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Quantum Deformations from Toric Geometry

Samuel Pinansky

TL;DR

This work establishes a direct correspondence between nonperturbative quantum corrections in a quiver gauge theory and complex-structure deformations of a toric Calabi–Yau singularity. By analyzing the cone over $dP_2$ with an ADS superpotential for deformation branes, the authors derive a deformed chiral ring governed by a single parameter $S$. Using Altmann's toric-geometry framework (Minkowski and tautological cones), they show that the resulting versal deformation of the toric singularity reproduces exactly the quantum-corrected relations found in the gauge theory, providing a geometry-driven, algorithmic route to IR dynamics that complements AdS/CFT insights. The results suggest a generalizable methodology for relating nonperturbative quiver dynamics to toric deformations, with potential applications to broader families of toric quivers and their gravity duals.

Abstract

We will demonstrate how calculations in toric geometry can be used to compute quantum corrections to the relations in the chiral ring for certain gauge theories. We focus on the gauge theory of the del Pezzo 2, and derive the chiral ring relations and quantum deformations to the vacuum moduli space using Affleck-Dine-Seiberg superpotential arguments. Then we calculate the versal deformation to the corresponding toric geometry using a method due to Altmann, and show that the result is equivalent to the deformation calculated using gauge theory. In an appendix we will apply this technique to a few other examples. This is a new method for understanding the infrared dynamics of certain quiver gauge theories.

Quantum Deformations from Toric Geometry

TL;DR

This work establishes a direct correspondence between nonperturbative quantum corrections in a quiver gauge theory and complex-structure deformations of a toric Calabi–Yau singularity. By analyzing the cone over with an ADS superpotential for deformation branes, the authors derive a deformed chiral ring governed by a single parameter . Using Altmann's toric-geometry framework (Minkowski and tautological cones), they show that the resulting versal deformation of the toric singularity reproduces exactly the quantum-corrected relations found in the gauge theory, providing a geometry-driven, algorithmic route to IR dynamics that complements AdS/CFT insights. The results suggest a generalizable methodology for relating nonperturbative quiver dynamics to toric deformations, with potential applications to broader families of toric quivers and their gravity duals.

Abstract

We will demonstrate how calculations in toric geometry can be used to compute quantum corrections to the relations in the chiral ring for certain gauge theories. We focus on the gauge theory of the del Pezzo 2, and derive the chiral ring relations and quantum deformations to the vacuum moduli space using Affleck-Dine-Seiberg superpotential arguments. Then we calculate the versal deformation to the corresponding toric geometry using a method due to Altmann, and show that the result is equivalent to the deformation calculated using gauge theory. In an appendix we will apply this technique to a few other examples. This is a new method for understanding the infrared dynamics of certain quiver gauge theories.

Paper Structure

This paper contains 17 sections, 1 theorem, 80 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Classical Properties of the del Pezzo 2 theory
  3. Chiral primaries
  4. Chiral ring relations
  5. Quantum Deformations of $dP_2$
  6. Branes on $dP_2$
  7. ADS superpotential
  8. Deformed chiral ring
  9. Deformations of $dP_2$ from Toric Geometry
  10. 14 relations that define $dP_2$
  11. Altmann's Construction of the Space of Versal Deformations
  12. Conclusions
  13. Some Techniques from Toric Geometry
  14. Counting relations for the $Y^{p,q}$
  15. Additional Examples
  16. ...and 2 more sections

Key Result

Theorem 4.1

The space of deformations of the complex structure of a toric Gorenstein singularity can be found by first lifting the defining relations to the tautological cone, and then restricting to the ideal of the base of the cone over the Minkowski summands.

Figures (4)

  • Figure 1: (a) The lattice cone defining the cone over the del Pezzo 2, (b) The dual lattice cone to the cone over the del Pezzo 2. The marked points are the extra generators needed to generate the lattice cone.
  • Figure 2: (a) Splitting the polytope into two Minkowski summands (b) Separating the p-q web
  • Figure 3: (a) The toric cone defining the conifold, and the corresponding polytope (simply a unit square) (b) The dual cone for the conifold
  • Figure 4: Sketch of the locations of the generators of the dual cone for $Y^{6,1}$, note that the b generators are beneath the plane corresponding to the lines of a and c generators.

Theorems & Definitions (1)

  • Theorem 4.1