#### 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.

#### Recommended Citation

Hasan, Azeem, "Physics and Mathematics of Graded Quivers" (2019). *CUNY Academic Works.*

https://academicworks.cuny.edu/gc_etds/3321