††institutetext: Department of Applied Mathematics and Theoretical Physics
(DAMTP)††institutetext: University of Cambridge, Wilberforce Road, CB3 0WA,
Cambridge, UK

# Moore-Tachikawa Varieties: Beyond Duality

Veronica Pasquarella<EMAIL_ADDRESS>
###### Abstract

We propose a generalisation of the Moore-Tachikawa varieties for the case in
which the target category of the 2D TFT is a hyperk$\ddot{\text{a}}$hler
quotient. The setup requires generalising the bordism operators of Moore and
Segal to the case involving lack of reparametrisation-invariance on the
Riemann surface, ultimately enabling to relate this to the issue of defining a
Drinfeld center for composite class ${\cal S}$ theories.

###### Keywords:

cochain level theories, symplectic varieties, topological field theories,
categorical duality, quiver gauge theories, higher categories, relative QFTs

##  1 Introduction

The present article is the first of a series by the same author meant to
provide a mathematical explanation of the claims made in two previous works,
Pasquarella:2023deo ; Pasquarella:2023exd . In doing so, we highlight
interesting connections with theoretical physics results, quoting part of the
most relevant literature wherever possible.

Our main aim is understanding the underlining mathematical structure
associated to the partition function, correlation functions, and spectrum of
operators of a given quantum field theory (QFT). Specifically, we focus on
supersymmetric gauge theories obtained by dimensional reduction of 6D ${\cal
N}=(2,0)$ SCFTs, Witten:1995zh ; Witten:2009at ; DBZ ; Witten:2007ct ; Moore .

Figure 1: Partial reproduction of a diagram displayed in Moore . The first
part of our treatment focuses on the functorial field theory description of
class ${\cal S}$ theories and their Higgs branches in terms of 2D TFT
cobordism constructions.

As explained in Pasquarella:2023deo ; Pasquarella:2023exd , for a theory,
${\cal T}$, to be absolute, the following triple needs to be defined

$\boxed{\ \ \ {\cal T}\ \longleftrightarrow\ ({\cal
F},\mu,\mathfrak{Z})\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bigg{]}\
},$ (1)

where ${\cal F}$ is the fiber functor, $\mu$ the moment map, and
$\mathfrak{Z}$ the Drinfeld center of a given theory111We refer to
Pasquarella:2023deo for a detailed explanation of this terminology..
Consistency of the underlying mathematical structure requires these three
quantities to be mutually related. Indeed, upon defining any one of them, the
other two should automatically follow. The purpose of Pasquarella:2023deo ;
Pasquarella:2023exd and the present work is to show that an apparent
shortcoming in defining such triple corresponds to the emergence of
interesting physics, rather than being a fault of the sought after absolute
theory. In particular, we will show that this can be used to explain the
emergence of non-invertible symmetries separating different class ${\cal S}$
theories, Pasquarella:2023deo ; Bashmakov:2022jtl ; Bashmakov:2022uek ; 9 ;
Kaidi:2021xfk ; Choi:2022zal ; Choi:2021kmx .

The crucial references we rely upon are the works of Moore and Segal,
Moore:2006dw , and Moore and Tachikawa, Moore:2011ee . As briefly reviewed in
the following sections, Moore:2011ee proposes a redefinition of class ${\cal
S}$ theories (cf. figure 1) in terms of a 2D TFT, namely the functor

$\boxed{\ \ \ \eta_{{}_{G_{\mathbb{C}}}}:\ \text{Bo}_{{}_{2}}\ \rightarrow\
\text{HS}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bigg{]}\
\ \ }$ (2)

with Bo${}_{{}_{2}}$ and HS denoting the bordism 2-category and the
holomorphic symplectic 2-category, respectively222For more details, we refer
the reader to section 3., associated to a given 4D ${\cal N}=2$ SCFT. The
definition of (2) strongly relies upon assuming, both, the source and target
categories, enjoy a duality structure which, in turn follows from the presence
of an identity element in both categories. In Moore:2011ee , the authors show
that, under the duality assumption, for the categories in (2) to be well-
defined, it is enough to specify their objects and 1-morphisms. Essentially,
the objects of Bo${}_{{}_{2}}$ are circles, $S^{{}^{1}}$, and the 1-morphisms
are cobordisms between different disjoint unions of circles and the empty set.
Their respective counterpart on the holomorphic symplectic side correspond to
the gauge group, Moore:2011ee ,

$\eta_{{}_{G_{{}_{\mathbb{C}}}}}\ \left(S^{{}^{1}}\right)\ \overset{def.}{=}\
G_{{}_{\mathbb{C}}},$ (3)

and the cobordism operators, Moore:2011ee ,

$\eta_{{}_{G_{{}_{\mathbb{C}}}}}\ \left(\text{Hom}\
\left(S^{{}^{1}},\emptyset\right)\right)\ \overset{def.}{=}\
U_{{}_{G_{{}_{\mathbb{C}}}}}$ (4) $\eta_{{}_{G_{{}_{\mathbb{C}}}}}\
\left(\text{Hom}\ \left(S^{{}^{1}}\ \sqcup\
S^{{}^{1}},\emptyset\right)\right)\ \overset{def.}{=}\
V_{{}_{G_{{}_{\mathbb{C}}}}}$ (5)

$\eta_{{}_{G_{{}_{\mathbb{C}}}}}\ \left(\text{Hom}\ \left(S^{{}^{1}}\ \sqcup\
S^{{}^{1}}\ \sqcup\ S^{{}^{1}},\emptyset\right)\right)\ \overset{def.}{=}\
W_{{}_{G_{{}_{\mathbb{C}}}}}.$ (6)

respectively.

Importantly for us, the Moore-Tachikawa varieties described by (3), (4), (5),
and (6) constitute the quiver gauge theory333Among the cobordism operators is
the Higgs branch of class ${\cal S}$ theories, (6). realisation of
Moore:2006dw , where Moore and Segal propose the mathematical formalism needed
for addressing the following question: given a certain closed string theory
background, what is its corresponding D-brane content444Note that this is
essentially the same question addressed in Aharony:2013hda .?

Our aim is that of explaining how and why one needs to generalise the
construction of Moore:2011ee from a higher-categorical point of view, and what
its implications are on the theoretical physics side. In doing so, we
highlight the crucial properties and axioms satisfied by the cobordism
operators outlined in Moore:2006dw ; Moore:2011ee , and how they can be
generalised to account for more general setups from, both, the mathematical
and theoretical physics perspectives.

In particular, we emphasise the dependence of (2) on the conformal structure
of the Riemann surface on which the compactification of the 6D ${\cal
N}=(2,0)$ SCFT has been performed to achieve a certain class ${\cal S}$ theory
and how lack of reparametrisation invariance, corresponding to the absence of
the identity element in its source and target categories, Moore:2011ee ,
signals the presence of (non-invertible) categorical symmetries separating
different absolute theories.

At the heart of this is the correspondence sketched in figure 2.

Figure 2: Adaptation of a correspondence first proposed in Pasquarella:2023deo
playing a key role towards generalising Moore:2011ee to the
hyperk$\ddot{\text{a}}$hler target category case. As explained in section 4,
this also requires the generalisation of cobordism operators, Moore:2006dw ,
due to the lack of reparametrisation-invariance of the Riemann surface on
which the compactification of the 6D ${\cal N}=(2,0)$ SCFT is performed.

In Pasquarella:2023deo we explained how gauging a Symmetry Topological Field
Theory (SymTFT) enables to change the boundary conditions of the fields living
in the absolute field theory resulting from the Freed-Moore-Teleman
construction, Freed:2012bs ; Freed:2022qnc ; Freed:2022iao ; TJF ;
Kong:2013aya ; Yu:2021zmu . To each gauging corresponds a choice of triples,
(2), and, for any absolute theory, it is enough to define one of the three
entries on the RHS of (2) to determine the other two. For the purpose of this
article, we will mostly focus on the second, namely the moment map, defined as
follows

$\mu:\ \mathfrak{G}\ \rightarrow\ {\cal A},$ (7)

where $\mathfrak{G}$ is an n-categorical structure, and ${\cal A}$ is the
algebra of invertible topological defects associated to the action $\mu\
(\mathfrak{G})$. Gauging means taking the categorical quotient with respect to
$\mathfrak{G}$ (in notation $//_{{}_{\mu}}\ \mathfrak{G}$), and projecting its
image under $\mu$ to the identity555We thank Nathan Seiberg for instructive
discussion regarding the appropriateness of the terminology to be adopted in
describing this formalism.. Practically, one could perform a total gauging of
the theory by choosing ${\cal A}$ such that the overall spectrum of the theory
of the gauged theory is only the (new) identity element. For the purpose of
our work, instead, we are interested in understanding mathematical structures
arising by gauging with respect to different subalgebras within ${\cal A}$,
that are mutually intersecting, albeit not contained within each other. The
ultimate aim is that of explaining the emergence of (non-invertible)
categorical symmetries in certain supersymmetric quiver gauge theories once
described in terms of Coulomb branches of magnetic quivers of 3D ${\cal N}=4$
quiver gauge theories, which is the main focus of an upcoming paper by the
same author, VP . In such analysis we will be applying some of the findings of
Braverman:2017ofm ; Benini:2010uu .

This article is structured as follows: section 2 is devoted to a brief
overview of cochain level theories as the most important generalisation of the
open and closed TFT construction. Mostly relying upon Moore:2006dw , we
highlight the importance of the construction of the cobordism operator,
highlighting its dependence on the conformal structure of the Riemann surface.
In section 3 we then turn to the discussion of a particular 2D TFT valued in a
symmetric monoidal category, namely the maximal dimension Higgs branch of
class ${\cal S}$ theories. After briefly reviewing the properties outlined in
Moore:2011ee , in section 4 we propose their generalisation for the case in
which the target category of the $\eta_{{}_{G_{{}_{\mathbb{C}}}}}$ functor is
a hyperk$\ddot{\text{a}}$hler quotient. We conclude outlining the possible
extension of this treatment towards a mathematical formulation of magnetic
quivers within the context of Coulomb branches of 3D ${\cal N}=4$ quiver gauge
theories which will be addressed in an upcoming work by the same author, VP .

