mu}$ are dominant coweights for
${\mathfrak{g}}$, both of level less than or equal to $k$, then
$(k,\overline{\lambda},n)\leq(k,\overline{\mu},n)$ if and only if
$\overline{\lambda}\leq\overline{\mu}$ in the partial order on
$\mathbb{Z}\Phi^{\vee}$. Thus (using the fact that the coroot lattice equals
the coweight lattice in type $E_{8}$), for any dominant coweight
$(k,\overline{\lambda},n)$ we have $(k,\overline{\lambda},n)\geq(k,0,n)$
(since $\overline{\lambda}$ is dominant, so is a non-negative linear
combination of the $\varpi_{i}^{\vee}$ for $i\neq 1$, hence is a non-negative
linear combination of the $\alpha_{i}^{\vee}$ too). By our remark in _ii_), it
follows that there is exactly one component of $\widehat{Y}_{\text{aff}}$ for
each positive value of $k$. This component is periodic with respect to the
$\lambda_{\text{Kac}}$ component.

####  3.2.3 Minimal degenerations for affine coweights

Let $\lambda=(\lambda_{\text{Kac}},n)$ be a dominant coweight in
$\widehat{Y}_{\text{aff}}$. As per the finite case, if $I$ is a connected
proper subdiagram of the affine Dynkin diagram then we define
$\widehat{\alpha}_{I}^{\vee}$ to be the highest coroot in the corresponding
(finite) root subsystem. As coweights, we have
$\widehat{\alpha}_{I}^{\vee}=\xi_{I}^{+}-\xi_{I}^{-}$. We have:

$\xi_{I}^{-}=\left\\{\begin{array}[]{cc}2\xi_{1}&\mbox{if $I=E_{8}$},\\\
\sum_{\text{edge}\,j-I}\xi_{j}&\mbox{otherwise},\end{array}\right.$

and $\xi_{I}^{+}=\xi_{2}$ if $I=E_{8}$ and follows exactly the same pattern as
$\varpi_{I}^{+}$ (see section 3.1.1) otherwise. We are interested in the
adjacent pairs of dominant coweights for the _affine_ Lie algebra of type
$E_{8}$. This combinatorial problem has been solved for arbitrary affine
Kac–Moody Lie algebras, by Roy [101]. The results for type $E_{8}$ (indeed for
any simply-laced case) are:

######  Theorem 3.1 (Roy).

For ${\mathfrak{g}}$ of type $E_{8}$, let $\lambda=(\lambda_{{\rm
Kac}},n)=(k,\overline{\lambda},n)$ be a dominant coweight for
$\mathfrak{g}_{\rm aff}$. If $k=1$ then $\lambda=\xi_{1}+nc$, and the only
minimal degeneration $\lambda<\mu$ is $\mu=\lambda+c$. For $k>1$, the minimal
degenerations are:

  1. i _)_

$\lambda<\lambda+\widehat{\alpha}_{I}^{\vee}$ where $I$ is a connected
component of the subdiagram of zeros of $\lambda_{\rm Kac}$;

  2. ii _)_

$\lambda<\lambda+\alpha_{i}^{\vee}$ where $\lambda(\alpha_{j})>0$ for all $j$
connected to $i$ by an edge.

Directly generalizing the cases in section 3.1.2, we will refer to these as
type _i_) and type _ii_) minimal degenerations. As an example, consider both
types in the framework of the previous subsection.

  1. _i_)

If $I$ does not contain the affine node, then
$\mu=(k,\overline{\lambda}+\widehat{\alpha}_{I}^{\vee},n)$, so this can be
identified with a degeneration in the ordinary affine Grassmannian of $E_{8}$.
If $I$ does contain the affine node then $c-\widehat{\alpha}_{I}^{\vee}$ is a
positive coroot and
$\mu=(k,\overline{\lambda}+c-\widehat{\alpha}_{I}^{\vee},n+1)$.

  2. _ii_)

If $i$ is not the affine node then
$\mu=(k,\overline{\lambda}+\alpha_{i}^{\vee},n)$. If $i$ is the affine node
then $\mu=(k,\overline{\lambda}-\widehat{\alpha}^{\vee},n+1)$.

####  3.2.4 Relationship between $N$ and $n$

The dominant coweights are the triples $(k,\overline{\lambda},n)$, with
$\overline{\lambda}$ a dominant coweight on the finite root system satisfying
$\overline{\lambda}(\widehat{\alpha})\leq k$. The subset of dominant coweights
with fixed values of $k$ and $n$ has a unique minimal element
($\overline{\lambda}=0$), hence is connected. For instance, for $k=4$ this
leads to a Hasse diagram such as [78, Fig. 12] (which is also identical to the
Higgs branch RG flow hierarchy of [127, Fig. 1] obtained via ’t Hooft anomaly
matching).

In [78, Fig. 11, Fig.s 13-25], different hierarchies were considered: in that
case the number of full and fractional instantons was kept constant. (Note
that $N_{\text{\cite[cite]{[\@@bibref{}{Fazzi:2022hal}{}{}]}}}=N_{6}\neq
N_{\text{here}}$, i.e. the number of full instantons, c.f. footnote 31.) Given
the full semi-infinite periodic poset of coweights of (_fixed_) level $k$, the
value of $n$ for a given coweight $\lambda$ is equal to the value of $n$ for
the maximal coweight $\nu\leq\lambda$ with $\overline{\nu}=0$. This
stratification (by value of $n$) is therefore very easy to understand in terms
of the combinatorics of coweights. On the other hand, we are interested in
Coulomb branches ${\mathcal{M}}_{\text{C}}(\mu,\lambda)$ with
$\overline{\mu}=0$, so, given any $\lambda$, we want to understand the minimal
coweight $\nu$ with $\overline{\nu}=0$ satisfying $\nu\geq\lambda$, i.e. we
want to find the minimum value of $n$ such that $(k,0,n)\geq\lambda$. Note
that the hierarchies in [78] each contained a unique maximal node, with Kac
diagram $[1^{k}]$, i.e. with $\overline{\lambda}=0$. (These were however
depicted with $\lambda$ at the bottom of the Hasse diagram. With this
convention, the direction of flow – i.e. smaller $a$ – is downwards; note that
the IR strata are of _larger_ dimension.)

In all of the following, integers $i^{\prime}$ are interpreted in formulas
“without primes” when required; the notation $\lceil x\rceil$, resp. $\lfloor
x\rfloor$ means the smallest integer greater than or equal to $x$, resp. the
greatest integer less than or equal to $x$.

