On the other hand, in absence of reparametrisation-invariance, the resulting class ${\cal S}$ theories would, by definition, be separated by an intrinsically non-invertible defect, with the latter being responsible for the lack of reparametrisation invariance of the Riemann surface [126].

Our main question is to find the Drinfeld center for a given Bo$_{_2}$

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

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

\begin{equation}
\mu:\ \mathfrak{G}\ \rightarrow\ {\cal A}_{_c}.
\end{equation}

with $\mathfrak{G}$ being Bo$_{_2}$ in this case, and ${\cal A}$ the algebra of invertible topological defects that projects to the identity $T^{^*}G_{_{\mathbb{C}}}$ under complete gauging of the theory. This is equivalent to stating that $T^{^*}G_{_{\mathbb{C}}}$ is the identity element once having projected over ${\cal A}_{_c}$

\begin{equation}
{\cal A}\ \dslash_{\mu}\ \mathfrak{G}\ \ \ \text{with}\ \ \ \mu:\ \mathfrak{G}\ \rightarrow\ {\cal A}_{_c}
\end{equation}

choosing the definition of the identity in the following way

\begin{equation}
T^{^*}G_{_{\mathbb{C}}}\ \simeq\ \mathbf{1}_{_{G_{_{\mathbb{C}}}}}\ \equiv\ {\cal A}\ \dslash_{\mu}\ \mathfrak{G}.
\end{equation}

For the purpose of our work, $\mathfrak{G}\ \overset{def.}{=}\ \text{Bo}_{_2}$, therefore

\begin{equation}
T^{^*}G_{_{\mathbb{C}}}\ \simeq\ \mathbf{1}_{_{G_{_{\mathbb{C}}}}}\ \equiv\ {\cal A}\ \dslash_{\mu}\ \text{Bo}_{_2}
\end{equation}

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

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

whose identity element

\begin{equation}
\tilde{\cal A}_{_c}\ \overset{def.}{=}\ {\cal A}_{_1}\ \otimes_{_{T^{^*}G_{_{\mathbb{C}}}}}{\cal A}_{_{2}}
\end{equation}

constitutes the identity of the composite theory. If $T^{^*}G_{_{\mathbb{C}}}$ is the identity that has been removed from a particular source category, it still exists, but is no longer the identity present in $\tilde{\text{Bo}}$$_{_2}$ of a given $\tilde\eta_{_{G_{_{\mathbb{C}}}}}$. From the RHS of (<ref>), the identity of $\tilde{\text{Bo}}_{_2}$ therefore reads

\begin{equation}
T^{^*}\tilde G_{_{\mathbf{C}}}\ \overset{def.}{=}\
T^{^*}G_{_{\mathbf{C}}}^{^{\prime}}\ \otimes_{_{T^{^*}G_{_{\mathbb{C}}}}}\ T^{^*}G_{_{\mathbf{C}}}^{^{\prime\prime}},
\label{eq:newidnow}
\end{equation}

such that its Drinfeld center can be determined.

Defining $\tilde\mu:\ \tilde{\text{Bo}}_{_2}\rightarrow\ \tilde{\cal A}$ as the moment map associated to the composite theory, the corresponding fiber functor can be explicitly rewritten as follows

\begin{equation}
\boxed{\ \ \ {\cal F}:\ \mathfrak{Z}\left(\tilde\eta_{_{\tilde G_{_{\mathbb{C}}}}}^{^{-1}}\left(T^{^*}\tilde G_{_{\mathbb{C}}}\right)\right)\ \rightarrow\ \tilde\eta_{_{\tilde G_{_{\mathbb{C}}}}}^{^{-1}}\left(T^{^*}\tilde G_{_{\mathbb{C}}}\right)\color{white}\bigg]\ \ }.
\label{eq:fiber}
\end{equation}

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

§ KEY POINTS

The main points are the following:

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

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

§.§ Back to the identity from moduli spaces

In this concluding section, we gather the key tools outlined in this work, highlighting some main features leading towards the relation between homological mirror symmetry and 3D mirror symmetry.

3D mirror symmetry is defined as a duality exchanging Coulomb and Higgs branches, [135]. Making use of their algebro-geometric definition, briefly overviewed in section <ref>, together with our previous work, [4], we will be arguing that ordinary 3D mirror symmetry, formulated as a 1-to-1 correspondence between a Coulomb and a Higgs branch, can be generalised to cases involving more theories (namely more Coulomb branches for a given Higgs branch). Specifically, we will be referring to the case of supersymmetric quiver gauge theories admitting a magnetic quiver description, and whose Higgs branch splits into two cones. This will ultimately provide further evidence to the claims made in [3].

As we shall see, the quantity that will enable us to probe such duality structure is the categorical interpretation of Hochschild cohomologies of block algebras, <ref>.

The present section is structured into three parts:

* At first, we briefly overview the definition of block algebras, their associated Hoschschild cohomology, [349], and the relation with the Drinfeld center, the latter following from our preliminary discussion in section <ref>.

* We then turn to Coulomb and Higgs branches arising from Hitchin system for class ${\cal S}$ theories, [120]. In particular, we highlight the fact that determining the Hitchin base, namely the Coulomb branch of the moduli space of vacua of class ${\cal S}$ theories, from the Higgs branch, is equivalent to specifying the identity in the embedding category, in complete analogy with the criterion outline din section <ref> and <ref>.

* Ultimately, we combine the tools outlined in the remainder of the present treatment, together with our previous work, [4], explaining how in presence of algebraic varieties related to Moore-Tachikawa varieties without categorical duality, admit a generalised notion of 3D mirror symmetry thanks to the analogy with algebraic gauging in SymTFT constructions leading to intrinsically-non-invertible defects.

§.§.§ Hilbert series of Hochschild cohomology of block algebras

In this first subsection, we will briefly overview a key result of [349], combining it with material outlined in section <ref>.

Let $p$ be a prime, $k$ an algebraically closed field of characteristic $p$, $G$ a finite group, and $B$ a block algebra of $kG$. A defect group of $B$ is a minimal subgroup $P$ of $G$ such that $B$ is isomorphic to a direct summand of $B\otimes_{_{kP}}B$ as a $B-B$-bimodule. The defect groups of $B$ form a $G$-conjugacy class of $p$-subgroups of $G$, and the defect of $B$ is the integer $d(B)$ such that $p^{^{d(B)}}$ is the order of the defect group of $B$.

Donovan's conjecture predicts that there should only be finitely-many Morita equivalence classes of block algebras of Hochschild cohomology algebras of block algebras with a fixed defect $d$. For any integer $n\ \in\ \mathbb{N}$, dim $HH^{^n}(B)$ is bounded in terms of the defect groups of the block algebra $B$.

The first result in [349] adds to this the fact that there is a bound on the degrees of the generators and relations of the Hochschild cohomology $HH^{\bullet}(B)$, which only depends on the defect of $B$. In [349], this comes in three theorems, ultimately claiming that, the Hilbert series of the Hochschild cohomology of a block and its defect determine each other up to finitely many possibilities. In particular,

\begin{equation}
HH^{^{{\bullet}\ 0}}(B)\ \equiv\ \mathfrak{Z}(B),
\end{equation}

where $\mathfrak{Z}(B)$ is the Drinfeld center.

§.§.§ Hitchin systems and mirror quivers

We now turn to the more specific setup of interest to us, namely that of class ${\cal S}$ theories. In particular, we are interested in their moduli space of vacua, as well as their description in terms of 2D TQFTs, thanks to the AGT correspondence.

§.§ Hitchin systems

A Hitchin systems is an integrable system depending on the choice of a complex reductive group and a compact Riemann surface. It lies at the interface of algebraic geometry, Lie algebra theory, and integrable systems. It also plays a crucial role in the geometric Langlands program.

The Hitchin equations consist in a system of PDEs on the Riemann surface, $C$, concerning a triple $(E, D, \varphi)$, [120], where:

* $E$ is a $G$-bundle on $C$.

* $D$ is a $G$-connection in $E$.

* $\varphi\ \in\ \Omega^{^1}\ (\text{End}\ E)$.

§.§ Spectral curves and Hitchin fibrations

The Higgs branch moduli space, ${\cal M}_{_H}$, is an example of one such complex integrable system. In particular, it is a fibration over a complex base space, ${\cal B}$, where the generic fiber is a compact complex torus.

For gauge groups $G=SU(N)$ or $G=PSU(N)$, given a Higgs bundle $(E_{_h}, \phi)$, then the eigenvalues of $\phi$ in the standard representation of $G$ give an $N$-sheeted branch cover of $C$, [120],

\begin{equation}
\Sigma\ \overset{def.}{=}\ \left\{\left(z\in C,\lambda\in T^{^*}_{_z}C\right):\text{det}(\phi(z)-\lambda)=0\right\}\ \subset\ T^{^*}C,
\end{equation}

which is the spectral curve corresponding to the Higgs bundle, $\left(E_{_h}, \phi\right)$.

The moduli space of harmonic bundles carries a natural family of complex structures $J_{_{\zeta}}$, parameterised by $\zeta\ \in\ \mathbf{CP}^1$. This follows form the fact that ${\cal M}$ carries a natural hyperk$\ddot{\text{a}}$hler metric, [120]. To see this, fix a $G$-bundle $E$. Then let ${\cal C}$ denote the space of pairs $(D,\varphi)$ without imposing the Hitchin equations. By definition, ${\cal C}$ is an infinite-dimensional affine space, with a natural hyperk$\ddot{\text{a}}$hler structure. Moreover, ${\cal C}$ is naturally acted on by the gauge group ${\cal G}$. This action preserves the hyperk$\ddot{\text{a}}$hler structure and has a moment map $\Vec{\mu}$. The Hitchin equations imply that all 3 components of $\Vec{\mu}$ must vanish. Denoting by ${\cal G}$ the smooth sections of Aut $E$, we therefore have

\begin{equation}
{\cal M}\ \equiv\ \mu^{^{-1}}(0)/{\cal G},
\end{equation}

namely the hyperk$\ddot{\text{a}}$hler quotient

\begin{equation}
{\cal M}\ \equiv\ {\cal C}\dslash {\cal G},
\label{eq:hkq}
\end{equation}

implying ${\cal M}$ is in turn hyperk$\ddot{\text{a}}$hler. Every hyperk$\ddot{\text{a}}$hler manifold carries a canonical family of complex structures parametrised by $\zeta\  \in\ \mathbf{CP}^1$, in this case denoted by $J_{_{\zeta}}$.

Let ${\cal B}$ denote the space of all $N$-sheeted branched covers $\Sigma\ \subset T^* C$ of $C$. In this case, ${\cal B}$ is a finite-dimensional complex vector space. Passing from the Higgs bundle to $\Sigma$ gives a projection known as the Hitchin fibration

\begin{equation}
{\cal M}_{_H}\ \rightarrow\ {\cal B},
\end{equation}

where ${\cal B}$ is the Hitchin base.

As thoroughly explained in [120], ${\cal B}$ is the Coulomb branch of the moduli space of vacua of the class ${\cal S}$ theory in question, conventionally denoted $X_{\mathfrak{g}}(C)$. At any nonsingular point in the Coulomb branch, there is an electromagnetic charge lattice

\begin{equation}
\Gamma\ \rightarrow\ {\cal B}_{_{reg}}\ =\ {\cal B}/{\cal B}_{sing}.
\end{equation}

At every point of base, $u\ \in\ {\cal B}$, there is a spectral curve, $\Sigma_{_u}$, thereby determining the local lattice system as the first cohomology, $H_{_1}(\Sigma_{_u},\mathbf{Z})$.

§.§.§ Looking for the identity

In this concluding part of the section, we are now ready to gather all the ingredients outlined in the previous part of this work, to show how it relates to the formalism outlined in [4].

In doing so, we mostly need to recall the notion of the identity in Moore-Tachikawa varieties, and their generalisation for the case in which the target 2-category is a hyperk${\dddot{\text{a}}}$hler quotient

\begin{equation}
T^{^*}\tilde G_{_{\mathbf{C}}}\ \overset{def.}{=}\
T^{^*}G_{_{\mathbf{C}}}^{^{\prime}}\ \otimes_{_{T^{^*}G_{_{\mathbb{C}}}}}\ T^{^*}G_{_{\mathbf{C}}}^{^{\prime\prime}}.
\label{eq:newidnow}
\end{equation}

As explained in [4], in such case, the identity is naturally removed by construction, and this clearly fits with taking the categorical quotient according to the GIT prescription, (<ref>). From the work of Moore and Tachikawa, we also know that such categories are non-dualisable, and, consequently, not associated to a fully-extended TQFT.

As explained in section <ref>, full-extendibility can be recovered only upon specifying the identity element in the fully extended mother category from which the GIT quotient descends, (<ref>). In particular, as suggested from (<ref>), the identity itself might be a composite object, as the one arising when performing double gauging of a SymTFT leading to non-invertible defects, [1].

This partial gauging and gluing of different theories is precisely of the type occurring in the context of supersymmetric quiver gauge theories whose Higgs branch admits a magnetic quiver description, [210], as well as 3D mirrors of Sicilian theories, [343]. We plan to report a more detailed treatment for these specific examples in due course, [549].

At first, we briefly overviewed cochain level theories as the most important generalisation of the open and closed TFT construction, emphasising its relation to the SymTFT construction leading to absolute theories. Mostly relying upon [126], we highlighted the importance of the construction of cobordism operators, emphasising their dependence on the conformal structure of the Riemann surface. In section <ref> 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 propose their generalisation for the case in which the target category of the $\eta_{_{G_{_{\mathbb{C}}}}}$ functor is a hyperk$\ddot{\text{a}}$hler quotient. We concluded outlining the possible extension of this treatment towards a mathematical formulation of magnetic quivers within the context of Coulomb branches of 3D ${\cal N}=4$ quiver gauge theories which will be addressed in an upcoming work by the same author.

§.§ Where to next?

With the discussion outlined in Part II, III and V, we have shown the importance of abelianisation in geometric representation theory. Specifically, we highlighted the common ground between [102, 550, 181].

As explained at the very beginning of our work, one of our key motivations is the need to deepen the mathematical toolbox adopted by Phenomenologists in answering open questions in Particle Physics. Recent developments in the Pheno community have started relying upon the calculation of the Hilbert series, most importantly [553, 554, 555], successfully encoding key fundamental processes, such as the Higgs mechanism. The crucial point of our analysis, though, is that, albeit Hilbert series calculations are indeed very useful for keeping track of the degrees of freedom and symmetries of a given theory, in principle they are not enough to determine the exact operator content of the theory in question.

On the other hand, ring homologies, on which Hilbert series are calculated, encode crucial properties of the QFT at hand, enabling a much deeper understanding of its underlying categorical structure. The key take-home message of this work is the crucial role played by abelianisation for identifying the underlying mathematical structure of a given QFT. This concept will be investigated further for the specific case of the SM and its extensions in future work, [549].

Additional complementary directions addressed by the same author will be developing ring homologies for Moore-Tachikawa varieties beyond categorical duality[Thereby following up from the analysis carried out in [4].], and understanding their implications for Koszul duality. We plan to report of any advancements in this regard in the near future.

§.§ Conclusions and Outlook

With this work, the aim of the author was that of continuing from past developments, bridging the gap in between the Pure Mathematics and the Theoretical Particle Physics. The purpose of this thesis was mainly:

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

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

* For those who are not experts in the field, serve as an oversimplified explanation of the powerful formalism put forward by BFN, and connecting it with Moore-Tachikawa varieties.

* Motivating Phenomenologists to engage more with Pure Mathematicians.

The main objective of this Part was that of explaining the higher categorical structures that are needed for describing the invariants associated to specific supersymmetric quiver gauge theories, with a particular focus on dualities and their mutual relations in terms of higher-categories. We plan of reporting on any future developments in this regard in the near future, [549].

§ CONCLUSIONS AND OUTLOOK OF PART V

Part V of this work is structured as follows:

* After a preliminary list of definitions, 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 algebraic varieties. This follows the treatment of section <ref>, which briefly overviewed cochain level theories as the most important generalisation of the open and closed TFT construction. Mostly relying upon [126], we highlight the importance of the construction of the cobordism operator, highlighting its 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.

Conclusions and Outlook

§ CONCLUSIONS AND OUTLOOK

String theory is a promising candidate unifying theory of fundamental interactions. In its fermionic formulation, it involves ten dimensions, of which nine are spatial and one is temporal.
Crucially, such formulation strongly relies upon Supersymmetry (SUSY), which is a correspondence in between bosonic and fermionic particles.

