Algebraic Singularities, Finite Graphs and D-Brane Theories by Y. He

By Y. He

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The reader is encouraged to consult such classic texts as [26, 27]. 6 A finite graph is a triple (V, E, I) such that V, E are disjoint finite sets (respectively the set of vertices and edges) with members of E joining those of V according to the incidence relations I. The graph is undirected if for each edge e joining vertex i to j there is another edge e′ joining j to i; it is directed otherwise. , edges joining a vertex to itself). The graph is connected if any two vertices can be linked a series of edges, a so-called walk.

A main theme therefore, is the construction of the various c = 9 so-called “internal” N = 2 superconformal theories. v. [18] for an excellent pedagogical review). These are 1. s ¯ i, 2. The Landau-Ginsberg (LG) theory, constructed from chiral superfields Ψi , Ψ and with a holomorphic polynomial superpotential W (Ψi) giving a unique vacuum. ). dz 2 dθ4 K(Ψi , Ψ We usually start with a non-conformal case and let it flow to a superconformal fixed point into IR; 3. The minimal models, being rational conformal field theories with a finite number of primary fields (and c < 1 in the bosonic case or c < 3/2 in N = 1), furnishing unitary highest-weight representations of the (super)-Virasoro algebra.

If the graph is simple, then we can also represent the graph by an incidence matrix, an n × m matrix dia in whose a-th columns there is a −1 (resp. 1) in row i (resp. row j) if there an a-th edge going from i to j. We emphasise that the graph must be simple for the incidence matrix to fully encapture its information. Later on in Liber III we will see this is a shortcoming when we are concerned with gauged linear sigma models. 1 Quivers Now let us move onto a specific type of directed graphs, which we shall call a quiver.

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