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The Gauge/Gravity Theory of Blown up Four Cycles

S. Benvenuti, M. Mahato, L. A. Pando Zayas, Y. Tachikawa

TL;DR

This work identifies a universal Kähler deformation of Calabi–Yau cones over Sasaki–Einstein manifolds, realized by blowing up a 4-cycle and interpreted in the dual quiver gauge theories as motion along non-mesonic directions in the moduli space. The authors construct explicit Calabi–Yau metrics and near-horizon gravity solutions, analyze a detailed example with a vanishing $ ext{P}^1\times\text{P}^1$ base, and show that the deformation is controlled by two parameters ($a$ global and $b$ local) with the global deformations producing dimension-2 operators and the local deformations producing dimension-6 operators in the dual CFT. They develop a toric description via GLSMs, classify global versus local deformations, and relate FI parameters to geometric moduli, establishing a precise dictionary between gravity modes and operators in the quiver theory. The paper further demonstrates that the $b$-deformation leads to area-law confinement as probed by strings, discusses the role of fractional branes, and argues for a broad universality of these deformations across toric Calabi–Yau cones, with potential extensions to KS-like baryonic branches. These results provide a robust framework for understanding non-mesonic moduli in AdS/CFT and reveal a rich interplay between geometry, toric data, and four-dimensional gauge dynamics.

Abstract

We present an explicit supersymmetric deformation of supergravity backgrounds describing D3-branes on Calabi-Yau cones. From the geometrical point of view, it corresponds to blowing up a 4-cycle in the Calabi-Yau and can be done universally. In the field theory, we identify this deformation with motion on non-mesonic directions in the full moduli space of vacua. For the case of a Z_2 orbifold of the conifold, we discuss an explicit gravity solution with two deformation parameters: one corresponding to blowing up a 2-cycle and one corresponding to blowing up a 4-cycle. The generic case where the Calabi-Yau is toric is also discussed in detail. Quite generally, the order parameter of these 4-cycle deformations is a dimension six operator. We also consider probe strings which show linear confinement and probe D7 branes which help in understanding the behavior far in the infrared.

The Gauge/Gravity Theory of Blown up Four Cycles

TL;DR

This work identifies a universal Kähler deformation of Calabi–Yau cones over Sasaki–Einstein manifolds, realized by blowing up a 4-cycle and interpreted in the dual quiver gauge theories as motion along non-mesonic directions in the moduli space. The authors construct explicit Calabi–Yau metrics and near-horizon gravity solutions, analyze a detailed example with a vanishing base, and show that the deformation is controlled by two parameters ( global and local) with the global deformations producing dimension-2 operators and the local deformations producing dimension-6 operators in the dual CFT. They develop a toric description via GLSMs, classify global versus local deformations, and relate FI parameters to geometric moduli, establishing a precise dictionary between gravity modes and operators in the quiver theory. The paper further demonstrates that the -deformation leads to area-law confinement as probed by strings, discusses the role of fractional branes, and argues for a broad universality of these deformations across toric Calabi–Yau cones, with potential extensions to KS-like baryonic branches. These results provide a robust framework for understanding non-mesonic moduli in AdS/CFT and reveal a rich interplay between geometry, toric data, and four-dimensional gauge dynamics.

Abstract

We present an explicit supersymmetric deformation of supergravity backgrounds describing D3-branes on Calabi-Yau cones. From the geometrical point of view, it corresponds to blowing up a 4-cycle in the Calabi-Yau and can be done universally. In the field theory, we identify this deformation with motion on non-mesonic directions in the full moduli space of vacua. For the case of a Z_2 orbifold of the conifold, we discuss an explicit gravity solution with two deformation parameters: one corresponding to blowing up a 2-cycle and one corresponding to blowing up a 4-cycle. The generic case where the Calabi-Yau is toric is also discussed in detail. Quite generally, the order parameter of these 4-cycle deformations is a dimension six operator. We also consider probe strings which show linear confinement and probe D7 branes which help in understanding the behavior far in the infrared.

Paper Structure

This paper contains 25 sections, 116 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Calabi-Yau metrics and D3 branes on blown up 4-cycles
  3. D3 branes
  4. Large radius expansion
  5. A detailed example: vanishing $\mathbb{P}^1 \times \mathbb{P}^1$
  6. Limit to Eguchi-Hanson
  7. The field theory dual
  8. The global deformation
  9. Global deformations from the metric: resolved conifold
  10. General geometric deformation
  11. Toric description of the blow up
  12. Comments on the 4-cycle deformation of the Klebanov-Strassler solution
  13. Blowing up $4$-cycles: the general case
  14. Deformations and Operators
  15. More on $a$-deformation
  16. ...and 10 more sections

Figures (7)

  • Figure 1: $(p,q)$ web description of the vanishing $\mathbb{P}^1 \times \mathbb{P}^1$ geometry as in Aharony:1997bh. The two possible deformations are shown as "global" and "local".
  • Figure 2: Quiver diagram for D3 branes probing the local $\mathbb{P}^1 \times \mathbb{P}^1$ singularity.
  • Figure 3: Toric description of the four-cycle in the orbifolded conifold.
  • Figure 4: $a$-type deformation
  • Figure 5: A four cycle in the orbifolded conifold.
  • ...and 2 more figures