Fixing $U\subset{\cal B}$ over which $\hat\Gamma$ can be trivialised, the holomorphic Darboux coordiantes, $\chi_{_{\gamma}}^{^{\vartheta,u_{_o}}}$, with $u_{_o}\in U\subset {\cal B}$, are labelled by sections $\gamma$ of $\hat\Gamma$ over $U$. These are functions on $\pi^{^{-1}}(U)\times\mathbb{H}_{_{\vartheta}}$, where

\begin{equation}
\mathbb{H}_{_{\vartheta}}\overset{def.}{=}\ \bigg\{\zeta:\vartheta-\frac{\pi}{2}<\text{arg}\ \zeta<\vartheta+\frac{\pi}{2}\bigg\}.
\end{equation}

By construction, these coordinates, $\chi_{_{\gamma}}^{^{\vartheta,u_{_o}}}$ experience a jump at specific values of $\vartheta$, and the latter are the phases of the central charges of the BPS states in the vacuum labelled by $u_{_o}$. Furthermore, these jumps are determined by the gauge charges of the BPS states.

such that the following properties are satisfied:

* $
\chi_{_{\gamma+\gamma^{^{\prime}}}}=\chi_{_{\gamma}}\chi_{_{\gamma^{^{\prime}}}}. $

* $\forall\ \xi\in\mathbb{C}^{^{\times}}$, $\chi_{_{\gamma}}(.;\xi)\in\mathbb{C}^{^{\times}}$, and is holomorphic.

* $\{\chi_{_{\gamma}},\chi_{_{\gamma^{^{\prime}}}}\}\ =\ <\gamma,\gamma^{^{\prime}}>\chi_{_{\gamma}}\chi_{_{\gamma^{^{\prime}}}}.

* $\forall (u,\theta)\in{\cal M}$, $\chi_{_{\gamma}}(u,\theta;\zeta)$ is holomorphic in $\zeta$.

* $\chi_{_{\gamma}}(.;\zeta)=\overline{\chi_{_{-\gamma}}\left(.;-\frac{1}{\zeta}\right)}$.

* $\underset{\zeta\rightarrow 0}{\lim}\ \chi_{_{\gamma}}(u,\theta;\zeta)\exp\left[-\frac{\pi RZ_{_{\gamma}}(u)}{\zeta}\right]$ exists.

* $\chi_{_{\gamma}}^{^{\vartheta,u_{_o}}}$ features discontinuities at pairs $(\vartheta, u_{_o})$ for which there is some $\gamma_{_{BPS}}$ with

\begin{equation}
\text{arg}(-Z_{_{BPS}}(u_{_o}))=\vartheta.
\end{equation}

* Fixing $\vartheta_{_o}\in\mathbb{R}/2\pi\mathbb{Z}$, $u_{o}\in{\cal B}$, one can define

\begin{equation}
\text{S}_{_{\vartheta, u_{_o}}}\overset{def.}{=}\prod_{_{\gamma_{_{BPS}}:\text{arg}(-Z_{_{\gamma_{_{BPS}}}}(u_{_o})=\vartheta_{_o}}}\text{K}_{_{\vartheta, u_{_o}}}^{^{\Omega(\gamma_{_{BPS}};u_{_o})}},
\end{equation}

\begin{equation}
\text{K}_{_{\vartheta, u_{_o}}}:\ \chi_{_{\gamma}}\ \mapsto\ \chi_{_{\gamma}}\left(1-\sigma(\gamma_{_{BPS}})\chi_{_{\gamma_{_{BPS}}}}\right)^{^{<\gamma,\gamma_{_{BPS}}>}},
\end{equation}

\begin{equation}
\Rightarrow \ \left(\underset{\vartheta\rightarrow\vartheta_{_o}^{^+}}{\lim}\chi_{_{\gamma}}^{^{\vartheta,u_{_o}}}\right)\ =\ \text{S}_{_{\vartheta_{_o}, u_{_o}}}\left(\underset{\vartheta\rightarrow\vartheta_{_o}^{^-}}{\lim}\chi_{_{\gamma}}^{^{\vartheta,u_{_o}}}\right).
\end{equation}

* When $\zeta\in\mathbb{H}_{_{\vartheta}}$, then

\begin{equation}
\chi_{_{\gamma}}\ \sim\ \chi_{_{\gamma}}^{^{approx.}}.
\end{equation}

If we now consider a point $u\in{\cal B}$ and two different phases $\vartheta_{_{\pm}}$, such that $\vartheta_{_+}-\vartheta_{_-}<\pi$, then, the two coordinate systems are related as follows

\begin{equation}
\chi_{_{\gamma}}^{^{\vartheta_{_+},u}}=\text{S}(\vartheta_{_-},\vartheta_{_+};u)\ \chi_{_{\gamma}}^{^{\vartheta,u}},
\end{equation}

\begin{equation}
\text{S}(\vartheta_{_-},\vartheta_{_+};u)\ \overset{def.}{=}\ \prod_{_{\gamma_{_{BPS}}:\vartheta_{_-}<\text{arg}(-Z_{_{\gamma_{_{BPS}}}}(u)<\vartheta_{_+}}}\text{K}_{_{\vartheta, u_{_o}}}^{^{\Omega(\gamma_{_{BPS}};u)}},
\end{equation}

where the product decomposition is uniquely defined.

§.§ The Alday-Gaiotto-Tachikawa (AGT) correspondence: a brief overview

This can be generalised to arbitrary class ${\cal S}$-theories of $\mathfrak{su}$(2)-type: the 4D ${\cal Z}_{_{S^{^1}\times S^{^{3}}}}$ associated to a Riemann surface of genus $g$ and $n$-punctures, where we place the holonomy $\left(a_{_{i}},\frac{1}{a_{_{i}}}\right)\ \in\ SU(2)_{_{i}}$ for the flavour symmetry for the $i^{^{th}}$ puncture, corresponds to the 2D $q$-deformed T=Yang-Mills theory on the same $\Sigma_{_{g}}$, where the punctures are described as holes with holonomy $\left(a_{_{i}},\frac{1}{a_{_{i}}}\right)\ \in\ SU(2)$, thereby resulting in the following

\begin{equation}
{\cal Z}_{_{4D, g,n}}(a_{_{i}})\ \equiv\ \frac{\prod_{i}K(a_{_{i}})}{K_{_{o}}^{^{2g-2+n}}}\ {\cal Z}_{_{2D,g,n,A\equiv0}}(a_{_{i}})
\end{equation}

where the prefactor can be reabsorbed under suitable redefinition of the handling operator. This correspondence was first found in [124].

Undeformed 2D YM

For a gauge theory on a cylinder, with holonomy

\begin{equation}
U\overset{def.}{=}\ {\cal P}\ \text{exp}\int_{_{0}}^{^{L}}dx^{^{1}}A_{_{1}}
\ \ \
, \ \ \
U\ \xrightarrow{\ \color{white}{}\color{black} \ G\ \color{white}{}\color{black}\ }\ g U g^{^{-1}}\ \ \ , \ \ \ g\in G
\end{equation}

the partition function behaves as

\begin{equation}
{\cal Z}_{_{A+A^{\prime}}}(U)\ \equiv\ e^{^{-Ac_{_{2}}}}\ {\cal Z}_{_{A^{\prime}}}(U)
\end{equation}

with $c_{_{2}}$ the operator acting on $\chi_{_{R}}(U)$ by $c_{_{2}}(R)$. From this follows that ${\cal Z}_{_{A\equiv o}}(U)$ is enough to determine ${\cal Z}$ for arbitrary $A$.

\begin{equation}
{\cal Z}_{_{A\equiv o}}(U)\ \equiv\ \alpha\ \delta(U)
\end{equation}

with $\delta(U)$ denoting the $\delta$-function at the ideal of $G$ and

\begin{equation}
\delta(U)\equiv\sum_{R}\ d_{_{R}}\chi_{_{R}}(U)  \ \ \ ,\ \ \ d_{_{R}}\equiv \int_{_{G}}dU\delta(U)\chi_{_{R}}(U)\equiv\text{Tr}\mathbb{1}\equiv \text{dim} R
\end{equation}

\begin{equation}
Z_{_{A}}(U)\equiv\alpha\sum_{R}e^{^{-Ac_{_{2}}(R)}}\ \text{dim} R\ \chi_{_{R}}(U)
\end{equation}

For a 2D surface with 2 boundary holonomies, $U,V$, instead, we have

\begin{equation}
Z_{_{A}}(U,V)\equiv\sum_{R}e^{^{-Ac_{_{2}}(R)}}\  \chi_{_{R}}(U) \chi_{_{R}}(V)
\end{equation}

Any 2D surface can be cut into pieces, each one being a sphere with 3 holes. This way, we can recover the cylinder by gluing a disk to a hole of a 3-holes sphere. The partition function of the latter reads

\begin{equation}
Z_{_{A}}(U,V,W)\equiv\frac{1}{\alpha}\sum_{R}\frac{e^{^{-Ac_{_{2}}(R)}}}{\text{dim} R}\  \chi_{_{R}}(U) \chi_{_{R}}(V)  \chi_{_{R}}(W)
\end{equation}

Proceeding iteratively, the final result for an arbitrary number of holonomies becomes

\begin{equation}
Z_{_{A}}(U_{_{i}})\equiv\alpha^{^{2-2g-n}}\sum_{R}e^{^{-Ac_{_{2}}(R)}}\  \frac{\ \prod_{i}\chi_{_{R}}(U_{_{i}})\  }{\ \ ( \text{dim} R)^{^{2g-2+n}}\ \ }
\label{eq:totZ}
\end{equation}

$q$-deformed 2D YM

The arguments outlined in the previous page are proper to the undeformed 2D YM theory.
In order to build the correspondence with the 4D theory, as outlined in [124], the 2D theory needs to be deformed. This takes place as follows: quantising $A$ into faces, $f$, to each edge $e$ one assigns a dynamical variable $U_{_{e}}\in G$. The total partition function of the system can therefore be expressed as follows

\begin{equation}
{\cal Z}_{_{tot}}\ \equiv\ \int\prod_{e}\ dU_{_{e}}\ \prod_{f}\ {\cal Z}_{_{f}}
\end{equation}

\begin{equation}
{\cal Z}_{_{f}}\ \equiv\ \sum_{R}e^{^{-A_{_{f}}c_{_{2}}(R)}}\text{Tr}_{_{R}}\prod_{e} U_{_{e}}
\end{equation}

The deformation occurs upon requiring $U_{_{e}}\ \in\ G_{_{q}}$, with $G_{_{q}}$ being the quantum group obtained from the ordinary gauge group $G$ upon making its matrix entries noncommutative. For the case of $SU(2)_{_{q}}$

\begin{equation}
U_{_{i}}^{^{j}}\ \overset{def.}{=}\ \left(\begin{matrix} \alpha\ \ \ \ \ \ \ \beta\\
\\
\gamma\ \ \ \ \ \ \delta\\
\end{matrix} \right)\ \ \ \ \ \ \ \ \ \ , \ \ \ \ \ \ \ \ \ \bar U_{_{_i}}^{^{j}}\ \overset{def.}{=}\ \left(\begin{matrix} \delta\ \ \ \ \ \  -q^{1/2}\gamma\\
\\
-q^{^{1/2}}\beta\ \ \ \ \ \ \alpha\\
\end{matrix} \right)
\end{equation}

\begin{equation}
\Rightarrow \ \ \ \alpha\beta\ \equiv\ q^{^{1/2}}\beta\alpha\ ,\ \alpha\delta-q^{^{1/2}}\gamma\beta \equiv \delta\alpha-q^{^{-1/2}}\gamma\beta \equiv 1
\end{equation}

\begin{equation}
\Rightarrow \ \ \ {\cal D}^{^{i\bar i}}U_{_{i}}^{^{j}}\bar U_{_{\bar i}}^{^{\bar j}} \ \equiv\ D^{^{j\bar j}}\ \neq\ \delta^{^{j\bar j}}
\end{equation}

\begin{equation}
{\cal D}^{^{i\bar i}}\ \overset{def.}{=}\ \left(\begin{matrix} q^{^{-1/2}}\ \ \ \ \ 0\\
\\
0\ \ \ \ \ \ \ q^{^{1/2}}\\
\end{matrix} \right)\ \ \ \ \ \ \ \ \ \ \Rightarrow\ \ \ \ \ \ \ \ \ \delta^{^{i\bar i}}\delta_{_{i\bar i}}\ \equiv\ 2\ \ \ \ \ \ \Rightarrow\ \ \ \ \boxed{\ \ \ D^{^{i\bar i}}\delta_{_{i\bar i}} \equiv q^{^{1/2}}+q^{^{-1/2}}  \color{white}\Bigg]\color{black}\ \ }
\label{eq:qd}
\end{equation}

The boxed expression in (<ref>) defines the quantum dimension of the 2D representation of $SU_{_{q}}(2)$ and is basically the character, $\chi_{_{SU(2)}}[q]$. For the $SU(N)$ case,   (<ref>) generalises as follows

\begin{equation}
\text{dim}_{_{q}}R\ \overset{def.}{=}\ \text{Tr}_{_{q}}\text{diag}\ \left(q^{^{(N-1)/2}}, q^{^{(N-3)/2}},...,q^{^{(1-N)/2}}\right)
\end{equation}

Upon substituting this to the dimension of the representation in (<ref>),

\begin{equation}
\boxed{\ \ \ \ Z_{_{A,g}}(U_{_{i}})\equiv\alpha^{^{2-2g-n}}\sum_{R}e^{^{-Ac_{_{2}}(R)}}\  \frac{\ \prod_{i}\chi_{_{R}}(U_{_{i}})\  }{\ \ ( \text{dim}_{_{q}} R)^{^{2g-2+n}}\ \ }  \color{white}\Bigg]\color{black}\ \ }
\label{eq:totZ1}
\end{equation}

The key point to notice is the fact that (<ref>) can be calculated for a surface with arbitrary number of holes just by knowing ${\cal Z}_{_{cyl}}, {\cal Z}_{_{3-holes}}$ and suitable gluing. This statement can be readily understood by saying that correlation functions of a given theory can be determined just by knowing 2- and 3-point functions.

For example, a 4-punctured sphere is obtained by connecting 2 3-punctured spheres.

4D class ${\cal S}$ theories of $SU(2)$-type

In a similar fashion, they can also be constructed from 3-punctured spheres and cylinders. In the 2D case, we saw how complex numbers are assigned to 2D surfaces. Once more, the main building blocks will be cylinders and punctured surface, with an $SU(2)$ flavour symmetry assigned to each puncture.

The 3-punctured sphere is equivalent to the ${\cal N}=1$ chiral multiplet in the $(2,2,2)$ representation of $SU(2)_{_{1}}\times SU(2)_{_{2}}\times SU(2)_{_{3}}$, constituting a half hypermultiplet.

The cylinder is associated to an ${\cal N}$=2 vector multiplet with gauge group $SU(2)$ whose complexified coupling is

\begin{equation}
\tau\ \overset{def.}{=}\ \frac{4\pi i}{g^{^2}}\ +\ \frac{\theta}{2\pi}  \ \ \ , \ \ \ g\ \in\ \mathbb{R}\ \ ,\ \ \theta\sim\theta+2\pi
\end{equation}

Given a collection of 3-punctured spheres, joining 2 of them with a cylinder, leads to a punctured torus. Parametrically, this corresponds to taking a cylinder with parameter $\tau$ and making the following identification

\begin{equation}
zw\ \equiv\ e^{^{2\pi i\tau}}\ \equiv\ q
\end{equation}

with $z, w$ indicating the positions of the 2 joined punctures. From the 4D g auge theory perspective, this corresponds to coupling 2 $SU(2)$ flavour symmetries to a dynamical $SU(2)$ gauge field with coupling constant $\tau$. Once having joined the 2 punctures, their flavour symmetry groups are identified, therefore the hypermultiplet $Q_{_{aiu}}$ can be relabelled into triplets and singlets, $\{A_{_{Iu}}\}, \{H_{_{u}}\}$, defining an ${\cal N}$=2 SU(2) gauge theory with enhanced ${\cal N}$=4 SUSY, corresponding to ${\cal N}$=4 SYM, namely a theory exhibiting S-duality.

S-duality states that, a theory with gauge group $G$ and coupling constant $\tau$ is equivalent to the theory with dual gauge group $G^{^{\text{V}}}$ and coupling constant $-\frac{1}{\tau}$. For the case of $SU(2)$, $G\equiv G^{^{\text{V}}}$.

For ${\cal N}$=2 Lagrangian theories, the partition function can be calculated performing a path integral on $S^{^{1}}\times S^{^{3}}$. In addition to the gauge holonomies parameterising the BPS configurations on which integration is performed, there might also be additional flavour holonomies. As previously mentioned, the holonomy $g\ \in\ G$ around $S^{^{1}}$ can be expressed as

\begin{equation}
g\ \equiv\ (z_{_{1}}, ... z_{_{r}})\ \in\ U(1)^{^{r}}\ \in G
\end{equation}

For an ${\cal N}=2$ hypermultiplet consisting of ${\cal N}=1$ chiral multiplets in a representation $R$ of a symmetry $G$, the partition function is

\begin{equation}
{\cal Z}_{_{R}}(q,g)\ \equiv\ \prod_{n\ge0}\prod_{w}\ \frac{1}{1 1-q^{^{n+1/2}}z^{^{w}}\ }
\end{equation}

with $w\equiv (w_{_{1}},...w-_{{r}})$ the weights of $R$ and

\begin{equation}
z^{^{w}}\ \equiv\ \prod_{i}\ z_{_{i}}^{^{w_{_{i}}}}
\end{equation}

A 3-punctured sphere, the partition function on $S^{^{1}}\times S^{^{3}}$ with holonomies of $SU(2)_{_{i}}$ $\left(a_{_{i}},\frac{1}{a_{_{i}}}\right) \in\ SU(2)_{_{i}}$

\begin{equation}
{\cal Z}(a_{_{1}},a_{_{2}},a_{_{3}})\ \equiv\ \frac{\ K(a_{_{1}})\ K(a_{_{2}})\ K(a_{_{3}})\ }{K_{_{o}}}\ \sum_{n\ge0}\ \frac{\chi_{_{n}}(a_{_{1}})\ \chi_{_{n}}(a_{_{2}})\ \chi_{_{n}}(a_{_{3}})\ }{\chi\left(q^{1/2}\right)}
\end{equation}

where the RHS is equivalent to the amplitude of a $q$-deformed $SU(2)$ YM theory on a 3-punctured sphere.

