\boxed{\ \ \ U_{_{G_{\mathbb{C}}}}\ \circ\ W_{_{G_{\mathbb{C}}}}\ \equiv\ T^{^*}G_{_{\mathbb{C}}} \color{white}\bigg]\ }\ \ \ \ , \ \ \ \
\boxed{\ \ \ \eta_{_{G_{\mathbb{C}}}}(V)\ \equiv\ T^{^*}G_{_{\mathbb{C}}}\color{white} \ \ }.
\end{equation}

and that

\begin{equation}
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}}},
\end{equation}

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

\begin{equation}
\text{dim}_{_{\mathbb{C}}}\ U_{_{G_{\mathbb{C}}}}\ \overset{def.}{=}\ \text{dim}_{_{\mathbb{C}}}\ G_{_{\mathbb{C}}}\ +\ \text{rank}\ G_{_{\mathbb{C}}}.
\end{equation}

\begin{equation}
\text{dim}_{_{\mathbb{C}}}\ W_{_{G_{\mathbb{C}}}}\ \overset{def.}{=}\ 3\ \text{dim}_{_{\mathbb{C}}}\ G_{_{\mathbb{C}}}\ -\ \text{rank}\ G_{_{\mathbb{C}}}.
\label{eq:dimhb}
\end{equation}

However, if the identity needs to be removed from Bo$_{_2}$, $T^{^*}G_{_{\mathbb{C}}}$ is not the identity and, in particular (<ref>) needs to be redefined precisely because the source is no longer a dual category.

As a concluding remark to what we have just said, in [126] they conjecture the following property for the moment maps associated to the $G^{^3}$ action on the Higgs branch $W_{_{G_{_{\mathbb{C}}}}}$

\begin{equation}
\boxed{\ \ \ \mu_{_i}:\ W_{_{G_{_{\mathbb{C}}}}}\rightarrow \mathfrak{g}_{_{\mathbf{C}}}^{^*}\ \ \ ,\ \ \ i=1,2,3.  \color{white}]\ \ }
\label{eq:HBB}
\end{equation}

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

\begin{equation}
\boxed{\ \ \ W_{_{G_{_{\mathbb{C}}}}}\ \equiv \ \mu^{^-1} /G^{^3}\color{white}\bigg]\ \ }.
\label{eq:HBtrue}
\end{equation}

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

§.§ Examples

These are exactly the examples we encountered in section <ref>, when dealing with Coulomb branches of star-shaped quivers.

§ KEY POINTS

The main points to bare in mind from this section are the following:

* Categorical duality ensures the presence of an identity object.

* 3D mirror symmetry requires reparametrisation-invariance.

§ DUALITIES AMONG ALGEBRAIC VARIETIES

In this section we describe how abelianisation fits in the algebro-geometric description of 3D mirror symmetry. In doing so, we explicitly explain the analogy in between different treatments, [125]. The main aim for doing so at this stage is that of introducing the reader to some crucial notions that will later be developed in more exotic setups, to which Part III and V are devoted.

The present section is structured as follows:

* At first, we overview some key features of equivariant cohomology and the convolution product, both within the context of quiver gauge theories, [134, 130].

* Then, making use of [550], we emphasise how and when 3D mirror symmetry manifests from a categorical perspective, specifically in terms of bordisms, making use of the Kostant-Whittaker symplectic reduction, [134].

* Last but not least, we close the section highlighting the importance of abelianisation for embedding the ring homology of the theory in question, explaining how this is in turn related to (un)gauging the categorical SymTFT introduced in section <ref>. In doing so, we also overview the work of Teleman, [181], thereby connecting it with Moore-Tachikawa varieties, and Coulomb branches ofstar-shaped quivers.

§.§ Equivariant cohomology, fixed point localisation and convolution product

Equivariant cohomology, [335], also known as Borel-Moore cohomology, [566], is a cohomology theory from algebraic topology which applies to topological spaces with a group action. It can be viewed as a generalisation of group cohomology. More explicitly, the equivariant cohomology ring of a space $X$ with action of a topological group $G$ is defined as the ordinary cohomology ring with coefficient ring $\Lambda$ of the homotopy quotient $EG\times_{_G}X$

\begin{equation}
H^{^{\bullet}}_{_G}(X;\Lambda)\ =\ H^{^{\bullet}}\left(EG\times_{_G} X;\Lambda\right).
\end{equation}

§.§ The Kirwan map

In section <ref> we saw that the Kirwan map, first introduced in [127], states

\begin{equation}
H^{^{\bullet}}_{_G}(M)\ \rightarrow\ H^{^{\bullet}}\left(M\dslash _{_p} G\right),
\end{equation}

where $M$ is a Hamiltonian $G$-space, i.e. a symplectic manifold acted on by a Lie group $G$ with a moment map

\begin{equation}
\mu:\ M\ \rightarrow\ \mathfrak{g}^{*}.
\end{equation}

$H^{^{\bullet}}(M)$ is an equivariant cohomology ring of $M$, i.e. the cohomology ring of the homotopy quotient $EG\times_{_G}M$ of $M$ by $G$.

\begin{equation}
M\dslash_{_p}G\ =\ \mu^{^{-1}}(p)/G
\end{equation}

is the symplectic quotient of $M$ by $G$ at a regular central value $p\in Z(\mathfrak{g}^{*})$ of $\mu$. It is defined as the map of equivariant cohomology induced by the inclusion

\begin{equation}
\mu^{^{-1}}(p)\ \hookrightarrow\ M
\end{equation}

followed by the canonical isomorphism

\begin{equation}
H^{^{\bullet}}_{_G}\left(\mu^{^{-1}}(p)\right)\ \simeq\ H^{^{\bullet}}\left(M\dslash_{_p} G\right).
\end{equation}

§ FIXED POINT LOCALISATION AND CONVOLUTION PRODUCT

In BFN, the embedding of algebras is performed under the so called convolution product. To see how this is defined, we start off with the convolution diagram for the affine Grassmannian, [134, 130]

\begin{equation}
\text{Gr}_{_G}\times\text{Gr}_{_G}\ \overset{p}{\leftarrow}\ G_{_{\cal K}}\ \times\ \text{Gr}_{_G}\ \overset{q}{\rightarrow}\ \text{Gr}_{_G}\tilde\times\ \text{Gr}_{_G}\ \overset{m}{\rightarrow}\ \text{Gr}_{_G},
\end{equation}

where $p,q$ are projections, $m$ is a multiplication, and

\begin{equation}
\text{Gr}_{_G}\tilde\times\ \text{Gr}_{_G}\ =\ G_{_{\cal K}}\ \times_{_{G_{_{\cal O}}}}\ \text{Gr}_{_G}.
\label{eq:convoldiagr}
\end{equation}

Hence, given any two $G_{_{\cal O}}$-equivariant perverse sheaves ${\cal A}_{_1}, {\cal A}_{_2}$, the pullback

\begin{equation}
p^{^*}\left({\cal A}_{_1}\boxtimes {\cal A}_{_2}\right) \ \equiv\ {\cal A}_{_1}\ \tilde\boxtimes\ {\cal A}_{_2}
\end{equation}

descends to $\text{Gr}_{_G}\ \tilde\times\ \text{Gr}_{_G}$ by equivariance, and we can define the convolution product as follows

\begin{equation}
{\cal A}_{_1}\star {\cal A}_{_2}  \ =\ m_{_*}\left({\cal A}_{_1}\ \tilde\boxtimes\ {\cal A}_{_2}\right).
\end{equation}

The latter defines a symmetric monoidal structure on the category of $G_{_{\cal O}}$-equivariant perverse sheaves on $\text{Gr}_{_G}$, and is equivariant to the monoidal category of finite-dimensional representations of the Langlands dual of $G$.

(<ref>) is the convolution diagram for the case in which $G$ is a complex reductive group and representation $\mathbf{N}=0$. For arbitrary $\mathbf{N}$, we can generalise the convolution diagram, achieving a noncommutative algebraic structure. In order to do so, we define a triple ${\cal R}=({\cal P},\varphi, s)$, where ${\cal P}$ is a $G$-bundle on a formal disk $D=\text{Spec}\ \mathbf{C}[[z]]$, $\varphi$ is its trivialisation over the punctured disk $D^{^*}=\text{Spec}\ \mathbf{C}((z))$, and $s$ is a section of the associated vector bundle ${\cal P}\times_{_G}\mathbf{N}$ such that it is sent to a regular section of the trivial bundle under $\varphi$.

§ COULOMB BRANCHES

Given a complex vector space, $V$, on a connected reductive algebraic group, $G$, with a fixed faithful linear action on $V$, the Higgs branch is defined as follows

\begin{equation}
{\cal M}_{_H}\ =\ \mu^{^{-1}}(0)\ \dslash\ G\ =\ \text{Spec}\left(\mathbf{C}\left[\mu^{^{-1}}(0)\right]^{^G}\right)
\end{equation}

\begin{equation}
\mu:\ T^{^*}V\ \rightarrow \ \mathfrak{g}
\end{equation}

denoting the moment map, whereas the Coulomb branch, ${\cal M}_{_C}$, is the spectrum of a ring constructed as a convolution algebra in the homology of the affine Grassmannian, where the choice of the representation $V$ is incorporated as certain quantum corrections to convolution in homology, which are kept track of by an auxiliary vector bundle, [134, 130]. ${\cal M}_{_H}$ and ${\cal M}_{_C}$ are conjectured to be symplectic duals to each other, with the most important relation in between them being Koszul duality between the categories of the quantised varieties, [552].

Let $G$ be a simply-connected complex semisimple group, and let $G_{_{ad}}$ be its adjoint form. The group $G_{_{ad}}$ acts on $G$ by conjugation, and $G$ contains a transverse slice $\Sigma$ for this action, which was first introduced by Steinberg, [567]. The resulting multiplicative universal centraliser is the smooth affine variety

\begin{equation}
Z_{_{G}}\ =\ \left\{(a,h)\ \in\ G_{_{ad}}\times\Sigma\ \bigg|\ a\in G_{_{ad}}^{^h}\right\}.
\end{equation}

When $G$ is simply-laced, Bezrukavnikov, Finkelberg, and Mirkovíc, [335], showed that its coordinate ring is isomorphic to the equivariant K-theory of the affine Grassmannian of the Langlands dual group $G^{^{\text{V}}}$, therefore, in this case, $Z_{_{G}}$ is an example of a Coulomb branch as defined by BFN [134, 130].

§.§ Mirror symmetry in Moore-Tachikawa varieties

We now turn to the specific case of interest to us, namely Moore-Tachikawa varieties, and their relation with the work of [102, 550, 181, 551]. First and foremost, we will briefly overview how the Higgs and Coulomb branches arise in this setup.

Given a reductive connected group, $G$, there is a functor defined as follows

\begin{equation}
\eta_{_{G_{_{\mathbf{C}}}}}:\ \ \text{Bo}_{_2}\ \rightarrow\ {\cal C}
\end{equation}

where ${\cal C}$ is a hyperbolic symplectic manifold, whose objects are affine algebraic groups, and whose morphisms

\begin{equation}
\text{Hom}_{_{\cal C}}(G_{_1}, G_{_2})
\end{equation}

are isomorphism classes of symplectic[By this we mean affine, Poisson, operationally symplectic, with a Hamiltonian chosen moment map.] varieties with Hamiltonian $G_{_1}\times G_{_2}$-action. $\eta_{_{G_{_{\mathbf{C}}}}}$ is fully specified by its action on objects and morphisms, namely

\begin{equation}
\begin{aligned}
&\eta_{_{G_{_{\mathbf{C}}}}}\left(S^{^1}\right)\ \mapsto\ G\ \ \ \ \ \ \ \ ,  \ \ \ \ \ \ \ \ \eta_{_{G_{_{\mathbf{C}}}}}\left(S^{^2}\right)\ \mapsto\ Z_{_G}\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \eta_{_{G_{_{\mathbf{C}}}}}\left(S^{^1}\times S^{^1}\right)\ \mapsto\ T^{^*}G\\
\end{aligned}
\label{eq:maps}
\end{equation}

where $Z_{_G}$ is the universal centraliser, consisting of all the pairs $(x,g)$ with  $x\in\mathfrak{g}$, $\mathfrak{g}$ Lie, regular, $g\in G$ and $ad(x)_{_{\mathfrak{g}}}=x$. Given $K\subset\mathfrak{g}$ a Kostant slice, and $(e, h,f)$ a principal $\mathfrak{sl}_{_2}$-triple in $\mathfrak{g}$, the Kostant slice can be rewritten as follows

\begin{equation}
\end{equation}

with $Z(f)$ the centraliser of $f$ in $\mathfrak{g}$, $\mathfrak{g}\simeq \mathfrak{g}^{^*}$. For a given Kostant slice, the universal centraliser can therefore be redefined as follows

\begin{equation}
Z_{_G}=\left\{(g, x)|g\in G, x\in K, \text{ad}_{_{\mathfrak{g}}}(x)=x\right\}.
\end{equation}

What we have outlined so far can be suitably adapted to the case of interest to us, namely when $\Sigma$ is a Riemann surface whose boundary has $n$ components, from which one gets a symplectic variety $X$ with a $G^{^n}$-action. This symplectic variety, $X$, is often times referred to as the Higgs branch.

§.§.§ The Kostant-Whittaker symplectic reduction

The action of the functor explicited in (<ref>), however, is not exhaustive. Indeed, there is one more piece of information that is needed in order for the 2D TFT to be fully determined, and this is its action on the bordism corresponding to the three-punctured sphere.

To see how this relates to quivers, let us consider the case in which $X$ is a symplectic variety with $G$-action. Let $N\subset G$ be a maximal map, and

\begin{equation}
\chi:\ \ N\ \longrightarrow\ \ \mathbf{C}
\end{equation}

a non-degenerate homomorphism such that

\begin{equation}
N\ \bigg/\ [N,N]\ \simeq\ \bigoplus_{_{{\alpha}_{_i}}}\ \mathbf{C}_{_{\alpha_{_i}}},
\end{equation}

where $\alpha_{_i}$ denote the simple roots. Then, the Kostant-Whittaker reduction on the symplectic variety corresponds to the quotient

\begin{equation}
KW(X)\ =\ \mu^{^{-1}}_{_N}(\chi)\ \bigg/\ N,
\end{equation}

and the universal centraliser becomes

\begin{equation}
Z_{_G}\ \simeq\ KW_{_{G\times G}}\left(T^{^*}G\right).
\end{equation}

Furthermore, we can take

\begin{equation}
G\times K = KW_{_{G^{^\text{V}}}}\left(T^{^*}G\right)  \ \ \ \ \ , \ \ \ \ \ KW_{_G}(G\times K)\ =\ Z_{_G},
\end{equation}

meaning that filling in a hole on the sphere corresponds to performing a $KW_{_G}$ reduction. Note that this is a statement of 3D mirror symmetry.

According to the Moore-Tachikawa conjecture, overviewed in slightly more detail in appendix <ref>, a 2D TFT is an example of a 2-categorical structure, that is the Alday-Gaiotto-Tachikawa dual of a supersymmetric 4D theory arising from dimensional reduction of 6D ${\cal N}=(2,0)$ SCFTs of type $A_{_{N-1}}$ on a sphere with $N$ maximal punctures. These theories, oftentimes denoted ${\cal T}_{_{N}}\left[\Sigma_{_{0,k}}\right]$, upon being further dimensionally reduced on a circle, ${\cal T}_{_{N}}\left[\Sigma_{_{0,k}}\times S^{^1}\right]$, are known to be related to Sicilian 3D ${\cal N}=4$ theories, the so-called star-shaped quivers, upon taking the $\underset{S^{^1}\rightarrow0}{\lim}$, [550].

In [560], the authors calculated the Hilbert series of the chiral ring

\begin{equation}
\mathbf{C}\left[{\cal M}_{_{C}}\right]\ \simeq\ \mathbf{C}\left[{\cal M}_{_{H}}^{^{4D}}\right]
\end{equation}

exaclty with the intention of obtaining new information about the structure of the Higgs branches of the Higgs branch of the 4D theory, ${\cal M}_{_{H}}^{^{4D}}$, via a direct analysis of the corresponding Coulomb branches ${\cal M}_{_{C}}$ of ${\cal T}_{_{N,k}}$, for general $N$ and $k$. In addition to this, our work wishes to emphasise the importance of the calculation of the Hilbert series together with the correct identification of the ring elements and relations in between them, which plays a crucial role in developing connection with Particle Physics and calculations therein. In achieving the second objective, a crucial role is played by the so-called abelianisation procedure, first introduced by [551], and subsequently emphasised also in [550].

§.§ Homological mirror symmetry

We now turn to a more general setup, first outlined by [181]. In our brief overview, we highlight the importance of its relation with our main example, <ref>, and how it fits within the context of abelianisation. As we shall see, the key expression that we need to consider is (<ref>).

At first, let us briefly overview the notion of homological mirror symmetry in its first formulation, [136].

Homological mirror symmetry, consists in a proposed agreement between two categories, namely:

* Fukaya's $A_{_{\infty}}$-category, ${\cal F}(X)$, on the symplectic side,

* The derived category with Yoneda structure, $D^{^b}\mathfrak{Coh}(X^{^{\text{V}}})$, on the complex side.

For completeness, we now turn to explain this terminology.

