The picture on the left is at a fixed time slice. The plane is where the CFT lives and the bulk is AdS, with $z$ realised holographically. The subregion of bulk spacetime lying between $A$ and $\Gamma_{A}$ defines the entanglement wedge of $A$. Global dS$_{_{D}}$ for $D>2$ (on the RHS). An observer at $r=0$ can only access degrees of freedom within the cosmological horizon $r=1/H$.

Equipped with this insight, we now turn to the case of the cosmological horizon, the main point of this digression being that of outlining the similarity between the unitarity issues in global dS and black hole evaporation in asymptotically flat spacetimes.

The cosmological horizon is placed at a finite distance[Each point along the red line in figure <ref> is a codimension-2 surface of radius $r_{h}$.], $r_{h}=1/H$, with respect to an observer located at the origin of the static patch, thereby implying that there is only a finite amount of spacetime that she/he can be in causal contact with. Because of this, one might be led to conclude that the area of the cosmological horizon should somehow be related to the entropy of the static patch itself, as the Holographic Principle would suggest. However, it is also known that the dS Hilbert space is infinite-dimensional, [272]. The reason for this follows from the fact that the dS isometry group is noncompact, and therefore has no finite dimensional unitary representations. This provides one of the main obstacles towards extending the holographic dictionary of [192] to the case of a positive cosmological constant.

The configuration just described seems to share similar unitarity issues as encountered for the case of the evaporating black hole. Assuming the horizon surrounding the static patch arises from partial tracing over the global spacetime, c.f. figure <ref>,  when taking $\underset{\Lambda\rightarrow 0}{\lim}$, the cosmological horizon will correspondingly be pushed towards ${\cal I}^{+}$, at which the entropy should vanish. Intuitively, this is in agreement with the partial-tracing argument outlined above, since, as the cosmological horizon is pushed further away from the observer in the static patch, the degrees of freedom which were previously “hidden” (i.e. traced-over), are expected to re-enter the horizon, thereby becoming accessible to the observer. However, this falls short from being true given the divergence of the entropy defined by the area-law, which, e.g., in 4D scales as $\sim{\cal O}\left(\frac{1}{H^2}\right)$. The area of the codimension-2 surface located at such distance can be interpreted by holographic arguments as the entropy of the static patch, which, in the von Neumann formulation, follows from having traced over the spacetime beyond the cosmological horizon, in analogy with the black hole picture. As such, it is not a QES.

Given the monotonic behaviour of, both, the background action for dS$\rightarrow$dS transitions (as found in [7]) and the von Neumann entropy, we suggest that the divergence issue arising from the flat spacetime limit of de Sitter might be solved by introducing a more suitable definition of the de Sitter entropy, in a similar fashion as $S_{gen}$ enables to recover unitarity within the context of black hole evaporation. As anticipated at the beginning of this section, this identification turns out to be possible[Interesting new developments towards realising a dS$_4$/CFT$_3$ correspondence have recently been presented in [538], where the interpretation of the maximal entanglement between the north and south pole in global dS shares the same motivation as the argument presented in our work.].

§.§ Holographic embedding of vacuum transitions

Here we present our proposal for the holographic description of the vacuum transitions. As highlighted throughout the previous sections, the three methods used for deriving transition amplitudes exhibit unexpected behaviours. In the present section, we wish to provide a unifying holographic framework describing all processes, mostly motivated by the fact that:

* The action for the system of 2D spacetimes joined by a wall is analogous to the JT-gravity coupled to a CFT$_2$ setup adopted to formulate the island proposal within the context of the information paradox.

* The bulk-brane-boundary description of the same phenomenon in the KR/HM setup accounts for complementary features of the same underlying process.

It is indeed in the spirit of the BCFT formulation of the information paradox, that we propose a holographic embedding of the transitions described analytically in the first three sections of the present work, after having implemented suitable adaptations due to the specifics of the configurations being analysed.

The LHS shows a generalisation of figure <ref> for vacuum transitions. A deformation drives the RG-flow on the boundary, assuming $c_{bulk}<<c_{bdy}$ at both endpoints. In the IR of the RG-flow (as shown on the RHS), the vacuum transition takes place in between subregions of different $AdS_{_{D}}$'s, each one lying on a different ETW. More details will be outlined in the concluding part of this section.

Figure <ref> shows our proposal for the holographic embedding of 2D vacuum transitions:

* Given that the process involves two different spacetimes, each one being associated to a different JT-action, the holographic embedding of the transition can be described by two ETW branes (each one corresponding to one of the two vauca, $ETW_{_{1,2}}$) with the brane separating them (mediating the decay process) being geometrically realised by a composite CFT (denoted by $\overline{CFT}_{_{A}}\otimes CFT_{_{B}}$ in figure <ref>), which can be achieved by performing the folding trick.

* The radiation region lies on the conformal boundary (denoted by the tensor product of CFTs arising from the folding trick), which in turn is brought at a finite cutoff, $\epsilon_{_{IR}}$, with nontrivial BCs, the latter denoted by $BCFT_{_{D-1}}, BCFT_{_{D-1}}^{\prime}$. The theory living on the gravitating bath interpolates between the values of the $D$-dimensional cosmological constants. As such, the wall plays the role of the entangling surface in between the spacetimes involved, with the entanglement being internal.

* Attached to the radiation region are the two ETW branes. Each one of them accommodates one of the two spacetimes, (A)dS$_{_{D}}$.

* The composite CFT on the gravitating bath (depicted on the LHS of figure <ref>) is ultimately driven by an irrelevant operator to the configuration on the RHS of figure <ref>, which is reminiscent of the wedge holographic construction, [225].

* As explained at multiple stages throughout our treatment, the total bounce and action for a given vacuum transition, result once having suitably accounted for background subtraction. As we shall see, it is particularly instructive to assign a corresponding holographic realisation of the background configuration as well. Our proposal is represented in figure <ref> on the RHS. We now turn to explaining it in more detail, albeit most arguments follow through from the ones outlined when explaining the RHS of figure <ref>. The background action is a pure CFT. As such, it can be equally described independently of the choice of the UV cutoff. Hence, starting from a pure AdS$_{_{D+1}}$/CFT$_{_{D}}$ setup, with the conformal boundary defined on the whole real line at $z=0$, we can perform a suitable conformal transformation enabling to restrict the boundary domain to an interval with BCs dual to two ETW branes characterised by the same $D$-dimensional cosmological constant. Given that the theory is still conformal, the width of the segment can be arbitrary; for the purpose of interest to us, we need to bring the conformal boundary at the same IR cutoff as the two-spacetime joined system on the RHS of figure <ref>, such that the background subtraction exactly overlaps with the instanton.

The background configuration, with respect to which the decay rate is normalised, is drawn on the right. From the boundary theory, CFT$_{D-1}$, this corresponds to the deconfined phase of the theory.

§.§.§ Concrete proposal for the vacuum transition from the CFT side

If there is a holographic dual formulation of vacuum transitions, a natural question to ask is: what is the corresponding physical effect that happens on the CFT side  that describes the vacuum transition in the bulk. Here we proposed that on the CFT side, the vacuum transition corresponds to a field theoretical phase transition such as a confinement/de-confinement phase transition. For this we need to identify the dual of the background spacetime and the dual of the final composite spacetime. For concreteness let us concentrate on a bulk AdS.
As we described before the vacuum transition  corresponds not to the decay of a full AdS but a portion of an AdS. For the dual we may represent this in terms of a double-well scalar potential that originally has two vacua related by $\phi \rightarrow -\phi$ separated by a domain wall. Modding out by the $\phi \rightarrow -\phi$ symmetry this reduces the bulk spacetime to a portion  with a boundary. This can provide a description of the dual of the original AdS. The description of the final spacetime can be seen in this way as a similar scalar potential but with the two minima being non-degenerate corresponding to two different spacetimes joined by the boundary. The question is how are the theories in the two boundaries related.

In figure <ref>  the background  can be thought of as being given by a degenerate double well potential, with the two vacua identified. In their holographic realisation in terms of ETW branes (as shown on the RHS of figure <ref>), only one of them will be undergoing the transition, leading to the configuration depicted on the RHS of figure <ref>.

The interpolation between the deconfined and the confined phase of the theory living on the boundary can be realised by adding to the deconfined phase corresponding to figure <ref>, a  deformation, leading to degeneracy breaking between the two vacua, as shown on the RHS of figure <ref>. Given that our analysis focuses on 2D, $T\bar T$-deformations will turn out being the relevant ones.

The reason why we can effectively interpret vacuum transitions in terms of a deconfinement/confinement transition can be understood making use of the  analysis carried out in [539] and [540]. In the former, it is shown that the difference between the two phases is dictated by the different dependence of the entanglement entropy on $N$, which is the key parameter in the holographic dictionary[Indeed, $N$ is related to the central charge and the AdS radius.]. In particular, following the RT prescription, this amounts to a diffference in the behaviour of the expectation value of the Wilson line connecting two arbitrary points on the conformal boundary: for the deconfined phase, $S_{_{EE}}\sim {\cal O}(N^{^2})$, wheras for the confined phase, $S_{_{EE}}\sim$ const. What this practically means is that, the former is actually associated to a conformal theory at $z=0$, for which the Wilson line diverges unless a UV-cutoff is introduced. Bulk reconstruction, in such setup, allows to recover the entire (infinite) AdS$_{_{D+1}}$ bulk volume upon removing the cutoff. On the other hand, the confined phase is characterised by a constant value of the entanglement entropy with respect to $N$, meaning that bulk reconstruction can effectively account for a fixed AdS$_{_{D+1}}$ bulk volume. In figure <ref>, this corresponds to having brought the conformal bundary at $\epsilon_{_{IR}}$, resulting only in a finite (A)dS$_{_{D}}$ volume being involved in the process.

In a similar, and somewhat complementary fashion, the CFT description of the processes dealt with in our treatment find realisation in a recent work [540], where scalar potentials of the kind depicted in figure <ref> have been identified with degenerate and non-degenerate vacua in studies of 2d QCD. In [540] the degenerate case can be identified with a deconfinement phase, whereas the non-degenerate case with a confined phase[See also [239] for an interesting discussion of confinement and cosmology.]. In their treatment, the expectation value of the Wilson loop separating different vacua exhibits, either, an area- or perimeter-law like behaviour according to, whether, the system is in the confined or deconfined phase, respectively. More efficiently, such classification can be further performed by analysing the ratio of partition functions of the gauged and ungauged theory in the infinite volume[Namely the large-$N$ limit.] limit. If such ratio is order unity, it means the system is deconfined. On the other hand, if the ratio vanishes it is confined.

Under the action of an irrelevant operator, the boundary theory can be smoothly interpolated from the deconfined (left) to the confined phase (right). The latter is the field theory realisation of the RHS of figure <ref>. In such description, the interpolating scalar, $\phi$, plays the role of a monopole field in a deconfinement/confinement transition, in a similar fashion as in a previous work by Komargodski et al.

In drawing a comparison between the processes dealt with in the present work and those of Komargodski et al., it is important to notice that, in such reference, their analysis embraces, both, vacua and universes. While often used interchangeably, the main difference between vacua and universes is that the latter are separated by infinite barriers, whereas the former might allow finite tension domain walls interpolating between different superselection sectors of the same universe, and are therefore the ones relevant for our treatment. The systems analysed in Komargodski et al.'s work exhibit multiple vacua, distributed between different universes, and study confinement and deconfinement of Wilson lines interpolating between them. Their counterpart for transitions of the kind associated to the RHS of figure <ref>,  are therefore those involving confined Wilson lines interpolating between vacua belonging to the same universe.

Clearly more work needs to be done in order to fully describe the vacuum transitions from the CFT side.

§.§.§ Local transitions,  entanglement and islands in wedge holography

As a first concrete application of the tools outlined in the first part of this section, we now turn to the holographic interpretation of the FMP results obtained in sections <ref> and <ref>. In particular, we prove that:

* Under suitable parametric redefinition, the results obtained by means of the FMP method are found to agree with the expression provided by [262] for describing mutual approximation of boundary states belonging to different CFTs.

* The corresponding expression for the transition rate in presence of gravity, and in absence of black holes, is given by the difference of entropies of $T\bar T$-deformed CFTs, hence proving the locality of the nucleation process. Given its interpretation as being an internal entanglement, such transitions provide an example of an AdS$_2$/CFT$_1\subset$ AdS$_3$/CFT$_2$. [The lower-dimensional holographic setup involved in these processes shares similar features as to the one involving spacetimes emerging from matrix QMs, as analysed, e.g. in [284], and their higher dimensional counterparts in [285].]

* Upon adding black holes, instead, the total action can be expressed as the difference of generalised entropies, with an island emerging beyond a critical value of the black hole mass. As such, this is an example of an  AdS$_2$/CFT$_1\not\subset$ AdS$_3$/CFT$_2$.

§.§ ICFTs and wedge holography

In this subsection, we prove that quantum transitions involving different subregions of spacetime can be described by means of dual CFTs interacting via an interface, [262]. The authors of the latter describe mutual approximation of boundary states, $\Psi_{_{A,B}}$, belonging to the Hilbert spaces of different CFTs, (as shown in figure <ref>).

The correspondence adopted in [262], specifying to the case of AdS$_3$/CFT$_2$, relates the following set of parameters on either side of the interface

\begin{equation}
\boxed{\ \ \ L_{1}\ ,\ L_{2}\ ,\ \kappa
\ \ \
\longleftrightarrow
\ \ \
c_{_{A}}\ ,\ c_{_{B}},\ c_{bdy}\sim\ln g,\color{white} \textcolor{white}{\Biggl[} \color{black} \ \ }
\end{equation}

with $L_{1,2}$ being the AdS radii, $\kappa$ the tension of the wall separating the bulk spacetimes, $c_{_{A,B}}$ the central charges, and $\ln g$ denoting the entropy of the ICFT, $S_{ICFT}$, with $g$ defining the degeneracy of the ground state.

Replacing a small portion of the background CFT$_{_{A}}$ with a different theory, CFT$_{_{B}}$ implies a change in the energy spectrum. Because of this, for a given boundary state, the corresponding $S_{_{EE}}$ leads to a different effective bulk-reconstruction of the AdS dual.

Our first finding is that AdS$_2\rightarrow$AdS$_2$ transitions are equivalent to the case where the mutual approximation described in [262] only involves the ground states. Indeed, under suitable parametric redefinition, the action for the transition coincides with the one obtained by [262], as explicitly shown below

\begin{equation}
\begin{aligned}
S_{tot}^{\ \text{AdS}\rightarrow\text{AdS}}
\frac{\eta}{G}\left[\ {\mu_{-}}\ln\bigg|\frac{{\mu_{-}}A_{1}+1}{{\mu_{-}}A_{1}-1}\bigg|-\mu_{+}\ln\bigg|\frac{{\mu_{+}}A_{2}+1}{{\mu_{+}}A_{2}-1}\bigg|\ \right]
\nonumber \\
& =
\frac{2\eta}{G}\bigg[\left({\mu_{+}}-{\mu_{-}}\right)\tanh^{-1}\left(\frac{\Delta_{-} }{\kappa}\right) -\left({\mu_{+}}+{\mu_{-}}\right)\tanh^{-1}\left(\frac{\kappa}{\Delta_{+}  }\right)
\bigg]  \nonumber \\
2\eta\ S_{_{ICFT}}
\
\
2\eta\ \ln g (\kappa) \ =\
-2\eta\ F_{\partial},
\label{eq:bricft}
\end{aligned}
\end{equation}

where we defined

\begin{equation}
\Delta_{\pm}
\
\overset{\text{def.}}{= }
\
\end{equation}

As shown in [262], the trajectory of the interface separating pure AdS$_{3}$ phases has no turning point (LHS), and, correspondingly, the degeneracy of the ground state is a monotonic function of the tension $\kappa$, as found in section <ref>. No island emerges in absence of conical defects (cf. picture on the RHS). The bulk spacetime is completely determined by means of wedge holography from the $CFT_{_{D-1}}$. $S_{_{EE}}$ is evaluated on the codimension-2 surface at the conformal boundary.

and $g (\kappa)$ denotes the ground state degeneracy and $F_{\partial}$ the boundary free energy, which is a monotonic function of $\kappa$. The entire transition is hence described by means of a codimension-2 theory as depicted in figure <ref>, with the bulk emerging via wedge holography.