######  Lemma 3.2.

Let $\lambda=(k,\overline{\lambda},n)$ be a dominant coweight for
${\mathfrak{g}}_{\rm aff}$. Let $n_{i}$ be the coefficients of $\lambda_{\rm
Kac}$, let $\overline{\lambda}=\sum_{i\neq 1}m_{i}\alpha_{i}^{\vee}$, and let

$N_{\rho}=\sum_{i=1}^{6}n_{i}+p\ ,\quad
p=\min\left(\left\lfloor\frac{n_{3^{\prime}}+n_{4^{\prime}}}{2}\right\rfloor,\left\lfloor\frac{n_{2^{\prime}}+n_{3^{\prime}}+2n_{4^{\prime}}}{3}\right\rfloor\right)$
(3.16)

as in (2.7). Then:

  1. a _)_

The minimum value of $m$ such that $(k,0,m)\geq(k,\overline{\lambda},n)$ is
$n+\max_{i\neq 1}\lceil\frac{m_{i}}{i}\rceil$;

  2. b _)_

in terms of the Kac coefficients, this value is $n+k-N_{\rho}$.

###### Proof.

We may clearly assume $n=0$. By the discussion in section 3.2.1, we have
$\overline{\lambda}=\sum_{i\neq 1}n_{i}\varpi_{i}^{\vee}$. Hence we have to
find the smallest value of $m$ such that $mc\geq\sum_{i\neq
1}n_{i}\varpi_{i}^{\vee}$. The equality $m=\max_{i\neq
1}\lceil\frac{m_{i}}{i}\rceil$ is clear, so we only have to prove (b). It is
easily observed from the inverse Cartan matrix that

$ic-\varpi_{i}^{\vee}=\left\\{\begin{array}[]{ll}c+\alpha_{i-1}^{\vee}+2\alpha_{i-2}^{\vee}+\ldots+(i-1)\alpha_{1}^{\vee}&\mbox{if
$i=2,\ldots,6$,}\\\
\alpha_{3^{\prime}}^{\vee}+\alpha_{4^{\prime}}^{\vee}+2\sum_{i=1}^{6}\alpha_{i}^{\vee}&\mbox{if
$i=2^{\prime}$,}\\\
\alpha_{2^{\prime}}^{\vee}+\alpha_{3^{\prime}}^{\vee}+2\alpha_{4^{\prime}}^{\vee}+3\sum_{i=1}^{6}\alpha_{i}^{\vee}&\mbox{if
$i=3^{\prime}$,}\\\
\alpha_{2^{\prime}}^{\vee}+2\alpha_{3^{\prime}}^{\vee}+2\alpha_{4^{\prime}}^{\vee}+4\sum_{i=1}^{6}\alpha_{i}^{\vee}&\mbox{if
$i=4^{\prime}$.}\end{array}\right.$

In particular, this implies that $m\leq
2n_{2^{\prime}}+3n_{3^{\prime}}+4n_{4^{\prime}}+\sum_{i=1}^{6}(i-1)n_{i}=k-\sum_{i=1}^{6}n_{i}$.
Then:

$\displaystyle\left(k-\sum_{i=1}^{6}n_{i}\right)c-\overline{\lambda}=$
$\displaystyle\
(n_{3^{\prime}}+n_{4^{\prime}})\alpha_{2^{\prime}}^{\vee}+(n_{2^{\prime}}+n_{3^{\prime}}+2n_{4^{\prime}})\alpha_{3^{\prime}}^{\vee}+(n_{2^{\prime}}+2n_{3^{\prime}}+2n_{4^{\prime}})\alpha_{4^{\prime}}^{\vee}\
+$
$\displaystyle+\sum_{i=1}^{6}\left(2n_{2^{\prime}}+3n_{3^{\prime}}+4n_{4^{\prime}}+n_{i+1}+2n_{i+2}+\ldots+(6-i)n_{6}\right)\alpha_{i}^{\vee}$
(3.17)

Note that
$\frac{n_{3^{\prime}}+n_{4^{\prime}}}{2}\leq\frac{n_{2^{\prime}}+2n_{3^{\prime}}+2n_{4^{\prime}}}{4}$
and
$\frac{n_{2^{\prime}}+n_{3^{\prime}}+2n_{4^{\prime}}}{3}\leq\frac{2n_{2^{\prime}}+3n_{3^{\prime}}+4n_{4^{\prime}}}{6}$.
It follows that the smallest value of $m$ such that $mc\geq\overline{\lambda}$
is
$k-\sum_{i=1}^{6}n_{i}-\min(\lfloor\frac{n_{3^{\prime}}+n_{4^{\prime}}}{2}\rfloor,\lfloor\frac{n_{2^{\prime}}+n_{3^{\prime}}+2n_{4^{\prime}}}{3}\rfloor)$,
which equals $k-N_{\rho}$, as required. ∎

######  Example 3.3.

  1. _a_)

Let $\lambda=(\lambda_{\text{Kac}},0)$ where $\lambda_{\text{Kac}}$ is the Kac
diagram $[3^{\prime},2^{\prime 2}]$. This can be thought of as
$\varpi_{3^{\prime}}^{\vee}+2\varpi_{2^{\prime}}^{\vee}=\xi_{3^{\prime}}+2\xi_{2^{\prime}}-7\xi_{1}$.
Then $k=7$ and $N_{\rho}=\sum_{i=1}^{6}n_{i}=p=0$. Concretely:

$\overline{\lambda}=\varpi_{3}^{\vee}+2\varpi^{\vee}_{2}=7\alpha^{\vee}_{2}+14\alpha^{\vee}_{3}+21\alpha^{\vee}_{4}+28\alpha^{\vee}_{5}+35\alpha^{\vee}_{6}+24\alpha^{\vee}_{4^{\prime}}+18\alpha^{\vee}_{3^{\prime}}+13\alpha^{\vee}_{2^{\prime}}\
,$ (3.18)

and this is the unique expression in $\mathbb{Z}\Phi^{\vee}$ for
$\overline{\lambda}$. The smallest multiple $mc$ satisfying
$mc\geq\overline{\lambda}$ is $m=7$, obtained from
$\lceil\frac{13}{2}\rceil=7$. This is clearly equal to $k-N_{\rho}$.