According to present-day evidence, SUSY is certainly broken in our world, and plenty of effort has been made in identifying traces of such symmetry breaking from the observational side, eventually leading to the extension of the Standard Model (SM) by encoding additional particles and interactions.
Understanding how the Standard Model can be embedded in String Theory is certainly a step forward towards identifying patterns of SUSY breaking in our world.
Albeit being broken, SUSY certainly plays a crucial role in understanding which String Theory model is most appropriate for the SM to arise as an effective field theory (EFT), precisely by studying its breaking.

String theory vacua, where the SM would eventually be embedded, are parametrised by the so-called moduli space, and the latter is known to admit a categorical algebro-geometric description. Crucially, great advancements in the latter took place simulatneously to the birth and development of SUSY and String Theory. In recent years, joint collaboration in between pure mathematicians and mathematical physicists, primarily Freed, Hopkins, Lurie, Moore, Segal, and Teleman, shed light to new interesting perspectives which happen to be particularly relevant for the objectives outlined above. The main feature of their approach is that of applying the powerful tool of higher-categorical structures within the context of gauge theories and their supersymmetric counterparts, with the ultimate aim being that of explaining what is the underlying mathematical structure of a quantum field theory (QFT). Their work is based upon identifying underlying topological structures from which partition functions, and correlation functions can be derived. Furthermore, their formalism happens to be particularly efficient for keeping track of the spectrum (i.e. the particle-field content) of a given QFT, and how the latter changes under gauging of the global symmetries of the theory one started from.

Up to now, most of the mathematical control in describing higher-dimensional field theories has been for descendants of maximally-supersymmetric 6D ${\cal N}=(2,0)$ superconformal field theories (SCFTs), from which a rich web of dualities arises upon dimensional reduction to, either, 4 or 3 spacetime dimensions. Such descendants feature fewer supersymmetries with respect to their non-Lagrangian parent theory. In particular, the most studied are 4D ${\cal N}=2$ (also known as class ${\cal S}$ theories), and 3D ${\cal N}=4$ SCFTs, exhibiting exactly half of the amount of supersymmetry of their parent 6D theory.

From the work of Alday, Gaiotto and Tachikawa (AGT), it is known that class ${\cal S}$ theories can be equally described in terms of a 2D topological field theory (TFT) living on the Riemann surface on which dimensional reduction has been performed. Such a 2D TFT is defined to be a functor in between 2-categories. Such 2-categories were first studied by Moore and Tachikawa, which, in turn, relied upon the pioneering work of Moore and Segal. The main point of their work is that, for a certain class ${\cal S}$ theory to be uniquely defined (i.e. for its spectrum of operators to be uniquely determined), one needs to fully specify the source and target 2-categories of the 2D TFT corresponding to the given 4D ${\cal N}=2$ SCFT in question. In technical terms, the objects of the source 2-category are circles, and the 1-morphisms between them are bordisms. The target 2-category is obtained by acting with the functor on these objects and 1-morphisms, resulting in the gauge group of the 4D dual theory, and the bordism operators (also known as the Moore-Segal bordism operators), with the latter being related to algebraic varieties defining moduli spaces in algebraic geometry, i.e. the Higgs and Coulomb branches.

Higgs and Coulomb branches of supersymmetric gauge theories are fascinating objects to explore, since they shed light on dualities characterising the theory they are associated with. From a pure mathematical point of view, they can be realised in terms of the so-called geometric invariant theory (GIT) construction, and quotienting. For the case of 3D ${\cal N}=4$ SCFTs, exchange in between Higgs and Coulomb branches corresponds to a statement of 3D mirror symmetry between two theories (first studied by Intriligator and Seiberg from a pure theoretical-physics point of view). This 1-to-1 correspondence extends to the calculation of invariants, most importantly the Hilbert Series, accounting for all the gauge-invariant operators of the theory.

However, as also pointed out by Moore, Segal and Tachikawa, their analysis lacked a rigorous mathematical treatment, most importantly for the case in which the target category of the functor is a hyperk$\ddot{\text{a}}$hler quotient, which is indeed the case when describing algebraic varieties such as the Higgs branch of a given theory. The work we have conducted so far provides a step forward towards achieving a rigorous mathematical formulation of such 2-categorical structures for the reason that we will now outline.

As previously explained, TFTs play a crucial role in bridging the gap between QFT and Mathematics. Because of this, the formalism outlined above suggests to always recast a given QFT to a TFT description. For example, 3D ${\cal N}=4$ SCFTs can be turned into 3D TFTs by performing a topological twist. Unlike ordinary 3D TFTs, which are usually trivial in nature in the sense that there are no local degrees of freedom, topologically-twisted TFTs come with nontrivial boundaries, called Lagrangian submanifolds. Importantly, the latter geometrise the boundary conditions of the field content of the theory in question, and, upon studying the intersection (i.e. the 1-morphisms) in between these Lagrangian submanifolds, one can reformulate the statement of 3D mirror symmetry of the parent 3D ${\cal N}=4$ SCFT as that of homological mirror symmetry, which is essentially a duality in between 2-categories. The steps described so far clearly show the advantage of bringing the description down to that the 2-category of Lagrangian submanifolds, which is exactly what we need to keep track of the field content of the theory we started with.

The 3D mirror symmetry/homological mirror symmetry correspondence has been extensively studied for the specific case in which the correspondence in between the Coulomb and Higgs branch in 3D is 1-to-1.  However, recent developments within the context of supersymmetric quiver gauge theories, mostly led by Hanany and collaborators, have shown that there are cases in which Higgs branches of fermionic theories can be broken down into several constituents, each one associated with the Coulomb branch of a certain 3D ${\cal N}=4$ SCFT (a.k.a. the cone structure), with the latter being described by the so-called magnetic quivers. Such correspondence was primarily identified by studying the Hilbert series as a geometric invariant. In one of our recent works [3], we explained how the emergence of this multi-cone structure in the Higgs branch is equivalent to a generalised statement of 3D mirror symmetry, in turn descending form relaxing constraining assumptions on which homological mirror symmetry is based. As a follow up, in [4], and [5], we proposed a generalisation of the Moore-Tachikawa varieties for the case in which the target category of the 2D TFT is a hyperk$\ddot{\text{a}}$hler quotient. As proved in the former of these two papers, achieving this requires generalising the bordism operators of Moore and Segal to the case involving lack of reparametrisation-invariance on the Riemann surface, ultimately enabling to relate this to the issue of defining a Drinfeld center for composite class ${\cal S}$ theories.

So far, our main focus has been on class ${\cal S}$ theories, and 3D ${\cal N}=4$ SCFTs, both featuring 8 supercharges.
To what extent the generalised 3D mirror symmetry/homological mirror symmetry correspondence can be extended to algebraic varieties associated with theories featuring different amounts of supersymmetries is part of my currently ongoing research. The ultimate aim would be to extend this formalism to quiver gauge theories describing the SM and its possible extensions, eventually shedding light on the mathematical description of SUSY breaking mechanisms leading to observable EFTs.

This work explored the importance of 2D structures towards describing higher-dimensional quantum field theories (QFTs). In doing so, we relied upon the joint role played by dualities and categories.

The present work was structured in six parts:

* At first, we introduced the notion of relative QFTs, with detailed explanation for the case of sigma models, gauge theories, and 6D SCFTs. We then turned to the main case of interest to us, namely Coulomb branches of star-shaped quiver gauge theories, introducing the Kirwan map, and the notion of abelianisation, naturally paving the way to the introduction of category theory, which played a crucial role in the remainder of our work.

* Part II introduced the main ingredients of our treatment, namely categories and dualities. After having introduced higher-categorical structures, we provided two specific examples that were recursively encountered throughout our treatment, specifying how they are realised in the main example outlined in Part II thanks to the work of Moore and Tachikawa, [125]. Category theory is particularly useful for describing dualities, which is another crucial ingredient characterising this thesis. The dualities we encountered are:

* Homological mirror symmetry.

* 3D mirror symmetry.

* The Alday-Gaiotto-Tachikawa (AGT) correspondence.

* Holographic duality.

Apart from revising well-known results in the literature, we highlight the importance of going beyond full categorical dualisability in the context of Moore-Tachikawa varieties, to which Part V was devoted. In doing so, we related their formalism to that of SymTFT constructions of Freed, Moore and Teleman, [102], briefly overviewed in Part III.

* Part III explained how QFTs arise from knot invariants. Specifically, we explained the Symmetry Topological Field Theory (SymTFT) construction of Freed, Moore, and Teleman. In particular, we focussed on the dimensional reduction of non-Lagrangian theories, specifically 6D ${\cal N}=(2,0)$ SCFTs, [108, 109], and the categorical structure of the resulting class ${\cal S}$ theories, [1]. Making use of the AGT correspondence, [124], we turned to a description in terms of 2D topological theories defined as functors between 2-categories.

This Part mostly centered around:

* The algebro-geometric formulation of moduli spaces, and their construction by means of the Geometric Invariant Theory (GIT) construction. Importantly, we relate this construction to the abelianisation procedure outlined in our main example in section <ref>, paving the way to further analysis in Part V.

* Turning from QM to Symplectic Geometry by means of the WKB approximation.

Both topics have thoroughly been addressed in the literature, and we therefore devoted some sections in the appendix to explaining them in greater depth.

* Part IV built from open questions pointed out in the first paper by the author and her supervisor, [7]. In addressing this topic, we restrict to lower-dimensional setups in order to deal with open issues usually arising in higher-dimensional settings, extensively addressed in the literature, [197, 198, 199]. In doing so, we built connection with setups involving the emergence of islands, as well as recent developments in the context of von Neumann algebras for calculating generalised entropies in gravitating systems and QFT, [200, 201, 202, 203, 204, 205, 206]. The interesting connection with the remainder of our work is twofold: firstly, the need to resort to a certain dual picture in order to interpret the analytic results of the calculations performed; secondly, the richer structure resulting from the lower-dimensional analysis, emphasises the role played by internal structures in higher-dimensional theories, therefore suggesting alternative perspectives might be needed in order to address other open issues in QFT.

* Part V results from the merge of three single-authored papers, [3, 4, 5]. The key findings in this Part are:

* Moore-Tachikawa varieties without dualities are naturally realised in supersymmetric quiver gauge theories, and we therefore put forward a correspondence in between such theories, magnetic quivers, and Drinfeld centers, [3]. As a result of the formalism adopted (namely the definition of Moore-Segal bordism operators, [4, 126]) applied to Moore-Tachikawa varieties without categorical duality, [125], we provide further evidence of our conjectured relation in between homological mirror symmetry and 3D mirror symmetry, first presented in [3].

* The last paper written by the author prior to submitting this Thesis, [5], nicely ties together the topics covered in [3, 4], while also preparing the stage for the future research the author will be conducting as a postdoc[Partly outlined in [6].]. In particular, this part takes 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, [208]. We will not attempt to provide an exhaustive definition of it in this introduction, and, instead, refer the reader to the related section in the core of this work. 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) quotient construction, [209]. As such, the GIT quotient provides a recipe for realising 3D mirror symmetry; we therefore show how it can be suitably extended for cases where mirror duals are not necessarily in a 1-to-1 correspondence. In relation to [210], 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 of class ${\cal S}$ theories.  We concluded outlining future directions that are currently under investigation by the same author.

* Part V closes outlining the main achievements of the present work, with an outlook on some key questions the author wishes to address in her future research.

We believe our treatment opens the way to further investigating the mutual relation in between homological mirror symmetry and 3D mirror symmetry with the ultimate aim of aiding at understanding the underlying mathematical structure of QFT. These are currently under investigation by the same author, and we plan to report about any advancement in the field in the near future.

§ ACKNOWLEDGEMENTS

These acknowledgements are way far from reflecting my sincere gratitude towards the many people who trusted my passion, instinct, and dedication towards research. The names listed below are only part of the ones I wish to thank, and to whom I will be forever grateful.

First and foremost, my supervisor, Professor Fernando Quevedo. To summarise my working experience with him, I would say our collaboration has been a unique blend of creativity and perseverance, from my side, and patience and wisdom, from his. Thank you for following my progress towards becoming an independent researcher: it has been an honour to be working with you.

I also wish to thank Ahmed Almheiri, Shanta de Alwis, Tudor Dimofte, I$\tilde{\text{n}}$aki García Etxebarria, Victor Gorbenko, David Jordan, Yolanda Lozano, Juan Maldacena, Gregory Moore, Hiraku Nakajima, Carlos Nu$\tilde{\text{n}}$ez, Fernando Quevedo, Jorge Santos, Nathan Seiberg, Marcus Sperling, Constantin Teleman, Mark Van Raamsdonk, and Edward Witten for insightful discussions and questions raised at different stages of this work.

A special acknowledgement goes to, both, the Hodge Institute in Edinburgh, and the Yau Center for Mathematical Sciences, Tsinghua University in Beijing, for their hospitality, where parts of this work were completed and first presented. In particular, many thanks to my respective hosts, Tudor Dimofte, David Jordan, Nicolai Reshetikhin and Shing-Tung Yau.

I also wish to thank the organisers of all the events that I attended throughout my PhD, allowing this work to be presented at different stages, and the partial support by the STFC Consolidated HEP theory grant

ST/T000694/1 through DAMTP towards covering tuition fees.

Being part of the DAMTP HEP Group made me feel at home while abroad, and I will be forever grateful for their warm welcome, which has certainly stimulated my research advancements in many ways. Many thanks to all other members of the HEP-Gang.

Last but not least, I will be forever grateful to the place that raised me during the first six years at university, the beautiful Trieste, and to the sailor whose dream made mine come true.

§ LOOP GROUPS

A loop group is a group of loops in a topological group $G$ with multiplication defined pairwise. It is a group of continuous mappings from a manifold $M$ to a topological group $G$. Assuming $M=S^1$, the loop group is therefore formally defined as follows

\begin{equation}
LG\ \overset{def.}{=}\ \{\gamma:S^1\rightarrow G\ |\ \gamma\in\ C(S^1,G)\}.
\end{equation}

Parametrising $S^1$ with $\theta$, we get that

\begin{equation}
\gamma:\ \theta\ \in\ S^1\ \mapsto\ \gamma(\theta)\ \in \ G ,
\end{equation}

from which multiplication in $LG$ is defined as follows

\begin{equation}
\left(\gamma_{_1}\gamma_{_2}\right)(\theta)\ =\ \gamma_{_1}(\theta)\gamma_{_2}(\gamma).
\end{equation}

Associativity follows from the associativity property in $G$. The inverse os given by

\begin{equation}
\gamma^{-1}:\ \gamma^{-1}(\theta)\ \equiv\ \gamma(\theta)^{-1},
\end{equation}

and the identity is defined as follows

\begin{equation}
\begin{aligned}
e:\ \theta\ \mapsto\ e\ \in\ G.
\end{aligned}
\end{equation}

An important example of loop groups is the group of based loops on $G$, $\Omega G$. It is the kernel of the evaluation map

\begin{equation}
\begin{aligned}
e_{_1}:\ &LG\ \rightarrow\ G  \\
&\ \gamma\ \mapsto\ \gamma(1),
\end{aligned}
\end{equation}

and as such it is a normal subgroup of $LG$. $G$ can also be embedded in $LG$, as the subspace of constant loops, leading to the split-exact sequence

\begin{equation}
1\ \rightarrow\ \Omega G\ \rightarrow\ LG\ \rightarrow\ G\ \rightarrow \ 1.
\end{equation}

The space $LG$, therefore naturally splits as a semi-direct product

\begin{equation}
LG=\Omega G\rtimes G.
\end{equation}

§ SUPERSYMMETRIC FIELD THEORIES

§.§ Physical Lagrangians

A theory admitting a Lagrangian formulation, requires the Lagrangian to be physical, namely it must be:

* real

* local

* Poincare'-invariant.

The Poincare' group, $P^{^n}$ is a subgroup of the global symmetry group of the theory, and, according to the Coleman-Mandula Theorem, the whole symmetry group of the theory takes the following form

\begin{equation}
P^{^n}\ \times\ G,
\label{eq:timesproduct}
\end{equation}

with $G$ denoting a compact Lie group. By definition, symmetries give rise to conserved quantities. For the case at hand, (<ref>) leads to conservation of energy and momentum, for the Poincare' group, and electric charges and quantum numbers for the compact Lie group.