Making use of the following parametric redefinitions

\begin{equation}
\mu
\
\overset{def.}{=}
\
\sqrt{\frac{\lambda c}{12}\  }
\ \ \
\ \ \
\phi_{o}
\
\overset{def.}{=}
\
\ \ \
\ \ \
\lambda
\
\overset{def.}{=}
\
2\sqrt{-{\cal C}\Lambda,\ }
\label{eq:2termssttb}
\end{equation}

the total action can be rewritten as

\begin{equation}
\boxed{\ \ S_{tot}^{\ \text{(A)dS}\rightarrow\text{(A)dS}}
\
\
2\pi\eta\left[\ S_{_{T\bar T}}^{^{-}}- S_{_{T\bar T}}^{^{+}}   \ \right]_{_{\text{univ.}}}
\
\
-2\pi\eta\   F_{\partial}
\
2\pi\eta\ S_{_{ICFT}}, \textcolor{white}{\Biggl[} \color{black} \ \ }
\label{eq:chofrel}
\end{equation}

\begin{equation}
\begin{aligned}
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)    \\
S_{_{T\bar T}}
\left(1-\frac{r}{2}\ \frac{\partial  }{\partial r}\right)\ln {\cal Z}_{S^{2}}^{^{\ \ dS,  T\bar T}}
\ =
\
\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)   +\text{const.}   \nonumber\\
S_{\epsilon} + S_{_{\text{univ.}}},
\label{eq:sttds}
\end{aligned}
\end{equation}

for the duals of AdS$_3$ and dS$_3$, respectively, [228, 229]. $S_{_{\text{univ.}}}$ and $S_{\epsilon}$ denote the universal and cutoff-dependent parts of (<ref>), respectively, with

\begin{equation}
\overset{def.}{=}
\frac{c}{3}\ \ln\left|\ \sqrt{\frac{\lambda c}{12 }}\frac{1}{r}\ \right|,
\ \ \ \ \ \
\text{and}\ \ \ \epsilon\overset{def.}{=}\sqrt{\frac{\lambda c}{12 }.\ }
\end{equation}

According to the Casini-Huerta-Myers prescription, the universal part is the one contributing to the definition of the boundary free energy for a BCFT, and therefore this is the only term we need to retain for our purposes. Indeed, $S_{\epsilon}$ can be removed by simply setting $r=\epsilon$, namely choosing the cutoff to be equal to the localisation radius $r$.
From (<ref>), we deduce that:

* This chain of equivalence relations, (<ref>), suggests an interesting direct correspondence between the mutual approximation of CFT states, [262], and $T\bar T$-deformed CFTs.
By virtue of the identification with $S_{gen}$, equation (<ref>) implies the absence of a QES, and therefore of an island, consistently with the fact that the spacetimes involved in the transitions have no event horizons.

* The direct relation between the extremised action and $S_{_{T\bar T}}$ in 2D, ensures the locality of the process being described. However, as also suggested from (<ref>), upon taking the flat limit on either side, the cutoff radius diverges ($r\rightarrow\infty$), and so too does the turning point, meaning the process is forbidden. We therefore conclude that, as long as the parameters of the theory are kept constant, no issue arises, and the transition remains local.

* The cutoff nature of the cosmological horizon suggested by (<ref>), proves that its contribution to $S_{gen}$ should be understood as being part of $S_{_{T\bar T}}$. The motivation that led us to connect this issue with the one encountered in the black hole information paradox, precisely resided in the need to correctly define the dS entropy. Indeed, its divergence resembles that of the monotonic von Neumann entropy in the BH evaporation process, and, holographically, this corresponds to the standard divergence of $S_{_{EE}}$ upon removing the UV-cutoff. Just as in the black hole evaporation process the event horizon is not the QES extremising $S_{gen}$, so too can be claimed for the cosmological horizon, therefore proving the absence of islands in pure de Sitter spacetimes.

* 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. In this section, we prove that these quantities are indeed proportional to each other.

§.§.§ The emergence of the island

In presence of a non-extremal black hole, the total action for AdS$_2\rightarrow$AdSBH$_2$ reads:

\begin{equation}
\begin{aligned}
S_{tot}^{\text{\ AdS$\rightarrow$AdSBH}}
\frac{2\pi\eta}{G}\ \left[\
\frac{{\cal B}}{2\Lambda_{-}}\ \left[1-\sqrt{1-\frac{4{\cal C}\ \Lambda_{-}}{{\cal B}^{2}}}- \ln\left|\frac{y_{2}}{y_{1}}\sqrt{\frac{1-y_{1}^{2}}{1-y_{2}^{2}}}\right| \right]+\  \right.
\nonumber\\
\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| +\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:next2}
\end{aligned}
\end{equation}

where the notation is the same as in section <ref>. (<ref>) coincides with the result by [262] for the case of an excited state of the new CFT approximating the ground state of the background theory under
the following identifications

\begin{equation}
\begin{aligned}
(1-\mu)\Delta t_{2} -\Delta t_{1} \nonumber\\
- \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| ,
\\
(1-\mu) \frac{\beta}{2}
\
\
- \frac{1}{2}\sqrt{\frac{\frac{{\cal B}^{2}}{4\Lambda_{-}}-{\cal C}}{\Lambda_{-}}} ,
\label{eq:shordt}
\end{aligned}
\end{equation}

with the latter, (<ref>),  being the 2$^{nd}$ term featuring in the total action (<ref>). The terms proportional to $\phi_{1,2}$ are $\sim{\cal O}({\cal B})$, and so too are the first and third terms featuring in the first line of the expression for the total action. The results in [262] are in agreement with ours upon expanding the total action with respect to the deformation parameter, ${\cal B}$. As also mentioned by the authors of [262], their expression does not contain terms that vanish upon removing the cutoff, thereby explaining the missing terms in performing the comparison. We also made a consistency check with the results obtained in section <ref>, and found that subleading terms in the l.o. ${\cal B}$-expansion vanish in the $\underset{{\cal B}\rightarrow 0}{\lim}$.

By adding a black hole in the 3D bulk (LHS), instead, a turning point emerges, resulting in the ETW brane closing up and rejoning the conformal boundary. Correspondingly, an island emerges in the wedge holographic picture (RHS). This is the key difference with respect to figure <ref> on the LHS, where there is no bulk turning point, and therefore, the RT surface doesn't intersects the ETW brane.

The integration variables in the descriptions outlined in section <ref> and [262] are related as follows

\begin{equation}
\boxed{\ \ y
\
\overset{\text{def.}}{=}
\
\sqrt{\frac{1-\mu}{\lambda}}\ r. \ \ }
\label{eq:newident1}
\end{equation}

In terms of which, the 3D BTZ geometry, defining an orbifold of the universal cover of the Poincarè metric, can be associated with the line element

\begin{equation}
\
\
-\frac{r^{2}-r_{h}^{2}}{L^{2}}\ dt^{2}+\frac{L^{2}}{r^{2}-r_{h}^{2}}\ dr^{2}+\frac{r^{2}}{L^{2}}\ dx^{2},
\label{eq:BTZM3}
\end{equation}

and becomes 2D conformally flat at the cutoff $r=r_{c}$, where the $T\bar T$-deformed CFT of inverse temperature $\beta\overset{def.}{=}2\pi L^{2}/r_{h}$ is located. Notice that $\underset{r_{h}\rightarrow 0}{\lim}\ \beta=\infty$, as expected for the T=0 theory.
Recall that the variable $y$, featuring in (<ref>), introduced in section <ref>
parametrises the value of the dilaton, $\phi_b$, on the brane interpolating between the 2D vacua involved in the decay process. As previously argued, in the holographic interpretation of the process, the 1D brane is mapped to a bulk CFT$_2$ on which $S_{_{T\bar T}}$ is evaluated.

In the same fashion, one might be led to believe that the role played by the cutoff $r_c$ is analogous to $\phi_b$, since all other parameters in the definition of $y$ are held fixed by the formalism. However, we will now be arguing that extra care is needed in building the correspondence between the 2 setups. For ${\cal B}=0$ , $y=\frac{\phi_b}{\mu}
$, and, due to the absence of horizons, $
\
=1$ is unique. On the other hand, in presence of a black hole, the following identification needs to be made

\begin{equation}
\tanh x
\
\overset{\text{def.}}{=}
\
\ \ \
\Rightarrow
\ \ \
\sinh x
\
\
\frac{r_{h}}{r_{c}}
\
\
\frac{\sqrt{1-y^{2}} }{y}
\ \ \ \Rightarrow\ \ \
\overset{\text{def.}}{= }
\
{r_{h}}\frac{y}{\sqrt{1-y^{2}} }
\ \ \
\overset{y\rightarrow 1}{\longrightarrow }
\ \ \
\infty,
\label{eq:divergcutoff}
\end{equation}

from which the following information can be extracted:

* If ${\cal B}=0$, then $y_{tp}=1$ is unique, and $r_c=\infty$. Correspondingly, (<ref>) reduces to a pure AdS$_3$ metric, and the bulk configuration is the same as the one drawn on the LHS of figure <ref>.

* As proved in section <ref>, $y_{tp}=1$ is also the limiting value of the turning points upon taking the flat limit on either side of the 2D transition.
The last passage in (<ref>) thereby shows that the effect of removing the cutoff in [262] is equivalent to taking $\underset{{\cal B}\rightarrow 0}{\lim}$ given the identification of the turning point $y_{tp}$ with $r_{c}$.

* These observations imply the need to add a black hole in the 3D bulk in order to to keep $r_{c}$ finite, thereby ensuring the ETW brane separating the two phases can actually close up, as shown on the RHS of figure <ref>.

In order to provide further justification for the claims made so far, we will now show that:

* Transition in between pure (A)dS spacetimes described by (<ref>) can also be obtained in presence of a suitable value of ${\cal B}\neq 0$, thereby justifying the identification of the black hole mass with the $T\bar T$-deformation ensuring the locality of the transition taking place[Our finding is hence in agreement with [288], where the authors show that dS black holes behave as localized, constrained, states, and therefore feature a lower entropy with respect to pure dS.].

* For arbitrary values of ${\cal B}$, the brane action undergoes a phase transition when the black hole mass decreases below a certain critical value.

§ HOW THE NUMBER OF TURNING POINTS DEPENDS ON ${\CAL B}$

For particular values of ${\cal B}$, the turning point reduces to

\begin{equation}
\boxed{\ \ \  \phi_{2}
\
\
\begin{cases}
\ 0\ \ \ \ \ \text{for  ${\cal B}=0$ \ \ \ \ \ \ \ \ \ \ \ \ \ $\equiv$ \ \ pure dS, 1 horizon, $y_{2}=0$} \\
\\
\ \mu_{\pm}\ \ \   \text{for ${\cal B}=-2\sqrt{|{\cal C}|\Lambda}$ \ \ $\equiv$ \ \ extremal dSBH, 2 degenerate horizons, $y_{2}=1$} \\
\\
\ \in\ \ ]0,\mu_{\pm}[\   \text{for  $-2\sqrt{|{\cal C}|\Lambda}<{\cal B}<0$  $\equiv$  non-extremal dSBH, 2 horizons, $y_{2}\neq\{0,1\}$.} \\
\end{cases} \textcolor{white}{\Biggl[} \color{black}}
\nonumber
\label{eq:cases1}
\end{equation}

This shows that, in terms of the coordinate $y$, there is no turning point in absence of a conformal symmetry breaking term in ${\cal L}$, thereby forbidding the transition to take place. On the other hand, for ${\cal B}_{\text{dS}}=-2\sqrt{|{\cal C}|\Lambda}$, the turning point is unique, and therefore coincides with the results obtained in section <ref>. For any other value ranging in between, i.e. ${\cal B}\ \in\ ]-2\sqrt{|{\cal C}|\Lambda},0[$, there are 2 physical turning points in terms of $\phi_b$, consistently with the constraint derived in section <ref>, (<ref>).

Having said this, we therefore conclude that:

* The role of ${\cal B}$ is effectively equivalent to that of an irrelevant operator on a CFT$_2$.

* The results obtained in section <ref> should be understood of as arising from the IR limit of the RG-flow of an irrelevant operator, whose imprint in the transition amplitude is given by the finiteness of the turning point at which nucleation takes place.

* The cosmological horizons and the cutoff radius at which localisation takes place are proportional to each other, implying the flat limit is equivalent to removing the cutoff altogether; this explains the divergences encountered in section <ref>.

* According to the value of ${\cal B}$, the nucleation process features a different number of physical turning points. Only in presence of two distinct physical values of the turning points, the flat limit can be taken (as explained in section <ref>).

* In light of further considerations made in the remainder of our work, it is important to notice that, due to the specifics of the formalism, the parameter ${\cal B}$ is not simply part of the theory, but, rather specifies the state involved in the transition. This follows from the Hamiltonian constraint and the definition of the transition amplitude.

§ RELATION TO  AND PHASE TRANSITION OF

The systems described in section <ref> experience a phase transition as the black hole mass decreases, thereby providing further evidence of the fact that, in some cases, 2D vacuum tranistion can only take place in presence of islands.
The actions obtained in sections <ref> and <ref> can be recast to the difference of two generalised entropies, (<ref>),

\begin{equation}
\boxed{\ \ \ \ S_{tot}^{\text{\ \ (A)dS}\rightarrow\text{(A)dS}}
\
\
2\pi\eta\ \left[S_{gen}-S_{gen}\right]
\
\
2\pi\eta\ \left[\frac{\phi_{h,+}}{G}-\frac{\phi_{h,-}}{G}+S_{_{EE,-}}^{^{(A)dS, T\bar T}}-S_{_{EE,+}}^{^{(A)dS, T\bar T}}\right].  \textcolor{white}{\Biggl[} \color{black}  \ \ }
\label{eq:phirho14}
\end{equation}

In agreement with arguments outlined in [264], the island only emerges beyond a certain threshold value of the black hole mass, which, as previously shown, is accounted for by the parameter ${\cal B}$. Each part of the brane action is an entanglement entropy of two disjoined intervals on a  CFT$_2$ with deformation parameter ${\cal B}$. Expanding to leading order in ${\cal B}$, both terms can be brought to the following form

\begin{equation}
\begin{aligned}
S_{_{EE}}^{^{(2)}}\bigg|_{{\cal B}=0}+\delta S_{_{EE}}^{\ s}\nonumber\\
-\frac{8c_-}{3}\ln\left|\frac{1}{\epsilon}\ \sinh\left(\frac{2\pi r_-}{\beta_-}\right)\right|-\left(1-\frac{\phi_{o}^2}{\mu_{-}^2}\right)\frac{{\cal B}c_{-}^2}{9\Lambda_-}\coth^2\left(\frac{2\pi r_-}{\beta_-}\right)+\frac{\frac{c_-}{3}}{\sinh^2\left(\frac{2\pi r_-}{\beta_-}\right)}.
\label{eq:sl}
\end{aligned}
\end{equation}

In the second line of (<ref>), we redefined the parameters featuring in our original expressions in order to make the comparison with the standard CFT calculation more explicit. Notice that this interpretation is fully legitimate, since the parameter ${\cal B}$ specifies the states on either vacuum being involved in the transition process.