  2. _b_)

Consider instead $\mu=([4^{\prime},3^{\prime}],0)$. Again $k=7$ and
$\sum_{i=1}^{6}n_{i}=0$, but here $N_{\rho}=1$. We have

$\overline{\mu}=\varpi_{4^{\prime}}^{\vee}+\varpi_{3^{\prime}}^{\vee}=7\alpha_{2}^{\vee}+14\alpha_{3}^{\vee}+21\alpha_{4}^{\vee}+28\alpha_{5}^{\vee}+35\alpha_{6}^{\vee}+24\alpha_{4^{\prime}}^{\vee}+18\alpha_{3^{\prime}}^{\vee}+12\alpha_{2^{\prime}}^{\vee}\
,$ (3.19)

with only the coefficient of $2^{\prime}$ changing. (We see here that
$\overline{\mu}<\overline{\lambda}=\overline{\mu}+\alpha_{2^{\prime}}^{\vee}$
is a minimal degeneration.) In this case we can read off that $m=6$.

  3. _c_)

Finally, consider $\nu_{\rm Kac}=[3^{\prime},2^{\prime},1^{2}]$. Then $k=7$
and $p=0$ as in (a), but now $N_{\rho}=2$. We have:

$\overline{\nu}=\varpi_{3^{\prime}}^{\vee}+\varpi_{2^{\prime}}^{\vee}=5\alpha_{2}^{\vee}+10\alpha_{3}^{\vee}+15\alpha_{4}^{\vee}+20\alpha_{5}^{\vee}+25\alpha_{6}^{\vee}+17\alpha_{4^{\prime}}^{\vee}+13\alpha_{3^{\prime}}^{\vee}+9\alpha_{2^{\prime}}^{\vee}\
,$ (3.20)

and the value of $m$ is $5$ (seen from the values of $m_{6}$,
$m_{4^{\prime}}$, $m_{3^{\prime}}$ and $m_{2^{\prime}}$).

Now we note that each time the number $n+k-N_{\rho}$ increases by $1$, the
number $N_{\text{\cite[cite]{[\@@bibref{}{Fazzi:2022hal}{}{}]}}}$ _decreases_
by 1. Recalling also that
$N_{\text{\cite[cite]{[\@@bibref{}{Fazzi:2022hal}{}{}]}}}=N_{\text{here}}+N_{\rho}$,
we therefore set $N_{\text{here}}=-n$.313131In the notation of [76], we
therefore have $n=-N_{3}=-N_{\text{here}}$, but
$N_{\text{\cite[cite]{[\@@bibref{}{Fazzi:2022hal}{}{}]}}}=N_{6}$. We remark
however that the $a$ anomaly formula in [78] was expressed in terms of
$N_{3}$.

We close this section with an explanation of the difference between the
hierarchies of Higgs branch RG flows that have already appeared in [78] for
any $k$ and in [127, 110] for a few chosen values of $k$, since they use
different notions of $N$.

  * •

In [78] we have kept the total number of instantons (full and fractional),
i.e. $N_{\rho}+N$ in our present notation, fixed (for ease of presentation,
and in analogy with what was done in [76]). Then, since in those flows the
number $N_{\rho}$ of fractional instantons increases along the Higgs branch RG
flow, to compensate this increment the number $N$ of full instantons has to
decrease. So, we are performing tensor branch flows down the hierarchy. As a
result, $n$ _increases_.

In that paper we have explicitly verified that $\Delta a>0$ for all such
flows. The allowed transitions (compatibly with the $a$-theorem) between orbi-
instantons are determined via 3d magnetic quiver subtraction. Such hierarchies
can be understood as connected subdiagrams of the Hasse diagrams at fixed
level $k$ that we will present below. They are obtained at fixed values of
$n+k-N_{\rho}$, so by slicing by minimal coweight with Kac diagram
$[1^{k}]\geq\lambda$.

  * •

In [127], which only analyzed the $k=4$ case, the number $N$ of full
instantons (and consequently $n$) is held fixed (so there are no tensor branch
flows involved), and the flow is between orbi-instantons from higher to lower
number of fractional instantons, i.e. $N_{\rho}$ decreases (as well as the
total number $N+N_{\rho}$, as a consequence).

Again, in [78] we have verified that $\Delta a>0$ for this second choice. The
allowed transitions are determined via a ’t Hooft anomaly matching
analysis.323232Essentially, one subtracts from the 6d anomaly polynomial of
the UV SCFT, the one defined by Kac diagram $[1^{k}]$, the anomaly polynomial
of all other IR SCFTs (defined by the other allowed $\rho_{\infty}$’s), and
checks which subtractions are perfect squares, according to a criterion of
[136, 137]. All such subtractions represent allowed flows from UV to IR SCFTs.

  * •

Finally, [110] constructed RG flows for a few values of $k$, and both $N$ and
$N_{\rho}$ are decreased along the flow. Again, this is done via 3d magnetic
quiver subtraction.

See figure 2 for a pictorial representation when $k=4$. See figure 3 for the
actual hierarchy of RG flows in the prescription of the first, respectively
second, bullet here above.

Figure 2: The hierarchy of Higgs branch RG flows for $k=4$ from [78, Fig.
11(c)] (left) and [127, Fig. 12] (right), which are also shown in figure 3 (on
the left and right respectively), and with red and blue nodes respectively in
the right panel of fig:k34notvec. (Notice that in the above diagrams we are
disregarding branchings between flows.) In the first the value of $n$
increases (as $-n$ decreases), whereas it remains constant in the second. The
$a$ conformal anomaly decreases from top to bottom along the yellow arrow.