§.§ 3D SCFTs on punctures

Higgs branch of $T^{^{\rho}}[\mathfrak{g}]$

Given a 4D ${\cal N}$=4 SYM with gauge group $G$ on $\frac{1}{2}$-space $\mathbb{R}^{^{2,1}}\times\mathbb{R}^{^{\ge0}}$, obtained by dimensional reduction of a 6D ${\cal N}$=(2,0) SCFT, admits $\frac{1}{2}$-BPS boundary conditions for the adjoint fields specified by a homomorphism

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

The adjoint orbit for each element $e\ \in\mathfrak{g}$, $O_{_{e}}^{^{\mathfrak{g}}}$, is such that

\begin{equation}
O_{_{e}}^{^{\mathfrak{g}}}\overset{def.}{=} G_{_{\mathbb{C}}}\cdot\ e.
\end{equation}

The classification of such $\rho$s, up to conjugation, is equivalent to that of nilpotent elements in $\mathfrak{g}$. When $\mathfrak{g}\equiv \mathfrak{su}(N)$, a nilpotent orbit is specified by the size of the Jordan blocks

\begin{equation}
n\equiv n_{_{1}}+...+n_{_{k}}
\end{equation}

or, equivalently, by the partition, $[n_{_{i}}]$ of $n$. Similar considerations hold for $\mathfrak{g}\equiv \mathfrak{so}(n), \mathfrak{sp}(n)$.

The boundary conditions for ${\cal N}$+4 SYM are given by Neumann boundary conditions for $\Phi_{_{1,2,3}}$ and Dirichlet BCs for $\phi_{_{4,5,6,}}$

\begin{equation}
\Phi_{_{1,2,3}}(s)\ \sim\ \frac{\rho(\tau_{_{1,2,3}})}{s}\ \ \ \ ,\ \ \ \ \Phi_{_{4,5,6}}(s)\bigg|_{_{s\equiv 0}}\ \equiv 0
\end{equation}

Under S-duality of the 4D bulk, the gauge group $G\rightarrow G^{^{\text{V}}}$, and at $s\equiv0$ lives a 3D SCFT, $T^{^{\rho}}[\mathfrak{g}]$, with $ G^{^{\text{V}}}$-flavour symmetry on the Coulomb branch, coupled to the restriction of the bulk gauge fields to the boundary.

For $\rho\equiv0$, the Coulomb and Higgs branches of $T[\mathfrak{g}]$ are ${\cal N}_{_{\mathfrak{g}^{^{\text{V}}}}},{\cal N}_{_{\mathfrak{g}}}$,  with the latter defining the nilpotent cone of $\mathfrak{g}$, namely the subset of $\mathfrak{g}$ consisting of its nilpotent elements. Its dimension reads

\begin{equation}
\text{dim}_{_{\mathbb{C}}}\ {\cal N}_{_{\mathfrak{g}}}\ \equiv\ \text{dim}\  G-\text{rank}\  G
\end{equation}

Note that this is always an even number due to the fact that ${\cal N}_{_{\mathfrak{g}}}$ is a hyperhähler cone.

More generally, turning on some Higgs vevs $e$, the low-energy limit of the theory at this point would feature singular moduli along the direction orthogonal to $O_{_{e}}$, whereas those along the nilpotent orbit will be smooth. Hence, their contribution as free hypermultiplets to the total Higgs branch quaternionic dimension of $T^{^{\rho}}[\mathfrak{g}]$ reads

\begin{equation}
\text{dim}_{_{\mathbb{H}}}\ H\left(T^{^{\rho}}[\mathfrak{g}]\right)\ \equiv\ \frac{1}{2}\left(\text{dim}\  G-\text{rank}\  G -\text{dim}_{_{\mathbb{C}}}O_{_{e}}\right)
\end{equation}

For classical Lie algebras, $\mathfrak{g}=\mathfrak{su}(N), \mathfrak{so}(2N), \mathfrak{sp}(N), \mathfrak{so(2N+1)}$, $O_{_{e}}$ is labelled by $[n_{_i}]$ and its dimensions are known.

The Coulomb branch and the Spaltenstein map

\begin{equation}
\text{CB}\left(T^{^{\rho}}[\mathfrak{g}]\right)\ \subset\ \text{CB}\left(T[\mathfrak{g}]\right)\
\end{equation}

implying the LHS is the union of closures of nilpotent orbits in $\mathfrak{g}^{^{\text{V}}}$. For $\mathfrak{g}$ a classical Lie algebra, $T^{^{\rho}}[\mathfrak{g}]$ can be constructed as a braneweb.

The Spaltstein map leads to

\begin{equation}
d:\ {\cal N}_{_{\mathfrak{g}}}/G\ \longrightarrow {\cal N}_{_{\mathfrak{g}^{^{\text{V}}}}}/G
\end{equation}

hence acting on the orbits as follows

\begin{equation}
d(O_{_{p}})\ \equiv\ O_{_{q}}  \ \ \ ,\ \ \ q\equiv p^{^{\dag}}\ \ \text{for}\ \ \mathfrak{su}(N)
\end{equation}

\begin{equation}
\text{if}\ \ \ O\ge O^{^{\prime}}\ \ \ \Rightarrow\ \ \ d(O)\le d(O^{^{\prime}})  \ \ \ \text{and}\ \ \ \text{CB}\left(T^{^{\rho}}[\mathfrak{g}]\right)\ \subset\ \text{CB}\left(T^{^{\rho^{\prime}}}[\mathfrak{g}]\right)\
\end{equation}

if $T^{^{\rho^{\prime}}}[\mathfrak{g}]$ can be Higgsed to $T^{^{\rho}}[\mathfrak{g}]$. Rephrasing this in terms of the mixed branches of a given $T[\mathfrak{g}]$: given the maximum Coulomb branch ${\cal N}_{_{\mathfrak{g}^{^{\text{V}}}}}$ and a maximum Higgs branch ${\cal N}_{_{\mathfrak{g}}}$, the mixed branches defining the total vacuum moduli space of $T[\mathfrak{g}]$ is

\begin{equation}
C\times H\ \equiv\ \left\{(x,y)\ \in\ {\cal N}_{_{\mathfrak{g}^{^{\text{V}}}}}\times {\cal N}_{_{\mathfrak{g}}}\ \bigg|\ d(O_{_{x}})\ge\ O_{_{y}}\ \ \text{and}\ \ d(O_{_{y}})\ge O_{_{x}}\ \right\}
\end{equation}

such that, upon assigning a vev $e\ \in\ {\cal N}_{_{\mathfrak{g}}}$ to the Higgs branch, the Coulomb branch is restricted to the closure of $d(O_{_{e}})$.

§.§ The Geometric Langlands Program (GLP)

The category of modules of a vertex operator algebra (VOA) is semisimple if and only if the VOA is rational. Any QFT has a trivial line operator. In a topological twist of a supersymmetric theory, it corresponds to an object $\mathbb{1}$ in the differential graded (dg) category. A dg-category is an additive category whose morphism spaces are dg vector spaces. Each space Hom$(L_{_{i}},L_{_{j}})$ has a cohomological $\mathbb{Z}$-grading and a nilpotent differential of degree 1, ${\cal Q}$, behaving as a derivation on a composition of morphisms. Equivalence relations are imposed by taking the ${\cal Q}$-cohomology.

A dg category, ${\cal D}$, can often be represented as a dg enhancement of the bounded derived category $D^{^{b}}({\cal C})$ of an abelian category ${\cal C}$

\begin{equation}
{\cal D}\simeq D^{^{b}}({\cal C})
\end{equation}

If ${\cal D}\equiv D^{^{b}}({\cal C})$ is represented as a derived category, then the complex describing local operators at a junction of objects in it are

\begin{equation}
\text{Hom}_{_{{\cal D}}}(L, L^{\prime})=\text{Hom}_{_{{\cal C}}}(L, L^{\prime})
\end{equation}

The degree of the complex corresponds to U(1) R-charge of the local operator. $R$-charges manifesting in terms of higher-extension groups follows from having chosen to represent the intrinsic category of line operators ${\cal D}$ as an enhancement of $D^{^{b}}({\cal C})$.

The category of lines in 3D TQFTs is expected to be a dg braided tensor category equipped with

* fusion rules

Fusion in ${\cal C}_{_{line}}$.

* and braiding

Braiding in ${\cal C}_{_{line}}$.

Non-semisimplicity affects the structure of state spaces on surfaces $\Sigma_{_{g}}$, which are closely related to the category of line operators. In a TQFT with semisimple dg category of line operators, ${\cal D}$, the torus Hilbert space si given by the Grothendieck group $K_{_{o}}({\cal D})$, with a basis labelled by simple objects. For higher genus, the Hilbert spaces are given by the fusion algebra of simple objects, with quantum dimensions provided by the Verlinde formula. When ${\cal D}$ is not semisimple, state spaces are more complicated, in particular, for the torus, it is given by the Hoschild homology of ${\cal D}$, $HH_{_{\circ}}({\cal D})$.

Admitting a semisimple category of line operators is a strong constrain to impose on a 3D TQFT; one such example being Chern-Simons theory with compact gauge group £G£ and level $k$. Indeed, CS is a TQFT of cohomological type, whose category of line operators ${\cal D}$ is finite semisimple, with trivial dg structure. The simple objects defining it are Wilson lines labelled by irreps of $G$, $\{S_{_{i}}\}$. There are only finitely many of them , due to an equivalence imposed by large gauge transformations. Furthermore, there are no local operators to define gauge-invariant junctions among different Wilson lines, namely, there are no local bulk operators apart from the identity, $\mathbb{1}$. On the other hand, topological twists of supersymmetric theories typically admit non-finite and non-semisimple categories of line operators.

3D ${\cal N}=$4 gauge theories admit 2 different topological twists:

* a reduction of Witten's 4D ${\cal N}=$2 Donaldson twist (i.e. the $A$-twist)

* an intrinsically 3D twist (the $B$-twist).

The two are related to the 2D $A/B$-twists under dimensional reduction on an $S^{^1}$. They also arise in 4D ${\cal N}=4$ SYM from the Kapustin-Witten $A/B$-twists. In such setups, semisimplicity is rare. Looking for a 3D TQFT that matches the structure of the CGP TQFT described above based on the non-semisimple category $U_{_{q}}(\mathfrak{g})$-mod at an even root of unity $q\equiv e^{^{i\pi/k}}$. The theory is CS-like, containing a subset of line operators labelled by the same irreps of $G$ at level $k$, matching the modules of $U_{_{q}}(\mathfrak{g})$ surviving semisimplicity.

The starting point ia the Gaiotto-WItten 3D ${\cal N}=$ 4 SCFTs, $T[G]$, defined as the S-duality interface of 4D ${\cal N}=4$ SYM. $G$ is a compact simple Lie group depending on the complexified Lie algebra, $\mathfrak{g}$. The theory has $\tilde G\times \tilde G^{^{\text{V}}}$ flavour symmetry, with $\tilde G, \tilde G^{^{\text{V}}}$ denoting the simply-connected forms of $G$ and its Langlands dual, respectively. The two factors of the flavour group act on the Coulomb and Higgs branches of the moduli space of vacua of $T[G]$, which are Langlands dual nilpotent cones

\begin{equation}
{\cal M}_{_{Coul}}[T[G]]\simeq {\cal N}^{^{\text{V}}}\in\ \mathfrak{g}^{^{\text{V}}}
\ \ \ ,\ \ \
{\cal M}_{_{Higgs}}[T[G]]\simeq {\cal N}\in\ \mathfrak{g}
\end{equation}

Requiring that $k\in\mathbb{Z}\simeq H^{^{4}}(B\tilde G)$, $k\ge h$, and $G\equiv\tilde G$ is simply connected, gauging the simply-connected $\tilde G$ symmetry of $T[G]$ by introducing a 3D ${\cal N}=2$ gauge multiplet and a SUSY CS term at level $K$, the resulting theory

\begin{equation}
{\cal T}_{_{\tilde G, k}}\ \overset{def.}{=}\ T[G]/G_{_{k}}
\end{equation}

which is still a 3D ${\cal N}=$ 4 SUSY theory, with WIlson line operators in the $G$-representation and $G^{^{\text{V}}}$ global symmetry. However, it is not topological due to the presence of superconformal matter in $T[G]$. To solve this issue, a topological twist is needed

\begin{equation}
{\cal T}_{_{\tilde G, k}}^{^{A}}\ \sim\ \text{CS}[G_{_{k-h}}]\ \otimes \ T[G]^{^{A}}
\end{equation}

with resulting Hilbert space

\begin{equation}
{\cal H}_{_{{\cal T}_{_{\tilde G, k}}^{^{A}}}}(\Sigma)\ \sim\ {\cal H}_{_{G_{_{k-h}}}}(\Sigma)\ \otimes \ {\cal H}_{_{{\cal N}^{^{\text{V}}}}}(\Sigma)
\end{equation}

§.§ 4D construction and 6D relations

The Gaiotto-Witten theories originate from 6D ${\cal N}=$(2,0) SCFTs of ADE-type, $\mathfrak{g}$, on a product of a 3-manifold and a twisted cigar, $M\times D^{^2}\times_{_{q}}S^{^{1}}$, where $q\equiv e^{^{2\pi\Psi^{^{\text{V}}}}}$. The theory is topologically twisted along $M$ and holomorphically topologically-twisted on $D^{^2}\times_{_{q}}S^{^{1}}$. At $\partial D^{^{2}}$, one places BCs labelled by a completely flat connection, ${\cal A}$, on $M$. Compactifying on the cigar circle and the noncontractible $S^{^{1}}$ in various orders, the resulting theory is 4D YM on $M\times \mathbb{R}_{_{+}}$ with different BCs.

As analysed by Kapustin and Witten within the context of the Geometric Langlands program, there is a prescription describing how to embed calculations in CS theory into GL-twisted 4D ${\cal N}=$4 SYM. A CS calculation on a 3-manifold ${\cal M}_{_3}$ maps to a 4D gauge theory calculation ${\cal M}_{_3}\times\mathbb{R}^{^+}$ with a specific BC deforming the standard Neumann BCs. The conventional notation for such generalised BCs is $B_{_{(p,q)}}$. The analytically-continued CS level $k$ is related to the coupling $\Psi$ of the GL-twisted 4D theory as

\begin{equation}
k+h\equiv\Psi  \ \overset{def.}{=}\ \frac{\theta}{2\pi}+\frac{4\pi i}{g_{_{YM}}^{^2}}\ \frac{t-t^{^-1}}{t+t^{^-1}}
\end{equation}

where $t\equiv\pm i$ corresponds to $\Psi\equiv\tau$.

$\tilde B_{_{1,0}}$ and $\tilde B_{_{0,1}}$ denote Neumann and Dirichlet boundary conditions, respectively. They are related by the action of the S-duality wall.

Gauging the $G$-flavour symmetry of $T[G]$ at CS level $k$, corresponds to replacing one of the Dirichlet boundary conditions with a deformed Neumann BC $B_{_{1,k}}$, obtained by placing an $T^{^{-k}}S$ interface between ordinary Neumann and Dirichlet BCs.

When $G\equiv (P)SU(N)$, the configuration of figure <ref> can be further lifted to branewebs in type-IIB string theory.

Physically, given an object $L$ in ${\cal C}_{_{ab}}$, one may perform a path-integral on a solid torus $D^{^2}\times S^{^1}$ with line operator $L$ inserted at its core to produce a state in the torus state space. This defines a map

\begin{equation}
\text{Ob}({\cal C}_{_{ab}})\ \xrightarrow{\ \kappa\ }\ {\cal H}(T^{^2})
\end{equation}

with the following properties:

* it is linear

\begin{equation}
\kappa(A\oplus B)\equiv \kappa(A)\oplus \kappa(B)
\end{equation}

\begin{equation}
\kappa(A\otimes B)\equiv \kappa(A)\cdot \kappa(B)
\end{equation}

it factors through the Grothendieck ring $K_{_{o}}({\cal C}_{_{ab}})$ because the states on the RHS don't see the difference between direct sums of linear operators $A\oplus B$ and nontrivial extensions (i.e. bound states)

* images of $\kappa$ belong to the cohomological degree-0 subspace of the state space ${\cal H}(T^{^2})$.

All this, together with the isomorphism ${\cal H}(T^{^2})\simeq HH_{_{o}}({\cal C})$, implies that $\kappa$ induces the map

\begin{equation}
K_{_{o}}({\cal C})\ \rightarrow\  HH_{_{o}}({\cal C})\ \subset\ {\cal H}(T^{^2})
\end{equation}

§.§ Identity stalk and the flag manifold

\begin{equation}
D^{^b}({\cal C}_{_{1}}^{^{ab}})\equiv{\cal C}_{_{1}}\ \simeq\ \text{Vect}^{^{\oplus 2}}\ \oplus\ [\text{Coh}(T^*[2]\mathbb{P}^{^1})]^{^\oplus k-1}
\end{equation}

where each of the two $\text{Vect}$s is an abelian semisimple category with a single simple object, whereas, $[\text{Coh}(T^*[2]\mathbb{P}^{^1})]^{^\oplus k-1}$, is a dg enrichment of the derived category. The direct sum  signals that there are no morphisms between objects belonging to the two summands. The abelian and derived categories decompose as a direct sum of $K+1$ blocks

\begin{equation}
{\cal C}_{_{1}}^{^{ab}}\ \simeq\ B_{_{k}}^{^{ab+}}\oplus B_{_{k}}^{^{ab-}}\oplus \ \bigoplus_{j=1}^{k+1} \ B_{_{j}}^{^{ab}}
\end{equation}

where $B_{_{k}}^{^{ab\pm}}$ denote the semisimple subcategories generated by the simple projectives $S_{_{k}}^{^{\pm}}$, wheras each $B_{_{j}}^{^{ab}}$ consists of a subcategory generated by the pair $(S_{_{j}}^{^+}, S_{_{k-j}}^{^-})$.

§ CATEGORIES, KNOTS, AND ENTANGLEMENT

§.§ Higher-categorical structure of (linked minimal models) knots

This appendix contains additional background material underlying section <ref>. In particular, the main focus is that of identifying fusion spaces associated with knots, [537]. A concrete calculation of knot multiplicity shows that the knot complement of a trefoil knot can store quantum information. Also spiral maps enabling to understand consistency relations for torus knots and spiral fusions of fluxes.

Immersion of $A$ in $M$ is defined by triples $(A_+, A, \mathfrak{p}_{_{A}})$. The projection $\mathfrak{p}_{_{A}}:A\rightarrow M$ may or may not be an embedding. In particular, the Hilbert space induced by $M$ on $A$ might be smaller w.r.t. the one with which $A$ is already equipped. Pictorially, this would appear as an overlap of $A$ on $M$. At the level of the Hilbert space, the overlap corresponds to a double copy in the tensor product structure. The correct definition can be achieved by suitably lifting the refined state on local disks contained in $\tilde\Gamma(A_+)$.