§.§ Fukaya categories

A symplectic manifold, $X$, is a smooth manifold of even dimension equipped with a non-degenerate symplectic form $\omega\ \in\ \Omega^{^2}_{_{cl}}(X)$. A Fukaya category, [142], of a symplectic manifold, $X$, is an $A_{_{\infty}}$-category with Lagrangian submanifolds $L\ \subset\ X$ as objects, whose intersections define the Hom-space.
An $A_{_{\infty}}$-category is a category with associativity condition ($A)$ relaxed without bound on degrees of homotopies ($\infty$). They are linear categories, i.e. their Hom-objects are chain complexes.

§.§ Yoneda structure

Yoneda structure, [345], provides in a 2-categorical setting the axiomatic description of the formal properties of the usual presheaf construction and Yoneda embedding of locally smaller categories.

The axioms of Yoneda structure capture the properties of the presheaf construction with CAT replaced by a general 2-category, ${\cal K}$. In order to handle size issues, a class of admissible or legitimate 0-cells is singled-out in $|{\cal K}|$. In order to handle size issues, as well as a class of 1-cells that behave well w.r.t. this class and the presheaf construction. In fact, it suffices to describe the admissible 1-cells, since one can identify the admissible 0-cells with the admissible identity 1-cells.

Let ${\cal K}$ be a 2-category, and $\mathbb{A}$ a class of 1-cells. The 1-cells of $f\ \in\ \mathbb{A}$ are called admissible if for all $f\ \in\ \mathbb{A}$ and composable 1-cells $g\in{\cal K}$, $f\circ g\in\mathbb{A}$. In this setup, a 0-cell $C\in|{\cal K}|$ is admissible if id$_{_C}$ is.

A Yoneda structure is therefore defined as a pair $({\cal P}, \mathbb{A})$, where ${\cal P}$ is a pre-sheaf construction for $\mathbb{A}$ assigns to every admissible object $A\in|\mathbb{A}|$ and object ${\cal P}A\in|{\cal K}|$ called its object of presheaves, and an admissible 1-cell

\begin{equation}
y_{_A}:\ A\ \rightarrow\ {\cal P}A
\end{equation}

called its Yoneda morphism subject to three conditions:

§.§.§ Relation to Factorisation Homology

As we shall see later on in section <ref>, this formulation is compatible with that of factorisation homology for class ${\cal S}$ theories, first outlined in our work, [5].

§.§.§ Relation to our main example

The main point, to which we will be coming back later on in Part V is that the morphisms in the category of compact contractible circles encountered in Moore-Tachikawa varieties are the 1-cells featuring in the Yoneda structure description.

§.§.§ Lagrangian submanifolds

A Lagrangian submanifold of a symplectic manifold[Note that here we are referring to Calabi-Yau 3-folds, hence $X$ is of even dimensions.], $L\ \subset\ X$, is a submanifold which is a maximal isotropic submanifold on which $\omega=0$. They constitute the leaves of real polarisations, and are, therefore, crucial elements of symplectic geometry.

There is an equivalence of categories between the category of $\mathbb{C}$-linear representations of a quiver ${\cal Q}$ and the category of left $\mathbb{C}{\cal Q}$-modules.

The upper and lower blue arrows correspond to geometric and algebraic resolutions of ${\mathfrak{M}^{0}}$, respectively.

§.§  Moment map and higher homologies

In general, homologies of derived representation schemes can be highly nontrivial. However, in this particular case, one can identify a necessary and sufficient condition for the vanishing of the higher homologies based on the flatness of $\mu$, [330]. In particular, in such reference, it was shown that the derived representation scheme DRep$_{_{v,w}}({\cal A})$ has vanishing higher homologies if and only if $\mu^{^{-1}}(0)\ \subset\ M({\cal Q},v,w)$ is a complete intersection, which happens only if the moment map is flat, [341]. As we shall see, the requirement of the algebraic variety to be a complete intersection is crucial for the purpose of our treatment. In particular, it ensures the emergence of a 2-categorical structure, whose importance will be the core topic of section <ref>.

Prior to that, we devote the following subsection to explain the role played by magnetic quivers for describing generalisations of homological mirror symmetry in quiver gauge theories with 8 supercharges.

§.§ 3D mirror symmetry

We now briefly overview one of the main dualities of interest to us, namely 3D mirror symmetry. The literature exploring this topic is extremely vast, and we are aware of the fact that our reference list is by no means complete. TO the extent possible, we will be referring to the works that are more closely related to the purpose and aims of our work. From the physical point of view, the pioneering work dates back to the joint work of Intriligator and Seiberg, [135], which in turn is related to that of CY-manifolds, developed in the mathematical community.

As initially quoted in section <ref>, CY-manifolds are the target of functors defining CY-categories.

The aim of this and the next section is that of geometrising the 3D mirror symmetry outlined in our main example in section <ref> as a duality exchanging Coulomb and Higgs branches of 3D ${\cal N}=4$ SCFTs. This aim was already achieved by Kapustin, Rozansky, and Saulina, [334], who were the first to address this in the context of Rozansky-Witten (RW) theories, [329].

The simplest topological BCs in the RW model with target $X$ correspond to complex Lagrangian submanifolds of $X$ corresponding to the fact that one associates a boundary condition to a fibration over a complex Lagrangian submanifold whose fibration is a CY-manifold. The set of BCs for the RW model with target $X$ should be thought of as a set of objects of a 2-category.

§.§.§ Categorical formulation

Gromov-Witten (GW) theory is among the most studied 2D TQFTs, [144]. It assigns to a compact symplectic manifold $X$ a space of states $H^{\bullet}(X)$. Surfaces with points labelled by states give correlators counting pseudo-holomorphic maps to $X$ with coincidence conditions.

Mirror symmetry reduces GW theory to calculations in the complex geometry of a mirror manifold, $X^{^V}$, introduced by Kontsevich, [136].

The key idea is that, including all surfaces with corners, and BCs (branes) forming a linear category with structure, this category should be able to determine all invariants.

Mathematically, the description starts with a group action acting on the generating category (the Fukaya category), which is factored through the topology of the group. Their mirror description involves holomorphic symplectic manifolds and Lagrangians related to the Langlands dual group. As an application: the complex mirrors of flag varieties proposed by Marsh and Rietsch, [562], [We refer to section <ref> for preliminary definitions. ].

The aim is that of studying equivariance in the higher-algebra surrounding TQFTs. The homological information lies in the neighborhood of a Lagrangian within $X$ (a symplectic manifold), whereas most of the interesting physics happens elsewhere. The emerging geometric picture is the following: representations admit a character theory, but characters are now coherent sheaves on a manifold related to the conjugacy classes, instead of functions. The manifold in question is the Bezrukavnikov-Finkelberg-Mirkovic (BFM) space of the Langlands dual Lie  group, $G^{^V}$, and is closely related to the space of conjugacy classes in the complex group $G_{_{\mathbb{C}}}^{^V}$.   Multiplicity spaces of $G$-invariant maps between linear representations are now replaced by multiplicity categories, whose dimensions are the Hom-spaces in the category of coherent sheaves.

There is a preferred family of simple representations that foliate the BFM space. Every such representation is symplectically-induced from a 1D representation of a certain Levi subgroup of $G$: it is the Fukaya category of a flag variety of $G$ (similarly to holomorphic induction, the Borel-Weil construction of irreducible representations of $G$). This is the same governing structure emerging in KRS, where BCs of the 3D TQFT are associated to a holomorphic Lagrangian submanifold.

Meta-statement, [181]: Pure topological gauge theory in 3D for a compact Lie group $G$ is equivalent to the Rozansky-Witten theory for the BFM space of the Langlands dual Lie group $G^{^V}$.

2D TQFT can be thought of as a higher analogue of cohomology (Fukaya-Floer theory of a symplectic manifold is like a refinement of  ordinary cohomology), according to which gauged TQFTs are described in terms of equivariant cohomology. This describes categorification of representation theory of a compact Lie group $G$, with representations being topological.

In the RW model, one must describe geometrically two functors from the 2-category of linear categories with $G$-action to linear categories:

* The forgetful functor, remembering the underlying category (describing the pre-gauged TQFT). This is represented by a regular representation of $G$, i.e. categories of coherent sheaves over the Lagrangians Spec$H_{*}(\Omega G)$.

* The invariant category, generating the gauged TQFT, represented by a trivial representation of $G$, $\mathfrak{t}_{_{\mathbb{C}}}/W\equiv T^*_{_{1}}T^{^V}_{_{\mathbb{C}}}/W$.

§.§ From Homological mirror symmetry to 3D mirror symmetry

The concluding part of this section is a very succinct overview of some key features emerging in the work of [181], which is of particular importance to us. As we shall see, his work is deeply related to that of Moore and Tachikawa, thanks to their relation to Coulomb branches of star-shaped quivers, briefly overviewed in section <ref>.

To the best of our knowledge, the correspondence in between these three setups has only been spelled out in our work, [3]. The crucial step in understanding these different setups is looking at how mirror symmetry is realised in each case. As we shall see, this builds a dictionary that can be used for furthering the study of more exotic underlying mathematical structures.

§.§ The 2-category of KRS and the Bezrukavnikov-Finkelberg-Mirkovic (BFM) space

The image of a point $Z$(pt) is an object in the 3D bordism 3-category. Lurie's generator for pure 3-dimensional gauge theory should have categorical depth 2. His proposal for such generator is a 2-category associated to a certain holomorphic symplectic manifold, whose existence had already been conjectured by KRS and KR. When $X$ is compact, this 2-category should generate the RW-theory of $X$. In particular, its Hochschild cohomology (a 1-category with braided tensor structure), should be a dg-refinement of the derived category of coherent sheaves on $X$, $\sqrt{\mathfrak{Coh}(X)\ }\equiv$ KRS$(X)$. Among objects in KRS$(X)$ are smooth holomorphic Lagrangians $L\subset X$. More general objects are coherent sheaves of ${\cal O}_{_{L}}$-linear categories on such $L$. Given two Lagrangians, $L,L^{\prime}\in X$, Hom$(L,L^{\prime})$ is a sheaf of categories supported on $L\cap L^{\prime}$, and a $(\mathfrak{Coh}(L),\otimes)-(\mathfrak{Coh}(L^{\prime}),\otimes)$ bimodule.

Localising at $L$, we choose a formal neighbourhood identified symplectically with $T^*L$ such that we can regard (locally) as the graph of a differential d$\Psi$ for a potential function $\Psi:L\rightarrow\mathbb{C}$. Locally, when this identification is valid, Hom$(L,L^{\prime})\equiv$MF$(L,\Psi)$. The corresponding figure is <ref>.

BFM space for an abelian gauge group, [181].

For the non-abelian case, instead, this description in terms of complex manifolds and potentials is inadequate, since the BFM space is not quite a cotangent bundle. The corresponding figure is <ref>.

BFM space for a non-abelian gauge group, [181].

§.§.§ Relation to our main example

The Hom category Hom$(L,L^{\prime})$ for two Lagrangians $(L,L^{\prime})\subset X$ with finite intersection supplies a 2D TFT for framed surfaces (following from matrix factorisations). $X$ itself aims at defining RW-theory, and each $L,L^{\prime}$ is a BC for it.

\begin{equation}
L,L^{\prime}:\text{Id}\rightarrow \text{RW}_{_{X}}
\end{equation}

are morphisms from the trivial 3D theory Id to RW$_{_{X}}$ viewed as functors from Bord$_2$ to the 3-category of linear 2-categories and Hom($L,L^{\prime})$ of natural transformations between these morphisms is the generator of the theory in [181]. Note that this is equivalent to the definition of the triple (<ref>) in our main example in section <ref>, since the $RW_{_X}$-theory lives on a Calabi-Yau manifold, and the Lagrangians $L, L^{^{\prime}}$ act as the left- and right- moment maps projecting the identity element Id. This basically shows that the method put forward by [181] is an abelianisation procedure enabling to foliate the BFM space for any given pair of Lagrangian submanifolds.

The Hochschild homology

\begin{equation}
\text{HH}_{_{\text{dim} L}}\text{Hom}(L,L^{\prime})\simeq\Gamma\left(L\cap L^{\prime};(\omega\otimes\omega^{\prime})^{^{1/2}}\right)
\end{equation}

A domain wall between different TQFTs generalises the notion of BCs. This is an adjoint pair of functors between the TQFTs meeting certain dualisability conditions. A BC is a domain wall with a trivial TQFT. Just as a holomorphic Lagrangian is expected to define a BC for RW$_{_{X}}$, a holomorphic Lagrangian correspondence

\begin{equation}
X\ \leftarrow\ C\ \rightarrow\ Y
\end{equation}

should define a domain wall between RW$_{_{X}}$ and RW$_{_{Y}}$. This can be used for comparing gauge theories of different groups.

\begin{equation}
\text{BFM}(G^{^V})\ \longleftarrow\ \text{BFM}(G^{^V})\times_{\mathfrak{t}_{_{\mathbf{C}}}/W}\mathfrak{t}_{_{\mathbb{C}}}\ \longrightarrow\ T^*T^{^V}_{_{\mathbb{C}}}
\end{equation}

The correspondence matches an adjoint pair of restriction-induction functors between categorical $T$- and $G$-representations. Induction from a category ${\cal C}$ with topological $T$-action is affected by string topology with coefficients in the flag variety $G/T$:

\begin{equation}
\text{Ind}({\cal C})\equiv C_{_{*}}\Omega G\otimes_{_{C_{_{*}}\Omega T}}\ {\cal C}
\end{equation}

Restricting the $G$-action to $T$, the flag manifold $G/L$ is a transformation from $L$-gauge theory to a $T$-gauge theory, given by composition of the symplectic induction and string topology domain walls

\begin{equation}
T(L^{^V})\ \overset{\text{SInd}}{\hookrightarrow}\ T(G^{^V})\ \underset{\text{Toda}}{\overset{~}{\rightarrow}}
\text{BFM}(G)\ \overset{\text{ST}}{\rightarrow} \ \text{BFM}(T^{^V})\ \equiv\ T^*T^{^V}_{_{\mathbf{C}}}
\end{equation}

The flag manifold $G/L$ is therefore a transformation from $L$-gauge theory to $T$-gauge theory given by the composition of symplectic induction and string topology domain walls.

One of the main aims of the work by KRS was that of furthering the understanding of 3D mirror symmetry in terms of homological mirror symmetry between categories of Lagrangian submanifolds. In the concluding part of this work, we show that this correspondence can equivalently be realised in terms of the magnetic quiver prescription, which can be further extended to the case involving non-invertible defects in presence of two cones[We will come back to this in section <ref>.].

§ HOLOGRAPHIC DUALITIES AND BEYOND

Main types of dualities arising in due course in the remainder of our treatment.

§.§ Holographic duality

Inspired by black hole mechanics, the Holographic Principle was first proposed by Gerard 't Hooft, and subsequently given a more precise string-theoretic interpretation by Susskind. It is a property of String Theory as well as a supposed property of QG, stating that, the description of a volume of space can be encoded in its boundary. The AdS/CFT correspondence provides a non-perturbative formulation of String Theory with certain boundary conditions, and constitutes one of the most successful realisations of the Holograpic Principle.

One of the original motivations for studying the AdS/CFT correspondence was developing a non-perturbative formulation of string theory.

§.§ The 4D  SYM/ string theory on AdS case

For the particular case of 4D ${\cal N}=4$ SYM/ string theory on AdS$_{_5}\times S^{^5}$, symmetries play a crucial role in establishing the holographic correspondence. Specifically, the isometry invariance group of AdS$_{_5}$, namely $SO(4,2)$, corresponds to the symmetry group of CFT$_{_4}$s, whereas the isometry group of $S^{^5}$, $SO(6)\simeq SU(4)$, corresponds to the R-symmetry group of ${\cal N}=4$ SYM.

This statement can be generalised to arbitrary dimensions, in the sense that the isometry group of AdS$_{_{D+1}}$ is $SO(D,2)$, corresponding to the conformal group in $(D-1,1)$ dimensions. The boundary of AdS$_{_{D+1}}$ is $\mathbb{R}^{^D}$ in Poincare' coordinates, and $\mathbb{R}_{_t}\times S^{^{D-1}}$ in global coordinates, which is conformal to $\mathbb{R}^{^D}$. Figure <ref> shows this.

A crucial feature of the correspondence is the fact that AdS acts as a quantum mechanical box, meaning light can travel from the bulk to the boundary in finite time, implying the two are in causal contact with each other. Consequently to this, unlike the case of Minkowski spacetime, the notion of S-matrix falls short from being well-defined in AdS. On the other hand, the well-defined observables at hand are correlation functions of fields, with sources placed on the conformal boundary.

For the purpose of our treatment, we only need to state what the key points of the holographic correspondence imply from the point view of the bulk-boundary interaction, which is what we will now turn to.

In its original formulation, one expects there to be a correspondence in between the theory of open strings living on $N$ $D$-branes, i.e. ${\cal N}=4$ SYM with $SU(N)$ gauge group, and the gravitating theory corresponding to fields of the Hawking radiation existing in the background curved by the D-branes. Indeed, the Hawking radiation process plays a crucial role in the AdS/CFT correspondence. Considering the $D$-branes as endpoints of open strings, then string theory with D3-branes has three ingredients:

* The open strings existing on the D3-branes, giving a theory that reduces to 4D ${\cal N}=4$ SYM in the low-energy limit.