The constraints of the Coleman-Mandula Theorem pertinent to bosonic theories can be relaxed in presence of fermions by extending the Poincare' group to the super-Poincare' group, $P^{^{n|s}}$, which is a super Lie group[Here, $P^{^{n|s}}$ denotes a supersymmetric extensions of the Poincare' algebra $P^{^n}\equiv\text{Lie}(P^{^n})$, whose odd part has dimension $s$.]. The supersymmetric extension of the Coleman-Mandula Theorem, results in stating that the only allowed super Lie groups are of the kind

\begin{equation}
P^{^{n|s}}\ \times\ G.
\label{eq:timesproduct}
\end{equation}

For the sake of completeness, we will briefly outline the main definitions of the terminology adopted in this section, [86].

A super Lie group is a group object in the category of supermanifolds. A super Lie algebra is a $\mathbb{Z}/2\mathbb{Z}$-graded vector space

\begin{equation}
\mathfrak{g}=\mathfrak{g}^{^0}\oplus\mathfrak{g}^{^1},
\end{equation}

equipped with a bracket operation

\begin{equation}
[.,.]\ :\ \mathfrak{g}\otimes\mathfrak{g}\ \rightarrow\ \mathfrak{g},
\end{equation}

which is skew-symmetric and satisfies the Jacobi identity, with

\begin{equation}
[\mathfrak{g}^{^1},\mathfrak{g}^{^1}]\ \in\ \mathfrak{g}^{^0}.
\end{equation}

The role of the super Poincare' group is that of restricting the possible vector spaces (i.e. particle content) and masses that can occur in a given theory. Mathematically speaking, an irreducible representation of $P^{^{n|s}}$ may be realised as the space of sections of a homogeneous $\mathbb{Z}/2\mathbb{Z}$-graded vector bundle over a mass shell. The extension to the case of interacting theories follows suite by noticing that one may use the free-field approximation at any vacuum, giving rise to a certain $P^{^{n|s}}$ representation, which, in turn, follows from the free theory constraints.

§.§ UV/IR flows and their parametrisation

§.§ The Witten index

Supersymmetric theories are characterised by the presence of supersymmetric charge operators, ${\cal Q}$, such that bosons, $|B>$, and fermions, $|F>$, of energy $E$, are transformed into each other

\begin{equation}
{\cal Q}|B>=\sqrt{E\ }|F>
\ \ \
\ \ \
{\cal Q}|F>=\sqrt{E\ }|B>.
\label{eq:fb}
\end{equation}

(<ref>) is equivalent to the statement that non-zero energy states come in Bose-Fermi pairs. This simple statement suggests how to establish whether SUSY is broken or not, namely by counting the zero-energy states. The key element for doing so was first proposed by Witten in [541], and reads

\begin{equation}
{\cal I}\overset{def.}{=}\text{Tr}(-1)^{F}\ e^{-\beta H}
\label{eq:WI}
\end{equation}

with $F, \beta$ and $H$ denoting the fermion number, the inverse temperature and the energy of the sate, respectively. Due to (<ref>), (<ref>) is topological in nature and, by definition, only depends on the number of ground states. In particular, if ${\cal I}\neq 0$, the theory is certainly supersymmetric. Furthermore, it is independent w.r.t. $\beta$ or any other parameter of the theory. Interestingly, though, there is a way in which ${\cal I}$ can be effectively influenced by the features of a specific theory, namely if the number of ground states changes. As we shall see, one way of achieving this is via spectral flow, [542]. The latter will be the main focus of the next sections. However, for consistency, we already provide some qualitative arguments for its importance already at this stage.

Essentially, the spectral flow is implemented by an element of the superconformal algebra, enabling to interpolate between states defined as path-integrals with antiperiodic (fermionic) and periodic (bosonic) boundary conditions, i.e. between the NS- and R-sectors of a certain SUSY theory. Because of this, the action of the spectral flow is equivalent to changing the ground state degeneracy of the original SUSY theory.

§.§.§ R-symmetry, supergroups, superalgebras and twisting

Throughout this work, we will often be referring to many types of dualities. Even though they might seem every different in nature, one of the aims of our analysis will be furthering the mutual relation in between them, highlighting how and what one can tell about the others.

Among the first examples of dualities relevant for our purposes are holography, [190], and modular transformations. The former was originally formulated as a correspondence in between type-IIB string theory on an AdS$_5\times S^5$, dual to 4D ${\cal N}=4$ SYM. The holographic dictionary comprises the equivalence of the partition function on the field theory and gravity sides as well as the correspondence in between their symmetries, given by the superalgebra $PSU(2,2|4)$. The bosonic subgroup is given by

\begin{equation}
SO(4,2)\times SO(6)\ \subset\ SU(2,2)\times SU(4)\ \subset\ PSU(2,2|4)
\end{equation}

consistently with the isometries of AdS$_5$ and $S^5$, as well as the conformal symmetry and the R-symmetry of the CFT$_4$. By definition, an $R-symmetry$ is one that does not commute with SUSY. As we shall see, its role will be essential throughout our treatment.

Modular transformations, instead, come in 2 kinds, denoted $S$ and $T$. In string theory setups, they amount to changing the Yang-Mills coupling, $\tau_{_{YM}}$, as follows

\begin{equation}
\tau_{_{YM}}\ \overset{S}{\longrightarrow}\ -\frac{1}{\tau_{_{YM}}}
\ \ \ \ \ \ \ \ \ \ \
\tau_{_{YM}}\ \overset{T}{\longrightarrow}\ \tau_{_{YM}}+1
\end{equation}

At the level of the objects living in a certain string theory setup, modular transformations enable to map one theory into the other by acting on the objects with $S, T$. In particular, $S$ relates strong and weak coupling, and therefore leads to a similar setup as the spectral flow mentioned earlier on <ref>.

One way of testing these dualities, is by looking for quantities that are protected by supersymmetry. As also mentioned in <ref>, one example is the matching of the spectrum of states characterising the Dp-branes. For example, the extended (i.e. ${\cal N}>1$) superalgebra of 4D massive particles at rest reads

\begin{equation}
\{{\cal Q}_{\alpha}^{I},{\cal Q}_{\beta}^{\dag J}\}=2M\delta^{IJ}\delta_{\alpha\beta}+2i Z\Gamma_{\alpha,\beta}^0
\label{eq:superconfalgebra}
\end{equation}

with $I, J=1,..., {\cal N}$, labelling the number of supersymmetries, $\alpha,\beta=1,2,3,4$, the components of the Majorana spinor supercharge, and $Z^{IJ}=-Z^{JI}$ denoting the central charge matrix. The components of the latter constitute a set of conserved quantities that commute with the other generators of the algebra. They re electric and magnetic charges coupling to the gauge fields belonging to the supergravity multiplet. Under the action of a unitary operator, $U$, it can be brought down to a block-diagonal form. The structure of the superconformal algebra therefore implies that the $2{\cal N}\times2{\cal N}$ matrix on the RHS of <ref>

\begin{equation}
\left(\ \begin{aligned}
&M\ \ \ Z\\
&Z^{\dag}\ \ \ M\\
\end{aligned}\ \right)
\end{equation}

should be positive-definite with nonnegative eigenvalues $M\pm|Z_i|$, whre $i=1,...$, and $|Z_1|\ge|Z_2|\ge...$. This results in the so-called $BPS-bound$

\begin{equation}
\boxed{\ \
|Z_1|\ \le M \  \color{white}{\bigg ]}}
\label{eq:BPS}
\end{equation}

Saturation of the BPS bound leads to a supermultiplet shortening, due to the emergence of zeros in the superalgebra when $M\equiv |Z_1|$. The bigger the number of equal central charges, the smaller the amount of SUSY left.
Interestingly, BPS states, i.e. those with $M\equiv |Z_1|$, are stable and independent from the parameters of the theory as long as supersymmetry is preserved, and therefore shows exactly the same property as the Witten index, (<ref>). As already argued in <ref>, this quantity, namely the BPS states, can be used for bridging the study of the theory from weak to strong coupling.

However, the independence of (<ref>) from the specifics of the theory is lost once another representation becomes degenerate with the BPS multiplet, such as occuring due to the Higgs mechanism.

For the purpose of our analysis, it is convenient to rewrite the superconformal algebra (<ref>) in terms of the $R$-charges

\begin{equation}
\{{\cal Q}_{\alpha}^{I},{\cal Q}_{\beta}^{\dag J}\}=2M_{\alpha\beta}\delta^{IJ}+2i \omega_{\alpha\beta}\epsilon^{IJ}R
\label{eq:superconfalgebra}
\end{equation}

with $\omega_{\alpha\beta}$ defining the symplectic form and $\epsilon^{IJ}$ the Levi-Civita symbol. For ${\cal N}=2$, $M_{\alpha\beta}\in USp(2|2)$ and $R$ being the $U(1)$ superconformal $R$-symmetry. Given that $R$ defines the eentral extension of the superalgebra, it follows that

\begin{equation}
[\ R,{\cal Q}_{\alpha}^{I}\ ]
\epsilon^{IJ}{\cal Q}_{\alpha}^{J}
\label{eq:noncomm}
\end{equation}

The lack of commutability in between the 2 sets of operators belonging to the extended superalgebra plays an essential role in the following analysis. In particular, it ensures that, as long as a relic R-symmetry is preserved, so too is SUSY (either totally or partially). These arguments therefore highlight the role played by the $R$-charges for protecting the index. As we shall see, (<ref>) is at the heart of the noncommutativity in between 2 operations: the spectral flow and twisting.

§.§.§ The importance of flavour to extremise

In [543, 544], for 2 and 4D respectively, it has been shown that the partition function of a CFT can be determined by means of localisation techniques starting from the knowledge of a UV-Lagrangian of the theory as long as the coupling to the background curvature of the sphere on which localisation is performed preserves the superconformal algebra. Equivalently, this statement amounts to ensuring the superconformal algebra contains the exact $R$-charges of the IR-theory, thereby ensuring that, along the spectral flow, the index accounts for the correct number of ground states of the theory.

The present work aims towards analysing the effect of partial gauging on, both, modular and holographic dualities in the attempt to provide a unique framework towards understanding the information loss paradox from a higher-dimensional realisation encompassing string theory and defect field theories. In pursuing such task, we will recast the issue of evaporating black hole microstate counting in the language of fusion of non-invertible defects.

§.§ SUSY enhancement from branes at orbifolds

Starting from a minimal model exhibiting the same SUSY enhancement w.r.t. the ABJM-theory, namely ${\cal N}=2$ Chern-SImons theory with gauge group $U(N)\times U(N)$, and $SU(2)\times U(1)$ gloal symmmetry. The field conetent consists of:

* 2 gauge fields: $A_{\mu}, \hat A_{\mu}$

* bi-fundamental bosons and fermions, $Z^A, \psi^A$, with $A=1,2$, and their hermitian conjugates.

Upper and lower indices denote the 2 and $\bar{\mathbf{2}}$-representations of the global $SU(2)$, respectively. Furthermore, the gauge and matter fields are equipped with gauge group indices, $a,b, \hat a, \hat b$, for $U(N)\times U(N)$. Explicitly, they would read $A^{a}_{\ \ \mu b}\ ,\  A^{\hat a}_{\ \ \mu \hat b}\ , \ Z^{A,a}_{\ \ \ \ \hat b}\ , \ \psi^{A,a}_{\ \ \ \ \hat b}$, etc.

For the given field content, the corresponding ${\cal N}=2$ action reads

\begin{equation}
S=\int d^3 x \left({\cal L}_o+{\cal L}_{CS}-V_{ferm}-V_{bos}\right)
\label{eq:SN2}
\end{equation}

The kinetic terms are

\begin{equation}
{\cal L}_o=\text{Tr}\left(-D_{\mu}Z^{\dag}_{A} D^{\mu}Z^A+i\psi^{\dag A}\gamma^{\mu}D_{\mu}\psi_A\right)
\end{equation}

with covariant derivative

\begin{equation}
D_{\mu}=\partial_{\mu}+i\left[A_{\mu},\ \right]
\end{equation}

whereas the Chern-Simons terms read

\begin{equation}
{\cal L}_{CS}=\epsilon^{\mu\nu\rho}\text{Tr}\left(A_{\mu}\partial_{\nu}A_{\rho}+\frac{2i}{3}A_{\mu}A_{\nu}A_{\rho}-\left(A_{\mu}\longleftrightarrow\hat A_{\mu}\right)\right)
\end{equation}

The total action (<ref>) is invariant under SUSY transformations. In particular, we have that

\begin{equation}
\delta Z^A=i\epsilon^{\dag}\epsilon^{AB}\psi_B
\end{equation}

meaning that supersymmetry interpolates between different representations of the fields. Furthermore, the action has an additional ${\cal N}=2$ SUSY dependent on the gauge group. In order to see this explicitly, we need to introduce a new local operator which enables for the representation to remain the same upon turning bosons to fermions and viceversa. As already shown in the literature, such local operator, $T_{ab}$ plays the role of a monopole operator ensuring gauge invariance of the fundamental fields of the theory. Correspondingly, the SUSY transformation rules turn into

\begin{equation}
\delta Z^{A a}_{\ \ b}=iT_{ac}\psi^{\dag A c}_{\ \ b}\epsilon
\end{equation}

Having introduced a new operator in the spectrum, gauge invariance is restored ensuring the supersymmetric transformations preserve the representation of the fields.

§.§.§ ABJM and mirror symmetry

ABJM is a specific example of generalised Gaiotto-Witten (GW) setups. Indeed, it can be thought of as arising from the RG-flow of a 4D ${\cal N}=4$ SYM on an interval with S-dual BCs.

However, its 1$^{st}$ formulation, was mainly motivated by the search for an AdS$_4$/CFT$_3$ correspondence, that could potentially be applied to string theory compactification setups. It was therefore As such, it constitutes a concrete setup where non-invertibles can be put to test.

§ HIGHER-CATEGORICAL SYMMETRIES FROM TOPOLOGICAL DEFECTS

To each $D$-dimensional QFT, ${\cal T}$, it is possible to assign a $(D-1)$-dimensional category, ${\cal C}_{_{{\mathfrak T}}}$, defined as higher-category if $D>2$. ${\cal C}_{_{{\mathfrak T}}}$ features $D$-levels:

* 1$^{st}$ level: 0-morphisms, namely the objects living in ${\cal C}_{_{{\mathfrak T}}}$

* 2$^{nd}$ level: 1-morphisms, defining the relations between the objects

* 3$^{rd}$ level: 2-morphisms between the 1-morphisms.

* iteratively up to $D-1$.

Fusion of objects in ${\cal C}_{_{\mathfrak T}}$.

Generalised fusion between 1-morphisms.

The objects are codimension-1 topological defects

\begin{equation}
D_{_{D-1}}\ \overset{def.}{=}\ \bigoplus_{i}\ n_i\ D_{_{D-1}}^{^{(i)}}\ \ \ ,\ \ \ n_i\ge0
\end{equation}

where $D_{_{D-1}}^{^{(i)}}$ are codimension-1 topological defects, and $2\le i\ \le D$ denotes the $(i-1)$-morphisms between $(i-2)$-morphisms. $\sum_i n_i$ defining the total number of vacua such that in $n_i$ of them, $D_{_{D-1}}$ behaves as $D_{_{D-1}}^{^{(i)}}$. For the case of a simple object, there is a unique vacuum, and the defect carries a single topological local operator on its worldvolume.

Under fusion, the objects give rise to a monoidal structure

\begin{equation}
D_{_{D-1}}^{^{(1)}}\otimes D_{_{D-1}}^{^{(2)}}\ \equiv \ D_{_{D-1}}^{^{(12)}}
\label{eq:fstr}
\end{equation}

as depicted on the LHS of figure <ref>.

If $D{_{p}}^{^{(1)}}$ and $D{_{p}}^{^{(2)}}$ are related by gauging, wrapping the $D_{_{p-1}}^{^{(1,2)}}$ defect around an $S^{^{p-1}}$, bridges the LHS and the RHS of this figure. The algebra can therefore be placed on the defect for defining interfaces.