##  2 Cochain level theories

This first section is devoted to a brief overview of cochain level theories as
the most important generalisation of the open and closed TFT construction. The
reason for doing so is that these mathematical structures are central to the
idea of D-branes, Moore:2006dw , enabling to determine the set of possible
D-branes given a closed string background. In their work, Moore:2006dw address
this problem from the point of view of a 2D TFT, (2), where the whole content
of the theory is encoded in a finite-dimensional commutative Frobenius
algebra.

The present section is therefore structured as follows:

  1. 1.

At first, we briefly overview cochain complexes as the essential mathematical
tools needed for translating the setup of our previous work,
Pasquarella:2023deo , in the formalism of Moore and Segal.

  2. 2.

We then turn to highlighting the construction of cobordism operators,
Moore:2006dw , emphasising its dependence on the conformal structure of the
Riemann surface.

###  2.1 Cochain complexes

A cochain complex, $(A^{{}^{\tiny\bullet}},d^{{}^{\tiny\bullet}})$, is an
algebraic structure that consists of a sequence of abelian groups (or
modules), $A^{{}^{\tiny\bullet}}$, and a sequence of homomorphisms between
consecutive groups, $d^{{}^{\tiny\bullet}}$, such that the image of each
homomorphism is included in the kernel of the next. To a chain complex,
$(A_{{}_{\tiny\bullet}},d_{{}_{\tiny\bullet}})$, there is an associated
homology, which describes how the images are included in the kernels. A
cochain complex is similar to a chain complex, except that its homomorphisms
are in the opposite direction. The homology of a cochain complex is called its
cohomology

$H({\cal C})\ \overset{def.}{=}\ \text{Ker}({\cal Q})/\ \text{Im}({\cal Q}).$
(8)

The nth cohomology group, $H_{{}_{n}}(H^{{}^{0}})$ is

$H_{{}_{n}}\ \overset{def.}{=}\ \text{Ker}\ d_{{}_{n}}/\ \text{Im}\
d_{{}_{n+1}}.$ (9)

The central object in closed string theory is the vector space ${\cal
C}\equiv{\cal C}_{{}_{S^{{}^{1}}}}$ of states of a single parametrised string.
${\cal C}$ denotes the cochain complex in this case, Moore:2006dw . The latter
comes equipped with a grading given by the ghost number, and an operator
${\cal Q}:\ {\cal C}\ \rightarrow\ {\cal C}$ called the BRST operator, raising
the ghost number by 1, and such that ${\cal Q}^{{}^{2}}\equiv 0$.

###  2.2 The Moore-Segal setup

The most general finite-dimensional commutative algebra over the complex
numbers is of the form

${\cal C}\ \overset{def.}{=}\ \underset{x}{\bigoplus}\ {\cal C}_{{}_{x}}\ \ \
,\ \ \ x\in\ \text{Spec}\ ({\cal C}),$ (10)

with

${\cal C}_{{}_{x}}\ \overset{def.}{=}\ \mathbb{C}\ {\cal E}_{{}_{x}}\ \oplus\
m_{{}_{x}},$ (11)

where ${\cal E}_{{}_{x}}$ is an idempotent, and $m_{{}_{x}}$ a nilpotent
ideal. If ${\cal C}$ is a Frobenius algebra, then so too is each ${\cal
C}_{{}_{x}}$.

In their treatment, Moore:2006dw restrict to the semisimple666Despite
appearing quite restrictive, committing to semisimplicity is enough to shed
light on the essential structure of the theory. According to Moore:2006dw , to
go beyond it, the appropriate objects of study, are cochain-complex valued
TFTs rather than non-semisimple TFTs in the usual sense. case. Semisimplicity
admits many equivalent definitions:

  1. 1.

The presence of simultaneously-diagonalisable fusion rules.

  2. 2.

There exists a set of basic idempotents ${\cal E}_{{}_{x}}$ such that

${\cal C}\ \overset{def.}{=}\ \underset{x}{\bigoplus}\ \mathbb{C}\ {\cal
E}_{{}_{x}}\ \ \ ,\ \ \ x\in\ \text{Spec}\ ({\cal C}),\ \ \ \text{with}\ \ \ \
\ {\cal E}_{{}_{x}}{\cal E}_{{}_{y}}\ \equiv\ \delta_{{}_{xy}}{\cal
E}_{{}_{y}}.$ (12)

  3. 3.

${\cal C}$ is the algebra of complex-valued functions on the finite set of
characters of ${\cal C}$, $X\in\ \text{Spec}\ ({\cal C})$.

For any pair of boundary conditions, $a,b$, the corresponding cochain complex
for a semisimple category is defined as follows, Moore:2006dw ,

${\cal O}_{{}_{aa}}\ \simeq\ \underset{x}{\bigoplus}\
\text{End}\left(W_{{}_{x,a}}\right),$ (13)

${\cal O}_{{}_{ab}}\ \simeq\ \underset{x}{\bigoplus}\
\text{Hom}\left(W_{{}_{x,a}};W_{{}_{x,b}}\right),$ (14)

where $W_{{}_{x,a}}$ is a vector space associated to every idempotent ${\cal
E}_{{}_{x}}$.

###  2.3 Cobordism operators

We now turn to the key elements for our analysis. In this first section we
will be using the definition provided by Moore:2006dw , highlighting the
crucial property that will be mostly needed in sections 3 and 4. A cobordism
$\Sigma$ from $p$ circles to $q$ circles gives an operator

$U_{{}_{\Sigma,\alpha}}:\ {\cal C}^{{}^{\otimes p}}\ \rightarrow\ {\cal
C}^{{}^{\otimes q}},$ (15)

which depends on the conformal structure $\alpha$ on $\Sigma$. This operator,
(15), is a cochain map, but its crucial feature is that, changing the
conformal structure $\alpha$ on $\Sigma$, changes $U_{{}_{\Sigma,\alpha}}$
only by a cochain homotopy.

To describe how $U_{{}_{\Sigma,\alpha}}$ varies with $\alpha$, if ${\cal
M}_{{}_{\Sigma}}$ is the moduli space of conformal structures on the cobordism
$\Sigma$ which are the identity on the boundary circles, there is a resulting
cochain map

$U_{{}_{\Sigma}}:\ {\cal C}^{{}^{\otimes p}}\ \rightarrow\ \Omega\left({\cal
M}_{{}_{\Sigma}};{\cal C}^{{}^{\otimes q}}\right),$ (16)

with the target denoting the de Rham complex of forms on ${\cal
M}_{{}_{\Sigma}}$ with values in ${\cal C}^{{}^{\otimes q}}$.

An alternative equivalent definition is that of the following cochain map

$U_{{}_{\Sigma}}:\ C_{{}_{{}_{{}_{\tiny\bullet}}}}\ \left({\cal
M}_{{}_{\Sigma}}\right)\ \rightarrow\ \left({\cal C}^{{}^{\otimes
p}}\right)^{{}^{*}}\ \otimes\ {\cal C}^{{}^{\otimes q}}.$ (17)

In the next sections, we will show that lack of reparametrisation-invariance
of the Riemann surface implies interesting mathematical and physical features
of the resulting theory of interest.

## Key points

The main points to keep in mind throughout the reminder of ourtreatment are
the following:

  * •

Cochain level theories provide the natural mathematical formalism for
describing absolute theories obtained by partial gaugings of the SymTFT in the
Freed-Moore-Teleman setup.

  * •

The definition of cobordism operators associated to such complex cochain
structure follows from the assumption that the Riemann surface is
reparametrisation-invariant.

##  3 Moore-Tachikawa varieties

Having outlined the importance of reparametrisation-invariance in the
definition of bordism operators, Moore:2006dw , we now turn to the particular
application in describing maximal dimensional Higgs branches of class ${\cal
S}$ theories, as first proposed by Moore:2011ee . Our major contribution in
the present section will be highlighting where upgrades to the categories
defined in Moore:2006dw are need for dealing with setups as the ones
associated to the correspondence depicted in figure 2, namely those leading to
the emergence of composite class ${\cal S}$ theories separated by a non-
invertible defect.

This section is structured as follows:

  1. 1.

At first, we briefly overview the source and target categorical structure
proposed in Moore:2011ee assuming duality.

  2. 2.

We then explain what categorical duality means from an algebraic perspective.

  3. 3.

We conclude the section indicating the relation between Moore-Tachikawa
varieties and Coulomb branches of quiver gauge theories as an interesting
realisation of 3D mirror symmetry, and how the categorical generalisation
proposed in this work suggests interesting applications to quiver varieties
that will be addressed in more detail in VP .

###  3.1 Categorical structure assuming duality

As already mentioned in the Introduction, to a given class ${\cal S}$ theory,
one can assign a 2D TFT valued in a symmetric monoidal category, Moore:2011ee
,

$\boxed{\ \ \ \eta_{{}_{G_{\mathbb{C}}}}:\ \text{Bo}_{{}_{2}}\ \rightarrow\
\text{HS}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bigg{]}\
\ \ }$ (18)

The existence of this 2D TFT relies on the the source and target categories
satisfying a certain list of properties, Moore:2011ee . We will not reproduce
all of them in our treatment, and refer the interested reader to the original
work of Moore and Tachikawa for a detailed explanation. In this first part of
the section, we will only point out some of the crucial assumptions made in
their work for reasons that will become clear in the following pages.