The information convex set, $\Sigma(\Omega)$, of a 2-hole disk can be completely characterised. Any element belonging to it can be expressed as

\begin{equation}
\rho\ =\ \bigoplus_{a,b,c\ \in\ {\cal C}}\ p_{_{ab}}^{^{c}}\ \rho^{^{abc}}
\end{equation}

with ${\cal C}$ denoting the set of topological charges, $\{p_{_{ab}}^{^{c}}\}$ a probability distribution, and $\{\rho^{^{abc}}\}$ a set of mutually orthogonal quantum states. For each triple $(a,b,c)$, $\{\rho^{^{abc}}\}$ defines the state space of a finite-dimensional Hilbert space, whose dimension is given by the fusion multiplicities of the underlying anyon theory. $(a,b)$ label the charges associated with the holes, and $c$ the total charge of the disk.

The information convex set for an immersed region is defined as $\Sigma(\Omega)=\{\rho_{_{\Omega}}\}$ such that

* $\rho_{_{\Omega}}\equiv\text{Tr}_{_{\Omega_+\diagdown \Omega}}\ \rho_{_{\Omega_+}}$.

* $\rho_{_{\Omega_+}}$ is consistent with $\sigma_{_{\omega}}^{^{\mathfrak{p}}}$  $\forall\ \omega\ \in\ \tilde\Gamma(\Omega_+)$.

Elements of the information convex sets are closed under the merging operation as long as a certain mild condition is satisfied. The merging theorem states that immersed spaces, $T_1, T_2$, with isomorphic ICSs, share the same number of elements. From this follows that, under the composition of density matrices belonging to each one of them, $\rho_{_{ABC}}\ \in\ \Sigma(ABC),\ \lambda_{_{BCD}}\ \in\ \Sigma(BCD)$, the resulting merged state reads

\begin{equation}
\tau\ \ \overset{def.}{=}\ \ \rho \bowtie \lambda \ \ \
, \ \ \
\tau\ \in\ \Sigma(ABCD)
\end{equation}

For the case in which $\Omega$ is sectorisable, it ICS can be expressed in the following way

\begin{equation}
\Sigma(\Omega)\equiv\ \bigg\{\ \sum_{i}\ p_{_{I}}\rho^{^{i}}_{_{\Omega}}\ \bigg|\ \sum_{_{i}}p_{_{i}}=1, p_{_{i}}\ge0\bigg\}
\end{equation}

with $\{\rho^{^{i}}_{_{\Omega}}\}$ denoting the set of mutually orthogonal density matrices. The set of labels $i$ forms the set of superselection sectors, ${\cal C}_{_{\Omega}}$, with $i=1$ defining the vacuum sector. For any given sectorisable region $\Omega$ embedded in $S^3$ or $B^3$, $\rho_{_{\Omega}}^{^{1}}\equiv\sigma_{_{\Omega}}$ is unique.

The quantum dimension can therefore be defined as follows

\begin{equation}
d_{_{i}}\ \overset{def.}{=}\ \exp\ \left(\frac{S\left(\rho_{_{\Omega}}^{^{i}}\right)-S\left(\sigma_{_{\Omega}}^{^{*}}\right)}{2}\right)
\end{equation}

where $\sigma_{_{\Omega}}^{^{*}}$ denotes the vacuum state.

A set of superselection sectors can be assigned to any sectorisable region. Conventionally, we will be denoting them as $\{{\cal C}_{_{I}}\}$, with the subscript $I$ specifying the sector. Among the simplest choices one could make are: sphere shells, ${\cal C}_{_{point}}$, solid tori, ${\cal C}_{_{flux}}$, and torus shells, ${\cal C}_{_{Hopf}}$. For the purpose of this work, the latter happens to be of particular importance. However, as we shall be arguing throughout, all 3 can be thought of as subcategories defining an $n$-category associated to an $(n+1)$-dimensional theory.

A sphere shell, (on the LHS). On the RHS, the topological charges of the 2 connected components are mapped to a particle/antiparticle pair joined by a string.

A solid torus, $\mathbb{T}$, (on the LHS). On the RHS, a visualisation of a flux operator operator, $\eta$.

A torus shell, $\mathbb{T}$, (on the LHS). On the RHS, a visualisation of the Hopf link, resulting from the mutual intersection of $\eta$ and $\eta^{^{\text{V}}}$.

Its corresponding superselection sector type is the Hopf fibration, ${\cal C}_{_{Hopf} } \equiv\{1, \eta, \zeta, ...\}$. Superselection sectors are associated with different classes of loop excitations. Each one of them can be created by applying a suitable type of membrane operator. The description in terms of the information convex set is somehow dual to that of the excitations, since it focuses on their complement when embedded in an $S^3$.

For a torus shell, $\Sigma(\mathbb{T})$ is a simplex with extreme points $\{\rho_{_{\mathfrak{T}}}^{^{\eta}}\}_{\eta\ \in\ {\cal C}_{_{Hopf}}}$. The set of Hopf excitations admit a natural decomposition

\begin{equation}
{\cal C}_{_{Hopf}}\ \equiv\ \bigcup_{\mu\ \in\ {\cal C}_{_{flux}}}\ {\cal C}_{_{Hopf}}^{^{[\mu]} }
\label{eq:emb}
\end{equation}

$\Sigma(\mathbb{T})$ comes equipped with nontrivial automorphisms, such as the one responsible for permuting the consituting loops in the Hopf link of its complement

\begin{equation}
v\ :\ {\cal C}_{_{Hopf}}\ \xrightarrow{\color{white}aaaa\color{black}}\ {\cal C}_{_{Hopf}}   \ \ \ , \ \ \ v(\eta)\ \equiv \eta^{^{\text{V}}}
\end{equation}

The set of shrinkable loops are not linked to any other by definition, and therefore provide a natural embedding for the fluxes

\begin{equation}
{\cal C}_{_{flux}}\ \overset{\varphi}{\hookrightarrow}\ {\cal C}_{_{loop}}\ \equiv\ {\cal C}_{_{Hpof}}^{^{[1}}\ \subset\ {\cal C}_{_{Hopf}}
\end{equation}

The embedding (<ref>) implies the matching of the quantum dimensions

\begin{equation}
\sum_{a\ \in\ {\cal C}_{_{loop}}}\ d_{_{a}}^2\ \equiv\ \sum_{\mu\ \in\ {\cal C}_{_{flux}}}\ d_{_{\mu}}^2
\end{equation}

as well as the presence of a well defined map ${\cal C}_{_{loop}}\rightarrow{\cal C}_{_{flux}}$. Indeed, given a torus shell $\mathbb{T}$, and $T\subset\mathbb{T}$ a solid torus, an extreme point $\rho_{_{\mathbb{T}}}^{^{\eta}}\ \in\ \Sigma(\mathbb{T})$,

\begin{equation}
\text{Tr}_{_{\mathbb{T}/T}}\rho_{_{\mathbb{T}}}^{^{\eta}}   \equiv \rho_{_{T}}^{^{\mu}} \ \in\ \Sigma(T)
\end{equation}

§.§ Fusion rules and knot complements

On the LHS, a solid torus with 2 tori removed. On the RHS, the fusion of 2 loops in ${\cal C}_{_{Hopf}}^{^{[mu]}}$, from which the multiplicities $\{N_{_{ij}}^{^{h}}\}$ can be extracted.

The Hilbert space theorem states that,

\begin{equation}
\Sigma_{_{I}}(\Omega)\ \simeq\ S\left(\mathbb{V}_{_{I}}\right)
\label{eq:HST}
\end{equation}

where $I$ labels the superselection sector and $S\left(\mathbb{V}_{_{I}}\right)$ defines the state space of a finite-dimensional Hilbert space $\mathbb{V}_{_{I}}$. From (<ref>), it follows that the information convex set is completely characterised by the fusion multiplicity

\begin{equation}
N_{_{I}}\ \equiv\ \text{dim}\ \mathbb{V}_{_{I}}
\end{equation}

For $N_{_{I}}=0$, $\Sigma_{_{I}}(\Omega)$ is empty, whereas for $N_{_{I}}=1$ it contains a single element, which is an isolated fixed point. For $N_{_{I}}=2$ it is isomorphic to  the Bloch ball and contains an infinite number of continuously parameterised extreme points. When $N_{_{I}}>1$, $\Omega$ can store quantum information.

the information convex sets of 3D regions with boundaries can be associated with a set of fusion spaces. The fusion of 2 loops, depicted on the RHS of figure <ref>, enables to derive the fusion multiplicities, $\{N_{_{ij}}^{^{h}}\}$,

\begin{equation}
h\times i\equiv\ \sum_j\ N_{_{ij}}^{^{h}} j
\ \ \
\ \ \
h,i,j \ \in\ {\cal C}_{_{Hopf}}^{^{[\mu]}}
\end{equation}

The complement of an unknot in $B^3$ is a shrinkable loop at a poin, with shrinking rule

\begin{equation}
\ell\ \equiv\ \sum_a N_{_{\ell}}^{^{a}}\ a
\ \ \
\ \ \
\ell \ \in\ {\cal C}_{_{loop}}\ \ \
\ \ \
a \ \in\ {\cal C}_{_{point}}
\end{equation}

$B^3\diagdown T$ on the LHS, with its corresponding shrinking rule on the right.

The next essential ingredient that needs to be introduced is the knot complement, which plays an important role in their classification. For the purpose of our treatment, their key role resides in the fact that their information convex sets are characterised by nontrivial fusion data. For a given knot, $K$, we will denote its complement on the 3-sphere as

\begin{equation}
\Omega_{_{K}}\ \overset{def.}{=}\ S^3/K
\end{equation}

with a thickened boundary $\partial\Omega_{_{K}}$, i.e. a knotted embedded torus shell. It therefore follows that

\begin{equation}
{\cal C}_{_{\partial\Omega_{_{K}}}}\ \simeq\ {\cal C}_{_{Hopf}}
\end{equation}

namely the 2 sets are identical up to permutations. The information convex set of the knot complement, $\Sigma(\Omega_{_{K}})$, is a convex hull of the subset $\Sigma_{_{\zeta}}(\Omega_{_{K}})$, with $\zeta\ \in\ {\cal C}_{_{Hopf}}$. In general, the set of superselection sectors that can exist on a knot $K$ of an $S^3$ alone is a subset of ${\cal C}_{_{Hopf}}$, implying that the knot excitations are a small number w.r.t. the Hopf link. However, they are more coherent, and therefore are able to store quantum information. The proof of this last statement was carried out in [537] and requires some additional elements we will now present.

§.§ Spiral maps, fusion multiplicities and holomorphic blocks

The knot multiplicity, $N_{_{\zeta}}(K)$ is the dimension of the Hilbert space of $\Sigma_{_{\zeta}}\left(\Omega_{_{K}}\right)$

\begin{equation}
\boxed{\ \ \ N_{_{\zeta}}(K)\ \overset{def.}{=}\ \text{dim}\ \mathbb{V}_{_{\zeta}}\left(\Omega_{_{K}}\right) \color{white}\bigg]\color{black} \ \ }
\label{eq:km}
\end{equation}

For the unknot, (<ref>) reduces to

\begin{equation}
N_{_{\eta}}(\text{unknot})\ \overset{def.}{=}\ \sum_{\mu\ \in\ {\cal C}_{_{flux}}}\ \delta_{_{\eta, \varphi(\mu)}}   \ \ \ , \ \ \ \forall \eta\ \in\ {\cal C}_{_{Hopf}}
\label{eq:km1}
\end{equation}

whereas for a general $(p,q)$-knot it reads as follows

\begin{equation}
\boxed{\ \ \ N_{_{\zeta}}^{^{\eta}}(p,q)\ \overset{def.}{=}\ \text{dim}\ \mathbb{V}_{_{\zeta}}^{^{\eta}}\left(T\diagdown K_{_{p,q}}\right) \color{white}\bigg]\color{black} \ \ \ , \ \ \ \eta, \zeta\ \in\ {\cal C}_{_{Hopf}}  \ \ }
\label{eq:km2}
\end{equation}

with $\eta, \zeta$ labelling the unknotted (outer) and knot boundaries, respectively.

Example of $S^3\diagdown K(2,3)$. With $\eta, \zeta$ denoting the unknotted and knotted boundaries, respectively.

$\Sigma(\Omega)$ is completely determined by the set of extreme points, $\text{ext}(\Sigma(\Omega))$. In order to define the latter, some preliminary definitions are of order.

$\Omega$ is not sectorisable, but $\partial\Omega$ is, since it has 3 connected components, labelled $i=a,b,c$. Each one of them corresponds to a different superselection sector, collectively denoted ${\cal C}_{_{\partial\Omega}}$.

For regions that are not sectorisable, such as the one depicted in figure <ref>, we can distinguish between $\Omega$ and its boundary defined as follows

\begin{equation}
\boxed{\ \ \ \partial\Omega\ \overset{def.}{=}\ {\cal M}_{_{D-1}}\ \times I  \color{white}\bigg]\color{black}\ \ }
\end{equation}

where ${\cal M}_{_{D-1}}$ is a $(D-1)$-dimensional closed manifold, and $I$ an interval. $\Sigma(\partial\Omega)$ is a simplex since $\partial\Omega$ is sectorisable, namely it is possible to assign to it a finite set of superselection sectors labelled by ${\cal C} _{_{\partial\Omega}}$. If $\partial\Omega$ contains multiple connected components, each component is sectorisable.

Under the action of partial tracing, every extreme point of $\Sigma(\Omega)$ reduces to one of $\Sigma(\partial\Omega)$.

Equipped with this machinery, it is now possible to define the extreme point criterion, stating that, for a given immersed $\Omega$, $\rho_{_{\Omega}}\ \in\ \Sigma(\Omega)$ is an extreme point if and only if

\begin{equation}
\left(S_{_{\Omega}}+S_{_{\Omega\diagdown\partial\Omega}}-S_{_{\partial\Omega}}\right)_{_\rho}\ \equiv\ 0
\end{equation}

Furthermore, $\forall i\ \in\ {\cal C}_{_{\partial\Omega}}$, corresponding to a nonempty $\Sigma_{_{i}}(\Omega)$, the quantum dimension is defined as

\begin{equation}
\ln\ d_{_{i}}\ \overset{def.}{=}\ S\left(\rho_{_{\Omega}}^{^{i,e}}\right)-S\left(\rho_{_{\Omega}}^{^{1,e}}\right)
\end{equation}

with $\rho_{_{\Omega}}^{^{i,e}}$ an extreme point of $\Sigma_i(\Omega)$, and $\rho_{_{\Omega}}^{^{1,e}}$ the vacuum reference state.

Once having identified the quantum dimensions associated to the different superselection sectors, we can calculate their fusion multiplicities, $N_{ab}^{c}$,

\begin{equation}
d_{_{a}}\ d_{_{b}}\ =\ \sum_{c}\ N_{ab}^{c}\ d_{_{c}}
\end{equation}

with $(a,b,c)$ denoting the different topological charges involved in the fusion process.

§.§ The total quantum dimension

The total quantum dimensions can therefore be extracted as follows:

* for points and fluxes

\begin{equation}
{\cal D}\ \overset{def}{=}\sqrt{\sum_{_{a\ \in\ {\cal C}_{_{point}}}}\ d_{_{a}}^2\ \  }\ \equiv\ \sqrt{\sum_{_{\mu\ \in\ {\cal C}_{_{flux}}}}\ d_{_{\mu}}^2\ \  }
\end{equation}

* for a knotted torus, instead, the definition can be relaxed, turning the equality into an upper bound on the topological entanglement entropy

\begin{equation}
\sum_{_{\zeta\ \in\ {\cal C}_{_{Hopf}}}}\ N_{\zeta}(K)\ d_{_{\zeta}}\ \equiv\ \sum_{_{\overset{\mu,\nu\ \in\ {\cal C}_{_{flux}}}{t_p(\mu)\equiv t_q(\nu)}}}\ \frac{d_{_{\mu}}^2 d_{_{\nu}}^2 }{d^2_{_{t_p(\mu)}}} \ \le\ {\cal D}^4
\end{equation}

For the case of $K$ with arbitrary $(p,q)$

\begin{equation}
\sum_{_{\zeta\ \in\ {\cal C}_{_{Hopf}}}}\ N_{\zeta}^{\eta}(p,q)\ d_{_{\zeta}}\ \equiv\ \sum_{_{\overset{\mu,\nu\ \in\ {\cal C}_{_{flux}}}{t_p(\mu)\equiv t_{(p,q)}(\nu)}}}\ \frac{d_{_{\varphi^{\text{V}}}} d_{_{\eta}} }{d^2_{_{t_p(\mu)}}} \ \le\ {\cal D}^4
\label{eq:fm}
\end{equation}

\begin{equation}
N_{\zeta}^{^{\eta}}(p,q)\ \equiv\ \sum_{_{\mu  \in\ {\cal C}_{_{flux}}}} N^{^{\eta}}_{\zeta\varphi^{^{\text{V}}}(\mu)}
\label{eq:SSN}
\end{equation}

Importantly, the embedding operation

\begin{equation}
\varphi^{^{\text{V}}}\ :\ {\cal C}_{_{flux}}\ \hookrightarrow\ {\cal C}_{_{Hopf}}
\end{equation}

really consists of 2 consecutive parts, namely

\begin{equation}
\varphi\ :\ {\cal C}_{_{flux}}\ \hookrightarrow\ {\cal C}_{_{loop}}\ \subset\ {\cal C}_{_{Hopf}}
\end{equation}

\begin{equation}
\eta\ \rightarrow\ \eta^{^{\text{V}}}
\end{equation}

from which follows that (<ref>) is equivalent to a gauging procedure.

In a 4D TQFT, the entanglement entropy is given by the area-law plus a topological subleading correction

\begin{equation}
S_{_{EE}}\ =\ \alpha\ \ell\ +\ \ln{\cal D}
\end{equation}

where $\alpha$ is a constant, $\ell$ is the area of the surface on which $S_{_{EE}}$ is being calculated, and ${\cal D}$ is the quantum dimension. Following the Casini-Huerta-Myers prescription, the latter corresponds to the so-called boundary free energy of the 3D boundary to the 4D TQFT. It is a universal quantity, and its importance is crucial for the study of holographic embeddings of such theories.

For $\Sigma(\Omega_{_{K}})$, with corresponding maximum entropy state $\rho_{_{\Omega_{_{K}}}}^{^{*}}$, the entropy w.r.t. the reference state can be extracted from the fusion multiplicities (<ref>) and therefore reads

\begin{equation}
S\left(\rho_{_{\Omega_{_{K}}}}^{^{*}}\right)-S\left(\sigma_{_{\Omega_{_{K}}}}^{^{}}\right)\ =\
\ln\left(\sum_{_{\zeta\ \in\ {\cal C}_{_{Hopf}}}}\ N_{\zeta}^{\eta}(p,q)\ d_{_{\zeta}}\right)
\label{eq;fm1}
\end{equation}

[analogy with Janus configurations]

§ SIGMA-MODELS

§.§ The sigma-model

By definition, [86], a sigma-model is a  field theory describing the field as a point particle confined to move on a fixed manifold. The latter could either be a Riemannian manifold, a Lie group, or a symmetric space.

