d-branes and giant gravitons in ads 4 xcp 3

19/02/09 (Imperial) Andrea Prinsloo (UCT)

D-Branes and Giant Gravitons in AdS4xCP3

Andrea Prinsloo*

in collaboration with

Alex Hamilton*, Jeff Murugan* and Migael Strydom*

(hep-th/0901.0009)

* University of Cape Town


OVERVIEW

1. Introduction

2. AdS4/CFT3

3. Giant Gravitons

4. Future Research


Introduction


M-Theory• Strong coupling limit of type IIA string theory.

• The long wavelength (low energy) limit of M-theory is 11D SUGRA, which contains

– metric g– 3-form potential A– fermion superpartners

• The potential A couples to M2 and M5-branes.

• These M-branes are the fundamental objects, but we cannot quantize them perturbatively, as we did for strings.

magneticallyelectrically


Compactification of M-Theory to Type IIA String Theory

• KK reduction on a circle S1, which shrinks to zero size, so we reduce the number of dimensions from

11D → 10D.

• How do we obtains strings?

• M2-branes wound around the S1 become open and closed strings in the type IIA string theory.


M2-Brane Actions• What is the worldvolume action of multiple coincident M2-

branes in flat space?

BL: scalar-spinor sector

– 2+1 dimensional

– SO(8)R symmetry amongst the 8 scalar fields (transverse directions)

– SUSY variations in terms of a 3-algebra – BUT the SUSY algebra did not close (no gauge fields),

BLG: gauge theory for 2 M2-branes

– N=8 superconformal 3-algebra Chern-Simons matter theory.

– Equivalent to a Chern-Simons theory with matter in the bifundamental representation of SU(2)k x SU(2)-k

Bagger & Lambert (hep-th/0711.0955);

Gustavsson (hep-th/0709.1260).

Bagger & Lambert (hep-th/0611108).


ABJM:

– N coincident M2-branes at a C4/Zk singularity (when k=1 these are simply M-branes in flat space).

– Described by 2+1 dimensional N=6 superconformal Chern-Simons-matter theory with a Uk(N) x U-k(N) gauge group.

– SO(8)R symmetry broken to SO(6)R ≡ SU(4)R

– (Complex) scalar fields

which transform in the bifundamental representation of the U(N) x U(N) gauge group and in the fundamental representation of the SU(4) R-symmetry group

Aharony, Bergman, Jafferis & Maldacena (hep-th/0806.1218).


AdS4/CFT3


• The coupling constant of the ABJM theory is 1/k

• In the large N limit, planar diagrams have the effective t’Hooft coupling = N/k

• Strongly coupled if k << N and weakly coupled if k >> N.

GRAVITY DUALS

k << N1/5

M-theory in AdS4 x S7/Zk

N1/5 << k << N

Type IIA string theory in AdS4 x CP3

Compactification on S1

radius of circle

small when k>>N1/5


M-Theory in AdS4xS7/Zk

• Start with the AdS4 x S7 background metric

• Parameterize S7 in terms of the coordinates

• We can write the S7 metric as a Hopf fibration of S1 over the complex projective space CP3 as follows:

with the total phase

magnitudes sum square to one

Fubini-Study metric of CP3


• We can mod out by Zk by identifying

and rewriting the metric in terms of

which parameterizes the new circle.

• The metric of AdS4 x S7/Zk is thus

where

new form fields under KK reduction on circle y


Type IIA String Theory in AdS4xCP3

• The metric of the AdS4 x CP3 background is

with new field strength forms

and a now non-zero, but constant, dilaton.

• The field strength forms in the M-theory now yield


Giant Gravitons


Gauge Invariant Operators• There are restrictions placed on how we can multiply the

scalar fields together due to the index structure

• It is possible, however, to construct composite fields

which now carry indices in the same SU(N).

1st SU(N)

2nd SU(N)


• In SYM, giant gravitons are not

single trace operators, but rather

Schur polynomials of scalar fields.

• We can use these composite fields to construct similar giant graviton operators in the Chern-Simons theory.

• In the special case of symmetric and totally anti-symmetric combinations of scalar fields, these operators are

totally symmetric and traceless

totally anti-symmetric

Balasubramanian et al (hep-th/0107119),

Corley, Jevicki & Ramgoolam (hep-th/0111222),

Berenstein (hep-th/0403110).


