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Gauge Theory RG Flows from a Warped Resolved Orbifold

Chethan Krishnan, Stanislav Kuperstein

TL;DR

This work analyzes gauge-string duality for D3-branes at a point on the resolution of ${\mathbb C}^3/{\mathbb Z}_3$, demonstrating a holographic RG flow from $AdS_5 \times S^5/{\bf Z}_3$ in the UV to $AdS_5 \times S^5$ in the IR. The authors construct the resolved ALE metric with a single parameter, solve the six-dimensional Laplacian to obtain the warp factor, and show that the gravity side realizes the RG flow; on the field theory side, they identify two RG-flow scenarios from the quiver gauge theory to ${\cal N}=4$ SYM via Higgsing and confinement. The two gauge-theory paths correspond to different patterns of VEVs in the bi-fundamental fields and lead to a consistent IR fixed point, with the resolution parameter mapped to a dimension-six operator (the precise operator remains unknown). The work thus clarifies how geometric resolutions in toric Calabi-Yau singularities induce controlled RG flows in holographic duals and highlights remaining open questions about the precise operator dictionary and nonperturbative effects.

Abstract

We study gauge-string duality of D3-branes localized at a point on the resolution of the ALE orbifold $\IC^3/\IZ_3$. By explicitly solving for the warp factor, we demonstrate the holographic RG flow from $AdS_5 \times S^5/\IZ_3$ (far away from the resolution) to the usual $AdS_5\times S^5$ (close to the stack). On the gauge theory side, this maps to the flow between the quiver gauge theory and ${\cal N}=4$ super Yang-Mills. We present two possible scenarios for this RG flow depending on the choice of the VEVs. In particular, one of the scenarios proceeds by two steps involving both Higgsing and confinement.

Gauge Theory RG Flows from a Warped Resolved Orbifold

TL;DR

This work analyzes gauge-string duality for D3-branes at a point on the resolution of , demonstrating a holographic RG flow from in the UV to in the IR. The authors construct the resolved ALE metric with a single parameter, solve the six-dimensional Laplacian to obtain the warp factor, and show that the gravity side realizes the RG flow; on the field theory side, they identify two RG-flow scenarios from the quiver gauge theory to SYM via Higgsing and confinement. The two gauge-theory paths correspond to different patterns of VEVs in the bi-fundamental fields and lead to a consistent IR fixed point, with the resolution parameter mapped to a dimension-six operator (the precise operator remains unknown). The work thus clarifies how geometric resolutions in toric Calabi-Yau singularities induce controlled RG flows in holographic duals and highlights remaining open questions about the precise operator dictionary and nonperturbative effects.

Abstract

We study gauge-string duality of D3-branes localized at a point on the resolution of the ALE orbifold . By explicitly solving for the warp factor, we demonstrate the holographic RG flow from (far away from the resolution) to the usual (close to the stack). On the gauge theory side, this maps to the flow between the quiver gauge theory and super Yang-Mills. We present two possible scenarios for this RG flow depending on the choice of the VEVs. In particular, one of the scenarios proceeds by two steps involving both Higgsing and confinement.

Paper Structure

This paper contains 12 sections, 67 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Toric Description of ${\mathbb C}^3/{\mathbb Z}_3$ and its Resolution
  3. Gauge-String Duality on the Unresolved Orbifold
  4. Supergravity with Brane Sources
  5. The Dual Quiver Theory
  6. Branes on the Resolved Orbifold
  7. Consistency Checks and Holographic RG Flow
  8. RG Flow in the Gauge Theory
  9. The $A_3=B_3=v$ RG flow
  10. The $C_3=v$ RG flow
  11. Discussion
  12. Acknowledgments

Figures (3)

  • Figure 1: Toric Diagram of ${\mathbb C}^3/{\mathbb Z}_3$.
  • Figure 2: The quiver for the gauge theory dual to ${\mathbb C}^3/{\mathbb Z}_3$.
  • Figure 3: The gauge theory quiver for the theory after the first "step" of the RG flow. The VEV of the field $C_3$ breaks $SU(N)_1 \times SU(N)_2$ down to $SU(N)_{1,2}$ and the massive fields $A_{1,2}$ and $B_{1,2}$ are integrated out. At the second step, node 3 confines.