$\overset{(N,N_{\rho}=0)}{[4^{\prime}]\leftrightarrow\mathfrak{f}={\mathfrak{su}({8})}\oplus{\mathfrak{su}({2})}}$$\overset{(N,N_{\rho}=0)}{[2^{\prime
2}]\leftrightarrow\mathfrak{f}={\mathfrak{so}({16})}}$$\overset{(N-1,N_{\rho}=1)}{[3^{\prime},1]\leftrightarrow\mathfrak{f}={\mathfrak{su}({8})}\oplus{\mathfrak{u}({1})}}$$\overset{(N-1,N_{\rho}=1)}{[2,2^{\prime}]\leftrightarrow\mathfrak{f}={\mathfrak{so}({12})}\oplus{\mathfrak{su}({2})}\oplus{\mathfrak{u}({1})}}$$\overset{(N-1,N_{\rho}=1)}{[4]\leftrightarrow\mathfrak{f}={\mathfrak{so}({10})}\oplus{\mathfrak{su}({4})}}$$\overset{(N-2,N_{\rho}=2)}{[2^{\prime},1^{2}]\leftrightarrow\mathfrak{f}={\mathfrak{so}({14})}}$$\overset{(N-2,N_{\rho}=2)}{[3,1]\leftrightarrow\mathfrak{f}=E_{6}\oplus{\mathfrak{su}({2})}\oplus{\mathfrak{u}({1})}}$$\overset{(N-2,N_{\rho}=2)}{[2^{2}]\leftrightarrow\mathfrak{f}=E_{7}\oplus{\mathfrak{su}({2})}}$$\overset{(N-3,N_{\rho}=3)}{[2,1^{2}]\leftrightarrow\mathfrak{f}=E_{7}\oplus{\mathfrak{u}({1})}}$$\overset{(N-4,N_{\rho}=4)}{[1^{4}]\leftrightarrow\mathfrak{f}=E_{8}}$$\mathfrak{a}_{1}$$\mathfrak{a}_{7}$$\mathfrak{d}_{8}$$\mathfrak{a}_{7}$$\mathfrak{d}_{6}$$A_{1}$$\mathfrak{a}_{3}$$\mathfrak{d}_{7}$$\mathfrak{a}_{1}$$A_{1}$$A_{3}$
(a) $N+N_{\rho}$ fixed

$\overset{(N,N_{\rho}=4)}{[1^{4}]\leftrightarrow\mathfrak{f}=E_{8}}$$\overset{(N,N_{\rho}=3)}{[2,1^{2}]\leftrightarrow\mathfrak{f}=E_{7}\oplus{\mathfrak{u}({1})}}$$\overset{(N,N_{\rho}=2)}{[2^{\prime},1^{2}]\leftrightarrow\mathfrak{f}={\mathfrak{so}({14})}}$$\overset{(N,N_{\rho}=2)}{[3,1]\leftrightarrow\mathfrak{f}=E_{6}\oplus{\mathfrak{su}({2})}\oplus{\mathfrak{u}({1})}}$$\overset{(N,N_{\rho}=2)}{[2^{2}]\leftrightarrow\mathfrak{f}=E_{7}\oplus{\mathfrak{su}({2})}}$$\overset{(N,N_{\rho}=1)}{[3^{\prime},1]\leftrightarrow\mathfrak{f}={\mathfrak{su}({8})}\oplus{\mathfrak{u}({1})}}$$\overset{(N,N_{\rho}=1)}{[2,2^{\prime}]\leftrightarrow\mathfrak{f}={\mathfrak{so}({12})}\oplus{\mathfrak{su}({2})}\oplus{\mathfrak{u}({1})}}$$\overset{(N,N_{\rho}=1)}{[4]\leftrightarrow\mathfrak{f}={\mathfrak{so}({10})}\oplus{\mathfrak{su}({4})}}$$\overset{(N,N_{\rho}=0)}{[4^{\prime}]\leftrightarrow\mathfrak{f}={\mathfrak{su}({8})}\oplus{\mathfrak{su}({2})}}$$\overset{(N,N_{\rho}=0)}{[2^{\prime
2}]\leftrightarrow\mathfrak{f}={\mathfrak{so}({16})}}$$\mathfrak{e}_{8}$$\mathfrak{e}_{7}$$\mathfrak{d}_{7}$$\mathfrak{a}_{1}$$\mathfrak{e}_{6}$$\mathfrak{e}_{7}$$\mathfrak{a}_{7}$$\mathfrak{d}_{6}$$\mathfrak{d}_{5}$$\mathfrak{a}_{1}$
(b) $N$ fixed

Figure 3: Hierarchies of flavor Higgsings for $k=4$: on the left $N+N_{\rho}$
fixed (both $N_{\rho}$ and $n$ increase), on the right we are keeping only $N$
fixed (both $N_{\rho}$ and $n$ decrease).

In sum, because of the semi-infinite periodic structure of the double affine
Grassmannian’s Hasse diagram, we can decrease $N_{\rho}$ and $N$
simultaneously: within a fixed-$N$ slicing (i.e. subdiagram) of it only
$N_{\rho}$ is reduced, but we can also “transition” from an orbi-instanton at
rank $N$ (i.e. with $N$ M5-branes) to another at $N-1$, which are generically
defined by two different Kac diagrams, by performing a tensor branch flow. In
the next two sections we will construct such Hasse diagram and prove that for
both types of transitions $\Delta a>0$.

####  3.2.5 Geometry of the double affine Grassmannian

As remarked earlier, the symplectic leaves in a slice
${\mathcal{M}}_{\text{C}}(\mu,\lambda)$ in the double affine Grassmannian
(when $k>1$) are expected to be indexed by two pieces of data:

  * •

coweights $\nu$ with $\lambda\leq\nu\leq\mu$;

  * •

for each such $\nu$, and for each $M$ such that $\nu-M\delta\geq\lambda$, a
partition $[m_{i}]$ of $M$.

Let us denote by ${\mathcal{M}}^{[m_{i}]}_{\text{C}}(\nu,\lambda)$ the
symplectic leaf corresponding to $\nu$ and the partition $[m_{i}]$. Note that
we count the case $M=0$ (i.e. the trivial partition), which is exactly
${\mathcal{M}}_{\text{C}}^{\text{smooth}}(\nu,\lambda)$. The partial order on
symplectic leaves restricts to the dominance order on coweights outlined in
section 3.2.2. The dominance order also replicates some of the partial order
for nontrivial partitions: if $\lambda\leq\xi\leq\nu\leq\mu$ and
$\xi-M\delta\geq\lambda$ then

$\overline{{\mathcal{M}}_{\text{C}}^{[m_{i}]}(\nu,\lambda)}\supset{\mathcal{M}}_{\text{C}}^{[m_{i}]}(\xi,\lambda)\
.$ (3.21)

There are three further transitions generating the full Hasse diagram of
symplectic leaves in ${\mathcal{M}}_{\text{C}}(\mu,\lambda)$, as we now
explain.

  1. _i_)

Firstly, let us subdivide the partition $[m_{i}]$ into two subpartitions
$[m_{i}^{(1)}]$, $[m_{j}^{(2)}]$ of $M_{1}$, $M_{2}$ respectively (with
$M_{1}+M_{2}=M$). Since $\nu-M\delta\geq\lambda$ implies a fortiori that $\nu-
M_{1}\delta\geq\lambda$, then the parts $m_{j}^{(2)}$ can be deleted,
producing the symplectic leaf
${\mathcal{M}}_{\text{C}}^{[m_{i}^{(1)}]}(\nu,\lambda)$. The closure of this
symplectic leaf is expected to contain
${\mathcal{M}}_{\text{C}}^{[m_{i}]}(\nu,\lambda)$. This aspect of the partial
order was considered in depth in [107, 102] (see e.g. [102, Fig. 5(a)]).

