AdS/CFT and Bubbling Geometries: Going Beyond the BPS Sector Sera - - PowerPoint PPT Presentation

AdS/CFT and Bubbling Geometries: Going Beyond the ½ BPS Sector Sera Cremonini

University of Michigan

Great Lake Strings, Madison, April 26 2007

• AdS/CFT and Bubbling Picture
• ½ BPS sector (LLM)
• Bubbling with less SUSY?
• The 1/4 and 1/8 BPS sectors
• Multi-matrix Models
• Interactions for two-matrix states
• Open Questions/Conclusions

Outline Outline

hep-th/0704.2233 hep-th/0712.4366

IIB string theory on AdS5 x S5 N=4 U(N) SYM in 4D

10 10th

th Anniversary of AdS/CFT Conjecture

Anniversary of AdS/CFT Conjecture

Gravity CFT on “boundary”

Holographic Duality:

passed many checks In its original incarnation:

Hints of relation between theories: AdS5 x S5 as an embedding :

• “Original” AdS/CFT: perturbations on AdS5 x S5
• Can one go beyond perturbative description?

(not just small perturbations of AdS?) Geometries that are asymptotically AdS5 x S5 are good candidates for dual states in CFT

LLM:

• constructed exact ½ BPS solutions in type IIB SUGRA
• identified them with the ½ BPS sector of N = 4 SYM

½ BPS Geometries in Type IIB ½ BPS Geometries in Type IIB

Lin, Lunin, Maldacena hep-th/0409174

S3 x S3 isometry and time-like Killing vector Only 3D really matter !

Z(x1,x2,y)

10 D spacetime of form Coordinate y plays special role

• measures volume of two 3-spheres

y=0 plane: either one or both S3 collapse to zero size Regularity demands certain boundary conditions on y=0 plane:

black and white color coding

• f solutions

unique 10 D geometry droplet

½ BPS states

(SYM chiral primaries)

matrix in a harmonic

• scillator potential

What about the CFT side?

known to describe ½ BPS sector of SYM Energy of SUGRA solutions (x1,x2) plane: phase space of fermions

Natural Questions:

• AdS/CFT dictionary for more complicated geometries?
• Bubbling with less SUSY?
• Is linearity crucial?

M-theory case not linear, (but integrable), and leads to bubbling

So far we have seen:

• Precise DICTIONARY between SUGRA and CFT in ½ BPS sector
• Unique map between fermion droplets and 10D geometries

But ½ BPS sector is “simple” (lots of symmetry)

B.Chen, S.C., A.Donos, F.Lin, H.Lin, J.Liu, D.Vaman, W.Wen, hep-th/0704.2233

• A. Donos

hep-th/0606199, hep-th/0610259

• N. Kim,

hep-th/0511029 But here no explicit solutions and no bubbling picture (highly non-linear systems)

Bubbling for More General States? Bubbling for More General States?

General SUGRA ansatz for 1/4 BPS and 1/8 BPS geometries done:

• Robust bubbling picture with less SUSY?
• Focus on 1/4 and 1/8 BPS sectors

Interested in geometries that are asymptotically AdS5 x S5

• Want to keep S3 inside AdS5
• Turning on R-charge (J1, J2 ,J3) breaks isometries of S5 :

Take s-wave states in AdS with no R-charge J1 J1 J2 J1 J2 J3

Gravity picture that has emerged :

1/2 BPS 1/4 BPS 1/8 BPS

Droplet picture originates from boundary conditions

For 1/2 BPS states ( S3 x S3 isometry):

BCs: metric remains smooth as either 1D curves (droplets) in 2D y=0 plane

For 1/4 and 1/8 BPS sectors many features survive:

• analog of LLM function z (crucial for BCs)
• hyperplanes where BCs are imposed DROPLETS
• But now equations are highly non-linear:
• general SUGRA solutions not known
• a number of subtleties arise

multi-matrix models!

locus of shrinking S3: For 1/4 BPS states (preserving S3 x S1): Potential singularity when either or BCs ensure regularity For 1/8 BPS states (preserving S3):

3D surfaces (droplets) in 4D (y=0) hyperplane 5D surfaces (droplets) in 6D base 6D base ENDS at these 5D surfaces - INTERIORS ARE UNPHYSICAL (matches gauge theory side numerical studies, Berenstein)

Summary of Bubbling Picture Summary of Bubbling Picture

schematic picture of 1/2 BPS configuration (four dual giant gravitons on AdS vacuum) 1/4 BPS configuration (five dual giant gravitons)

1/8 BPS configuration filling 6D plane

With even less SUSY:

Main challenge with less SUSY: system is highly non-linear (Monge-Ampere for 1/4 BPS) Still possible to develop robust bubbling picture

(even w/out complete knowledge of SUGRA solutions)

Some explicit evidence:

• Embedded several known and some new solutions

Some open issues:

• Given a droplet, is the 10D geometry unique? (yes for LLM)
• Require asymptotically AdS x S, and regularity near

droplets, but is that enough?

• Is 1/4 BPS sector integrable?
• Need better understanding of regularity conditions

(e.g. singular superstar solutions)

Multi-Matrix Models Multi-Matrix Models

Need to understand more than one matrix challenging program

(e.g. Berenstein’s numerical simulations

• f multi-matrix models)

It turns out that symmetries can help:

Take vertex for three chiral primaries (1/2 BPS states) By using SL(2,R) generators of AdS (RAISING AND LOWERING OPERATORS) you can “raise” the vertex and relate it to: SC, A. Jevicki, R. de Mello Koch hep-th/0712.4366

Simple strategy:

Ingredients:

• Wavefunctions for two-matrix states
• Map from matrix model variables to AdS x S variables

(Kernel)

• SL(2,R) Raising/Lowering operators

End result: Identity that reconstructs the full two-matrix vertex from the simpler, one-matrix vertex

• Much progress recently in AdS/CFT
• dictionary extended to more complicated geometries
• Nice free fermion picture (bubbling picture) for

½ BPS SUGRA solutions

• Robust Bubbling picture even with less SUSY
• We can exploit symmetries to deal with multi-matrix models

(at the interacting level)

• Many open questions/possible directions:
• Can we better understand regularity of bubbling solutions?
• Uniqueness of solutions?
• Integrability in 1/4 and 1/8 BPS sectors?
• Breaking SUSY completely? Horizon formation? (with J. Simon)
• Time dependent geometries and CFT interpretation?

Conclusion Conclusion