## Duality

For the purpose of our work, the crucial assumption made in Moore:2011ee is
the duality structure of the source category Bo${}_{{}_{2}}$. As explained in
Moore:2011ee , duality implies that the 2-category Bo${}_{{}_{2}}$ is fully
specified by its objects, $S^{1}$, and 1-morphisms, namely the bordisms
depicted in figure 3. The middle bordism, i.e. the one labelled $V$, is the
identity bordism. One can easily see this by noticing that $V$ is
topologically equivalent to a cylinder whose edges are the red circles, i.e.
the object of 2-category Bo${}_{{}_{2}}$ (the closed string we were referring
to in section 2).

For $\eta_{{}_{G_{\mathbb{C}}}}$ to be well defined, the source and target
categories are required to satisfy certain sewing relations, Moore:2006dw ;
Moore:2011ee . This practically means that, compositions between morphisms
should close. In particular, the identity itself can be defined in terms of
composite homomorphisms as follows,

$\boxed{\ \ \ U_{{}_{G_{\mathbb{C}}}}\ \circ\ W_{{}_{G_{\mathbb{C}}}}\ \equiv\
T^{{}^{*}}G_{{}_{\mathbb{C}}}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bigg{]}\
},$ (19)

where $T^{{}^{*}}G_{{}_{\mathbb{C}}}\ \equiv\ V{{}_{G_{\mathbb{C}}}}$. Indeed,
one can easily see that combining the first and third bordims in figure 3, is
topologically equivalent to $V$.

777Indeed, $V$ is topologically equivalent to the cylinder, i.e. the cobordism
between $S^{{}^{1}}$ and itself.

Figure 3: Basic bordisms assuming duality of, both, the source and target
categories leading to the definition of the identity element,
$V_{{}_{G_{{}_{\mathbb{C}}}}}$, and the maximal dimensional Higgs branch,
$W_{{}_{G_{{}_{\mathbb{C}}}}}$.

We therefore wish to highlight the following

Main point: Duality ensures the presence of an identity associated to a
certain gauge group, $G_{{}_{\mathbb{C}}}\equiv\eta_{{}_{G_{\mathbb{C}}}}\
(S^{{}^{1}})$, (3).

### Key axiom

(19) is essential for us in relating the formalism of Moore:2006dw ;
Moore:2011ee to the setup of figure 2. In particular, it is what leads to the
definition of the triple featuring on the RHS of (1). To see this explicitly,
let us recall a crucial axiom required to be satisfied by (18), and,
therefore, in turn by (19), Moore:2011ee .

For $X\in$ Hom $(G_{{}_{\mathbb{C}}}^{{}^{\prime}},G_{{}_{\mathbb{C}}})$ and
$Y\in$ Hom $(G_{{}_{\mathbb{C}}},G_{{}_{\mathbb{C}}}^{{}^{\prime\prime}})$,
their composition

$Y\ \circ\ X\ \in\ \text{Hom}\
(G_{{}_{\mathbb{C}}}^{{}^{\prime}},G_{{}_{\mathbb{C}}}^{{}^{\prime\prime}})$
(20)

is identified with the holomorphic symplectic quotient

$\displaystyle Y\ \circ\ X$ $\displaystyle\overset{def.}{=}\ X\ \times\ Y\ //\
G_{{}_{\mathbb{C}}}$ (21) $\displaystyle\ =\\{(x,y)\ \in\ X\times Y|\
\mu_{{}_{X}}(x)+\mu_{{}_{Y}}(y)=0\\}\ /\ G_{{}_{\mathbb{C}}},$

where

$\mu_{{}_{X}}:\ X\ \longrightarrow\ \mathfrak{g}_{{}_{\mathbb{C}}}^{*}\ \ \ ,\
\ \ \mu_{{}_{Y}}:\ Y\ \longrightarrow\ \mathfrak{g}_{{}_{\mathbb{C}}}^{*}$
(22)

are the moment maps of the action of $G_{{}_{\mathbb{C}}}$ on $X$ and $Y$,
with $\mathfrak{g}_{{}_{\mathbb{C}}}$ the Lie algebra associated to
$G_{{}_{\mathbb{C}}}$. The identity element

$T^{{}^{*}}G_{{}_{\mathbb{C}}}\ \overset{def.}{=}\
\text{id}_{{}_{G_{{}_{\mathbb{C}}}}}\ \in\ \text{Hom}\
(G_{{}_{\mathbb{C}}},G_{{}_{\mathbb{C}}})$ (23)

comes with a Hamiltonian $G_{{}_{\mathbb{C}}}\ \times\ G_{{}_{\mathbb{C}}}$
action. As also explained in Moore:2011ee , to see that
$T^{{}^{*}}G_{{}_{\mathbb{C}}}$ acts as the identity, it is enough to consider
a composition of homomorphisms, $T^{{}^{*}}G_{{}_{\mathbb{C}}}\ \circ\ X$.
Identifying $T^{{}^{*}}G_{{}_{\mathbb{C}}}\ \simeq\ G_{{}_{\mathbb{C}}}\
\times\ \mathfrak{g}_{{}_{\mathbb{C}}}$, and identifying an element of
$T^{{}^{*}}G_{{}_{\mathbb{C}}}$ as $(g,a)$. The moment map condition, (21),
reduces to

$a+\mu(x)=0,$ (24)

from which $a$ can be removed. Consequently, the induced 2-form on the
solution space is $G_{{}_{\mathbb{C}}}$-invariant and basic. Upon taking the
quotient with respect to $G_{{}_{\mathbb{C}}}$, we can gauge $g$ to 1, leading
to a holomorphic isomorphism with the original $X$ space with its symplectic
form.

The categorical quotient taken in defining the composition (21) is equivalent
to the one that a given absolute theory should be equipped with to potentially
gauge away its entire operator content, while leaving only the identity in the
spectrum888cf. explanation in the Introduction. This is basically what leads
to the definition of the fiber functor, moment map and Drinfeld center..
Indeed, this is true as long as the embeddings of the subalgebras associated
to $G_{{}_{\mathbb{C}}}^{{}^{\prime}}$ and
$G_{{}_{\mathbb{C}}}^{{}^{\prime\prime}}$ are subsets of each other within the
mother algebra $\mathfrak{g}_{{}_{\mathbb{C}}}$. However, we are interested in
describing more general setups, where the embeddings of the algebras are
intersecting albeit not one included within the other. In the remainder of our
treatment, we will explain that, for this to be described in the formalism of
Moore:2006dw ; Moore:2011ee , the standard identity element associated to the
gauge group $G_{{}_{\mathbb{C}}}$ and embedding Lie algebra
$g_{{}_{\mathbb{C}}}$ needs to be removed from Bo${}_{{}_{2}}$, while being
replaced by a new composite bordism, and propose the definition of a new
functor.

###  3.2 Duality from an algebraic perspective

Before turning to explaining what are the changes that the soruce and target
categories should undergo999Which will be the core topic of section 4., we
will briefly pause for a digression explaining how reparametrisation-
invariance of the Riemann surface involved in the definition of the bordism
operators, outlined in section 2, is strongly related to the aforementioned
duality assumption.

As explained in Moore:2011ee , the crucial point is that, thanks to the
duality propriety of the source 2-category Bo${}_{{}_{2}}$, the identity
element in the target category, $T^{{}^{*}}G_{{}_{\mathbf{C}}}$, is
reparametrisation-invariant. In particular, one could compose the identity
morphisms as follows101010Making use of the axiom (21).

$T^{{}^{*}}G_{{}_{\mathbf{C}}}^{{}^{a}{{}^{{}^{\prime}}}}\ \circ\
T^{{}^{*}}G_{{}_{\mathbf{C}}}^{{}^{a}}\equiv\left(T^{{}^{*}}G_{{}_{\mathbf{C}}}^{{}^{a}}\
\times\
T^{{}^{*}}G_{{}_{\mathbf{C}}}^{{}^{a}{{}^{{}^{\prime}}}}\right)//G_{{}_{\mathbf{C}}}.$
(25)

From the considerations made above, it therefore follows that one could
rephrase (25) as the definition of the Drinfeld center for the composite
system made up of two class $\cal S$ theories (associated to the two gauge
groups involved) separated by an invertible defect, with the latter ensuring
reparametrisation invariance of the Riemann surface Moore:2006dw .

Concretely, under the duality assumption, one could gauge away either of the
two groups, while being left with the following

$\mathbb{1}_{{}_{G_{{}_{\mathbf{C}}}^{{}^{a^{{}^{\prime}}}}}}\ \circ\
T^{{}^{*}}G_{{}_{\mathbf{C}}}^{{}^{a}}\equiv\left(T^{{}^{*}}G_{{}_{\mathbf{C}}}^{{}^{a}}\
\times\
\mathbb{1}_{{}_{G_{{}_{\mathbf{C}}}^{{}^{a^{{}^{\prime}}}}}}\right)//G_{{}_{\mathbf{C}}}.$
(26)

If the conformal structure were the same on the two sides, then it would be
the same, either with or without the composition rule. If

$G_{{}_{\mathbf{C}}}^{{}^{a}}\ \times\
G_{{}_{\mathbf{C}}}^{{}^{a^{{}^{\prime}}}}\ \equiv\
G_{{}_{\mathbf{C}}}^{{}^{a+a^{{}^{\prime}}}}\ \equiv\ G_{{}_{\mathbf{C}}},\ \
\ \forall a,a^{\prime},$ (27)

then (25) can be recast to the following

$\boxed{\ \ \ T^{{}^{*}}G_{{}_{\mathbf{C}}}^{{}^{a}}\equiv
T^{{}^{*}}G_{{}_{\mathbf{C}}}^{{}^{a}}\ \ //\ \
G_{{}_{\mathbf{C}}}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bigg{]}\
\ }\ ,$ (28)

which is equivalent to a statement of S-duality. Once more, we highlight that
this is possible because the source category for the 2D TFT associated to the
group $G_{{}_{\mathbf{C}}}$ contains the identity element. But, in case this
is not true111111Such as the case in which the Riemann surface is no longer
reparametrisation-invariant., (28) needs to be changed accordingly, which one
could think of as a generalisation of an S-duality statement. Indeed, if the
group composition rules don’t hold,

$G_{{}_{\mathbf{C}}}^{{}^{a+a^{{}^{\prime}}}}\ \neq\ G_{{}_{\mathbf{C}}},\ \ \
\forall a,a^{\prime},$ (29)

we get something that is not simply the ordinary S-dual theory, (28).

