We de ne the quantum cluster character assigning an element of a quantum torus to each representation of a valued quiver (Q; d) and investigate its relationship to external and internal mutations of a quantum cluster algebra associated to (Q; d). We will see that the external mutations are related to re ection functors and internal mutations are related to tilting theory. Our main result will show the quantum cluster character gives a cluster monomial in this quantum cluster algebra whenever the representation is rigid, moreover we will see that each non-initial cluster variable can be obtained in this way from the quantum cluster character.

Given a quiver flag variety Y equipped with a tilting bundle E, a construction ofCraw, Ito and Karmazyn [CIK18] produces a closed immersion f_E : Y -> M(E), where M(E) is the fine moduli space of cyclic modules over the algebra End(E).In this thesis we present two classes of examples where f_E is an isomorphism. Firstly, when Y is toric and E is the tilting bundle from [Cra11]; this generalises the well-known fact that P^n can be recovered from the endomorphism algebra of \oplus_{0\leq i \leq n} O_{P^n}(i). Secondly, when Y = Gr(n, 2), the Grassmannian of 2-dimensional quotients of k^n and E is the tilting bundle from [Kap84]. In each case, we give a presentation of the tilting algebra A = End(E) by constructing a quiver Q' such that there is a surjective k-algebra homomorphism \Phi: kQ' -> A, and then give an explicit description of the kernel.

We study a family of three-dimensional Lie algebras that depend on a continuous parameter. We introduce certain quivers and prove that idempotented versions of the enveloping algebras of the Lie algebras are isomorphic to the path algebras of these quivers modulo certain ideals in the case that the free parameter is rational and non-rational, respectively. We then show how the representation theory of the introduced quivers can be related to the representation theory of quivers of affine type A, and use this relationship to study representations of the family of Lie algebras of interest. In particular, though it is known that this particular family of Lie algebras consists of algebras of wild representation type, we show that if we impose certain restrictions on weight decompositions, we obtain full subcategories of the category of representations that are of finite or tame representation type.

Quiver gauge theories are widely studied in the context of AdS/CFT, which establishes a correspondence between CFTs and string theories. CFTs in turn offer a map between quantum states and Gauge Invariant Operators (GIOs). This thesis presents results on the counting and correlators of holomorphic GIOs in quiver gauge theories with flavour symmetries, in the zero coupling limit. We first give a prescription to build a basis of holomorphic matrix invariants, labelled by representation theory data. A fi nite N counting function of these GIOs is then given in terms of Littlewood-Richardson coefficients. In the large N limit, the generating function simpli fies to an in finite product of determinants, which depend only on the weighted adjacency matrix associated with the quiver. The building block of this product has a counting interpretation by itself, expressed in terms of words formed by partially commuting letters associated with closed loops in the quiver. This is a new relation between counting problems in gauge theory and the Cartier-Foata monoid. We compute the free fi eld two and three point functions of the matrix invariants. These have a non-trivial dependence on the structure of the operators and on the ranks of the gauge and flavour symmetries: our results are exact in the ranks, and their expansions contain information beyond the planar limit. We introduce a class of permutation centraliser algebras, which give a precise characterisation of the minimal set of charges needed to distinguish arbitrary matrix invariants. For the two-matrix model, the relevant non-commutative algebra is parametrised by two integers. Its Wedderburn-Artin decomposition explains the counting of restricted Schur operators. The structure of the algebra, notably its dimension, its centre and its maximally commuting sub-algebra, is related to Littlewood-Richardson numbers for composing Young diagrams.

