Topological Holography
and Chiral Algebras
Work in progress with Kevin Costello

General Motivations•

Topological twist of sugra/superstrings asholographic dual of twisted SQFT (Costello)•

Topological Holography: a consistent subset ofprotected correlation functions and dual dynamics•

It may be rigorously provable and yet rather rich,including loop effects in supergravity or string
theory.

Possible payoffs•

Fully solvable examples of holography•

Insights on perturbative and non-perturbative
aspects of supergravity and superstrings inRamond backgrounds•

Non-perturbative definition of topological stringtheories•

Mathematics

The chiral subsector•

2d chiral algebra hidden in 4d N=2 gauge theories(Beem, Lemos, Liendo, Peelaers, Rastelli, van Rees)•

Cohomology of “Q+S” supercharge••

Gauged beta-gamma system:
Alternative constructions?

Sch =Z

ChZ, D̄A Zi

The N=4 chiral algebra••

Adjoint beta-gamma:C

TrX D̄A Y

Hidden supersymmetry: N=4 super-Virasoro
nmsTr[X Y ]sTr[X @XYnmnsTr[X Y @c + · · · ]•N =4Sch =Znm+ ···]

msTr[X Y b + · · · ]Action is dimensional reduction of Holomorphic
Chern-Simons (hCS). Same as B-model D-branes!

Chiral algebra fromHolography
5•

Objective: twist sugra/superstrings on AdS5 ⇥ S•

Lowest KK modes on AdS5 localize to ChernSimons on AdS3(Bonetti, Rastelli)•Direct twist is challenging.

•Shortcut: twist flat space and compute topologicalback reaction

AdS/CFT from open/closedduality
N D3 branes in flat spaceBlack brane geometry

Near horizon/decoupling limit(Maldacena)

A quick refresh on B-model•

Topological sector of IIB superstrings on R4 ⇥ CY3(Antoniadis, Gava, Narain, Taylor)•Closed B-model: Kodaira-Spencer theory of complex
structure deformations(Bershadsky, Cecotti, Ooguri, Vafa)•

Topological twist of D-branes on R2 ⇥ ⌃(Ooguri-Vafa)•

Open B-model: (dimensionally reduced) hCS (Witten)

Back to the (B)asics,continued•

Topological twist of D3 branes on•

B-model topological open string on••

Worldvolume action: U(N)
N=4 Chiral algebra!

N =4Sch =ZC

TrX D̄A Y

Back-reaction and nearhorizon limit
N topological branes on C ⇢ C33C! SL(2, C)
Backreaction:

Decoupling limit is automaticComplex structuredeformation•

Backreaction:•

New holomorphic functions:•Deformed conifold, aka SL(2, C) withZ⌦=NS3

Some geometry•

Original SU(2) acts from the right•

Emergent SL(2,C) acts from the left!

•The space has topology•

Projection to AdS3:

KK reduction•Projection to AdS3 helps understand chiral algebra•KK reduction of KS theory: infinite tower oftopological higher spin fields in AdS3•“Good” boundary conditions give a chiral algebra.
Easy to match quantum numbers.

Back to 6d•

Lift boundary conditions to KS theory on SL(2,C)
11CPz ⇥ CPx•

Boundary is (a circle bundle over)•Deform boundary conditions at a point in CP1z
1with spherical harmonics on CPx

Status report•

Tree level 2pt and 3pt functions can be matched tolarge N chiral algebra•

Systematic 1/N expansion?

•Giant gravitons as generating functions?

Some simple extensions•“Standard” Dijkgraaf-Vafa is a further twist.

•4d N=4 with flavor: breaks conformal invariance,
but we can fix with ghosts.

•k flavors and k ghost flavors give u(k|k) hCS theory•``Orbifold’’ 4d N=2 quivers and quotients ofdeformed conifold

Open questions•Find a way to do all-loop calculations•

New interpretation of local Calabi-Yau. Which ones
have boundary chiral algebras?

•New non-perturbative definition of B-model?

•Relation to integrability?
(Aganagic, Dijkgraaf, Klemm, Marino, Vafa)