* The closed strings existing in the bulk spacetime, giving a theory that is SUGRA coupled to the massive modes of the string. In the low-energy limit, only SUGRA will remain.

* The interactions in between the two theories, of which Hawking radiation is a manifestation.

§.§ Generalisations

To better understand the quantum aspects of our 4D world, some have considered lower-dimensional mathematical models, where spacetime is assumed to be 3-dimensional. The advantage of such a lower-dimensional setup is the fact that the description of the gravitational field is greatly simplified, and one can therefore use standard QFT techniques, circumventing the need to resort to string theory in 4D.

Two of the most important proposals for such lower-dimensional correspondence are the following:

* In 1986, Brown and Henneaux, [196], realised that quantum gravity in 3D is closely related to a particular CFT$_{_2}$, namely Liouville Field Theory.

* Edward Witten related 3D gravity in AdS to a CFT with monster group theory.

Both cases provide examples of the AdS/CFT correspondence that do not require the full apparatus of string or M-theory.

§.§.§ Liouville Theory

Liouville Theory describes the dynamics of a field, $\varphi$ on a 2D space, subject to the potential

\begin{equation}
\end{equation}

where $b$ plays the role of the coupling constant. Its action reads

\begin{equation}
S[\varphi]=\frac{1}{4\pi}\int d^{^2}x\sqrt{g\ }\left(g^{^{\mu\nu}}\partial_{_{\mu}}\varphi\partial_{_{\nu}}\varphi+Q{\cal R}\varphi+\lambda^{^{\prime}}e^{^{2b\varphi}}\right),
\label{eq:liouville}
\end{equation}

where $g^{^{\mu\nu}}$ is the 2D metric, ${\cal R}$ the Ricci scalar, and $\lambda^{^{\prime}}$ is often referred to as the cosmological constant.

The equations of motion from (<ref>) read

\begin{equation}
\frac{1}{\sqrt{g\ }}\partial_{_{\mu}}\left(\sqrt{g\ }g^{^{\mu\nu}}\partial_{_{\nu}}\right)\varphi=\frac{1}{2}Q{\cal R}+\lambda^{^{\prime}}be^{^{2b\varphi}}.
\label{eq:liouville}
\end{equation}

In Euclidean signature, (<ref>) reduces to

\begin{equation}
\left(\frac{\partial^{^2}}{\partial x^{^2}}+\frac{\partial^{^2}}{\partial y^{^2}}\right)\varphi=\frac{1}{2}Q{\cal R}+\lambda^{^{\prime}}be^{^{2b\varphi}}.
\label{eq:liouvilleeucl}
\end{equation}

Assuming the Euclidean metric takes the following form

\begin{equation}
ds^{^2}=g_{_{\mu\nu}}dx^{^{\mu}}dx^{^{\nu}} = dzd\bar z,
\label{eq:liouvilleeuclmetric}
\end{equation}

the components of the energy-momentum tensor read

\begin{equation}
T_{_{z\bar z}}=T_{_{\bar z z}}=0
\end{equation}

\begin{equation}
T_{_{z z}}=\left(\partial_{_z}\varphi\right)^{^2}+Q\partial_{_z}^{^2}\varphi
\ \ \ \ \ \ \ \ , \ \ \ \ \ \ \ \
T_{_{\bar z \bar z}}=\left(\partial_{_{\bar z}}\varphi\right)^{^2}+Q\partial_{_{\bar z}}^{^2}\varphi,
\label{eq:nonv}
\end{equation}

\begin{equation}
\partial_{_{\bar z}} T_{_{z z}}=0
\ \ \ \ \ \ \ \ , \ \ \ \ \ \ \ \
\partial_{_z}T_{_{\bar z \bar z}}=0.
\end{equation}

Each one of the non-vanishing components (<ref>) generates a Virasoro algebra with central charge

\begin{equation}
\end{equation}

For such Virasoro algebras, $e^{^{2\alpha\varphi}}$ is a primary field with conformal dimension

\begin{equation}
\Delta=\alpha(Q-\alpha).
\end{equation}

For the theory (<ref>) to be conformally-invariant, the field $e^{^{2b\varphi}}$ must be exactly marginal, i.e. it must have conformal dimension such that

\begin{equation}
\Delta(b)=1\ \ \ \ \ \ \Rightarrow\ \ \ \ \ \ Q=b+\frac{1}{b}.
\label{eq:confinv}
\end{equation}

The last condition in (<ref>) ensures $(c, \Delta)$ is invariant under the duality

\begin{equation}
b\ \longrightarrow\ \frac{1}{b}.
\label{eq:confinvduality}
\end{equation}

§.§ Relation to other theories

Liouville theory admits two different supersymmetric extensions, namely with ${\cal N}=1$ and ${\cal N}=2$. Furthermore, it can be obtained from the WZW model.

Liouville theory also appears in the context of bosonic string theory. In its semiclassical limit it coincides with Jackiw-Teitelboim (JT)-gravity, which is another formulation of 2D gravity. To see how JT-gravity can be recovered from Liouville theory, we start from a double Liouville theory built combining a spacelike and a timelike Liouville theory.

To begin with, we start by defining the following CFT action

\begin{equation}
S_{_L}=\int d^{^2}x\sqrt{g\ }\left[\left(\partial_{_{\mu}}\phi\right)^{^2}-\left(\partial_{_{\mu}}\chi\right)^{^2}+\mu e^{^{2b\phi}}-\mu e^{^{-2b\chi}}\right] +2\int d\theta\  (Q\phi+q\chi) {\cal K}^{^{(0)}}.
\label{eq:doubleliouville}
\end{equation}

Under the field redefinition

\begin{equation}
\Phi\overset{def.}{=}-\frac{1}{2b}(\phi+\chi)\ \ \ ,\ \ \ \rho\overset{def.}{=}\frac{b}{2}(\phi-\chi),
\end{equation}

(<ref>) reduces to

\begin{equation}
S=\int d^{^2}x\sqrt{g\ }\left[-4\left(\partial\rho\right)\left(\partial\Phi\right)-2\mu e^{^{2\rho}}-\sinh\left(2b^{^2}\Phi\right)\right] -4\int d\theta\  \Phi{\cal K}^{^{(0)}}.
\label{eq:doubleliouville1}
\end{equation}

Taking $\rho$ to be the conformal factor of the JT-gravity metric

\begin{equation}
\end{equation}

(<ref>) ultimately becomes

\begin{equation}
S=-2\int d^{^2}x\sqrt{g_{_{JT}}\ }\left[\Phi{\cal R}+\mu e^{^{2\rho}}-\sinh\left(2b^{^2}\Phi\right)\right] -4\int d\theta\ \sqrt{\gamma_{_{JT}}\ } \Phi{\cal K},
\label{eq:JTfromL}
\end{equation}

which in the semiclassical limit $\left(\text{namely}\  \underset{b\rightarrow 0}{\lim}\  \text{with}\  \mu b^{^2}=1 \right)$ is exactly equivalent to JT-gravity.

§.§ The Chern-Simons/Wess-Zumino-Witten (CS/WZW) correspondence

One realisation of the holographic principle in QFT between 3D $G$-CS theory as the bulk field theory, and the 2D WZW model on a suitable Lie group $G$ as the boundary theory. This case was known and understood well before the holographic principle was first formulated.

Unlike the AdS/CFT correspondence, this is an actual theorem, stating the normal equivalence in between the space of quantum states of CS theory on a surface $\Sigma$, and the space of conformal blocks on the WZW model.

The FRS construction/classification provides a rigorous classification of all rational full CFT$_{_2}$ s, where full means that these theories are defined at all genera, namely on all 2D cobordisms.

The construction of CFT$_{_2}$s relies on the following:

* A local geometric aspect.

* A global topological aspect. Taking in the spaces of conformal blocks one element such that this choice globally fits together under gluing of 2D cobordisms defined the sewing constraint.

Because of this, the FRS formalism provides a mathematical formulation of the holographic correspondence.

Given a vertex operator algebra (VOA) that controls the CFT locally, then, for RCFTs, their representation category is a MTC (in turn obtained from, either, a Hopf algebra, or loop groups representations), and the Reshetikhin-Turaev or Turaev-Viro construction defines the 3D TQFTs, namely the 3D CS theory.

Full CFT$_{_2}$ s are relevant in String Theory, since these correspond to the vacua that string perturbation theory series are perturbing about. The FRS classification is part of the mathematical identification of the landscape of string theory vacua,

RCFT$_{_2}$s are a small part of all possible CFT$_{_2}$s (indeed, usually, any deformed CFT$_{_2}$ is non-rational).
However, RCFT$_{_2}$s (namely the WZW-model) plays an important role as the internal parts of string theory KK-compactifications.

§.§ Jackiw-Teitelboim (JT)-gravity from CS-theory

In its first-order formulation, the Einstein-Hilbert action for 3D $SL(2,\mathbf{R})$ CS gravity is defined as the difference of two CS actions

\begin{equation}
S_{_{EH}}\ =\ S_{_{CS}}[A]-S_{_{CS}}[\bar A].
\label{eq:ehcs3d}
\end{equation}

For AdS$_{_3}$ in Lorentzian signature, the connections $A$ and $\bar A$ are $\mathfrak{sl}(2,\mathbf{R})$-valued 1-forms. In the fundamental representation, the generators of the Lie algebra $\mathfrak{sl}(2,\mathbf{R})$ read

\begin{equation}
\frac{1}{2}\ \ \ \ \ 0\\
0\ \ -\frac{1}{2}\\
\end{matrix}\right)\ \ \ \  ,  \ \ \ \    L_{_1}=\left(\begin{matrix}
\ 0\ \ \ \ \ \ \ 0\\
-1\ \ \ \ \ \ 0\\
\end{matrix}\right)\ \ \ \  ,  \ \ \ \ L_{_{-1}}=\left(\begin{matrix}
0\ \ \ \ 1\\
0\ \ \ \ 0\\
\end{matrix}\right)\ \ \ ,
\end{equation}

obeying the commutation relation

\begin{equation}
\left[L_{_m}, L_{_n}\right]=(m-n)\ L_{_{m+n}}.
\end{equation}

The e.o.m. for the connections in (<ref>) read

\begin{equation}
F=dA+A\wedge A=0\ \ \ \ \ \ \ ,\ \ \ \ \ \ \ \bar F=d\bar A+\bar A\wedge \bar A =0
\end{equation}

whose solution is pure gauge

\begin{equation}
A=g^{^{-1}}dg\ \ \ \ \ ,\ \ \ \ \ \bar A=\bar g^{^{-1}}d\bar g  \ \ \ \ \ \, \ \ \ \ g, \bar g\in\ \mathfrak{sl}(2,\mathbf{R}).
\label{eq:solAAbar}
\end{equation}

In terms of $(t,z,r)$-coordinates, where $r$ denotes the holographic direction, upon choosing suitable boundary conditions, (<ref>) reads as follows

\begin{equation}
A=b(r)(d+a(z))b(r)\ \ \ \ \ \ \ ,\ \ \ \ \ \ \ \ b(r)=r^{^{L_{_{o}}}}
\end{equation}

\begin{equation}
\bar A=\bar b(r)(d+\bar a(z))\bar b(r)^{^{-1}}\ \ \ \ \ \ \ ,\ \ \ \ \ \ \ \bar b(r)=r^{^{\bar L_{_{o}}}}.
\end{equation}

In presence of a boundary, the CS action needs to be equipped with a suitable boundary term, ensuring  admits a well-defined variational principle

\begin{equation}
\delta S_{_{EH}}\ =\ \delta S_{_{CS}}[A]-\delta S_{_{CS}}[\bar A]
\end{equation}

\begin{equation}
\Rightarrow\ \delta S_{_{EH}}+\delta S_{_{bdy}}\ =\ \frac{1}{4\pi G}\int_{_{\partial_{_{M_{_{3}}}}}}\ \epsilon_{_{ab}}f^{^a}\wedge\delta e^{^b},
\end{equation}

with $e^{^a}$ the source and $f^{^a}$ the expectation value of the dual operator. In particular, the operator dual to the boundary vielbein is the stress tensor

\begin{equation}
\delta S\ =\ 4\int_{_{\partial_{_{M_{_{3}}}}}}\ d^{^2}x\ (\text{det}\  e)\ T^{^i}_{_a}\delta e^{^b}.
\end{equation}

Interestingly, upon choosing a particular boundary term and subsequently taking the dimensional reduction of either CS action, we end up with a BF-theory

\begin{equation}
\begin{aligned}
8\pi G S_{_{CS}}\ &=\ \frac{1}{2}\int_{_{M_{_{3}}}}d^{^3}x\ \epsilon^{^{\mu\nu\rho}}\ \text{Tr}\left(A_{_{\mu}}\partial_{_{\nu}}A_{_{\rho}}+\frac{2}{3}A_{_{\mu}}A_{{\nu}}A_{_{\rho}}\right) \\
&=\int_{_{M_{_{3}}}}d^{^3}x\ \text{Tr}\left(A_{_{\varphi}}F_{_{tr}}+A_{_r}\partial_{_{\varphi}}A_{_t}\right)+\frac{1}{2}\oint_{_{\partial M_{_{3}}}}\text{Tr} A_{_t}^{^2}.
\label{eq:bdyth}
\end{aligned}
\end{equation}

The corresponding 2D theory living on the conformal boundary is obtained upon taking the $\underset{r\rightarrow 0}{\lim}$ of (<ref>). Imposing the following boundary conditions

\begin{equation}
A_{_t}=A_{_{\varphi}}\bigg|_{_{\partial M_{_{3}}}}\ \ \ \ \text{such that}\ \ \ \phi\equiv A_{_{\varphi}}  \ \ \ \text{and}\ \ \ \partial_{_{\varphi}}=0,
\end{equation}

(<ref>) turns into

\begin{equation}
\boxed{\ \ S_{_{BF}}\ =\ \int_{_{M_{_{2}}}}\ \text{Tr}\left(\phi F\right) +\frac{1}{2}\oint_{_{\partial M_{_{2}}}}\text{Tr} \left(\phi^{^2}\right)\color{white}\bigg]\ },
\label{eq:bdythBF}
\end{equation}

where the first term is the usual BF-term for the gauge group $G=SL(2, \mathbf{R})$, whereas the second term is the Schwarzian-type boundary term.

The degrees of freedom in this resulting action, (<ref>), are: a gauge field $A_{_{\mu}}$, with associated field strength $F_{_{\mu\nu}}$, along with an $SL(2, \mathbf{R})$-valued scalar field $\phi$. The latter corresponds to a boundary degree of freedom (also known as the particle-on-a-group).

s̄ubsectionBranes in QFTs and String Theory

§.§ D-branes

In the cobordism formulation of a higher-dimensional QFT, a D-brane is a type of data assigned to the QFTs boudaries of cobordisms.

Key examples include the following:

* RCFT$_{_2}$s (the full rational CFT$_{_2}$, are classified using the FRS-formalism[FRS stands for Fuchs, Runke, and Schweigert, who provided a rigorous construction and classification of all rational full 2D CFTs, where full means that these theories are defined at all genera, namely on all 2D cobordisms.], by means of:

* A MTC, ${\cal C}$, the category of representations of the VOA of the CFT$_{_2}$.

* A special symmetric Frobenius algebra object, $A$, internal to ${\cal C}$.

Hence, a D-brane of the theory is an $A$-module in ${\cal C}$.

* The $(\infty,1)$-categorical formulation of TFT$_{_2}$s, leads to the A- and B-models. The A-model's branes form the Fukaya category of the target space. The B-model's branes form a stable $(\infty,1)$-category of complexes. The WZW-model is a particular example of this, where the target space is a simple Lie group $X\equiv G$, and the background field is a circle 2-bundle with connection on $G$, representing the background field, known as the Kapustin-Rozansky (KR) field.
The study of Kontsevich's homological algebra reformulation of mirror symmetry, and the study of derived D-brane categories constituted a field of pure maths on its own.

* In type-IIA SUGRA, they are the D0-, D2-, D4-, D6-, D8-, F1-, NS5-branes.

* In type-IIB SUGRA, they are D1-, D3-, D5-, D7-, F1-, NS5-, $(p,q)$-5, $[p,q]$-7 branes.

* In M-theory, M2-, M5-, M9-, M-wave.

The topological part of D-brane actions behave as generalised symmetry operators. In particular, they exhibit:

* Confinement/deconfinement transition, via the Hanany-Witten effect via D3-brane nucleation.

The Hanany-Witten mechanism, [557], is realised in brane web setups. As shown in figure <ref>, it consists in the extension of F1-branes between D3- and D5-branes as they pass across each other. This behaviour is naturally arising within the context of 4D QFTs with mixed t'Hooft anomaly. We will briefly outline this example in Part III of the present treatment.

* Fusion rules via the Myers effect. This is an additional feature connecting braneweb constructions with symmetry generators in 4D QFTs. In particular, the Myers effect, well known in the context of D0-D2 branes, is analogous to the fusion rules arising from gauging higher-form symmetries in QFT.