When coupling it to a gauge field, we get the Landau-Ginzburg (LG)-model, [444].

The Lagrangian density for the sigma-model can be expressed in many different ways. One of the simplest expressions is the following

\begin{equation}
{\cal L}=\frac{1}{2}\sum_{i=1}^{n}\sum_{_j=1}^{n}\ g_{_{ij}}(\phi)\partial^{^{\mu}}\phi_{_i}\partial_{_{\mu}}\phi_{_j},
\end{equation}

where $g_{_{ij}}(\phi)$ denotes the metric tensor on the field space $\phi\ \in\ \Phi$, where $\Phi$ can be any Riemannian manifold.

In a more fully-geometric notation, it can be written as a fiber bundle with fibers $\Phi$ over a differentiable manifold $M$. Given a section

\begin{equation}
\phi:\ M\ \rightarrow\ \Phi,
\end{equation}

fix a point $x\ \in\ M$. Then, the pushforward at $x$ is a map of tangent bundles

\begin{equation}
d_{_{x}}\phi:\ T_{_{x}}M\ \rightarrow\ T_{_{\phi(x)}}\Phi,
\end{equation}

\begin{equation}
\partial_{_{\mu}}\ \mapsto\ \frac{\partial \phi^{^i}}{\partial x^{^{\mu}}}\partial _{_i}.
\end{equation}

The sigma model action then reduces to

\begin{equation}
S=\frac{1}{2}\ \int_{_M}d\phi\ \wedge\ \star\ d\phi.
\label{eq:clsm}
\end{equation}

Among the most important interpretation of the classical sigma-model is that of non-interacting QM. Taking $\Phi\equiv \mathbf{C}$, (<ref>) becomes

\begin{equation}
S=\frac{1}{2}\ <<\phi\ ,\ \Delta\ \phi>>,
\label{eq:clsm1}
\end{equation}

\begin{equation}
\phi:\ M\rightarrow\mathbb{C}
\end{equation}

interpreted as a wavefunction, whose Laplacian corresponds to its associated kinetic energy. Hence, the classical sigma-model on $\mathbf{C}$ can be interpreted as the QM of a free particle.

§.§ Riemann surfaces

The geodesic structure of a Riemannian manifold is described by the Hamilton-Jacobi equations. As previously mentioned, the $M$ and $\Phi$ defining the sigma-model can both be taken to be Riemannian.

The cotangent bundle $T^*\Phi$ can always be locally trivialised, namely

\begin{equation}
T^*\Phi\bigg|_{_U}\ \simeq\ U\times\ \mathbf{R}^{^n}.
\end{equation}

Given $g_{_{ij}}$ on $\Phi$, the Hamiltonian function is defined as follows

\begin{equation}
H(q,p)\ \overset{def.}{=}\ \frac{1}{2}\ g^{^{ij}}(q)\ p_{_i}p_{_j},
\end{equation}

from which the Hamilton-Jacobi equations define the geodesic (or Hamiltonian) flow on $\Phi$

\begin{equation}
\dot q^{^i}=\frac{\partial H}{\partial p_{_i}}\ \ \ ,\ \ \ \dot p_{_i}=-\frac{\partial H}{\partial q^{^i}}.
\end{equation}

The Hamiltonian description therefore enables us to interpret the sigma-model as the gluing of two energy functionals, namely the momenta in $T^*\Phi$ and $T^*M$.

§.§ Moduli spaces

In algebraic geometry, a moduli space is a geometric space (a scheme or an algebraic stack) whose points represent algebro-geometric objects of some kind, or their isomorphic classes. The main motivation for introducing moduli spaces is that of finding solutions of geometric problems.

A scheme is a mathematical structure that enlarges the notion of an algebraic variety, accounting for multiplicities, and allowing varieties defined on any commutative ring. Scheme theory was introduced by A. Grothendieck in 1960 for addressing deep problems in algebraic geometry.

A scheme is a topological space together with commutative rings for all of its open sets, which arises from gluing together spectra of commutative rings along their open subsets, i.e. ringed space which is locally a spectrum of a commutative ring.

Algebraic surfaces can be studied making use of morphisms of schemes. In many cases, the family of all varieties of a given type can itself be viewed as a variety of schemes, known as moduli space.

§.§ The nonlinear sigma-model

Considering Minkowski spacetime, and denoting with ${\cal F}$ the space of fields, $L$ the Lagrangian, and $\gamma$ the variational 1-fomr, ${\cal M}$ the space of classical solutions, and $\omega$ the local symplectic form. In its classical formulation, the Lagrangian encodes a classical Hamiltonian system once a specified time direction has been selected, thereby breaking Poincare' invariance.

Upon taking a real-valued scalar field

\begin{equation}
\phi: M^{^n}\ \rightarrow\ \mathbb{R}.
\end{equation}

The kinetic term in the Lagrangian would therefore read

\begin{equation}
\begin{aligned}
L_{_{kin}} &=\frac{1}{2}|d\phi|^{^2}\left|d^{^n}x\right| \\
&=\frac{1}{2}g^{^{\mu\nu}}\partial_{_{\mu}}\phi\partial_{_{\nu}}\phi \left|d^{^n}x\right| \\
\label{eq:lkin}
\end{aligned}
\end{equation}

Adding a potential term of degree $\le 2$ of the kind

\begin{equation}
V: \mathbb{R}\ \longrightarrow\ \mathbb{R}
\end{equation}

to (<ref>), and assuming $V$ is bounded below by choosing suitable coefficients of th e polynomial, then one can eliminate the linear and constant terms by assuming the minimum of $V$ is at the origin with $V=0$ there. The total Lagrangian therefore becomes

\begin{equation}
\end{equation}

In non-free theories, the potential is no longer quadratic. The simplest example of an interacting theory is the case of the quartic potential. In either case, in a pure mathematical formulation, the non-linear $\sigma$-model is characterised by the following data:

* A Riemannian manifold, $X$.

* A potential energy function

\begin{equation}
V:\ X\ \longrightarrow\ \mathbb{R}.
\end{equation}

In this formulation, the space of fields is th emapping space

\begin{equation}
{\cal F}=\left(\text{Map}(M^{^n}, X\right).
\end{equation}

The total Lagrangian therefore reads

\begin{equation}
\end{equation}

from which the energy density reads

\begin{equation}
{\cal E}(\phi)=\left[\frac{1}{2}|\left|\partial_{_0}\phi\right|^{^2}+\sum_{_{1=1}}^{^{n-1}}\frac{1}{2}\left|\partial_{_i}\phi\right|^{^2}+V\right]\left|d^{^{n-1}}x\right|,
\end{equation}

The space of solutions, namely the moduli space of vacua, ${\cal M}_{_vac}$, is

\begin{equation}
{\cal M}_{_vac}\ \subset\ {\cal F}{\cal E}_{_N},
\end{equation}

with ${\cal F}{\cal E}_{_N}$ denoting the space of static fields of finite energy.

For the case of a real scalar field with potential $V$, one usually assumes that the minimum value is at $V=0$, and therefore that the solutions lie at

\begin{equation}
{\cal M}\ \overset{def.}{=}\ V^{^{-1}}(0).
\end{equation}

For example, if $X\equiv\mathbb{R}$, and $V=0$, the resulting theory is that of a massless real scalar field. For this case, one finds that ${\cal M}_{_{vac}}\simeq\mathbb{R}$. On the other hand, for the case of a scalar field with potential

\begin{equation}
\end{equation}

${\cal M}_{_{vac}}$ is a single point $\phi=0$. For a quartic potential, instead, ${\cal M}_{_{vac}}$ is constituted by two points, i.e. the values of $\phi$ where $V$ is minimised.

§.§ Principle bundles and connections

Formulating gauge theories in Lagrangian formulation that is suitable for a QFT, some preliminary differential geometry notatioons are needed.

The initial ingredients we start from are a manifold, $M$, and a Lie group, $G$. Then, a principal G-bundle, $P\rightarrow M$ is defined as a manifold $P$ on which $G$ acts freely on the right with quotient

\begin{equation}
P/G\simeq M,
\end{equation}

such that there exist local sections. If $P^{^{\prime}}, P$ are principal $G$-bundles over $M$, then an isomorphism

\begin{equation}
\varphi^{^{\prime}}:P^{^{\prime}}\ \longrightarrow\ P,
\end{equation}

is a smooth diffeomorphism which commutes with $G$, and induces the identity map on $M$. If $P\equiv P^{^{\prime}}$, such automorphism defines a gauge transformation of $P$.

For every $M$, there is a category of principal $G$-bundles and isomorphisms.

A connection on a principal $G$-bundle $\pi: P\rightarrow M$ is a $G$-invariant distribution in $TP$ which is transverse to the vertical distribution kerd$\pi$. A connection can therefore be expressed as a 1-form $A\in\Omega^{^1}(P;\mathfrak{g})$ whose value at $p$ is the projection

\begin{equation}
T_{_p}P\ \longrightarrow\ V_{_p}\simeq\mathfrak{g},
\end{equation}

with kernel $H_{_p}$.

Connections form a category, arranging in a set of equivalence classes under isomorphisms.

§.§ Mathematical Gauge theory

The prototypical example of a gauge theory is Maxwell's theory. In its quantised formulation, one needs to introduce Dirac's charge quantisation, replacing the group $\mathbb{R}$ by the compact group $\mathbb{Z}/2\mathbb{Z}$. This statement can be generalised by replacing $\mathbb{Z}/2\mathbb{Z}$ with any compact Lie group $G$.

Hence, we are drawn to the conclusion that the Lagrangian formulation of the field theory in question is simply based on the following data:

* A compact Lie group with Lie algebra, $\mathfrak{g}$.

* A bi-invariant inner product on $\mathfrak{g}$, $<.,.>$.

From this, the total Lagrangian for the gauge field reads

\begin{equation}
L=-\frac{1}{2}\left<F_{_A}\wedge\star F_{_A}\right>
\end{equation}

from which the energy density reads

\begin{equation}
{\cal E}(A)=\left[\ \sum_{_{\mu<\nu}}]\frac{1}{2}\left|F_{_{\mu\nu}}\right|^{^2}\right]\ \left|d^{^{n-1}}x\right|,
\end{equation}

whose moduli space is simply

\begin{equation}
{\cal M}_{_{vac}}=\text{pt},
\end{equation}

where the calculation is understood to be performed on the space of equivalence classes of connections.

§.§ The gauged sigma-model

This is a very general bosonic field theory, combining pure gauge theory with the $\sigma$-model, both of which are greatly used in QFT. The data specifying the model in question are:

* A Lie group, $G$, with Lie algebra, $\mathfrak{g}$.

* A bi-invariant scalar product, $<,>$, on $\mathfrak{g}$.

* A Riemannian manifold, $X$, on which $G$ acts by isometries.

* A potential function

\begin{equation}
V:\ X\ \longrightarrow\ \mathbb{R},
\end{equation}

invariant under $G$.

The space of fields, ${\cal F}$, consists of pairs

\begin{equation}
{\cal F}\ \overset{def.}{=}\{(A,\phi)\},
\end{equation}

with $A$ denoting a collection of principal $G$-bundles $P\rightarrow M$, and $\phi$ a section of the associated bundle over $P\times_{_G}X\rightarrow M$. In most cases, it is convenient to view $\phi$ as an equivariant map

\begin{equation}
\phi:\ P\ \longrightarrow\ X.
\end{equation}

The space of fields can be thought of as a category, with an isomorphism

\begin{equation}
\varphi:\ P^{^{\prime}}\ \longrightarrow\ P
\end{equation}

of principal bundles, inducing an isomorphism of fields

\begin{equation}
\left(A^{^{\prime}},\phi^{^{\prime}}\right)\ \longrightarrow\ (A,\phi).
\end{equation}

The global symmetry group of the theory is the subgroup of isometries of $X$ commuting with the $G$-action, and preserving the potential, $V$.

The Lagrangian density, therefore reads

\begin{equation}
{\cal L}=-\frac{1}{2}|F_{_A}|^{^2}+\frac{1}{2}|{\cal D}_{_A}\phi|^{^2}-\phi^{^*}V,
\label{eq:Lagrangiandensity}
\end{equation}

where ${\cal D}_{_A}$ denotes the covariant derivative, coupling $\phi$ with $A$. The energy density of the pair $(A,\phi)$, can readily be extracted from (<ref>), and reads

\begin{equation}
{\cal E}(A,\phi)=\left[\ \sum_{_\mu<\nu}\frac{1}{2}|F_{_{\mu\nu}}|^{^2}+\sum_{_{\mu}}\frac{1}{2}\left|\left(\partial_{_A}\right)_{_{\mu}}\phi\right|^{^2}+\phi^{^*}V\ \right]\ \left|d^{^{D-1}}x\right|,
\label{eq:energyd}
\end{equation}

where $D$ denotes the total number of spacetime dimensions.

Assuming $V$ has a minimum at 0, one can look for soulitions with 0-energy. By direct inspection of (<ref>), it is possible to deduce that:

* The first term implies that $A$ is flat, hence, up to equivalence, it is the trivial connection with zero-curvature.

* The second term tells us that $\phi$ has to be a constant for it to be a zero-energy solution.

* The third term constraints the constant value of $\phi$ to lie in the $V^{^{-1}}(0)$ set.

A trivial connection $A$ has a group of automorphisms that are isomorphic to $G$, i.e. the group of global gauge transformations. For a vacuum solution, $\phi$ must be a constant function with values in $V^{^{-1}}(0)$, and, therefore, $V^{^{-1}}(0)$ inherits the same $G$-action from $\phi$. Hence, the moduli space of vacua is a subquotient space of $X$

\begin{equation}
\boxed{\ \ \ {\cal M}_{_{vac}}\ \overset{def.}{=}\ V^{^{-1}}(0)\bigg/G\color{white}\bigg]\ \ }.
\end{equation}

For the case of certain supersymmetric field theories, $X$ is K$\ddot{\text{a}}$hler of hyperk$\ddot{\text{a}}$hler, and $V^{^{-1}}(0)$ is the norm square of an appropriate moment map, in which case ${\cal M}_{_{vac}}$ reduces to a K$\ddot{\text{a}}$hler of hyperk$\ddot{\text{a}}$hler quotient (with the latter actually being the setting where this had first been studied).

§ (A)DS-DEFORMED CFTS

As previously argued, the entropy of pure AdS is divergent, compatibly with the fact that bulk reconstruction enables to probe the IR bulk at arbitrary depth by evaluating $S_{_{EE}}$ on ever-increasing subregions of the conformal boundary.

Within the AdS/CFT correspondence, finiteness of bulk-reconstruction can be achieved by shifting the conformal boundary at a finite cutoff by deforming the CFT Lagrangian density by means of an irrelevant operator. In 2D, this is known as the $T\bar T$-deformation, initially introduced by Zamolodchikov, [227]. The RG-flow equations driven by this operator interpolate between the UV and the IR.

For the 2D dual to an (A)dS$_3$, [228, 229], the action associated to the deformed theory reads

\begin{equation}
S_{T\bar T}^{^{AdS}}
\
\
S_{CFT}+\lambda\int d^{2} x\ \sqrt{-\gamma} \ T\bar T
\label{eq:TTADS}
\end{equation}

\begin{equation}
S_{T\bar T}^{^{dS}}
\
\
S_{CFT}+\lambda\int d^{2} x\ \sqrt{-\gamma} \  \left[\ T\bar T +\frac{1}{2\lambda^2}\  \right]
\label{eq:TTDS}
\end{equation}

respectively, with

\begin{equation}
T\bar T\ \overset{\text{def.}}{=}\ \frac{1}{8}(T^{ab}T_{ab}-(T^{a}_{a})^{2})
\end{equation}

The deformation of the CFT$_2$ Lagrangian induces a change in the partition functions,  which, for AdS$_3$ reads

\begin{equation}
\delta \ln {\cal Z}
\
\
\ln {\cal Z}_{T\bar T}-\ln {\cal Z}_{CFT}
\
\
-\frac{1}{2}\ \int d^{2}x\ \sqrt{\gamma}\ <T^{ab}>\ \delta \gamma_{ab}
\label{eq:zeta}
\end{equation}

and trace flow equations, [227],

\begin{equation}
<T^{\ a}_{\ a}>
\
\
-\frac{c}{24}{\cal R}-\frac{\lambda}{4}\left(\ <T^{ab}><T_{ab}>-<T^{a}_{a}>^{2}\ \right)
\end{equation}

where the 1$^{st}$ term defines the trace anomaly associated with a CFT$_{2}$.

The first application of these deformations for describing the gluing of bulk spacetimes from a holographic perspective, was carried out by H. Verlinde et al. in [228] for AdS$_{3}$ spacetimes, and further developed by Silverstein et al. [229] for local realisations of dS$_3$s in an asymptotically AdS$_3$ background, with the latter relying upon the dS$_{_{D+1}}$/dS$_{_{D}}$-correspondence (dS/dS for brevity). The fact that the latter defines a duality in between 2 dS theories follows from the fact that the deformed Lagrangian driving the RG-flow contains an additional term w.r.t. the AdS$_3$/CFT$_2$ case which is playing the role of a positive 2D cosmological constant. We will come back to this in greater detail when deriving the IR $S_{_{EE}}$s in both setups. Before doing so, we will briefly outline the key features characterising the dS/dS-correspondence, motivating that the gluing of different spacetimes in the IR can be naturally accounted for in both correspondences alike.

According to the dS/dS correspondence, [232], a D+1-dimensional dS spacetime with curvature radius $1/H_{D+1}$ is equivalent to 2 interacting CFTs of D-dimemsions with a cutoff $H_{D+1}$ and coupled to gravity (cf. figure <ref>).

The conformal boundary is depicted in blue. The RT-surface is still a codimension-2 surface, in complete analogy with the AdS/CFT setup. For $D=2$, it reduces to 2 antipodal points on an $S^{1}$.

The analogy in between the 2 AdS/CFT and the dS/dS correspondences already shows at the level of the metric, which read[$\sin$ is for dS and $\sinh$ for AdS.]

\begin{equation}
ds^{2}_{_{D+1} }
\
\
dw^{2}+\sin(\text{h})^{2}\left(w\ H\right)\left[-d\tau^{2}+\frac{1}{H^{2}}\cosh^{2}\left(\tau\ H\right)\ d\Omega_{_{D-1}}^{2}\right]
\end{equation}

with the key difference being the change in the scale factor $\sin\rightarrow\sinh$, implying that the 2 match in the IR-region of the bulk, i.e. for small values of the arguments of the trigonometric/hyperbolic functions. From the CFT point of view this corresponds to the fact that the CFT on the dS/dS side has to be deformed by an irrelevant operator.