The main purpose of our work is basically to go backwards, starting from the
LHS of (25) and determining what the RHS should be. Most importantly, we need
to:

  1. 1.

Identify $G_{{}_{\mathbf{C}}}$ in the new theory obtained by composing the two
theories on the LHS, each one characterised by a different choice of conformal
structure on the Riemann surface.

  2. 2.

Equivalently to 1., reconstruct $T^{{}^{*}}G_{{}_{\mathbf{C}}}$, namely the
identity of the composite theory.

  3. 3.

We highlight that the most important generalisation of the 2D TFT (21) that
one should really be using for the case of interest to us is instead the
following

$\boxed{\ \ \ \tilde{\eta}_{{}_{G_{\mathbb{C}}}}:\
\text{Bo}_{{}_{2}}\backslash\ V\rightarrow\
\text{HK}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bigg{]}\
\ \ }$ (30)

which, as already pointed out in Moore:2011ee , requires removing the identity
element from the source category. Its effect on the target is to turn it into
a hyperk$\ddot{\text{a}}$hler quotient. Its connection with theoretical
physics121212Already presented in Pasquarella:2023exd . is the main focus of
VP .

From step 1., an important observation is in order. $G_{{}_{\mathbf{C}}}$ acts
on the two factors on the RHS of (25) in separate ways. This is part of the
meaning of the generalisation of S-duality that we were previously referring
to. Indeed, the categorical quotient $//G_{{}_{\mathbf{C}}}$ tells us what the
identity is as a result of gauging a certain subalgebra. This is obtained by
taking the 1-morphisms on either side of the correspondence and taking their
nontrivial composition w.r.t. $T^{{}^{*}}G_{{}_{\mathbf{C}}}$, with the latter
being the identity in the target category. But the latter was assumed to be
removed. The immediate suggestion to circumvent this shortcoming is that the
functors defining the identity element of each individual theory on the LHS of
(25) is different w.r.t. the one on the RHS. In section 4 we therefore propose
the generalisation of S-duality as the need to define two different 2D TFT
functors associated to the left and right hand sides of (25).

###  3.3 Algebraic Varieties

In the concluding part of this section, we highlight an interesting
application of (30) in the context of quiver gauge theories, which will be
explained in more detail in an upcoming work by the same author, VP , in
particular towards generalising 3D mirror symmetry.

The Higgs branches described by Moore and Tachikawa are known to have been
reproduced by Braverman:2017ofm as the Coulomb branches of 3D ${\cal N}=4$
supersymmetric quiver gauge theories. Such correspondence is therefore
equivalent to a statement of 3D mirror symmetry. The purpose of VP is to
explain what the 3D dual of a theory described by (30) is in terms of Coulomb
branches of 3D ${\cal N}=4$ quiver gauge theories. In this way, we expect to
be able to prove the statements made in Pasquarella:2023exd .

If $V$ has been removed from the source, one should expect there to be more
than one 2D TFT of the kind (18) associated to two different gauge groups
whose embedding in the gauge group associated to the original TFT with
identity element $V$ is not simply a cochain complex. Correspondingly, this
also means that there is more than one 1-morphism $U_{{}_{G_{\mathbb{C}}}}$.

Given that the identity of the embedding theory is defined as follows

$\boxed{\ \ \ U_{{}_{G_{\mathbb{C}}}}\ \circ\ W_{{}_{G_{\mathbb{C}}}}\ \equiv\
T^{{}^{*}}G_{{}_{\mathbb{C}}}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bigg{]}\
}\ \ \ \ ,\ \ \ \ \boxed{\ \ \ \eta_{{}_{G_{\mathbb{C}}}}(V)\ \equiv\
T^{{}^{*}}G_{{}_{\mathbb{C}}}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bigg{]}\
\ }.$ (31)

and that

$U_{{}_{G_{\mathbb{C}}}}\ \overset{def.}{=}\ G_{{}_{\mathbb{C}}}\times
S_{{}_{n}}\ \subset\ G_{{}_{\mathbb{C}}}\ \times\
\mathfrak{g}_{{}_{\mathbb{C}}}\ \simeq\ T^{{}^{*}}G_{{}_{\mathbb{C}}}.$ (32)

with $S_{{}_{n}}$ is the Slodowy slice at a principal nilpotent element $n$.
The physical theories of class ${\cal S}$ predict the existence of a variety
$W_{{}_{G_{{}_{\mathbb{C}}}}}$ satisfying the properties needed to define a
TFT $\eta_{{}_{G_{\mathbb{C}}}}$. From the duality assumption, it follows that
the dimensionalities of the two varieties are related as follows

$\text{dim}_{{}_{\mathbb{C}}}\ U_{{}_{G_{\mathbb{C}}}}\ \overset{def.}{=}\
\text{dim}_{{}_{\mathbb{C}}}\ G_{{}_{\mathbb{C}}}\ +\ \text{rank}\
G_{{}_{\mathbb{C}}}.$ (33)

$\text{dim}_{{}_{\mathbb{C}}}\ W_{{}_{G_{\mathbb{C}}}}\ \overset{def.}{=}\ 3\
\text{dim}_{{}_{\mathbb{C}}}\ G_{{}_{\mathbb{C}}}\ -\ \text{rank}\
G_{{}_{\mathbb{C}}}.$ (34)

However, if the identity needs to be removed from Bo${}_{{}_{2}}$,
$T^{{}^{*}}G_{{}_{\mathbb{C}}}$ is not the identity and, in particular (34)
needs to be redefined precisely because the source is no longer a dual
category. How to rederine (34) will be explained in VP .

As a concluding remark to what we have just said, in Moore:2006dw they
conjecture the following property for the moment maps associated to the
$G^{{}^{3}}$ action on the Higgs branch $W_{{}_{G_{{}_{\mathbb{C}}}}}$

$\boxed{\ \ \ \mu_{{}_{i}}:\
W_{{}_{G_{{}_{\mathbb{C}}}}}\rightarrow\mathfrak{g}_{{}_{\mathbf{C}}}^{{}^{*}}\
\ \ ,\ \ \
i=1,2,3.\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bigg{]}\
\ }$ (35)

This is crucial to our analysis since (35) can be inverted to obtain the Higgs
branch as a hyperk$\ddot{\text{a}}$ler quotient

$\boxed{\ \ \ W_{{}_{G_{{}_{\mathbb{C}}}}}\ \equiv\
\mu^{{}^{-}1}/G^{{}^{3}}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bigg{]}\
\ }.$ (36)

However, for the case in which the identity is removed from the source
category, (36) does not hold anymore precisely because of the lack of
permutational symmetry arising in the quotient. In VP we will be explaining
how to define $W_{{}_{G_{{}_{\mathbb{C}}}}}$ and its dimensionality for the
case involving categories without a duality structure.

## Key points

The main points are the following:

  * •

Categorical duality ensures the presence of an identity object.

  * •

S-duality requires reparametrisation-invariance.

##  4 Moore-Tachikawa varieties beyond duality

In this concluding section, we piece together several considerations made
throughout our treatment, ultimately showing how apparent shortcomings in
mathematical descriptions might lead to interesting physical realisations.

This section is structured as follows:

  1. 1.

We show why the formalism of Moore:2006dw needs to be generalised if
reparametrisation-invariance of the Riemann surface falls short from being
satisfied, and how this opens up to interesting generalisations of the
proposal of Moore:2011ee for 2D TFTs describing maximal dimensional Higgs
branches of class ${\cal S}$ theories131313The latter will be the core topic
of section 4.. In particular, we build the relation with the definition of the
fiber functor and Drinfeld center, (1).

  2. 2.

We conclude highlighting interesting features of the formalism of Moore:2006dw
; Moore:2011ee in absence of reparametrisation-invariance of the Riemann
surface, connecting them to the emergence of intrinsically non-invertible
symmetries separating different class ${\cal S}$ theories, Pasquarella:2023deo
.