A 3D plot of $S_{_{EE}}({\cal B}, x)$. $x$ denotes the portion of the CFT$_2$ on which $S_{_{EE}}$ is evaluated. As ${\cal B}$ changes, so too does the slope of (<ref>), as shown on the LHS. The figure on the RHS shows the smooth interpolation in between the monotonic behaviour of $S_{_{EE}}$ as one would expect if evaluated on the disconnected channel over 2 intervals, to a profile admitting a minimum. The phase transition occurs at ${\cal B}<0$, corresponding to a upper bound to the black hole mass below which an island needs to be taken into account.

In section <ref>, we showed that ${\cal B}<0$ and, from section <ref>, we know that $\phi_{o}^2>\mu_{-}^2$, therefore $y>0$ is the range of interest for the decay process described by (<ref>). As shown on the RHS of figure <ref>, the entanglement entropy, $S_{_{EE}}$, experiences a phase transition, for $y>0$. In particular, the latter takes place at $y=1$. It is possible to argue that this is indeed consistent with our finding in section <ref> that the black hole mass features in the Lagrangian density as a $T\bar T$-deformation, which needs to be constrained between a certain range, to ensure the existence of two turning points, (<ref>).

Indeed, figure <ref> further supports our argument that, the phase tranistion undergone by the brane action corresponds to a Page transition beyond a suitable value of ${\cal B}$, with the latter playing a similar role as $t_{Page}$ within the context of the information loss paradox. Consequently, this justifies the emergence of the island in (<ref>) upon interpreting the total action as being the difference of generalised entropies.

§.§ An emergent holographic embedding of 2D vacuum transitions with gravity

In summary, from the holographic interpretation of the FMP results we proved that:

* The flexibility of the FMP method relies upon the fact that such kinds of exactly integrable deformations ($T\bar T$s) can be accounted for at the level of the Lagrangain density, as explicitly outlined in section <ref>. Because of this, this setup can potentially accommodate an island, as long as the black hole mass lies within a certain range.

* In absence of black holes, the total action for the transition is proportional to the defect entropy, $S_{_{ICFT}}$, which is codimension-2 with respect to the auxiliary AdS$_3$ bulk. This can also be re-expressed as a difference of $S_{_{EE}}^{^{T\bar T}}$s, thereby ensuring the locality of the transition as well as the fact that $S_{_{TOT}}$  is given in terms of internal entanglement entropies evaluated on a gravitating bath.

* In presence of black holes, $S_{_{brane}}$ behaves as a quantity experiencing a phase transition beyond a certain value of ${\cal B}$, which can be thought of as playing the role of $t_{_{Page}}$ within the island formulation. Furthermore, given the fact that the $S_{_{EE}}$s defining $S_{_{brane}}$ are basically describing an internal entanglement, due to the role of the dilaton in the given setup, our findings are compatible with the claims of [269].

Having said this, we are now able to further motivate our proposal, i.e. figure <ref>, as being an appropriate adaptation of figure <ref> for describing vacuum transitions in presence of gravity.

* The total action for AdS$_2\rightarrow$AdS$_2$ transitions calculated by means of the FMP method, is a codimension-2 quantity with respect to its AdS$_3$ embedding, and therefore is to be asssociated to the IR-limit of an RG-flow, in agreement with wedge-holography. The same picture can be drawn for transitions involving dS spacetimes, given their realisation via $T\bar T$-deformations in the IR of an AdS.

* The portions of spacetime involved in the transition lie on different ETW branes, which, in turn, are holographically dual of the defects BCFT$_{D-1}$s. The interval in between the two defect theories now accommodates a composite CFT$_{D}$ with nontrivial BCs provided by the defect theories themselves.

* Under the assumption that $c_{bdy}>>c_{bulk}$, the RG-flow will drive the pair of defect CFT$_{D-1}$s in the UV to a single CFT$_{D-1}$ in the IR. An island emerges if the resulting CFT$_{D-1}$ alone is unable to realise the bulk AdS$_3$ by wedge holograhy.

§.§.§ Implications for up-tunnelling

Given the chain of relations (<ref>), which in turn relies upon the parametric identification (<ref>), we can now appreciate the fact that the obstructions to up-tunneling outlined in section <ref> share a common origin. In particular, we notice that:

* The vanishing brane action in the flat limit, is compatible with the fact that $S_{_{EE}}^{^{T\bar T}}\rightarrow 0$ as $\epsilon\equiv\frac{r}{\sqrt{c\lambda}}\rightarrow0$, thereby further justifying (<ref>).

* The divergence of the bulk action in the flat limit of dS can be recast to the need for redefining the universal part of $S_{_{EE}}^{^{T\bar T}}$ for dS including the $\frac{c}{6}$ term, which, for the decay process analysed in section <ref> corresponds to $\mu$, namely the value of the dilaton at the horizon. Given the cutoff nature of the cosmological horizon for pure dS spacetimes (due to (<ref>)), it therefore follows that, upon taking the flat limit, the transition is no longer local.

* AdS$_2\rightarrow$AdS$_2$ up-tunneling, AdS$_2\rightarrow$Mink$_2$ and Mink$_2\rightarrow$dS$_2$ can only be achieved as long as we are able to ensure the locality of the decay processes, or, equivalently, the finiteness of the turning point associated to them. For this to happen, the localisation radius, $r$, and the cosmological constant of a given vacuum need to disentangle. This can be achieved by introducing an additional scale, ultimately ensuring the finiteness of the transition under arbitrary change of the cosmological constant.

* This new scale, namely the black hole mass, is responsible for an additional contribution to the total action which cannot be reabsorbed within the definition of $S_{_{EE}}^{^{T\bar T}}$, thereby signalling the emergence of a QES in (<ref>), and proving the need for an island to be present for up-tunneling to take place.

§.§.§ A BCFT perspective: transitions between end-of-the-world branes

In section <ref> we proved agreement between the BT and FMP results in absence of black holes. This subsection is meant to provide further supportive evidence of their matching from a holographic point of view, showing that transitions described via the BT formalisms provide an example of an AdS$_2$/CFT$_1 \subset\ $AdS$_3$/CFT$_2$. In doing so, we will be focussing on AdS$_2\rightarrow$ AdS$_2$ processes, showing that the total bounce can be mapped to the entanglement entropy on a BCFT$_2$.

As already explained in section <ref>, in the BT setup, the starting point is analogous to that of a type-II Einstein-Maxwell-dilaton gravity theory in 2D, which was proved to arise from dimensional reduction of a BTZ black hole in [286].

In performing this comparison, the key quantity of interest is the boundary free energy, $F_{\partial}$, [276], which, for a BCFT$_2$ reads

\begin{equation}
\frac{T}{\ \sqrt{1-T^2\ }\ } - \ln\left|\frac{\ 1-T\ }{\ 1+ T\ }\right| ,
\label{eq:freeE}
\end{equation}

where $T$ is the tension of the ETW brane and is related to its opening angle with respect to the Cardy brane as $\Theta\overset{def.}{=}\tan^{-1} T$, as depicted on the LHS of figure <ref>.

The CFT$_1$s are dual to the AdS$_2$ spacetimes involved in the transition living on the two ETW branes and the wedges of opening angles $\Theta_{1,2}$. Their corresponding tensions, result in the value of $\Lambda_{\pm}$ on the two sides of the wall.

For type-1 instantons, (<ref>) is nontrivial for both vacua, i.e. $T,\Theta\neq0$, and the total bounce can be re-expressed as follows:

\begin{equation}
\boxed{\ \ \ B_{type-1}
2\pi\ S_{BCFT_2}=2\pi\ \left[S_{CFT_2}+S_{bdy}^{-}-S_{bdy}^{+}\right] \textcolor{white}{\Biggl [},\ \ }
\label{eq:BThol}
\end{equation}

\begin{equation}
\begin{aligned}
r_o^{-1}\overset{def.}{=}\bar\rho\ \ \ ,\ \ \  r_H\overset{def.}{=}\sqrt{\Lambda_{o,I}\ }\ \ \ ,\ \ \
\overset{def.}{=}
\sqrt{1-\frac{\ r_H^2\ }{r_o^2}\ }
\ \ ,\ \ \
S_{bdy}     =
- 2\ln\left|\frac{1-\sqrt{1-\Lambda\ \bar\rho^2\ }  }{\ \sqrt{\Lambda\ }\ \bar\rho\ }\right|.\nonumber
\label{eq:sbdy}
\end{aligned}
\end{equation}

Equation (<ref>) indicates that either vacuum can be associated to a defect CFT$_1$ living at the endpoints of a bulk CFT$_2$. The overall system therefore defines a BCFT$_2$ as depicted on the RHS of figure <ref>, with $S_{bdy}$ being evaluated on the lower-dimensional defect CFTs.  In particular, there are two different values of $T$, one for each vacuum. The mismatch in between the two gives rise to the holographic central charge, which counts the degrees of freedom of the ground state of the codimension-2 theory living on the boundary.

§.§.§ An internal CFT

We now turn to the holographic interpretation of the results obtained in section <ref> by means of the CDL method, further supporting the emergent analytic complementarity between the CDL and BT formalisms in 2D. In doing so, two main features will become manifest:

* The entanglement along the dilatonic direction is effectively internal.

* The CDL method (in 2D) describes vacuum decays in absence of gravity.

A CFT$_2$ is uniquely defined by its central charge and the conformal dimensions of the operators living in the theory, $(c,\Delta_i)$. Combined together, they define the energy spectrum of the states of a given CFT.

From the fundamental parameters of the CFT$_2$, the entropy of an excited state is defined in terms of the Cardy formula,

\begin{equation}
S_{_{Cardy}  }
2\pi \sqrt{ \frac{c}{6}\ \left(\Delta-\frac{c}{24}\right),\ }
\label{eq:CardyS}
\end{equation}

where $\Delta$ and $\Delta_o=\frac{c}{24}$ denote the black hole and ground state energies, respectively.

For the case of a CFT$_2$ where the degeneracy of the ground state is 1, the following definitions hold

\begin{equation}
\frac{c}{3}\ \sqrt{\mu-1\ },
\ \ \
\ \ \
\frac{c}{12}\ \mu,
\ \ \ \ \ \
6\frac{\ \partial S_{CFT}^2\ }{\partial E_o},
\label{eq:entr1}
\end{equation}

with $\mu\overset{def.}{=}\frac{\Delta}{c}$ being related to the black hole mass in the excited state.
Focussing on the AdS$_2\rightarrow$ AdS$_2$ process, with associated extremised bounce that can be rewritten as follows

\begin{equation}
\begin{aligned}
B_{tot}^{\ \text{AdS}_2\rightarrow \text{AdS}_{2}}
&=      4\pi\ S_\pm^2 =
4\pi\ c_{hol}^{\ {^{CFT_2}}}\ M_{_{BH}},
\label{eq:fc1}
\end{aligned}
\end{equation}

\begin{equation}
c_{hol}^{\ ^{CFT_2}}
\overset{def.}{=}
\phi_-^2+2\phi_+^2 ,
\ \ \ \ \ \
\overset{def.}{=}
\frac{\phi_-^2}{\phi_+^2}-1,
\ \ \ \ \ \
\overset{def.}{=}
\sqrt{\phi_-^2+2\phi_+^2 \ }\ \sqrt{\frac{\phi_-^2}{\phi_+^2}-1\ },
\label{eq:ep11}
\end{equation}

with $M_{_{BH}}  $ being a conserved charge along the flow and $c_{hol}^{\ ^{CFT_2}}$ its conjugate chemical potential. The last passage in (<ref>) follows from the relation between the entropy product $S_+S_-$ and the central charge.  Our result (<ref>) goes beyond the original formulation of $c$ by Cardy because it is associated to the defect lying in between the two vacua.

A key feature of (<ref>) is the fact that it defines an entropy product, and (apparently) not a difference in between entropies, as one might have expected to start with. The main reason for this is that the background configuration in CDL is providing a reference state, the ground state of the CFT$_2$, with respect to which the newly-nucleated spacetime behaves as an excited state, namely an extremal black hole. Comparison with (<ref>) enables us to deduce that

\begin{equation}
\boxed{\ \ \ \frac{\phi_-^2}{\phi_+^2}\overset{def.}{=}\mu,\textcolor{white}{\Biggl[}\ \ \ }
\label{eq:boxed1}
\end{equation}

indicating that the newly nucleated vacuum is to be identified with an extremal black hole spacetime embedded in the background vacuum.
Extremality follows from the fact that the mass of the BH is related to the cosmological constant itself. Indeed, from (<ref>), $M_{_{BH}}$ is a function of $\Lambda_{\pm}$.

Relying upon arguments inspired by black hole microstate cosmology, [289], a further remark is in order. We have shown that transitions of the kind AdS$_2\rightarrow$Mink$_2$ are forbidden. This is perfectly in agreement with the identification we have just made, namely (<ref>), since, from the CFT point of view, we can think of Mink$_2$ as being obtained from $\underset{c\rightarrow \infty}{\lim}$ of a CFT$_2$. The nucleation process would therefore correspond to the ground state of a CFT$_2$ insertion attempting to mimic an excited state of the background CFT$_2$. However, by definition, there is no horizon in the new vacuum that could attempt to screen the infinite number of degrees of freedom associated to a divergent central charge, and therefore the relative entropy (<ref>) cannot be defined. In conclusion, due to the extremality nature of the spacetime suggested by (<ref>), such transition is forbidden, since it corresponds to an overcounting of the microstates defining the entropy.

§ FINAL REMARKS

To finish this section  we briefly collect some relevant points and mention the potential relation of our results  with other recent developments.

In summary, in absence of black holes, the three methods lead to the following results in 2D

\begin{equation}
\begin{cases}
B_{_{CDL/AP}}^{\ \ wo.kt}
2\pi\ S_{_{CFT_2}}^2
2\pi\ c_{hol}^{\ {^{CFT_2}}}\ M_{_{BH}}\\    \\
S_{_{FMP}}^{^{\ {\cal B}=0}}=B_{_{BT} }
B_{_{CDL/AP}}^{\ \ w.kt}+M\bigg|^-_+
2\pi\ S_{_{BCFT_2}}
=2\pi\ \left[S_{CFT_2}+S_{bdy}^{-}-S_{bdy}^{+}\right]
\end{cases}     ,
\label{eq:BT1BT1}
\end{equation}

where $B_{_{CDL/AP}}^{\ \ wo.kt}$ and $B_{_{CDL/AP}}^{\ \ w.kt}$ denote the bounces calculated in absence and in presence of the kinetic term for $\phi$, respectively, whereas

\begin{equation}
=    \ln\  <0|{\cal B}>\overset{def.}{=}
\ln g=
\ln\left|\frac{1+T}{1-T}\right|  ,
\end{equation}

in turn highlighting the consistency and mutual complementarity of their holographic interpretations.

* The identifications featuring in (<ref>), signal the emergence of an entropic hierarchy, resulting in an interesting correspondence between our findings and those of [204, 205] within the realm of von Neumann algebras (VNAs)[See also [545] for an alternative perspective in the study of the Hilbert space of dS in arbitrary dimensions, mostly motivated by the bootstrap program. We thank the anonymous referee for bringing this article to our attention.]. In particular, the fact that the CDL calculation describes transitions in absence of gravity and the total bounce can be recast in the form of a relative entropy, is compatible with the fact that a type-III VNA only admits a relative entropy definition, rather than a von Neumann entropy. Turning on gravity, namely going beyond large-$N$, we have the BT and FMP results (in absence of black holes), therefore corresponding to type-II$_{\infty}$ VNAs. Upon adding a black hole, instead, which at the level of the Lagrangian density corresponds to turning on an irrelevant deformation, the UV theory is different with respect to that of the case with ${\cal B}=0$, leading us to conclude that the projection operator ensuring the reduction from type- II$_{\infty}$ to type-II$_1$ is mapped to the choice of the deformation in the FMP setup, hence to the choice of the black hole mass.