  2. _ii_)

On the other hand, one also has $(\nu-M_{2}\delta)-M_{1}\delta\geq\lambda$,
hence there is a symplectic leaf
${\mathcal{M}}_{\text{C}}^{[m_{i}^{(1)}]}(\nu-M_{2}\delta,\lambda)$. Any such
stratum is in the closure of
${\mathcal{M}}_{\text{C}}^{[m_{i}]}(\nu,\lambda)$.

  3. _iii_)

Finally, if we fix the coweight $\nu$ and the integer $M$ such that
$\nu-M\delta\geq\lambda$, then the strata are in one-to-one correspondence
with the partitions of $M$. Joining any of the parts of the partition
$[m_{i}]$ together (always possible unless $[m_{i}]=[M]$), one obtains a
coarser partition $[m^{\prime}_{i}]$ of $M$; then the closure of
${\mathcal{M}}_{\text{C}}^{[m_{i}]}(\nu,\lambda)$ contains
${\mathcal{M}}_{\text{C}}^{[m^{\prime}_{i}]}(\nu,\lambda)$.

Considering the last two types of transition, and taking all pairs
$(\nu-l\delta,[m^{\prime}_{i}])$ where $[m^{\prime}_{i}]$ is a partition of
$(M-l)$, we obtain a subhierarchy similar to [102, Fig. 5(b)]. In fact, a
transverse slice from
${\mathcal{M}}_{\text{C}}^{\text{smooth}}(\nu-M\delta,\lambda)$ to
$\overline{{\mathcal{M}}_{\text{C}}^{[1^{M}]}(\nu,\lambda)}$ is expected to be
isomorphic to $\operatorname{Sym}^{M}({\mathbb{C}}^{2}/{\mathbb{Z}}_{k})$.
This symplectic singularity is well understood (for example, it is known to
have a symplectic resolution).

The above considerations lead us to the classification of minimal
degenerations of symplectic leaves (i.e. edges in the Hasse diagram),
according to four basic types. We can also describe the associated
singularities. Recalling that the slice from
${\mathcal{M}}_{\text{C}}^{[m_{i}]}(\nu,\lambda)$ is independent of $\lambda$,
it follows that the singularities associated to each minimal degeneration are
also independent of the choice of $\lambda$.

In the following, $[m_{i}]$ is a partition of $M$, with the parts taken in any
order:

  1. _i_)

Degenerations
$\overline{{\mathcal{M}}_{\text{C}}^{[m_{i}]}(\nu,\lambda)}\supset{\mathcal{M}}_{\text{C}}^{[m_{i}]}(\xi,\lambda)$,
where $\nu>\xi$ is a minimal degeneration of coweights and
$\xi-M\delta\geq\lambda$. The singularity can be determined by an extension of
the algorithm of [111] in finite types, or using quiver subtraction: if
$\nu=\xi+\widehat{\alpha}_{I}^{\vee}$ where $I$ is a connected subdiagram of
the affine Dynkin diagram, then we obtain a minimal singularity of type $I$;
if $\nu=\xi+\alpha_{i}^{\vee}$ then we obtain the surface singularity
${\mathbb{C}}^{2}/{\mathbb{Z}}_{n_{i}+2}$.

  2. _ii_)

Degenerations
$\overline{{\mathcal{M}}_{\text{C}}^{[m_{i}]}(\nu,\lambda)}\supset{\mathcal{M}}_{\text{C}}^{[m_{i},1]}(\nu,\lambda)$,
where $\nu-(M+1)\delta\geq\lambda$. The singularity is a union of $l$ copies
(branches) of the minimal singularity ${\mathfrak{e}}_{8}$, where $l$ is the
multiplicity of $1$ in the partition $[m_{i}]$ (see e.g. [102, Fig. 5(a)]).
(Note the presence of branching, which does not occur in _i)_.)

  3. _iii_)

Degenerations
$\overline{{\mathcal{M}}_{\text{C}}^{[m_{i}]}(\nu,\lambda)}\supset{\mathcal{M}}_{\text{C}}^{[\ldots,m_{i-1},m_{i+1},\ldots]}(\nu-
m_{i}\delta,\lambda)$ (including the case
$\overline{{\mathcal{M}}_{\text{C}}^{[M]}(\nu,\lambda)}\supset{\mathcal{M}}_{\text{C}}^{\text{smooth}}(\nu-M\delta,\lambda)$).
The singularity in all such cases is ${\mathbb{C}}^{2}/{\mathbb{Z}}_{k}$
(where $k$ is the level of $\lambda$).

  4. _iv_)

Degenerations
$\overline{{\mathcal{M}}_{\text{C}}^{[m_{i}]}(\nu,\lambda)}\supset{\mathcal{M}}_{\text{C}}^{[\ldots,m_{i}+m_{i+1},\ldots]}(\nu,\lambda)$
(where there exist at least two parts of the partition $[m_{i}]$). Let $l$ be
the multiplicity of $m_{i}+m_{i+1}$ in $[\ldots,m_{i}+m_{i+1},\ldots]$. Then
the singularity associated to this degeneration has $l$ isomorphic branches:
these are ${\mathbb{C}}^{2}/{\mathbb{Z}}_{2}$ if $m_{i}=m_{i+1}$ and are the
non-normal singularity $m$ from [109] otherwise.

Note that a mathematical proof of these results depends on extending [95, Thm.
7.26] to affine $E_{8}$. However, the classification of the singularities is
not needed for our proof of the $a$-theorem.

##  4 Hierarchy of flows as Hasse of dominant coweights

We can now bring the abstract lessons learned in the previous section to
fruition, and construct the hierarchy of flavor Higgsings among A-type orbi-
instantons as the Hasse diagram of strata in the double affine Grassmannian of
$E_{8}$ corresponding to dominant coweights $\lambda,\mu$ connected by minimal
degenerations $\lambda<\mu$ (transverse slices). We will do so explicitly for
$k=1,\ldots,7$, even though it is clear the algorithm presented above can be
applied at any arbitrary level $k$.