We can view quiver representations of a fixed dimension vector as an algebraic variety over an algebraically closed field $K$. There is an action of the product of general linear groups on each of these varieties where the orbits of the action correspond to isomorphism classes of quiver representation. A $K$-algebra $A$ is said to have the dense orbit property if for each dimension vector, the product of the general linear group acts on each irreducible component of the module variety with a dense orbit. Under certain conditions, a $K$ algebra $A$ is representation finite if and only if it $A$ has the dense orbit property. The implication representation finite implies the dense orbit property is always true. The converse is not true in general, as shown by Chindris, Kinser, and Weyman in \cite{ryan}. Our main theorem of this thesis builds on their work to give a family of representation infinite algebras with the dense orbit property. We also give a conjectured classification of indecomposables with dense orbits. \par In the future, we hope the work presented here can be used to find even more examples of representation infinite algebra with the dense orbit property to then develop deeper theory to classify algebras with the dense orbit property that are representation infinite.

We explore supersymmetric quantum field theories in three and four dimensions via an analysis of their BPS spectrum. In four dimensions, we develop the theory of BPS quivers which provides a simple picture of BPS states in terms of a set of building block atomic particles, and basic quantum mechanical interactions. We develop efficient techniques, rooted in an understanding of quantum-mechanical dualities, for determining the spectrum of bound states, and apply these techniques to calculate the spectrum in a wide class of field theories including ADE gauge theories with matter, and Argyres-Douglas type theories. Next, we explore the geometric content of quivers in the case when the four-dimensional field theory can be constructed from the six-dimensional (2; 0) superconformal field theory compactified on a Riemann surface. We find that the quiver and its superpotential are determined by an ideal triangulation of the associated Riemann surface. The significance of this triangulation is that it encodes the data of geodesics on the surface which in turn are the geometric realization of supersymmetric particles. Finally we describe a class of three-dimensional theories which are realized as supersymmetric domain walls in the previously studied four-dimensional theories. This leads to an understanding of quantum field theories constructed from the six-dimensional (2; 0) superconformal field theory compactified on a three-manifold, and we develop the associated geometric dictionary. We find that the structure of the field theory is determined by a decomposition of the three-manifold into tetrahedra and a braid which species the relationship between ultraviolet and infrared geometries. The phenomenon of BPS wall-crossing in four dimensions is then seen in these domain walls to be responsible for three-dimensional mirror symmetries. / Physics

Doctor of Philosophy / Department of Mathematics / Zongzhu Lin / For any given quiver [Gamma], there is a preprojective algebra and deformed preprojective algebras associated to it. If the ground field is of characteristic 0, then there are no finite dimensional representations of deformed preprojective algebras with special weight 1. However, if the ground field is of characteristic p, there are many dimension vectors with nonempty representation spaces of that deformed preprojective algebras. In this thesis, we study the representation category of deformed preprojective algebra with weight 1 over field of characteristic p > 0. The motivation is to count the number of rational points of the numbers X[subscript [lambda]] =[mu]⁻¹([lambda]) of moment maps at the special weights [lambda] [element of] K[superscript x] over finite fields, and to study the relations of the Zeta functions of these algebraic varieties X[subscript [lambda]] which are defined over integers to Betti numbers of the manifolds X[subscript [lambda]](C). The first step toward counting is to study the categories of representations of the deformed preprojective algebras [Pi][superscript [lambda]]. The main results of this thesis contain two types of quivers. First result shows that over finite field, the category of finite dimensional representations of deformed preprojective algebras [Pi]¹ associated to type A quiver with weight 1 is closely related to the category of finite dimensional representations of the preprojective algebra associated to two different type A quivers. Moreover, we also give the relations between their Zeta functions. The second result shows that over algebraically closed field of characteristic p > 0, the category of finite dimensional representations of deformed preprojective algebras [Pi]¹ associated to Jordan quiver with weight 1 has a close relationship with the category of finite dimensional representations of preprojective algebra associated to Jordan quiver.