* Vacuum transitions are defined in terms of amplitudes, and we have gathered supportive evidence throughout this work that such processes actually resemble a statement of mutual approximation of CFT states, [262]. We emphasised that the emergence of the island can only be achieved in presence of non-extremal black holes, and that, because of this, upon taking the flat limit, the corresponding action remains finite and experiences a phase transition beyond a critical value of the black hole mass. Rephrasing this in the language [263, 264], the flat limit corresponds to the large-$N$ limit of the dual gravitational theory at hand. Inspired by the arguments of [264], regarding the importance of scale separation for reconciling connected amplitudes from disconnected boundaries (CADBs) with holography, we therefore claim that, below a critical value of the black hole mass, the flat limit leads to a divergent action due to the lack of scale separation between the cosmological constant and the black hole mass.

§ CONCLUSIONS AND OUTLOOK OF PART IV

Part IV of the present work was devoted to understanding the subject of vacuum transitions in the simple setting of 2-dimensional gravity. Due to the peculiarities of the latter, the general analysis is not a straightforward extension of the 4D cases. Our main results are the calculation of transition amplitudes obtained following three known approaches, namely the two Euclidean methods of CDL and BT,  and the Lorentzian one of FMP. We presented explicit expressions for the transition rates in each method, and for the different signs of the vacuum energies, and took the first steps towards  understanding the results. Interestingly, in comparing the Euclidean and Lorentzian prescriptions, we find many similarities, as well as differences.

Specifying to the case of (A)dS$_2\rightarrow$(A)dS$_2$ transitions, we derived the corresponding expressions for the total bounce or action in the AP setup following the CDL method in thick, (<ref>), and thin-wall approximation, (<ref>), the BT, (<ref>), and FMP method without black holes, (<ref>). As emphasised in section <ref>, the BT and FMP results agree under suitable parametric redefinition, thereby proving expected agreement in between the 2 methods. Such redefinition, though, is highly nontrivial, and can be recast to the different way in which the cosmological constants are being defined in the 2 setups, which in turn is due to the difference in between the Lagrangian densities defining the 2 theories.

On the other hand, the result obtained by means of the CDL setup in the thin-wall approximation, (<ref>), is equivalent to part of the BT result. In particular, all terms can be identified (under suitable redefinition), apart from the ones arising from the boundary terms associated to the ADM mass. As also argued in section <ref>, these provide a nontrivial contribution to the total bounce upon turning on gravity. However, as a consequence of the particular nature of the 2D theory, the equation of motion for the AP setup (when removing the kinetic term for the dilaton from the Lagrangian density), imply $M_{_{ADM}}=0$. Therefore, 2D transitions described by means of the thin-wall approximated CDL method, should really be interpreted as describing nucleation processes in absence of gravity.

The 3 holographic descriptions are complementary to each other.

Furthermore, the thick-wall analysis leads to an expression for $B_{_{TOT}}$ with unique features, in the sense that there seems to be no suitable reparametrisation enabling to recast it in any of the expressions obtained by means of other procedures. However, as argued in section <ref>, under suitable parametric redefinitions, we can still provide a holographic interpretation of our result, which is complementarily in agreement with the others.

We may summarise the holographic interpretation of our results as follows:

* In section <ref> we provide a possible holographic interpretation of the total bounces and actions calculated throughout our work, showing mutual compatibility and complementarity. Figure <ref> shows how these processes can be understood  as taking place in a KR/HM setup with two ETW branes and one internal gravitating bath, geometrising the brane mediating the nucleation process. In particular, we find that the corresponding expression for the transition rate in presence of gravity, and in absence of black holes, is given by the difference of entropies of $T\bar T$-deformed CFTs, hence illustrating the locality of the nucleation process.

* Upon adding black holes of suitable mass, (<ref>), instead, the total action can be expressed as the difference of generalised entropies, with an island emerging beyond a critical value of the black hole mass. Furthermore, under suitable parametric redefinition, the results obtained by means of the FMP method are found to agree with the expressions derived in [262] for describing mutual approximation of boundary states belonging to different CFTs.

* We propose that the vacuum transitions in the bulk correspond to a deconfinement/confinement transition on the CFT side although further study needs to be made towards this correspondence.

* The BT results, which were shown to be equivalent to those obtained through the FMP method in absence of black holes, can be expressed in terms of entropies of BCFT$_2$'s with 2 nontrivial boundary conditions dual to ETW branes. Furthermore, the CDL result can be recast in the form of an entropy product of a CFT$_2$ thereby showing agreement with the expectations following from the analytic behaviour encountered in section <ref>.

* One of our  results is that the total action (or bounce) associated to the decay process carries an internal entropic interpretation. In particular, for the BT and FMP cases, they can always be expressed as the difference of generalised entropies. Only the latter, however, provides the right setup for an island to emerge. In particular, the black hole mass, which in 2D is an explicit parameter of the theory, is, in a way, playing the role of time in the Hawking evaporation process, and is responsible for $S_{_{brane}}$ undergoing a phase transition, as explained in section <ref>.

Understanding the behaviour of the same physical process by means of different formalisms is equally important in all fields of research, and appears to be particularly promising within the context of quantum gravity. Recent progress towards understanding the behaviour of the partition function for QG within, either, the canonical or microcanonical ensemble, [267, 268], as well as the study of the phases of gauge theories and their holographic duals, [293], are certainly among the most promising directions deserving more investigation. Clearly our work leaves many open questions. We hope that our results  will be useful for addressing further questions related to vacuum transitions, early universe cosmology, and holography.

Understanding the behaviour of the same physical process by means of different formalisms is equally important in all fields of research, and appears to be particularly promising within the context of quantum gravity. Recent progress towards understanding the behaviour of the partition function for QG within, either, the canonical or microcanonical ensemble, [267, 268], as well as the study of the phases of gauge theories and their holographic duals, [293], are certainly among the most promising directions deserving more investigation. We hope the results found through this investigation will be useful for addressing further questions related to vacuum transitions, early universe cosmology, and holography.

Beyond Cetegorical Dualisability

§ SUMMARY AND MOTIVATIONS OF PART V

§.§ Motivations

For a theory, ${\cal T}$, to be absolute, the following triple needs to be defined

\begin{equation}
\boxed{\ \ \ {\cal T}\ \longleftrightarrow\ ({\cal F}, \mu, \mathfrak{Z})\color{white}\bigg]\  },
\label{eq:HB1}
\end{equation}

where ${\cal F}$ is the fiber functor, $\mu$ the moment map, and $\mathfrak{Z}$ the Drinfeld center[We refer to section <ref> for a detailed explanation of this terminology.]. Consistency of the underlying mathematical structure requires these three quantities to be mutually related. Indeed, upon defining any one of them, the other two should automatically follow. The purpose of [1, 3] and the present work is to show that an apparent shortcoming in defining such triple corresponds to the emergence of interesting physics, rather than being a fault of the sought after absolute theory. In particular, we will show that this can be used to explain the emergence of non-invertible symmetries separating different class ${\cal S}$ theories, [1, 312, 313, 316, 317, 320, 321].

The crucial references we rely upon are the works of Moore and Segal, [126], and Moore and Tachikawa, [125]. As briefly reviewed in the following sections, [125] proposes a redefinition of class ${\cal S}$ theories (cf. figure <ref>) in terms of a 2D TFT, namely the functor

\begin{equation}
\boxed{\ \ \ \eta_{_{G_{\mathbb{C}}}}:\ \text{Bo}_{_2}\ \rightarrow\ \text{HS} \color{white}\bigg]\ \ \ }
\label{eq:etaGC11}
\end{equation}

with Bo$_{_2}$ and HS denoting the bordism 2-category and the holomorphic symplectic 2-category, respectively[For more details, we refer the reader to section <ref>.], associated to a given 4D ${\cal N}=2$ SCFT. The definition of (<ref>) strongly relies upon assuming, both, the source and target categories, enjoy a duality structure which, in turn follows from the presence of an identity element in both categories. In [125], the authors show that, under the duality assumption, for the categories in (<ref>) to be well-defined, it is enough to specify their objects and 1-morphisms. Essentially, the objects of Bo$_{_2}$ are circles, $S^{^1}$, and the 1-morphisms are cobordisms between different disjoint unions of circles and the empty set. Their respective counterpart on the holomorphic symplectic side correspond to the gauge group, [125],

\begin{equation}
\eta_{_{G_{_{\mathbb{C}}}}}\ \left(S^{^1}\right)\ \overset{def.}{=}\ G_{_{\mathbb{C}}},
\label{eq:gaugegroup}
\end{equation}

and the cobordism operators, [125],

\begin{equation}
\eta_{_{G_{_{\mathbb{C}}}}}\ \left(\text{Hom}\ \left(S^{^1}, \emptyset\right)\right)\ \overset{def.}{=}\ U_{_{G_{_{\mathbb{C}}}}}
\label{eq:UG1}
\end{equation}

\begin{equation}
\eta_{_{G_{_{\mathbb{C}}}}}\ \left(\text{Hom}\ \left(S^{^1}\ \sqcup\ S^{^1}, \emptyset\right)\right)\ \overset{def.}{=}\ V_{_{G_{_{\mathbb{C}}}}}
\label{eq:VG1}
\end{equation}

\begin{equation}
\eta_{_{G_{_{\mathbb{C}}}}}\ \left(\text{Hom}\ \left(S^{^1}\ \sqcup\ S^{^1}\ \sqcup \ S^{^1},\emptyset\right)\right)\ \overset{def.}{=}\ W_{_{G_{_{\mathbb{C}}}}}.
\label{eq:HBCS}
\end{equation}

Importantly for us, the Moore-Tachikawa varieties described by (<ref>), (<ref>), (<ref>), and (<ref>) constitute the quiver gauge theory[Among the cobordism operators is the Higgs branch of class ${\cal S}$ theories, (<ref>).] realisation of [126], where Moore and Segal propose the mathematical formalism needed for addressing the following question: given a certain closed string theory background, what is its corresponding D-brane content[Note that this is essentially the same question addressed in [342].]?

Our aim is that of explaining how and why one needs to generalise the construction of [125] from a higher-categorical point of view, and what its implications are on the theoretical physics side. In doing so, we highlight the crucial properties and axioms satisfied by the cobordism operators outlined in [126, 125], and how they can be generalised to account for more general setups from, both, the mathematical and theoretical physics perspectives.

Adaptation of a correspondence first proposed in [1] playing a key role towards generalising [125] to the hyperk$\ddot{\text{a}}$hler target category case. As explained in the following sections, this also requires the generalisation of cobordism operators, [126], due to the lack of reparametrisation-invariance on the Riemann surface on which the compactification of the 6D ${\cal N}=(2,0)$ SCFT is performed.

In particular, we emphasise the dependence of (<ref>) on the conformal structure of the Riemann surface on which the compactification of the 6D ${\cal N}=(2,0)$ SCFT has been performed to achieve a certain class ${\cal S}$ theory and how lack of reparametrisation invariance, corresponding to the absence of the identity element in its source and target categories, [125], signals the presence of (non-invertible) categorical symmetries separating different absolute theories.

At the heart of this is the correspondence sketched in figure <ref>.

In [1] we explained how gauging a Symmetry Topological Field Theory (SymTFT) enables to change the boundary conditions of the fields living in the absolute field theory resulting from the Freed-Moore-Teleman construction, [103, 102, 323, 324]. To each gauging corresponds a choice of triples, (<ref>), and, for any absolute theory, it is enough to define one of the three entries on the RHS of (<ref>) to determine the other two. For the purpose of this article, we will mostly focus on the second, namely the moment  map, defined as follows

\begin{equation}
\mu:\ \mathfrak{G}\ \rightarrow\ {\cal A},
\end{equation}

where $\mathfrak{G}$ is an n-categorical structure, and ${\cal A}$ is the algebra of invertible topological defects associated to the action $\mu\ (\mathfrak{G})$. Gauging means taking the categorical quotient with respect to $\mathfrak{G}$ (in notation $//_{_{\mu}}\ \mathfrak{G}$), and projecting its image under $\mu$ to the identity[We thank Nathan Seiberg for instructive discussion regarding the appropriateness of the terminology to be adopted in describing this formalism.]. Practically, one could perform a total gauging of the theory by choosing ${\cal A}$ such that the overall spectrum of the theory of the gauged theory is only the (new) identity element. For the purpose of our work, instead, we are interested in understanding mathematical structures arising by gauging with respect to different subalgebras within ${\cal A}$, that are mutually intersecting, albeit not contained within each other. The ultimate aim is that of explaining the emergence of (non-invertible) categorical symmetries in certain supersymmetric quiver gauge theories once described in terms of Coulomb branches of magnetic quivers of 3D ${\cal N}=4$ quiver gauge theories, which is the main focus of an upcoming paper by the same author, [549]. In such analysis we will be applying some of the findings of [134, 343].

In [3], we argued in favour of the following statements:

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

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

Mostly relying upon [126], we highlighted the importance of the construction of cobordism operators, emphasising their dependence on the conformal structure of the Riemann surface. We then turned to the discussion of a particular 2D TFT valued in a symmetric monoidal category, namely the maximal dimension Higgs branch of class ${\cal S}$ theories. After briefly reviewing the properties outlined in [125], we proposed their generalisation for the case in which the target category of the $\eta_{_{G_{_{\mathbb{C}}}}}$ functor is a hyperk$\ddot{\text{a}}$hler quotient. We concluded outlining the possible extension of this treatment towards a mathematical formulation of magnetic quivers within the context of Coulomb branches of 3D ${\cal N}=4$ quiver gauge theories.

In this new work, we take a more algebro-geometric approach in explaining the mathematical structure underlying supersymmetric quiver gauge theories, with a particular focus on dualities and their mutual relations in terms of higher-categories.

As will be explained in depth, the crucial role is played by factorisation homology. We will not attempt to provide an exhaustive definition here, and refer the reader to the related section in the core of this work, instead. What we can anticipate, though, is that factorisation homology is related with the counting of ground state degeneracy, and, when applied to the AGT correspondence, provides a tool for evaluating the 4-sphere partition function of class ${\cal S}$ theories. Specifically, as we shall see, factorisation homology allows us to relate categorical dualisability and full-extendibility[And is therefore in line with the connection between dualities and categorical structures.]. Within the context of non-Lagrangian theories, such as 6D ${\cal N}=(2,0)$ SCFTs, this is particularly relevant since it introduces the notion of ring objects, of which Hilbert series are one of the examples of most interest to us.

As an algebraic variety, we focus on the calculation of the Hilbert series on Coulomb and Higgs branches, with the latter being related by the geometric invariant theory (GIT) construction, [209]. GIT being a recipe for realising 3D mirror symmetry, [135], we show how it can be suitably extended for cases where mirror duals are not necessarily in a 1-to-1 correspondence. In relation to Part II, this nicely fits with the magnetic quiver prescription for describing the Higgs branch of certain supersymmetric quiver gauge theories with non-primitive ideals[Namely the 2-cone structure, [211].].

We then show how the 2D description of disk algebras can be suitably adopted for describing Hitchin systems associated to joint class ${\cal S}$ theories.  We conclude outlining future directions that are currently under investigation by the same author.