###  4.1 Categorical structure without duality

As explained in the previous section, removing the identity element in
Bo${}_{{}_{2}}$ requires having to introduce at least two different functors
$\eta_{{}_{G_{{}_{\mathbb{C}}}^{{}^{\prime}}}},\eta_{{}_{G_{{}_{\mathbb{C}}}^{{}^{\prime\prime}}}}$,
whose action on the circle and its bordism reads as follows

$\eta_{{}_{G_{{}_{\mathbb{C}}}^{{}^{\prime}}}}\left(\ S^{{}^{1}}\ \right)\
\equiv\ G_{{}_{\mathbb{C}}}^{{}^{\prime}}\ \ \ ,\ \ \
\eta_{{}_{G_{{}_{\mathbb{C}}}^{{}^{\prime\prime}}}}\left(\ S^{{}^{1}}\
\right)\ \equiv\ G_{{}_{\mathbb{C}}}^{{}^{\prime\prime}}$ (37)

$\eta_{{}_{G_{{}_{\mathbb{C}}}^{{}^{\prime}}}}\left(\ U\ \right)\ \equiv\
U_{{}_{G_{{}_{\mathbb{C}}}^{{}^{\prime}}}}\equiv\
G_{{}_{\mathbb{C}}}^{{}^{\prime}}\times S_{{}_{n}}\ \subset\
G_{{}_{\mathbb{C}}}^{{}^{\prime}}\times\mathfrak{g}_{{}_{\mathbf{C}}}^{{}^{\prime}}\
\simeq\ T^{{}^{*}}G_{{}_{\mathbf{C}}}^{{}^{\prime}},$ (38)

$\eta_{{}_{G_{{}_{\mathbb{C}}}^{{}^{\prime\prime}}}}\left(\ U\ \right)\
\equiv\ U_{{}_{G_{{}_{\mathbb{C}}}^{{}^{\prime\prime}}}}\equiv\
G_{{}_{\mathbb{C}}}^{{}^{\prime\prime}}\times S_{{}_{n}}\ \subset\
G_{{}_{\mathbb{C}}}^{{}^{\prime\prime}}\times\mathfrak{g}_{{}_{\mathbf{C}}}^{{}^{\prime\prime}}\
\simeq\ T^{{}^{*}}G_{{}_{\mathbf{C}}}^{{}^{\prime\prime}},$ (39)

$\eta_{{}_{G_{{}_{\mathbb{C}}}^{{}^{\prime}}}},\left(\ V\ \right)\ \equiv\
V_{{}_{G_{{}_{\mathbb{C}}}^{{}^{\prime}}}}\overset{def.}{\equiv}\
T^{{}^{*}}G_{{}_{\mathbb{C}}}^{{}^{\prime}},$ (40)

$\eta_{{}_{G_{{}_{\mathbb{C}}}^{{}^{\prime\prime}}}}\left(\ V\ \right)\
\equiv\
V_{{}_{G_{{}_{\mathbb{C}}}^{{}^{\prime\prime}}}}\overset{def.}{\equiv}\
T^{{}^{*}}G_{{}_{\mathbb{C}}}^{{}^{\prime\prime}},$ (41)

where we are assuming

$\mathfrak{g}_{{}_{\mathbf{C}}}^{{}^{\prime}}\ \cap\
\mathfrak{g}_{{}_{\mathbf{C}}}^{{}^{\prime\prime}}\ \neq\ \\{\emptyset\\}\ \
,\ \ \text{and}\ \ \ \mathfrak{g}_{{}_{\mathbf{C}}}^{{}^{\prime}}\ \cup\
\mathfrak{g}_{{}_{\mathbf{C}}}^{{}^{\prime\prime}}\ \equiv\
\mathfrak{g}_{{}_{\mathbf{C}}},$ (42)

$\mathfrak{g}_{{}_{\mathbf{C}}}^{{}^{\prime}}\ \not\subset\
\mathfrak{g}_{{}_{\mathbf{C}}}^{{}^{\prime\prime}}\ \ ,\ \ \text{and}\ \ \ \
\mathfrak{g}_{{}_{\mathbf{C}}}^{{}^{\prime\prime}}\ \not\subset\
\mathfrak{g}_{{}_{\mathbf{C}}}^{{}^{\prime}}.$ (43)

(42) and (43) imply that the two subalgebras involved,
$\mathfrak{g}_{{}_{\mathbf{C}}}^{{}^{\prime}},\mathfrak{g}_{{}_{\mathbf{C}}}^{{}^{\prime\prime}}$
are associated to different subgroups,
$G_{{}_{\mathbb{C}}}^{{}^{\prime}},G_{{}_{\mathbb{C}}}^{{}^{\prime\prime}}$,
and that the identity elements differ,
$T^{{}^{*}}G_{{}_{\mathbf{C}}}^{{}^{\prime}}\ \neq\
T^{{}^{*}}G_{{}_{\mathbf{C}}}^{{}^{\prime\prime}}$, even under
reparametrisation of the Reimann surface. For each one of the 2D TFTs,
$\eta_{{}_{G_{{}_{\mathbb{C}}}^{{}^{\prime}}}},\eta_{{}_{G_{{}_{\mathbb{C}}}^{{}^{\prime\prime}}}}$
one could use the formalism of Moore:2006dw ; Moore:2011ee , describing two
different class ${\cal S}$ theories, both descending from 6D ${\cal N}=(2,0)$
by dimensionally reducing on a Riemann surface without reparemetrisation
invariance. However, given the assumption that
$(G_{{}_{\mathbb{C}}}^{{}^{\prime}},\mathfrak{g}_{{}_{\mathbf{C}}}^{{}^{\prime}}),(G_{{}_{\mathbb{C}}}^{{}^{\prime\prime}},\mathfrak{g}_{{}_{\mathbf{C}}}^{{}^{\prime\prime}})$
can be embedded in a unique
$(G_{{}_{\mathbb{C}}},\mathfrak{g}_{{}_{\mathbb{C}}})$, it is natural to ask
what should the triple on the RHS of (1) be for the resulting theory to be
absolute?

We know that the fiber functor and Drinfeld centers for a given absolute
theory are defined as follows

${\cal F}\ :\ \mathfrak{Z}\left(\text{Bo}_{{}_{2}}\right)\ \rightarrow\
\text{Bo}_{{}_{2}},$ (44)

$\mathfrak{Z}\left(\text{Bo}_{{}_{2}}\right)\equiv\text{End}_{{}_{\text{Bo}_{{}_{2}}}}\left(T^{{}^{*}}G_{{}_{\mathbf{C}}}\right)\equiv\text{Hom}_{{}_{\text{Bo}_{{}_{2}}}}\left(T^{{}^{*}}G_{{}_{\mathbf{C}}},T^{{}^{*}}G_{{}_{\mathbf{C}}}\right),$
(45)

both of which crucially rely upon the presence of an identity in the source
and target of $\eta_{{}_{G_{{}_{\mathbb{C}}}}}$. Apparently, we run into a
contradiction, since the identity element $T^{{}^{*}}G_{{}_{\mathbf{C}}}$ has
been removed by assumption, thereby implying
$\mathfrak{Z}\left(\text{Bo}_{{}_{2}}\right)$ cannot be defined in the
ordinary way. On the other hand, one could define the new identity as being a
composite object defined in the following way

$\boxed{\ \ \ T^{{}^{*}}\tilde{G}_{{}_{\mathbf{C}}}\ \overset{def.}{=}\
T^{{}^{*}}G_{{}_{\mathbf{C}}}^{{}^{\prime}}\
\otimes_{{}_{T^{{}^{*}}G_{{}_{\mathbb{C}}}}}\
T^{{}^{*}}G_{{}_{\mathbf{C}}}^{{}^{\prime\prime}}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bigg{]}\
\ },$ (46)

and therefore our proposal for (44) and (45) reads as follows

$\boxed{\ \ \ {\cal F}\ :\
\mathfrak{Z}\left(\tilde{\text{Bo}_{{}_{2}}}\right)\ \rightarrow\
\tilde{\text{Bo}_{{}_{2}}}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bigg{]}\
\ },$ (47)

$\boxed{\ \ \ \mathfrak{Z}\left(\tilde{\text{Bo}_{{}_{2}}}\right)\ \equiv\
\text{End}_{{}_{\tilde{\text{Bo}_{{}_{2}}}}}\left(T^{{}^{*}}G_{{}_{\mathbf{C}}}^{{}^{\prime}}\
\otimes_{{}_{T^{{}^{*}}G_{{}_{\mathbb{C}}}}}\
T^{{}^{*}}G_{{}_{\mathbf{C}}}^{{}^{\prime\prime}}\right)\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bigg{]}\
\ },$ (48)

with

$\tilde{\text{Bo}_{{}_{2}}}\ \overset{def.}{=}\ \text{Bo}_{{}_{2}}\ /\ V.$
(49)

In the following subsection, we will explain why (47) and (48) are reasonable
proposals for defining an absolute theory in absence of a categorical duality
structure.

###  4.2 Interesting shortcomings

As explained in Pasquarella:2023deo , the loss of reparametrisation invariance
of the Riemann surface signals the presence of an intrinsically-non-invertible
defect between different class ${\cal S}$ theories. We will now show that the
setup described in section 4.1 is equivalent to that of Pasquarella:2023deo .

The starting point in our argument is the conjecture (35) and (36). In absence
of categorical duality of, both, source and target, the 2D TFT associated to
the maximal dimensional Higgs branch of Moore:2011ee is no longer associated
to a moment map that is equivalent for all the constituent $S^{{}^{1}}$s,
thereby violating the conjecture made by Moore:2011ee . Explicitly,

$\boxed{\ \ \ \ W_{{}_{G_{{}_{\mathbb{C}}}}}\ \neq\
\mu^{{}^{-}1}/G^{{}^{3}}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bigg{]}\
\ }.$ (50)

This is because the algebraic variety associated to the Higgs branch
$W_{{}_{G_{{}_{\mathbb{C}}}}}$ is not a hyperk$\ddot{\text{a}}$ler quotient.
In particular, it is associated to a non-primitive ideal.

Expanding further on this topic, the crucial point is that, when giving up the
duality propriety, there is no longer the identity element in the source
category, $T^{{}^{*}}G_{{}_{\mathbf{C}}}$, but, rather, there is one identity
for each underlying constituent, (38) and (39).