###  4.1 $k=1$

The $k=1$ case is somewhat degenerate and deserves a separate discussion.
(Since there is no orbifold to begin with, the theories in this class are just
E-strings.) It falls into the class of diagrams studied in point _ii_) of
section 3.2.2. The only allowed diagram is $\lambda_{\text{Kac}}=[1]$ so
$(\lambda_{\text{Kac}},n)=([1],n)$ is the only dominant coweight at level
$k=1$. Therefore the only possible minimal degeneration between diagrams is
$([1],n)<([1],n+1)$, and is of type $\mathfrak{e}_{8}$ according to theorem
3.1. That is, it is a minimal $\mathfrak{e}_{8}$ singularity, which is the
closure of the minimal nilpotent orbit of the $E_{8}$ algebra, which has
$\dim_{\mathbb{H}}\overline{\min_{E_{8}}}=29$ and $E_{8}$ isometry as a
hyperkähler space. $N_{\rho}=1$ and the number $N$ of full instantons is given
by $n=-N$. Therefore, we can equivalently label the dominant coweights by $N$.
Suppose we start with $([1],N)$, then each $\mathfrak{e}_{8}$ degeneration
removes a 1, i.e. we land on $([1],N-1)$, and so on. That is, this flow is a
tensor branch one. (From the geometric perspective, this was already analyzed
in [137]; from the 6d anomaly polynomial perspective, in [136].)

On the other hand, we could also have Higgs branch flows corresponding to
separating vertically (along the M9) some of the M5’s, or even dissolving some
(or all) back into flux. These transitions are still captured by the
degenerations in section 3.2.5.

Focusing on the tensor branch flows, the double affine Grassmannian at level 1
has a “degenerate” Hasse (sub)diagram:

$\cdots\xrightarrow{\mathfrak{e}_{8}}([1],N+1)\xrightarrow{\mathfrak{e}_{8}}([1],N)\xrightarrow{\mathfrak{e}_{8}}([1],N-1)\xrightarrow{\mathfrak{e}_{8}}\cdots\xrightarrow{\mathfrak{e}_{8}}([1],1)\
.$ (4.1)

Just as easily we can understand the transverse slices of type
$\mathfrak{e}_{8}$ from a 3d Coulomb branch, or 6d Higgs branch, perspective.

If $k=1$ there is no orbifold in the first place: the $N$ M5’s are just
probing the M9, and this engineers a rank-$N$ E-string rather than an orbi-
instanton, with electric quiver

$[E_{8}]\underbrace{12\cdots 2}_{N}\ .$ (4.2)

Each tensor branch flow contracts the $-1$ curve to zero size, and F-theory
requires [138] that the adjacent $-2$ in turn blows down to $-1$, leaving
behind a rank-$(N-1)$ E-string. This process can be iterated all the way down
to $[E_{8}]1$, the rank-1 E-string corresponding to $([1],1)$, or even to
empty theory corresponding to $([1],0)$ (i.e. with 0 M5’s).

What does this correspondence mean in practice? If we include the center-of-
mass hypermultiplet degrees of freedom, the Higgs branch of the rank-$N$
E-string has $\dim_{\mathbb{H}}\text{HB}=30N$, which for $N=1$ gives
$30=29+1=\dim_{\mathbb{H}}\overline{\min_{E_{8}}}+\dim_{\mathbb{H}}\mathbb{H}$.
This agrees with the dimension of the Coulomb branch of the magnetic quiver
for the “$k=1$ $N=1$ orbi-instanton” (i.e. the rank-1 E-string) in [132, Eq.
(2.20)],333333This quiver is the “over-extended” affine $E_{8}$ Dynkin quiver
introduced in [139, 140].

$1-1-2-3-4-5-\overset{\underset{|}{\displaystyle 3}}{6}-4-2\ ,$ (4.3)

given by the total sum of the gauge ranks minus 1. (See appendix A for general
definitions and results on magnetic quivers, in particular the dimension
formula in (A.8), when $k>1$.) This is not a coincidence, as the Coulomb
branch of the above quiver is by construction the same as the Higgs branch of
the E-string SCFT (via compactification on $T^{3}$ followed by mirror
symmetry). Decoupling the center-of-mass hypermultiplet,343434A free
hypermultiplet, corresponding to the center-of-mass motion of the $N$ M5’s
decouples from the dynamics. It corresponds to a translational mode of the
codimension-4 instantons on $\mathbb{C}^{2}$ [136]. the Higgs branch of the
_interacting_ part of the E-string theory is given by the reduced (i.e.
centered) moduli space of $N$ $E_{8}$-instantons on $\mathbb{C}^{2}$ [141],
i.e. $\dim_{\mathbb{H}}\text{HB}_{\text{E-str}}=30N-1$. For $N=1$, this has
quaternionic dimension 29, as does $\overline{\min_{E_{8}}}$ or the Coulomb
branch of

$1-2-3-4-5-\overset{\underset{|}{\displaystyle 3}}{6}-4-2\ ,$ (4.4)

which is the magnetic quiver originally proposed in [121, Eq.
(2.27)].353535Here
$k_{\text{\cite[cite]{[\@@bibref{}{Hanany:2018uhm}{}{}]}}}=4$.

In other words, in the language of (1.6)-(1.7) there are no flavor Higgsings
(slices between dominant coweights at level $k$) but only
$\prescript{\text{c}}{}{\mathcal{U}_{1}}$ slices between the stratum of the
symmetric product of $N$ objects (M5’s) identified with the partition $[N]$
and the analog stratum for $N-1$ objects. This Higgs branch flow is an
instanton transition dissolving one M5 (or more) into flux. Notice that these
symmetric products have nontrivial Hasse diagrams themselves, corresponding to
separating the $N$ M5’s into substacks (according to a partition $[n_{i}]$ of
$N$). Here, we are neglecting slices between strata of a given symmetric
product (i.e. different partitions $[n_{i}]$) as well as slices between strata
of two different symmetric products except for the one identified with
$\prescript{\text{c}}{}{\mathcal{U}_{1}}$. (The schematic form of the full
Higgs branch of the rank-$N$ E-string can be seen in [108, Fig. 3.6] and was
given already in [120, Fig. 45] for $N=2$.363636The full semi-infinite
periodic Hasse diagram should also be recoverable using the decorated quiver
subtraction algorithm of [107, 102], though the complexity of this procedure
increases dramatically as we increase $k$, as we should also include slices
among the symmetric product strata themselves, corresponding to the
“reunification” of substacks of separated M5’s into larger and larger
substacks. We would like to thank Antoine Bourget and Julius Grimminger for
illuminating discussions on this point. The new algorithm of [142, 143] should
remedy this situation.)