§.§.§ (P)SU(N) in 4D  Super Yang Mills

To put in practice what we have said up to now, we briefly reproduce an example outlined in [561], in the context of symmetries of 4D QFTs in relation to D-branes structures. The 4D theory is a gauge theory with a 0-form symmetry, $\mathbb{Z}_{_M}$, with background field $A_{_1}$, and a 1-form symmetry, $Z_{_M}$, with background field $B_{_2}$, related by a mixed 't Hooft anomaly

\begin{equation}
{\cal A}=-\frac{2\pi}{M}\int A\cup\frac{{\cal P}(B_{_2})}{2},
\label{eq:anomaly}
\end{equation}

where ${\cal P}(B_{_2})$ is the Pontrjagin square of $B_{_2}$.

The generators of the 0- and 1-form symmetries are generated by 3D and 2D topological defects $D_{_3}^{^g}(M_{_3}), D_{_2}^{^h}(M_{_2})$, respectively, such that their fusion rules obey group-like structure relations

\begin{equation}
D_{_p}^{^{g_{_1}}}(M_{_p})\ \otimes\ D_{_p}^{^{g_{_2}}}(M_{_p}) = D_{_p}^{^{g_{_1} g_{_2}}}(M_{_p}).
\end{equation}

Due to the anomaly (<ref>), the generators of the 0-form symmetry transform nontrivially, namely

\begin{equation}
D_{_3}^{^g}(M_{_3})\ \rightarrow\ D_{_3}^{^g}(M_{_3})\ \exp\left(\int_{_{M_{_4}}}-\frac{2\pi i}{M}\frac{{\cal P}(B_{_2})}{2}\right),
\label{eq:transform0}
\end{equation}

with $\partial M_{_4}=M_{_3}$. In particular, when gauging the 1-form symmetry, (<ref>) is inconsistent. To solve this problem, we need to dress $D_{_3}(M_{_3})$ with a minimal TQFT ${\cal A}^{^{M,p}}$ featuring a 1-form symmetry $Z_{_{M}}$ that cancels the anomaly. For this example, this is a $U(1)_{_M}$ theory, and the dressed defect is redefined as follows

\begin{equation}
{\cal N}_{_3}^{(1)}\ =\ D_{_3}^{^{(1)}}\ \otimes\ {\cal A}^{^{M,1}},
\end{equation}

with non-invertible fusion rules

\begin{equation}
{\cal N}_{_3}^{^{(1)}}\ \otimes\ {\cal N}^{^{(1)}}\ =\ {\cal A}^{^{M,2}}{\cal N}_{_3}^{^{(2)}}  \ \ \ \ \ \ \ ,\ \ \ \ \ \ \ {\cal N}_{_3}^{^{(1)}}\ \otimes\ {\cal N}^{^{(1)}\dag}\ =\ \sum_{_{M_{_2}\in H_{_2}(M_{_3},\mathbb{Z}_{_M})}}\frac{(-1)^{^{Q(M_{_2})}}D_{_2}(M_{_2})}{|H^{^0}(M_{_3},\mathbb{Z}_{_M})|},
\end{equation}

\begin{equation}
D_{_2}(M_{_2})\ =\ e^{^{i 2\pi\ \int_{_{M_{_2}}} \frac{b_{_2}}{M}}},
\end{equation}

with $b_{_2}$ the gauge field for the 1-form symmetry. In relation to type-IIB string theory, these symmetry operators ${\cal N}_{_3}^{^{(1)}}$ are associated to D5-branes, whereas 't Hooft operators are associated to D3-branes, [561]. We will be referring to this example for later convenience in Part III when introducing Symmetry Topological Field Theories (SymTFTs).

§.§ From brane webs to quiver gauge theories

Quivers encode the Lagrangian of a supersymmetric gauge theory. They take different forms according to the number of supercharges. For the case in which there are 8 of them, quiver diagrams encode the gauge symmetries, flavour symmetries, and matter field content. The fields featuring in a certain theory are contained in vector multiplets and hypermultiplets. The field content changes according to the dimensionality of the theory in question.

The quiver diagram encodes the representations of the gauge and matter fields under which they transform. Specifically, the vector multiplets transform under the adjoint representation of the gauge group representation, whereas the hypermultiplets transform under the bifundamental representation of the gauge group and the flavour group.

Importantly, from the quiver, we can read the superpotential. In figure <ref> we show the example of a framed Nakajima quiver variety. In this case, the corresponding superpotential reads as follows

\begin{equation}
{\cal W}=\text{Tr}\left(A^{^{\alpha}}_{_i}\Phi^{^i}_{_j}B^{^{j\beta}}\right),
\end{equation}

where $A^{^{\alpha}}_{_j}, \Phi^{^i}_{_j}$ and $B^{^{j\beta}}$ denote the chiral multiplets, the anti-chiral multiplets, and the adjoint chirals, respectively. The indices $i,j$ are related to the gauge group, whereas the indices $\alpha, \beta$ are related to the flavour group.

§.§ Beyond holographic duality: adding boundaries

§.§ Boundary Conditions (BCs) of  SCFTs

$AdS_{_4}\times S_{_2}^{^{(A)}}\times S_{_2}^{^{(B)}}$ fibered over a Riemann surface, $\Sigma$, the corresponding metric solution reads

\begin{equation}
\end{equation}

where the metric functions $f_{_i}, \rho$ are functions of $h_{_1}, h_{_2}$ defined on $\Sigma$, via $(r,\theta)$. As explained in [277], the field theory in question is uniquely defined by the following set of parameters

\begin{equation}
N_{_{D5}}^{^{(A)}}\overset{def}{=}\frac{1}{\sqrt{g\ }}c_{_A}\ \ \ \ \ \ \ \ \ \ \ \ , \ \ \ \ \ \ \ \ \ \ \ \ N_{_{BS5}}^{^{(B)}}\overset{def}{=}\sqrt{g\ }d_{_B}
\end{equation}

\begin{equation}
L_{_A}\overset{def}{=}\sqrt{g\ }l_{_A}+\frac{2}{\pi}\sum_{_B}N_{_{D5}}^{^{(B)}}\tan^{^{-1}}\left(\frac{l_{_A}}{k_{_B}}\right)\ \ \ , \ \ \ K_{_B}\overset{def}{=}\frac{1}{\sqrt{g\ }}k_{_B}+\frac{2}{\pi}\sum_{_A}N_{_{NS5}}^{^{(B)}}\tan^{^{-1}}\left(\frac{k_{_B}}{l_{_A}}\right).
\end{equation}

At this point, one could further redefine $\hat L_{_i}$ as the number of D3-branes ending from the right on the $i^{^{th}}$ D5-brane plus the number of NS5-branes to the left of this D5-brane, and with $\hat K_{_i}$ the number of D3-branes ending on the $i^{^{th}}$ NS5-brane plus the number of D5-branes to the left of this NS5-brane, leading to the brane-web intersection reported in figure <ref>.

In a SCFT with ${\cal N}=4$ supersymmetry, the Higgs and Coulomb branch are both conical and meet at their common origin. In general, there might also be mixed branches.

§.§ -BPS BCs in  SYM

$\frac{1}{2}$-BPS BCs in ${\cal N}=4$ SYM with gauge group $G$ admit a general classification in terms of a triple, $(\rho, H, {\cal B})$, [117], where

\begin{equation}
\rho:\ \mathfrak{su}(2)\ \rightarrow\ \mathfrak{g}
\label{eq:rho}
\end{equation}

denotes the embedding of the Lie algebra of $SU(2)$ in the Lie algebra of $G$, $H$ is a subgroup of $G$ resulting from the symmetry breaking induced by (<ref>), whereas ${\cal B}$ denotes the 3D boundary field theory with symmetry $H$.

§.§.§ Islands in JT-gravity

The holographic description of the entanglement entropy for Hawking radiation in arbitrary spacetime dimensions can be described in 3 different ways. This follows from the extension of the 2d JT-gravity model to arbitrary dimensions relying upon the Randall-Sundrum model with a brane embedded in AdS$_{d+1}$. Maldacena et al.'s treatment [200] is equivalent to the brane (EFT) description. The bulk perspective shows that it is indeed reasonable to extend JT-gravity results to higher-dimensions relying upon dualities. In particular, wedge holography [224] looks very promising for addressing vacuum transitions in higher dimensions.

*  Boundary description

Both of the QMs and field theory systems are prepared in independent thermofield double states.

* Brane description

Introducing a 2D holographic geometry (JT-gravity), dual to the entangled state of QM$_{L}$ and QM$_{R}$. The CFT$_{2}$ coupling to JT-gravity is the same as the one featuring in the bath.

* Bulk description

The doubly-holographic description, where the holographic CFT$_{2}$ is replaced by a bulk region with geometry of an AdS$_{3}$ black hole.

§.§.§ Conformal interfaces as boundary states

The idea of the replica trick, at first applied within the context of statistical field theory and condensed matter physics, basically consists in taking $n$-copies of a certain system and relating its behaviour to that of the original system, i.e for $n=1$. The reason for doing so is that, in principle, there might be noncommutative behaviour emerging from permutations of different subregions of the $n$-copied system, which in turn would highlight the presence of interesting entangling features that are in turn scale-dependent, i.e. their behaviour is determined by the size of one copy.

Translating this idea in a suitable language for our purposes, i.e. describing quantum transition in between different spacetimes,[and as it will become clear later on in the treatment.] there is one additional key point to bare in mind. Quantum transitions are really meant to describe local spacetime transformations. As an example, describing uptunneling from AdS would really mean considering a portion of the domain at a given instant of time (in global coordinates) and then defining its corresponding density matrix by post-selection according to Takayanagi et al.'s prescription discussed in the previous section. The final state $|\varphi>$ will be defined such that the covariant holographic entanglement entropy satisfies the 2-step minimisation-extremisation (min-extr) of the expectation value of $Area(\Gamma_{A})$. $\rho_{A}$ is therefore the density matrix associated to the states defined on such subregion at the boundary of AdS. Being this a portion of the entire domain at $r_{\infty}$, we can think of $A$ as arising by quotienting w.r.t. $Z_{n}$-symmetry, with $n$ determined by the particular choice of the domain $A$. However, as also explained in the appendix, even for the simplest $U(1)$-case, an anomaly emerges, since the symmetry is broken in the quantum theory by the state and the partition function. Restoring it at the level of the partition function requires adding Wilson loops, i.e. additional operators to the quantum theory. In turn, these operators can be thought of as the lower-dimensional counterparts of background fields living in one spatial dimension higher. Their coupling will ultimately lead to the definition of Chern-Simons terms, i.e. defects added to the bulk theory in order to compensate for the anomaly featuring in the original QM theory. This procedure is known as the anomaly-inflow mechanism.

Our main interest is addressing the issue form the quantum gravity perspective. In order to do so, we rely upon the holographic dictionary of the AdS$_{d+1}$/CFT$_{d}$ correspondence, at first starting with $d=2$, from which some major considerations can be extracted and later on extended to higher-dimensional setups.

We start by considering 2 CFTs separated by a conformal interface, which basically provide a special example of extended operators, such as Wilson lines, D-branes, or other extended sources. Upon exchanging the role of space and time, these conformal boundaries can be mapped to boundary states, following the so-called folding trick. As a result of this procedure, we end up with a BCFT. The boundary conditions defined at the interface can be suitably chosen accordingly to the setup we wish to describe. 2 particular examples are provided by the totally-reflecting and the topological BCs.

The boundary arising from the folding trick can really be thought of as a special case of a defect, for which an entropy can be derived from the entanglement entropy of a $(1+1)$ CFT. Specifically, the entanglement entropies with and without the boundary defect resepctively read

\begin{equation}
\
\
\frac{c}{6}\ln\frac{L}{\epsilon}\ +\ \ln(g_{B})\ +\ \alpha_{0}
\ \ \
\ \ \
\
\
\frac{c}{6}\ln\frac{L}{\epsilon}\ +\ \alpha_{0}
\label{eq:entent}
\end{equation}

where $c$ is the central charge of the CFT$_{_2}$, $\epsilon$ is the UV cutoff introduced for regularisation purposes, $L$ is the length of the spatial region on which the entanglement entropy is being evaluated, $\alpha_{_0}$ is an overall constant, and $g_{_B}$ is the ground state degeneracy. Upon taking the difference in between the two quantities in (<ref>), we are left with the entropy of the defect

\begin{equation}
\
\
S_{1}\ -\ S_{2}
\
\
\ln(g_{B})
\ \ \
\end{equation}

The defect, therefore, contributes with an additional term to the partition function, which asymptotically reads

\begin{equation}
\
\
\ln\ g_{B}
\ \ \
\ \ \
\
\
\
\
<0|\ D\ |0>
\ \ \
\end{equation}

with $D$ defining the defect operator

\begin{equation}
\
\
g|0><0|\ +\ ...
\ \ \
\end{equation}

In string theory, e.g., the role of the $g$-factor is played by the tension of a D-brane. It is interesting to note that this is exactly the term entering in the Hamiltonian formulation of quantum transitions by implementing the junction conditions, with the brane tension separating different spacetimes.

The degeneracy of the ground state is therefore defined by the overlap between the boundary state and the vacuum

\begin{equation}
\
\
\ \ \
\end{equation}

The g-theorem states that the value of $g_{B}$ must decrease under the boundary RG-flow [195].

Given $g_{B}$, the distance in between the two original CFTs is defined as follows

\begin{equation}
\
\
\text{min}\sqrt{\ln g_{B}}
\ \ \
\end{equation}

From this, we therefore understand that BCFTs, such as the case of interest for us being $\rho_{A}=\text{Tr}_{B}\rho$, are incredibly rich in structure and carry many interesting and hidden features that will turn out essential in the forthcoming sections. In particular, the boundaries of the BCFT carry d.o.f. associated to the degeneracy of the ground state and and therefore signal whether a defect needs to be added to the path-integral evaluation when taking n-copies of the given subregion where the BCFT lives. Such defect is, in turn associated to an anomaly induced by the anomaly-inflow mechanism of a bulk background field which needs to be added to the quotiented geometry.

In terms of the von Neumann formulation, the entanglement entropy reads

\begin{equation}
\
\
- \text{Tr}_{A}\ (\rho_{A}\ln(\rho_{A}) )
\
\
\
\lim_{n\rightarrow 1}\ \frac{d}{dn}\text{Tr}_{A}(\rho_{A}^{n})
\ \ \
\label{eq:VNE}
\end{equation}

where in the last step of (<ref>) we implemented the replica trick.

For integer values of $n$-replicas, $\text{Tr}(\rho_{A}^{n})$ is the partition function on a Riemann surface, i.e.

\begin{equation}
\text{Tr}_{A}(\rho_{A}^{n})
\
\sim
\
{\cal Z}_{n}
\
\
\int\ [d\varphi_{1}...d\varphi_{n}]_{{\cal M}_{n}}\ \exp\left[-\int_{{\cal M}_{n}}\ d^{2}x\ {\cal L}^{(n)}[\varphi_{1},...,\varphi_{n}](x)\right]
\ \ \
\label{eq:pf}
\end{equation}

Invariance of the boundary theory under $Z_{n}$-symmetry corresponds to the Lagrangian being preserved under permutations of the n-copies of the QFT, i.e.

\begin{equation}
{\cal L}^{(n)}[\varphi_{1},...,\varphi_{n}](x)...,\sigma\varphi_{n}](x)
\
\
{\cal L}[\varphi_{1}](x)\ +\ ...\ +\ {\cal L}[\varphi_{n}](x)
\ \ \
\end{equation}

with $\sigma\varphi_{i}\ =\ \varphi_{i+1 \text{mod n}}$. The action of $\sigma$ belonging the symmetry group ${\cal Z}_{n}$ and its inverse $\sigma^{-1}$ are associated to a twist fields, whose correlation function is proportional to (<ref>). The partition function of the n-copied system can therefore be used for redefining the entanglement entropy in the $\lim_{n\rightarrow 1}$, and (<ref>) therefore reduces to

\begin{equation}
\
\
\
\lim_{n\rightarrow 1} \ \frac{d}{d n}\ {\cal Z}_{n}
\ \ \
\end{equation}

With this setup, we are now ready to derive the generalised expression for the entanglement entropy, where the contribution from quantum extremal surfaces adds to the von Neumann entropy.

From knot invariants to QFTs

§ SUMMARY AND MOTIVATIONS OF PART III

§.§ Motivations

Since first encountered, [108], 6D ${\cal N}=(2,0)$ SCFTs, also referred to as theory X,[Please see appendix <ref> for some brief overview of its importance in pure mathematics and the Geometric Langlands Program.], have attracted great interest in, both, the mathematics and high-energy physics community alike. From the high-energy physics point of view, they are essential for deriving lower-dimensional SCFTs, [109, 296, 123, 112], whose features are much better understood[We review some key features of its 4D descendants in appendix <ref>.]. On the other hand, from a purely mathematical perspective, they provide an extremely interesting and rich setup where to exploit the present-day knowledge of representation theory, [110, 111].

In recent years, the two communities have been brought together by fascinating developments on either side, [298], highlighting the importance of bridging the gap in between different formalisms, with category theory playing a key role in pursuing this task. Indeed, the pioneering works of [299, 300, 301] led to the discovery of modular tensor categories in anyon gauging theory. Since then, application of such deep mathematical language inspired applicability to higher-dimensional QFTs, by identifying their symmetries with topological defects, [302]. Of particular importance to us is the richness of higher-categorical theory applied to topological orders (TOs), [303, 304, 305, 306, 307, 308, 309, 310, 311], and symmetry topological field theories (SymTFTs), [103, 102].