Following our treatment in [5] we explain the higher categorical structures that are needed for describing the invariants associated to specific supersymmetric quiver gauge theories admitting a magnetic quiver description.

The key tool we will be using in achieving such aim is factorisation homology, of which a more detailed definition will be provided in the core of the paper. For the purpose of the introduction, we might just say that its role is that of assigning invariants to topological manifolds.

§.§ Outline and main results

Part V of this work is structured as follows:

* section <ref> introduces factorisation homology and its applicability within the context of class ${\cal S}$ theories by means of the AGT correspndence. The section ends recapitulating the interplay between dualisability and full-extendibility of TQFTs, with a particular focus of Moore-Tachikawa varieties[In section <ref>, we briefly overviewed cochain level theories as the most important generalisation of the open and closed TFT construction. Mostly relying upon [126], we highlighted the importance of the construction of the cobordism operator, highlighting their dependence on the conformal structure of the Riemann surface. We then turn to the discussion of a particular 2D TFT valued in a symmetric monoidal category, namely the maximal dimension Higgs branch of class ${\cal S}$ theories.].

* Section <ref> explains the relation in between the Drinfeld center and magnetic quivers, combining sections <ref> and <ref>.

* In section <ref> we propose the generalisation of Moore-Tachikawa varieties for the case in which the target category of the $\eta_{_{G_{_{\mathbb{C}}}}}$ functor is a hyperk$\ddot{\text{a}}$hler quotient. We conclude outlining the possible extension of this treatment towards a mathematical formulation of magnetic quivers within the context of Coulomb branches of 3D ${\cal N}=4$ quiver gauge theories which will be addressed in an upcoming work by the same author, [549]. The concluding part of this section 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. We conclude showing how Coulomb and Higgs branches from class ${\cal S}$ Hitchin systems carry the same information as the gaugeable algebras leading to SymTFT constructions, discussed in Part III. We conclude outlining interesting open questions and future directions of investigation by the author.

§ CATEGORICAL DUALISABILITY AND FULL EXTENDIBILITY

The present section is structured as follows:

* Section <ref> is devoted to explaining the role played by factorisation homology in describing the invariants of a given TFT. In particular, this prepares the stage for the following sections, where we will be dealing with the applicability of this formalism to the specific case of class ${\cal S}$ theories[Note the importance of having introduced the notion of contractible cells in section <ref> in relation to abelianisation and equivariant cohomology.].

* In section <ref>, we then turn to explaining the relation between categorical dualisability and full-extendibility. In particular, this nicely relates to the notion of Drinfeld centers and gauging categorical structures.

§.§ Factorisation Homology in context

The present section is mostly revision of well-known literature[Specifically, we suggest the interested reader to consult the following reference [126].]. It provides a brief mathematical digression, needed for introducing factorisation homology, and its utility for evaluating the index of a 2D TQFT. In doing so, we explain the notions of Hochschild cohomology, stratified spaces, and disk algebras. In the following sections we will see that, combining this formalism to algebro-geometric techniques, leads to interesting new findings.

§.§ Cohomology and Homotopy Equivalence

In homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain[Cochains are functions on the group of chains in the homology theory.] complex. Cohomology can be viewed as a way of assigning richer algebraic invariants to a space with respect to homology.

Some versions of cohomology arise by dualising the homological construction. Some of their properties are formulated in a similar fashion with respect to their homological counterpart. However, cohomology enjoys a richer structure, which is lacking in homology, which we now turn to explain.

For a given topological space $X$, and a commutative ring, $R$, the cup product is a bilinear map

\begin{equation}
H^{^i}(X,R)\ \times \ H^{^j}(X,R)\ \rightarrow\ H^{^{i+j}}(X,R).
\end{equation}

This product defines the cohomology ring of $X$ as follows

\begin{equation}
H^{^{\bullet}}(X,R)\ \equiv\ \bigoplus_{_i} \ H^{^j}(X,R).
\end{equation}

It is graded commutative in the sense that, given $u\ \in\ H^{^i}(X,R)$, $v\ \in\ H^{^j}(X,R)$,

\begin{equation}
uv\ =\ (-1)^{^{ij}}\ vu.
\end{equation}

It follows that, if two spaces are homotopy equivalent, their cohomology rings are isomorphic.

Homotopy equivalence can be expressed in terms of looping by representation spheres. In particular, a homotopy equivalence between topological spaces $X$ and $Y$ is a morphism $f:X\rightarrow Y$ which has a homotopy inverse, hence such that there exists a morphism $g: Y\ \rightarrow X$ and homotopies

\begin{equation}
g\ \circ\ f\ \sim\ \mathbf{1}_{_X}\ \ \ \text{and} \ \ \ f\ \circ\ g\ \sim\ \mathbf{1}_{_Y},
\end{equation}

thereby making homotopy equivalence very similar to the notion of categorical equivalence.

For a given topological space, $X$, elements of $H^{^i}(X)$ can be thought of as represented by codimension-$i$ subspaces of $X$ that can move freely on $X$. For the case at hand, an element of $H^{^i}(X)$ is defined by a continuous map

\begin{equation}
f:\ X\ \rightarrow\ M,
\end{equation}

with pullback

\begin{equation}
f^*:\ H^{{\bullet}}(M,R)\ \rightarrow\ H^{\bullet}(X,R),
\end{equation}

and a closed codimension-$i$ submanifold $N$ of $M$ with an orientation in the normal bundle. As such, the class $f^*([N])\ \in H^{^i}(X)$ can be thought of to lie in the subspace $f^{^{-1}}(N)$ of $X$, since $f^*([N])$ restricts to $0$ in the cohomology of the open subset $X-f^{^{-1}}(N)$. The cohomology class $f^*([N])$ is free to move in $X$ in the sense that $N$ could be replaced by any continuous deformation of $N$ inside $M$.

§.§.§ Hochschild cohomology and cochain complexes

Given a finite group action on a suitably enhanced triangulated category linear over a field, [347], Hochschild cohomology and group actions establish a formula for the Hochschild cohomology of the category of invariants, assuming the order of the group is coprime to the characteristic of the base field.

For a $k$-linear category with $G$-action,

\begin{equation}
\underset{g\in G}{\bigoplus}\ \phi_{_g}:\ {\cal C}\ \rightarrow\ {\cal C}
\end{equation}

induces the norm functor

\begin{equation}
\begin{aligned}
&N_{_m}: \ {\cal C}_{_G}\rightarrow{\cal C}^{G} \\
&\ \ \ \ \ \ \ \ x\ \mapsto\ \underset{g\in G}{\sum}\ g(x)
\end{aligned}
\end{equation}

characterised by the existence of a factorisation property

\begin{equation}
N_{_m}\circ q: \ {\cal C}\rightarrow {\cal C}^{^G}.
\end{equation}

Importantly, $N_{_m}$ is an equivalence.

If ${\cal C, D}$ are $k$-linear categories, then the $k$-linear exact functors from ${\cal C}$ to ${\cal D}$ form the objects of the $k$-linear category Fun$_{_k}({\cal C},{\cal D})$.

Let ${\cal C}$ be a $k$-linear category, and let $\phi:\ {\cal C}\rightarrow{\cal C}$ be an endofunctor. The Hochschild cochain complex of ${\cal C}$ with coefficients in $\phi$ is defined as

\begin{equation}
HC^{^{\bullet}}({\cal C},\phi)\ \overset{def.}{=}\ \text{Map}_{_{\text{Fun}_{_k}({\cal C,C})}}(\mathbf{1}_{_{\cal C}},\phi)\ \in\ \text{Vect}_{_k}.
\end{equation}

The Hochschild cohomology $HH^{^{\bullet}}({\cal C},\phi)$ of ${\cal C}$ with coefficients in $\phi$ is the cohomology of this complex. For the case in which $\phi\equiv\mathbf{1}_{_{\cal C}}$,

\begin{equation}
HC^{^{\bullet}}({\cal C})\ \overset{def.}{=}\ HC^{^{\bullet}}({\cal C},\mathbf{1}_{_{\cal C}})
\ \ \ ,\ \ \
HH^{^{\bullet}}({\cal C})\ \overset{def.}{=}\ HH^{^{\bullet}}({\cal C},\mathbf{1}_{_{\cal C}})
\end{equation}

define the Hochschild cochain complex and the Hochschild cohomology of ${\cal C}$, respectively. The latter comes equipped with a natural algebra structure, induced by composition in $\text{Map}_{_{\text{Fun}_{_k}({\cal C,C})}}(\mathbf{1}_{_{\cal C}},\mathbf{1}_{_{\cal C}})$.

Note that, given a $k$-linear category $\cal C$ with an action by a finite group $G$, there is an induced action of $G\times G$ on $\text{Fun}_{_k}({\cal C,C})$ consisting in the following. An element $(g_{_1},g_{_2})\ \in\ G\times G$ acts on $\text{Fun}_{_k}({\cal C,C})$ by sending $F:\ {\cal C}\rightarrow {\cal C}\ \in\ \text{Fun}_{_k}({\cal C,C})$ to $\phi_{_{g_{_2}}}\circ F\circ \phi^{^{-1}}_{_{g_{_1}}}$. Via the diagonal embedding $G\ \subset\ G\times G$, this restricts to the conjugation action of $G$ on $\text{Fun}_{_k}({\cal C,C})$.

A main theorem in [347] states that, given $\cal C$ a $k$-linear category with an action by a finite group $G$, and assuming the order of $G$ is coprime to the characteristic of $K$, then there is an isomorphism

\begin{equation}
HH^{^{\bullet}}({\cal C}^{^G})\ \simeq\ \left(\underset{g\in G}{\bigoplus}\ HH^{^{\bullet}}({\cal C},\phi_{_g})\right)^{^G},
\end{equation}

with $\phi_{_g}: {\cal C}\rightarrow {\cal C}$ an autoequivalence corresponding to $g\in G$, and the $G$-action on the right side induced by the conjugation action of $G$ on $\text{Fun}_{_k}({\cal C,C})$. In particular,

\begin{equation}
HH^{^{\bullet}}({\cal C})^{^G}\ \hookrightarrow\ HH^{^{\bullet}}({\cal C}^{^G}).
\end{equation}

We will come back to this in section <ref> when discussing the GIT quotient and the Kirwan map.

§.§.§ Factorisation homology of stratified spaces

Factorisation homology satisfies a defining property called $\otimes$-excision, which determines $\int_{_M}{\cal A}$ up to equivalence,

\begin{equation}
\boxed{\ \ \ \int_{_M}{\cal A}\ \simeq\ (H,u_{_{\Sigma}})  \color{white}\bigg]\  \ },
\label{eq:facthom1111}
\end{equation}

with such property being a special case of the pushforward property. The RHS of (<ref>) consists of the Hilbert space of the TO, $H$, and the ground state degeneracy, $u_{_{\Sigma}}$. (<ref>) allows to reduce the computation of the factorisation homology of a surface to that of a lower-dimensional manifold, and, eventually, to that of a 0-dimensional manifold.

The only known global observable on $\Sigma$ for an anomaly-free TO is the ground state degeneracy. The theory of factorisation homology gives a powerful tool for calculating it. When a closed stratified surface is decorated by anomaly-free topological defects of codimension 0,1,2, the theory of factorisation homology gives us a powerful tool for calculating the degeneracy of the ground state. As an important bi-product, FH provides a powerful tool for calculating global observables on surfaces for anomalous TOs. Indeed, relation between FH and TOs enables to make direct contact with the theoretical physics setup of interest to us.

There is a well-defined fully faithful functor

\begin{equation}
\mathfrak{Z}:\ \text{UMTC}^{^{\text{ind}}}\ \rightarrow\ \text{UMTC}
\label{eq:drinf}
\end{equation}

acting on objects and morphisms as follows

\begin{equation}
{\cal C}\ \rightarrow\ \mathfrak{Z}({\cal C})\ \ \ ,\ \ \ _{_{{\cal C}}}{\cal M}_{_{{\cal D}}}\ \mapsto\ \mathfrak{Z}({\cal M})\ \overset{def.}{=}\ \text{Fun}_{_{{\cal C}|{\cal D}}}({\cal M},{\cal M}).
\end{equation}

For ${\cal C}$ a UMTC, (<ref>) defines the Drinfeld center of ${\cal C}$, $\mathfrak{Z}({\cal C})$, a canonical braided monoidal category such that

\begin{equation}
\mathfrak{Z}({\cal C})\ \simeq\ {\cal C}\ \boxtimes\ \overline{\cal C}.
\end{equation}

More explicitly, Th.2.5 in [324] implies that, given ${\cal C}, {\cal D}, {\cal E}$ UMFCs, ${\cal M}$ a multi-fusion ${\cal C}-{\cal D}$-bimodule, and ${\cal N}$ a multi-fusion ${\cal D}-{\cal E}$-bimodule, the assignment

\begin{equation}
f\boxtimes_{_{{\mathfrak{Z}({\cal D})}}}g\ \mapsto\ f\boxtimes_{_{\cal D}}g
\end{equation}

defines an equivalence between two multi-fusion $\mathfrak{Z}({\cal C})-\mathfrak{Z}({\cal E})$-bimodules:

\begin{equation}
\text{Fun}_{_{{\cal C|D}}}({\cal M},{\cal M})\boxtimes_{_{{\mathfrak{Z}({\cal D})}}}\text{Fun}_{_{{\cal D|E}}}({\cal N},{\cal N})\ \simeq\ \text{Fun}_{_{{\cal C|E}}}({\cal M}\boxtimes_{_{{\cal D}}}{\cal N},{\cal M}\boxtimes_{_{{\cal D}}}{\cal N})
\end{equation}

from which the following two corollaries follow:

\begin{equation}
{\cal C}\boxtimes_{_{{\mathfrak{Z}({\cal C})}}}{\cal D}^{^{rev}}\ \simeq\ \text{Fun}_{_{H}}({\cal M},{\cal M})
\end{equation}

\begin{equation}
\boxtimes_{_{({\cal C}_{_{0}},...,{\cal C}_{_{n-1}})}}^{^{\circlearrowright}}({\cal M}_{_{1}},...,{\cal M}_{_{n}})\ \simeq\ \text{Fun}_{_{H}}(\cal P,P),
\end{equation}

with ${\cal P}$ a unique unitary category such that this correspondence holds.

Next, consider Mfld$_{_{n}}^{^{or}}$ to be the topological category whose objects are oriented n-manifolds whose objects are oriented n-manifolds without boundary. For any two oriented manifolds $M,N$, the morphism Hom$_{_{\text{Mfld}_{_{n}}^{^{or}}}}(M,N)$ is the space of all orientation-preserving embeddings $e: M\rightarrow N$ endowed with the compact-open topology.

Denoting with ${\cal M}\text{fld}_{_{n}}^{^{or}}$ the symmetric monoidal $\infty$-category associated to the topological category Mfld$_{_{n}}^{^{or}}$, the symmetric monoidal $\infty$-category Disk$_{_{n}}^{^{or}}$, is the full subcategory of ${\cal M}\text{fld}_{_{n}}^{^{or}}$ whose objects are disjoint union of finitely-many n-dimensional Euclidean spaces equipped with standard orientation.