• A spherical D2-brane blown up in AdS4 and moving on a trajectory in CP3.

• Supported by coupling to C(3).

• The AdS giant has no maximum size.

AdS Giant Graviton

p

x

r

AH, JM, AP & MS (hep-th/0901.0009)


• More specifically,

• Set

angular direction of motion

2-sphere with radius r

Nishioka & Takayanagi (hep-th/0808.2691)


• The bosonic part of the D2-brane action is

• There exist solutions with any constant radius r0 and (related) angular velocity .

• The energy of this D2-brane solution is

with

We can integrate out the sphere degrees of freedom.

supported by the 3-form


point graviton giant graviton


Fluctuation Spectrum

• Consider the spectrum of small fluctuations about the D2-brane solution.

Describe the two 2-spheres embedded in CP3 in terms of cartesian coordinates

Das, Jevicki & Mathur (hep-th/0204013)


• The D2-brane action can be expanded in orders of .

• The 0th order action is just the original D2-brane action, while the 1st order action vanishes up to total derivatives.

• We impose the condition that the 2nd order action vanishes.

• Decompose the fluctuation in terms of

and

where the latter are the spherical harmonics, which satisfy the eigenvalue equation

with


• We find that

1) The spectrum of eigenfrequencies is entirely real. The AdS giant is thus a stable configuration.

2) There is a zero mode corresponding to radial fluctuations.

3) This spectrum is independent of the size of the AdS giant graviton

↓The giant graviton does not ‘see’ the AdS geometry.


• We can attach words to the operators dual to the AdS giant graviton in various ways.

Attaching Words to Operators

remove composite field and replace with word

remove one scalar field Z or Z† and replace with word


• These words can be constructed from scalar fields or derivatives of scalar fields.

• We are particularly interested in words constructed from derivatives of scalar fields.

• These correspond to excitations in the AdS directions.


Attaching Open Strings• Consider the open string excitations

of the AdS giant graviton.

• These open strings cannot be quantized in general in the full background spacetime, so we consider two limits:

– short pp-wave strings– long semiclassical strings

• Open strings in these limits were studied for D3-branes in AdS5 x S5.

Berenstein, Correa & Vazquez (hep-th/0604123)

Correa & Silva (hep-th/0608128)


Short pp-wave Strings

• We take a Penrose limit to zoom in on a null geodesic (great circle) on the AdS giant, which is described by

• The metric of this pp-wave background is

modes corresponding to the two 2-sphere degrees of freedom.


• We can quantize these pp-wave strings in the light-cone gauge (with the relevant Dirichlet and Neumann B.C.)

• The string spectrum was obtained

with m = -pv.

• The approximation is valid when

which is the BMN scaling limit in the gauge theory.


Long Semiclassical Strings• Consider strings in a semiclassical limit propagating in

which corresponds to attaching words formed from derivatives of our composite scalar fields.

• The metric becomes

• This is the same problem as for D3-brane AdS giant in AdS5 x S5 – the extra CP3 structure has been lost.

describes trajectory on giant graviton with

p = L.


• The semiclassical limit involves

– Choosing a the non-diagonal uniform gauge in which the angular momentum L has been spread evenly along the string.

– Taking the fast motion limit, in which the angular momentum of the string along trajectory described by is large, and the time derivatives of the other coordinates are small by comparison.

– Expanding to leading order in /L2.

• The leading order semiclassical action is

with


Future Research


Dibaryons / Wrapped D4-branes

• Consider an M5-brane wrapped on the Hopf fibre and a CP2 inside CP3.

• Descend to D4-branes wrapped on non-contractible

cycle after the compactification.

• One can, again, construct the spectrum of small fluctuations.

Hoxha, Martinez-Acosta & Pope (hep-th/0005172)


• Dual to a dibaryon operators.

• Dibaryons in Klebanov-Witten theory / wrapped D3-branes in AdS5 x T1,1 previously compared.

– Gauge group SU(N) x SU(N)

– Baryon number symmetry U(1)b NOT gauged

• In the ABJM theory the gauge group is U(N) x U(N), so the baryon number symmetry is gauged.

• It appears one must modify the dibaryon operators, which are simply constructed out of one type of scalar field, to obtain gauge invariant operators.

Berenstein, Herzog & Klebanov (hep-th/0202150)

d-branes and giant gravitons in ads 4 xcp 3

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