§.§ ZSTbT and  2  for AdS

For the case of a CFT$_{2}$, the $S^{2}$-partition function can be determined upon integrating (<ref>) over the metric defined by the line element

\begin{equation}
\
\
r^{2}(d\theta^{2}+\sin^{2}\theta\ d\phi^{2})
\end{equation}

The partition function changes under a change in $r$, [228, 229], hence, from (<ref>)

\begin{equation}
\frac{\partial\ \ln {\cal Z}_{S^{2}}^{^{\ \ AdS,  T\bar T}} }{\partial r}
\
\
-\frac{1}{r}\ \int d^{2}x\ \sqrt{\gamma}\ T^{a}_{a}
\
\
-\frac{1}{r}\int\ d\theta\ d\phi\ \sin\theta\ r^{2}\ \frac{2}{\lambda}\left(1-\sqrt{1+\frac{c\lambda}{12 r^{2}}}\right)
\nonumber
\end{equation}

\begin{equation}
\Rightarrow
\ \ \
\frac{\partial\ \ln {\cal Z}_{S^{2}}^{^{\ \ AdS,  T\bar T}} }{\partial r}
\
\
-\frac{8\pi}{\lambda}\ \left(r-\sqrt{r^{2}+\frac{c\lambda}{12 }}\right)
\label{eq:t2}
\end{equation}

\begin{equation}
\Rightarrow
\ \ \
\ln {\cal Z}_{S^{2}}^{^{\ \ AdS,  T\bar T}}
\
\
-\frac{8\pi}{\lambda}\ \int \ dr\ \left(r-\sqrt{r^{2}+\frac{c\lambda}{12 }}\right)
\
\
\frac{8\pi}{\lambda}\ \left[\ -\frac{r^{2}}{2}+\frac{c\ \lambda}{12}\int\ dx\ \ \cosh^{2}x\ \right]
\nonumber
\end{equation}

where $r(x)\overset{\text{def.}}{=}\sqrt{\frac{c\lambda}{12}}\ \sinh x$; the partition function becomes

\begin{equation}
\ln {\cal Z}_{S^{2}}^{^{\ \ AdS, T\bar T}}
\
\
\pi \frac{c}{3} \sinh^{-1}\ \left(\sqrt{\frac{12}{c\lambda}}\ r\ \right)-\frac{4\pi\  r^{2}}{ \lambda}\left[\ 1-\ \sqrt{\frac{c\lambda}{12\  r^{2}}+1}\ \right]
\label{eq:AdSPF2}
\end{equation}

up to an overall integration constant. The entanglement entropy can be determined from (<ref>) and (<ref>), [228],

\begin{equation}
\boxed{\ \ S_{_{T\bar T}}
\
\
\left(1-\frac{r}{2}\frac{\partial}{\partial r}\right) \ln {\cal Z}_{S^{2}}^{^{\ \ AdS, T\bar T}}
\
\
\pi \frac{c}{3}\  \sinh^{-1}\ \left(\sqrt{\frac{12}{c\lambda}}\ r\ \right)    \ \ }
\label{eq:finalresult1}
\end{equation}

For later purposes, it is important to highlight that, (<ref>) and (<ref>) differ by means of the following term

\begin{equation}
-\frac{r}{2}\frac{\partial}{\partial r}  \ln {\cal Z}_{S^{2}}^{^{\ \ AdS, T\bar T}}
\frac{4\pi\  r^{2}}{ \lambda}\left[\ 1-\ \sqrt{\frac{c\lambda}{12\  r^{2}}+1}\ \right]
\label{eq:diffttb}
\end{equation}

and therefore, one could argue that, under a suitable choice of $\lambda$, (<ref>) can be set to vanish. In such case, $S_{_{T\bar T}}$ would be uniquely determined by $\ln{\cal Z}_{S^{2}}^{^{\ \ AdS, T\bar T}}$. We will be coming back to the importance of this observation later on in section <ref>.

For $r>>\sqrt{\lambda c}$,

\begin{equation}
\begin{aligned}
\sinh^{-1} x
\ln\ \left|\ x\ +\ \sqrt{1+x^{2}}\ \right |
\
\
\ln\ \left|\ x\ +\ x\ \sqrt{1+\frac{1}{x^{2}}}\ \right |
\
\sim \nonumber\\
\sim
\ln\ \left|\ x\ \right| +\left(1+\frac{1}{2 x^{2}}\right)
\
\
\ln\ \left|\ x\ \right| +\frac{2x^{2}+1}{2 x^{2}}
\end{aligned}
\end{equation}

and (<ref>) therefore turns into

\begin{equation}
S_{_{T\bar T}}
\
\
\frac{c}{3}\ \ln\ \left|\ \sqrt{\frac{12}{c\lambda}}\ r\ \right|\ \ +  \frac{c}{3}\ \frac{2\left(\frac{12}{c\lambda}\ r^{2}+1\right)}{2 \frac{12}{c\lambda}\ r^{2}} \ - \ \frac{c}{3}\ \frac{1}{\frac{24}{c\lambda}\ r^{2}}
\label{eq:lee1}
\end{equation}

By further substituting $r(x)\overset{\text{def.}}{=}\sqrt{\frac{c\lambda}{12}}\ \sinh x$, (<ref>) can be brought down to the more familiar expression

\begin{equation}
S_{_{T\bar T}}
\
\
\frac{c}{3}\ \ln\ \left|\ \sinh x\right|\ \ +   \frac{c^{2}\lambda}{36}\ \coth^{2}x \ - \ \frac{c}{6}\ \frac{1}{ \sinh^{2}x }
\label{eq:lee2}
\end{equation}

where the 1$^{st}$ term is the entanglement entropy for an interval of length $r$ at finite temperature $\beta\sim\sqrt{12/c\lambda}$. Equation (<ref>), to leading order in the coupling $\lambda$, therefore reads

\begin{equation}
S_{_{T\bar T}}
\
\
\frac{c}{3}\ \ln\ \left|\ \sqrt{\frac{12}{c\lambda}}\ r\ \right|\ \ +  \frac{c^{2}\ \lambda}{72}\ \frac{1}{ r^{2}} \ + \ \frac{c}{3}
\label{eq:lee3}
\end{equation}

From (<ref>) we can now see that in the UV-limit, i.e. as we take $\underset{r\rightarrow 0}{\lim} S_{_{T\bar T}}$, the first two terms cancel each othert out, thereby preventing the entanglement entropy form diverging. This can be checked explicitly through de l'Hopitàl's rule for the $\infty-\infty$ case.

§.§  2 and  2    for dS

For dS$_{3}$/dS$_{2}$, the corresponding geodesic in the space of 2D theories is given by, [230],

\begin{equation}
\boxed{\ \ \ \frac{\delta{\cal L}}{\delta \lambda}
\
\
2\  T\bar T-\frac{1}{ \lambda^{2}}  \textcolor{white}{\Biggl[} \color{black}\ \ }
\end{equation}

which differs from the AdS case by means of the 2$^{nd}$ term, taking into account that the CFT$_{2}$ is now coupled to gravity. The corresponding change to the sphere partition function now reads

\begin{equation}
\frac{\partial}{\partial \lambda}\ \ln {\cal Z}_{S^{2}}^{^{\ \ dS, T\bar T}}
\
\
-2\int\ d^{2}x\ \sqrt{\gamma}\ <T\bar T> +\ \frac{1}{ \lambda^{2}}\ \int\ d^{2}x\ \sqrt{\gamma}
\
\
-8\pi r^{2}\ <T\bar T> +\frac{4\pi\ r^{2}}{ \lambda^{2}}
\end{equation}

From the factorisation of the $T\bar T$ expectation value,

\begin{equation}
<T\bar T>
\
\
\
\
-\frac{1}{4\ \lambda^{2}}\ \left(\frac{\lambda c}{12r^{2}}-2\sqrt{\frac{\lambda c}{12r^{2}}-1}\right)
\end{equation}

\begin{equation}
\Rightarrow
\ \ \
\frac{\partial}{\partial \lambda}\ \ln {\cal Z}_{S^{2}}^{^{\ \ dS,  T\bar T}}
\
\
\frac{2\pi\ r^{2}}{\lambda^{2}}\ \left(\frac{\lambda c}{12r^{2}}-2\sqrt{\frac{\lambda c}{12r^{2}}-1}\right) +\frac{4\pi\ r^{2}}{\lambda^{2}}
\end{equation}

which can be integrated w.r.t. $\lambda$ resulting in

\begin{equation}
\ln {\cal Z}_{S^{2}}^{^{\ \ dS,  T\bar T}}
\
\
\pi\frac{c}{6}\ \ln\left|\ \frac{\lambda c}{12 r^{2}}\ \right| -\frac{4\pi\ r^{2}}{\lambda}- 4\pi\ r^{2}\ \int d\lambda\ \frac{1}{\lambda^{2}} \sqrt{\frac{\lambda c}{12r^{2}}-1}
\end{equation}

The last term can be integrated by parts

\begin{equation}
\begin{aligned}
- 4\pi\ r^{2}\ \int d\lambda\ \frac{1}{\lambda^{2}} \sqrt{\frac{\lambda c}{12r^{2}}-1}
- 4r^{2} \left(\ \-\frac{1}{\lambda}\  \sqrt{\frac{\lambda c}{12r^{2}}-1}  +\int\ d\lambda\ \frac{1}{2\lambda}\ \frac{1}{ \sqrt{\frac{\lambda c}{12r^{2}}-1}  }\ \frac{c}{12 r^{2}} \right)
= \nonumber\\
- 4\pi\ r^{2} \left(\ \-\frac{1}{\lambda}\  \sqrt{\frac{\lambda c}{12r^{2}}-1}  +\frac{c}{12 r^{2}}\ \tan^{-1}\ \left(\frac{1}{\sqrt{\frac{\lambda c}{12r^{2}}-1}  }\right) \right)
\end{aligned}
\end{equation}

Gathering all the terms, the partition function thus reads

\begin{equation}
\ln {\cal Z}_{S^{2}}^{^{\ \ dS, T\bar T}}
\
\
\pi \left[\frac{c}{3}\ \ln\left|\ \sqrt{\frac{\lambda c}{12 }}\frac{1}{r}\ \right| -\frac{4\ r^{2}}{\lambda}\left(1-  \sqrt{\frac{\lambda c}{12r^{2}}-1} \right) +\frac{c}{3}\ \tan^{-1}\ \left(\frac{1}{\sqrt{\frac{\lambda c}{12r^{2}}-1}  }\right)\right]
\label{eq:dSPF1}
\end{equation}

from which the entanglement entropy thus follows, [230],

\begin{equation}
\boxed{\ \ S_{_{T\bar T}}
\
\
\left(1-\frac{r}{2}\frac{\partial}{\partial r}\right) \ln {\cal Z}_{S^{2}}^{^{\ \ dS,  T\bar T}}
\
\
\pi\left[\frac{c}{6}+\frac{c}{3}\ \ln\left|\ \sqrt{\frac{\lambda c}{12 }}\frac{1}{r}\ \right|+\frac{c}{3}\ \tan^{-1}\ \left(\frac{1}{\sqrt{\frac{\lambda c}{12r^{2}}-1}  }\right)   \right] \ \ }
\label{eq:dSEE11}
\end{equation}

Notice that (<ref>) and (<ref>) differ by means of the following terms

\begin{equation}
-\frac{r}{2}\frac{\partial}{\partial r}  \ln {\cal Z}_{S^{2}}^{^{\ \ dS,  T\bar T}}
-\frac{c}{6}+\frac{4\ r^{2}}{\lambda}\left(1-  \sqrt{\frac{\lambda c}{12r^{2}}-1} \right)
\end{equation}

which, as also argued for the AdS case, (<ref>), can be set to vanish under suitable choice of $\lambda$.

§.§ Comparing  for AdS and dS

Comparing the expressions for the partition functions on the dS and AdS side, i.e. (<ref>) and (<ref>) respectively, we can rewrite the $\sinh^{-1}$ term in the latter as a $\ln$,

\begin{equation}
\sinh^{-1} \left(\ \sqrt{\frac{12}{c\lambda}}\ r\ \right)
\
\
\ln\ \left|\sqrt{\frac{12}{c\lambda}}\ r+\sqrt{\frac{12 r^{2}}{c\lambda}+1}\ \right|
\
\
\ln\ \left|\sqrt{\frac{12}{c\lambda}}\ r\ \left(1+\sqrt{\frac{c\lambda}{12 r^{2}}+1}\ \right) \right|
\nonumber
\end{equation}

which is reminiscent of the usual structure of $S_{_{EE}}$ for a finite-T CFT. The AdS partition function therefore becomes

\begin{equation}
\ln {\cal Z}_{S^{2}}^{^{\ \ AdS, T\bar T}}
\
\
\pi \frac{c}{3} \ln\ \left|\sqrt{\frac{12}{c\lambda}}\ r\ \left(1+\sqrt{\frac{c\lambda}{12 r^{2}}+1}\ \right) \right| +\frac{4\pi}{ \lambda}\left[\ r\ \sqrt{r^{2}+\frac{c\lambda}{12}}\  -r^{2}\ \right]
\label{eq:AdSPF3}
\end{equation}

suggesting the factor of $\sqrt{\frac{12}{c\lambda}}=\frac{1}{\epsilon}$ can be interpreted as the cutoff at which the dS can be glued in the IR of the original AdS. By means of such regularisation, (<ref>) can be rewritten in terms of $\epsilon$ as follows

\begin{equation}
\ln {\cal Z}_{S^{2}}^{^{\ \ dS,  T\bar T}}
\
\
\pi\left[\frac{c}{3}\ \ln\left|\ \sqrt{\frac{\lambda c}{12 }}\frac{1}{\epsilon}\ \right| -\frac{4r^{2}}{\lambda}\left(\ 1- \frac{1}{\pi}\ \sqrt{\frac{\lambda c}{12r^{2}}-1}\ \right) +\frac{c}{3}\ \tan^{-1}\ \left(\frac{1}{\sqrt{\frac{\lambda c}{12r^{2}}-1}  }\right)   \right]
\label{eq:dSPF2}
\end{equation}

Notice that only the $\epsilon$-dependent term diverges one upon taking the UV-limit with $\underset{r\rightarrow 0}{\lim}$, i.e. when probing short-distance scales. From the redefinition (<ref>),  the entanglement entropy for the $T\bar T$ deformed dS-theory now reads, [230],

\begin{equation}
\boxed{\ \ S_{_{T\bar T}}
\
\
\left(1-\frac{r}{2}\frac{\partial}{\partial r}\right) \ln {\cal Z}_{S^{2}}^{^{\ \ dS,  T\bar T}}
\
\
\pi\left[\frac{c}{6}+ \frac{c}{3}\ \ln\left|\ \sqrt{\frac{\lambda c}{12 }}\frac{1}{\epsilon}\ \right|+\frac{c}{3}\ \tan^{-1}\ \left(\frac{1}{\sqrt{\frac{\lambda c}{12r^{2}}-1}  }\right)   \right] \ \ }
\label{eq:dSEE1}
\end{equation}

In the work of Silverstein et al., [230], the 2$^{nd}$ term in (<ref>) is set to vanish by appropriately tuning the value of the CFT parameters and consequently of their corresponding bulk duals.
For our purposes, though, this cutoff dependence plays a key role, and will therefore be kept explicit. As a further remark, notice that, for the case in which the dS well approximates the original AdS, the universal parts of the entanglement entropies on the AdS and dS side agree, since the  $\tan^{-1}$ term in (<ref>) can be rewritten as $\sinh^{-1}$ by making the argument imaginary

\begin{equation}
\tan^{-1}\ \left(\frac{1}{\sqrt{\frac{\lambda c}{12r^{2}}-1}  }\right)
\
\sinh^{-1}\ \left(\sqrt{\frac{12}{c\lambda}}\ r\ \right)
\end{equation}

thereby proving that, under a suitable definition of $\epsilon$, dS can arise from the RG-flow of a bulk AdS. We will be coming back to this when comparing localisation calculation with the actions for vacuum transitions.

§.§ Cutoff dependence of

The divergence of $S_{_{EE}}$ for pure AdS when evaluated at $z=0$ follows from removing the UV-cutoff $\epsilon$ acting as a regulator, [275]. When deforming the CFT with an irrelevant operator, $S_{_{EE}}$ will still be dependent on the cutoff. However, its explicit dependence can be removed by appropriately tuning the parameters in the UV and IR of the theory, thereby leading to finite $S_{_{EE}}$s.

§.§ Key points of -deformations

* They constitute an example of UV/IR-mixing, and therefore provide an interesting setup where to address unitarity issues similar to the ones arising within the context of the information loss paradox.

* dS can arise as the IR-limit of an RG-flow driven by an irrelevant operator starting from a UV-theory dual to AdS.

* Localisation techniques play a key role in evaluating the partition function for a given CFT. Given its key role in determining $S_{_{EE}}$, the extremisation procedure rooted in the formalism is compatible with the method defining $S_{_{TOT}}$ by means of the FMP method. It is therefore reasonable to argue, as we shall see in section <ref>, that these quantities must be proportional to each other.

* The cutoff radius ensures the finiteness of $S_{_{T\bar T}}$ on both sides, explicitly featuring only universal terms.

* According to the dimensionality of the CFTs being involved, the behaviour of the universal terms is predicted by the $g,c$ and $a-theorems$.

* The main advantage of having introduced $\lambda\neq 0$ is that, in the limit $r/\sqrt{c\lambda }\rightarrow 0$, $S_{EE}\rightarrow 0$ and does not diverge upon taking the UV-limit, unlike the usual CFT result. This proves that, removing the regulator, unitarity is still preserved. As we will see, removing the CFT-cutoff is equivalent to taking the limit of the vanishing cosmological constant in the dS$\rightarrow$ dS transition in absence of a black hole.

§ VON NEUMANN ALGEBRAS (VNAS)

§.§ Types of VNAs

A von Neumann algebra (VNA) is a $*$-subalgebra $M$ of ${\cal B}({\cal H})$ that coincides with its bicommutant

\begin{equation}
\end{equation}

A VNA $M\subset {\cal B}({\cal H})$ is called a factor if $M\cap M^{^{\prime}}=\mathbb{C}\mathbf{1}_{_{{\cal B}({\cal H})}}$.

VN factors can be of different types:

* I$_{_n},$ or I$_{_{\infty}}$, classified by their dimension.

* II$_{_1}$ or II$_{_{\infty}}$, with

\begin{equation}
II_{_{\infty}}\simeq II_{_1}\otimes {\cal B}({\cal H}).
\end{equation}

* III$_{_{\lambda}}$ with $\lambda\in[0,1]$.

