Reconstructing BPS Geometries from 4 SYM

1
Reconstructing ½ BPS Geometries from ? 4 SYM

• Samuel E. Vázquez
• Physics Department, University of California at
  Santa Barbara

Based on S.E.V hep-th/0612014 Work in progress
with Sean Hartnoll (KITP) See also D.
Berenstein, D. H. Correa S.E.V hep-th/0502172
D. Berenstein, D. H. Correa, S.E.V
hep-th/0604123
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Motivation

• We want a non-perturbative definition of Quantum
  Gravity
• String Theory itself will be emergent from such a
  theory
• Example AdS/CFT
• Want to study how gravity and String Theory
  emerges from a large N, SU(N) QFT
• Problem strong/weak coupling duality
• Solution investigate BPS or near-BPS states
  (protected by SUSY)
• In particular, how do we reconstruct physical
  information such as distances in space-time?

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Plan of Talk

• Review ½ BPS states on SUGRA
• Probe Strings on ½ BPS geometries and large spin
  limits (SU(2) sector)
• SU(2) sector of SYM and probe strings
• Matrix Quantum Mechanics and Random Matrix Theory
• Reconstructing spacetime metrics
• Using boundary to boundary geodesics (work in
  progress)
• Again. Random Matrix Theory
• Probing spacetime singularities?
• Conclusions

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½ BPS Geometries

• Preserve 16 supersymmetries and have R x SO(4) x
  SO(4) isometries. (Lin, Lunin, Maldacena, 2004)

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SU(2) Probe Strings

• At first, we will focus on an SU(2) sector which
  correspond to strings that live inside the
  droplets and rotate on an S1 ? S3 fiber.

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SU(2) Probe Strings

• Write Polyakov action in momentum space (M.
  Kruczenski, A.V. Ryzhov, and A.A. Tseytlin)

• A and B implement the Virasoro constraints

• Use constraints directly in the action

• Can systematically eliminate time derivatives in
  favor of spatial derivatives

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SU(2) Probe Strings

• Pick a gauge in the Polyakov action, where the
  angular momentum in ? is distributed uniformly
  along string.

• Take the limit L ?? with ?/L2 fixed can show that the Polyakov action becomes,

Example S3
Unit disk distribution
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Dual Gauge Theory

• SU(2) Scalar Sector of ? 4 SYM on R x S3

String States with two angular momenta on
(asymp.) S5
States with two UR(1) ? SU(4) charges
?
String sees a reduced R x S3 geometry
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Hamiltonian

• Hamiltonian in the SU(2) sector

One loop truncation
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Coherent States
Random Matrix Model!
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Closed Strings on R x S3

• Single-trace states Single string state

• In the large N limit the Hamiltonian only acts
  on nearest neighbors.

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Closed Strings on R x S3

• The Hamiltonian is (periodic boundary cond.)

hopping
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Emergent World Sheet

• Action for coherent states

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D-branes on S5

• Consider Heavy states with E JZ ? p N

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Open Strings on D-branes

• Add a word to the state

Z exchange
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Open Strings on D-branes

• In coherent states, and taking the L ?? limit

• At large ?, these terms localize the ends of the
  open string on the Giant Graviton just like in
  String Theory!

?
?
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Emergent ½ BPS Geometries

• The Hamiltonian
• has a very degenerate ground state any function
  of (say) Z only

• Some heavy" ground states behave classically in
  the large N limit
• Example

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Emergent ½ BPS Geometries

• Normalization in terms of eigenvalues of Z

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Emergent ½ BPS Geometries

• At large N, we have a Landscape of constant
  density droplets on the complex plane

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(Closed) Probe String

• Probe string state
• where, in the large N limit

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(Closed) Probe String

• Define the following operators

Basis

• Assuming that ?n are holomorphic, one finds an
  interesting algebraic structure (large N)