If ${\cal V}$ is a symmetric monoidal $\infty$-category, $M$ an oriented manifold, and

\begin{equation}
\boxed{\ \ \ {\cal A}:\ \text{Disk}_{_{n}}^{^{or}}\ \rightarrow\ {\cal V} \color{white}\bigg]\ \ }
\end{equation}

an n-disk algebra in ${\cal V}$, the factorisation homology of $M$ with coefficients in ${\cal A}$ is an object of ${\cal V}$ given by

\begin{equation}
\int_{_{M}}\ {\cal A}\ \overset{def.}{=}\ \text{Colim.}\left(\left(\text{Disk}_{_{n}}^{^{or}}\right)_{_{/M}}\ \rightarrow\ \text{Disk}_{_{n}}^{^{or}}\ \xrightarrow{\cal A}\ {\cal A}\right),
\end{equation}

with $\left(\text{Disk}_{_{n}}^{^{or}}\right)_{_{/M}}$ denoting the category of $n$-disks embedded in $M$. If $\mathbf{1}_{_{n}}$ is the trivial $n$-disk albegra assigning to each $\mathbf{R}^{^n}$ $\mathbf{1}_{_{{\cal V}}}\ \in\ {\cal V}$, it follows that

\begin{equation}
\int_{_{M}}\mathbf{1}_{_{n}}\ \simeq\ \mathbf{1}_{_{{\cal V}}}.
\end{equation}

The canonical morphism $\mathbf{1}_{_{n}}\rightarrow{\cal A}$ then induces a morphism $\mathbf{1}_{_{{\cal V}}}\rightarrow\int_{_{M}}{\cal A}$, implying the factorisation homology $\int_{_{M}}{\cal A}$ is not just an object of ${\cal V}$, but also comes equipped with a 0-disk algebra structure, the latter playing a crucial role for ground state degeneracy calculations.

Explicit calculation of factorisation homology as a 0-disk algebra is very useful. Indeed, factorisation homology on a closed stratified surface $\Sigma$ with anomaly-free coefficient ${\cal A}$, $(H,u_{_{\Sigma}})$, exactly gives the ground state degeneracy. of the same surface decorated by anomaly-free topological defects of codimension 0, 1, and 2 that are associated to ${\cal A}$, namely $u_{_{\Sigma}}$.

§.§ Examples of stratified factorisation homology

Given a stratified 2-disk, $M_{_1}$, with ${\cal A}$ a 2-disk algebra assigned to 2-cells, 1-disk algebras associated to 1-cells, and ${\cal P}$ as the unique 0-algebra assigned to the unique 0-cell, examples of its factorisation homology read as follows

\begin{equation}
\int_{_{M_{_1}}}{\cal A}_{_1}\ \simeq\ {\cal A}\ \ \ ,\ \ \ \int_{_{M_{_1}\diagdown \{0\}}}{\cal A}_{_1}\ \simeq\ HH_{_{\bullet}}({\cal A})\ \overset{def.}{\equiv}\ {\cal A}\otimes_{_{{\cal A}\otimes{\cal A}^{^{op}}}}{\cal A},
\end{equation}

with $HH_{_{\bullet}}({\cal A})$ denoting the Hochschild cohomology.

For anomaly-free coefficient systems,

A stratified 2-disk with 2-cells labelled by UMTCs ${\cal C}$ and ${\cal D}$. The unique 1-cell is labelled by a closed multi-fusion ${\cal C}-{\cal D}$-bimodule, ${\cal M}$. The two 2-cells are labelled by $({\cal D},d), ({\cal M},m)$, for $d\in{\cal D}$ and $m\in{\cal M}$, respectively.

\begin{equation}
\boxtimes_{_{({\cal C}_{_{0}},...,{\cal C}_{_{n-1}})}}^{^{\circlearrowright}}({\cal M}_{_{1}},...,{\cal M}_{_{n}})\ \simeq\ \text{Fun}_{_{H}}(\cal P,P).
\end{equation}

The unique 1-cell labelled by a closed multi-fusion ${\cal D}-{\cal C}$-bimodule ${\cal M}$. The two 0-cells are labelled by $({\cal D}, d)$ and $({\cal M},m)$, for $d\in{\cal D}$, and $m\in{\cal M}$, respectively. The closed multi-fusion ${\cal C-D}$-bimodule structure on ${\cal M}$ induces two monoidal functors

\begin{equation}
{\cal C}\ \xrightarrow{L}\ {\cal M}\ \xleftarrow{R}\ {\cal D}.
\end{equation}

Given $(\mathbb{R}^{^2};\Gamma_{_0})$ a stratified 2-disk, the FH reads as follows

\begin{equation}
\int_{_{(\mathbb{R}^{^2};\Gamma_{_0})}}({\cal C},{\cal D};{\cal M};({\cal D},d),({\cal M},m))\ \simeq\ ({\cal C},L^{^{\text{V}}}(m\otimes R(d))),
\label{eq:FH1}
\end{equation}

with $L$ the right-adjoint functor of $L$. If ${\cal D}\equiv{\cal C}^{^0}_{_{{\cal A}}}, {\cal M}\equiv{\cal C}_{_{{\cal A}}}$ for a connected commutative separable algebra ${\cal A}$ in ${\cal C}$, $d\equiv\mathbf{1}_{_{{\cal D}}}$, and $m\equiv\mathbf{1}_{_{\cal M}}$, then (<ref>) reduces to

\begin{equation}
\int_{_{(\mathbb{R}^{^2};\Gamma_{_0})}}\left({\cal C},{\cal C}^{^0}_{_{{\cal A}}};{\cal C}_{_{{\cal A}}}^{^0};\left({\cal C}_{_{{\cal A}}}^{^0},d\right),\left({\cal C}_{_{{\cal A}}},\mathbf{1}_{_{{\cal C}^{^0}_{_{{\cal A}}}}}\right)\right) \ \simeq\ ({\cal C},{\cal A}).
\label{eq:FH}
\end{equation}

§.§.§ Non-contractible loops

In presence of non-contractible loops,

\begin{equation}
\int_{_{(S^{^2};S^{^1})}}({\cal C},{\cal D};{\cal K};\emptyset)\ \simeq\ (H, \text{Hom}_{_{{\cal C}\boxtimes{\cal D}}}(\mathbf{1}_{_{{\cal C}}\boxtimes_{_{\cal D}}} {\cal A})
\end{equation}

\begin{equation}
{\cal A}\ \overset{def.}{=}\ (L\ \boxtimes\ R)^{^V} (\mathbf{1}_{_{\cal K}})
\end{equation}

a commutative separable algebra in $\bar {\cal C}\ \boxtimes\ {\cal D}$. If ${\cal K}$ is not fusion, then

\begin{equation}
\text{dim Hom}_{_{{\cal C}\boxtimes{\cal D}}}(\mathbf{1}_{_{{\cal C}}}\boxtimes \mathbf{1}_{_{{\cal D}}}, {\cal A})\ =\ \text{dim Hom}_{_{k}}(\mathbf{1}_{_{{\cal K}}},  \mathbf{1}_{_{\cal K}})\ >\ 1.
\end{equation}

When this happens, the corresponding topological order is not stable, and triggers an RG-flow.

Via factorisation homology, one defines the $W$-matrix, $W^{^{\cal M}}$, for ${\cal M}$ as follows

\begin{equation}
(L_{_{\cal M}}\ \boxtimes\ R_{_{\cal M}})^{^V}(\mathbf{1}_{_{{\cal M}}})\ \simeq\ \underset{i\ \in\ O({\cal C}), \ j\ \in\ O({\cal D})}{\bigoplus}\ W^{^{\cal M}}_{_{ij}}\ \ i\ \boxtimes\ j^{^*}
\end{equation}

\begin{equation}
(L_{_{\cal M}}\ \boxtimes\ R_{_{\cal M}})^{^V}(\mathbf{1}_{_{{\cal M}\boxtimes_{_{{\cal D}}}{\cal N}}})\ \simeq\ \underset{i\ \in\ O({\cal C}), \ j\ \in\ O({\cal E})}{\bigoplus}\ (W^{^{\cal M}}W^{^{\cal N}})_{_{ij}}\ \ i\ \boxtimes\ j^{^*},
\end{equation}

given $(\mathbf{R}\times S^{^1}; S^{^1}\ \cup\ ...\ \cup\ S^{^1})$ a stratified open cylinder with 2-cells labelled by UMTCs ${\cal C}_{_{0}}, ..., {\cal C}_{_{n}}$, and 1-cells (namely non-contractible loops) labelled by closed multi-fusion ${\cal C}_{_{i-1}}-{\cal C}_{_i}$-bimodules ${\cal M}_{_i}$, $i=1,...,n$, and no 0-cells.

Crucially, the above leads the definition of the ground state degeneracy, namely the index we are looking for

\begin{equation}
\boxed{\ \ \ \text{dim}\  u \ \overset{def.}{=}\ \text{Tr}\ \left(W^{^{\cal M}}_{_1}W^{^{\cal M}}_{_2}...W^{^{\cal M}}_{_n}\right) \color{white}\bigg] \ }.
\end{equation}

From the AGT correspondence, we know that this is equal to the partition function of a class ${\cal S}$ theory.

As we will see in the upcoming sections, we can map the issue of gluing different class ${\cal S}$ theories separated by a non-invertible defect, [313, 1], to that of defining the ground state degeneracy of a 2D TO in presence of non-contractible loops.

§.§ Dualisability and full-extendibility

We now turn to overviewing the relation between categorical dualisability and full-extendibility of a 2D TQFT. As we shall see, when applying this to  a specific algebraic variety, namely the Hilbert series, homological mirror symmetry, in its categorical formulation, results as a biproduct of full-extendibility.

§.§.§ Extending TQFTs

Lurie proved that a TQFT is fully determined by what it assigns to a point. Furthermore, fully extended TQFTs with values in a symmetric monoidal $(\infty, n)$-category, ${\cal C}$, are equivalent to fully dualisable objects in ${\cal C}$.

For a 2D TQFT, there are two main cases, [351]:

* The TQFT is fully-extended, in which case the 2-functor reads as follows:

\begin{equation}
{\cal Z}:\ \text{Bord}_{2,1,0}^{^{\sigma}}\ \rightarrow\ {\cal B},
\end{equation}

where $\text{Bord}_{2,1,0}^{^{\sigma}}$ denotes the bicategories of points, 1-manifolds with boundaries, and 2-manifolds with corners. The target, ${\cal B}$, is most commonly associated with the bicategory of finite-dimensional $\textbf{k}$-algebras, bimodules, and bimodule maps, Alg$_{_{\textbf{k}}}$. From the cobordism hypothesis, it follows that the extended framed TQFTs with values in Alg$_{_{\textbf{k}}}$are classified by finite-dimensional separable $\textbf{k}$-algebras.

* The TQFT is not fully-extended, instead, if

\begin{equation}
{\cal Z}:\ \text{Bord}_{2,1}^{^{\text{or}}}\ \rightarrow\ {\cal V},
\end{equation}

with ${\cal V}$ a symmetric monoidal 1-category.

A non-extended 2D TQFT can be extended to the point if there is a symmetric monoidal bicategory, $\cal B$, and an extended TQFT such that

\begin{equation}
{\cal V}\ \simeq\ \text{End}_{_{\cal B}}\left(\mathbf{1}_{_{\cal B}}\right)\ \ \ \ \text{and} \ \ \ \ {\cal Z}_{_{ne}}\ \simeq\ {\cal Z}\bigg|_{{\text{End}_{_{\text{Bord}_{_{2,1,0}}^{^{\sigma}}(\emptyset)}}}},
\label{eq:prescr}
\end{equation}

where $\mathbf{1}_{_{\cal B}}\ \in\ {\cal B}$ and $\emptyset\ \equiv\ \mathbf{1}_{_{\text{Bord}_{_{2,1,0}}^{^{\sigma}}(\emptyset)}}$.

The prescription (<ref>) is clearly not unique. In particular, it depends on the target ${\cal B}$.

If a non-extended 2D TQFT is a restriction of an appropriate defect TQFT ${\cal Z}_{_{ne}}^{^{\text{def.}}}$, then the non-extended TQFT can be extended to the point as long as we associate it with the bicategory ${\cal B}_{_{{\cal Z}_{_{ne}}^{^{\text{def.}}}}}$.

§.§ Comparison with our main example

We now turn to explaining why this closely resembles the gauging prescription of SymTFTs, [102]. In order to do so, we build connection with our treatment and main exaple outlined in Part I and Part II of this work, namely algebraic varieties and ring homologies.

In building the understanding for how the setup of subsection <ref> is of use for relating 3D mirror symmetry and homological mirror symmetry, we first need a brief mathematical digression, presented in the remainder of this section. Combining these tools with those of section <ref>, will be the main focus of sections <ref> and <ref>.

§.§.§ Dualisability in categorical structures

A dualisable object in a symmetric monoidal $(\infty,n)$-category, ${\cal C}$, is fully dualisable if the structure maps of the duality unit and counit each admit adjoints, as well as higher-morphisms between them, up to level $(n-1)$.

By the Cobordism Hypothesis Theorem, Lurie proved that symmetric monoidal $(\infty, n)$-functors out of the $(\infty, n)$-category of cobordisms are characterised by their value at the point, which is a fully-dualisable object.

The notion of dualisability and full dualisability can be extended to the category of chain complexes, where an object is said to be dualisable if and only if it is a bounded chain complex of dualisable modules.

§.§.§ The case of the Hilbert series

We now turn to the higher-categorical structure most relevant for our treatment, whose dualisability is the core topic of the present and following sections.

In [347], the author explores many applications of representation theory of categories to sequences of groups arising in topology, algebra, and combinatorics.

Given $(V_{_n})_{_{n\in\mathbb{N}}}$ a sequence of representations of groups $G_{_n}$ over a field $k$, it can be lifted to a category ${\cal C}$ whose objects (or isomorphism classes) are indexed by natural numbers, such that the automorphism group of the $n^{th}$ object of ${\cal C}$ is $G_{_n}$. A ${\cal C}$-representation, or ${\cal C}$-module, is a functor ${\cal C}\rightarrow$ Mod$k$. One attempts to use the representation theory of ${\cal C}$ to show that the whole sequence of representations is determined by a finite amount of data, and to discover universal patterns which the sequence must satisfy.

These patterns are often expressed in terms of generating functions. For example, one invariant of a sequence of representations is its sequence of dimensions, (dim $V_{_n})_{_n\in\ \mathbb{N}}$. Such data can be recorded as a generating function, called the Hilbert series of $(V_{_n})_{_{n\in\mathbb{N}}}$

\begin{equation}
\text{HS}_{_V}(t)\ \overset{def.}{=}\ \underset{n}{\sum}\ \text{dim}_{_k}V_{_n}t^{^n}\ \in\ \mathbb{Z}[[t]].
\end{equation}

In most cases, theorems from the representation theory of ${\cal C}$ imply that HS$_{_V}(t)$ is rational, with denominator of a certain type. For example, considering the category of finite dimensional vector spaces over $\mathbb{F}_{_q}$ and linear injections between them, if $V$ is a finitely generated $VI_{_q}$ module, and char $k \neq$ char $\mathbb{F}_{_q}$, then HS$(t)$ is rational, with denominator

\begin{equation}
w_{_d}(t)\ \overset{def.}{=}\ \overset{d-1}{\underset{j=0}{\prod}}\ \left(1-q^{^j}t\right).
\label{eq:wp}
\end{equation}

Each factor in (<ref>) is a Whitney polynomial of the poset of subsets of a $d$-element set.

Building connection with categorical structures, we the notion of posets. In order theory, a poset is a partially-ordered set, namely an arrangement such that, for certain pairs of elements, one precedes the other. The adjective partial denotes the fact that not every pair of elements needs to be comparable; indeed, there may be pairs for which neither element precedes the other. Partial order, thus generalises total orders, in which ever pair is comparable. Given this, one can further define the order complex associated to a poset $(S,\le)$ which has the set $S$ as vertices, and the finite chains of $(S,\le)$ as faces. To a poset, one can associate a poset topology, namely a topological structure on the poset of finite chains of $(S,\le)$, ordered by inclusion. Using methods from poset topology, it is possible to construct a chain complex of ${\cal C}$-modules $k_{_d}(V), \forall\ d\in\mathbb{N}$, such that exactness of $k_{_d}(V)$ categorifies the equation

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

\begin{equation}
w_{_d}(t)\ \text{HS}_{_V}(t)\ \equiv\ p(t),
\end{equation}

for $p(t)$ a polynomial. If $V$ is finitely generated, either $k_{_d}$ or a power of it applied to $V$ is exact. In particular, categorification explains why the denominator (<ref>) appears. This goes back to the construction outlined in our main example when integrating over the Kirwan map, (<ref>).