The fusion structure (<ref>) descends to a fusion structure of objects of a higher-category of symmetries localised along $D_{_{D-1}}$

\begin{equation}
D_{_{D-2}}^{^{(1,2)}}\otimes_{_{D_{_{D-1}}^{^{(2)}}}} D_{_{D-2}}^{^{(2,3)}}\ \equiv \ D_{_{D-2}}^{^{(2,3)}}\circ   D_{_{D-2}}^{^{(1,2)}}
\label{eq:fstr1}
\end{equation}

For the fusion (<ref>) to define a 1-morphism, ${\cal C}_{_{\mathfrak T}}$ is required to admit the existence of 2-morphisms, namely the theory has to be, at least, 3D. However, this does not prevent the possibility of defining other types of fusion rules, independently of the existence of higher-morphisms, such a as the one depicted in figure <ref>.

§.§ Generalised gauging and localised symmetries

Let us consider the defects $D_{_{p}}^{^{(1)}}, D_{_{p}}^{^{(2)}}, D_{_{p-1}}^{^{(1,2)}},$ with the latter being defined such that, when wrapping an $S^{^{D-1}}$, $D_{_{p}}^{^{(2)}}$ vanishes. This corresponds to saying that $D_{_{p}}^{^{(1)}}$ and $D_{_{p}}^{^{(2)}}$ are related by gauging. Practically,  this relation implies that, for any given representative of $D_{_{p}}^{^{(1)}}$, we obtain a representative of $D_{_{p}}^{^{(2)}}$ upon gauging the worldvolume theory of $D_{_{p}}^{^{(1)}}$. Furthermore, all topological subdefects of $D_{_{p}}^{^{(2)}}$ can be obtained from those of $D_{_{p}}^{^{(1)}}$.

This observation enables one to define a generalised gauging construction for any subcategory ${\cal C}_{_{{\mathfrak T}, D_{p}}}\ \subset\ {\cal C}_{_{\mathfrak T}}$. W.r.t. the original categorical theory ${\cal C}_{_{\mathfrak T}}$, the $D_{_{p}}$-defect is a $(D-p-1)$-morphism. The objects of the subcategory ${\cal C}_{_{{\mathfrak T}, D_{p}}}$ are $(D-p)$-morphisms of ${\cal C}_{_{\mathfrak T}}$ from $D_{_{p}}$ to itself. Its 1-morphisms are $(D-p+1)$-morphisms of ${\cal C}_{_{\mathfrak T}}$.   The fusion structure $\otimes$ of ${\cal C}_{_{{\mathfrak T}, D_{p}}}$ descends from $\otimes_{_{D_{p}}}$ on ${\cal C}_{_{\mathfrak T}}$, and, consequently, its fusion rules as well.

For $p=2$, the gauging is performed by acting with an algebra, $A^{^{(1,2)}}$, living in the 1-category ${\cal C}_{_{{\mathfrak T}, D_{2}^{(1)}}}$. The algebra itself consists of the following elements

\begin{equation}
\
\overset{def.}{=}\ \left\{A_{_{1}}^{^{(1,2)}}, A_{_{0}}^{^{i}}\right\}
\ \ \
\ \ \
i\equiv \alpha,\beta,\gamma,\delta
\label{eq:defectalgebra}
\end{equation}

namely an object[Constituting a gaugeable 1-form symmetry. This is because the algebra described here is an example of an algebra that can be projected to the identity to achieve a gauged theory, namely quotienting the higher-category of topological defects in the SymTFT of the QFT in question, [323].]

\begin{equation}
A_{_{1}}^{^{(1,2)}}\ \overset{def}{=}\ D_{_{1}}^{^{(1,2)}}\otimes D_{_{1}}^{^{(2,1)}}
\end{equation}

and canonical morphisms

\begin{equation}
\begin{aligned}
&A_{_{0}}^{^{\alpha}}\ :\ A_{_{1}}^{^{(1,2)}}\otimes A_{_{1}}^{^{(1,2)}}\ \rightarrow\ A_{_{1}}^{^{(1,2)}}\ \ \ \ \ \ , \ \ \ \ \ \ A_{_{0}}^{^{\beta}}\ :\ A_{_{1}}^{^{(1,2)}}\ \rightarrow\ A_{_{1}}^{^{(1,2)}}\otimes A^{^{(1,2)}}\ \\
\\
&A_{_{0}}^{^{\gamma}}\ :\ A_{_{1}}^{^{(1,2)}}\ \rightarrow\ 1_{D_{2}^{^{(1)}}}\ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ , \ \ \ \ \ \  A_{_{0}}^{^{\delta}}\ :\ 1_{D_{2}^{^{(1)}}}\ \rightarrow\ A_{_{1}}^{^{(1,2)}}.
\\
\\
\end{aligned}
\end{equation}

§.§ Category of lines after gauging

After gauging, the resulting defect can be expressed in terms of (<ref>)

\begin{equation}
D_{2}^{^{(2)}}\ \equiv\ \frac{D_{2}^{^{(1)}}}{A^{^{(1,2)}}}.
\end{equation}

The gauging of ${\cal C}_{_{{\mathfrak T}, D_{2}^{(1)}}}$ is performed upon inserting a mixture of topological defects, constructed from $A^{^{(1,2)}}$, along the entire domain of the line defect $D_{_{1}}^{^{(1,2)}}$.

For the case of arbitrary $p$, the gauging is described by a $(p-1)$-algebra in the $(p-1)$-category ${\cal C}_{_{{\mathfrak T}, D_{p}^{(1)}}}$

\begin{equation}
\
\overset{def.}{=}\ \left\{A_{_{p-1}}^{^{(1,2)}}, A_{_{p-2}}^{^{i}}\right\}
\ \ \
\ \ \
1\le\ i\le\ p-1
\label{eq:defectalgebrap}
\end{equation}

where the morphisms (denoted by $i$) describe all possible creation and annihilation channels. $D_{p}^{^{(2)}}$ is obtained by placing a mixture of elements of $A_{_{p}}^{^{(1,2)}}$ along $D_{p}^{^{(1)}}$.

§.§ Fusion rules

The importance of having introduced the algebra to implement the gauging, manifests when looking at the fusion rules between the defects characterising a given theory. For ease of notation, we will now argue on the basis of a category ${\cal C}_{_{\text{id}, {\mathfrak T}}}$, and the fusion of 2D objects

\begin{equation}
D_{2}^{^{(O)}}\otimes D_{2}^{^{(O^{\prime})}}  \ = \ \bigoplus_{i} D_{2}^{^{(i)}},
\label{eq:fd}
\end{equation}

with $O, O^{\prime}$ denoting different orbits. The line operators describing the 1-morphisms

\begin{equation}
D_{2}^{^{(O)}}\otimes D_{2}^{^{(O^{\prime})}}  \ \rightarrow \  D_{2}^{^{(i)}}
\end{equation}

arrange into orbits, $O^{\prime\prime}$, under the action of the gauge group $G$, such that the fusion can be re-expressed in terms of them

\begin{equation}
D_{2}^{^{(O)}}\otimes D_{2}^{^{(O^{\prime})}}  \ = \ \bigoplus_{O^{\prime\prime}} D_{2}^{^{(O^{\prime\prime})}}
\end{equation}

where the RHS is a representative of the equivalence class of the fusion on the LHS in the quotient theory ${\mathfrak T}/G$. The generalised gauging on top of each $D_{2}^{^{(O^{\prime\prime})}}$ term is encoded by an algebra $A^{^{(O^{\prime\prime}),(OO^{\prime})}}$ in the 1-category ${\cal C}_{_{\text{id}, {\mathfrak T}/G,D_{2}^{^{(O^{\prime\prime})}}}}$, with object

\begin{equation}
A_{_{1}}^{^{(O^{\prime\prime}),(OO^{\prime})}}  \ \equiv \  \bigoplus_{i} n_{i} D_{1}^{^{(i)}}
\end{equation}

where $n_{i}$ denotes the dimension of the vector space constituted by local operators living at the end of $D_{1}^{^{(i)}}$ along $D_{1}^{^{(OO^{\prime}),(O^{\prime\prime})}}$. Consequently, after gauging, the fusion of surface defects, (<ref>), turns into

\begin{equation}
D_{2}^{^{(O)}}\otimes D_{2}^{^{(O^{\prime})}}  \ = \ \bigoplus_{O^{\prime\prime}} \frac{D_{2}^{^{(O^{\prime\prime})}}}{A_{_{1}}^{^{(O^{\prime\prime}),(OO^{\prime})}}}
\end{equation}

with the dependence on the algebra describing the gauging on top of each elementary surface term. This is referred to as the dressing of the topological operators. Similar considerations can be made for the case of line defects.

For the purpose of this work, we will be mostly focussing on duality, triality and $N$-ality defects. By definition, the first 2 are associated to theories with $\mathbb{Z_{_{N}}^{^(1)}}$ 1-form symmetries, whereas the latter emerge in presence of a $\mathbb{Z}_{_{N}}^{^{(1)}}\times\mathbb{Z}_{_{N}}^{^{(1)}}$ symmetry. It is worth mentioning already at this stage that their fusion rules are basically providing a generalisation of (<ref>). Specifying to the case of (3+1) TQFTs, the fusion of 3D duality/triality defects, ${\cal N}({\cal M}_{3})\overset{def.}{=}D_3({\cal M}_{3})A_3$ reads

\begin{equation}
{\cal N}({\cal M}_{3})\times\bar{\cal N}({\cal M}_{3})\ \equiv\ {\cal C}({\cal M}_{3})
\label{eq:squareof}
\end{equation}

with ${\cal C}({\cal M}_{3})$ denoting the gauging defect defined as

\begin{equation}
{\cal C}({\cal M}_{3})\ \overset{def.}{=}\ \frac{1}{{|H^0({\cal M}_{3}, G)|}}\ \sum_{\Sigma_2\ \in\ H_2 ({\cal M}_{3}, G)}  L(\Sigma_2)
\end{equation}

TWR as an example of holographic duality.

From (<ref>), ${\cal N}({\cal M}_{3})$ is often referred to as the square root of the gauging. As a further major consideration worth mentioning is that the quantum dimensions of the duality and triality defects on $S^3$ coincide.

§ TOPOLOGICAL WICK ROTATION AND HOLOGRAPHIC DUALITIES

The AdS/CFT correspondence is an example of holographic duality. Two main features characterising such correspondence are that:

* Gauge symmetries in AdS $\leftrightarrow$ global symmetries of the CFT.

* Correlation functions are only defined when boundary conditions of the AdS theory are specified.

These same properties are shared by other holographic dualities. A new type of holographic duality is the one given by topological Wick rotation.

The holographic dictionary encoded in figure <ref> reads as follows:

* an $N+1$ dimensional topological order with a gapped boundary is dual to an $N$ dimensional liquid with an internal symmetry of finite type. ${\cal S}$ is the category of topological defects in the bulk. On the RHS it is the superselection sector of states. $\mathfrak{Z}({\cal S})$ is the category of topological defects in the bulk and naturally acts on ${\cal S}$. $\mathfrak{Z}({\cal S})$ is the category of topological operators (non-local operators invariant under LOA).

* $^{^{\mathfrak{Z}({\cal S})}}{\cal S}$ is an enriched category.

Consider gapped/gapless edges of a 2+1 dimensional topological order $({\cal C},c)$, with $a,b,c\in{\cal S}$ labelling the topological line defects. The macroscopic observables of the 1+1 dimensional edge form an enriched fusion category $^{^{{\cal B}}}{\cal S}$. The spaces of fields living on the 0 dimensional defect junction $\{M_{_{a,b}}\}$ lead to the OPE

\begin{equation}
M_{_{b,c}}\ \otimes_{_{\mathbb{C}}}\ M_{_{a,b}}\ \xrightarrow{\ \ \circ\ \ }\ M_{_{a,c}}
\end{equation}

$\{M_{_{a,b}}\}_{_{a,b\in{\cal S}}}$ is similar to a category $\{\text{hom}(a,b)\}_{_{a,b\in{\cal C}}}$. $M_{_{a,b}}\ \circ\ \in\ {\cal B}$ defines an enriched fusion category.nIntroducing an abstract MTC ${\cal B}$ and a braided equivalence $\phi:\ \text{Mod}_{_{V}}\ \xrightarrow{\ \ \simeq\ \ }\ {\cal B}$, $(\phi,\ ^{^{{\cal B}}}{\cal S})$ can encode different enrichments $^{^{\text{Mod}_{_{V}}}}_{_{\phi}}{\cal S}$. Correlation functions and OPEs among defect fields can be recovered from the triple $(V,\phi, ^{^{\cal B}}{\cal S})$. The gapped/gapless boundaries of a 2+1 dimensional topological order $({\cal C}, c)$ can be completely characterised or classified by the triple $(V,\phi, ^{^{\cal B}}{\cal S})$, where:

* $V$ is the chiral/non-chiral symmetry which

* for a chiral gapless boundary, corresponds to a rational VOA of central charge $c$

* for a non-chiral gapless boundary, it is a rational full field algebra

* when $V\equiv \mathbb{C}$, it is described by a gapped boundary.

* $\phi:\ \text{Mod}_{_{V}}\ \xrightarrow{\ \ \simeq\ \ }\ {\cal B}$ is a braided equivalence.

* ${\cal S}$ is a fusion category equipped with a braided equivalence ${\cal C}\boxtimes\bar {\cal B}\simeq\mathfrak{Z}({\cal S})$, which determines $^{^{\cal B}}{\cal S}$ via the so-called canonical construction.

All gapped/gapless edges of 2+1 dimensional topological orders can be determined by performing a TWR and adding the chiral/non-chiral symmetry data by hand $(V,\text{Mod}_{_{V}}\ \xrightarrow{\ \ \phi\ \ }\ {\cal B}, ^{^{\cal B}}{\cal S})$. $^{^{\cal B}}{\cal S}$ is the topological skeleton (encoding the topological or categorical data), whereas $(V,\phi)$ constitutes the local quantum symmetry of the edge (e.g. the VOA or the full field algebra or the conformal net of local operator algebras (LOA)). $^{^{\cal B}}{\cal S}$ is an abstract enriched FC, acquiring precise meaning only once ${\cal B}$ is identified with $\text{Mod}_{_{V}}$ via the braided equivalence $\phi:\ \text{Mod}_{_{V}}\ \xrightarrow{\ \ \simeq\ \ }\ {\cal B}$, suggesting that a gapped/gapless quantum liquid $\chi$ can be described by a 2-step data $\chi\equiv(\chi_{_{lqs}}, \chi_{_{sk}})$, since the same $^{^{\cal B}}{\cal S}$ can be realised by different gapped phases. Examples of these constructions are 2D WZW CFTs from 3D Chern-Simons theories. Constructing correlators in 2D CFTS from 3D TFTs. There is a correspondence between Lagrangian algebras in 3D anyon gauging and modular-invariant partition functions in 2D CFTs. Dualities are explicitly constructed from classical statistical models. Correlators mapping 3D TQFTs to 2D RCFTs. String-net construction of 2D correlators.

Theorem  (Kong, Huang)

There is a holographic duality between $(\mathfrak{Z}({\cal S}), {\cal S})$ and $^{^{\cal B}}{\cal S}$. An N+1 dimensional topological order with higher-dimensional symmetry ${\cal R}$ can be completely characterised:

* a fusion n-category ${\cal A}$ equipped with a braided embedding ${\cal R}\ \hookrightarrow\ \mathfrak{Z}_{_{1}}({\cal A})$

* a braided equivalence $\phi:\ \mathfrak{Z}_{_{1}}({\cal R})\ \xrightarrow{\ \ \simeq\ \ }\ \mathfrak{Z}_{_{1}}({\cal A})$.

Generalising ${\cal R}$ to any fusion $(N-1)$-category, one obtaine=s a characterisation of $N$ dimensional gapped quantum liquids by an $N+1$ dimensional topological order with gapped boundary. TWR suggests that this holographic duality is actually functorial. TWR gives an equivalence between the category of $N+1$ dimensional non-chiral topological orders with gapped boundary and the category of $N$ dimensional topological liquids

\begin{equation}
(\mathfrak{Z}_{_{1}}({\cal A}), {\cal A})  \ \ \longrightarrow\ ^{^{\mathfrak{Z}_{_{1}}({\cal A})}}{\cal A}
\end{equation}

This holographic dictionary automatically includes a dictionary map of dualities because many dualities can be realised by invertible DWs, e.g. e-m, and K-W duality.

One of the main goals behind all these studies is the mathematical theory of higher-dimensional CFTs.