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(Closed) Probe String

• One can show that, in the large N limit, the
  Hamiltonian in this basis becomes (
  ), (hep-th/0612014)

• Representation of the Algebra assume ?n(Z) is a
  polynomial of degree n. They will obey a
  recursion relation

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(Closed) Probe String

• One finds,

Determined by the algebra

• Note that cross terms in H will not preserve
  boson number (can also be visualized as
  dynamical spin chain)

hopping
source/sink
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Classical Limit

• In this representation, the coherent states are

• Action (in the large angular momentum limit)

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Examples

• Circular Droplet (?(z) 0) recover usual
  bosonic lattice

Normalized only inside unit disk
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Examples

• Elliptical Droplet (?(z) t2 z2 z2/2 )

Chebyshev Polynomials of Second Kind
Normalizable only inside ellipse
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Examples

• Hypotrochoid (?(z) a z3) One can solve the
  algebra perturbatively around a 0.

• One finds

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String Theory Interpretation

• We discover that the full metric on the droplet
  can be reconstructed from the orthogonal
  polynomials

• All cases match perfectly with SUGRA result

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String Theory Interpretation

• Chen, Correa and Silva (hep-th/0703068) have
  generalized the procedure for concentric rings,
  which correspond to geometries with different
  topologies

Penrose Limits in the space-time, correspond to
strings traveling close to the edges
Their energies can be matched with the spectrum
of the one loop Hamiltonian (in the BMN limit)
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Outside the Droplet

• One would like to learn how to probe the ½ BPS
  geometries outside the droplets.
• Original motivation look for signatures of the
  time-like singularity of the Superstar (the
  droplet is the singularity!)

• Options
• Use different reduced sectors such as SL(2),
  SU(1,1), etc. (difficult combinatorics!!)
• Analyze analytic structure of boundary
  correlation functions causal structure of
  spacetime
• Used in the context of Black Holes Kraus,
  Ooguri, Shenker, etc.

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Basic Idea
geodesic length
O(t,?)

• Expect poles and zeroes when diverges
  (geodesic is almost null)

• Example AdS5

O(t,?)

• Can match with a geodesic calculation (Hubeny,
  Liu and Rangamani 2006)

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Time Dilation

• For non-singular geometries one generically
  expect a time dilation in the arrival of null
  geodesics (Hubeny, Liu and Rangamani 2006)

Naked singularity
Time delay
?t ? ? as gedesics approach singularity
?t (AdS)
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For ½ BPS geometries

• Naively, we expect that if we use ½ BPS probes
  such as Tr Zn on ½ BPS backgrounds, the result
  will still be protected by SUSY.

• We believe that ½ BPS states are dual to zero
  modes of Z(t,?) on S3. For example (elliptical
  droplet)

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For ½ BPS geometries

• So perhaps we should calculate,
• where

Drops out because
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For ½ BPS geometries

• One can show that in the free field theory
• where

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For ½ BPS geometries

• We can shift integration variables Z X ? , Y
  X - ? , and then (in principle) integrate out
  ?. For example, for elliptical droplet (?
  exp?/2 Tr(Z2)) we can explicitly do integration

Convergence Conditions
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Partial Results

• For AdS (? 0) we get that the singularities are
  at (as expected)

• For (? ? 0) we find the same singularities plus
  some others that do not quite agree with the
  geodesic calculations.

• It seems that the BPS correlation functions are
  renormalized!

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Conclusions

• Free field and one-loop calculations in ? 4 SYM
  permit the extraction of highly non-trivial
  geometric information for ½ BPS backgrounds.
• We can reconstruct the metric inside the
  droplets in terms of orthogonal matrix
  polynomials.
• These are constructed from an interesting algebra
• Sigma model of a probe string has the form of a
  dynamical bosonic lattice.
• One can also try to probe the causal structure of
  the backgrounds by studying poles of correlation
  functions in ? 4 SYM on R x S3. (free field
  theory)
• Discrepancy?