For the case of type-II$_{_1}$, a projection in a VNA $M$ is an element $p\in M$ such that

\begin{equation}
\end{equation}

The type of factor depends on the existence of small projections. Specifically, a linear functional $\tau$ on a VNA $M$ is

* Positive, if $\tau(x^{^*}x)\ge 0 \forall x\in M$.

* Faithful, if $\tau(x^{^*}x)=0\Rightarrow x=0$.

* Stable, if $\tau (1)=1$.

* Tracial, if $\tau(xy)=\tau(yx)\  \forall x,y\in M$. A tracial state is simply referred to as trace.

§.§ Standard form of a II factor

Let $M$ be a factor with continuous and faithful trace $\tau$. The pairing

\begin{equation}
<x,y>=\tau(y^{^*}x)\ \in\ \mathbb{C}
\end{equation}

defines an inner product on $M$. Let $L^{^2}(M,\tau)$ be the associated completion of $M$. Then there is an embedding

\begin{equation}
\begin{aligned}
&M\ \longrightarrow\ L^{^2}(M)  \\
&x\ \mapsto\ \hat x\\
&1\ \mapsto\ \Omega.
\end{aligned}
\end{equation}

$M$ is represented on a Hilbert space $L^{^2}(M)$ by considering

\begin{equation}
\pi_{_\tau}(x)\hat y=\hat{xy}\equiv x\hat y
\end{equation}

and extending it to a morphism

\begin{equation}
\pi_{_{\tau}}:\ M\ \longrightarrow\ {\cal B}\left(L^{^2}(M)\right).
\end{equation}

This representation is called the standard form of $M$, implying that

\begin{equation}
\hat x=\hat{x1}=x\hat 1=x\Omega,
\end{equation}

such that

\begin{equation}
\tau(x)=<x\Omega,\Omega.
\end{equation}

§.§ Representations and von Neumann dimensions

A representation of a factor $M$ is a Hilbert space ${\cal H}$ with a structure of an $M$-module.

There is a Theorem, stating that any representation ${\cal H}$of a II$_{_1}$ factor $M$ is equivalent to

\begin{equation}
\end{equation}

where $p$ is a projection

\begin{equation}
p=vv^*    \ \in\ (M\times 1)^{\prime}
\end{equation}

whose trace is the size of the representation ${\cal H}$

\begin{equation}
\text{dim}_{_M}{\cal H}\ \equiv\ \text{Tr}\left(vv^*\right)\ \in\ [0,\infty].
\end{equation}

§.§ Properties of the VN dimension

If $M$ is a type-II$_{_1}$ factor, then

\begin{equation}
\text{dim}_{_M}\bigoplus_{_{j\in J}}{\cal H}_{_j}\ \equiv\ \sum_{_{j\in J}}\text{dim}_{_M}{\cal H}_{_j}.
\end{equation}

If $p\in M^{\prime}$ is a projection, then

\begin{equation}
\text{dim}_{_M}L^{^2}(M)_{_p}\ =\ \tau(p).
\end{equation}

Representations are classified by their $M$-dimension, and it therefore follows that

\begin{equation}
\text{dim}_{_M}{\cal H}\ =\ \text{dim}_{_M}{\cal K}\ \ \ \iff\ \ \ {\cal H}\ \simeq\ {\cal K}.
\end{equation}

§.§ The Jones index

Let $N\subset M$ be type II$_{_1}$-factors. Then $N$ is a representation on $L^{^2}(M)$.

The Jones index of the subfactor $N\subset M$ is defined as follows

\begin{equation}
[M:N]\ \overset{def.}{=}\ \text{dim}_{_N}L^{^2}(M).
\end{equation}

If $N$ is represented on ${\cal H}$ with dim$_{_N}{\cal H}<\infty$, then

\begin{equation}
[M:N]\ =\ \frac{{dim}_{_N}{\cal H}}{\text{dim}_{_M}{\cal H}}.
\end{equation}

§ OTHER APPENDICES TO PART IV

§.§ Twist operators and Rényi entropies

The definition of entanglement entropy involves a path integral over a Riemann surface, where the boundary conditions dictated by the twist field operators[Twist field operators are located at the endpoints of the intervals w.r.t. which $S_{_{EE}}$ is being evaluated.], ${\cal T}$, encode the nontrivial topological features of the theory, [275]. This is of particular interest whenever dealing with theories with no 1:1 correspondence between the solutions to the e.o.m. and the original Lagrangian density.

A similar issue arises when performing coordinate transformations in 2D gravity theories, such as maximal extensions, ultimately leading to a rich web of dualities. The bulk-boundary correspondece of AdS/CFT, as well as the factorisation problem, provide further examples. Before delving into a more detailed analysis of concrete examples, we wish to highlight first the importance of one major quantity relating all these topics together, which in turn prompted the question of understanding its relation with non-invertible symmetries, namely the Rényi entropies.

For later convenience, we will briefly overview the main features of the arguments outlined by Cardy and Calabrese for proposing this quantity as the suitable candidate to study CFT$_2$s in all their richness.  In doing so, we will also recapitulate how the emergence of the island becomes manifest from a pure $CFT/BCFT$ calculation as the change in dominance of connected/disconnected channels within correlation functions of twist operators, stressing the importance of the fact that $S_{_{EE}}$ for a single interval in a BCFT shares the same behaviour as that of 2 intervals on a CFT.

4-point functions in a $CFT$

In this digression, we briefly outline the derivation of $S_{_{EE}}$, following the techniques of [275].
In presence of conformal symmetry breaking, $<T^{\mu}_{\mu}>\neq 0$, and the boundary-to-boundary propagator picks up a correction of the kind $<{\cal T}{\cal T} T_{\mu}^{\mu}>$. The map from the Riemann surface[... where the $n$-point functions of the twist operators are evaluated.] to the complex plane is defined as, [275],

\begin{equation}
\
\
\xi^{1/n}
\
\
\left(\frac{w-u}{w-v}\right)^{1/n}
\ \ \
\Rightarrow
\ \ \
{\cal R}_{n,1}\rightarrow\mathbb{C}
\end{equation}

and the trace of the energy-momentum tensor transforms accordingly as follows

\begin{equation}
\
\
\left(\frac{dz}{dw}\right)^{2}T(z)+\frac{c}{12}\{z,w\}
\label{eq:Scw}
\end{equation}

with the 2$^{nd}$ term in <ref> denoting the Schwarzian derivative,  from which its expectation value becomes

\begin{equation}
<T(w)\underset{{\ \cal R}_{n,1} }{>\ \ \ }
\
\
<T(z)\underset{\mathbb{C} }{>\ \ \ }\ +\ \frac{c(1-n^{-2})}{24}\frac{(v-u)^{2}}{(w-u)^{2}(w-v)^{2}}
\frac{ <{\cal T}_{n}(u,0)\tilde{\cal T}_{n}(v,0)T(w)\underset{{\cal L}^{(n)}, \mathbb{C} }{>\ \ \ } }{<{\cal T}_{n}(u,0)\tilde{\cal T}_{n}(v,0)>}
\end{equation}

with twist operators ${\cal T}_{n}, \tilde{\cal T}_{n}$ being primary fields of conformal dimension $\Delta_{n},\bar \Delta_{n}$ determined from the 2-point function

\begin{equation}
<{\cal T}_{n}(u,0)\tilde{\cal T}_{n}(v,0)>_{{\cal L}^{(n)},\mathbb{C}}
\
\
\ \ \
\Rightarrow
\ \ \
\Delta_{n}
\
\
\frac{c}{12}\left(n-\frac{1}{n}\right)
\ \ \
\end{equation}

Following the work of Cardy and Calabrese, [275], the 4-point function of the twist operators characterising the Riemann surface on which a field theory lives, defines the partition function for 2 disjoined intervals on the FT side. The $n^{th}$-Rènyi entropy is defined as

\begin{equation}
\
\overset{\text{def.}}{=}
\
\text{Tr}(\rho_{A})^{n}
\
\
\frac{{\cal Z}_{n}  }{{\cal Z}^{n}}
\ \ \
\end{equation}

\begin{equation}
\begin{aligned}
{\cal Z}_{n}    \
&    \overset{\text{def.}}{=}      \
<{\cal T}(z_{1})\bar{\cal T}(z_{2})\bar{\cal T}(z_{3}){\cal T}(z_{4})> _{cyl}
\
= \nonumber\\
\left(\frac{2\pi}{\beta}\right)^{\frac{c}{3}\left(n-\frac{1}{n}\right)} |w_{1}w_{2}w_{3}w_{4}|^{\frac{c}{12}\left(n-\frac{1}{n}\right)}<{\cal T}(w_{1})\bar{\cal T}(w_{2})\bar{\cal T}(w_{3}){\cal T}(w_{4})> _{\mathbb{C}}
\label{eq:2ch}
\end{aligned}
\end{equation}

with the complex coordinates $w_{i}$ being related to the spacetime coordinates $(t,r)$ via the following change of variables

\begin{equation}
\begin{aligned}
\
\
e^{-2\pi(r_{I}+t_{I})/\beta}e^{i\pi/2 }
\ \ \
\ \ \
\
\
e^{2\pi(r_{I}-t_{I})/\beta}e^{i\pi/2 }
\nonumber\\
\
\
e^{-2\pi(r_{R}+t_{R})/\beta}e^{-i\pi/2 }
\ \ \
\ \ \
\
\
e^{2\pi(r_{R}-t_{R})/\beta}e^{-i\pi/2 }
\end{aligned}
\end{equation}

Given a subregion $A$ obtained by partial tracing over the density matrix of the total system, the Rényi entropies $S^{(n)}$, the entanglement entropy $S_{_{EE}}$, the reduced denisty matrix $\rho_{_{A}}$ and the partition function ${\cal Z}$ are related according to the following

\begin{equation}
\boxed{\ \ S_{A}^{^{EE}}
\
\overset{\text{def.}}{=}
\
\underset{n\rightarrow1}{\lim}\  S_{A}^{(n)}
\
\
-\underset{n\rightarrow1}{\lim}\ \left[\ \frac{1}{1-n}\ln\text{Tr}\rho_{A}^{n}\ \right]
\
\
-\underset{n\rightarrow1}{\lim}\  \frac{\partial}{\partial n}\ \text{Tr}\rho_{A}^{n}
\
\
-\underset{n\rightarrow1}{\lim}\ \frac{\partial}{\partial n}\ \frac{{\cal Z}_{n}(A)}{{\cal Z}^{n}} \ \ }
\label{eq:ententr}
\end{equation}

§.§ Key points:

* $n$ is assumed to be analytically continued in order for the derivative to be defined in (<ref>).

* The need to consider 1 interval rather than 2 for the case of a BCFT, suggests one should be looking for a more generic approach towards describing the behaviour of the system for $n\ \in \mathrm{R}$.

* The twist operators are playing the same role as the non-invertible defects of a QFT featuring global internal symmetries.

In this appendix we will briefly argue how to deal with transitions involving a black hole only on one side of the brane. At the level of the Lagrangian density, this amounts to turning on the mass deformation on, either, the interior or the exterior vacua. The procedure to obtain the total action turns out to be a combination of the cases outlined in sections <ref> and <ref>, and we therefore quote the key steps, referring the reader to the aforementioned parts of this work for further details and intermediate steps.

The extrinsic curvatures on the 2 sides of the brane would now be

\begin{equation}
\frac{\phi^{\prime}}{L}\bigg|_{\pm}
\
\
\frac{\Lambda_{+}-\Lambda_{-}\mp\frac{\kappa^{2}}{4}}{\kappa}\ \phi_{b}+\frac{{\cal B}}{\kappa}
\nonumber
\end{equation}

Assuming the black hole lies in the inner spacetime[Labelled by the “-”  sign.], the turning point can be determined from the potential

\begin{equation}
\phi_{1,2}
\
\
-\frac{\left(\Lambda_{+}-\Lambda_{-}\ -\frac{\kappa^{2}}{4}\right)\ {\cal B}}{\kappa}\ \phi_{o}^{2}\ \left[1\pm\sqrt{1-\frac{4\kappa^{2}\left(\frac{{\cal B}^{2}}{\kappa^{2}}+{\cal C}\right)}{\left(\Lambda_{+}-\Lambda_{-}-\frac{\kappa^{2}}{4}\right)^{2}\ {\cal B}^{2}\ \phi_{o}^{2}}}\right]
\nonumber
\end{equation}

The bulk horizon contribution will only come from the AdSBH,

\begin{equation}
\
\
\frac{\eta}{G}\frac{{\cal B}}{\Lambda_{-}}\left[\ 1-\sqrt{1-\frac{4{\cal C}\Lambda_{-}}{{\cal B}^{2}}}\ \right]
\nonumber
\end{equation}

whereas the brane action would now read

\begin{equation}
\begin{aligned}
\frac{2\eta}{G}\int_{\phi_{1}}^{\phi_{2}}d\phi_{b} \left[\cosh^{-1}\left[\frac{\left(\Lambda_{+}-\Lambda_{-}+\frac{\kappa^{2}}{4}\right)\phi_{b}+{\cal B}}{\kappa\sqrt{-{\cal C}-{\cal B}\phi_{b}-\Lambda_{-}\phi_{b}^{2}}}\right]-\cosh^{-1}\left[\frac{\left(\Lambda_{+}-\Lambda_{-}-\frac{\kappa^{2}}{4}\right)\phi_{b}+{\cal B}}{\kappa\sqrt{-{\cal C}-\Lambda_{+}\phi_{b}^{2}}}\right]\right]
\nonumber \\
\frac{2\eta}{G}\ \left[\
-\frac{{\cal B}}{2\Lambda_{-}}\  \ln\left|\frac{y_{2}}{y_{1}}\sqrt{\frac{1-y_{1}^{2}}{1-y_{2}^{2}}}\right| -\phi_{2}\ln\left|\frac{A_{2}\ y_{2}\sqrt{1-y_{4}^{2}}}{A_{4}\ y_{4}\sqrt{1-y_{2}^{2}}}\right|+\phi_{1}\ln\left|\frac{A_{2}\ y_{1}\sqrt{1-y_{3}^{2}}}{A_{4}\ y_{3}\sqrt{1-y_{1}^{2}}}\right| +\  \right.
\nonumber\\
\left.\frac{1}{2}\sqrt{\frac{\frac{{\cal B}^{2}}{4\Lambda_{-}}-{\cal C}}{\Lambda_{-}}}\ \ln\ \left|\frac{1-y_{2}}{1+y_{2}}\frac{1+y_{1}}{1-y_{1}}\right| +\mu_{+}\ \ln\ \left|\frac{1-y_{4}}{1+y_{4}}\frac{1+y_{3}}{1-y_{3}}\right| \  \right]
\label{eq:2VR}
\end{aligned}
\end{equation}

§.§ No obstructions from the Generalised Second Law (GSL)

Wormhole traversability has been amply studied in the last 2 decades thanks to the development of the AdS/CFT correspondence, [192]. The birth of an inflating universe from an asymptotically flat background has been argued to violate wormhole traversability as a consequence of the  generalised 2$^{nd}$ law (GSL), [295]. However, we would like to highlight that the results obtained in the present paper by means of the Hamiltonian formalism show that this does not hold in our case for several reasons that we will now outline. In light of the closing remarks in section <ref>, the total action for the transition can be recast in the form of the difference of 2 $S_{_{EE}}^{T\bar T}$. Relying upon the holographic understanding of such kinds of theories, we can deduce that the transition is only involving a subregion of the entire spacetime, and is, therefore, compatible with the traversable wormhole picture first outlined by Gao, Jafferis and Wall (GAO) by means of double-trace deformations, [294].  In summary, in our setup, there is no obstruction to up-tunnelling due to the following features:

* Transitions are local

* The instanton is singular and the metric is degenerate

* The transition amplitude is related to the Entanglement entropy, $S_{_{EE}}$, rather than the Von Neumann entropy. This justifies the necessity for topology change to take place whenever describing vacuum transitions. Notice that, in the Hamiltonian formalism, transitions are only described on spatial slices and not on the entire manifold. For wormhole saddles of the kind expected to emerge in the path integral of quantum gravity, the entire spacetime should really be understood of as being multiply-connected.

As long as the manifold is simply connected, the action is related to a monotonic $S_{_{EE}}$, and therefore behaves as the ordinary von Neumann entropy of a bipartite system. In any other case, though, such as for when describing up-tunnelling, the von Neumann entropy needs to be upgraded to $S_{_{EE}}$, meaning spacetime is multiply-connected. The singular metric involved in up-tunnelling is unsuitable for applying the GSL, given that the latter takes into account the causal dependence in between different spacetimes at ${\cal I}^{\pm}$. But this is different from the local transitions we are dealing with. The Hamiltonian constraint defines the energy of the system, i.e. not of the entire spacetime, but only that of the local subregion involved in the transition.

§.§ Singular instantons

The present appendix is meant to be a short digression providing further insight into the singular instanton solution obtained by FGG. As also argued by the authors in [222], there are 2 possible ways in which the singularity might arise:

* Either in the past of the spacetime, or

* In the wormhole configuration mediating the tunnelling process (double analytic continuation).

The former can be avoided via the quantum tunnelling interpolating between 2 classical solutions, defined as type-a and type-b trajectories. The latter is unavoidable given the change in sign of the lapse function in the forbidden region, as already noticed in [199].

\begin{equation}
\begin{aligned}
\\
\\
\\
\\
\\
\\
\\
\\
\\
\\
\\
\\
\end{aligned}
\ \
\begin{aligned}
\text{dS}
\ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \           \text{Schw    }    \ \ \ \ \ \ \ \ \
\\
\includegraphics[scale=0.5]{M3.pdf}
\ \ \ \
\includegraphics[scale=0.5]{M1.pdf}
\\
\\
\includegraphics[scale=0.5]{M4.pdf}
\ \ \ \
\includegraphics[scale=0.5]{M2.pdf}
\end{aligned}
\ \
\vcenter{\hbox{\includegraphics[scale=0.45]{M5new.pdf} } }
\nonumber
\end{equation}

For the quantum process to take place, and effectively interpolate in between the 2 classical trajectories depicted in the figure above, the brane's mass, $M$, must be at least smaller w.r.t, $M_{S}$. i.e. the Schwarzschild mass. This ensures that the wall's trajectory intersects the horizon and the extrinsic curvature changes sign. In particular, [222] choose $M<M_{D}$. The reason for this is that, being $M$ smaller than any other scale of the theory, it can be tuned to be arbitrarily small, ultimately enabling to circumvent the emergence from the initial singularity[[222] also argue in favour of this by means of the so-called trapped and anti-trapped surfaces, with the latter being those to which the Penrose Theorem applies.] as we will now briefly outline. Given a 2-surface, $P$, its past light-cone will inevitably intersect the singularity with at least one light ray. However, in the weak gravity limit, it is possible to create a bubble of suitable radius to be at the 1$^{st}$ turning point of the potential barrier, treating the nucleation process as a pure quantum fluctuation of a flat background.

