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Twisted Holography

Kevin Costello, Davide Gaiotto

TL;DR

This work develops a concrete topological holographic duality between B-model strings on Calabi–Yau backgrounds with SL$_2(\mathbb{C})$ symmetry and 2d chiral algebras defined as gauged $\beta\gamma$ systems, with the deformed conifold as a central geometry. A detailed dictionary is established: local boundary insertions correspond to boundary-condition modifications in the gravity theory, while the bulk KS/hCS system generates a holographic chiral algebra whose global symmetry algebra matches the large-$N$ chiral algebra's symmetry algebra. The authors prove an isomorphism between the holographic global symmetry algebra $\mathfrak{a}_{\infty}^{hol}$ and the large-$N$ global symmetry algebra $\mathfrak{a}_{\infty}$, and show that this symmetry data fixes all two- and three-point functions (up to a few low-dimension data) and determines higher OPE coefficients. The framework extends to affine quivers, boundary conditions from holomorphic data, and a KK-reduction perspective yielding higher-spin-like towers, connecting to IIB duals via holomorphic twists and suggesting a robust, perturbatively testable topological holography beyond the original AdS/CFT setting.

Abstract

We derive and test a novel holographic duality in the B-model topological string theory. The duality relates the B-model on certain Calabi-Yau three-folds to two-dimensional chiral algebras defined as gauged $βγ\,$ systems. The duality conjecturally captures a topological sector of more familiar $\mathrm{AdS}_5 / \mathrm{CFT}_4$ holographic dualities.

Twisted Holography

TL;DR

This work develops a concrete topological holographic duality between B-model strings on Calabi–Yau backgrounds with SL symmetry and 2d chiral algebras defined as gauged systems, with the deformed conifold as a central geometry. A detailed dictionary is established: local boundary insertions correspond to boundary-condition modifications in the gravity theory, while the bulk KS/hCS system generates a holographic chiral algebra whose global symmetry algebra matches the large- chiral algebra's symmetry algebra. The authors prove an isomorphism between the holographic global symmetry algebra and the large- global symmetry algebra , and show that this symmetry data fixes all two- and three-point functions (up to a few low-dimension data) and determines higher OPE coefficients. The framework extends to affine quivers, boundary conditions from holomorphic data, and a KK-reduction perspective yielding higher-spin-like towers, connecting to IIB duals via holomorphic twists and suggesting a robust, perturbatively testable topological holography beyond the original AdS/CFT setting.

Abstract

We derive and test a novel holographic duality in the B-model topological string theory. The duality relates the B-model on certain Calabi-Yau three-folds to two-dimensional chiral algebras defined as gauged systems. The duality conjecturally captures a topological sector of more familiar holographic dualities.

Paper Structure

This paper contains 70 sections, 6 theorems, 378 equations, 1 figure.

Table of Contents

  1. Introduction and Conclusions
  2. Relations to physical dualities
  3. Global symmetry algebras
  4. Open questions and future directions
  5. The ${\mathcal{N}}=4$ chiral algebra
  6. The large $N$ limit
  7. Large $N$ single-trace operators
  8. Adding Flavor
  9. The affine quivers
  10. A quick review of B-model topological strings
  11. Coupling to holomorphic Chern-Simons theory
  12. An alternative formulation of Kodaira-Spencer theory, and a hidden $SL_2(\mathbb C)$ symmetry
  13. Coupling to holomorphic Chern-Simons revisited
  14. The B-brane backreaction
  15. The deformed geometry
  16. ...and 55 more sections

Key Result

Lemma 1

In this situation, the Beltrami differential satisfies equation eqn_source. Moreover, this is the unique solution to this equation which in addition :

Figures (1)

  • Figure 1: Concatenating open string global symmetries. The dotted lines represent Wick contractions that are allowed in the planar limit. Note we contract one $I$ field with one $J$ field and a varying number of fields $Z_i$.

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Theorem 1
  • Proposition 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof