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

Outline Outline  AdS/CFT and Bubbling Picture  ½ BPS sector (LLM)  Bubbling with less SUSY?  The 1/4 and 1/8 BPS sectors hep-th/0704.2233  Multi-matrix Models  Interactions for two-matrix states hep-th/0712.4366  Open Questions/Conclusions

th Anniversary of AdS/CFT Conjecture 10 th Anniversary of AdS/CFT Conjecture 10 Holographic Duality: passed many checks Gravity CFT on “boundary” In its original incarnation: IIB string theory on AdS 5 x S 5 N=4 U(N) SYM in 4D

Hints of relation between theories: AdS 5 x S 5 as an embedding :  “Original” AdS/CFT: perturbations on AdS 5 x S 5  Can one go beyond perturbative description? (not just small perturbations of AdS?) Geometries that are asymptotically AdS 5 x S 5 are good candidates for dual states in CFT

½ BPS Geometries in Type IIB ½ BPS Geometries in Type IIB Lin, Lunin, Maldacena hep-th/0409174 LLM:  constructed exact ½ BPS solutions in type IIB SUGRA  identified them with the ½ BPS sector of N = 4 SYM 10 D spacetime of form S 3 x S 3 isometry and Only 3D really matter ! time-like Killing vector Z(x 1 ,x 2 ,y) Coordinate y plays special role  measures volume of two 3-spheres

y=0 plane: either one or both S 3 collapse to zero size Regularity demands certain boundary conditions on y=0 plane: black and white color coding of solutions unique droplet 10 D geometry

What about the CFT side? ½ BPS states (SYM chiral primaries) Energy of SUGRA solutions matrix in a harmonic known to describe ½ BPS sector of SYM oscillator potential (x 1 ,x 2 ) plane: phase space of fermions

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) 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

Bubbling for More General States? Bubbling for More General States? B.Chen, S.C., A.Donos, F.Lin, H.Lin, J.Liu, D.Vaman, W.Wen, hep-th/0704.2233  Robust bubbling picture with less SUSY?  Focus on 1/4 and 1/8 BPS sectors General SUGRA ansatz for 1/4 BPS and 1/8 BPS geometries done: A. Donos N. Kim, hep-th/0606199, hep-th/0610259 hep-th/0511029 But here no explicit solutions and no bubbling picture (highly non-linear systems)

Interested in geometries that are asymptotically AdS 5 x S 5 Take s-wave states in AdS with  Want to keep S 3 inside AdS 5  Turning on R-charge (J 1 , J 2 ,J 3 ) breaks isometries of S 5 : no R-charge J 1 J 1 J 2 J 1 J 2 J 3

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 ( S 3 x S 3 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 multi-matrix models!  a number of subtleties arise

For 1/4 BPS states (preserving S 3 x S 1 ): Potential singularity when either or BCs ensure regularity 3D surfaces (droplets) in 4D (y=0) hyperplane For 1/8 BPS states (preserving S 3 ): locus of shrinking S 3 : 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)

With even less SUSY: 1/8 BPS configuration filling 6D plane

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 of multi-matrix models)

It turns out that symmetries can help: SC, A. Jevicki, R. de Mello Koch hep-th/0712.4366 Simple strategy: 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:

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

Conclusion Conclusion  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?