§ QUIVER GAUGE THEORIES

§.§ Graphs

A graph, $\Gamma$, gives a category, ${\cal C}_{_{\Gamma}}$, with 3 objects: $A, B$, and $C$. ${\cal C}_{_{\Gamma}}(A,B)$ is just the single arrow from $A$ to $B$.

§.§ Dynkin diagrams

Examples of type-A and type-D quivers.

Examples of type-E quivers.

The main focus of our analysis will be the understanding of the behaviour of 4D ${\cal N}=2$ theories. In doing so, we wish to bring the attention or the reader towards the construction of the 4D theory itself, which already accounts for very useful insight towards understanding major issues we wish to address.

The 4D ${\cal N}=2$ theories that are usually being dealt with, are assumed to possess $S$-duality as inherited from their mother theory, namely 6D ${\cal N}=(2,0)$ SCFT theories, upon dimensional reduction and topological twist on a Riemann surface $\Sigma_2$. Albeit not being this the unique way of deriving such 4D theories, yet they provide the most generic way, which, in turn, allows for a plethora of different descending theories. These 6D SCFTs are very particular UV theories. They were discovered as the decoupling limit of type-IIB on an orbifold singularity. The resolution of the singularity leads to a 6D SCFT theory with characterised according to the ADE classification, with A, D, E denoting the corresponding types of Dynking diagrams.

§.§ Framed Nakajima quiver varieties

In this first subsection, we briefly overview the features of framed Nakajima quiver varieties, [330], and their geometric and algebraic resolutions.

An example of a framed Nakajima quiver variety. This will be a prototypical example of the theories we will be addressing in the present work, namely for quiver gauge theories with 8 supersymmetries. The unframed quiver would be the gauge node without matter. Here we have set $G_{_v}\equiv SU(N)$. However, in principle, other choices could have been taken.

§.§ Geometric (Nakajima) resolution

Quiver varieties are varieties of quiver representations of a quiver: one fixes a vector space at each vertex, then considers the linear space of representations associated to each arrow of the quiver linear map. A framed version of this was initially introduced by Kronheimer and Nakajima, amounting to doubling the set of vertices, and drawing a new arrow from each new vertex to its corresponding old one. An example of the resulting quiver (which is of the prototypical type of interest for us) is depicted[The nomenclature featuring in the framed quiver will be explained in section <ref>, but we emphasise that this is standard notation in the quiver literature.] in figure <ref>.

Framed representations appear naturally in ADHM quiver construction of self-dual or anti-self-dual YM on $S^{^4}$. This is also interesting from the point of view of representation theory of Lie algebras because dimension vectors of the framed vertices appear as highest weights of the representations. There are different notions of framing. Using Nakajima's version for quiver varieties, the framed quiver is doubled, meaning each arrow gets doubled by an arrow that goes in the opposite direction. The linear space of representations becomes a linear cotangent bundle

\begin{equation}
M({\cal Q},v,w)\ \overset{def.}{=}\ T^{^*} L({\cal Q}^{^{\text{fr}}},v,w),
\end{equation}

where ${\cal Q}, {\cal Q}^{^{fr}},v,w$ respectively denote the original quiver, the framed quiver, the number of original vertices, and the number of framed vertices.
The gauge group is a general linear group on the original vertices  $G\equiv G_{_v}$, and there is a moment map

\begin{equation}
\boxed{\ \ \ \ \ \mu:\ M({\cal Q},v,w)\ \rightarrow\ \mathfrak{g}^{^*} \color{white}\bigg] \ \ }.
\end{equation}

Nakajima quiver varieties are Hamiltonian reductions of the following action

\begin{equation}
G\  \circlearrowright\ M({\cal Q},v,w),
\end{equation}

and come into two types:

* the affine Nakajima variety, defined as the partial character variety

\begin{equation}
\mathfrak{M}^{^0}({\cal Q},v,w)\ \equiv\ \mu^{^{-1}}(0)//G,
\label{eq:first}
\end{equation}

* quasi-projective

\begin{equation}
\mathfrak{M}^{^{\chi}}({\cal Q},v,w)\ \equiv\ \mu^{^{-1}}(0)//_{_{\chi}}G.
\label{eq:second}
\end{equation}

The gauge group by which we take the quotient is

\begin{equation}
G\ \equiv\ G_{_v}\ \overset{def.}{=}\ \prod_{a}\ GL_{_{v_{a}}} (\mathbb{C})\ \subset G_{_{v}}\ \times G_{_{w}}
\end{equation}

For any choice of the nontrivial character

\begin{equation}
\chi:\ G\ \rightarrow\ \mathbb{C}^{^{\times}}
\end{equation}

there is a proper Poisson morphism

\begin{equation}
p: \mathfrak{M}^{^{\chi}}({\cal Q},v,w)\ \rightarrow\ \mathfrak{M}^{^0}({\cal Q},v,w),
\end{equation}

which is as a symplectic resolution of the singularities of $\mathfrak{M}^{^0}$.

In both cases, (<ref>) and (<ref>), $\mu^{^{-1}}(0)$ denotes the fiber of zero through the moment map. The latter can be used as a representation scheme for the path algebra, ${\cal A}$, modulo the ideal, ${\cal I}_{_{\mu}}$,

\begin{equation}
{\cal A}\ \overset{def.}{=}\  \mathbb{C}\overline{{\cal Q}^{^{fr}}}/{\cal I}_{_{\mu}},
\end{equation}

of the framed doubled quiver

\begin{equation}
\mu^{^{-1}}(0)\ \equiv\ \text{Rep}_{_{\mathbb{C}^{^v}\otimes\mathbb{C}^{^w}}}({\cal A})\ \overset{def.}{=}\  \text{Rep}_{_{v,w}}({\cal A}),
\end{equation}

where we made use of the following shorthand notation

\begin{equation}
\mathbb{C}^{^v}\ \overset{def.}{=}\ \bigoplus_{a}\ \mathbb{C}^{^{v_{_a}}}\ \ \ , \ \ \ \mathbb{C}^{^w}\ \overset{def.}{=}\ \bigoplus_{a}\ \mathbb{C}^{^{w_{_a}}},
\end{equation}

For the purpose of our work, the main result of [330] is being able to relate such varieties to derived representation schemes, with the latter being obtained by means of an alternative resolution of the affine Nakajima quiver variety. The underlying reason for the importance of this resides with it being a sample realisation of homological mirror symmetry[We will be explaining this in due course.].

Prior to explaining the alternative resolution leading to derived representation schemes, we wish to emphasise that the correspondence that we are looking for, summarised in figure <ref>, [330], emerges from comparing invariants, and the required conditions for them to match at the two endpoints of the red arrow. From the symplectic resolution point of view, this requires flatness of the moment map, as explained in [330], following [341]. We refer the interested reader to such references for more detailed explanation regarding the definition and properties of flat moment maps. For our purposes, the crucial point is that when the invariants in the symplectic and derived scheme description match, we are dealing with a complete intersection[The meaning of the latter will be explained in section <ref>.].

§.§ Algebraic resolution

The technique of algebraic resolution was first put forward by [331], and consists in resolving the singularities of the representation schemes by introducing homological algebra.

Prior to explaining derived representation schemes, let us first briefly overview the notion of a differential graded (dg)-schemes. A dg-scheme is a pair $X\overset{def.}{=}\left(X_{_o},{\cal O}_{_{X,\bullet}}\right)$, with $X_{_o}\equiv M({\cal Q},v,w)$ denoting the vector space of linear representations, and ${\cal O}_{_{X,\bullet}}$ a sheaf of dg-algebras such that their zeroth homology reads as follows

\begin{equation}
\pi_{_0}(X)\ \overset{def.}{\equiv}\ \text{Spec}\left(H_{_0}\left({\cal O}_{_{X,\bullet}}\right)\right)\ \equiv\ \mu^{^{-1}}(0).
\end{equation}

The derived representation scheme of the relative algebra[Associated to the choice of a path in the framed quiver.], ${\cal A}$, in a vector space $V$, is the object, [330],

\begin{equation}
\text{DRep}_{_V}({\cal A})\ \in\ \text{Ho}\left(\text{DGAff}_{_k}\right),
\end{equation}

obtained applying the following composition of functors

\begin{equation}
\text{DRep}_{_V}(-):\ \text{Alg}_{_S}\ \longrightarrow\ \text{Ho}(\text{DGA}_{_S}^{^+})\ \xrightarrow{L(-)_{_V}}\ \text{Ho}(\text{CDA}_{_k}^{^+})\ \xrightarrow{\text{RSpec}}\ \text{Ho}(\text{DGAff}_{_k}),
\label{eq:totalmap}
\end{equation}

\begin{equation}
L(-):\ \text{Ho}\left(\text{DGA}_{_S}\right)\ \rightarrow\ \text{Ho}\left(\text{CDGA}_{_S}\right)
\end{equation}

is the derived representation functor acting on the homotopy category, Ho, whose homology is the representation homology of a differential graded algebra, ${\cal A}$,

\begin{equation}
H_{_{\bullet}}({\cal A},V)\ \overset{def.}{=}\ H_{_{\bullet}}\left(L({\cal A})_{_V}\right).
\end{equation}

On the other hand, RSpec in (<ref>) denotes the derived spectrum, defining an equivalence on the homotopy categories

Altogether, (<ref>) leads to

\begin{equation}
\text{DRep}_{_V}({\cal A})\ =\ \text{RSpec}\left(L({\cal A}_{_V})\right) =\ \text{Rep}_{_V}\left({\cal A}_{_{cof}}\right)
\label{eq:DRS}
\end{equation}

\begin{equation}
{\cal A}_{_{cof}}\ \overset{\sim}{\twoheadrightarrow}\ {\cal A}
\end{equation}

a cofibrant replacement. Different choices of cofibrant replacements lead to different models of (<ref>). For the specific case of framed Nakajima quiver varieties, the derived representation scheme reads, [330],

\begin{equation}
\text{DRep}_{_{v,w}}({\cal A})\ =\ \text{Spec}\left(L({\cal A}_{_{v,w}})\right) \ \in\ H_{_0}\left(\text{DGAff}_{_{\mathbb{C}}}\right),
\label{eq:DRSDA}
\end{equation}

with representation homology

\begin{equation}
H_{_{\bullet}}({\cal A},v,w)\ \equiv\ H_{_{\bullet}}\left(L({\cal A})_{_{v,w}}\right)\ \in\ \text{CDGA}_{_{\mathbb{C}}}^{^+},
\label{eq:DRSDA1}
\end{equation}

a graded commutative algebra. Denoting by $\{U_{_{\lambda}}\}$ the irreducible representations of $G$, (<ref>) explicitly reads as follows

\begin{equation}
H_{_{\bullet}}({\cal A},v,w)\ \overset{def.}{=}\ \underset{\lambda}{\bigoplus}\ \text{Hom}_{_G}\left(U_{_{\lambda}}, H_{_{\bullet}}({\cal A},v,w)\right)\otimes U_{_{\lambda}},
\label{eq:DRSDA2}
\end{equation}

with $\text{Hom}_{_G}\left(U_{_{\lambda}}, H_{_{\bullet}}({\cal A},v,w)\right)$ denoting its isotypical components, namely modules over

\begin{equation}
H_{_0}({\cal A},v,w)^{^G}\ \equiv\ {\cal O}\left(\mu^{^{-1}}(0)\right)^{^G}.
\label{eq:DRSDA3}
\end{equation}

For each irreducible representation of $G_{_v}$, the Euler character is defined as follows

\begin{equation}
\begin{aligned}
\chi_{_T}^{^{\lambda}}({\cal A},v,w)\ &\overset{def.}{=}\ \overset{\infty}{\underset{i=0}{\sum}}\ (-1)^{^i}\left[\text{Hom}_{_G}\left(U_{_{\lambda}}, H_{_i}({\cal A},v,w)\right)\right]\\
&=\overset{\infty}{\underset{i=0}{\sum}}\ (-1)^{^i}\left[H_{_i}(L({\cal A}))_{_{\lambda,v,w}}^{^{G_{_v}}}\right]\ \ \ \in\ \ K_{_T}\left(\mathfrak{M}^{^0}\right),
\end{aligned}
\end{equation}

where $T\overset{def.}{=}T_{_w}\times T_{_l}$ accounts for the framing, and

\begin{equation}
H_{_i}(\pi):\ H_{_i}\left({\cal A}_{_{cof}}\right)\ \rightarrow\ H_{_i}({\cal A})
\end{equation}

is an isomorphism such that ${\cal A}_{_{cof}}$ has no higher homologies, namely,

\begin{equation}
\begin{cases}
&H_{_0}(\pi):\ H_{_i}\left({\cal A}_{_{cof}}\right)\ \xrightarrow{\sim}\ {\cal A}\\
&H_{_i}\left({\cal A}_{_{cof}}\right)\ \equiv\ 0, \ \forall i\ge1.\\
\end{cases}
\end{equation}

\begin{equation}
\pi:\ {\cal A}_{_{cof}}\ \longrightarrow\ {\cal A}
\end{equation}

defines an acyclic fibration in DGA$_{_{\mathbb{C}}}^{^+}$.

§.§ Hilbert series

One of the key results of [330] is that agreement in between invariants calculated from the Nakajima and derived representation schemes requires flatness of the moment map on the symplectic side, and vanishing higher homologies in the derived case. Importantly for us, such agreement involves integrated characters, also known as the Hilbert series (HS). The latter is a map which reads

\begin{equation}
\text{HS}_{_T}:
\begin{cases}
&K_{_T}\left(\mathfrak{M}^{^0}\right)\ \longrightarrow\ K_{_T}(\text{pt}),\  \text{if $\mathfrak{M}^{^0}$ is compact}\\
&K_{_T}\left(\mathfrak{M}^{^0}\right)\ \longrightarrow\ \text{Frac}(T),\  \text{otherwise}.\\
\end{cases}
\end{equation}

From this, the resulting Weyl integral formula reads

\begin{equation}
\text{HS}_{_T}\left(\chi_{_T}^{^G}({\cal A},v,w)\right)\ \equiv\ \frac{1}{|G|}\ \int_{_G}dg\ \text{HS}_{_{G\times T}}\left(\chi_{_T}^{^G}({\cal A},v,w)\right).
\label{eq:hs}
\end{equation}

Upon choosing the variables in the maximal torus of the gauge group, $x\ \in\ T_{_v}\ \subset\ G$, and equivariant variables $t\equiv\ (a,l)\ \in\ T\ \equiv\ T_{_w}\times T_{_l}$, (<ref>) can be re-expressed as follows

\begin{equation}
\text{HS}_{_T}\left(\chi_{_T}^{^G}({\cal A},v,w)\right)\ \equiv\ \frac{1}{|W|}\ \int_{_{T_{_v}}}dx\ WF(x)\ \frac{\ \underset{i}{\prod}\ \left(1-l_{_1}l_{_2}b_{_i}\right)\ }{\underset{j}{\prod}\ \left(1-a_{_j}\right)},
\label{eq:hs1}
\end{equation}

with $b_{_i}, a_{_i}$ respectively denoting the weights of $M({\cal Q},v,w)^{^*}$ and $\mathfrak g$.

§ DERIVATION OF GENERALISED BLACK HOLE (BH) ENTROPY

§.§ "Replica trick" derivation

§.§ Rènyi and von Neumann entropies

Given 2 systems A and B both equipped with a field theory which, for the latter it is assumed to be, in turn, coupled to JT-gravity,

\begin{equation}
\
\
\ln Z_{B}
\
\
\frac{\phi_{o}}{4\pi}\left[\int_{B}R+2\int_{\partial B}K\right]+\frac{\Phi}{4\pi}\int_{B}(R-\Lambda)+\frac{\Phi_{b}}{2\pi}\int_{\partial B}K+\ln Z_{CFT}[g]
\end{equation}

the Rènyi entropies read

\begin{equation}
\
\
\text{tr}\rho_{A}^{n}
\
\
\sum_{i_{1}...i_{n}}p_{i_{1}}...p_{i_{n}}<\psi_{i_{1}}|\psi_{i_{2}}><\psi_{i_{2}}|\psi_{i_{3}}>...<\psi_{i_{n}}|\psi_{i_{1}}>
\ \ \
\end{equation}

Taking n-copies of the gravitating universe over which the path integral is performed. The pictures at the centre and on the RHS, instead, show the completely disconnected and completely connected saddles, the latter being associated to Euclidean wormholes.

