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Twisted holography from the B-model on a 7-fold

Seraphim Jarov

TL;DR

This work extends twisted holography to higher-dimensional Calabi–Yau geometries by studying the B-model on CY7-folds with branes that realize twistor uplifts of self-dual $N=4$ and $N=2$ theories. By solving the Kodaira–Spencer equations with brane sources and localizing via carefully chosen bulk superpotentials, the author obtains backreacted geometries that collapse to $AdS_3\times S^3$ (described by $SL(2,\mathbb{C})$ or $SL(2,\mathbb{C})/\mathbb{Z}_2$) and boundary theories that reduce to 2d chiral-algebra subsectors of the corresponding SYM theories. The paper then connects these twisted holography constructions to matrix-model frameworks, including Pestun localization on $S^4$ and the Dijkgraaf–Vafa matrix model, showing how Gaussian subsectors arise from Beltrami deformations and how these relate to the chiral algebras in the twisted setting. Together, these results illuminate how higher-dimensional topological strings encode lower-dimensional holographic correspondences and provide new avenues to test and extend AdS/CFT ideas within a twistor/topological-string context.

Abstract

I study a topological string construction of the holographic duality between Kodaira-Spencer gravity on the Calabi-Yau 7-fold $\mathcal{O}(-1)^4\to\mathbb{PT}$ in the presence of a stack of $N$ backreacted D5 branes wrapping twistor space, $\mathbb{PT}$. The theory on the stack of branes is the twistor uplift of self-dual $\mathcal{N}=4$ gauge theory. I show that turning on a bulk superpotential and twisting the brane theory by the dual supercharge reduces the duality to twisted holography which relates the B-model on AdS$_3\times S^3\cong SL(2,\mathbb{C})$ to the 2d chiral algebra subsector of $\mathcal{N}=4$. I do an analogous computation for the twistor uplift of self-dual $\mathcal{N}=2$ by working on the Calabi-Yau 7-fold $\mathcal{O}(-2,-2)\oplus \mathcal{O}(0,-1)^2\to\mathbb{CP}^1\times\mathbb{PT}$. I also connect twists of the twistor uplift of self-dual $\mathcal{N}=4$ with the matrix model found by supersymmetric localization on $S^4$ and the Dijkgraaf-Vafa matrix model construction.

Twisted holography from the B-model on a 7-fold

TL;DR

This work extends twisted holography to higher-dimensional Calabi–Yau geometries by studying the B-model on CY7-folds with branes that realize twistor uplifts of self-dual and theories. By solving the Kodaira–Spencer equations with brane sources and localizing via carefully chosen bulk superpotentials, the author obtains backreacted geometries that collapse to (described by or ) and boundary theories that reduce to 2d chiral-algebra subsectors of the corresponding SYM theories. The paper then connects these twisted holography constructions to matrix-model frameworks, including Pestun localization on and the Dijkgraaf–Vafa matrix model, showing how Gaussian subsectors arise from Beltrami deformations and how these relate to the chiral algebras in the twisted setting. Together, these results illuminate how higher-dimensional topological strings encode lower-dimensional holographic correspondences and provide new avenues to test and extend AdS/CFT ideas within a twistor/topological-string context.

Abstract

I study a topological string construction of the holographic duality between Kodaira-Spencer gravity on the Calabi-Yau 7-fold in the presence of a stack of backreacted D5 branes wrapping twistor space, . The theory on the stack of branes is the twistor uplift of self-dual gauge theory. I show that turning on a bulk superpotential and twisting the brane theory by the dual supercharge reduces the duality to twisted holography which relates the B-model on AdS to the 2d chiral algebra subsector of . I do an analogous computation for the twistor uplift of self-dual by working on the Calabi-Yau 7-fold . I also connect twists of the twistor uplift of self-dual with the matrix model found by supersymmetric localization on and the Dijkgraaf-Vafa matrix model construction.

Paper Structure

This paper contains 21 sections, 1 theorem, 85 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Future work
  3. Background
  4. The backreaction
  5. The theory on the brane
  6. The twistor string
  7. The self-dual $\mathcal{N}=4$ cosntruction
  8. The backreaction
  9. The theory on the brane
  10. The self-dual $\mathcal{N}=2$ construction
  11. The brane construction
  12. The backreaction
  13. The theory on the brane
  14. Supersymmetric localization and matrix models
  15. Supersymmetric localization on $S^4$
  16. ...and 6 more sections

Key Result

Proposition 5.2.1

Introducing the Beltrami differential deforms the holomorphic vector bundle $T^*\mathbb{CP}^1\times\mathbb C$ to $\mathcal{O}(-1)^2\to\mathbb{CP}^1$.

Figures (2)

  • Figure 1: The holographic duality studied in this paper at the top of the figure, where KS stands for Kodaira-Spencer and hCS stands for holomorphic Chern-Simons. The vertical arrows denote twisting by the superpotential/supercharge $W$/$Q$. The bottom row is the duality studied in twisted holography costello2021twistedholography.
  • Figure 2: An illustration of the steps taken to arrive at the Pestun matrix model localization from the twistor uplift of self-dual $\mathcal{N}=4$. We start with the twistor uplift of self-dual $\mathcal{N}=4$ SYM, which we twist by the two choices of $Q$ to get two copies of the 2d chiral algebra subsector of $\mathcal{N}=4$ at the north and south poles of $S^4$. Then we KK compactify the 2d chiral algebras to arrive at Gaussian matrix models located at the north and south poles of $S^4$.

Theorems & Definitions (2)

  • Proposition 5.2.1
  • proof