Embracing the thrust of recent developments in the field, the present work is meant to be the first of a series of papers where the author applies the techniques of category theory to furthering the understanding of 6D ${\cal N}=(2,0)$ SCFTs and theories derived from them. In doing so, we will attempt, wherever possible, to provide an understanding of the procedure as well as of the outcome of our findings, from, both, a mathematician's and a theoretical physicist's point of view. Hopefully, this will further motivate both communities to benefit form each other's influence.

This part of our work aims at furthering the understanding of the categorical structure arising from dimensional reduction of 6D ${\cal N}=(2,0)$ SCFTs. In particular, our aim is that of identifying a criterion to distinguish between intrinsic and non-intrinsic non-invertible symmetries[See, for example, [312, 313, 314, 315, 316, 317, 318, 319, 320, 321] for explanation on the current state of the art.] in terms of the quantum dimension[The total quantum dimension is defined from the braiding and fusion rules of the underlying categorical structure of the topological symmetries of the theory in question. Further explanation will be provided in due course.]. In doing so, we rely upon their description in terms of relative field theories, as described in [313] in terms of the Freed-Moore-Teleman setup, [103, 102].

As previously argued, class ${\cal S}$ theories are special in the sense that they are absolute, and admit a quiver gauge theory description. Their TFT counterparts are the so-called minimal models, which, in turn are characterised by a higher-categorical structure, as reviewed in section <ref>.
The multiplicities of the different superselection sectors might exceed unity, thereby signalling the possibility of information storage, correspondingly leading to a decrease in ${\cal D}$. Given that the definition of the entanglement entropy involves a partial tracing of the total boundary state, it might be that the Rényi entropy shows a characteristic periodicity $\mathfrak{p}$ signalling that the state defining the reduced density matrix can be factorised under suitable redefinition of the modular operators involved in the definition of the link state itself.

Our main proposal is that the total quantum dimension of the gaugeable algebra implementing the gauging in the bulk Sym TFT is a key quantity to distinguish in between non-invertibility being intrinsic or non-intrinsic. Specifically, for the intrinsic case, multiplicity is greater w.r.t. the non-intrinsic case, thereby signalling the possibility for additional d.o.f. to be stored in certain superselection sectors of the resulting absolute theory.

Our results extend arguments proposed in [319], where the authors also proposed a way of distinguishing intrinsic from non-intrinsic non-invertibility in 2D by means of the quantum dimension of the non-invertible defect.

§.§ Outline and main results

The present Part contains the main findings first presented in an original work by the same author, and serves as a bridge between the preliminary sections of the present work and the remainder of this thesis. The content of the following two sections is as follows:

* In section <ref> we introduce the symmetry topological field theory (SymTFT) construction proposed by Freed, Moore, and Teleman for describing supersymmetric gauge theories descending from dimensional reduction of 6d ${\cal N}=(2,0)$ SCFTs. In particular, we focus on the non-invertible defects arising in between class ${\cal S}$ theories. Our main focus is on the following:

* We highlight the importance of defining the Drinfeld center for the case in which non-invertibility is non-intrinsic, and relate this to the need of generalising the 3D mirror symmetry/homological mirror symmetry correspondence explained in section <ref>. This prepares the stage for the more detailed analysis to which sections <ref>, <ref>, <ref>, and <ref> are devoted.

* Exploiting the relation in between the fusion rules of such defects and the partition function on the absolute 4D theory, we show that the quantum dimension of the gaugeable algebra in the intrinsic case always exceeds that featuring in the non-intrinsic case. We thereby propose this as a general criterion to be applied to class ${\cal S}$ theories descending from 6D ${\cal N}=(2,0)$ SCFTs.

* In section <ref>, we then turn to one of the key topics bridging with the remainder of our treatment, namely instantons. After having overviewed how they arise in QQM, QFT, 4D supersymmetric gauge theories, and oriented string theory, we explain their relation to the ADHM construction, and the Higgs branch moduli space. In the concluding part of the same section, we then introduce the WKB approximation as a method used for determining approximate solutions to wall-crossing setups, with particular focus on two settings of interest for us, namely Hitchin systems and vacuum transitions.

§ QUANTUM FIELD THEORY FROM CATEGORY THEORY

We now turn to explaining how QFT can be mathematically described in terms of Category Theory.

To begin with, we start off by listing some of the ingredients that are most relevant to our analysis[As we shall see, more richness will be added in due course in our treatment. The aim of the table is simply that of helping the reader in gaining intuition for the setup and the motivations behind the procedure adopted.], briefly outlined in the table below.

\begin{equation}
\begin{aligned}
&\text{\underline{Particle Physics}} \ \ \ \color{white}spacespacespacespace\color{black} \ \ \text{\underline{Category Theory}} \\
&\text{Gauge Theory}\ \ \ \color{white}spacespacespacespaces\color{black} \ \ \text{Algebras}\\
&\text{\color{red}Operators}\ \ \ \color{white}spacespacespacespacespace\color{black} \ \ \text{Homologies}\\
&\text{Quivers}\ \ \ \color{white}spacespacespacespacespacesp\color{black} \ \ \text{Quivers/Algebraic varieties}\\
&\text{Hilbert Series}\ \ \ \color{white}spacespacespacespacece\color{black} \ \ \text{Hilbert Series}\\
&\text{\color{red}SUSY}\ \ \ \color{white}spacespacespacespacecespacesp\color{black} \ \ \text{SUSY}\\
\end{aligned}
\nonumber
\end{equation}

This shows some quantities/entities encountered in the realm of Particle Physics and their Category Theory counterpart. We have highlighted two of them in red. The reason for this will be more clear in Part V, but the familiar reader will certainly notice that some motivation was already hinted at in section <ref>. For the moment, we simply recall what was mentioned in the Introduction, namely that the Hilbert series and ring homologies are key complementary ingredients towards understanding a given QFT. Among the two, though, the latter is the one that is more directly related to understanding the underlying categorical structure of the theory in question, and therefore constitutes the ultimate objective one wishes to obtain. On the other hand, the Hilbert series, is obtained from ring homologies, but hides some key categorical features that are only accessible via the homological perspective.[Indeed, it is known that different ring homologies could lead to the same Hilbert series, [560].]

This intermediate section is mostly an overview of [102, 103, 102], and is needed in order to understand the analysis carried out in the concluding part of our work, <ref>. This section is structured as follows:

* At first, we review mathematical gauge theory and the definition of moduli spaces therein, [102].

* Then, we recall the notion of relative field theories, [103], which can also be adapted to gauge theories.

* In conclusion, we explain how this fits with the higher-categorical prescription of Freed, Moore and Teleman, [102], and how gauging can be implemented in terms of gaugeable homomorphisms in the Symmetry Topological Field Theory categorising the symmetries of a given QFT, [323].

For reasons that will become clear by the end of this section, the crucial part of the higher-categorical perspective can be summarised as follows:

* Keeping track of the identity.

* Embedding the gauged algebra.

Most importantly, they both lie within the notion of abelianisation. In section <ref> we will see explicitly how they arise within the context of supersymmetric quiver varieties, [550, 551].

§.§ The Freed-Moore-Teleman construction

§.§.§ Gauging in relative QFTs

In the formulation of [103], a relative field theory, $\tilde F$, requires additional topological data in order to be fully specified. Such data is encoded in a pair $(\sigma,\rho)$, referred to as quiche. $\sigma$ is the symmetry topological field theory (SymTFT), whereas $\rho$ geometrises the choice of boundary conditions for the fields defining the relative theory $\tilde F$. The overall system, depicted in figure <ref>, gives rise to an absolute QFT, $F_{_{\rho}}$.

The Freed-Moore-Teleman setup, with $\tilde F$ denoting a relative QFT. Specifying the topological data $(\sigma,\rho)$, the resulting theory, $\tilde F_{_{\rho}}$ is absolute.

In mathematical terms, the description outlined above can be formulated in terms of bordism in the following way. Fixing $N\in\mathbb{Z}^{^{\ge 0}}$, then a quiche is a pair $(\sigma, \rho)$ in which $\sigma:\text{Bord}_{_{N+1}}(F)\rightarrow{\cal C}$ is an $N+1$-dimensional TFT and $\rho$ is a right topological $\sigma$-module. The quiche is $N$-dimensional, hence it shares the same dimensionality as the theory on which it acts. Let $F$ be an $N$-dimensional field theory. A $(\sigma,\rho)$-module structure on $F$ is a pair $(\tilde F, \theta)$, in which $\tilde F$ is a left $\sigma$-module and $\theta$ is an isomorphism

\begin{equation}
\theta\ : \rho\ \otimes_{_{\sigma}}\ \tilde F\xrightarrow{\ \ \simeq\ \ }\ F_{_{\rho}},
\label{eq:LHS}
\end{equation}

of absolute $N$-dimensional theories, with (<ref>) defining the dimensional reduction leading to the absolute theory. $\sigma$ needs only be a once-categorified $N$-dimensional theory, whereas $\rho$ and $\tilde F$ are relative theories.

In theories admitting a higher-categorical structure as (<ref>), topological defects in different superselection sectors have different dimensionality. Consequently, the fusion rules are categorical, in the sense that they usually don't follow group-like composition laws. Because of this these defects are also called categorical or non-invertible.

Prior to delving into the specific case of interest to us, namely relative QFTs, we first wish to provide some further explanation for what gauging a categorical structure actually means. In doing so, we refer to the work of many experts in the field, and, in particular [323]. As explained in such reference, for any fusion n-category $\mathfrak{G}$, any fiber functor

\begin{equation}
{\cal F}:\ \mathfrak{G}\ \rightarrow\ \text{nVec},
\label{eq:functor}
\end{equation}

selects nVec as the image of a gaugeable algebra living in $\mathfrak{G}$, corresponding to a projection on the identity. The gauging process, can therefore be defined as a map

\begin{equation}
\mu:\ \mathfrak{G}\ \rightarrow \ {\cal A}_{_C},
\label{eq:mu}
\end{equation}

with ${\cal A}_{_C}$ the algebra of invertible topological operators in $\mathfrak{G}$. In terms of the example reproduced in section <ref>, ${\cal A}_{_C}$ refers to the algebra that dresses the symmetry operators once having implemented the gauging to ensure gauge invariance. Hence, $\mu$ really corresponds to the projection of the identity to a subalgebra of the total algebra, ${\cal A}$, featuring in the pre-gauged theory, with ${\cal A}_{_C}\ \subset\ {\cal A}$. For later purposes, we highlight the relation in between this description with that introduced in section <ref> in relation to Moore-Tachikawa varieties. As also defined there, the crucial role from the category theory point of view really consists in defining the cotangent bundle as a triple of elements, two of which are left- and right- acting moment maps, mutually constrained by the main axiom outlined by Moore and Tachikawa. As explained in section <ref>, this follows from full-dualisability of the 2-categories involved in defining the 2D TFT associated to a given class ${\cal S}$ theory[As explained in Part V, this follows from the fact that the category of contractible cycles associated to a given algebraic variety admits a Kirwan map ensuring abelianisation in terms of equivariant cohomology.]. Given (<ref>), the norm element

\begin{equation}
N\overset{def}{=}\bigoplus_{g\in\mathfrak{G}}\ \mu(g).
\label{eq:idemp}
\end{equation}

carries the structure of an n-categorical idempotent, also known as gaugeable algebra, depicted in black in figure <ref>. The requirement for (<ref>) to be a higher-idempotent is needed to ensure the flooding doesn't depend on the specific features of the network being adopted to perform the gauging. The algebra of topological operators that are left are denoted by ${\cal A}//^{^{\mu}}\mathfrak{G}$. The equivalence of the second and third picture from the left in figure <ref> follows from $N$ being a higher-gaugeable algebra. As we shall see, this pattern emerges when gauging the 5D SymTFT of class ${\cal S}$ theories, where the objects of the 2-category in question will be Wilson surfaces charged under the 1-form symmetry being gauged.

Gauging corresponds to condensing an algebra in a TFT. Idempotency ensures the resulting theory can be effectively thought of as featuring a unique defect, as shown on the RHS.

Condensing two different subalgebras, ${\cal A}_{_1}, {\cal A}_{_2}\ \subset\ {\cal A}$, the resulting theory corresponds to one with a changed phase with a gauged defect resulting from a relative gaugeable algebra, ${\cal A}_{_{12}}$ ending in the bulk. The defect at the endpoint is nontrivial, and can therefore be thought of as a Hom$(\mathbf{1}_{_{{\cal C}}},{\cal A}_{_{12}})$.

Unlike <ref>, <ref> does not admit a straightforward expression as (<ref>).

Adaptation of Freed-Moore-Teleman to the case involving twisted gauged defects. The figure on the far right corresponds to the case of interest to us, namely a configuration involving two different absolute 4D gauge theories separated by a defect. As we shall see, such defect is intrinsically non-invertible, corresponding to the presence of a relative uncondensed subalgebra, dressing ${\cal N}_{_3}$ in a nontrivial way. In higher-categorical terms, it corresponds to a fusion tensor category implementing the morphisms between the operators charged under the gauged symmetry.

§.§.§ Absolute 4D theories from 6D  SCFTs

The main features of 6D ${\cal N}$=(2,0) SCFTs are that:

* They admit no Lagrangian formulation, <cit.>.

* The theory depends on the Lie algebra rather than the Lie group.

* Ordinary[By this we mean that the lower-dimensional theories depend on the specific choice of the Lie group.] 5D and 4D QFTs can be obtained by dimensional reduction on a Riemann surface, $\Sigma_{_{g,n}}$.

* It is a relative w.r.t. an extended QFT in 1-dimension higher.

Examples of relative theories are models whose fields form a fibration, <cit.>,

\begin{equation}
{\cal F}\ \longrightarrow\  {\cal M}\ \xrightarrow{\ \ \pi\ \ }\ {\cal B}
\end{equation}

where ${\cal B}$ and ${\cal F}$ denote the base and the fiber, respectively. Relative fields are the fibers of $\pi$. The path integral over the base reduces a finite sum. For the theory to be absolute, the same should hold for ${\cal F}$ as well.

When compactifying the 6D theory on $\Sigma_{_{g,n}}$ without punctures:

* If $g\equiv 1$ $\Sigma_{_{g}}\equiv T^{^{2}}$, the resulting class ${\cal S}$ theories are ${\cal N}$=4 SYM with gauge algebra $\mathfrak{su}(N)$.

* For $g>1$, the resulting theories are models with an $S$-duality frame, interpreted as conformal gaugings of $2g-2$ trinion $T_{_{N}}$ theories coupled to $3g-3$ gauge groups with $\mathfrak{su}(N)$ gauge algebras.

§.§.§ From relative to absolute

The procedure described in section <ref> cannot be applied to the case of maximally supersymmetric 6D theories. The reason being that, while a bulk SymTFT can be defined, thanks to the Milnor-Moore theorem, [297], specifying any topological boundary conditions would inevitably imply supersymmetry breaking[To the best of our knowledge, the only exception to this is dealt with in [563].], [103]. Nevertheless, such 6D theories provide the natural realm from which one can realise lower-dimensional absolute theories, as well as constituting an extremely rich setting where to explore and put to practice mathematical techniques provided by representation theory, [111].

In this subsection we briefly overview the work of [313], and propose a criterion for distinguishing intrinsic from non-intrinsic non-invertible defects.

Assuming the 6D ${\cal N}=(2,0)$ SCFT is defined on $Y=X\times \Sigma_{_{g}}$, with $X$ a compact 4D spacetime, torsion-free, and with trivial 1$^{^{st}}$ cohomology, $H^{^{1}}(X,\mathbb{Z}_{_{N}})$, the 6D self-dual fluxes belong to[$\mathbb{Z}_{_{N}}$ denotes the defect group of the 6D theory.]

\begin{equation}
H^{^{3}}(Y,\mathbb{Z}_{_{N}})\ \simeq\ H^{^{1}}(\Sigma_{_{g}},\mathbb{Z}_{_{N}})\ \otimes\ H^{^{2}}(X,\mathbb{Z}_{_{N}}).
\end{equation}

As far as the 6D theory is concerned, this is the only information at our disposal, given that the theory is non-Lagrangian, [103, 110]. However, for the 4D theory to be fully defined, additional data is needed. In particular, one needs to identify a maximal isotropic sublattice $L$ of $H^{^{1}}(\Sigma_{_{g}},\mathbb{Z}_{_{N}})$ w.r.t. the canonical pairing induced by the intersection pairing on $\Sigma_{_{g}}$, thereby leading to

\begin{equation}
{\cal L}\ \simeq\ L\ \otimes\ H^{^{2}}(X,\mathbb{Z}_{_{N}})
\label{eq:maxissub}
\end{equation}

as the maximal isotropic lattice of $H^{^{3}}(Y,\mathbb{Z}_{_{N}})$. (<ref>) enables to distinguish between different global forms of class ${\cal S}$ theories. Having fixed ${\cal L}$, the fluxes in its complement

\begin{equation}
{\cal L}^{\perp}\  \overset{def.}{=}\ H_{_{3}}(X_{_{6}},\mathbb{Z}_{_{N}})/{\cal L}
\end{equation}

parametrise the possible partition functions for the 4D theory with inequivalent 1-form symmetry backgrounds along $X$.