The path integral is assumed to be performed over all the possible topologies consistent with the same boundary conditions.

\begin{equation}
\text{tr}\rho_{A}^{n}
\
\
\frac{1}{Z_{1}^{n}}\left[e^{-I_{grav}[M_{disc}]}Z_{CFT}[M_{disc}]+e^{-I_{grav}[M_{conn}]}Z_{CFT}[M_{conn}]\right]
\ \ \
\end{equation}

\begin{equation}
\
\
e^{-I_{grav}[M_{conn}]} Z(\beta)
\end{equation}

being the saddle point geometry for the gravitating system $B$.

\Rightarrow
\ \ \
\text{tr}\rho_{A}^{n}
\
\
\frac{1}{Z_{1}^{n}}\left[e^{-nI_{grav}[M_{1}]}\sum_{i}e^{-n\beta E_{i}}+e^{-I_{grav}[M_{conn}]}Z_{CFT}[M_{conn}]\right]
\

\
\frac{1}{Z_{1}^{n}}\left[Z(n\beta)+e^{-I_{grav}[M_{conn}]}Z_{CFT}[M_{conn}]\right]
\ \ \

These pictures shows the emergence of fixed points under $Z_{n}$-symmetry of the boundary. On the right, the fixed point is depicted in red. Under the action of the generators of $Z_{n}$-symmetry, the green subregion is mapped to equivalent subregions. The nontrivial fixed point indicates the emergence of a nontrivial boundary for the CFT, which in turn carries d.o.f. associated to the degeneracy of the ground state, as already outlined in previous subsection.

To build $M_{conn}$, we define $C_{x}$ as a potion of a Cauchy slice of $B$ with linear size 2$\pi$ x. Then, by gluing cyclically n-disconnected components of $M_{disc}$ along $C_{x}$ forming a ${\textbf{Z}}_{n}$-symmetric Euclidean wormhole topology obeying the gravitational e.o.m., the Rènyi entropy can be rewritten as follows

\begin{equation}
\text{tr}\rho_{A}^{n}
\
\
\frac{Z(n\beta)}{Z_{1}^{n}(\beta)}\left[Z(n\beta)+e^{-I_{grav}[M_{conn}]}Z_{CFT}[M_{conn}]\right]
\end{equation}

In this subsection, we apply the machinery outlined in the previous pages for deriving the explicit expression of the generalised entanglement entropy, showing the emergence of the contribution of the quantum extremal surface resulting from the quotiented boundary geometry.

As already mentioned, AdS/CFT and, more generally, the Holographic Principle, claims that

\begin{equation}
\
\ Z_{bulk}(M,J)
\ \ \
\end{equation}

i.e., the bulk string partition function is equivalent to the generating functional of its boundary CFT dual.

Expecting a UV completion of the bulk theory, i.e. not pure gravity. However, when $G_{N}$ is small, it is possible to approximate it as

\begin{equation}
\
\
\sum_{g} e^{-I(g)}\ Z_{matter}(g)
\label{eq:PF}
\end{equation}

where $g$ is a classical background with geometry $g$ and

\begin{equation}
\frac{\delta}{\delta g}\left[ e^{-I(g)}\ Z_{matter}(g)  \right]
\
\
\ \ \
\Longleftrightarrow
\ \ \
R_{\mu\nu}\ -\ \frac{1}{2}g_{\mu\nu}R
\
\
8\pi G<T_{\mu\nu}>
\ \ \
\label{eq:extr}
\end{equation}

The partition function is determined by the saddle point extremising (<ref>). In general, these solutions are hard to find even for AdS$_{3}$, and even more for higher-dimensions. Finding $g$ for arbitrary $n$ turns out to be possible through the aid of the replica trick, which in turn relies upon the assumption that the boundary has a $Z_{n}$-symmetry, which in turn corresponds to a cyclic permutation of the $\hat\rho$ featuring in the boundary path-integral.

* in the dominant saddle, this $Z_{n}$-symmetry is also a symmetry of the bulk, i.e. not spontaneously-broken

* the dominant saddles are all we need, i.e. it provides a good analytic continuation, but there might be jumps in $n$ leading to discontinuities.

As a matter of fact, we can therefore restrict our treatment to quotiented geometries. In addition, we need to take into account the fixed points[uniquely identified] of the quotieneted geometry (i.e. the branch cuts). In the bulk a surface with a similar behaviour must also be defined, such that the bulk geometry is smooth everywhere except at the fixed points.

Quotienting by $Z_{n}$, we end up with a conical singularity. Writing up the semi-classical action in terms of the data of the quotiented geometry, equation (<ref>) becomes

\begin{equation}
\
\
e^{-I_{n}(g^{(n)},X^{(n)})}\ Z_{matter,n}(g^{(n)},X^{(n)})
\ \ \
\ \ \
\
\
\text{Tr}_{_{\bar b}}(\rho_{b}^{n})
\ \ \
\end{equation}

The full unquotiented geometry is a saddle-point of the gravitational action, therefore satisfies the extremisation

\begin{equation}
\frac{\delta}{\delta g}\left[ e^{-I(g)}\ Z_{matter}(g)  \right]
\
\
\label{eq:c}
\end{equation}

\begin{equation}
\Longleftrightarrow
\ \ \
\frac{\delta}{\delta g^{(n)}}\left[ e^{-I(g^{(n)})}\ Z_{matter,n}(g^{(n)})  \right]
\
\
\ \ \
\ \ \
\frac{\delta}{\delta X^{(n)}}\left[ e^{-I_{n}(g^{(n)})}\ Z_{matter,n}(g^{(n)})  \right]
\
\
\ \ \
\label{eq:constr}
\end{equation}

A possible solution to (<ref>) is of the form

\begin{equation}
\
\
\exp\left[-nI_{1}(g^{(n)})-(n-1)\frac{A(X^{(n)})}{4G_{N}}\right]\ \text{Tr} (\rho_{b}^{n}) \ \ \
\end{equation}

At this point, we can ease the assumption that $n$ is an integer, in particular $n=1+\epsilon$, with $\epsilon<<1$. Working to leading-order in $\epsilon$, the partition funcion becomes

\
\
e^{-I(g^{(1)})}\ Z_{matter}\ [1-\epsilon I_{1}(g^{(1)})-\epsilon\frac{\partial g^{(n)}}{\partial N}\frac{\delta I_{1}(g^{(1)}}{\delta g^{(n)}}-\epsilon\frac{A(X^{(1)})}{4G_{N}}\ +\

\begin{equation}
\
\frac{\epsilon}{Z_{matter}}\frac{\partial g^{(n)}}{\partial N}\frac{\delta Z_{matter}(g^{(1)}}{\delta g^{(n)}}\ +\ \frac{\epsilon}{Z_{matter}}\text{Tr}(\rho_{b}\text{ln}\rho_{b})]
\ \ \
\label{eq:1e}
\end{equation}

At $n=1$, i.e. with one copy of the geometry, the action does not depend on the fixed point and $\text{Tr} \rho_{b}$ doesn't depend on $b$. However, the location of $X^{(1)}$ is not determined at $n=1$, but at leading-order-$\epsilon$ corrections. (<ref>) can be further simplified by applying the semi-classical Einstein equations that read

\begin{equation}
\frac{\delta I^{(1)}}{\delta g} \ +\ \frac{1}{Z_{matter}}\frac{\delta Z_{matter}}{\delta g} \
\
\ \ \
\end{equation}

therefore, we are left with

\begin{equation}
\
\
e^{-I_{1}(g^{(1)})}\ Z_{matter}\left[1-\epsilon\ I_{1}(g^{(1)})\ +\ \epsilon\ \text{ln}\ \left(Z_{matter}\right) \right]
\ \ \
\end{equation}

For the von Neumann entropy

\begin{equation}
\frac{Z_{1+\epsilon}}{Z_{1}^{1+\epsilon}}
\
\
1\ -\ \epsilon\frac{A(X)}{4G_{N}}\ +\ \epsilon\ \text{Tr}\ \left(\hat\rho_{b}\ \text{ln}\ \rho_{b}\right)-\epsilon\ \text{Tr}\left(\hat\rho_{b}\ \text{ln}\ Z_{matter}\right)
\end{equation}

\begin{equation}
\
1\ -\ \epsilon\frac{A(X)}{4G_{N}}\ +\ \epsilon\ \text{Tr}\ \left(\hat\rho_{b}\ \text{ln}\ \hat\rho_{b}\right)
\end{equation}

\begin{equation}
\Rightarrow
\ \ \
\
\
\lim_{\epsilon\rightarrow\ 0}\ \left[ -\ \frac{1}{\epsilon}\ \ln\ \frac{Z_{1+\epsilon}}{Z_{1}^{1+\epsilon}}\right]
\
\
\frac{A(X)}{4G_{N}}\ +\ S_{bulk}(X)
\ \ \
\label{eq:1l}
\end{equation}

For $n=1$, $X$ is nothing but the quantum extremal surface satisfying (<ref>). Furthermore, among all possible $X$s choose the smallest. This two-step (extr-min) procedure is what defines the quantum extremal surface prescription. Notice that the deep point of the replica trick consists in treating $n$ as an arbitrary parameter and expanding it around $n=1$, such that $g^{(1)}$ parametrises the background metric without backreactions dominates the path-integral. On the other hand, $Z_{matter}$ across the whole derivation, is really meant to be the full path-integral for the matter content. It is therefore takes into account of all fluctuations in the matter sector, which is what enables to achieve the 1-loop correction to the pure gravity result in (<ref>).

§ TOPOLOGICAL QUANTUM FIELD THEORIES (TQFTS)

The present section introduces topological field theories and their String Theory counterparts, namely Topological String Theory. The reason for introducing this is multifold. As already mentioned in the introduction, they constitute an essential tool for probing the mathematical structure of QFTs. The most important advancements in the field took place in 2 and 3-dimensions. Most importantly, the pioneering works of Atiyah, Schwarz, Segal and Witten[The pioneering works on the subject are the following [159, 160, 161, 162, 163, 164, 165, 166].]. Sticking to such lower-dimensional setups might seem reductive, especially when having in mind higher-dimensional field theories. However, the great advantage of the 2 and 3D settings have proved to give rise to an extremely rich variety of studies, whose findings can be interestingly applied to higher dimensions as well. As mentioned in the Introduction, the AGT correspondence is a clear example of the utility of 2D TFTs for describing 4D ${\cal N}=2$ SCFTs[We will come back to this in greater detail in Part <ref>.].

§.§ Attempts and advancements

Topological field theories (TFTs) are a bridge between QFT and maths. In gauge theory and mathematical physics, a TQFT is a QFT calculating topological invariants. They are related to knot theory, the theory of 4-manifolds, and the theory of moduli spaces in algebraic geometry [405, 136, 160, 522].

The adjective topological refers to the fact that correlation functions are independent from the spacetime metric. On Minkowski spacetime, topological theories are trivial when contracted to a point. Because of this, they need to be placed on a Riemann surface instead. They come in two main types:

* Schwarz-type TQFTs, e.g. the BF-model

\begin{equation}
\end{equation}

where $B$ is an auxiliary field, whereas $F$ is the derivative of a connection, $A$. This action is explicitly topological, given that the metric does not feature at all. The alternative formulation by Schwarz reads as follows

\begin{equation}
S=\int_MA\ \wedge\ dA.
\end{equation}

An example of this is Chern-Simons theory, applied to knot invariants, to which we shall turn in the following section.

* Witten-type TQFTs [topological 4D Yang Mills theory in 4D]. For a given 2-form, $B$, the corresponding action reads as follows

\begin{equation}
S=\int_MB\ \wedge\ \delta B,
\end{equation}

* $S$ has a symmetry, i.e. $\delta S=0$ under it.

* $\delta^2=0$, i.e. the symmetry is exact.

* There are observables, $\{{\cal O}_{_i}\}$, such that $\delta {\cal O}_{_i}=0$.

* $T^{\alpha\beta}=\delta G^{\alpha\beta}$ for any arbitrary tensor $G^{\alpha\beta}$.

2D TFTs are mostly related to mirror symmetry.
On the other hand, 3D TFTs are much less understood, apart from Chern-Simons theory, which has no nontrivial local observables given that there are no boundary conditions. The only topological observables are Wilson loops (localised on 1D closed submanifolds). Kapustin, Rozansky and Saulina (KRS) were the first to have introduced nontrivial boundary conditions for a 3D TFT. This theory as a 3D sigma-model, first introduced by Rosanzky and Witten (a.k.a. the RW-model). The target of the TFT is required to be hyperk$\ddot{\text{a}}$hler, $X$, or, more generically, a holomorphic symplectic manifold. Upon dimensional reduction on an $S^1$, it yields the B-model.

A physical motivation for studying the RW-model is its relation with 3D mirror symmetry. In particular, understanding the RW-model, is a step forward towards understanding 3D mirror symmetry.

As we will see in greater detail, the boundaries of KRS are Langrangian submanifolds of $X$, with the latter building relation with derived algebraic geometry.

§.§ The Atiyah-Segal axioms

The Atiyah-Segal axioms are particularly useful for understanding the mathematical structure of Schwarz-type QFTs. The key point of such axioms is that a TQFT is a functor from a bordism category to the category of vector spaces.

§.§.§ Segal's Conformal Field Theory (CFT) axioms

The object of Segal's work, is that of presenting a mathematical formulation of CFT$_{2}$s, and describing basic examples already known from previous literature. Furthermore, he also presents five examples of areas of pure mathematics where CFTs play a crucial role:

* The monster group.

* The representation theory of loop groups, and the group Diff$(S^1)$ of diffeomorphisms on the circle.

* How representations of Diff$(S^1)$ are related to the geometry of moduli spaces of the Riemann surfaces. (Mumford's classification of holomorphic line bundles descends from this.)

* Some of V. Jones' new representations of braid groups, and his classification of subfactors in von Neumann algebras.

* Elliptic cohomoogy theory of Lendweber-Stong, [180].

§.§.§ The Atiyah-Segal axioms

The key point of such axioms is that a TQFT is a functor from a bordism category to the category of vector spaces.

Take a commutative ring, $\Lambda$, with the identity, $\mathbf{1}$. Then, a $D$-dimensional TQFT[Where $D=d+1$, $d$ denoting the number of spatial dimensions.] is defined from $\Lambda$ according to the following axioms:

* The Homotopy axiom A finitely-generated $\Lambda$-module, ${\cal Z}(\Sigma)$ associated to each oriented closed smooth $d$-dimensional manifold, $\Sigma$.

* The Additive axiom An element ${\cal Z}(M)\in{\cal Z}(\partial M)$ associated to each oriented smooth $(d+1)$-dimensional manifold $M$.

* ${\cal Z}$ is functorial with respect to orientation-preserving diffeomorphisms of $\Sigma$ and $M$.

* ${\cal Z}$ is involutory, i.e. ${\cal Z}(\Sigma^*)={\cal Z}(\Sigma)^*$.

* ${\cal Z}$ is multiplicative.

* ${\cal Z}(\emptyset)=\Lambda$ for the $d$-dimensional manifold, and ${\cal Z}(\emptyset)=\mathbf{1}$ for the $(d+1)$-dimensional empty manifold.

* The Hermitian axiom ${\cal Z}(M^*)=\overline{{\cal Z}(M)}$.

In practice, $\Sigma$ and ${\cal Z}(\Sigma)$ respectively correspond to the physical space and the Hilbert space of the QFT, with associated vanishing Hamiltonian, $H=0$. The latter property follows from the topological nature of the theory in question, implying there is no real dynamics in the $\Sigma\ \times \ I$-cylinder. However, there can still be nontrivial propagation from $\Sigma_{_o}$ to $\Sigma_{_I}$ through an intermediate, topologically nontrivial, manifold $M$ such that

\begin{equation}
\partial M\ =\ \Sigma_{_o}^{*}\ \cup\ \Sigma_{_I}.
\end{equation}

If $\partial M\equiv\Sigma$, then the distinguished vector in the Hilbert space ${\cal Z}(M)$ is the vacuum state defined by $M$. For a closed manifold $M$, ${\cal Z}(M)$ is nothing but the partition function, namely the expectation value of the identity operator.

