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The Refined Topological Vertex

Amer Iqbal, Can Kozcaz, Cumrun Vafa

Abstract

We define a refined topological vertex which depends in addition on a parameter, which physically corresponds to extending the self-dual graviphoton field strength to a more general configuration. Using this refined topological vertex we compute, using geometric engineering, a two-parameter (equivariant) instanton expansion of gauge theories which reproduce the results of Nekrasov. The refined vertex is also expected to be related to Khovanov knot invariants.

The Refined Topological Vertex

Abstract

We define a refined topological vertex which depends in addition on a parameter, which physically corresponds to extending the self-dual graviphoton field strength to a more general configuration. Using this refined topological vertex we compute, using geometric engineering, a two-parameter (equivariant) instanton expansion of gauge theories which reproduce the results of Nekrasov. The refined vertex is also expected to be related to Khovanov knot invariants.

Paper Structure

This paper contains 34 sections, 204 equations, 23 figures.

Table of Contents

  1. Introduction
  2. GV Formulation and Topological Vertex
  3. Topological string amplitudes and GV reformulation
  4. Partition function from the topological vertex
  5. The Refined Topological Vertex
  6. Open String Partition Function and $\hbox{Sym}^{\bullet}(\mathbb{C})$
  7. Stack of Branes
  8. Refined vertex and open string partition function
  9. Brane orientation and the gluing rule
  10. Refined Partition Functions from the Refined Vertex
  11. ${\cal O}(-1)\oplus {\cal O}(-1)\mapsto \mathbb{P}^{1}$
  12. $\chi_{y}$-genus, $\hbox{Sym}^{\bullet}(\mathbb{C}^2$) and the refined topological vertex
  13. ${\cal O}(0)\oplus {\cal O}(-2)\mapsto \mathbb{P}^{1}$
  14. Another toric geometry: $\widetilde{{\mathbb C}^{3}/\mathbb{Z}_{2}\times\mathbb{Z}_{2}}$
  15. Local $\mathbb{P}^{1}\times \mathbb{P}^{1}$
  16. ...and 19 more sections

Figures (23)

  • Figure 1: The holomorphic maps wrapping the disks along the degeneration loci with boundaries on the Lagrangian branes.
  • Figure 2: Slices of the 3$D$ partitions are counted with parameters $t$ and $q$ depending on the shape of $\nu$.
  • Figure 3: Toric diagram of ${\cal O}(-1)\oplus {\cal O}(-1)\mapsto \mathbb{P}^{1}$. The vertices are glued along the preferred direction $\nu$.
  • Figure 4: Toric diagram of ${\cal O}(-1)\oplus {\cal O}(-1)\mapsto \mathbb{P}^{1}$. The vertices are glued along the unpreferred direction $\lambda$
  • Figure 5: Toric diagram of partially compactified ${\cal O}(-1)\oplus {\cal O}(-1)\mapsto \mathbb{P}^{1}$.
  • ...and 18 more figures