In order to determine how the resulting composite identity element should look
like, we need to briefly recall what was outlined in section 3.2. Given two
different morphisms, and taking their composition

$T^{{}^{*}}G_{{}_{\mathbf{C}}}^{{}^{a^{{}^{\prime}}}}\ \circ\
T^{{}^{*}}G_{{}_{\mathbf{C}}}^{{}^{a}}\equiv\left(T^{{}^{*}}G_{{}_{\mathbf{C}}}^{{}^{a}}\
\times\
T^{{}^{*}}G_{{}_{\mathbf{C}}}^{{}^{a^{{}^{\prime}}}}\right)//G_{{}_{\mathbf{C}}},$
(51)

we know that, under the duality assumption, the axiom (25) comes with two
moment maps, (21), each one describing the embedding of the individual
morphisms within the algebra of the mother theory. As explained in section
3.2, one actually uses the mutual relation in between such moment maps to
prove that $T^{{}^{*}}G_{{}_{\mathbf{C}}}$ behaves as the identity. As also
claimed in the previos section, (51) is expresses the need to define a
Drinfeld center for the composite system made up of two class $\cal S$
theories. In presence of reparametrisation invariance, such theories can be
thought of as being separated by an invertible defect, from which (51) can be
reduced to a statement of S-duality.

On the other hand, in absence of reparametrisation-invariance, the resulting
class ${\cal S}$ theories would, by definition, be separated by an
intrinsically non-invertible defect, with the latter being responsible for the
lack of reparametrisation invariance of the Riemann surface Moore:2006dw .

Our main question is to find the Drinfeld center for a given Bo${}_{{}_{2}}$

${\cal F}:\ \mathfrak{Z}(\text{Bo}_{{}_{2}})\ \rightarrow\
\text{Bo}_{{}_{2}},$ (52)

and we know that, to a given fiber functor, ${\cal F}$, there is an associated
moment map

$\mu:\ \mathfrak{G}\ \rightarrow\ {\cal A}.$ (53)

with $\mathfrak{G}$ being Bo${}_{{}_{2}}$ in this case, and ${\cal A}$ the
algebra of invertible topological defects that projects to the identity
$T^{{}^{*}}G_{{}_{\mathbb{C}}}$ under complete gauging of the theory. This is
equivalent to stating that $T^{{}^{*}}G_{{}_{\mathbb{C}}}$ is the identity
element once having projected over ${\cal A}$

${\cal A}\ //_{\mu}\ \mathfrak{G}\ \ \ \text{with}\ \ \ \mu:\ \mathfrak{G}\
\rightarrow\ {\cal A}$ (54)

choosing the definition of the identity in the following way

$T^{{}^{*}}G_{{}_{\mathbb{C}}}\ \simeq\ \mathbb{1}_{{}_{G_{{}_{\mathbb{C}}}}}\
\equiv\ {\cal A}\ //_{\mu}\ \mathfrak{G}.$ (55)

For the purpose of our work, $\mathfrak{G}\ \overset{def.}{=}\
\text{Bo}_{{}_{2}}$, therefore

$T^{{}^{*}}G_{{}_{\mathbb{C}}}\ \simeq\ \mathbb{1}_{{}_{G_{{}_{\mathbb{C}}}}}\
\equiv\ {\cal A}\ //_{\mu}\ \text{Bo}_{{}_{2}}$ (56)

However, in the case of section 4.1, there are two different moment maps
involved, one for each choice of conformal structure on the Riemann surface,
that are not mutually related by the moment map constraint following from the
axiom (25). We therefore need to find the moment map (and corresponding gauge
group $\tilde{G}_{{}_{\mathbb{C}}}$)

$\tilde{\mu}:\ \tilde{\mathfrak{G}}\ \rightarrow\ \tilde{\cal A},$ (57)

whose identity element

$\tilde{\cal A}\ \overset{def.}{=}\ {\cal A}_{{}_{1}}\
\otimes_{{}_{T^{{}^{*}}G_{{}_{\mathbb{C}}}}}{\cal A}_{{}_{2}}$ (58)

constitutes the identity of the composite theory. If
$T^{{}^{*}}G_{{}_{\mathbb{C}}}$ is the identity that has been removed from a
particular source category, it still exists, but is no longer the identity
present in $\tilde{\text{Bo}}$${}_{{}_{2}}$ of a given
$\tilde{\eta}_{{}_{G_{{}_{\mathbb{C}}}}}$. From the RHS of (51), the identity
of $\tilde{\text{Bo}}_{{}_{2}}$ therefore reads

$T^{{}^{*}}\tilde{G}_{{}_{\mathbf{C}}}\ \overset{def.}{=}\
T^{{}^{*}}G_{{}_{\mathbf{C}}}^{{}^{\prime}}\
\otimes_{{}_{T^{{}^{*}}G_{{}_{\mathbb{C}}}}}\
T^{{}^{*}}G_{{}_{\mathbf{C}}}^{{}^{\prime\prime}},$ (59)

such that its Drinfeld center can be determined.

Defining $\tilde{\mu}:\ \tilde{\text{Bo}}_{{}_{2}}\rightarrow\ \tilde{\cal A}$
as the moment map associated to the composite theory, the corresponding fiber
functor can be explicitly rewritten as follows

$\boxed{\ \ \ {\cal F}:\
\mathfrak{Z}\left(\tilde{\eta}_{{}_{\tilde{G}_{{}_{\mathbb{C}}}}}^{{}^{-1}}\left(T^{{}^{*}}\tilde{G}_{{}_{\mathbb{C}}}\right)\right)\
\rightarrow\
\tilde{\eta}_{{}_{\tilde{G}_{{}_{\mathbb{C}}}}}^{{}^{-1}}\left(T^{{}^{*}}\tilde{G}_{{}_{\mathbb{C}}}\right)\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bigg{]}\
\ }.$ (60)

ultimately enabling us to reformulate the problem of finding the Drinfeld
center to that of identifying ${\cal F}$ for a given $\mu$.

## Key points

The main points are the following:

  * •

Lack of reparametrisation-invariance of bordism operators signals the presence
of intrinsic non-invertible defects separating different class ${\cal S}$
theories.

  * •

Defining the Drinfeld center for a system of composite class ${\cal S}$
theories separated by non-invertible defects constitutes a nontrivial
generalisation of an S-duality statement.

##  5 Conclusions and Outlook

The present article is the first of a series of works by the same author
providing mathematical support of the claims made in Pasquarella:2023deo ;
Pasquarella:2023exd .

At first, we briefly overviewed cochain level theories as the most important
generalisation of the open and closed TFT construction, emphasising its
relation to the SymTFT construction leading to absolute theories. Mostly
relying upon Moore:2006dw , we highlighted the importance of the construction
of cobordism operators, emphasising their dependence on the conformal
structure of the Riemann surface. In section 3 we then turned to the
discussion of a particular 2D TFT valued in a symmetric monoidal category,
namely the maximal dimension Higgs branch of class ${\cal S}$ theories. After
briefly reviewing the properties outlined in Moore:2011ee , we propose their
generalisation for the case in which the target category of the
$\eta_{{}_{G_{{}_{\mathbb{C}}}}}$ functor is a hyperk$\ddot{\text{a}}$hler
quotient. We concluded outlining the possible extension of this treatment
towards a mathematical formulation of magnetic quivers within the context of
Coulomb branches of 3D ${\cal N}=4$ quiver gauge theories which will be
addressed in an upcoming work by the same author.

## Acknowledgements

I wish to thank Hiraku Nakajima and Constantin Teleman for insightful
discussions on related topics. My work is partly supported by an STFC
studentship through DAMTP.

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# Cosmological constraints from HSC Y1 lensing convergence PDF

Leander Thiele<EMAIL_ADDRESS>Department of Physics, Princeton
University, Princeton, NJ 08544, USA  Center for Data-Driven Discovery, Kavli
IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan
Gabriela A. Marques  Fermi National Accelerator Laboratory, Batavia, IL 60510,
USA  Jia Liu  Center for Data-Driven Discovery, Kavli IPMU (WPI), UTIAS, The
University of Tokyo, Kashiwa, Chiba 277-8583, Japan  Masato Shirasaki
National Astronomical Observatory of Japan (NAOJ), National Institutes of
Natural Sciences, Osawa, Mitaka, Tokyo 181-8588, Japan  The Institute of
Statistical Mathematics, Tachikawa, Tokyo 190-8562, Japan

###### Abstract

We utilize the probability distribution function (PDF) of convergence maps
reconstructed from the Subaru Hyper Suprime-Cam (HSC) Y1 shear catalogue, in
combination with the power spectrum, to measure the matter clustering
amplitude $S_{8}=\sigma_{8}\sqrt{\Omega_{m}/0.3}$. The large-scale structure’s
statistical properties are incompletely described by the traditional two-point
statistics, motivating our investigation of the PDF — a complementary higher-
order statistic. By defining the PDF over the standard deviation-normalized
convergence map we are able to isolate the non-Gaussian information. We use
tailored simulations to compress the data vector and construct a likelihood
approximation. We mitigate the impact of survey and astrophysical systematics
with cuts on smoothing scales, redshift bins, and data vectors. We find
$S_{8}=0.852^{+0.086}_{-0.094}$ from the PDF alone and
$S_{8}=0.827^{+0.033}_{-0.044}$ from the combination of PDF and power spectrum
($68\,\%\,\text{CL}$). The PDF improves the power spectrum-only constraint by
about $10\,\%$.

##  I Introduction

Weak lensing surveys are rapidly catching up with the precision afforded by
cosmic microwave background (CMB) experiments [e.g., 1, 2, 3]. Weak lensing
directly probes the clustering of matter through statistical measurements of
distortion of galaxy shapes. In recent years, weak lensing surveys have
produced increasingly precise measurements of the matter clustering amplitude
$S_{8}$ [e.g., 4, 5, 6, 7, 8], showing hints of a discrepancy between lensing
and CMB determinations, dubbed the $S_{8}$-tension. Typically, these studies
adopt two-point statistics — the two-point correlation function or the power
spectrum.