O Modelo Padrão das partículas elementares descreve com sucesso as interações eletrofracas e fortes da natureza, quando comparado com todos os dados experimentais que temos ate hoje. Porém, ele apresenta problemas relacionados a origem da quebra da simetria eletrofraca assim como da hierarquia das massas dos fermions. A solução de ambos esses problemas requer a geração de grandes hierarquias estáveis. Essas hierarquias podem ser obtidas em uma classe de teorias quadridimensionais chamadas de teorias quiver de hierarquia completa, que são relacionadas a teorias de dimensões extras em AdS no limite de grande número de sítios. Mostramos que, assim como em teorias de dimensões extras curvas, existe um grau de liberdade leve associado com a quebra da invariância de escala, que pode ser identicado com um dílaton. Partindo da teoria extra-dimensional em um fundo, mostramos como esse dílaton leve também pode ser obtido em teorias quiver de hierarquia completa. / The standard model of particle physics successfuly describes the electroweak and strong interactions when compared with all the experimental data we have until now. However, it has problems regarding the origin of electroweak symmetry breaking as well as the hierarchy of fermion masses. The solutions of both these problems require the generation of large stable hierarchies. These can be obtained in a class of four-dimensional quiver theories called full-hierarchy quiver theories, which are related to extra dimensional theories in AdS, in the large-number-ofsites limit. We show that, just as in curved extra dimensional theories, there is a light degree of freedom associated with the breaking of scale invariance, which can be identied with a dilaton. Starting from an extra dimensional theory in an AdS5 background, we show how this light dilaton can be obtained in full-hierarchy quiver theories as well.

O Modelo Padrão das partículas elementares descreve com sucesso as interações eletrofracas e fortes da natureza, quando comparado com todos os dados experimentais que temos ate hoje. Porém, ele apresenta problemas relacionados a origem da quebra da simetria eletrofraca assim como da hierarquia das massas dos fermions. A solução de ambos esses problemas requer a geração de grandes hierarquias estáveis. Essas hierarquias podem ser obtidas em uma classe de teorias quadridimensionais chamadas de teorias quiver de hierarquia completa, que são relacionadas a teorias de dimensões extras em AdS no limite de grande número de sítios. Mostramos que, assim como em teorias de dimensões extras curvas, existe um grau de liberdade leve associado com a quebra da invariância de escala, que pode ser identicado com um dílaton. Partindo da teoria extra-dimensional em um fundo, mostramos como esse dílaton leve também pode ser obtido em teorias quiver de hierarquia completa. / The standard model of particle physics successfuly describes the electroweak and strong interactions when compared with all the experimental data we have until now. However, it has problems regarding the origin of electroweak symmetry breaking as well as the hierarchy of fermion masses. The solutions of both these problems require the generation of large stable hierarchies. These can be obtained in a class of four-dimensional quiver theories called full-hierarchy quiver theories, which are related to extra dimensional theories in AdS, in the large-number-ofsites limit. We show that, just as in curved extra dimensional theories, there is a light degree of freedom associated with the breaking of scale invariance, which can be identied with a dilaton. Starting from an extra dimensional theory in an AdS5 background, we show how this light dilaton can be obtained in full-hierarchy quiver theories as well.

Quivers (directed graphs) and species (a generalization of quivers) as well as their representations play a key role in many areas of mathematics including combinatorics, geometry, and algebra. Their importance is especially apparent in their applications to the representation theory of associative algebras, Lie algebras, and quantum groups. In this thesis, we discuss the most important results in the representation theory of species, such as Dlab and Ringel’s extension of Gabriel’s theorem, which classifies all species of finite and tame representation type. We also explain the link between species and K-species (where K is a field). Namely, we show that the category of K-species can be viewed as a subcategory of the category of species. Furthermore, we prove two results about the structure of the tensor ring of a species containing no oriented cycles that do not appear in the literature. Specifically, we prove that two such species have isomorphic tensor rings if and only if they are isomorphic as “crushed” species, and we show that if K is a perfect field, then the tensor algebra of a K-species tensored with the algebraic closure of K is isomorphic to, or Morita equivalent to, the path algebra of a quiver.