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Lectures on D-branes, Gauge Theories and Calabi-Yau Singularities

Yang-Hui He

TL;DR

The lectures develop a concrete framework connecting D-brane gauge theories to local Calabi–Yau singularities, using forward and inverse algorithms to map between gauge data (quivers, ranks, superpotentials) and geometric data (toric diagrams, moduli spaces). They systematically build from the flat-space ${N}=4$ theory on D3-branes to ${N}=1$ theories arising from orbifolds, toric singularities, and del Pezzo cones, with the McKay correspondence and quiver representations playing central roles. A core theme is the emergence of the Calabi–Yau moduli space as the vacuum of the worldvolume gauge theory, and the rich duality structure (toric, Seiberg, mirror) that relates seemingly different quivers to the same geometry. The material emphasizes an algorithmic, geometry-driven approach to engineering gauge theories from string theory, highlighting applications to phenomenology, dualities, and the interplay between physics and algebraic geometry. These insights lay a foundation for constructing and classifying ${N}=1$ quiver gauge theories from Calabi–Yau singularities and for understanding the geometric content of gauge dynamics in string theory.

Abstract

These lectures, given at the Chinese Academy of Sciences for the BeiJing/HangZhou International Summer School in Mathematical Physics, are intended to introduce, to the beginning student in string theory and mathematical physics, aspects of the rich and beautiful subject of D-brane gauge theories constructed from local Calabi-Yau spaces. Topics such as orbifolds, toric singularities, del Pezzo surfaces as well as chaotic duality will be covered.

Lectures on D-branes, Gauge Theories and Calabi-Yau Singularities

TL;DR

The lectures develop a concrete framework connecting D-brane gauge theories to local Calabi–Yau singularities, using forward and inverse algorithms to map between gauge data (quivers, ranks, superpotentials) and geometric data (toric diagrams, moduli spaces). They systematically build from the flat-space theory on D3-branes to theories arising from orbifolds, toric singularities, and del Pezzo cones, with the McKay correspondence and quiver representations playing central roles. A core theme is the emergence of the Calabi–Yau moduli space as the vacuum of the worldvolume gauge theory, and the rich duality structure (toric, Seiberg, mirror) that relates seemingly different quivers to the same geometry. The material emphasizes an algorithmic, geometry-driven approach to engineering gauge theories from string theory, highlighting applications to phenomenology, dualities, and the interplay between physics and algebraic geometry. These insights lay a foundation for constructing and classifying quiver gauge theories from Calabi–Yau singularities and for understanding the geometric content of gauge dynamics in string theory.

Abstract

These lectures, given at the Chinese Academy of Sciences for the BeiJing/HangZhou International Summer School in Mathematical Physics, are intended to introduce, to the beginning student in string theory and mathematical physics, aspects of the rich and beautiful subject of D-brane gauge theories constructed from local Calabi-Yau spaces. Topics such as orbifolds, toric singularities, del Pezzo surfaces as well as chaotic duality will be covered.

Paper Structure

This paper contains 35 sections, 49 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Minute Waltz on the String
  3. The D3-brane in $\mathbb{R}^{1,9}$
  4. D3-branes on Calabi-Yau threefolds
  5. The Simplest Case: $S = \mathbb{C}^3$
  6. Orbifolds and Quivers
  7. Projection to Daughter Theories
  8. Quivers
  9. The McKay Correspondence
  10. McKay, Dimension 2 and ${\mathcal{N}}=2$
  11. ${\mathcal{N}}=1$ Theories and $\mathbb{C}^3$ Orbifolds
  12. Quivers, Modular Invariants, Path Algebras?
  13. More Games
  14. Gauge Theories, Moduli Spaces and Symplectic Quotients
  15. Quiver Gauge Theory
  16. ...and 20 more sections

Figures (3)

  • Figure 1: A pictorial representation of our motivations. We need to reduce the 10 dimensions of superstring theory with various gauge groups down to a 4 dimensional world with ${\mathcal{N}}=1$ supersymmetry and $SU(3) \times SU(2) \times U(1)$ gauge group with specific matter content and interactions. Various techniques have been adopted. The one we will study here is that of D3-branes probing a transverse Calabi-Yau threefold singularity.
  • Figure 2: Open strings stretched between parallel D-branes. On the world-volume of each brane is a $U(1)$ gauge bundle. As the two coincide, the gauge group is enhanced to $U(2)$.
  • Figure 3: Our paradigm is to place a stack of $n$ parallel coincident D3-branes on an affine Calabi-Yau threefold singularity $S$. The geometry of $S$ will project the $U(n)$ gauge group on the branes to product gauge groups with bi-fundamental matter and interactions. The resulting theory is conveniently represented as a quiver diagram. Examples of $S$ thus far investigated have been orbifolds, toric singularities and cones over del Pezzo surfaces, as shewn in the Venn diagram.