The 6D ${\cal N}$=(2,0) theory, assigns to the 6D manifold $Y$ a partition vector rather than a partition function, with the former belonging to the Hilbert space constituting a representation of a Heisenberg algebra of non-commuting discrete 3-form fluxes in $\mathbb{Z}_{_{N}}$.

For fixed ${\cal L}$, the partition vector takes the form

\begin{equation}
|{\cal Z}(Y)>\ \equiv\ \sum_{v\ \in\ {\cal L}^{^{\perp}}}\ {\cal Z}_{_{v}}(Y)\ |{\cal L}; v>,
\label{eq:chofb}
\end{equation}

with ${\cal Z}_{_{v}}(Y)$ denoting the 6D conformal blocks, which also encode the global structures of the 4D theory. Under a change of basis, (<ref>) turns into

\begin{equation}
|{\cal Z}(Y)>\ \equiv\ \sum_{v^{\prime}\ \in\ {\cal L}^{^{\prime\perp}}}\ {\cal Z}_{_{v}}(Y)\ \sum_{v\ \in\ {\cal L}^{^{\perp}}}\ R_{_{v}}^{^{v^{\prime}}}\ |{\cal L}; v>,
\label{eq:chofb1}
\end{equation}

leading to the following relation between conformal blocks

\begin{equation}
{\cal Z}_{_{v}}(Y)\ \equiv\ \sum_{v^{\prime}\ \in\ {\cal L}^{^{\prime\perp}}}\ R_{_{v}}^{^{v^{\prime}}}\ {\cal Z}_{_{v}}(Y).
\label{eq:chofb2}
\end{equation}

§.§ The 7D and 5D SymTFTs

The Milnor-Moore Theorem states that, the universal enveloping algebra of a Lie algebra is a Hopf algebra whose primitive elements are the elements of the original Lie algebra. A Hopf algebra on an associative algebra turns the category of modules into a monoidal category equipped with a fiber functor. This is a statement of Tannaka duality.

Putting this together for the case of 6D ${\cal N}=(2,0)$ SCFTs, the bulk SymTFT is a pointed braided tensor category where the braiding is in bijection with 3-cocycles.

Following the arguments outlined in section <ref>, one can take the 7D TQFT in the bulk of the relative 6D ${\cal N}=$(2,0) theory, to be a CS theory with action

\begin{equation}
S_{_{7D}}=\frac{N}{4\pi}\ \int_{_{W_{_{7}}}}\ dc\ \wedge\ c\ \ \ ,\ \ \ c\in H^{^{3}}(W_{_{7}},U(1)), \label{eq:7D}
\end{equation}

and Wilson surfaces

\begin{equation}
\Phi_{_{q}}({\cal M}_{_{3}})\overset{def.}{=}\ e^{^{iq\oint_{_{{\cal M}_{_{3}}}} c}}\ \ \ ,\ \ \ q\in\mathbb{Z}_{_{N}}\ \ \ ,\ \ \ {\cal M}_{_{3}}\ \in H^{^{3}}(W_{_{7}},U(1)).
\end{equation}

Taking two such Wilson surfaces $\Phi_{_{q}}({\cal M}_{_{3}}), \Phi_{_{q^{\prime}}}({\cal M}_{_{3}}^{\prime})$, with ${\cal M}_{_{3}}, {\cal M}_{_{3}}^{\prime}$ forming a Hopf link in $W_{_{7}}$, and inserting them in the path integral, amounts to changing the action (<ref>) by adding the holonomy terms associated to the operator insertions

\begin{equation}
\begin{aligned}
S_{_{7D}}&=\frac{N}{4\pi}\ \int_{_{W_{_{7}}}}\ dc\ \wedge\ c  + q\int_{_{{\cal M}_{_{3}}}} c\ +q^{\prime} \int_{_{{\cal M}_{_{3}}^{\prime}}} c\\
&=\frac{N}{4\pi}\ \int_{_{W_{_{7}}}}\ dc\ \wedge\ c +\ \int_{_{W_{_{7}}}}\ \left(q\omega_{_{{\cal M}_{_{3}}}}+q^{\prime}\omega_{_{{\cal M}_{_{3}}^{\prime}}}\right)\ \wedge\ c,
\label{eq:CS7D}
\end{aligned}
\end{equation}

with $\omega_{_{{\cal M}_{_{3}}}}, \omega_{_{{\cal M}_{_{3}}^{\prime}}}$ denoting the Poincaré duals of ${\cal M}_{_{3}}, {\cal M}_{_{3}}^{\prime}$, respectively. Integrating-out $c$, and imposing

\begin{equation}
dc\equiv-\frac{2\pi}{N}\left(q\omega_{_{{\cal M}_{_{3}}}}+q^{\prime}\omega_{_{{\cal M}_{_{3}}^{\prime}}}\right),
\end{equation}

defining $V_{_{4}}$ such that

\begin{equation}
\partial V_{_{4}}\equiv\left(q\omega_{_{{\cal M}_{_{3}}}}+q^{\prime}\omega_{_{{\cal M}_{_{3}}^{\prime}}}\right),
\end{equation}

\begin{equation}
c\equiv -\frac{2\pi}{N}\ \text{PD}(V_{_{4}}).
\end{equation}

From this, (<ref>) reduces to

\begin{equation}
\begin{aligned}
&=\frac{2\pi}{N}\ q q^{\prime}\ \text{link}\left({\cal M}_{_{3}}, {\cal M}_{_{3}}^{\prime}\right),
\label{eq:linkact7}
\end{aligned}
\end{equation}

where the linking is the Hopf link occurring in $W_{_{7}}$. It features in (<ref>) due to the fact that $V_{_{4}}$ is a Seifert surface for the combination $q{\cal M}_{_{3}}+q^{\prime}{\cal M}_{_{3}}^{\prime}$, meaning that, every time ${\cal M}_{_{3}}$ pierces $V_{_{4}}$, it links ${\cal M}_{_{3}}^{\prime}$ once. It thereby follows that, the Hopf link of Wilson surfaces supported on ${\cal M}_{_{3}}$ and ${\cal M}_{_{3}}^{\prime}$ in $W_{_{7}}$ can be thought of as an operator insertion in the 7D path integral,

\begin{equation}
<\Phi_{_{q}}({\cal M}_{_{3}})\Phi_{_{q^{\prime}}}({\cal M}_{_{3}}^{\prime})...>\equiv e^{^{\frac{2\pi i}{N}q q^{\prime}\text{link}({\cal M}_{_{3}},{\cal M}_{_{3}}^{\prime})}}\ <...>,
\end{equation}

where the phase on the RHS being the analog of the $S$-matrix element $S_{_{ab}}$ between anyons of charges $a,b$ in $U(1)_{_{k}}$ CS theory in 3D, namely $S_{_{ab}}\equiv e^{^{2\pi iab/k}}$.

For a given $X_{_{6}}\equiv \Sigma_{_{g,n}}\times X_{_{4}}$, there is a particular class of 7-manifolds $W_{_{7}}$ obtained by taking $W_{_{7}}\equiv V_{_{g,n}}\times X_{_{4}}$, with $V_{_{g,n}}$ a 3-manifold with $\partial V_{_{g,n}}\equiv\Sigma_{_{g,n}}$.

For any Riemann surface $\Sigma_{_{g,0}}$, there are many inequivelent 3-manifolds with $\Sigma_{_{g,0}}$ as its boundary. One such example are handlebodies. To construct a handlebody, choose a set $g$ of meridians, $\{\mu_{_{i}}\}, i\equiv1,...g$, i.e. a set of generators of $H_{_{1}}(\Sigma_{_{g,0}}, \mathbb{Z})$ which become trivial as elements of $H_{_{1}}(V_{_{g,0}}, \mathbb{Z})$.  The remaining $g$ generators of $H_{_{1}}(\Sigma_{_{g,0}}, \mathbb{Z})$ lift to generators of $H_{_{1}}(V_{_{g,0}}, \mathbb{Z})$, and are referred to as longitudes, $\{\lambda_{_{i}}\}, i\equiv1,...g$. Not every choice of $g$ generators gives rise to a legitimate set of meridians (i.e. vanishing cycles). For a specific choice of meridians to be legitimate, it must correspond to a maximal isotropic sublattice

\begin{equation}
L\ \in\ H_{_{1}}(\Sigma_{_{g,0}},\mathbb{Z}).
\end{equation}

The handlebody specified by the choice of meridians $L$ will be denoted by $V_{_{g,0}}^{^{L}}$. Given $V_{_{g,0}}^{^{L}}$, there are multiple choices of longitudes, differing by shifts in meridians

\begin{equation}
\lambda_{_{i}}^{\prime}\equiv\lambda_{_{i}}+\sum_{j=1}^{3}k_{_{ij}}\mu_{_{j}}\ \ \ ,\ \ k_{_{ij}}\in\mathbb{Z},
\end{equation}

satisfying the following constraint

\begin{equation}
<\mu_{_{i}},\lambda_{_{j}}>\equiv-<\lambda_{_{j}},\mu_{_{i}}>\equiv\delta_{_{ij}} .
\end{equation}

7D CS theory on a handlebody can be formulated by starting with a definition of the fields which reads

\begin{equation}
b_{_{i}}\overset{def.}{=}\oint_{_{\mu_{_{i}}}} c\ \ \ ,\ \ \ \hat b_{_{i}}\overset{def.}{=}\oint_{_{\lambda_{_{i}}}} c\ \ \ ,\ \ \ i=1,...,g.
\end{equation}

The meridians are naively contractible by definition. However, this does not necessarily imply the vanishing of $b_{_{i}}$, since there might be Wilson surfaces supported on $\lambda_{_{i}}$ and $\mu_{_{i}}$ that are nontrivially linking, effectively making $\mu_{_{i}}$ non-contractible.

The Wilson lines wrapping meridians and longitudes can be expressed as follows

\begin{equation}
\Phi_{_{i}}({\cal M}_{_{2}})\ \overset{def.}{=} \Phi_{_{i}}({\cal M}_{_{2}}\times\mu_{_{i}})\equiv e^{^{\frac{2\pi i}{N}\oint_{_{{\cal M}_{_{2}}}}b_{_{i}}}}
\ \ \ ,\ \ \ \hat \Phi_{_{i}}({\cal M}_{_{2}})\ \overset{def.}{=} \hat \Phi_{_{i}}({\cal M}_{_{2}}\times\lambda_{_{i}})\equiv e^{^{\frac{2\pi i}{N}\oint_{_{{\cal M}_{_{2}}}}\hat b_{_{i}}}},
\end{equation}

which might be nontrivially linking in the 5D theory resulting from dimensional reduction of the 7D TFT on $\Sigma_{_{g,0}}$. Compactifying the theory on $\Sigma_{_{g,0}}$ in presence of a series of Wilson 3-surfaces on $\lambda_{_{j}}\times {\cal M}_{_{2,j}}^{\prime}$, the linking equation becomes an operator equation in 4D

\begin{equation}
\Phi_{_{i}}({\cal M}_{_{2}})\ \equiv\ e^{^{\frac{2\pi i}{N}\sum_{j=1}^{g}<\mu_{_{i}},\lambda_{_{i}}>\oint_{_{{\cal M}_{_{2}}}}B_{_{j}}}},
\label{eq:extraphase}
\end{equation}

where $B_{_{j}}\in H^{^{2}}(X_{_{4}},\mathbb{Z}_{_{N}})$ denotes the Poincaré dual of ${\cal M}_{_{2,j}}^{\prime}$. Hence, in any 4D calculation involving $\Phi_{_{i}}$, the latter can be moved to the right of $\hat\Phi_{_{j}}$, and subsequently be shrunk to a point by introducing the phase (<ref>). Hence, the insertion of longitudinal Wilson lines allows us to make $b_{_{i}}\neq 0$, while still keeping it constant, thereby defining a background field.

At this point, the Sym TFT can be obtained by splitting the handlebody in the following way:

\begin{equation}
V_{_{g,0}}\ \overset{def.}\ \begin{cases} V_{_{g,0}}^{^{L,in}}\ \ \ \text{for}\ \ \ y\ge y_{_{*}}\\
\\
V_{_{g,0}}^{^{ext}}\overset{def}{=}\Sigma_{_{g,0}}\times[0,y_{_{*}}]\ \ \ \text{for}\ \ \ 0\le y\le y_{_{*}}\\
\end{cases}.
\label{eq:splitting}
\end{equation}

Importantly, it features two boundaries (cf. figure <ref>):

* One, at $y=0$, is where the 6D ${\cal N}=$(2,0) theory lives.

* The other, at $y=y_{_{*}}$, is where the longitudinal Wilson surfaces fixing the background fields are placed.

In presence of the splitting (<ref>), the state $<L|$ separates the anomaly TFT from the Sym TFT. The picture above is valid for $\forall g$, but has been drawn with $g=1$ for simplicity.

§.§ Fiber functors from non-invertible symmetries

The main motivation of the present section is that of studying the defects interpolating different 4D class ${\cal S}$ theories descending from dimensional reduction of 6D ${\cal N}$=(2,0) SCFTs. In particular, explaining how such defects emerge under gauging of multiple subalgebras, we propose the quantum dimension of the relative gaugeable subalgebra as the quantity needed to distinguish whether such interpolating defect is intrinsically or non-intrinsically non-invertible, thereby proposing the 4D counterpart of arguments proposed in [319] for the 2D case.

The present section is therefore structured into 4 main parts:

* At first, we describe the procedure, [323, 324, 325], and its adaptation to the Freed-Teleman, [103], and Freed-Moore-Teleman construction, [102], of relative field theories and topological symmetries of QFTs, respectively, with particular emphasis on the case of multiple simultaneous gauging of different subalgebras.

* We then briefly overview the procedure adopted by [313] for realising absolute 4D SCFTs from ${\cal N}=(2,0)$ SCFTs, explicitly rephrasing it in terms of the multiple gauging procedure described in section <ref>.

* When describing such interpolating defects in terms of gauging of homomorphisms, they naturally admit an interpretation in terms of fusion tensor categories interpolating between different 2-categories, to a total quantum dimension can be assigned. Section <ref> closes with the relation between the issue in defining a fiber functor in presence of non-invertible defects interpolating between different absolute theories.

* Making use of the gauging procedure explained in section <ref>, we propose a criterion for distinguishing between the two types of non-invertible defects in terms of the total quantum dimension of the relative gauged algebra implementing the gauging.

§.§.§ Intrinsic non-invertible symmetries and fiber functors

On the handlebody, one can perform two different operations:

* A modular transformation of $\Sigma_{_{g,0}}$, acting on the entire handlebody and generically changing the period matrix of $\Sigma_{_{g,0}}$ at each cross section. The theory in the bulk of the handlebody is topological, implying only $\Sigma_{_{g,0}}$ is affected by such transformations, which is where the 6D ${\cal N}=$(2,0) theory lives.

* Excising an inner handlebody, in the setup where the splitting (<ref>) takes place. The operation of interest consists in gluing together the 2 parts $V_{_{g,0}}^{^{L,in}}$ and $V_{_{g,0}}^{^{ext}}$ with a nontrivial element of the modular group $Sp(2g, \mathbb{Z}_{_{N}})$, corresponding to a surgery operation on the 3-manifolds.

Given that the latter takes place in the interior of the handlebody, it doesn't affect $\Sigma_{_{g,0}}$ where the 6D theory lives, hence, the period matrix remains unchanged in the region the theory is sensitive to. However, this surgery operation changes the global form of $L$, since it changes which are the contractible cycles in the handlebody. Hence, surgery is implemented by an element $F$ of the modular group which only affects the inner boundary of the handlebody, thereby changing the global form. This operation may in turn be combined with a modular transformation acting on the entire handlebody, reverting the interior of the handlebody back to its original form, but now changing the period matrix of the Riemann surface.

Choosing a period matrix that remains invariant under the action of $F$, the geometry with the combined surgery plus modular transformation has the same boundaries as the original geometry. Importantly, though, the full geometry is no longer the same, due to the internal twist.

Under dimensional reduction on $\Sigma_{_{g,0}}$, (<ref>) becomes

\begin{equation}
S_{_{5D}}\equiv\frac{2\pi}{N}\sum_{i=1}^{g}\int_{_{X_{_{4}\times[0,y_{_{*}}]}}}b_{_{i}}\ \cup\ \delta\hat b_{_{i}}.
\label{eq:newBFnew}
\end{equation}

The fact that $b_{_{i}}, \hat b_{_{i}}$ are dynamical, implies $\lambda_{_{i}}, \mu_{_{i}}$ are both nontrivial in $V_{_{g,0}}^{^{ext}}$. This follows from the fact that $L$ specifies which directions of the Riemann surface become trivial in the handlebody, namely the meridians, and which of the dynamical fields of the $BF$-theory become background. To each $L$ corresponds a different choice of Dirichlet boundary conditions for different sets of fields in the $BF$-theory. As explained in [313], the 7D CS theory on a Riemann surface can only capture the full Sym TFT iff all the non-invertible defects are non-intrinsic.

