Transcript Slide 1

D-Branes and Giant
3
Gravitons in AdS4xCP
Andrea Prinsloo*
in collaboration with
Alex Hamilton*, Jeff Murugan* and Migael Strydom*
(hep-th/0901.0009)
19/02/09 (Imperial)
* University
of Cape(UCT)
Town
Andrea Prinsloo
OVERVIEW
1. Introduction
2. AdS4/CFT3
3. Giant Gravitons
4. Future Research
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Introduction
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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 gmn
– 3-form potential Amnl
– fermion superpartners
electrically
magnetically
• The potential Amnl couples to M2 and M5-branes.
• These M-branes are the fundamental objects, but we
cannot quantize them perturbatively, as we did for strings.
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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.
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M2-Brane Actions
• What is the worldvolume action of multiple coincident M2branes in flat space?
BL: scalar-spinor sector
– 2+1 dimensional
Bagger & Lambert
(hep-th/0611108).
– 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.
Bagger & Lambert
(hep-th/0711.0955);
Gustavsson
(hep-th/0709.1260).
– Equivalent to a Chern-Simons theory with
matter in the bifundamental representation of SU(2)k x SU(2)-k
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ABJM:
Aharony, Bergman,
Jafferis & Maldacena
(hep-th/0806.1218).
– 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 ChernSimons-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
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AdS4/CFT3
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• The coupling constant of the ABJM theory is 1/k
• In the large N limit, planar diagrams have the effective
t’Hooft coupling l = 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
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Compactification
on S1
radius of circle
small when k>>N1/5
N1/5 << k << N
Type IIA string theory
in AdS4 x CP3
Andrea Prinsloo (UCT)
M-Theory in
7
AdS4xS /Zk
• Start with the AdS4 x S7 background metric
• Parameterize S7 in terms of the coordinates
magnitudes sum square to one
• We can write the S7 metric as a Hopf fibration of S1 over
the complex projective space CP3 as follows:
Fubini-Study
metric of CP3
with the total phase
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• 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
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new form fields under
KK reduction on circle y
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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
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Giant Gravitons
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Gauge Invariant Operators
• There are restrictions placed on how we can multiply the
scalar fields together due to the index structure
1st SU(N)
2nd SU(N)
• It is possible, however, to construct composite fields
which now carry indices in the same SU(N).
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• In SYM, giant gravitons are not
single trace operators, but rather
Schur polynomials of scalar fields.
Balasubramanian et al
(hep-th/0107119),
Corley, Jevicki & Ramgoolam
(hep-th/0111222),
Berenstein (hep-th/0403110).
• 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
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totally anti-symmetric
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AdS Giant Graviton
• A spherical D2-brane blown up in AdS4 and moving on a
trajectory in CP3.
• Supported by coupling to C(3).
AH, JM, AP & MS
(hep-th/0901.0009)
p
r
x
• The AdS giant has no maximum size.
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• More specifically,
2-sphere with radius r
angular direction of motion
• Set
Nishioka & Takayanagi
(hep-th/0808.2691)
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• The bosonic part of the D2-brane action is
We can integrate out the
sphere degrees of freedom.
supported by
the 3-form
• There exist solutions with any constant radius r0 and
(related) angular velocity
.
• The energy of this D2-brane solution is
with
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point graviton
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giant graviton
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Fluctuation Spectrum
• Consider the spectrum of small fluctuations about the
D2-brane solution.
Das, Jevicki & Mathur
(hep-th/0204013)
Describe the two 2-spheres
embedded in CP3 in terms of
cartesian coordinates
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• The D2-brane action can be expanded in orders of e.
• 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
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•
We find that
1) The spectrum of eigenfrequencies w 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.
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Attaching Words to Operators
• We can attach words to the operators dual to the AdS
giant graviton in various ways.
remove composite field
and replace with word
remove one scalar field Z or
Z† and replace with word
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• 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.
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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.
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Berenstein, Correa & Vazquez
(hep-th/0604123)
Correa & Silva
(hep-th/0608128)
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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
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modes corresponding to the two
2-sphere degrees of freedom.
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• 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.
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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
describes trajectory
on giant graviton with
pj = L.
• This is the same problem as for D3-brane AdS giant in
AdS5 x S5 – the extra CP3 structure has been lost.
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• 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 j is large, and the time
derivatives of the other coordinates are small by comparison.
– Expanding to leading order in l/L2.
• The leading order semiclassical action is
with
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Future Research
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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
Hoxha, Martinez-Acosta & Pope
(hep-th/0005172)
cycle after the compactification.
• One can, again, construct the spectrum of small
fluctuations.
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• 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)
Berenstein, Herzog & Klebanov
(hep-th/0202150)
– 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.
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