While two-point statistics are sufficient to describe Gaussian fields such as
the CMB, the large-scale structure that sources the weak lensing signal has
undergone nonlinear growth and is thus far from Gaussian. Therefore, weak
lensing non-Gaussian statistics have been proposed to extract complementary
information.111Popular non-Gaussian statistics include minima, peaks [9, 10,
11, 12, 13, 14, 15, 16, 17], voids [18], Minkowski functionals [19, 20, 21,
22, 23, 24, 25], Betti numbers [26, 27], persistent homology [28, 29],
wavelets [30], scattering transform [31, 32], moments [33, 34, 35, 21, 22,
36], higher-order correlation functions [37, 38, 39, 40, 41, 42], and
convolutional neural networks [43, 44, 45]. A comparison of some of these in a
forecast setting for Euclid has been carried out by [46], finding about
similar performance for many of these statistics but limited to the Fisher
approximation. Equally importantly, non-Gaussian statistics are affected by
systematics differently than the two-point statistics. This is especially
relevant for the $S_{8}$-tension, since whether its origin stems from new
physics or systematics is currently under hot debate. Non-Gaussian statistics
will be particularly beneficial for upcoming surveys such as _Rubin_ LSST
[47], Euclid [48], and _Roman_ [49]. These surveys will provide a high source
density and hence probe deeper into the nonlinear regime.

In this work, we focus on the probability distribution function [PDF, 50, 51,
52, 53] of the lensing convergence map222Also called mass map. from Subaru
Hyper Suprime-Cam (HSC) Y1 data release [54, 55]. HSC is currently the deepest
large-scale lensing survey and is thus considered a precursor of Stage-IV
surveys. The PDF collects the amplitudes — but not shapes — of correlation
functions of all orders. Thus, it is a highly non-Gaussian statistics and
contains distinct information “far” from the power spectrum. Forecasts have
shown promise in tightening the constraints on not only $S_{8}$ but also the
neutrino mass sum and the dark energy equation of state [56, 57, 58, 59].

Our work is the first to obtain cosmological constraints from the lensing PDF
with observations. While there exist analytic models of the PDF based on
deviation theory [60, 61, 62] and the halo model [63, 64, 59], they are not
yet at the level of accuracy and flexibility required to incorporate the
complex survey configurations and systematics. Therefore, we use a large set
of cosmological simulations tailored to HSC Y1 data to model the lensing PDF
and its likelihood.

##  II Methods

In this section, we provide a brief overview of the HSC Y1 data, the PDF and
power spectrum measurements, the simulations suite, the likelihood, our
blinding procedure, and systematics. More detail may be found in our companion
paper on counts of lensing peaks and minima [65].

###  II.1 HSC Y1 Data

We use lensing convergence maps estimated from the HSC Y1 galaxy shapes
catalogue [66]. After applying masks, the shear maps span
$136.9\,\text{deg}^{2}$ in six spatially disjoint fields. We adopt galaxy
redshifts determined using the MLZ code [67]. After applying redshift cuts of
$0.3<z_{s}<1.5$ and restricting to sources with reliable shape measurements,
we obtained a total number density of $\sim 17\,\text{arcmin}^{-2}$. We split
source galaxies into four tomographic redshift bins with edges [0.3, 0.6, 0.9,
1.2, 1.5] and construct convergence maps via Kaiser-Squires inversion [68].

###  II.2 PDF and power spectrum

Figure 1: Example of the non-Gaussian lensing convergence PDFs from the HSC Y1
data (orange) and the fiducial simulation (blue), for the tomographic redshift
bin $0.6<z_{s}<0.9$ and with a smoothing scale $\theta_{s}=5\,\text{arcmin}$.
For most of the data points the error bars are invisible. Unfilled data points
are not used in the analysis. The PDF’s deviation from a Gaussian (black)
provides the cosmological information to be exploited in this work.

The primary summary statistic we consider is the lensing PDF, calculated as
histograms of pixels in normalized convergence maps. One crucial step in our
computation is that we define the PDF over the signal-to-noise ratio, i.e.,
each convergence map is divided by its standard deviation before its pixels
are histogrammed. This is important to remove information duplicated in the
power spectrum. The PDF’s non-Gaussian character becomes apparent in Fig. 1,
where the tails deviate from a Gaussian (which is dominated by galaxy shape
noise).

We measure PDFs on maps smoothed with Gaussian filters
$\propto\exp(-\theta^{2}/\theta_{s}^{2})$, where we choose
$\theta_{s}=\\{5,7,10\\}\,\text{arcmin}$ to mitigate systematics while
retaining non-Gaussian information. For each smoothing scale, we histogram in
19 equally spaced bins between -4 and 4 (cf. Fig. 1). Of these, we remove the
first three bins to minimize contamination by baryons [cf. 69]. Finally, we
remove the 10th bin as otherwise the linear constraint from fixing the first
three moments would render the PDF’s auto-covariance almost singular.

We also consider the auto-power spectrum for each tomographic redshift bin,
deconvolving the survey mask using the pseudo-$C_{\ell}$ method as implemented
in NaMaster [70, 71]. We measure $C_{\ell}^{\kappa\kappa}$ in 14
logarithmically spaced bins in angular multipole $81<\ell<6580$. Of these, we
remove $\ell<285$ as Ref. [72] found unmodeled systematic errors for these
scales in HSC Y1 data. Furthermore, we remove $\ell>1000$ due to possible
contamination by baryons. We note that $\ell_{\text{max}}\sim 1000$ is also
comparable to the minimum smoothing scale of $5\,\text{arcmin}$ used in the
PDF data vector. This leaves us with four $\ell$-bins. In contrast to previous
two-point analyses [73, 74] we do not include cross-spectra between
tomographic redshift bins. Upon unblinding the power spectrum-only posteriors
we found that the highest tomographic redshift bin causes significant shifts
in $S_{8}$.333This may be due to deficiencies in our simulations; for example,
the simulations use lower resolution at high redshifts which may be not
sufficient. Alternatively, the highest source redshift bin may be subject to
photo-$z$ calibration errors [7, 8]. Thus, we exclude the highest redshift bin
in our analysis.

In summary, our raw data vector consists of the PDF with 135 dimensions (3
smoothing scales $\times$ 3 redshifts $\times$ 15 bins) and the power spectrum
with 12 dimensions (3 redshifts $\times$ 4 bins).

###  II.3 Simulations

We adopt two sets of $N$-body simulations in our analysis:

  1. 1.

To model the covariance, we use 2268 realizations of the HSC Y1 footprint at a
fiducial point ($S_{8}=0.791$, $\Omega_{m}=0.279$), generated from the 108
quasi-independent [75] full-sky maps of Ref. [76].

  2. 2.

To model our statistics, we use simulations at 100 different cosmological
models with varying values of $S_{8}$ and $\Omega_{m}$, as introduced in Ref.
[77]. For each cosmology, 50 quasi-independent realizations are generated by
randomly placing observers within the periodic box. These simulations are used
to construct emulators for the mean theory prediction; thus, they are designed
to have better accuracy than the fiducial simulations.

For each of these 7768 (=2268+100$\times$50) realizations, mock galaxy shapes
are generated following Refs. [78, 79, 80]. Convergence map generation and
summary statistic measurements are performed on these simulated mocks
identically to as done for the real data.

###  II.4 Likelihood and inference

For the PDF, to reduce the size of the data vector and Gaussianize its
likelihood, we score-compress the logarithmic PDF under the approximation of a
Gaussian likelihood [MOPED, 81].444 We note that upon combining the PDF with
the power spectrum, the compression is likely sub-optimal; future work could
investigate more rigorous ways to preserve a maximum of information. Of
course, the choice of likelihood to compute the score cannot induce biases,
only information loss. This reduces the number of PDF bins from 135 to only 2,
corresponding to the number of parameters ($S_{8}$ and $\Omega_{m}$). We then
construct at each of the 100 cosmological models a 2$\times$2 covariance
matrix, using the 50 realizations. Finally, we build emulators of both the
compressed PDF and the cosmology-dependent inverse covariance using Gaussian
processes. If we use a cosmology-independent covariance the $S_{8}$-posterior
tightens and its peak is almost unchanged. However, with that choice the ranks
plot discussed below would indicate an overconfident posterior.

The power spectrum data vector is small enough and its distribution is known
to be close to Gaussian. Therefore, we apply no data compression to the power
spectrum. We build a Gaussian process emulator to model the power spectrum and
estimate a cosmology-independent covariance matrix using the fiducial
simulations. To jointly analyze the compressed PDF and the power spectrum, we
approximate the cross-correlation between them as constant and estimate it
from the fiducial simulations.

We adopt uniform priors on our parameters $S_{8}$ and $\Omega_{m}$, in
intervals $[0.5,1.0]$ and $[0.2,0.4]$, respectively. Our prior is well-covered
by the available simulations. Markov chain Monte Carlo (MCMC) sampling is
performed using emcee [82, 83].

We validate our likelihoods with a “ranks plot” [e.g., 84] shown in Fig. 2. To
do so, we run MCMC on realizations drawn from the 30 cosmology-varied
simulations within our prior. To cleanly separate our training and test sets,
all emulators are rebuilt without the test cosmology. We then order each
Markov chain by its $S_{8}$ values and find the rank of the true $S_{8}$. For
a valid likelihood, it should be impossible to statistically distinguish the
true $S_{8}$ from randomly selected Monte Carlo samples, so the ranks should
follow a uniform distribution. If the posterior is over-confident (under-
confident), the histogram exhibits a U-shape (inverse U). However, because we
only have a relatively small number of simulated cosmologies available within
our prior, a perfectly uniform distribution may not be possible to attain.
Indeed, Fig. 2 shows an approximately uniform distribution for all data vector
choices (PDF, power spectrum, and them jointly), except for spikes at the
edges for the power spectrum and joint statistic. These spikes are
attributable to a small number of simulations near the prior boundary where
the power spectrum emulation appears to work less well. We do not expect these
to affect our results as they have little overlap with our final posterior and
the problem is mostly due to underestimated tails which are less important for
the confidence levels typically considered.