Finally, for $k=1$ we can already give a direct computational proof of the
$a$-theorem, given the simplicity of the transverse slices between the strata
considered. This is done as follows (we repeat the argument of [66] for the
convenience of the reader). The eight-form 6d anomaly polynomial of the
rank-$N$ E-string plus free hypermultiplet $I=I_{\text{E-str}}+I_{\text{free
hyper}}$ was computed in [141] (to which we refer the reader for the
notation), and reads:

$\displaystyle I$
$\displaystyle=\frac{4N^{3}+6N^{2}+3N}{24}c_{2}(R)^{2}-\frac{6N^{2}+5N}{48}p_{1}(T)c_{2}(R)+\frac{7N}{192}p_{1}(T)^{2}-\frac{N}{48}p_{2}(T)\
,$ (4.5) $\displaystyle I_{\text{E-str}}$
$\displaystyle=\frac{4N^{3}+6N^{2}+3N}{24}c_{2}(R)^{2}-\frac{6N^{2}+5N}{48}p_{1}(T)c_{2}(R)$
$\displaystyle\quad+\frac{7(30N-1)}{5760}p_{1}(T)^{2}-\frac{120N-4}{5760}p_{2}(T)\
,$ (4.6) $\displaystyle I_{\text{free hyper}}$
$\displaystyle=\frac{7p_{1}(T)^{2}-4p_{2}(T)}{{5760}}\ .$ (4.7)

In particular, from the relation
$I\supset\dim_{\mathbb{H}}\text{HB}\,\frac{7p_{1}(T)^{2}-4p_{2}(T)}{5760}$ one
can read off the dimension of the Higgs branch of the theory, since
$\frac{7p_{1}(T)^{2}-4p_{2}(T)}{5760}$ is the contribution to the 6d anomaly
polynomial of a single hypermultiplet. Then, (4.5) gives
$\dim_{\mathbb{H}}\text{HB}=30N$, while (4.6) gives
$\dim_{\mathbb{H}}\text{HB}=30N-1$, as expected. The $a$ anomaly can now be
computed applying the well-known relations [66, Eqs. (1.6) & (1.7)] to (4.5),
yielding [66, Eq. (5.2)], namely:

$\displaystyle a_{\text{E-str $+$ free hyper}}(N)$
$\displaystyle=\frac{64}{7}N^{3}+\frac{144}{7}N^{2}+\frac{99}{7}N\ ;$ (4.8)
$\displaystyle a_{\text{E-str}}(N)$
$\displaystyle=\frac{64}{7}N^{3}+\frac{144}{7}N^{2}+\frac{99}{7}N-\frac{11}{7\cdot
30}\ .$ (4.9)

It is easy to verify the $a$-theorem for tensor branch flows:

$\displaystyle\Delta a_{\text{tensor}}$ $\displaystyle=a_{\text{E-str $+$ free
hyper}}(N+1)-a_{\text{E-str $+$ free
hyper}}(N)=a_{\text{E-str}}(N+1)-a_{\text{E-str}}(N)$
$\displaystyle=\frac{192}{7}N^{2}+\frac{480}{7}N+\frac{307}{7}>0$ (4.10)

for any $N\geq 1$. Similarly, for a Higgs branch flow dissolving $M$ M5’s into
the M9 we have

$\displaystyle\Delta a_{\text{Higgs}}$ $\displaystyle=a_{\text{E-str $+$ free
hyper}}(N)-30Ma_{\text{free hyper}}=a_{\text{E-str}}(N)-29Ma_{\text{free
hyper}}$ (4.11)
$\displaystyle=\frac{64}{7}N^{3}+\frac{144}{7}N^{2}+\frac{99}{7}N-29\frac{11}{7\cdot
30}M\ ,$ (4.12)

which is positive for

$\frac{1920}{319}N^{3}+\frac{4320}{319}N^{2}+\frac{270}{29}N>M\ ,$ (4.13)

i.e. for any $0\leq M\leq N$.

###  4.2 $k=2$

The $k=2$ case is the first for which actual orbi-instantons exist (i.e. we
have a $\mathbb{C}^{2}/\mathbb{Z}_{2}$ orbifold in M-theory).

In figure 4 we can see the repeating pattern of the semi-infinite Hasse
diagram: in red we have highlighted the hierarchy of RG flows at fixed number
$N$ of full instantons,373737The red slicing can also be seen in the leftmost
column of [108, Fig. 3.1], when read from bottom to top. (That is, the $a$
anomaly decreases when climbing _up_ that diagram: the top leaf is the
largest, the bottom one the smallest, and the opposite is true for the
associated $a$ anomalies.) Only the bottom three entries in that column
correspond to orbi-instantons, namely Kac diagrams $[1^{2}],[2],[2^{\prime}]$
respectively, with the others corresponding to either (decoupled collections
of) E-strings (indicated by $\sqcup$) or the empty theory (top entry). whereas
in blue the hierarchy at fixed total number of instantons (full plus
fractional), which coincides with [78, Fig. 11(a)].

${\scriptstyle([1^{2}],N)}$
---
${\scriptstyle([2],N)}$
---
${\scriptstyle([2^{\prime}],N)}$
---
${\scriptstyle([2],N-1)}$
---
${\scriptstyle([1^{2}],N-1)}$
---
${\scriptstyle([2^{\prime}],N-1)}$
---
${\scriptstyle([1^{2}],N-2)}$
---
${\scriptstyle([2^{\prime}],N-2)}$
---
${\scriptstyle([2],N-2)}$
---
$\mathfrak{e}_{8}$
---
$\mathfrak{e}_{7}$
---
$\mathfrak{d}_{8}$
---
$A_{1}$
---
$\mathfrak{d}_{8}$
---
$A_{1}$
---
$\mathfrak{e}_{8}$
---
$\mathfrak{e}_{7}$
---
$\mathfrak{e}_{8}$
---
$\mathfrak{e}_{7}$
---
Figure 4: A cutout from the semi-infinite periodic Hasse diagram of dominant
coweights at level $k=2$, that is the hierarchy of flavor Higgsings between
orbi-instantons for $k=2$ and reducing numbers of full ($N$) and fractional
($N_{\rho}$) instantons. The labels on the transitions are the minimal
degeneration types from section 3, and correspond to the subtraction of the 3d
magnetic quivers associated with the UV and IR orbi-instantons. The $a$
conformal anomaly decreases along all allowed oriented paths (RG flows).

