Lecture notes on quivers with superpotential and their representations

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cture notes on quivers with superpotential and their representations Alexander Quintero Vélez1 · Fabricio Valencia2 Accepted: 26 September 2020 © Instituto de Matemática e Estatística da Universidade de São Paulo 2020

Abstract These lecture notes are based on a mini-course presented at the fifth version of the Workshop Geometry in Algebra and Algebra in Geometry held in Medellín–Colombia in October 2019. The aim is to provide the background necessary to understand the theory of quivers with relations given by superpotentials. A heavy emphasis is placed throughout on examples to illustrate the applicability of the theory. The motivations for the lectures come from several sources: superpotentials in physics, Calabi–Yau algebras, and noncommutative resolutions. Keywords  Representations of quivers · Superpotentials · Moduli spaces

1 Introduction The notion of quiver with superpotential has appeared prominently in both the mathematics and the physics literature as a tool that underlies many topics of current interest. In mathematics, they appear in the study of noncommutative crepant resolutions [22], Calabi–Yau algebras [11, 23], the homological minimal model program [10], motivic Donaldson-Thomas invariants [17] and cluster algebras [8, 9]. In physics, they arise in connection with the study of N = 1 supersymmetric gauge theories on D-brane probes at singularities [2, 3, 15] and the BPS spectrum of N = 2 supersymmetric field theories [1]. Communicated by Pedro Hernandez Rizzo. * Alexander Quintero Vélez [email protected] Fabricio Valencia [email protected] 1

Escuela de Matemáticas, Universidad Nacional de Colombia Sede Medellín, Carrera 65 # 59A– 110, Medellín, Colombia

2

Instituto de Matemáticas, Universidad de Antioquia, Calle 70 # 52–21, Medellín, Colombia



13

Vol.:(0123456789)



São Paulo Journal of Mathematical Sciences

In these lectures we introduce the reader to the subject of quivers with superpotentials and their representations by providing the very basic definitions and properties. In this way we introduce the tools and techniques which are central for any further study in the topic. For a much more detailed account of many of the ideas presented here we recommend the reader consult [5] and [24]. The lecture notes are structured as follows. In Sect. 2 we provide the basic definitions concerning quivers and their representations. In Sect.  3 we introduce the notion of superpotential and explore many examples. Section 4 is devoted to a brief discussion of dimer models, as one of the most notable examples of quivers with superpotential. In Sect. 5 we give a brief overview of quiver GIT that allows us to extract geometry from quivers with superpotential and, in general, any quiver with relations. Finally, in Sect.  6 we sketch how quivers with superpotentials arise in the context of N = 1 supersymmetric gauge theories living on the worldvolume of D-branes.

2 Quivers and their representations A quiver Q is a directed graph which is specified by a finite set of vertices Q0 , a finite set of

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