Dissertations, Theses, and Capstone Projects

Date of Degree

9-2019

Document Type

Dissertation

Degree Name

Ph.D.

Program

Physics

Advisor

Sebastian Franco

Committee Members

Daniel N. Kabat

Dimitra Karabali

V. Parameswaran Nair

Alexios Polychronakos

Subject Categories

Elementary Particles and Fields and String Theory

Keywords

D-branes, Calabi-Yau, Toric geometry, Topological Strings, Graded Quivers

Abstract

A graded quiver with superpotential is a quiver whose arrows are assigned degrees c ∈ {0, 1, · · · , m}, for some integer m ≥ 0, with relations generated by a superpotential of degree m − 1. For m = 0, 1, 2, 3 they often describe the open string sector of D-brane systems; in particular, they capture the physics of D(5 − 2m)-branes at local Calabi-Yau (CY) (m + 2)- fold singularities in type IIB string theory. We introduce m-dimers, which fully encode the m-graded quivers and their superpotentials, in the case in which the CY (m + 2)-folds are toric. A key result is the generalization of the concept of perfect matching, which plays a central role in this map, to arbitrary m. We also introduce a simplified algorithm for the computation of perfect matchings, which generalizes the Kasteleyn matrix approach to any m. We also explore various algorithms for constructing dimer models. We give a physical realization to m-dimers for m > 3, showing that for any m they describe the open string sector of the topological B-model on Xm+2. We illustrate these ideas explicitly with a few infinite families of toric singularities indexed by m ∈ N, for which we derive graded quivers associated to the geometry, using several complementary perspectives developed in this thesis.

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