The additive axiom can be understood by taking into account the fact that finite dimensional vector spaces for a compact closed category. For genuine TFTs, all spaces of quantum states are finite-dimensional, hence one could equivalently consider the linear dual spaces, implying  the propagator (or correlator) map

\begin{equation}
Z(M):\ Z\left(\partial_{_{in}}M\right)\ \rightarrow\ Z\left(\partial_{_{out}}M\right)
\end{equation}

becomes a linear map

\begin{equation}
\mathbb{C}\ \rightarrow\ Z\left(\partial_{_{out}}M\right) \ \otimes\ Z\left(\partial_{_{in}}M\right)^{^*}\ =\ Z(\partial M).
\end{equation}

§.§ Schwarz-type TQFTs

§.§.§ Chern-Simons (CS) Theory

A 3D TQFT of Schwarz-type, developed by Edward Witten, and named after Chern and Simons who introduced the Chern-Simons 3-form. The latter is defined as follows: given a manifold and a Lie algebra valued 1-form, $A$, over it, one can define a family of $p$-forms as

\begin{equation}
d\omega_{_{2k-1}}\ =\ \text{Tr}\left(F^{^k}\right),
\label{eq:CSF}
\end{equation}

where the right-hand-side is proportional to the $k^{th}$ Chern-character of the connection $A$. For the specific case where $p=3$, (<ref>) reads

\begin{equation}
\text{Tr} \left[F\wedge A-\frac{1}{3}A\wedge A\wedge A\right]\ =\ \text{Tr} \left[dA\wedge A+\frac{2}{3}A\wedge A\wedge A\right].
\end{equation}

In gauge theory, the integral of the Chern-Simons form is a global geometric invariant, modulo addition of an integer.

§.§ Topological String Theory

Topological String Theory (TST) is a version of String Theory that follows the idea of TQFTs. There are 2 main versions of it, conventionally referred to as the topological A- and B-model.

Calculations in TST encode all holomorphic quantities that are protected by spacetime SUSY. Various such calculations are related to CS-theory, Gromov-Witten (GW)-invariants, mirror symmetry, and the Geometric Langlands Program, just to name a few.

The operators in a TST represent the algebra of operators in the full string theory preserving a certain amount of SUSY.

Abelian gauge theories usually appear in String and M-theory, where abelian means that, in these theories, the space of gauge inequivalent fields forms an abelian group.

A central extension of a group $G$ by an abelian group $A$ is a group $\tilde G$ such that

\begin{equation}
0\ \rightarrow\ A\ \rightarrow\ \tilde G\ \rightarrow\ G\ \rightarrow\ 0,
\end{equation}

with $A$ being in the center of $\tilde G$. $\tilde G$ can be thought of as a set of pairs $(g,a)$ with twisted multiplication

\begin{equation}
\left(g_{_1}, a_{_1}\right) \left(g_{_2}, a_{_2}\right)    = \left(g_{_1}g_{_2}, c\left(g_{_1},g_{_1}\right) a_{_1} a_{_2}\right).
\end{equation}

From a purely mathematical perspective, it is natural to question about the implications of this for the formulation of topological D-branes in terms of derived categories.

§.§.§ The A- and B-model

* The A-model

The topological A-model comes with a 6-dimensional K$\ddot{\text{a}}$hler target space. Such theory is equipped with fundamental strings wrapping around 2D holomorphic curves, as well as D2-branes wrapping Lagrangian submanifolds of spacetime. The worldvolume theory on a stack of $N$ $D2$-branes is the string field theory of the open strings of the A-model,namely a $U(N)$-CS-theory.

The fundamental strings may, in turn, end on the D2-branes. Additional (coisotropic) branes in various dimensions might also be added [Kapustin, Orlov].

* The B-model

The B-model also contains fundamental strings. However, scattering amplitudes here only depend on the complex structure, and are completely independent w.r.t. the K$\ddot{\text{a}}$hler structure. Furthermore, the theory admits D(-1)-,D1-,D3-, and D5-branes, wrapped around holomorphic 0, 2, 4, and 6-submanifolds, respectively.

Mirror symmetry relates the amplitudes calculated on the two sides.

§ QFT AND THE JONES POLYNOMIAL

§.§ QFT and the Jones polynomials

Among the many useful applications of CS-theory as a 3D TQFT of Schwarz-type, is that it interprets the Jones polynomials (JPs).

JPs assign a Laurent polynomial in $t^{^{1/2}}$ to any knot, $V(knot)$. Each JP can thus be completely determined by the unknot, $V(unknot)=1$, and the Skein relations

\begin{equation}
\left(t^{^{1/2}}-t^{^{-1/2}}\right)\ V(L_{_{o}})\ =\ t^{^{-1}}\ V\left(L_{_+}\right)-t\ V\left(L_{_-}\right),
\end{equation}

where $L_{_{\pm}}, L_{_o}$ denote knots differing in between them only at a point, as shown in figure <ref>.

§.§ Schemes or algebraic stacks

A scheme is a mathematical structure that enlarges the notion of an algebraic variety, accounting for multiplicities, and allowing varieties over any commutative ring. Scheme theory was originally introduced by A. Grothendieck in 1960 as a key tool for solving open issues in algebraic geometry.

A scheme is a topological space, as well as a ringed space which is locally a spectrum of a commutative ring.

A morphisms of schemes can be used for defining algebraic surfaces. In many cases, the family of all varieties of a given type can itself be viewed as a variety of schemes.

§.§ Morse Theory and Classifying Spaces

The main idea of Morse Theory is to associate to a Morse functor

\begin{equation}
f:\ M\rightarrow\mathbf{R}
\end{equation}

on a closed Riemannian manifold, $M$, a category, ${\cal C}_{_f}$, whose objects are the critical points of $f$, whereas the morphisms, $\overline{\cal M}(a,b)$, between two critical points, $a$ and $b$, are piece-wise flow lines of the gradient flow of $f$ connecting $a$ and $b$. One can recover from the topological category ${\cal C}_{_f}$ the topology of $M$. As such, ${\cal M}(a,b)$ denotes the the moduli space of lines from $a$ to $b$.

From local data, one cannot tell whether a given critical point is required by the topology or if, instead, it is removable. For the system in question to be nontrivial, we need to perform calculations that are sensitive of the presence of more than one critical point on the manifold in question. For this to be the case, we need to ensure that the potential energy in the problem, $V(\phi)$, has more than one minimum, specifically, one for each critical point, allowing for the possibility of tunnelling from one critical point to another.

Tunnelling can be calculated in the WKB-approximation, namely by means of instantons (using Witten's terminology), [404]. Tunnelling effects often remove spurious degeneracies arising in perturbation theory, as is precisely the case here. From the instanton analysis, we know that the relevant instantons or tunnelling paths are the paths of steepest descent leading from one critical point to another. As such, they are therefore the solutions of the following equation

\begin{equation}
\frac{d\phi}{d\lambda} \ =\ \gamma^{^{ij}}\frac{\partial h}{\partial\phi_{_j}},
\end{equation}

where $\lambda$ denotes time in the supersymmetric nonlinear sigma-model, assuming $\phi$ is only a function of time.

The instanton calculation shows that the only relevant solutions of this equation are the ones connecting two critical points whose Morse indices differ by unity.

Instanton solutions or tunnelling paths in a given theory are extrema of this Lagrangian, written with a Euclidean metric, neglecting the fermionic content. The relevant action would therefore read

\begin{equation}
\begin{aligned}
S&=\frac{1}{2}\ \int d\lambda\ \left(\gamma^{^{ij}}\frac{d\phi^{^i}}{d\lambda}\frac{d\phi^{^j}}{d\lambda}+t^{^2}\gamma^{^{ij}}\frac{\partial h}{\partial\phi^{^i}}\frac{\partial h}{\partial\phi^{^j}}\right)  \\
&=\frac{1}{2}\ \int d\lambda\ \bigg|\frac{d\phi^{^i}}{d\lambda}\pm t \gamma^{^{ij}}\frac{\partial h}{\partial\phi^{^j}}\bigg|^{^2}\ \mp\ t\int d\lambda\frac{dh}{d\lambda}.
\end{aligned}
\end{equation}

The minimum action paths between any two critical points $a$ and $b$ are paths of steepest descent, and the action for such paths reduces to

\begin{equation}
S=t\ \bigg|\ h(B)-h(A)\ \bigg|.
\end{equation}

An example of a classifying space for the $\infty$-cyclic group $G$ is the circle as $X$. Moduli space is essentially a synonym for representing object and classifying space. Usually, the term classifying space is used in topology, whereas moduli space is used in algebraic geometry.

§.§ Donaldson invariants

Donaldson theory (DT), [405], is the study of the topology of 4-manifolds using moduli spaces of anti-self-dual instantons. Many of the theorems in DT can be proved by a much more convenient formalism, namely that of Seiberg-Witten (SW) Theory.

Even though this has been amply studied in the literature, there are still some open questions that remain to be addressed, phrased in terms of the following conjectures:

* The Witten conjecture[The Witten conjecture was later extended by Virasoro.]: in Algebraic Geometry, it is related to the intersection numbers of stable classes on the moduli space of curves. In its formulation, it is a special case of a more general relation between integrable systems of Hamiltonian PDEs, and the geometry of certain 2D TFTs, also known as cohomological field theories[The latter were first studied by Kontsevich and Manin.].

* The Atiyah-Floer conjecture: connecting the instanton Floer homology with the Lagrangian intersection Floer homology[The latter is, in turn, related to homological mirror symmetry.].

Witten showed that Donaldson invariants are correlation functions in a supersymmetric gauge theory, and that the infinite-dimensional Morse theory developed by Floer is part of a 2D QFT.

Mathematical gauge theory was later revolutionised when Seiberg and Witten deduced the long-distance behaviour of the supersymmetric gauge theory calculating the Donaldson invariants. Donaldson invariants are topological in nature, and are therefore scale-independent. Because of this, they found an alternative way of calculating them, this time by means of the so-called Seiberg-Witten equations.

The original motivation that led Witten to propose such conjecture was the existence if two different 2D models of quantum gravity that should be exhibiting the same partition function. These are: on the one hand, the intersection of number on the moduli stack of algebraic curves, and, on the other, it is the ln of of the T-function of the KdV-hierarchy, with $T$ a translation operator such that

\begin{equation}
\end{equation}

§.§ Seiberg-Witten theory

A Seiberg-Witten theory is an ${\cal N}=2$ supersymmetric gauge theory whose low-energy effective action is exact in the massless limit. Such theory is also referred to as ${\cal N}=2$ SYM theory, whose field content is a single ${\cal N}=2$ vector supermultiplet, in complete analogy with the single vector gauge field (or connection) in ordinary YM theory. Its corresponding Lagrangian density in 2D reads

\begin{equation}
{\cal L}_{_{SYM}}=\text{Im}\text{Tr}\left(\frac{1}{4\pi}\int d^{^2}\theta d^{^2}\vartheta\ {\cal F}(\Psi)\right),
\end{equation}

with $\theta,\vartheta$ denoting the coordinates for the spinor directions of superspace, whereas ${\cal F}$ is the prepotential. The minimal theory is defined by

\begin{equation}
{\cal F}(\Psi)=\frac{1}{2}\tau\Psi^{^2},
\end{equation}

with $\tau$ the complex coupling constant.

§.§ From CS to Morse theory

§.§.§ Morse Theory

Morse Theory relates to the description of closed geodesics on smooth Riemannian manifolds. Since the first studies of R. Bott, the theory underwent several reformulations, which we will briefly overview in this section. The crucial feature of Morse Theory is its relation to the so-called critical-point theory.

The main idea is that of associating a Morse function

\begin{equation}
f:\ M\ \rightarrow\ \mathbf{R}
\end{equation}

on a closed Riemannian manifold, $M$, a category, ${\cal C}_{_f}$, whose objects, $a, b$, are the critical points of $f$, whereas the morphisms between two critical points of $f$ are denoted by $\overline{\cal M}(a,b)$. The latter are piecewise flow lines of the gradient flow of $f$ connecting $a$ and $b$. One can recover the topology of $M$ from the category ${\cal C}_{_f}$. ${\cal M}(a,b)$ is the moduli space of flow-lines from $a$ to $b$. This setup can be rephrased in terms of the space of loops, <ref>.

Morse Theories are classified in terms of:

* Their dimensionality, which can either be finite or infinite.

* The role of symmetry groups.

* Their connection to higher or lower-dimensional TQFTs.

Its finite formulation, due to Bott, followed from approximating the description in terms of a finite set of polygons (i.e. piecewise geodesics). Furthermore, he restricted his studies on manifolds with Lie group acting on the manifold while preserving the Morse function, also known as equivariant Morse Theory.

On the other hand, Morse Theory in its infinite-dimensional version has many fascinating applications in lower-dimensional topology, to which we will come back momentarily.

§.§.§ Relation to Moduli Spaces of SQFTs

The most important feature of Morse Theory is its applicability to extended setups going beyond the work of Atiyah and Bott on Riemann surfaces.

Take a compact Lie group $G$, and $M$ a smooth manifold. A $G$-connection on $M$ gives a notion of parallel transport along paths, which can be considered as an element of $G$ itself. For example, if $M\equiv S^{^2}$, then parallel transport on a half-circle defines a map

\begin{equation}
S^{^1}\ \longrightarrow \ G,
\end{equation}

relating the loop space of $G$ and connections on the Riemann surfaces, the latter being the original setup investigated by Atiyah and Bott, namely the equivariant Morse Theory of the Yang-Mills (YM) functional

\begin{equation}
YM(A)=\frac{1}{2}\int_{_M}\left|F_{_A}\right|^{^2}\ d^{^D}x.
\label{eq:YMAB}
\end{equation}

To determine the critical points of (<ref>), one needs to solve the associated Euler-Lagrange equation

\begin{equation}
\end{equation}

The minima of the energy functional (<ref>) are flat connections with zero-curvature, which , from the Narasimhan-Seshadri Theorem corresponds to the space of stable bundles. Later developments in 4D by Atiyah, [159], Bott, [511], and Donaldson, [405], eventually lead to the study of instanton equations in 4D, further developed by Atiyah, Hitchin, and Singer, [98].

A great turnaround came when Floer started investigating 3D manifolds, [437], in which case the functional to be extremised is CS theory. The critical points of the functional are still flat connections on the given Riemannian manifold, but a crucial difference arises with respect to the 2D and 4D cases: the number of negative eigenvalues of the Hessian at the critical points is infinite-dimensional, implying traditional Morse Theory could not be applied straightforwardly. Importantly, Floer realised that one could define a relative index in between two critical points, by making use of a curve interpolating in between them. In such way, one is able to define chain complex, whose homology defines the Floer homology of the 3-manifold in question.

Furthermore, the latter can be related to the Donaldson invariants of a 4D manifold whose boundary is the 3D manifold whose Floer homology has just been defined. Indeed, Floer homology groups and Donaldson invariants follow some of the defining properties of a TQFT, such as the gluing property.

2 main lines of development in Morse Theory, [404]:

* Witten was first motivated by supersymmetry models in lower-dimensional QFT.

* The second line of development has been, instead, that of understanding that Morse functions decompose smooth manifolds into simple constituents.

Mathematical constructions of TQFTs usually start with algebraic data which is then assigned to elementary bordisms between manifolds of fixed dimension. Invariants are obtained by decomposing arbitrary bordisms into elementary ones by means of Morse functions. The fact that this defines an invariant follows from comparing the results achieved from different Morse functions.

For the case of a 2D TFT, such algebraic data is a commutative Frobenius ring, first studied by Moore and Segal.

§.§ SQFTs from Morse Theory

We now briefly outline how Witten's approach shed light on the relation in between supersymmetric QFT (SQFT) and Morse Theory, from a mathematical perspective[See appendix <ref> for descriptions of gauge theories in mathematical language.].

Let us start by taking a smooth compact manifold $M$, and a smooth function

\begin{equation}
f:M\ \longrightarrow\ \mathbb{R}.
\end{equation}

Then, introducing a deformation

\begin{equation}
0\ \longrightarrow\ \Omega^{^1}(X)\ \xrightarrow{d+t\epsilon(df)}\ \Omega^{^2}(X)\ \  \ \xrightarrow{d+t\epsilon(df)}\ \Omega^{^3}(X)\ \xrightarrow{d+t\epsilon(df)}\ ...
\end{equation}

of the ordinary de Rham complex, if $f$ is a Morse function, and $M$ is endowed with a Riemannian metric, then, in the $\underset{t\rightarrow\infty}{\lim}$, there is a close relation between the spectrum of the associated Laplace operator, $\Delta_{_t}$, and the critical points of $f$. From a physical point of view, these correspond to the different vacuum states of the quantum system. Quantum tunnelling in between them can be encoded as a differential on the $\mathbb{Z}$-graded  abelian group generated by the vacuum states, or the critical points, namely where the grading is given by the Morse index. Smale was the the first to realise this from a pure mathematical perspective, but Witten was the first to start the physical studies behind this, with the relevant system in question being that of supersymmetric quantum mechanics (SQM).