§.§ Poset homology

Poset homology are chain complexes classifying rationality.
Extended literature on the subject explains there is a method for assigning a chain complex to a module over a combinatorial category, such that, if the complex is exact, the module has a rational Hilbert series. Furthermore, these complexes are known to satisfy Homology-vanishing theorems. Hasse diagrams, discussed in [3], are an example of chain complexes associated to poset representations. Homological mirror symmetry as an equivalence in between Hilber series on two categories, and rationality, are a direct biproduct of the defining features of poset homology, and constitutes a particular example of dualisable categorical structures, [136].

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

§.§ Moment map and higher homologies

In general, homologies of derived representation schemes can be highly nontrivial. However, in this particular case, one can identify a necessary and sufficient condition for the vanishing of the higher homologies based on the flatness of $\mu$, [330]. In particular, in such reference, it was shown that the derived representation scheme DRep$_{_{v,w}}({\cal A})$ has vanishing higher homologies if and only if $\mu^{^{-1}}(0)\ \subset\ M({\cal Q},v,w)$ is a complete intersection, which happens only if the moment map is flat, [341]. As explained in [3, 4], the requirement of the algebraic variety to be a complete intersection is crucial for the purpose of our treatment. In particular, it ensures the emergence of a 2-categorical structure, whose importance will be the core topic of sections <ref> and <ref>.

§ DRINFELD CENTERS FROM MAGNETIC QUIVERS

We now turn to address a key issue highlighted in section <ref>, namely how to define a fiber functor associated to multiple gaugings not isomorphic to each other. For completeness, prior to explaining how this can be solved combining magnetic quivers and Drinfeld centers[The definition of the latter will be provided in due course.], we will first review the features of the setup outlined in section <ref> needed in the present treatment. This section is structured into three parts:

* Relying upon the description of the HB and CBs as algebraic varieties (cf. 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 turn to explaining how the identification of such moment map ensures the quiver gauge theory enjoys a generalisation of homological mirror symmetry, with the latter corresponding to the presence of a Drinfeld center and associated fiber functor for a given 2-categorical structure, related to Rozansky-Witten theory, [329].

* We conclude opening a connection between the topics outlined in the present work and those of [181, 182].

The relation between posets, complex cohomology, and Drinfeld centers, will be explored in greater mathematical detail in section <ref>. This relation is intrinsic in this section, thanks to the connections outlined in section <ref>; however, we are confident that a more mathematical treatment could also lead to further interesting insights that are worth exploring.

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

§.§ Gauging and Hasse diagrams

We first start by recalling the configuration of interest outlined in section <ref>, motivating the importance of the Drinfeld center in geometric representation theory, [182].

§.§ Fiber functors

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

This picture reproduces an example of a correspondence, explained in section <ref>.

In our previous work, [1], we already argued that, unlike figure <ref>, the third picture from the left in figure <ref> does not admit a straightforward expression for the fiber functor as (<ref>). Nevertheless, we claimed that a composite fiber functor could still be assigned thanks to the identification of the underlying algebraic structure of the composite theories

\begin{equation}
\boxed{ \color{white}{bla} \color{black}\large _{_{{\cal A}_{_1}}}{\cal A}_{_{{\cal A}_{_2}}}\ \overset{def.}{\equiv}\ {\cal A}_{_1}\ \otimes_{_{\cal A}}\ {\cal A}_{_2}\ \ \color{white}\bigg]\  }
\end{equation}

Our proposal in [1] was that the corresponding fiber functor should really be defining the partition function of a 3D theory, whose identification we anticipated being the core topic of the present work. The remainder of this section is devoted to prove that the resulting 3D theory is a 3D ${\cal N}=4$ quiver gauge theory whose HB, when viewed as a complete intersection algebraic variety, enables to define the Drinfeld center of its underlying 2-categorical structure. In doing so, we need to completely specify its 1- and 2-morphisms, namely the generators and the relations in between them, both needed in the evaluation of the HB Hilbert series. As explained in section <ref>, the HS is nothing but the sum of gauge-invariant algebraic operators living in the chiral ring of the theory.

The Drinfeld center is defined as the 0$^{th}$-Hochschild cohomology, namely the endomorphisms of the identity element of a given category, ${\cal C}$:

\begin{equation}
\boxed{\color{white}{blan} \color{black}\mathfrak{Z}({\cal C})\ \color{black}\overset{def.}{=}\ \text{End}(\mathbf{1}_{_{{\cal C}}})\ \equiv\ \text{Hom}(\mathbf{1}_{_{{\cal C}}},\mathbf{1}_{_{{\cal C}}})\ \equiv\ HH^{^{{\bullet}\  0}}({\cal C})\ \ \large\color{white}\bigg]\  }
\end{equation}

The existence of the Drinfeld center and that of the fiber functor are mutually related by definition, since

\begin{equation}
\boxed{\color{white}{blan} \color{black}{\cal F}:\ \mathfrak{Z}({\cal C})\ \longrightarrow\ {\cal C}\ \ \large\color{white}\bigg]\  }.
\end{equation}

From what we have just outlined, it clearly follows that, the issue of identifying the Drinfeld center is equivalent to that of assigning a fiber functor projecting to the category ${\cal C}$. For the purpose of our treatment, namely building connection with geometric representation theory, ${\cal C}$ is really meant to be the category of representations, with the latter being described in terms of 2D TFTs, namely Lagrangian submanifolds living in a symplectic manifold $X$, playing the role of boundary conditions for 3D Rozansky-Witten theory, $RW_{_X}$. the meaning of this will be explained in section <ref>.

§.§ Relation to Hasse diagrams

The relation in between gauging and the construction of the Hasse diagram basically follows from the underlying role played by the moment map. In the former, it comes hand-in-hand with the definition of the fiber functor, meaning a partition function can be assigned to the resulting absolute theory, whereas, in the context of quiver gauge theories, the flatness of the moment map ensures a Hilbert series can be given, thereby fully specifying the generators and relations leading to the Hasse diagram construction itself. Importantly, the poset structure of the Hasse diagram admits a complex cochain description, thereby opening the possibility of studying this from a more mathematically rigorous point of view.

§.§ Drinfeld centers and mirror symmetry

As a key result of our work, we therefore wish to highlight the following statement as being applicable to the theories specified in the previous section <ref>, namely quiver gauge theories with 8 supercharges,

Key point: identifying the Drinfeld center is equivalent to assigning a Hilbert series with irreducible representations, i.e. specifying the cones in the Hasse diagram, and their intersection.

The remainder of the present section is devoted to proving such assertion. In doing so, we mostly refer to [181, 182], which was one of the first motivations of this work. A more detailed mathematical description of the correspondence with such references, as well as with [332], is the core topic of the remaining sections. For the moment, we wish to highlight that such a correspondence exists, and can be understood from a mathematical physicist's perspective. As a bi-product, we show how magnetic quivers can be combined with higher categories to extend the setup of  [181, 182] for describing higher dimensional QFTs admitting a quiver gauge theory description.

Prior to doing so, we will first briefly recapitulate the notion of homological mirror symmetry, introduced in section <ref>

§.§ Homological mirror symmetry

For our purposes, we are mostly interested in a geometrical representation theory realisation of homological mirror symmetry, for the same motivations outlined in [181]. As previously explained in [340], 3D mirror symmetry can be rephrased in terms of topological representation theory thanks to the Drinfeld center, [182]. We now briefly overview such argument, in order to make contact with the setting we are dealing with.
For the case in which the two Lagrangians are related by mirror symmetry, such as is the case between Neumann and Dirichlet boundary conditions, the LHS of figure <ref> is such that the intersection is unable to account for the Drinfeld in a straightforward manner, signalling the presence of a non-invertible defect separating the resulting absolute theories.

\begin{equation}
\boxed{ \color{white}{bla} \color{black} \text{Vect}<G>\ \otimes_{_{\mathfrak{Z}}}\ \text{Rep}(G)\ \equiv\ \text{Vect}\ \ \color{white}\bigg]\  }
\end{equation}

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

The black arrow in the diagram on the RHS denotes the fiber functor mapping from the Drinfeld center to the category of representation. Correspondingly, on the LHS, the Drinfeld center is the green shaded region whose boundaries are the regular and trivial representations, denoted in blue and red, respectively. For the case in which the two are related by ordinary mirror symmetry, their intersection contains the same information as $\mathfrak Z$, and therefore a fiber functor which is fully forgetful can be assigned.

§.§ Drinfeld centers from magnetic quivers of 3D ${\cal N}=4$ gauge theories

In this concluding section, we combine the tools encountered throughout our treatment, drawing additional major conclusions, as well as setting the stage for the core topic of the next section, first presented by the author in [5].

§.§ 2-fiber products

Some preliminary mathematical tools. If $\cal C$ is a 2-category of categories, then there is a notion of a 2-fiber product, $\cal C$$_{_1}\ \times_{_{\cal D}}\ \cal C$$_{_2}$, also denoted by a diagram

\begin{equation}
{\cal C}_{_{1}}\ \xrightarrow{f}\ {\cal D}\ \xleftarrow{g}\ {\cal C}_{_2}
\end{equation}

is the category of triples $(c_{_1}, c_{_2},\phi)$, where $c_{_1}\in{\cal C}_{_1}$, $c_{_2}\in{\cal C}_{_2}$, and $\phi:f(c_{_1})\ \xrightarrow{\sim}\ g(c_{_2})$ is an isomorphism of their images $\cal D$. For an object $P\in\cal C$, a 2-fiber product $X$$\ \times_{_S}\ $$Y$ associated to the diagram

\begin{equation}
X\ \xrightarrow{f}\ S\ \xleftarrow{g}\ Y
\end{equation}

is a quadruple $(P,p,q,\phi)$, with 1-morphisms $p:P\rightarrow X, q:P\rightarrow Y$, and a 2-isomorphism

\begin{equation}
\phi: f\ \circ\ p\ \simeq\ g\ \circ\ q
\end{equation}

such that $\forall Z\in\cal C$, the natural functor

\begin{equation}
\text{Hom}_{_{\cal C}}\left(Z,P\right)\ \rightarrow\ \text{Hom}_{_{\cal C}}\left(Z,X\right)\times_{_{\text{Hom}_{_{\cal C}}(Z,S)}}\ \text{Hom}_{_{\cal C}}(Z,Y)
\end{equation}

is an equivalence of categories.

§.§ Rozansky-Witten Theory and the Drinfeld center

One of the examples encountered in the previous section, namely $SU(2)$ with $N_{_f}=2$, is the simplest case involving two cones with nontrivial intersection in its Hasse diagram. As such, it is an example where ordinary mirror symmetry in the sense of (<ref>) does not apply for the 3D ${\cal N}=4$ theory arising under dimensional reduction. Nevertheless, a suitable mirror dual can still be assigned once the generators of the two cones featuring in the Hasse diagram, together with their intersection, are correctly accounted for in the calculation of the Hilbert series (HS), i.e. the partition function counting the gauge-invariant operators living in the chiral ring.
In the previous section we explained how the magnetic quiver prescription enables to identify the HS correctly for cases where ordinary mirror symmetry does not apply.

To make contact with the calculation of Drinfeld centers, let us remind ourselves of the case of 4D $SU(2)$ ${\cal N}=2$, first studied by Seiberg and Witten within the context of its 3D reduction. Its low-energy limit is described as a sigma-model in the space of vacua, corresponding to the Atiyah-Hitchin moduli space of charge 2 monopoles. The resulting 3D $SU(2)$ theory was then identified with the Rozansky-Witten theory (RW) of the hyper-kah$\ddot{\text{a}}$ler Atiyah-Hitchin manifold. Later developments of this by Kapustin, Rozansky, [333], and later with Saulina, [334], addressed their 2-category of branes, with the latter containing smooth holomorphic Lagrangians, $L$.

Adaptation of the RW$_{_X}$ theory (on the LHS) to the quiche of a relative field theory (RHS).

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

* the forgetful functor, keeping track of the underlying category (describing the pre-gauged TQFT), represented by the regular representation of $G$,

* the invariant category, generating the gauged TQFT, associated to the trivial representation of $G$.

This already encodes the idea of mirror symmetry, whose generalisation is realised thanks to the composite fiber functor described in [181], and overviewed in the concluding part of this section in relation to our analysis.

The objects in the 2-category of KRS are branes in the BFM space whose boundaries are Lagrangians $L,L^{^{\prime}}\ \subset\ X$, $X$ a symplectic manifold, specifying the RW$_{_X}$ according to the choice of $X$. $L,L^{^{\prime}}$ are 1-morphisms generating the theory upon acting on the trivial theory

\begin{equation}
\boxed{\ \ \  L,L^{^\prime}:\ \text{Id}\ \longrightarrow\ RW_{_X}\color{white}\bigg]\ \ }.
\end{equation}

The BFM space for an abelian (LHS) and non-abelian (RHS) gauge theory, [181]. The KRS objects living in them are the objects in the 2-category referred to in the text and the trivial and regular representations are the Lagrangian submanifolds $L,L^{^{\prime}}$. Their intersection are Hom-spaces.

The 2-morphisms are denoted by Hom$(L,L^{^{\prime}})$. For the case in which $L, L^{^{\prime}}$ are related by ordinary mirror symmetry, the 2-morphisms encode all the information, and the 2-category can be completely reconstructed from them. For the case of non-ablelian gauge groups, a generalisation of the notion of mirror symmetry is needed. Practically, this means we need to specify, both, the 1-morphisms and the 2-morphisms to reconstruct the bulk of the BFM space, and completely describe the KRS branes living in them.

In connection with our previous work, [1], it is instructive to visualise what we have just said as shown in figure <ref>. Fully specifying the topological boundary conditions leads to an absolute theory. In case Hom$(L,L^{^\prime})$ leads to a non-invertible defect, the fiber functor, hence the Drinfeld center in figure <ref> needs a 2-fiber product structure to be fully specified, as also explained in [1]. In particular, this highlights the importance of specifying the entire underlying 2-categorical structure, namely the Lagrangian submanisfolds as well as the homomorphisms between them. This ensures the objects in the 2-category of KRS, can be fully specified. From the arguments outlined in the preliminary part of this work, we are therefore led to conclude that the configuration on the RHS of figure <ref> is dual to another theory with the same Drinfeld center and only one Lagrangian submanifold on the topological boundary condition, thereby providing a sample realisation of generalised homological mirror symmetry.

§.§ Symplectically induced representations

In conclusion, we outline more explicitly the connection with the work of [181], setting the stage for a more detailed mathematical treatment, which is the main focus of [5].

The composition of fiber functors associated to double symplectic inductions leading to the regular and trivial representations.