§.§ Roots of a quartic

In this appendix we outline the procedure for determining the extremising value of $\bar\rho$ for an (A)dS$_2\rightarrow$(A)dS$_2$ in the thin-wall approximation, (<ref>), outlined in section <ref>. Upon extremising $B_{_{TOT}}$ w.r.t. $\bar\rho$, we end up with a polynomial with all even powers of $\bar\rho$. In particular, redefining $x\overset{def.}{=}\bar\rho^2$, it can be brought down to the following form

\begin{equation}
x^4+a_3x^3+a_2x^2+a_1x+a_o =0
\label{eq:quart1}
\end{equation}

By further redefining $y\overset{def.}{=}x+\frac{a_3}{4}$, (<ref>) becomes

\begin{equation}
y^4+py^2+qy+r =0
\label{eq:quart2}
\end{equation}

Adding and subtracting $y^2u+\frac{u^2}{4}$, (<ref>) becomes

\begin{equation}
\left(y^2+\frac{1}{2}u\right)^2-\left[(u-p)x^2-qx+\left(\frac{u^2}{4}-r\right)\right]  =0
\end{equation}

After a suitable manipulation, it is possible to factorise it into the product of 2 quadratics subject to satisfying a cubic constraint equation, namely

\begin{equation}
\ \ \ \text{with}\ \ \
\begin{cases}
q\overset{def.}{=}a_1-\frac{1}{2}a_2a_3+\frac{1}{8}a_3^3  \\
\end{cases}
\end{equation}

The corresponding solution $u_o=t_o-\frac{p}{3}$, with

\begin{equation}
t_o\overset{def.}{=}\frac{2}{3}\sqrt{p^2+12r\ }\ \cos\left[\frac{1}{3}\cos^{-1}\left(\frac{2p^3+9(3q^2-8rp)}{18}\left(\frac{3}{p^2+12r}\right)^{3/2}\right) \right]
\end{equation}

Once having obtained such factorisation, the corresponding value of the turning point can be extracted. In the 4D case, extremisation leads to a relation which is equivalent to the junction conditions in BT, and given that the turning point can be analytically computed in such case, it also corresponds to the value of the bubble radius extremising $B_{tot}$. In this case, though, it is not so straightforward, therefore we need to calculate it explicitly from the extremising procedure. However, we anticipate that (<ref>) prior to extremisation already shares the same structure as the total bounce for type-1 instantons as found by BT in 2D. The turning point reads

\begin{equation}
\bar\rho^2
-\frac{2A_3+A_5}{2}\left[1\pm\sqrt{1-\frac{4\left(A_3^2+A_4+A_5A_6\right)}{(2A_3+A_5)^2}\ }\right]
\end{equation}

\begin{equation}
\ \ , \ \
A_4\overset{def.}{=}\frac{1}{2}\left(t_o-\frac{p}{3}\ \right)
\ \ , \ \
A_5\overset{def.}{=}\sqrt{t_o-\frac{p}{3}\ }
\ \ , \ \
A_6\overset{def.}{=}-\frac{1}{2}\sqrt{\frac{\left(t_o-\frac{p}{3}\right)^2-4r}{t_o-\frac{4p}{3}}\ }
\end{equation}

§ GEOMETRIC INVARIANT THEORY (GIT)

Delving in the explicit description of the duality structure arising from class ${\cal S}$ theories, to which Part <ref> is devoted, requires some key preliminary algebro-geometric tools, specifically the definition of Coulomb and Higgs branches from geometric invariant theory, [128, 332, 133, 130, 129, 344, 331, 131, 132, 134].

This appendix recapitulates the essential tools required for performing GIT quotient constructions, highlighting its importance for realising 3D mirror symmetry. In particular, we focus on the calculation of the Hilbert series as specific examples of algebraic varieties of crucial interest in the study of categorical dualities for supersymmetric quiver gauge theories.

Key algebraic varieties and dualities between invariants.

§.§ Equivariant cohomology

Equivariant cohomology, [335], also known as Borel cohomology, 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

The Kirwan map, first introduced in [127], states

\begin{equation}
H^{^{\bullet}}_{_G}(M)\ \rightarrow\ H^{^{\bullet}}\left(M//_{_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//_{_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)\ =\ H^{^{\bullet}}\left(M//_{_p} G\right).
\end{equation}

§.§ The GIT construction

Geometric invariant theory (GIT), [209] is a recipe for constructing quotients in algebraic geometry. An affine GIT quotient, or affine geometric invariant theory quotient of an affine scheme $X=\text{Spec} {\cal A}$ with an action by a group scheme $G$, is the affine scheme Spec$({\cal A}^{^G})$, namely the prime spectrum of the ring of invariants of ${\cal A}$, and is denoted by $X//G$.

The input of the GIT construction consists of:

* A complex reductive group, $G$.

* A representation of $G$

\begin{equation}
X \ =\ \mathbf{A}^{^n}.
\end{equation}

* A character

\begin{equation}
\theta:\ G\ \rightarrow\ G.
\end{equation}

GIT takes this data and gives an open subset of $X$, namely the semistable locus

\begin{equation}
X_{_{\theta}}^{^{ss}}(G)\ \overset{\text{open}}{\subset}\ X,
\end{equation}

from which one can define the stack quotient as follows

\begin{equation}
X//_{_{\theta}}(G)\ \overset{def.}{\equiv}\ \left[X_{_{\theta}}^{^{ss}}(G)\ /\ G\right].
\label{eq:ssq}
\end{equation}

As (<ref>) suggests, a GIT quotient is therefore a categorical quotient of the locus of semistable points, i.e. the quotient of the semistable locus.

Examples of GIT are Grassmannians, whose definition goes as follows. Take $G$ a complex (connected) reductive group. Let ${\cal O}$ denote the formal power series ring $\mathbf{C}[[z]]$ and ${\cal K}$ its fraction field $\mathbf{C}((z))$. The Grassmannian is defined by the group quotient

\begin{equation}
\text{Gr}_{_G}\ \overset{def.}{=}\ G_{_{\cal K}}/G_{_{\cal O}},
\end{equation}

where $G_{_{\cal K}}$ and $G_{_{\cal O}}$ denote the groups of ${\cal K}$ and ${\cal O}$-valued points of $G$, respectively. More formally, Gr$_{_G}$ is defined as being the moduli space of pairs $({\cal P}, \varphi)$ of a $G$-bundle, ${\cal P}$, on the formal disk

\begin{equation}
D\ \overset{def.}{=}\ \text{Spec}({\cal O}),
\end{equation}

and its trivialisation $\varphi$ over the punctured disk

\begin{equation}
D^{^*}\ \overset{def.}{=}\ \text{Spec}({\cal K}).
\end{equation}

Similarly, for the case of a complex representation, N, the moduli space is now parameterised by a triple

\begin{equation}
{\cal R}\ \overset{def.}{=}\ ({\cal P}, \varphi, s),
\end{equation}

where $({\cal P}, \varphi)$ is in Gr$_{_G}$, and $s$ is a section of an associated vector bundle ${\cal P}_{_N}={\cal P}\times_{_G}\textbf{N}$, such that it is sent to a regular section of the trivial bundle under $\varphi$.

For a given triple ${\cal R}$, there is a corresponding projection

\begin{equation}
\pi:\ {\cal R}\ \rightarrow \text{Gr}_{_G},
\end{equation}

where Gr$_{_G}$ is the affine Grassmannian, and $G$ is a complex reductive group. There is a natural commutative ring object, ${\cal A}$, in the derived $G_{_{\cal O}}$-equivariant constructible category on Gr$_{_G}$, $D_{_G}\left(\text{Gr}_{_G}\right)$, such that $H_{_{\bullet}}^{^{G_{_{\cal O}}}}\left({\cal R}\right)$ can be constructed from $H_{_{\bullet}}^{^{G_{_{\cal O}}}}\left(\text{Gr}_{_G},{\cal A}\right)$ equipped with a commutative ring structure.

In presence of a collection of such commutative ring objects, $\{{\cal A}_{_i}\}$, in $D_{_G}\left(\text{Gr}_{_G}\right)$, one can perform a gluing construction by defining a new commutative ring object as

\begin{equation}
\iota _{_{\Delta}}\left(\ \boxtimes\ {\cal A}_{_i}\right),
\end{equation}

\begin{equation}
\iota _{_{\Delta}}:\ \text{Gr}_{_G}\ \rightarrow\ \prod_{_i} \text{Gr}_{_G}
\end{equation}

is the diagonal embedding. One such example are star-shaped quivers, [343].

Its equivariant Borel-Moore homology, [566], can be defined by the $G_{_{\cal O}}$-equivariant Borel-Moore homology of

\begin{equation}
{\cal R}_{_{\le\lambda}}\ \equiv\ {\cal R}\ \cap\ \pi^{^{-1}}\left(\bar{\text{Gr}}_{_G}^{^{\lambda}}\right)
\end{equation}

in such a way that an embedding

\begin{equation}
{\cal R}_{_{\mu}}\ \hookrightarrow\ {\cal R}_{_{\lambda}}
\end{equation}

will induce a map

\begin{equation}
H_{_{\bullet}}^{^{G_{_{\cal O}}}}\left({\cal R}_{_{\le\mu}}\right) \ \rightarrow\ H_{_{\bullet}}^{^{G_{_{\cal O}}}}\left({\cal R}_{_{\le\lambda}}\right)
\end{equation}

for $\mu\le\lambda$.

More generally, the euivariant Borel-Moore homology can be expressed as a graded sum

\begin{equation}
H_{_{\bullet}}^{^{G_{_{\cal O}}}}\left({\cal R}\right) \ \simeq\ \underset{\gamma}{\bigoplus}\  H_{_{\bullet}}^{^{G_{_{\cal O}}}}\left({\cal R}^{^{\gamma}}\right),
\end{equation}

with ${\cal R}^{^{\gamma}}$ the connected component corresponding to $\gamma\ \in\ \pi_{_1}(G)$.

§.§ Moduli spaces: Coulomb and Higgs branches

The Coulomb branch is an affine algebraic variety whose coordinate ring is the equivariant Borel-Moore homology group, $H_{_{\bullet}}^{^{G_{_o}}}(\cal R)$ of a certain space ${\cal R}$, defining the variety of triples.

The Coulomb branch is defined as, [130],

\begin{equation}
{\cal M}_{_C}\ \overset{def.}{=}\ \text{Spec}\ {\cal A} (G, \mathbf{N})\ \equiv\ \text{Spec}\left(H_{_*}^{^{G_{_o}}}({\cal R}), *\right).
\end{equation}

Similarly, one can define the quantised Coulomb branch as follows

\begin{equation}
{\cal M}_{_C}\ \overset{def.}{=}\ \text{Spec}\ {\cal A}_{_{\hbar}} (G, \mathbf{N})\ \equiv\ \text{Spec}\left(H_{_*}^{^{G_{_o}\ \rtimes\mathbf{C}^{^{\times}}}}({\cal R}), *\right),
\end{equation}

which, in terms of quantum homology, can also be re-expressed as follows,

\begin{equation}
{\cal M}_{_C}\ \overset{def.}{=}\ \text{Spec}\left(QH_{_G}^{^{\bullet}}\left(G;HF^{^{\bullet}}(X)\right)\right).
\end{equation}

On the other hand, the Higgs branch is defined as  the spectrum of the categorical quotient, namely, [130],

\begin{equation}
{\cal M}_{_H}\ \overset{def.}{=}\ \text{Spec}\left(QH^{^\bullet}\left(X//G\right)\right)\ \simeq\ QH^{^{\bullet}}_{_{LG}}(X).
\end{equation}

§.§ The monopole formula

The monopole formula contains all the gauge-invariant chiral operators that acquire a non-zero expectation value[Their specific VEV depends on their respective dimensions and quantum numbers.] along the CB. It is particularly useful for investigating expected properties of Coulomb branches, and reads as follows

\begin{equation}
P_{_t}^{^{G_{_{\cal O}}}}({\cal R})\ =\ \sum_{_k}\ t^{^{-k}}\ \text{dim} \ H_{_k}^{^{G_{_{\cal O}}}}({\cal R}).
\label{eq:MF}
\end{equation}

As already remarked in [130], (<ref>) is the same as the Hilbert series on the Coulomb branch of a 3D ${\cal N}=4$ supersymmetric quiver gauge theory associated with the pair $(G_{_{\mathbf{C}}}, \mathbf{N}\ \oplus\ \mathbf{N}^{^*})$.

§ MAGNETIC QUIVERS

§.§ Magnetic Quivers: a unifying framework

This section outlines some further preliminary background tools needed in Part <ref> for addressing the formulation of mirror symmetry from the perspective of geometric representation theory, [3]. Mostly inspired by [181] and further upcoming work by Teleman, [182], we argue how constructions as the ones addressed in [1], involving multiple gauging, can still be assigned a fiber functor, as in the Freed-Moore-Teleman symmetry TFT (SymTFT) setup, [102]; however, such fiber functor should not be associated to either of the absolute theories separated by the non-invertible defect, but, rather to a 3D theory whose partition function correctly accounts for the symplectic projection leading to either of the two absolute theories.

Our approach consists in a combination of the following:

* Defining a fiber functor for different absolute theories connected by non-invertible defects, [1].

* Relating mirror symmetry with the identification of a Drinfeld center, [182].

* Recent advancements in the understanding of the Higgs branch (HB) structure of higher-dimensional quiver gauge theories with 8 supercharges by means of Coulomb branches (CBs) of magnetic quivers (MQs) associated to 3D ${\cal N}=4$ gauge theories, [210, 184, 185, 186, 187, 211].

In the current and next sections we mostly outline how the topics listed above combine together, and in the following sections we will provide a more detailed mathematical correspondence with the work of [181], highlighting how his constructions can effectively be generalised to higher dimensions and be naturally embedded in the setup of [102] precisely thanks to the understanding in terms of magnetic quivers of 3D ${\cal N}=4$ theories addressed here.

One could see our proposal as further supporting the idea that higher-categorical symmetries probe representation theory structures.

In the present section, we review the correspondence between geometric and algebraic resolutions of framed Nakajima quiver varieties, [330, 133, 130], highlighting it as an interesting example of homological mirror symmetry. In particular, we emphasise the property the moment map and higher homologies need to satisfy to ensure agreement in between the calculation of the two Hilbert series. We conclude the section with a brief overview of Hasse diagram constructions via magnetic quivers for quiver gauge theories with 8 supercharges, highlighting an interesting 2-categorical structure arising when dealing with complete intersections.

Section <ref> is meant to provide a brief overview of some key elements we will be using throughout our treatment, emphasising the ones that are needed for building the connection with the work of [181, 182][Throughout the entire treatment, we will be assuming the basic knowledge of higher categories and ADE quivers. We refer the reader to the extended literature on both topics for detailed definitions and examples. Specific additional tools will be explained in due course when needed.], which will be the core focus of section <ref>. The first part of the present article is structured as follows:

* At first we review the correspondence between geometric and algebraic resolutions of framed Nakajima quiver varieties, [330, 133, 130], highlighting it as an interesting example of homological mirror symmetry[The main explanation for this statement will become manifest in Part III of our treatment.]. In particular, we emphasise the property the moment map and higher homologies need to satisfy to ensure agreement in between the two resolutions. Consequently, this is mapped to an equivalence in between the Hilbert series resulting from summing over characters.

* We then turn to a brief overview of Higgs and Coulomb branches[Simply denoted by HB and CB for convenience.] of quiver gauge theories as algebraic varieties, pointing out the role of magnetic quivers (MQs) for describing the HB of a quiver gauge theory with 8 supercharges and arbitrary flavour, [210, 184, 185, 186, 187, 211]. In doing so, we highlight the importance of the role played by, both, the Hilbert series (HS) and highest weight generating function (HWG) in identifying different intersecting cones in the HB Hasse diagram, [210, 184, 185, 186, 187, 211], suggesting a 2-categorical structure.

In section <ref> we explain how gauging can be related to the poset ordering leading to the construction of Hasse diagrams, thanks to the unifying role of the moment map. We then explain how the identification of such moment map ensures the quiver gauge theory enjoys a generalised notion of homological mirror symmetry, with the latter corresponding to the presence of a Drinfeld center and a corresponding fiber functor for a 2-categorical structure, related to Rozansky-Witten theory, [329]. We conclude the section highlighting connections between the topics outlined in the present work and those of [181, 182], thereby opening the scene to the more mathematical treatment to which Part III is devoted.

As we shall see, the underlying motivation for building the connections explained in section <ref> is the importance of identifying a unifying framework for realising mirror symmetry and its generalisations.

§.§ Intersecting cones

From the more mathematically-inclined overview of section <ref>, we know how to obtain a configuration admitting homological mirror symmetry, and the conditions that must be satisfied for the Hilbert series to match on the GIT and derived representation scheme side. For the purpose of what we will be addressing in section <ref>, we will now overview a setup where homological mirror symmetry can be described in an interestingly generalised and controlled manner, suggesting this as a probe for investigating its underlying mathematical structure.

Moduli space with Coulomb branch (CB), ${\cal M}_{_C}$, Higgs branch (HB), ${\cal M}_{_H}\equiv{\cal M}_{_B}\cup{\cal M}_{_M}$, with its nontrivially intersecting mesonic and baryonic branch, denoted by ${\cal M}_{_B}$ and ${\cal M}_{_M}$, respectively.

§.§ Higgs and Coulomb branches as algebraic varieties

The main focus of this section are quiver gauge theories with 8 supercharges, classical flavour and gauge groups, whose vacuum moduli space is related to the nilpotent orbits, ${\cal O}_{_{\rho}}$, of classical Lie algebras, $\mathfrak{g}$. For a given gauge group $G$, the nilpotent orbits of an algebra $\mathfrak{g}$ are defined by equivalence classes of nilpotence conditions on representation matrices, [188]. From the theoretical physics point of view, the nilpotent conditions describe the way in which the scalar fields in the $F$-term equations vanish at the  SUSY vacuum, and can be specified by a quiver gauge theory.

According to the Jacobson Morozov Theorem, these nilpotent orbits are
in 1 to 1 correspondence with equivalence classes of embeddings of $\mathfrak{su}(2)$ into $\mathfrak{g}$. Each such embedding

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

is a homomorphism, labelled, by, either, the partition of the representation of $G$, or by a characteristic, using Dynking labels to specify the mapping of the roots and weights of $\mathfrak{g}$ onto $\mathfrak{su}(2)$.

