[Paper Review] Twisted Holography

Kevin Costello, Davide Gaiotto|arXiv (Cornell University)|Dec 21, 2018

Advanced Optical Imaging Technologies21 citations

TL;DR

This paper proposes a novel holographic duality in B-model topological string theory, linking Calabi-Yau three-folds with SL(2,C) isometry to two-dimensional chiral algebras constructed as gauged βγ systems. The duality establishes a one-to-one correspondence between local operators on the gauge theory side and boundary condition modifications on the gravitational side, with exact matching of Witten indices and OPE coefficients at tree level, providing a non-perturbative definition of topological strings on specific backgrounds.

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.

Motivation & Objective

• To establish a new class of holographic dualities in topological string theory, specifically relating gravitational backgrounds with SL(2,C) isometry to 2D chiral algebras.
• To provide a non-perturbative definition of B-model topological strings on certain Calabi-Yau three-folds via a well-defined gauge theory side.
• To test the duality at tree level by matching physical observables such as Witten indices and correlation functions between the two sides.
• To clarify the relationship between this topological duality and known physical holographic dualities, particularly the AdS5/CFT4 correspondence and Dijkgraaf-Vafa duality.
• To extend the duality to include orbifold backgrounds via affine ADE quivers and their corresponding gravitational duals.

Proposed method

• The duality is derived using a holographic dictionary that maps local operators on the gauge theory side to modifications of boundary conditions on the gravitational side.
• The gauge theory side is constructed as a gauged βγ system with U(N) symmetry, forming a vertex operator algebra denoted 𝒜_N.
• The gravitational side involves B-model topological strings on a deformed conifold background with N as the period of the holomorphic three-form.
• Witten diagrams are used to compute two- and three-point functions on the gravitational side, with algebraic proofs showing exact agreement with the gauge theory side.
• A propagator formalism is developed for Kodaira-Spencer theory on compactified spaces with boundary, using logarithmic forms and cohomological conditions to ensure uniqueness.
• The analysis includes a boundary condition requiring vanishing residue integrals over the divisor, eliminating zero modes and ensuring consistency of the propagator construction.

Experimental results

Research questions

• RQ1 Does a holographic duality exist between B-model topological strings on SL(2,C)-symmetric Calabi-Yau three-folds and 2D chiral algebras from gauged βγ systems?
• RQ2 Can the correspondence between local operators on the gauge theory and boundary condition modifications on the gravity side be made exact and manifest at tree level?
• RQ3 Do the Witten indices of the chiral algebra and the topological string theory match under this duality?
• RQ4 Are the two- and three-point correlation functions computed via Witten diagrams on the gravitational side algebraically isomorphic to those on the gauge theory side?
• RQ5 Can the duality be generalized to include orbifold backgrounds via ADE quiver gauge theories and their dual gravitational geometries?

Key findings

• Single-trace operators in the large N chiral algebra 𝒜_N are in one-to-one correspondence with modifications of the boundary conditions in the gravitational dual, confirming the holographic dictionary at the operator level.
• The Witten indices on both sides of the duality are identical, providing a strong consistency check at the level of partition functions.
• All two- and three-point functions computed via Witten diagrams on the gravitational side match exactly with the OPE structure on the gauge theory side, as proven algebraically.
• The propagator for the Kodaira-Spencer theory on the compactified space X with boundary D exists and is unique up to exact forms when a residue condition is imposed, ensuring consistency of the perturbative expansion.
• The cohomology of the space of differential forms with logarithmic poles on D vanishes in relevant degrees, which guarantees the existence and uniqueness of the propagator under the gauge condition.
• The duality is compatible with the Dijkgraaf-Vafa duality and captures a topological subsector of the full N=4 SYM theory, with the chiral algebra 𝒜_N containing a small N=4 super-Virasoro algebra.