The 5D Wilson lines are defined from the 7D Wilson surfaces as follows

\begin{equation}
\Phi_{\overset{\rightarrow}{n}}\left({\cal M}_{_{2}}\right)\ \overset{def.}{=}\  \Phi\left({\cal M}_{_{2}}\times\gamma_{_{\overset{\rightarrow}{n}}}\right),
\label{eq:phim2}
\end{equation}

with 1-cycles

\begin{equation}
\gamma_{_{\overset{\rightarrow}{n}}}  \ \overset{def.}{=}\ \sum_{j=1}^{g} e_{_{j}}\lambda_{_{j}}+\sum_{j=1}^{g}\ m_{_{j}}\mu_{_{j}}\ \ \ ,\ \ \ \overset{\rightarrow}{n}\ \overset{def.}{=}\ \left(e_{_{1}},..., e_{_{g}};m_{_{1}},...,m_{_{g}}\right).
\end{equation}

Explicitly, (<ref>) can be expressed as follows

\begin{equation}
\Phi_{\overset{\rightarrow}{n}}\left({\cal M}_{_{2}}\right)\ \equiv\  e^{^{\frac{2\pi i}{N}\ \frac{1}{2}e\cdot m({\cal M}_{2},{\cal M}_{2})}}\ \prod_{i=1}^{g}\Phi^{^{e_{_{i}}}}\left({\cal M}_{_{2}}\times\gamma_{_{\overset{\rightarrow}{n}}}\right) \ \ \prod_{i=1}^{g}\Phi^{^{m_{_{i}}}}\left({\cal M}_{_{2}}\times\gamma_{_{\overset{\rightarrow}{n}}}\right),
\label{eq:phim21}
\end{equation}

with each electric and magnetic component, in turn descending from a 3D Wilson surface in 7D associated to a cycle $\gamma_{_{\overset{\rightarrow}{n}}}$.

Surgery defects in 7D correspond to loci across which $F$ acts on $\gamma_{_{\overset{\rightarrow}{n}}}$ as follows

\begin{equation}
F\left(\gamma_{_{\overset{\rightarrow}{n}}}\right)\ \equiv\ \gamma_{F\overset{\rightarrow}{n}}.
\end{equation}

In 5D, they reduce to gaugeable defects implementing morphisms in a higher-categorical structure

\begin{equation}
{\cal C}_{_{F}}(X_{_{4}})\ :\ \Phi_{\overset{\rightarrow}{n}}\left({\cal M}_{_{2}}\right)\ \mapsto\ \Phi_{F\overset{\rightarrow}{n}}\left({\cal M}_{_{2}}\right)\ \ \ \ ,\ \ \ {\cal M}_{_{2}}\ \subset\ X_{_{4}}.
\label{eq:morphism}
\end{equation}

The picture on the LHS shows the identification of a morphism between objects as a defect in a higher-categorical structure. On the RHS, the interface turns into a boundary defect once having performed the folding trick.

As suggested by (<ref>), $F$ acts on the labellings of the objects living in the theory. These objects are dyonic, with nontrivial linking. An explicit expression for the morphism (<ref>) as a defect follows by performing the folding trick on the surgery point in the 5D configuration depicted in figure <ref>

\begin{equation}
{\cal C}(X_{_{4}}) \ =\ \frac{\ |H^{^{0}}(X_{_{4}},\mathbb{Z}_{_{p}})|\ }{|H^{^{1}}(X_{_{4}},\mathbb{Z}_{_{p}})|}\ \sum_{_{{\cal M}_{_{2}} \in H_{_{2}}\left(X_{_{4}}, \mathbb{Z}_{_{p}}^{^{2g}}\right)}}\ \exp\left(\frac{2\pi i}{p}\ <F{\cal M}_{_{2}},{\cal M}_{_{2}}>\right) \Phi \left((\mathbb{1}-F){\cal M}_{_{2}} \right),
\label{eq:conddef}
\end{equation}

implying that, $\forall \ F\in\ Sp(2g, \mathbb{Z}_{_{p}})$, there is a codimension-1 defect implementing the symmetry associated to $F$ in the bulk 5D TQFT. Such defect is invertible, and therefore its fusion rules simply read

\begin{equation}
\boxed{\ \ \ {\cal C}(X_{_{4}})\ \times\ \bar {\cal C}(X_{_{4}})\ \equiv\ {\cal A}\ \equiv\ \mathbf{1}\color{white}\bigg]\ \ }.
\label{eq:identity}
\end{equation}

where ${\cal A}$ corresponds to the condensed algebra implementing the gauging. Notice that the defect (<ref>) is equivalent to the one arising from gauging by algebraic gauging in a topological order, as explained in section <ref> and depicted in figure <ref>. The crucial meaning of equation (<ref>) is the fact that, gauging is effectively acting as a redefinition of the identity from the mother theory to the gauged one. In particular, the last equality claims that the condensed algebra in the mother theory is projected to the identity of the gauged theory. Pictorially, this corresponds to the configuration on the LHS of <ref>.

Gauging the defects living in the SymTFT upon implementing an outer automorphisms, changes the boundary conditions of the resulting absolute theory. In presence of a unique gauging, the resulting configuration is depicted on the LHS. On the other hand, upon condensing two different subalgebras of the original bulk theory, leads to a bulk gauging defect effectively ending in the bulk, as shown in the figure in the middle. This is equivalent to realising a configuration comprising two different boundary conditions separated by a topological defect (as shown on the RHS).

Gauged defects associated to $F$, can in turn give rise to $|F|$-ality defects in the boundary 4D gauge theory. Such defects can be constructed from (<ref>) by admitting Dirichlet boundary conditions for the gauged defects, allowing them to terminate in the bulk, [319, 327, 328]. Practically, this corresponds to a 3D-manifold attached to a 4D manifold, which, combined together, give rise to a twist defect, $T_{_{F}}({\cal M}_{_{3}}, {\cal M}_{_{4}})$, whose expression is obtained from (<ref>) once rewritten in terms of relative cohomology. From the perspective of the gauging procedure, this amounts to considering simultaneous gauging of two different subalgebras within the gaugeable subalgebra of the mother theory. In terms of its SymTFT characterisation, this would correspond to the gaugeable defect effectively ending in the bulk SymTFT as shown in the middle of figure <ref>. Equivalently, this is equivalent to describing a relative field theory with two boundary conditions interpolated by a topological defect.

The gauged defect ${\cal C}_{_{F}}(X_{_{4}})$ implements the outer automorphism in the 5D TQFT. Applying twisted BCs, the gauged defect can effectively terminate in the 5D bulk, and is therefore topological. At this point, the width of the 5D theory can be taken to vanish, resulting in the configuration on the RHS, with the two phases of the 4D theory separated by a topological non-invertible defect, ${\cal N}_{_{F}}({\cal M}_{_{3}})$.

The gauged defect can be reexpressed in terms of the quantum dimensions of the underlying anyon description by noticing that the action of $F$ is equivalent to that of the outer automorphism keeping track of the anyonic symmetry being gauged[As explained in section <ref> and <ref>.]. This follows from the fact that the elements of the sum appearing in the gauged defect, (<ref>), can be broken down into two main parts, namely

\begin{equation}
\exp\left(\frac{2\pi i}{p}\ <F{\cal M}_{_{2}},{\cal M}_{_{2}}>\right) \Phi \left((\mathbf{1}-F){\cal M}_{_{2}} \right)\ \equiv\ S_{_{\overset{\rightarrow}{a}\ F\overset{\rightarrow}{a}}}\ {\cal G}_{_{F\overset{\rightarrow}{a}}},
\end{equation}

with $S_{_{\overset{\rightarrow}{a},\ F\overset{\rightarrow}{a}}}$ and ${\cal G}_{_{F\overset{\rightarrow}{a}}}$ denoting the $S$-matrix entries for the associated anyonic charges, and the exceptional fibration operator, respectively. Extending these arguments, the sum over the 2-manifolds ${\cal M}_{_{2}}$ can be replaced by a sum over the anyonic charges compatible with the symmetries of the given theory. This way, (<ref>) can be rewritten as follows

\begin{equation}
T_{_{F}}({\cal M}_{_{3}}, {\cal M}_{_{4}}) \ \overset{def.}{=}\ \frac{\ |H^{^{0}}(X_{_{4}},\mathbb{Z}_{_{p}})|^{^{2g}}\ }{|H^{^{1}}(X_{_{4}},\mathbb{Z}_{_{p}})|^{^{2g}}}\ \sum_{_{\overset{\rightarrow}{a}}} \ S_{_{\overset{\rightarrow}{a},\ F\overset{\rightarrow}{a}}}\ {\cal G}_{_{F\overset{\rightarrow}{a}}},
\label{eq:conddef1}
\end{equation}

where $\overset{\rightarrow}{a}\ \overset{def.}{=}\ \left([a], \rho(a)\right)$, with $[a], \rho(a)$ denoting the conjugacy class and the representation of the corresponding anyon charges, respectively.
Given that the defect is now topological, it can be moved to the 4D boundary, defining an interface for the 4D gauge theory

\begin{equation}
{\cal N}({\cal M}_{_{3}})\ \overset{def.}{=}\ \underset{y\rightarrow0}{\lim}\ T_{_{F}}({\cal M}_{_{3}}, {\cal M}_{_{4}}),
\end{equation}

obeying non-invertible fusion rules. From the 4D perspective of the absolute theory on the RHS of figure <ref>, ${\cal N}({\cal M}_{_{3}})$ is intrinsically-non-invertible, namely, it cannot be rendered invertible by acting with $SL(2,\mathbb{Z})$-transformations. This is equivalent to stating that the action of $F$ cannot be reversed simply by acting with a modular transformation. Non-invertibility implies its fusion rules now read

\begin{equation}
\boxed{\ \ \ {\cal N}({\cal M}_{_{3}})\ \times \bar{\cal N}({\cal M}_{_{3}})\ =\ {\cal A}_{_{X_{_{4}}}}\color{white}\bigg]\ \ },
\label{eq:NNA}
\end{equation}

with ${\cal A}_{_{X_{_{4}}}}$ denoting the relative gaugeable algebra in the absolute theory.

Crucially, the RHS of equation (<ref>) is not simply the identity of a 4D absolute theory, corresponding to the fact that there is no direct counterpart of equation (<ref>) upon performing a double gauging involving two distinct subalgebras of the gaugeable algebra characterising the mother SymTFT. Given that the latter is basically equivalent to the existence of a well-defined partition function for the resulting absolute theory, we claim that, if might be able to define a counterpart of the functor (<ref>) for the case depicted at the centre and RHS of figure <ref>, by composing the following chain of functors

A qualitative diagram depicting the counterpart of the fiber functor's domain for the case of figure <ref>. The black arrows indicate the choice of different gaugeable subalgebras, leading to different 4D absolute theories, labelled by 2-categories ${\cal C,D}$.

where ${\cal A}_{_{4D}}$ is the gaugeable algebra leading to a single 4D absolute theory, whereas ${\cal A}_{_1}$ and ${\cal A}_{_2}$ correspond to different subalgebras of ${\cal A}_{_{4D}}$ leading to different 2-categories of underlying charges associated to the gauged symmetry characterising the two distinct 4D absolute theories. This is current work in progress by the same author, and we plan to report about it in a forthcoming work. The main message we want to convey with this is that the fiber functor that can be defined for the configuration at the centre and RHS of figure <ref> leads to the partition function of a 3D theory.

§.§.§ Quantum dimension from relative gaugeable algebra

In the concluding part of this section, we make use of the tools outlined in the previous parts of the present treatment, leading to the proposal of a probe quantity enabling to distinguish whether the defect interpolating in between class ${\cal S}$ theories is intrinsically or non-intrinsically non-invertible. The starting point will be the setting of section <ref>, where we revised the gauging procedure, implemented by flooding the topological order corresponding to the 2-category of Wilson surfaces living in the bulk SymTFT with a tensor network of an idempotent element, namely the norm element (<ref>).

Practically, the two phases (prior and after gauging) can be thought of as arising from a phase transition in between two MTCs, ${\cal B}_{_{ung}}$ and ${\cal B}_{_g}$, corresponding to the ungauged and the gauged phases, respectively, [324]. Imposing natural physical requirements, it is possible to derive a relation between the anyons in the ${\cal B}_{_{g}}$ phase and those in the ${\cal B}_{_{ung}}$ phase. The vacuum, or tensor unit ${\cal A}$ in ${\cal B}_{_g}$ is necessarily a connected commutative separable algebra in ${\cal B}_{_{ung}}$, and the category ${\cal B}_{_g}$ is equivalent to the category of local ${\cal A}$-modules as MTCs. This gauging produces a gapped domain wall (DW) with wall excitations given by the category of ${\cal A}$-modules in ${\cal B}_{_{ung}}$. The domain wall separating them, is, in turn a category, ${\cal A}\equiv{\cal C}(X_{_4})$, as shown in figure <ref>.

This picture shows the categories of anyons associated to the ungauged and gauged theories, denoted by ${\cal B}_{_{ung}}$ and ${\cal B}_{_g}$, respectively. Gauging can be visualised as the domain wall effectively separating two different topological phases.

For any pair of elements, $M, N$,

\begin{equation}
\text{Hom}_{_{{\cal B}_{_g}}}(M,N)\ \hookrightarrow \ \text{Hom}_{_{{\cal B}_{_{ung}}}}(M,N).
\end{equation}

Assuming vacuum degeneracy being trivial,

\begin{equation}
\text{Hom}_{_{{\cal A}}}(\mathbf{1}_{_{\cal A}},\mathbf{1}_{_{\cal A}}) \ \simeq \ \mathbb{C},
\end{equation}

${\cal A}$ must be a UFC with a unique spherical structure. All objects in ${\cal A}$ should come from objects in ${\cal B}_{_{ung}}$, and should satisfy the following properties:

* ${\cal A}$ is a subcategory of ${\cal B}_{_{ung}}$

\begin{equation}
\text{Hom}_{_{{\cal A}}}(M,N)\ \hookrightarrow\   \text{Hom}_{_{{\cal B}_{_{ung}}}}(M,N)
\end{equation}

* $\mathbf{1}_{_{ung}}$ should condense into $B$, such that

\begin{equation}
\iota_{_B}:\ \mathbf{1}_{_{{\cal B}_{_{ung}}}}\ \rightarrow\ B\ \in\ {\cal B}_{_{ung}}
\end{equation}

$\mathbf{1}_{_{{\cal B}_{_g}}}\equiv{\cal A}$ should fuse into the vacuum on the wall when moving ${\cal A}$ close to the wall, such that, in ${\cal B}_{_{ung}}$, the corresponding map reads

\begin{equation}
\iota_{_A}^{^B}:\ A\ \rightarrow\ B
\end{equation}

What we have just said can be succinctly re-expressed as follows

\begin{equation}
\boxed{\ \ \mathbf{1}_{_{{\cal B}_{_g}}}\ \equiv\ {\cal A}\ \subset\ {\cal B}_{_{ung}} \color{white}\bigg]\ \ }.
\label{eq:identityoperator}
\end{equation}

After what we have just said, the key point to focus on is the fact that the gauging procedure, the identity operator changes, in the sense that, from being a simple element in the category ${\cal B}_{_{ung}}$, it is nontrivial in the category of the gauged theory, ${\cal B}_{_g}$, since $\mathbf{1}_{_{{\cal B}_{_g}}}\equiv{\cal A}$.

The analogy we have drawn so far with the example shown in figure (<ref>) taken from [324], corresponds to the case of a single gauging of the 5D SymTFT, leading to the LHS of our figure <ref>. For the case involving a double gauging, instead, such prescription needs to be suitably adapted. The motivation still follows the arguments presented in [324]. We now argue why this is indeed the case specifying to the example described in section <ref>.

Recall that ${\cal C}(X_{_4})$ is the operator implementing the outer automorphism on the category of defects defining the 5D SymTFT, and corresponds to the idempotent object that is constituted of the algebra elements that are flooding the SymTFT in order to implement the gauging. Denoting by ${\cal C}_{_g}, {\cal C}_{_{ung}}$ the 2-category of defects characterising the 4D absolute theories, if the two phases are related by a single bulk gauging, it follows that

\begin{equation}
\text{dim}_{_{{\cal C}_{_g}}}\ \mathbf{1}_{_{{\cal C}_{_g}}}\ \equiv\ \frac{\ \text{dim}_{_{{\cal C}_{_{ung}}}} {\cal A}_{_{4D}}\ }{\text{dim}_{_{{\cal C}_{_{ung}}}} {\cal A}_{_{4D}}}\ \equiv\ 1.
\label{eq:BBungg5}
\end{equation}

But this is only from the point of view of the gauged theory, ${\cal C}_{_g}$. On the other hand, ${\cal A}_{_{4D}}$ is a graded object in ${\cal C}_{_{ung}}$, and therefore

\begin{equation}
\text{dim}_{_{{\cal C}_{_{ung}}}}\ {\cal A}_{_{4D}}\ \neq\ 1.
\label{eq:BBungg6}
\end{equation}

However, our aim is that of describing a composite configuration for the absolute theory of the kind depicted on the LHS of figure <ref>, where, as explained in sections <ref> and <ref>, both phases really descend form the gauging of different subalgebras within the gaugeable algebra of the original ungauged SymTFT[This is our proposal on the basis of [323, 324].]. The different 4D absolute gauge theories are interpolated by a non-invertible defect defined as

\begin{equation}
{\cal N}({\cal M}_{_3})\ \overset{def.}{\equiv}\ {\cal D}_{_3}\ {\cal A}_{_{\epsilon\rho}}
\label{eq:noninv}
\end{equation}

where ${\cal A}_{_{\epsilon\rho}}$ plays the role of a relative gaugeable algebra that cannot be identified with the vacuum on either side.