###  4.3 $k=3,\ldots,7$

In this section we simply showcase the Hasse diagrams of dominant coweights of
affine $E_{8}$ at level $k=3,\ldots,7$. Once again, the $a$ anomaly decreases
along any allowed path, and by slicing at fixed $N+N_{\rho}$ we recover the
hierarchies of [78].

(a) $k=3$ (b) $k=4$

Figure 5: A cutout from the semi-infinite periodic Hasse diagram at level
$k=3,4$. The magenta node is shared between blue and red slicing (connected
subdiagram). The blue slicing coincides with [78, Fig. 11(b,c)].

(a) $k=5$ (b) $k=6$

Figure 6: A cutout from the semi-infinite periodic Hasse diagram at level
$k=5,6$. The magenta node is shared between blue and red slicing (connected
subdiagram). The blue slicing coincides with [78, Fig. 11(d), Fig. 3].

Figure 7: A cutout from the semi-infinite periodic Hasse diagram at level
$k=7$. The full diagram is too long to present here, so it has been cut into
three pieces here corresponding to (a), (b), (c), with each beginning where
the previous ends. Some overlapping regions have been presented for clarity.
The magenta node is shared between blue and red slicing (connected
subdiagram). The blue slicing coincides with [78, Fig. 13].

##  5 Proving the $a$-theorem

We are finally in a position to prove the 6d $a$-theorem for Higgs branch RG
flows of the flavor Higgsing type between A-type orbi-instantons. To prove
that $\Delta a>0$ for all allowed flows we use the partial order on the Hasse
diagram of dominant coweights of affine $E_{8}$, and express $a$ in terms of
combinatorial data of the latter.

###  5.1 The exact $a$ anomaly of A-type orbi-instantons

We retain all the notation
$\lambda=(\lambda_{\text{Kac}},n)=(k,\overline{\lambda},n)$, $n_{i}$,
$N_{\rho}$ (defined in Lemma 3.2) and the relation $N=-n$ determined in
section 3.2.4. In particular, the condition that the total number of full and
fractional instantons is at least equal to $k$ is given in our notation by
$n\leq N_{\rho}-k$. Recall that there is a bilinear form $\langle.\,,.\rangle$
on $Y(T)$ which is invariant with respect to the Weyl group and which
satisfies
$\langle\overline{\lambda},\alpha^{\vee}\rangle=\overline{\lambda}(\alpha)$
for any root $\alpha$. For distinct simple coroots
$\alpha_{i}^{\vee},\alpha_{j}^{\vee}$, we therefore have
$\langle\alpha_{i}^{\vee},\alpha_{i}^{\vee}\rangle=2$ and
$\langle\alpha_{i}^{\vee},\alpha_{j}^{\vee}\rangle$ is $-1$ if $i-j$ is an
edge of the Dynkin diagram, and is zero otherwise. Expressing
$\overline{\lambda}$ as $\sum_{i\in\Delta}b_{i}\alpha_{i}^{\vee}$, we
therefore have:

$\langle\overline{\lambda},\overline{\lambda}\rangle=2(b_{2}^{2}-b_{2}b_{3}+b_{3}^{2}-\ldots+b_{6}^{2}-b_{6}b_{3^{\prime}}+b_{3^{\prime}}^{2}-b_{6}b_{4^{\prime}}+b_{4^{\prime}}^{2}-b_{4^{\prime}}b_{2^{\prime}}+b_{2^{\prime}}^{2})\
.$ (5.1)

The quadratic form $\langle\overline{\lambda},\overline{\lambda}\rangle$ is
the length function; it is simply the bilinear form obtained from the Cartan
matrix for $E_{8}$. When
$\overline{\lambda}=\sum_{i\in\Delta}n_{i}\varpi_{i}^{\vee}$, then the length
function is obtained from the inverse Cartan matrix. One important property is
that $\langle\overline{\lambda},\overline{\lambda}\rangle=2$ if and only if
$\overline{\lambda}$ is a coroot.

An equally important function is obtained from the Weyl vector
$\rho=\frac{1}{2}\sum_{\alpha\in\Phi^{+}}\alpha=\sum_{i\in\Delta}\varpi_{i}\in
X(T)$. This satisfies $\alpha_{i}^{\vee}(\rho)=1$ for all simple coroots
$\alpha_{i}^{\vee}$, hence we obtain the height function:
$\overline{\lambda}(\rho)=\sum_{i\in\Delta}b_{i}$ where
$\overline{\lambda}=\sum_{i\in\Delta}b_{i}\alpha_{i}^{\vee}$.

The $a$ anomaly of orbi-instantons can be extracted from the anomaly inflow
calculation in [76, Sec. 3.4]. In our notation, i.e. when expressed in terms
of the data $\overline{\lambda},k,n$, it reads:

$\begin{split}a=&-\frac{3}{2}\overline{\lambda}(\rho)-\frac{16}{7}\langle\overline{\lambda},\overline{\lambda}\rangle\Big{[}-15n+k\left(6n^{2}-6kn+2k^{2}+13\right)\Big{]}+\frac{48}{7}\langle\overline{\lambda},\overline{\lambda}\rangle^{2}\Big{[}k-n\Big{]}\\\
&+\frac{1}{420}\Big{[}-502+14415k+251k^{2}+3680k^{3}+576k^{5}\\\
&-(18900+12960k^{2}+2880k^{4})n+2880k\left(5+2k^{2}\right)n^{2}-3840k^{2}n^{3}\\\
&+400\sum_{{\alpha}\in\Phi^{+}}\lambda(\alpha)^{3}-96\sum_{\alpha\in\Phi^{+}}\overline{\lambda}(\alpha)^{5}\Big{]}.\
\end{split}$ (5.2)

Since $N=-n$, we see that $a$ can only depend on $N$ and the choice of Kac
diagram $\lambda_{\text{Kac}}=(k,\overline{\lambda})$, and its large-$N$
leading term ($\tfrac{64}{7}N^{3}k^{2}$) scales like $N^{3}$ and is universal,
as it should.383838See [73, 74] for analogous calculations in T-brane
theories. This leading term was also obtained via holography in [144, 74]. For
example, it is straightforward to verify that for $k=1$ (which implies
$\overline{\lambda}=0,\overline{\lambda}(\rho)=\langle\overline{\lambda},\overline{\lambda}\rangle=0$)
the expression in (5.2) correctly reduces to (4.8).

###  5.2 The proof