Once having a consistent understanding of SQM, one could lift the description by, either, increasing the number of spacetime dimensions, and/or the amount of supersymmetries at play, and subsequently studying its impact on the associated Morse Theory.

For the sake of completeness, we briefly outline a few examples of the statement we have just made.

* Starting from SQM, doubling the amount of supersymmetry while preserving the dimensionality implies removing flows between the critical points of the associated Morse function. Upon raising the theory to 2D QFT (namely Landau-Ginzburg theory), one recovers nontrivial interpolating soliton solutions, with the latter being related to Picard-Lefschetz theory.

* For the case of 4D supersymmetric gauge theories, instead, one can study the tunnelling between vacua on a 3D manifold in terms of Floer homology.

* In higher dimensions, with increasing amount of supersynmetry, namely 5D and 6D SCFTs, Witten proposed a new definition of the Jones polynomial and its categorification in terms of Khovanov homology. In this case, the critical point equations are special cases of the Kapustin-Witten equations introduced in the context of the Geometric Langlands Program (GLP), whereas the flow equations turn out being new equations in a 5D mathematical gauge theory.

§.§.§ The Cobordism Hypothesis

More generally to this has been the formulation of the so-called cobordism hypothesis. The Cobordism Hypothesis was first formulated by Baez and Dolan, and later proved by Hopkins and Lurie in 2D, and eventually in $D>$2 from Lurie.

The Cobordism Hypothesis states that, given a symmetric monoidal $(\infty,n)$-category, ${\cal C}$, which is fully dualisable and every $k$-morphism, for $1\le k\le n-1$, there is a bi-jection between the ${\cal C}$-valued symmetric monoidal functors of the cobordism category and the objects of ${\cal C}$.

The motivation for this is a statement we have already introduced in our treatment, namely that symmetric monoidal functors from the cobordism category correspond to TQFTs. In practice, the Cobordism Hypothesis is very similar in nature to the Eilenberg-Steenrod axioms for homology theories, with the latter essentially stating that, a homology theory is uniquely determined by its value from the point. In the cobordism case, this corresponds to saying that the bijection between ${\cal C}$-valued symmetric monoidal functors and the objects of ${\cal C}$ is uniquely determined by its value for the point.

§ 4D DESCENDANTS OF 6D  SCFTS

§.§ Class

First, we briefly overview the properties of class ${\cal S}$ theories, the calculation of the dimension of the Coulomb and Higgs branches, as well as the contribution from punctures to the 4D central charges, and how 3D SCFTs can be accommodated on nontrivial punctures. Then, we recall the basic motivations underlying the 2D/4D correspondence resulting from dimensional reduction of 6D ${\cal N}=$(2,0) SCFTs, needed in core sections of the present work. We end the section by recalling that the 3D theories resulting by twisted compactification on the non-invertible defects of 4D ${\cal N}$=4 SYM are always of the ABJ-type, namely gauge theories with $U(M)\times U(N)$ gauge group, [318].

§.§ =2 data

* The moduli space: Coulomb, Higgs and mixed branches.

* The charge lattice, $\Gamma$, with

\begin{equation}
\text{dim}\ \Gamma\ \equiv\ 2r+f
\end{equation}

where $r$ denotes the rank of the gauge group and $f$ the rank of the gauge flavour symmetry. BPS states sre points in $\Gamma$.

* An antisymmetric electromagnetic inner product on $\Gamma$

\begin{equation}
\circ: \Gamma\times\Gamma\ \longrightarrow\ \mathbb{C}
\label{eq:emip}
\end{equation}

* A central charge function

\begin{equation}
Z_{_{u}}:\Gamma\ \longrightarrow\ \mathbb{C}
\label{eq:ZU}
\end{equation}

where $u$ denotes a generic point in the Coulomb branch.

Such data is conventionally encoded in a family of Riemann surfaces, $\{\Sigma_{_{u}}\}$, known as Seiberg-Witten curve, which varies over the Coulomb branch, and whose homology lattice is identified with $\Gamma$. By definition, (<ref>) is linear in the charges, and can therefore be expressed as a 1-form integral of the Seiberg-Witten differential, $\lambda_{_{u}}$

\begin{equation}
Z_{_{u}}(\gamma)\ \overset{def.}{=}\ \int_{_{\gamma}}\lambda_{_{u}}
\ \ \ \
\ \ \ \
\forall\ \gamma\ \in\ \Gamma
\end{equation}

Consequently, the electro-magnetic inner product (<ref>) is naturally identified with the intersection of 1-cycles. Jointly, $(\Sigma_{_{u}}, \lambda_{_{u}})$ define the complete solution to the low-energy physics. On the Coulomb branch, this description is provided by a special Kähler geometry.

§.§.§ BPS quivers

The purpose of the present work work requires restricting to complete theories. These include:

* Theories associated to triangulation surfaces.

* The quivers $E_{_{n}}, \hat E_{_{n}}, \hat\hat E_{_{n}}$.

* The Derksen-Owen quivers, $X_{_{6}}, X_{_{7}}$.

The natural relation between quivers and triangulations is manifest in terms of BPS state counting.

Assuming the theory of interest admits a set of $2r+f$ hypermultiplett states, $\{\gamma_{_{i}}\}$, forms a positive integral basis for all particles, such as is the case for class ${\cal S}$ theories, a BPS quiver is constructed by drawing a node $\forall \gamma_{_{i}}$, and an arrow in between them, $\gamma_{_{i}}\rightarrow\gamma_{_{j}}$ if $\gamma_{_{i}}\circ\gamma_{_{j}}>0$. If a site of the charge lattice

\begin{equation}
\gamma\ \equiv\ \sum_{i}n_{_{i}}\gamma_{_{i}}  \ \in\ \Gamma
\end{equation}

is occupied by a BPS state, it is possible to determine its degeneracy and spin. The intarections of the BPS elementary particles are encoded in the 4-supercharge quiver QM of the given configuration, i.e. a 1D supersymmetric gauge theory with gauge and matter groups

\begin{equation}
G\overset{def.}{=}\prod_{\text{nodes}}U(n_{_{i}})\ \ \ , \ \ \ \text{Matter}\overset{def.}{=}\bigoplus_{\text{arrows}} B_{_{ij}}^{^a}
\end{equation}

where $B_{_{ij}}^{^a}$ denotes the bifundamental field charged under $G$. To find the BPS states, we need to study the moduli space of supersymmetric ground states, ${\cal M}_{_{\gamma}}$,

\begin{equation}
{\cal M}_{_{\gamma}}\ \overset{def.}{=}\ \left\{R\equiv\left\{B_{_{ij}}^{^{a}}:\ \mathbb{C}^{^{n_i}}\rightarrow\mathbb{C}^{^{n_i}}\right\}\ \bigg|\ \frac{\partial {\cal W}}{\ \partial B_{_{ij}}^{^{a}}\ }\ \equiv\ 0, R\  \Pi-\text{stable}\ \right\}\bigg/\ \prod_{i}\ Gl\left(n_{_{i}}, \mathbb{C}\right)
\label{eq:modsp}
\end{equation}

Hence, ${\cal M}_{_{\gamma}}$ is the space of solutions to the $F$-term equations

\begin{equation}
\frac{\partial {\cal W}}{\ \partial B_{_{ij}}^{^{a}}\ }\ \equiv\ 0
\end{equation}

subject to an additional stability constraint, also known as $\Pi$-stability, modulo the action of the complexified gauge group $Gl\left(n_{_{i}}, \mathbb{C}\right)$. $\Pi$-stability is the relevant condition for describing moduli space. In quiver representation theory, it means that, a representation $R$ of a quiver $Q$ is a complex vector space $\mathbb{C}^{^{n_{i}}}$ $\forall$ node $i$ and a linera map $B_{_{ij}}^{^{a}}: \mathbb{C}^{^{n_{i}}}\rightarrow \mathbb{C}^{^{n_{j}}}$ $\forall$ arrow $a$ from $i$ to $j$, corresponding to a choice of expectation values of matter fields in the corresponding quiver QM.

A subrepresentation $S$ of $R$ is a choice of vector subspaces $\mathbb{C}^{^{m_{i}}}\ \subset\ \mathbb{C}^{^{n_{i}}}$ for each node, and maps $b_{_{ij}}^{^{a}}: \mathbb{C}^{^{m_{i}}}\rightarrow \mathbb{C}^{^{m_{j}}}$ for each arrow, such that all diagrams look like the following

For a given charge associated to a representation $R$

\begin{equation}
\gamma_{_{R}}\ \equiv\ \sum_{j}n_{_{j}}\gamma_{_{j}}
\label{eq:pi}
\end{equation}

the stabilisation condition corresponds to

\begin{equation}
\text{arg} Z_{_{u}}(S)\ <\ \text{arg}Z_{_{u}}(R)
\end{equation}

for $S$ a subrepresentation of $R$. Upon identifying $S$ with a parton state, $\Pi$-stability corresponds to forbidding decay channels of the kind $\gamma_{_{R}}\rightarrow\gamma_{_{S}}$, from which the definition (<ref>) results.

The nodes of a BPS quiver are  always $\Pi$-stable representations, due to the absence of destabilising subrepresentations. In terms of (<ref>), this corresponds to setting $n_{_{i}}\equiv\ \delta_{_{ij}}$, meaning each note contributes with a multiplicity 1 hypermultiplet BPS state. States of higher spin or multiplicity do not turn into nodes in the quiver.

If ${\cal M}_{_{\gamma}}$ is nonempty, there is a BPS particle in the spectrum with charge $\gamma$. Since ${\cal M}_{_{\gamma}}$ has 4 supercharges, it is a Kähler manifold, and its cohomology forms representations of Lefschetz $SU(2)$. If ${\cal M}_{_{\gamma}}$ is a point, it defines a hypermultiplet. If, instead, ${\cal M}_{_{\gamma}}\simeq\mathbb{P}^{^{1}}$, it defines a vector multiplet.

BPS-wall crossing arises from the $\Pi$-stability condition if at some point $u$ in ${\cal M}_{_{\gamma}}$, $Z_{_{u}}(S)\equiv Z_{_{u}}(R)$, with $S$ a subrepresentation of $R$.

§.§ Vacuum structure: Coulomb, Higgs and mixed branches

The full moduli space is unknown in most cases. For this reason, in the present work we will only be focussing on class ${\cal S}$-theories obtained by twisted compactification on a Riemann surface of 6D ${\cal N}$=(2,0) on $A_{_{N-1}}$, with generic moduli space

\begin{equation}
{\cal M}\ \overset{def.}{=}\ \bigcup_{\alpha}\ C_{_{\alpha}}\ \times\ H_{_{\alpha}}
\end{equation}

with $C_{_{\alpha}}, H_{_{\alpha}}$ denoting the Coulomb and Higgs branches of the theory.

The former is defined by the Seiberg-Witten curve

\begin{equation}
x^{^{N}}+\sum_{i=2}^{N}\phi_{_{i}}(z)x^{^{N-i}} = 0
\end{equation}

where $z$ is the coordinate of the base Riemann surface, and $x$ is the one on the fiber of the cotangent bundle $K\equiv T^*C$. $\phi_{_{i}}(z)$ is a section of the line bundle, $K^{^{\otimes i}}$, namely it is the $i^{^{th}}$ differential on the Riemann surface $C$. In absence of punctures, it s contribution to the dimension of the Coulomb branch reads

\begin{equation}
\text{dim}(\phi_{_{i}})\ \equiv\ \text{dim} H^{^{0}}\left(C, K^{^{\otimes i}}\right)\ =\ (2i-1)(g-1)
\end{equation}

The general expression in presence of regular punctures reads

\begin{equation}
\text{dim}_{_{{\mathbb{C}}}} C_{_{\alpha}}\ \equiv\ (k-1)g + \sum_{s=1}^{k}(n_{_{s}}^2-1)(g-1)+\sum_{p}\text{dim}_{_{\mathbb{C}}} C_{_{\alpha}}(p)
\end{equation}

where $k$ denotes the number of elements featuring in the partition $X\equiv\left[n_{_{1}},...,n_{_{k}}\right]$, and

\begin{equation}
\text{dim}_{_{{\mathbb{C}}}} C_{_{\alpha}}(p)\ \equiv\ \sum_{s=1}^{k} d(Y_{_{s}}^{^{D}})\ =\    \sum_{s=1}^{k}\sum_{i=2}p_{_{i}}\equiv \frac{1}{2}\sum_{s=1}^{k} \left(N^2-\sum_{a=1}^{\ell}m_{_{a}}^2\right)
\end{equation}

where $Y^{^{D}}$ denotes the local contribution from a regular puncture.

The total dimension along the Higgs branch, instead, reads

\begin{equation}
\text{dim}_{_{{\mathbb{H}}}} H_{_{\alpha}}\ \equiv\ k-1 + \sum_{p}\text{dim}_{_{\mathbb{H}}} H_{_{\alpha}}(p)
\end{equation}

\begin{equation}
\text{dim}_{_{\mathbb{H}}} H_{_{\alpha}}(p)
\equiv d(Y^{^{\prime}})-d(Y)
\end{equation}

The vacuum structure of ${\cal N}=2$ class ${\cal S}$ theories, comprising a Coulomb ($C$), Higgs ($H$), and mixed branch.

Upon choosing a partition of $N$: $X\equiv\left[n_{_{1}},...,n_{_{k}}\right]$, the maximally factorised SW curve can be expressed as follows

\begin{equation}
\prod_{s=1}^k\left(x^{^{n_{_{s}}}}+\sum_{i=2}^{n_{_{s}}}\phi_{_{s,i}}(z)x^{^{n_{_{s}}-i}}\right) = 0
\end{equation}

§.§ Metrics

We now briefly overview the definition of the metric on the Riemann surface, following the work by Gaiotto, Moore, and Nietzke.

Take ${\cal B}$ to be the Coulomb branch of a certain 4D ${\cal N}=2$ supersymmetric gauge theory. At any point $u\in{\cal B}$, the gauge group is broken to a maximal torus $U(1)^{^r}$. Denoting by $\hat\Gamma_{_u}$ the charge lattice equipped with a pairing $<,>$, with radical being the flavour charge lattice, $\left(\Gamma_{_{flav}}\right)_{_u}$, the quotient charge sublattice of gauge charges can be defined as follows

\begin{equation}
\Gamma_{_u}\ \overset{def.}{=}\ \hat\Gamma_{_u}\bigg/\left(\Gamma_{_{flav}}\right)_{_u},
\end{equation}

with rank$\Gamma_{_u}=2r$.

The central charge homomorphism is

\begin{equation}
Z:\ \hat\Gamma_{_u}\ \longrightarrow\ \mathbb{C}.
\end{equation}

Then, for every particle, one can define $Z_{_{\gamma}}(u)$, with $\gamma$ a local section of $\hat\Gamma$, as denoting the central charge of a particle of charge $\gamma$.

For a theory placed on $\mathbb{R}^{^3}\times S^{^1}$, with $R$ denoting the radius of the $S^{^1}$, for $R<<1$, the 4D theory is effectively 3D, and its moduli space is locally a product of hyperk$\ddot{\text{a}}$hler manifolds

\begin{equation}
{\cal M}_{_{tot}}\ =\ \text{HB}_{_{4D}}\times \text{CB}_{_{3D}}
\end{equation}

At generic nonsingular points of the 3D CB, the HB is absent, so one can just restrict to the 3D CB

\begin{equation}
{\cal M}\ \overset{def.}{=}\ \text{CB}_{_{3D}},
\end{equation}

such that one can define the map

\begin{equation}
\pi:\ {\cal M}\ \longrightarrow\ {\cal B}.
\end{equation}

That the metric on ${\cal M}$ is hyperk$\ddot{\text{a}}$hler follows from the theory being supersymmetric.

The main idea proposed by [121, 122, 122] is that of describing the metric by introducing holomorphic Darboux coordinates for ${\cal M}$ (when viewing the latter as a holomorphic symplectic manifold).