Composite 4D absolute theories separated by an intrinsic non-invertible defect on the LHS. Rephrased in terms of the higher-categorical structure underlying the resulting theory, the algebra dressing the interpolating defects ${\cal N}_{_3}$ is a homomorphism between the objects of a 2-category.

Consistency of the configuration associated to the LHS of figure <ref>, is mapped to a statement of Witt equivalence between the underlying 2-categories depicted on the RHS of the same figure, [324],

\begin{equation}
{\cal C}\ \boxtimes\ \bar{\cal D}\ \simeq\ \mathfrak{Z}({\cal {\cal A}_{_{\epsilon\rho}}}),
\label{eq:inclusion}
\end{equation}

where ${\cal A}_{_{\epsilon\rho}}$ denotes the relative gaugeable algebra featuring in (<ref>).

The relative gaugeable algebra, ${\cal A}_{_{\epsilon\rho}}$, constitutes a fusion tensor category[With ${\cal A}_{_1}$ and ${\cal A}_{_0}$ denoting the object and the morphisms, respectively.] defining morphisms between any pair of elements $L, L^{\prime}\ \in\ {\cal C,D}$.

\begin{equation}
{\cal A}_{_{\epsilon\rho}}^{^{L,L^{^{\prime}}}} \ \overset{def.}{=}\ \left\{{\cal A}_{_1}^{^{L,L^{^{\prime}}}}, {\cal A}_{_0}^{^{L,L^{^{\prime}}}}\ \right\},
\end{equation}

whose total quantum dimension can be compared to that of the identity, i.e. of the algebra that has been condensed to achieve the reference gauge theory, with 2-category ${\cal C}$.

The algebra for the underlying category, denoted by ${\cal A}$ can be partially condensed, under suitable choice of subalgebras ${\cal A}_{_1}, {\cal A}_{_2}\ \subset\ {\cal A}$, such that they lead to different gauge theories, whose 2-categories are labelled by ${\cal C, D}$.

The composite system (<ref>) can be thought of as describing a thickened domain wall, $_{_{{\cal A}_{_1}}}{\cal A}_{_{{\cal A}_{_2}}}\ \overset{def.}{\equiv}\ {\cal A}_{_1}\ \otimes_{_{\cal A}}\ {\cal A}_{_2}$.

From the point of view of the absolute 4D theory, the non-invertible defect can be thought of as a thickened domain wall obtained by condensing two different subalgebras within the original condensing algebra ${\cal A}$, as shown in figure <ref>. Given that the target absolute theory we want to achieve is that of figure <ref>, then the corresponding 2-categories of defects can be obtained starting from

\begin{equation}
{\cal A}\ \equiv\ {\cal D}\ \boxtimes\ \mathfrak{Z}({\cal E}),
\end{equation}

\begin{equation}
G:\ {\cal C}\ \boxtimes\ \mathfrak{Z}({\cal D})\ \simeq\ {\cal D}\ \boxtimes\ \mathfrak{Z}({\cal E}),
\end{equation}

and the  fiber functors

\begin{equation}
{\cal F}_{_{\cal E}}\ :\ \mathfrak{Z}({\cal E})\ \rightarrow\ {\cal E}
\ \ \ ,\ \ \
{\cal F}_{_{\cal D}}\ :\ \mathfrak{Z}({\cal D})\ \rightarrow\ {\cal D}
\end{equation}

with corresponding duals ${\cal F}_{_{\cal E}}^{^V}, {\cal F}_{_{\cal D}}^{^V}$.

The fact that ${\cal N}({\cal M}_{_{3}})$ is intrinsically non-invertible corresponds to saying that the outer automorphism implemented by the action of $F$ cannot be undone by a combination of modular transformations, as described in section <ref>. In particular, ${\cal A}_{_{\epsilon\rho}}$ is different from the identity on either side of the defect, and, therefore, ${\cal D}_{_{{\cal A}_{_{\epsilon\rho}}}}\neq {\cal D}_{_{\mathbb{1}}}$.

§ PROOF

Take ${\cal B}$ to be a semisimple abelian category over $\mathbb{C}$; assuming it is a rigid braided tensor category. $I$ is the set of isomorphism classes of irreducible objects in ${\cal B}$, with representatives $V_{_i}, \ \forall i\in I$. Assuming the spaces of morphisms are finite-dimensional. $\mathbb 1\in {\cal B}$ is assumed to be a simple object corresponding to the labelling $i\equiv0\in I\ \Rightarrow\ V_{_0}\overset{def.}{\equiv}\mathbf{1}$. A ${\cal B}$-algebra is an object ${\cal A}\in{\cal C}$ with morphisms $\mu:\ {\cal A}\otimes{\cal A}\rightarrow{\cal A}$ and $\iota_{_{{\cal A}}}:\mathbf{1}\hookrightarrow{\cal A}$, with

\begin{equation}
\text{dim \ Hom}_{_{{\cal B}}}(\mathbf{1},{\cal A})\ \equiv\ \mathbf{1}.
\end{equation}

Given ${\cal B}$ and a ${\cal B}$-algebra ${\cal A}$, Rep${\cal A}$ is defined as pairs $(V,\mu_{_{V}})$, where $V\in{\cal B}$ and $\mu_{_{V}}:{\cal A}\otimes V\rightarrow V$ morphisms in ${\cal B}$.

For example, if $G$ is a finite group and ${\cal B}$ is the category of finite-dimensional complex representations of $G$, semisimple ${\cal B}$-algebras correspond to different semigroups in $G$ (this is what we need to implement gauging in the Freed-Moore-Teleman setup).

Here, in Rep${\cal A}$, ${\cal A}$ is the unit object. For the case where there are two different absolute theories separated by an intrinsically non-invertible defect, the setup of joint absolute theories results from a double gauging procedure [Liang Kong] and it makes sense to consider the composite object $X\otimes_{_{{\cal A}}}Y$ with ${\cal A}$ the gaugeable algebra in ${\cal C}$; so, we have that $X,Y\in$ Rep${\cal A}$. But, at the same time, $X$ and $Y$ are also elements of two different subalgebras within ${\cal A}$ corresponding to two different gaugings. E.g. $X\in$Rep${\cal A}_{_1}$ and $Y\in$Rep${\cal A}_{_2}$, with ${\cal A}{_{_1}}, {\cal A}_{_{2}}\subset{\cal A}$. In turn, ${\cal A}_{_1}$ is the unit element in Rep${\cal A}_{_1}$ and ${\cal A}_{_2}$ id the unite element in Rep${\cal A}_{_2}$.

Hence, if we consider the original mother algebra ${\cal A}$ with its characteristic elementary objects $V_{_i}$, and assuming every one of them has multiplicity $1$ within ${\cal A}$, we get

\begin{equation}
\text{dim}_{_{\cal A}}(X\otimes_{_{{\cal A}}}Y) \ \equiv\ \text{dim}_{_{\cal A}}(X)\ \text{dim}_{_{\cal A}}(Y)
\label{eq:first}
\end{equation}

\begin{equation}
\text{dim}_{_{\cal A}}(X) \ \equiv\ \frac{\text{dim}_{_{\cal B}}({ X})}{\text{dim}_{_{\cal B}}({\cal A})}.
\label{eq:second}
\end{equation}

Furthermore, we have that

\begin{equation}
\text{dim}_{_{\cal A}}(X\otimes_{_{\cal A}}Y) \ \equiv\ \frac{\text{dim}_{_{\cal B}}({ X})\ \text{dim}_{_{\cal B}}({ Y})}{\text{dim}_{_{\cal B}}({\cal A})}.
\label{eq:third}
\end{equation}

Extending equation (<ref>) to the case in which $X, Y$ are replaced by the ${\cal A}_{_1}, {\cal A}_{_2}$ subalgebras themselves, we can look at the following ratio

\begin{equation}
\frac{\text{dim}_{_{\cal B}}({\cal A}_{_1}\otimes_{_{\cal A}}{\cal A}_{_2})}{\text{dim}_{_{\cal B}}({\cal A}_{_1})}\ \equiv\ \frac{{\cal D}^{^{intr.}}_{{\cal A}}}{{\cal D}^{^{non-intr.}}_{{\cal A}}},
\label{eq:third1}
\end{equation}

where on the RHS we identified the numerator and denominator with the quantum dimension of algebras associated to the intrinsic or non-intrinsic non-invertible defects separating absolute theories resulting from the Freed-Moore-Teleman construction discussed in the first part of section <ref>. Making use of (<ref>) and (<ref>), (<ref>), we get

\begin{equation}
\frac{\text{dim}_{_{\cal A}}({\cal A}_{_1}\otimes_{_{\cal A}}{\cal A}_{_2})\ \text{dim}_{_{\cal B}}({\cal A})}{ \text{dim}_{_{\cal A}}({\cal A}_{_1})\ \text{dim}_{_{\cal B}}({\cal A})}\ \equiv\ \frac{\text{dim}_{_{\cal A}}({\cal A}_{_1})\ \text{dim}_{_{\cal A}}({\cal A}_{_2})}{\text{dim}_{_{\cal A}}({\cal A}_{_1})}\ \equiv\ \text{dim}_{_{\cal A}}({\cal A}_{_2}).
\label{eq:third1}
\end{equation}

Given that

\begin{equation}
\text{dim}_{_{\cal A}}({\cal A}_{_2})\ \overset{def.}{=}\ \sum_{_i}d_{_i}V_{_i}
\end{equation}

with $V_{_i}\in {\cal A}_{_2}$ being simple objects in ${\cal A}_{_2}$ (similar arguments hold for ${\cal A}_{_1}$, since both are assumed to be subalgebras in ${\cal A}$, whereas

\begin{equation}
d_{_i}\ \overset{def.}{=}\ \text{dim}_{_{\cal A}}V_{_i},
\end{equation}

it therefore follows that

\begin{equation}
\boxed{\ \ \ \frac{\text{dim}_{_{\cal B}}({\cal A}_{_1}\otimes_{_{\cal A}}{\cal A}_{_2})}{\text{dim}_{_{\cal B}}({\cal A}_{_1})}\ \equiv\ \frac{{\cal D}^{^{intr.}}_{{\cal A}}}{{\cal D}^{^{non-intr.}}_{{\cal A}}}\ >\ 1\color{white}\bigg]\ \ }.
\label{eq:third2}
\end{equation}

For the configuration of interest, namely two different absolute theories obtained by gauging of two different subalgebras ${\cal A}_{_1}, {\cal A}_{_2}\subset{\cal A}$, separated by an intrinsic non-invertible defect, the numerator in (<ref>) corresponds to the nontrivial fusion.

We are therefore lead to conclude that the total quantum dimension of the relative gaugeable algebra ${\cal A}_{_{\epsilon\rho}}$ is greater w.r.t. that of the condensed algebra leading to the single gauged phase, ${\cal C}$, and we propose the following

\begin{equation}
{\cal N}({\cal M}_{_{3}})\ \times\ \bar {\cal N}({\cal M}_{_{3}})
\equiv{\cal A}_{_{4D}}
\end{equation}

for class ${\cal S}$ theories, whereas the denominator in (<ref>) corresponds to the quantum dimension of the algebra that has been condensed in the SymTFT to obtain a specific absolute theory, namely

\begin{equation}
{\cal D}_{_3}\ \times\ \bar {\cal D}_{_3} \equiv\ \mathbf{1}\
\equiv\ {\cal A}_{_1}
\end{equation}

Gathering these results together, we can rewrite equation (<ref>) in terms of the notation adopted in the first part of this section, leading to the following result

\begin{equation}
\begin{aligned}
\boxed{\ \ \ \frac{<{\cal N}({\cal M}_{_{3}})\ \times\ \bar {\cal N}({\cal M}_{_{3}}) >_{_{{\cal B}}}}{<{\cal D}_{3}\ \times\ \bar{\cal D}_{3}>_{_{{\cal B}}}}\  \equiv\ \frac{ <{\cal A}_{_{\epsilon\rho}} >_{_{{\cal B}}} }{ \ <\mathbf{1}_{_{{\cal B}}} >_{_{{\cal B}}} \ }\ \equiv\ \left(\frac{{\cal D}^{^{intr.}}_{{\cal A}}}{{\cal D}^{^{non-intr.}}_{{\cal A}}} \right)_{_{X_{4} }}\ >\ 1\color{white}\bigg]\ \ },
\label{eq:4DNN3}
\end{aligned}
\end{equation}

with the last relation indicating that such ratio can be used as a parameter probing whether the two gauge theories are separated by an intrinsic non-invertible symmetry. Consequently, the following relation

\begin{equation}
\boxed{\ \ \ {\cal D}_{{\cal A}_{_{\epsilon\rho}}{(X_{_{4}})}}^{^{intr.}} \ >  \ {\cal D}_{{\cal A}_{_{\epsilon\rho}}{(X_{_{4}})}}^{^{non-intr. }}\color{white}\bigg]\color{black}\ \  }
\label{eq:BFE}
\end{equation}

implies that the fusion category associated to intrinsic non-invertible defects is characterised by a bigger quantum dimension, implying such configuration (arising from double gauging) is able to store a higher number of degrees of freedom w.r.t. the case in which there is a unique gauging taking place[Note that this criterion is perfectly in agreement with Remark 2.5 in [564].].

§ KEY POINTS

* For class ${\cal S}$ theories, ${\cal D}_{{\cal A}_{_{\epsilon\rho}}}$ enables to probe whether a defect is intrinsically or non-intrinsically non-invertible.

* ${\cal D}_{{\cal A}_{_{\epsilon\rho}}{(X_{_{4}})}}^{^{intr.}} \ >  \ {\cal D}_{{\cal A}_{_{\epsilon\rho}}{(X_{_{4}})}}^{^{non-intr. }}$, and therefore the former is a theory equipped with a higher-categorical structure enabling to store more d.o.f. in certain superselection sectors.

* For theories descending from 6D ${\cal N}=(2,0)$ SCFTs, if the gauging procedure leads to the presence of intrinsic non-invertible defects, the total quantum dimension of the algebra implementing the non-invertibility of such defects exhibits an increase in its total quantum dimension, signalling the relative gaugeable algebra ${\cal A}_{_{\epsilon\rho}}$ is characterised by superselection sectors admitting a richer structure w.r.t. the condensed algebra leading to a single absolute 4D gauged theory.

At first sight, the title chosen for this article might appear provocative, and the length of the paper is certainly not fair towards the rich and deep advancements made by the phenomenological community so far[For a detailed overview of the state of the art and major developments in the field, we refer the interested reader to the rich literature at our disposal, including [546, 547, 548, 20, 21, 22, 23, 24, 25, 26, 27], and [19, 32, 48, 50, 58, 60, 61] for slightly more theoretically-oriented perspectives in this regard, especially in relation with string theory, [8, 9, 10, 11, 12, 13, 14, 15, 17].]. However, we stress that our aim is that of introducing important open questions in the realm of Particle Physics in a mathematical language approachable by categorical algebraic geometers.

Actually, this work is really meant to be the first of a series of papers by the same author continuing from past developments, bridging the gap between Pure Mathematics and Theoretical Particle Physics achievements. The author's engagement with experts from both lines of research is what motivated this project in the first instance. In upcoming work, [549], we will be developing this description in terms of algebraic and Hilbert series calculation in greater detail. For the moment, we anticipate that the purpose of this introductory article is mainly fourfold:

* To illustrate (some) key fundamental problems in Theoretical Particle Physics necessitating further mathematical understanding.

* Opening the scene to well-established techniques in representation theory and categorical algebraic geometry to help in this regard.

* For those who are not experts in the field, serve as an oversimplified explanation of the powerful formalism put forward by Braverman, Finkelberg, and Nakajima (BFN), [134, 130]. In doing so we will explicitly show the relation in between their work, that of Freed, Moore, and Teleman (FMT), [102], and Teleman's, [181], which, to the best of our knowledge, has only previously been referred to in [3, 5].

* Motivating Phenomenologists to engage more with Pure Mathematicians, fostering mutual inspiration for common aims.

The main objective of this paper is that of explaining the higher categorical structures that are needed for describing the invariants associated to specific supersymmetric quiver gauge theories, with a particular focus on dualities and their mutual relations in terms of higher-categories.  We believe that deepening its understanding and generalisations thereof, could lead to uncharted corners of mathematics that could be used for embedding of the Standard Model (SM) of Particle Physics, [28] in possible supersymmetric unifying theories, [18].

The present work is structured as follows:

* Section <ref>  explains the motivations and outline of the present and forthcoming papers [549]. In doing so, we briefly introduce scattering amplitudes as crucial tools within the context of particle physics, explaining their connection with Hilbert series. We then briefly explain some of the key open questions in Theoretical Particle Physics necessitating further mathematical understanding, specifically the hierarchy problem, and the lightness of the Higgs mass (and its relation to Supersymmetry (SUSY), [18]).

* Section <ref>   is mostly a revision of [102, 103], and we use it at this stage for multiple reasons. First of all because it allows to introduce gauge theories and Lagrangians in a mathematical language, which in turn is suitable to make connection with the higher-categorical Symmetry Topological Field Theory (SymTFT) prescription put forward by Freed, Moore, and Teleman, (FMT), [102], in the context of relative quantum field theories (QFTs).