Figure 2: Validations of our likelihoods. The histograms are over hundreds of
Markov chains, run with realizations from the cosmo-varied simulation set as
“observation”. The $x$-axis is the position of the simulation’s true $S_{8}$
value in the ordered chain. A uniform distribution is expected for a valid
likelihood. The grey band represents the $2\sigma$ interval approximating the
bin counts as Poissonian.

###  II.5 Systematics

Figure 3: _Left_ : Estimate of systematic errors. We contaminate the
PDF+$C_{\ell}^{\kappa\kappa}$ data vector with various systematics as
described in the text. No systematic exceeds $0.45\sigma(S_{8})$. _Right_ :
Assessment of training set and emulator quality. Each point is one of the
cosmologies for which cosmo-varied realizations exist. The bias indicated is
estimated by averaging over chains run on 25 realizations, while the emulator
is trained with the other 25 realizations and without including the cosmology
where the inference is performed.

We study potential biases caused by effects not included or wrongly described
in the simulated model by running inference on fiducial mocks contaminated
with the following methods:

  * •

We simulate miscalibration of multiplicative bias by shifting it by $\pm
1\,\%$, corresponding to the uncertainty on the mean quoted in Ref. [85].

  * •

We assess the impact of photometric redshift uncertainty by generating
fiducial mocks with source redshifts from two other codes, frankenz and
mizuki.

  * •

We model the impact of baryonic effects using the $\kappa$TNG simulations [86]
— two sets of convergence maps with and without baryons. We contaminate the
fiducial mocks with the ratio of hydrodynamic to dark matter-only data
vectors.

  * •

We estimate the impact of intrinsic alignments similarly to the baryons case
by multiplying the fiducial data vectors with a fixed ratio which we obtain
from mocks with nonlinear intrinsic alignments [87]. We choose alignment
amplitudes, $A_{\text{IA}}=\\{-0.32,1.18\\}$, comparable to the $1\sigma$
constraints in Refs. [73, 74].

We show the resulting biases on $S_{8}$ for these systematics in the left
panel of Fig. 3 (also see the more comprehensive discussion in Ref. [65]). The
most constraining joint PDF+$C_{\ell}^{\kappa\kappa}$ data vector is used. We
do not find biases exceeding $0.45\sigma$ in $S_{8}$.

In addition, we also consider potential biases caused by imperfections in the
simulations and emulators:

  * •

The mean recovered $S_{8}$ on the fiducial simulations is about $1\sigma$
higher than the input, indicating a systematic difference with the cosmology-
varied simulations. This is likely due to the lower fidelity of the fiducial
simulations. With this in mind, the observed $1\sigma$ bias can be considered
an upper bound on biases caused by resolution effects in the $N$-body
simulations.

  * •

To assess biases caused by the limited training set size, at each cosmology we
divide the realizations into 25 samples each for training and testing. We
match the realization indices (i.e., observer positions) between cosmologies
so as to maximize statistical independence between training and testing
samples. In addition, all realizations at the test cosmology are removed from
the training. The right panel of Fig. 3 shows the average biases on $S_{8}$
within our prior range, in units of the standard deviation. The bias is
computed using the mode of the marginalized $S_{8}$ posterior, to minimize the
effect of the prior which can shift the mean. The bias is a few ten percent
for most of the points, except at the edges where the emulator quality
decreases. Since the described test aggressively reduces the training set
size, our bias estimates are likely a conservative upper bound.

###  II.6 Blinding

To help build best practice for non-Gaussian statistical analyses with Stage-
IV surveys [e.g., 46], we follow a three-step blinding procedure in our
analysis:

  1. 1.

We build the mock generation pipeline using one random realization from the
fiducial model as “observation”. This includes shear bias correction,
convergence map reconstruction, masking, in-painting, and summary statistics
measurements.

  2. 2.

We construct the inference pipeline using the ranks plot and similar tests.
This includes data compression, covariances, emulators, and MCMC sampling.

  3. 3.

We select smoothing scales, redshift bins, and cuts on data vectors to
minimize the impact of systematics.

  4. 4.

First unblinding: B-modes. We compare the power spectra and PDFs measured in
B-mode maps of the HSC Y1 data and our fiducial simulation.555B-mode maps are
built by rotating galaxy shapes by 45 degrees. We validate that the two
distributions are statistically consistent.

  5. 5.

Second unblinding: power spectrum. We apply inference to the measured HSC Y1
power spectrum and compare our results internally between different redshift
bins and to the official HSC Y1 analyses by Refs. [73, 74]. We discover
systematics in the highest tomographic redshift bin and remove it from our
final analysis.

  6. 6.

Third unblinding: PDF. We unblind the PDF-only posterior during an invited
presentation at the HSC weak lensing working group telecon. The initial
posterior was approximately uniform. This failure was due to a bug in
computing the standard deviations of the HSC data maps.666 Thanks to the
feedback from the HSC weak lensing working group, we learned a valuable lesson
of comparing the PDFs before moving on to inferences. They also pointed out
that a more sophisticated blinding policy, as typically employed by
collaborations, would have flagged this issue before unblinding. The bug did
not affect our previous steps, so once we resolved it we did not change any
other parts of the pipeline.

Since the HSC Y1 data is already public our blinding procedure is rather an
honor system.777During our study, we also became aware of other recently
completed non-Gaussian statistical analyses using the same dataset [45, 88].
To avoid confirmation bias, we refrained from reading these papers until we
finalized our results.

##  III Results

Figure 4: $S_{8}$ posteriors from different data vector choices. Since the
posteriors are slightly asymmetric, we quote their modes as well as the
symmetric $68\,\%$ confidence interval. For comparison, we include $68\,\%$
intervals from Refs. [73, 74]. Note that these analyses differ in various
details. The inset compares the best-fit model to the data
($\text{PTE}=0.92$). We remind the reader that the PDF does not contain
information on the variance.

We show the $S_{8}$ posteriors obtained from the power spectrum, the PDF, and
the two combined in Fig. 4. In all cases, the constraint on $\Omega_{m}$ is
prior-dominated and thus not shown here for brevity. For reference, we also
show the $68\,\%$ intervals from the official HSC Y1 two-point analyses with
the power spectrum [73] and the two-point correlation function [74].

Our two-point result (blue) agrees well with the previous analyses in terms of
mean value. However, our posterior is wider: $\sigma\simeq 0.042$ compared to
$\sigma\simeq 0.03$ in the previous two-point analyses. This is expected as
these analyses differ in many respects, such as additional cross-spectra
between tomographic redshift bins, choice of priors,treatment of systematic
errors, data vector binning, and scale cuts.

Our PDF-only posterior (orange) shows a clear detection of $S_{8}$, or
specifically nonlinear clustering, at $9\,\sigma$. This is after removing
variance information from the PDF and thus almost independent of the two-point
results. The mean $S_{8}$ value is slightly higher than the one from
$C_{\ell}^{\kappa\kappa}$ but statistically consistent. Upon combining the PDF
and $C_{\ell}^{\kappa\kappa}$, we observe a slight upwards shift in the mean
$S_{8}$ and a $\approx\,10\,\%$ tightening of the posterior. This shows that
the PDF indeed contains complementary information to the power spectrum.

In many aspects, our results match the CNN analysis from Ref. [45], including
the $S_{8}$ posterior and an $\mathcal{O}(10\,\%)$ improvement compared to the
power spectrum.

##  IV Conclusions

In this work, we obtained the first cosmological constraints using the
probability distribution function of weak lensing convergence. We built
convergence maps using the HSC Y1 shear catalogue, constructed a likelihood
based on tailored numerical simulations, and validated that known survey and
astrophysical systematics are under control after applying cuts on smoothing
scales, redshift, and data vectors. We designed and followed a three-step
blinding procedure to minimize confirmation bias. We make our inference code
publicly available at this URL.

The PDF improves the power spectrum-only constraint by about $10\,\%$. For the
clustering amplitude, we obtained $S_{8}=0.852^{+0.086}_{-0.094}$ and
$S_{8}=0.827^{+0.033}_{-0.044}$ from the lensing PDF alone and the combination
of PDF and power spectrum, respectively ($68\,\%\,\text{CL}$). We computed the
PDF on convergence maps normalized by their standard deviation, hence
maximally removed two-point information from the PDF. Our results are
consistent with previous analyses on the same data and show that the PDF
provides additional information not contained in the power spectrum.

We find no tension between the $S_{8}$ inferred from HSC Y1 lensing and from
primordial CMB measurements.

Future work could investigate alternatives to the probably lossy compression
step as well as the forward modeling of systematics to allow inclusion of
smaller scales. Additionally, Stage-IV data will be sensitive to cosmological
parameters beyond $S_{8}$, for example, the neutrino mass sum. The PDF could
be instrumental in complementing two-point statistics in order to optimally
constrain such model extensions.

_Acknowledgements._ We thank Sihao Cheng, Will Coulton, Daniela Grandón,
Surhud More, Ken Osato, David Spergel, Sunao Sugiyama, and Masahiro Takada for
useful discussions. We thank Joachim Harnois-Déraps for sharing the IA mocks.

We thank the organizers of the Kyoto CMBxLSS workshop, during which this work
was completed.

This work was supported by JSPS KAKENHI Grant Number 23K13095 (to JL),
19K14767 and 20H05861 (to MS). This work was supported in part by the UTokyo-
Princeton Strategic Partnership Teaching and Research Collaboration. This
manuscript has been authored by Fermi Research Alliance, LLC under Contract
No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science,
Office of High Energy Physics. The authors are pleased to acknowledge that the
work reported on in this paper was substantially performed using the Princeton
Research Computing resources at Princeton University which is consortium of
groups led by the Princeton Institute for Computational Science and
Engineering (PICSciE) and Office of Information Technology’s Research
Computing.

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