One of the main aims of [181] is that of identifying a preferred family of simple representations, foliating the BFM space, ultimately describing the space of possible representations. The prescription outlined in such reference is motivated by the realisation of homological mirror symmetry. The construction of the preferred family of foliations also requires making use of a 2-fiber product, defined as follows

\begin{equation}
\boxed{ \large\ \ \ {\cal T}\ \times_{_{T^{^*}_{_{reg}}G_{_{\mathbb{C}}}}}\ Z_{_{reg}}\ \ \color{white}\bigg] }
\label{eq:composite}
\end{equation}

with ${\cal T}$ and $Z_{_{reg}}$ denoting the trivial and regular fibers, respectively. (<ref>) results from the composition outlined in figure <ref>, where

\begin{equation}
BFM(G)\ \overset{def.}{=}\ Z_{_{reg}}/G_{_{\mathbb{C}}}\ \ \ \ \ \ ,\ \ \ \ \ T(G)\ \overset{def.}{=}\ (N, \chi) \backslash\backslash T^*G_{_{\mathbb{C}}} // (N, \chi)\ \equiv\ N\ \backslash\ {\cal T}\ /N,
\end{equation}

\begin{equation}
\chi:\ n\ \longrightarrow\ \mathbb{C}^{^{\times}}
\end{equation}

the regular character. Every such representation is symplectically induced from a 1D representation of a certain Levi subgroup $G$. As such, the preferred family of foliations resulting from (<ref>) is the Fukaya category of a flag variety of $G$.

Given the considerations made when describing figure <ref> in the context of RW-theory, we therefore notice that the aim of describing objects in the bulk of figure <ref> is equivalent to that of defining the Drinfeld center for two absolute theories separated by a non-invertible defect, as encountered in [1]. Our main assertion is that, thanks to the prescription outlined in section <ref>, in presence of multiple symplectically-induced representations, the resulting composite fiber functor defines the Drinfeld center of a mirror 3D ${\cal N}=4$ gauge theory, resulting from the union of multiple Coulomb branches of magnetic quivers associated to 3D ${\cal N}=4$ theories, thereby constituting a generalised realisation of homological mirror symmetry.

A more mathematical reformulation of this is the core topic of upcoming sections, [5].

§ KEY POINTS

We can summarise the main results of sections <ref> and <ref> as follows:

* The 2-categorical structure description of the HB Hasse diagram of quiver gauge theories with 8 supercharges is equivalent to that of KRW.
Importantly, this follows from the dimensional reduction of 4D ${\cal N}=2$ $SU(2)$ gauge theory with $N_{_f}=2$, as the most basic case with HB Hasse diagram consisting of two intersecting cones.

* Importantly, the flatness of the moment map, ensuring the vanishing of higher homology, [330], corresponds to the possibility of fully specifying the HS associated to the HB of a given theory. We emphasise that this is equivalent to stating the existence of a fiber functor in a 2-category, ultimately leading to the definition of a partition function of an absolute theory, according to the prescription of [102].

* The identification of the MQs whose CBs equal the HB of the original theory, corresponds to the identification of the Drinfeld center of a 3D theory, with the latter being a RW theory, $RW_{_X}$, whose BCs are Lagrangian submanifolds within a certain symplectic manifold, $X$. The complete reconstruction of the BFM space, where the objects of the 2-category live, corresponds to defining the Drinfeld center w.r.t. the given Lagrangian submanifolds and their intersection, i.e. the 1- and 2-morphisms characterising the 2-category in question.

* The fiber functor defining the partition function of interest is really obtained as a 2-fiber product, and is therefore compatible with the result of [181], motivated by generalising homological mirror symmetry.

§ MOORE-TACHIKAWA VARIETIES: BEYOND DUALITY

We now have all the tools needed for addressing an open question raised in section <ref>, namely extending Moore-Tachikawa varieties beyond categorical duality. In doing so, we will be exploiting tools from algebraic geometry and higher-category theory.

In particular, we wish to stress the importance of dealing with the hyperk$\ddot{\text{a}}$hler structure of the target 2-category of 2D TFTs describing Moore-Tachikawa varieties, and how this is related to the emergence of new higher-categorical structures requiring generalisation of the ordinary homological mirror symmetry/3D mirror symmetry correspondence.

The starting point is the following relation in between class ${\cal S}$ theories and the gluing of their AGT dual 2D TFTs, [125],

\begin{equation}
\boxed{\ \ \ S_{_L}\ \times_{_{G,q}}\ S_{_R}\ =\ S\left(C_{_L}\ \times \ C_{_R}, D_{_{LR}}\right)\ \color{white}\bigg] },
\label{eq:cclS}
\end{equation}

where $(C_{_L}, D_{_L}), (C_{_R}, D_{_R})$ are decorated 2-surfaces. The AGT correspondence states the equivalence holds in general. However, Moore and Tachikawa pointed out some interesting shortcomings for the case in which the 2-surface $C$ is a sphere with two punctures. In such case, they claim that one does not achieve a genuine 4D class ${\cal S}$ theory. The main aim of the present section is that of proving that this issue could be overcome by making use of a generalised notion of homological mirror symmetry/3D mirror symmetry correspondence, which, from the perspective of Moore-Tachikawa varieties, amounts to accounting for the hyperk$\ddot{\text{a}}$hler structure of the target 2-category of the 2D TFT AGT-dual to the composite class ${\cal S}$ theory on the LHS of (<ref>).

Prior to delving to delving into explaining a possible way of constructing a 2D TFT whose target category is a hyperk$\ddot{\text{a}}$hler quotient, we briefly overview the work of [125] who first pointed out the hyperk$\ddot{\text{a}}$hler structure needed generalisation of the target category of the 2D TFT $\eta_{_{G_{_{\mathbb{C}}}}}$.  We briefly overview some specific examples they outlined, explaining the relation with our treatment in Part I when dealing with star-shaped quivers.

Upon removing the identity from the source category, a morphism is now defined as a pair $(C,a)$, when $C$ is an orientable 2-manifold with boundaries, and $a$ is a positive real number which can be thought of as the area of $C$.

The composition of the two morphisms adds the area. For example, we see that

\begin{equation}
\underset{a\rightarrow 0}{\lim}\ (W,a)\ =\ \mathbb{C}^{^2}\ \otimes\ \mathbb{C}^{^2}\ \otimes\ \mathbb{C}^{^2},
\end{equation}

which is equivalent to the Coulomb branch encountered in Part I. Similarly, for $\mathfrak{g}=A_{_2}$, $\eta_{_{A_{_2}}}(W,a)$ is equivalent to the minimal nilpotent orbit of E$_{_6}(\mathbb{C})$ as the hyperk$\ddot{\text{a}}$hler manifold in the $\underset{a\rightarrow 0}{\lim}$. At finite $a$, the hyperk$\ddot{\text{a}}$hler metric is only invariant under $SU(3)^{^3}$ and not the whole $E_{_6}$.

§.§ Categorical structure without duality

As explained in section <ref>, removing the identity element in Bo$_{_2}$ requires having to introduce at least two different functors $\eta_{_{G_{_{\mathbb{C}}}^{^{\prime}}}}, \eta_{_{G_{_{\mathbb{C}}}^{^{\prime\prime}}}}$, whose action on the circle and its bordism reads as follows

\begin{equation}
\eta_{_{G_{_{\mathbb{C}}}^{^{\prime}}}}\left(\ S^{^1}\ \right)\ \equiv\ G_{_{\mathbb{C}}}^{^{\prime}}\ \ \ , \ \ \ \eta_{_{G_{_{\mathbb{C}}}^{^{\prime\prime}}}}\left(\ S^{^1}\ \right)\ \equiv\ G_{_{\mathbb{C}}}^{^{\prime\prime}}
\label{eq:morph0}
\end{equation}

\begin{equation}
\eta_{_{G_{_{\mathbb{C}}}^{^{\prime}}}}\left(\ U\ \right)\ \equiv\ U_{_{G_{_{\mathbb{C}}}^{^{\prime}}}} \equiv \  G_{_{\mathbb{C}}}^{^{\prime}}\times S_{_n}\ \subset \ G_{_{\mathbb{C}}}^{^{\prime}}\times\mathfrak{g}_{_{\mathbf{C}}}^{^{\prime}}\ \simeq \ T^{^*}G_{_{\mathbf{C}}}^{^{\prime}},
\label{eq:morph1}
\end{equation}

\begin{equation}
\eta_{_{G_{_{\mathbb{C}}}^{^{\prime\prime}}}}\left(\ U\ \right)\ \equiv\ U_{_{G_{_{\mathbb{C}}}^{^{\prime\prime}}}} \equiv \  G_{_{\mathbb{C}}}^{^{\prime\prime}}\times S_{_n}\ \subset \ G_{_{\mathbb{C}}}^{^{\prime\prime}}\times\mathfrak{g}_{_{\mathbf{C}}}^{^{\prime\prime}}\ \simeq \ T^{^*}G_{_{\mathbf{C}}}^{^{\prime\prime}},
\label{eq:morph2}
\end{equation}

\begin{equation}
\eta_{_{G_{_{\mathbb{C}}}^{^{\prime}}}}, \left(\ V\ \right)\ \equiv\ V_{_{G_{_{\mathbb{C}}}^{^{\prime}}}} \overset{def.}{\equiv} \  T^{^*}G_{_{\mathbb{C}}}^{^{\prime}},
\label{eq:id1}
\end{equation}

\begin{equation}
\eta_{_{G_{_{\mathbb{C}}}^{^{\prime\prime}}}}\left(\ V\ \right)\ \equiv\ V_{_{G_{_{\mathbb{C}}}^{^{\prime\prime}}}} \overset{def.}{\equiv} \  T^{^*}G_{_{\mathbb{C}}}^{^{\prime\prime}},
\label{eq:id2}
\end{equation}

where we are assuming

\begin{equation}
\mathfrak{g}_{_{\mathbf{C}}}^{^{\prime}}\ \cap\ \mathfrak{g}_{_{\mathbf{C}}}^{^{\prime\prime}}\ \neq\ \{ \emptyset \}\   \ , \ \ \text{and}\ \ \   \mathfrak{g}_{_{\mathbf{C}}}^{^{\prime}}\ \cup\ \mathfrak{g}_{_{\mathbf{C}}}^{^{\prime\prime}}\ \equiv\ \mathfrak{g}_{_{\mathbf{C}}},
\label{eq:fa}
\end{equation}

\begin{equation}
\mathfrak{g}_{_{\mathbf{C}}}^{^{\prime}}\ \not\subset\ \mathfrak{g}_{_{\mathbf{C}}}^{^{\prime\prime}}\    \ , \ \ \text{and}\  \  \ \ \mathfrak{g}_{_{\mathbf{C}}}^{^{\prime\prime}}\ \not\subset\ \mathfrak{g}_{_{\mathbf{C}}}^{^{\prime}}.
\label{eq:sa}
\end{equation}

(<ref>) and (<ref>) imply that the two subalgebras involved, $\mathfrak{g}_{_{\mathbf{C}}}^{^{\prime}}, \mathfrak{g}_{_{\mathbf{C}}}^{^{\prime\prime}}$ are associated to different subgroups, $G_{_{\mathbb{C}}}^{^{\prime}}, G_{_{\mathbb{C}}}^{^{\prime\prime}} $, and that the identity elements differ, $T^{^*}G_{_{\mathbf{C}}}^{^{\prime}}\ \neq\ T^{^*}G_{_{\mathbf{C}}}^{^{\prime\prime}}$, even under reparametrisation of the Reimann surface. For each one of the 2D TFTs, $\eta_{_{G_{_{\mathbb{C}}}^{^{\prime}}}}, \eta_{_{G_{_{\mathbb{C}}}^{^{\prime\prime}}}}$ one could use the formalism of [126, 125], describing two different class ${\cal S}$ theories, both descending from 6D ${\cal N}=(2,0)$ by dimensionally reducing on a Riemann surface without reparemetrisation invariance. However, given the assumption that $(G_{_{\mathbb{C}}}^{^{\prime}}, \mathfrak{g}_{_{\mathbf{C}}}^{^{\prime}}), (G_{_{\mathbb{C}}}^{^{\prime\prime}}, \mathfrak{g}_{_{\mathbf{C}}}^{^{\prime\prime}}) $ can be embedded in a unique $(G_{_{\mathbb{C}}}, \mathfrak{g}_{_{\mathbb{C}}})$, it is natural to ask what should the triple on the RHS of (<ref>) be for the resulting theory to be absolute?

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

\begin{equation}{\cal F}\ :\ \mathfrak{Z}\left(\text{Bo}_{_2}\right)\ \rightarrow\ \text{Bo}_{_2},
\label{eq:F}
\end{equation}

\begin{equation}
\mathfrak{Z}\left(\text{Bo}_{_2}\right)\equiv\text{End}_{_{\text{Bo}_{_2}}}\left(T^{^*}G_{_{\mathbf{C}}}\right)\equiv\text{Hom}_{_{\text{Bo}_{_2}}}\left(T^{^*}G_{_{\mathbf{C}}},T^{^*}G_{_{\mathbf{C}}}\right),
\label{eq:Z}
\end{equation}

both of which crucially rely upon the presence of an identity in the source and target of $\eta_{_{G_{_{\mathbb{C}}}}}$. Apparently, we run into a contradiction, since the identity element $T^{^*}G_{_{\mathbf{C}}}$ has been removed by assumption, thereby implying $\mathfrak{Z}\left(\text{Bo}_{_2}\right)$ cannot be defined in the ordinary way.   On the other hand, one could define the new identity as being a composite object defined in the following way

\begin{equation}
\boxed{\ \ \ T^{^*}\tilde G_{_{\mathbf{C}}}\ \overset{def.}{=}\ T^{^*}G_{_{\mathbf{C}}}^{^{\prime}}\ \otimes_{_{T^{^*}G_{_{\mathbb{C}}}}}\ T^{^*}G_{_{\mathbf{C}}}^{^{\prime\prime}}\color{white}\bigg]\ \ },
\label{eq:newid}
\end{equation}

and therefore our proposal for (<ref>) and (<ref>) reads as follows

\begin{equation}
\boxed{\ \ \ {\cal F}\ :\ \mathfrak{Z}\left(\tilde{\text{Bo}_{_2}}\right)\ \rightarrow\ \tilde{\text{Bo}_{_2}}\color{white}\bigg]\ \ },
\label{eq:Fnew}
\end{equation}

\begin{equation}
\boxed{\ \ \ \mathfrak{Z}\left(\tilde{\text{Bo}_{_2}}\right)\ \equiv\ \text{End}_{_{\tilde{\text{Bo}_{_2}}}}\left(T^{^*}G_{_{\mathbf{C}}}^{^{\prime}}\ \otimes_{_{T^{^*}G_{_{\mathbb{C}}}}}\ T^{^*}G_{_{\mathbf{C}}}^{^{\prime\prime}}\right)\color{white}\bigg]\ \ },
\label{eq:Znew}
\end{equation}

\begin{equation}
\tilde{\text{Bo}_{_2}}\ \overset{def.}{=}\ \text{Bo}_{_2}\ /\ V.
\end{equation}

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

§.§ Interesting shortcomings

As explained in [1], the loss of reparametrisation invariance of the Riemann surface signals the presence of an intrinsically-non-invertible defect between different class ${\cal S}$ theories. We will now show that the setup described in section <ref> is equivalent to that of [1].

The starting point in our argument is the conjecture (<ref>) and (<ref>). In absence of categorical duality of, both, source and target, the 2D TFT associated to the maximal dimensional Higgs branch of [125] is no longer associated to a moment map that is equivalent for all the constituent $S^{^1}$s, thereby violating the conjecture made by [125]. Explicitly,

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

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

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

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

\begin{equation}
T^{^*}G_{_{\mathbf{C}}}^{^{a^{\prime}}}\ \circ\ T^{^*}G_{_{\mathbf{C}}}^{^a}\equiv\left(T^{^*}G_{_{\mathbf{C}}}^{^a}\ \times\ T^{^*}G_{_{\mathbf{C}}}^{^{a^{\prime}}} \right)\dslash G_{_{\mathbf{C}}},
\label{eq:compths}
\end{equation}

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