A Slodowy slice, $S_{_{\rho}}\overset{def.}{\equiv}{\cal O}_{_{\rho}}^{^{\perp}}$, is a space transverse to a nilpotent orbit, and therefore commuting with it, while living within the adjoint orbit of the ambient group $G$. These transverse spaces can be further restricted to their intersections with the closure of any enclosing nilpotent orbit $\overline{\cal O}_{_{\sigma}}$, thereby leading to spaces labelled by pairs of nilpotent orbits, whose elements are Slodowy interesections

\begin{equation}
S_{_{{\sigma,\rho}}}\ \overset{def.}{\equiv}\ \overline{\cal O}_{_{\sigma}}\ \cap\ S_{_{\rho}}
\label{eq:SlInt}
\end{equation}

encoding, as particular examples, nilpotent orbits, Slodowy slices and Kraft-Procesi transitions[The connection between the 3D boundary conditions on type-II brane systems in 4D ${\cal N}=4$  CFTs and Slodowy intersections was highlighted in [114].].

(<ref>) are the algebraic varieties of which a Hilbert series can be calculated[Hilbert series for Slodowy intersections can also be constructed by means of purely group theoretic methods, making use of localisation formulae related to the Hall Littlewood polynomials.], as we will explain later on. Prior to turning to that we outline some preliminary definitions that will be recursively used throughout the remainder of this work.

The nilcone is defined as

\begin{equation}
\overline{\cal O}_{_{max}}\ \overset{def.}{\equiv}\  {\cal N}.
\end{equation}

with dimension

\begin{equation}
\left|{\cal N}\right|\ \equiv\ \left|\mathfrak{g}\right|-\text{rank}\left[\mathfrak{g}\right].
\end{equation}

Nilpotent orbits can be arranged as a Hasse diagram according to the inclusion relations of their closures:

\begin{equation}
{\cal N}\ \equiv\ \overline{\cal O}_{_{max}}\ \supset\  \overline{\cal O}_{_{sub-reg}}\ \supset\ ...\ \overline{\cal O}_{_{min}}\ \supset\ \overline{\cal O}_{_{trivial}}\ \equiv\ \{0\}.
\end{equation}

From the definition of the Slodowy intersection, it follows that

\begin{equation}
S_{_{{{\cal N},trivial}}}\ \equiv\ \overline{\cal O}_{_{max}}   \equiv\ {\cal N}
\end{equation}

By means of such terminology, the definition of the Higgs and Coulomb branches can be expressed as follows

\begin{equation}
\overline{\cal O}_{_{max}}\   \overset{def}{\equiv}\ \text{HB}\left[{\cal M}_{_A}(\rho,0)\right]
\ \ \ , \ \ \
S_{_{{{\cal N},\rho^{^T}}}}\   \overset{def}{\equiv}\ \text{CB}\left[{\cal M}_{_A}(\rho,0)\right]
\ \ \ , \ \ \
{\cal M}_{_A}(\rho,0)\   \overset{def}{\equiv}\ {\cal L}_{_{A}}\left(\rho^{^T}\right)
\end{equation}

with ${\cal M}_{_A}(\rho,0)$ denoting a single-flavoured linear quiver.

§.§.§ Magnetic quivers and Hasse diagrams

Having said this, we now briefly overview the prescription for constructing magnetic quivers and Hasse diagrams from quiver subtraction, [189][Such terminology is explained in due course. We refer to the extensive literature, especially the one in the references, for more detailed explanations as well as several examples explicitly analysed.]. The procedure can be summarised by the following steps:

* Start from a certain quiver gauge theory with 8 supercharges. Focussing on SQCD theories, namely a 3-parameter family of theories labelled by $(N_{_c}, N_{_f}, k)$, denoting number of colours, flavours, and Chern-Simons level, respectively. As a framed Nakajima quiver variety, we assign to it an electric quiver of the kind <ref>. Its HB is an algebraic variety associated to a 5-brane web configuration.

* Identify the number of maximal decompositions of the 5-brane web.

* To each maximal decomposition associate the magnetic quiver (MQ) of a 3D ${\cal N}=4$ gauge theory.

* The HB of the SQCD theory is preserved under dimensional reduction, and is equivalent to the union of the CBs of the MQs identified in step 3.

* Construct the Hasse diagram associated to the HB by implementing quiver subtraction[We are assuming familiarity with the notion of ADE quivers. We refer to the extended literature on the topic if needed.] on the MQs.

Most importantly, this procedure generalises the following relation:

\begin{equation}
\boxed{\ \ \ \ \text{HB}\left(T\right)\ \equiv\ \text{CB}\left(T^{^{\text{V}}}\right)\color{white}\bigg]\ \ },
\label{eq:mirrordual}
\end{equation}

with $T$ a 3D ${\cal N}=4$ quiver gauge theory, and $T^{^{\text{V}}}$ its mirror dual. (<ref>) can also be re-expressed in terms of the electric and magnetic quivers

\begin{equation}
\boxed{\ \ \ \ \text{HB}\left(\text{EQ}\right)\ \equiv\ \text{CB}\left(\text{MQ}\right)\color{white}\bigg]\ \ }.
\label{eq:mirrordualquivers}
\end{equation}

The advantage of the procedure put forward by [210, 184, 185, 186, 187, 211] is that, when $T$ does not admit a unique $T^{^{\text{V}}}$, its generalised mirror symmetric theory can still be recovered by specifying the MQs, or, equivalently, the generators of the cones featuring in the Hasse diagram, together with their intersections. For concreteness, we briefly overview some key examples of the procedure summarised above, encompassing the main features we will be needing to consider in section <ref>.

§.§.§ Rank-1 and rank-2 examples

The examples reproduced in this subsection are for 5D ${\cal N}=1$ SQCD, with varying parameters $(N_{_c},N_{_f},k)$. Notice that, for $SU(2)$, $k=0$, reason why it is conventionally omitted in the quiver representation. However, for $N_{_c}>2$ it might be $k\neq0$, reason why in such case we specify this in the corresponding quiver.

For each case we draw brane intersections (NS5s-branes and (p,q)-branes are always present). When adding flavour, D5-branes also contribute to the intersection. On the right of each brane intersection, we show the Hasse diagram constructed by implementing quiver subtraction on the magnetic quiver. Conventionally, the node at the bottom of the Hasse diagram denotes the original magnetic quiver, prior to implementing any quiver subtraction. At each step of the iteration, we draw a vertical line, ending at another node, which therefore denotes a reduced magnetic quiver. When a bifurcation takes place, it signals that there are multiple possible quiver subtractions that could be performed. This corresponds to the emergence of multiple cones in the Hasse diagram. In case no bifurcations are present, it means the cone is unique.

§.§ 5D  SU(N

\begin{equation}
\begin{aligned}
&\vcenter{\hbox{\includegraphics[scale=0.8]{nf=0.pdf} }}\\
\end{aligned}
\qquad\qquad\qquad\qquad\qquad
\begin{aligned}
\end{aligned}
\label{}
\nonumber
\end{equation}

In absence of flavour degrees of freedom, the 5-braneweb decomposition is unique (LHS) leading to a HB Hasse diagram characterised by a unique cone (RHS).

\begin{equation}
\text{HB}_{_{\infty}}^ {\ \text{5D}}\ \left(\ \raisebox{-30pt}{\includegraphics[ scale=0.75]{elq1}}\ \right) \ \equiv\ \mathbb{C^{^3}}/\mathbb{Z}_{_{N_{_c}}}\ \equiv \ \text{CB}^{\ \text{3D}}\ \left(\ \raisebox{-15pt}{\includegraphics[scale=0.75]{mq}}\ \right)
\end{equation}

§.§ 5D   SU(2),  N

5-braneweb with added flavour (LHS) and HB Hasse diagram with two cones.

Adding flavour, there are two possible decompositions of the braneweb. Each decomposition is mapped to a different magnetic quiver. The union of their CBs is equivalent to the HB of the original 5D quiver gauge theory we started from. The intersection of the two cones is given by a single symplectic leaf.

\begin{equation}
\text{HB}_{_{\infty}}^ {\ \text{5D}}\ \left(\ \raisebox{-30pt}{\includegraphics[scale=0.75]{elq2}}\ \right) \ \equiv \ \text{CB}^{\ \text{3D}}\ \left(\ \raisebox{-23pt}{\includegraphics[scale=0.6]{three}}\ \right) \ \bigcup\ \text{CB}^{\ \text{3D}}\ \left(\ \raisebox{-10pt}{\includegraphics[scale=0.6]{11}}\ \right)
\end{equation}

§.§ 5D   SU(2),  N

\begin{equation}
\begin{aligned}
\end{aligned}
\qquad\qquad\qquad\qquad
\begin{aligned}
\includegraphics[scale=1]{d5.pdf}  }}
\end{aligned}
\nonumber
\end{equation}

5-braneweb and corresponding Hasse diagram for 5D $SU(2)$, $N_{_f}=4$.

\begin{equation}
\text{HB}_{_{\infty}}^ {\ \text{5D}}\ \left(\ \raisebox{-30pt}{\includegraphics[scale=0.75]{elq3}}\ \right) \ \equiv\ \text{CB}^{\ \text{3D}}\ \left(\ \raisebox{-25pt}{\includegraphics[scale=0.6]{mq1}}\ \right)
\end{equation}

§.§ 5D   SU(3)

\begin{equation}
\begin{aligned}
&\vcenter{\hbox{\includegraphics[scale=0.6]{su3n6k0.pdf} }}\\
\end{aligned}
\qquad\qquad
\begin{aligned}
&\vcenter{\hbox{\includegraphics[scale=0.8]{su3nf3k0.pdf}  }}\\
\end{aligned}
\nonumber
\end{equation}

5-braneweb for 5D $SU(3)_0$, $N_{_f}=6$. The cone is unique since $k=0$.

\begin{equation}
\text{HB}_{_{\infty}}^ {\ \text{5D}}\ \left(\ \raisebox{-30pt}{\includegraphics[scale=0.75]{elq4}}\ \right) \ \equiv\ \text{CB}^{\ \text{3D}}\ \left(\ \raisebox{-25pt}{\includegraphics[scale=0.55]{onecone}}\ \right)
\end{equation}

§.§ 5D   SU(6),  N

Adding a CS level $k\neq0$ leads the coexistence of two maximal decompositions of the 5-braneweb. The corresponding Hasse diagram is therefore constituted by 2 intersecting cones. The intersection in this case goes along the vertical direction where the blue and red lines are parallel to each other.

\begin{equation}
\begin{aligned}
\text{HB}_{_{\infty}}^ {\ \text{5D}}\ \left(\ \raisebox{-30pt}{\includegraphics[scale=0.75]{elq5}}\ \right) \ &\equiv \ \text{CB}^{\ \text{3D}}\ \left(\ \raisebox{-25pt}{\includegraphics[scale=0.35]{mq5}}\ \right) \ \bigcup\
\text{CB}^{\ \text{3D}}\ \left(\ \raisebox{-20pt}{\includegraphics[scale=0.35]{mq4}}\ \right)
\end{aligned}
\end{equation}

From the point of view of the HB of quiver gauge theories with 8 supercharges in 5, 4, and 3 dimensions share a similar structure. In particular, the ramification of the HB due to the presence of multiple cones is preserved when considering the derived category of BPS objects in the lower-dimensional theory.

§.§ The 2-categorical structure of complete intersections

As previously mentioned, there are quiver gauge theories with 8 supercharges that do not admit a mirror symmetric 3D ${\cal N}=4$ theory description. The reason for this can be recast to the fact that its HB is not a single kyperk$\ddot{\text{a}}$ler cone, but, rather, the union of two, with nontrivial intersection, such as the cases shown in figure <ref> and <ref>.

For classical gauge groups $G\equiv U(N), Sp(N)$ or $USp(2N), SO(2N)$, the moduli space is a complete intersection, meaning it is an algebraic variety whose dimension, $d$, is given by the following expression

\begin{equation}
d\ \overset{def.}{=}\ g-r,
\end{equation}

where $g, r$ respectively denote the generators and relations between them.

For the case in which a 3D ${\cal N}=4$ gauge theory, $T$, with given gauge group and suitable choice of matter content, admits a mirror dual theory, $T^{^{\text{V}}}$, the Hilbert series evaluated on the Coulomb branch (CB) of the former equals that of the Higgs branch (HB) of the latter,

\begin{equation}
\text{HS}\left[\text{CB}\  \left(T\right)\right]\ \equiv\ \text{HS}\left[\text{HB} \ \left(T^{^{\text{V}}}\right)\right],
\label{eq:perfectmirror}
\end{equation}

\begin{equation}
\text{HB}\left(\text{EQ}\right)\ \equiv\ \text{CB}\left(\text{MQ}\right).
\label{eq:pm1}
\end{equation}

Perfect realisation of ordinary mirror symmetry in (<ref>) follows from the HB and CB of the dual theory both being hyperk$\ddot{\text{a}}$hler quotients. Recent developments in terms of magnetic quivers and Hasse diagrams highlighted the fact that setups where (<ref>) effectively takes place involve Hasse diagrams featuring a single cone, such as in figure <ref> and <ref>. This feature naturally emerges from the HS calculation in the sense that (<ref>) would only account for the contribution from the generators of the unique cone involved in the construction of the diagram. On the other hand, for theories whose Hasse diagram is the union of two intersecting cones, (<ref>) splits into terms associated to the generators of each individual cone, together with additional subtractive terms accounting for the nontrivial intersection in between them.

Extended calculation of HS for arbitrary gauge theories with 8 supercharges has been the main focus of several recent works, mostly [186, 187, 211]. As shown in [187], there are some cases in which one could trade a cone for an intersection, thereby indicating that the generators and the relations in between them encode the same information. This turns out to be of particular importance for our purposes, in particular in building the correspondence[A more detailed explanation of this will be provided in the next section, when comparing with [181].] with the Drinfeld center[As stated in the introduction, the focus of [3] was outlining the main proposal of connecting Drinfeld centers and magnetic quivers from a theoretical point of view, while reserving a more formal mathematical treatment of the same correspondence to followup work by the same author, [4, 5].].

Within the context of quiver gauge theories with 8 supercharges, the Hilbert series (HS) is a partition function counting chiral gauge-invariant operators, encoding the variety of vacua generated by such operators. The highest weight generating function (HWG), encodes the same information in a more succinct way, that is more useful to be dealing with for cases involving higher rank, [211].

The general expression reads

\begin{equation}
\begin{aligned}
\text{HS}_{_{N_{_{f}}}}(t;x_{_1},,...x_{_k})\ &\equiv\ \underset{j}{\sum}\ f_{_j}(x_{_1},,...x_{_k})\ t^{^j},
\end{aligned}
\label{eq:HS1}
\end{equation}

with $t$ denoting a fugacity for the highest weight of the $SU(2)_{_R}$ $R$-symmetry group providing the grading for the ring of functions, while $f_{_i}(x_{_1},,...x_{_k})$ are sums of characters for irreducible representations of the global symmetry group. From this follows the equivalence with (<ref>) and (<ref>). In the notation of [211], this reads

\begin{equation}
\begin{aligned}
\text{HS}_{_{N_{_{f}}}}(t;x_{_1},,...x_{_k})\ &\equiv\ \int d\mu_{_{G}}\ \text{HS}\left(\frac{\mathbb{C}[{\cal Q}, \tilde{\cal Q}]}{\sqrt{\text{$F$-terms}\ }}\right) \ \equiv \text{HS}\left(\frac{\mathbb{C}[M, {\cal Q}, \tilde{\cal Q}]}{\sqrt{\text{Rel.s}\ }}\right)
\end{aligned}
\label{eq:HS}
\end{equation}

with $M, {\cal Q}$ denoting mesons and baryons, respectively.

For the case of 4D ${\cal N}=2$ SQCD with gauge group $G=SU(N_{_c})$ and $N_{_f}$ hypermultiplets in the fundamental representation of $SU(N_{_c})$, the calculation of <ref> requires:

* solving the $F$-term equations, setting the derivatives of the superpotential to zero

* identifying the gauge-invariant operators.

For ${\cal N}=2$ SU(2) $N_{_f}=2$, the HS is a sum of rational functions, [211],

\begin{equation}
\begin{aligned}
\text{HS}(t;x_{_1},x_{_2})\ &\equiv\ \int d\mu_{_{SU(2)}}\ F(t,z,x_{_1},x_{_2})\\
\ &\equiv\ \text{HS}(\mathbb{C}^{^2}/\mathbb{Z}_{_2};t,x_{_1})+\text{HS}(\mathbb{C}^{^2}/\mathbb{Z}_{_2};t,x_{_2})-1
\label{eq:hs3}
\end{aligned}
\end{equation}

with the first two terms corresponding to the two cones and the last term denoting the intersection in between them at the origin, as shown in figure <ref>. When expanding it in terms of powers of $t$, we get the Plethystic logarithm (PL),

\begin{equation}
\begin{aligned}
\text{PL}(t;x_{_1},x_{_2})\ &\equiv\ ([2;0]+[0;2])t^{^2}-([2;2]+2[0;0])t^{^4}+...
\label{eq:hs4}
\end{aligned}
\end{equation}

with $[;]$ denoting the characters of the corresponding representations of $SO(4)$.
The character corresponding to a certain representation can be encoded in the corresponding Dynkin label. We can therefore choose a set of fugacities to keep track of them, enabling us to rewrite (<ref>) and (<ref>) as a highest weight generating function

\begin{equation}
\text{HWG}\left(t;\mu_{_1},\mu_{_2}\right)\ \equiv\ \text{PE}\left[\left(\mu_{_1}^{^2}+\mu_{_2}^{^2}\right)t^{^2}-\mu_{_1}^{^2}\mu_{_2}^{^2}t^{^4}\right],
\end{equation}

with the first two terms corresponding to the generators of the two cones, and the last (subtractive) term to their intersection in the Hasse diagram.

§ KEY POINTS

The main message of this section can be summarised as follows:

* Quiver gauge theories with 8 supercharges are characterised by a structure fully specified by knowing the generators and intersections of the cones in their HB Hasse diagram. The latter is obtained by implementing quiver subtraction on MQs associated to maximal decompositions of the 5-brane webs associated to the electric quiver. As such, it admits a 2-categorical structure description.

* The moment map plays a key role in ensuring the identification of the underlying MQs. The flatness condition it is required to satisfy defines the HS, hence the HWG function, [330]. Combining the works of [210, 184, 185, 186, 187, 211], this constitutes a sample realisation of homological mirror symmetry.

* In presence of more than one cone in the Hasse diagram, the identification of the MQs whose CBs' union equals the HB of the original theory, is a generalised statement of homological mirror symmetry.

We wish to emphasise that, while most of the above statements have been individually addressed in the literature, to the best of our knowledge, their combined application towards identifying the Drinfeld center in connection to CBs constitutes an original approach, to which Part <ref> is devoted.

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