1. D-Branes and Giant Gravitons in AdS4xCP3
Andrea Prinsloo*
in collaboration with
Alex Hamilton*, Jeff Murugan* and Migael Strydom*
(hep-th/ )
* University of Cape Town
19/02/09 (Imperial)
Andrea Prinsloo (UCT)

2. OVERVIEW Introduction AdS4/CFT3 Giant Gravitons Future Research
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Andrea Prinsloo (UCT)

3. Introduction
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4. 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
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.
electrically
magnetically
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5. 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.
19/02/09 (Imperial)
Andrea Prinsloo (UCT)

6. 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/ ).
Bagger & Lambert (hep-th/ );
Gustavsson (hep-th/ ).
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7. Aharony, Bergman, Jafferis & Maldacena (hep-th/0806.1218).
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
Similar to Klebanov-Witten theory, BUT 1) CS-matter NOT SYM; 2) SU(4) NOT SU(2)xSU(2) R-symmetry group and 3) U(N)xU(N) gauge group NOT SU(N)xSU(N) so that the baryon number symmetry is now gauged.
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Andrea Prinsloo (UCT)

8. AdS4/CFT3
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Andrea Prinsloo (UCT)

9. 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
N1/5 << k << N
Type IIA string theory in AdS4 x CP3
Compactification on S1
The M-theoretic AdS4/CFT3 correspondence may allow us to gain insight into non-perturbative aspects of M-branes; The string theoretic version, however, is easier to test.
radius of circle
small when k>>N1/5
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10. Fubini-Study metric of CP3
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
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Andrea Prinsloo (UCT)

11. new form fields under KK reduction on circle y
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
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Andrea Prinsloo (UCT)

12. 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
C(3) and C(5) couple to D2 and D5-branes respectively
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13. Giant Gravitons
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14. 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)
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15. In SYM, giant gravitons are not single trace operators, but rather
Balasubramanian et al (hep-th/ ),
Corley, Jevicki & Ramgoolam (hep-th/ ),
Berenstein (hep-th/ ).
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
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16. AdS Giant Graviton
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.
AH, JM, AP & MS (hep-th/ ) p x r
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17. More specifically, Set 2-sphere with radius r
angular direction of motion
Nishioka & Takayanagi (hep-th/ )
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18. We can integrate out the sphere degrees of freedom.
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
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19. point graviton giant graviton 19/02/09 (Imperial)
Andrea Prinsloo (UCT)

20. Fluctuation Spectrum
Consider the spectrum of small fluctuations about the D2-brane solution.
Das, Jevicki & Mathur (hep-th/ )
Describe the two 2-spheres embedded in CP3 in terms of cartesian coordinates
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21. 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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Andrea Prinsloo (UCT)

22. There is a zero mode corresponding to radial fluctuations.
We find that
The spectrum of eigenfrequencies w is entirely real. The AdS giant is thus a stable configuration.
There is a zero mode corresponding to radial fluctuations.
This spectrum is independent of the size of the AdS giant graviton ↓
The giant graviton does not ‘see’ the AdS geometry.
Mention about 2) zero modes in radial directions – any constant r is a solution to the equations of motion, and 3) not ‘seeing’ geometry. If they ‘saw’ the geometry there would be some imprint of the geometry in the operators – same thing happens in AdS5xS5.
19/02/09 (Imperial)
Andrea Prinsloo (UCT)

23. 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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24. These correspond to excitations in the AdS directions.
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.
19/02/09 (Imperial)
Andrea Prinsloo (UCT)

25. 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/ )
Correa & Silva (hep-th/ )
19/02/09 (Imperial)
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26. modes corresponding to the two 2-sphere degrees of freedom.
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.
19/02/09 (Imperial)
Andrea Prinsloo (UCT)

27. The string spectrum was obtained
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.
19/02/09 (Imperial)
Andrea Prinsloo (UCT)

28. 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 pj = L.
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Andrea Prinsloo (UCT)

29. 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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Andrea Prinsloo (UCT)

30. Future Research
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31. 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/ )
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32. 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/ )
Work in progress. If it is, indeed, necessary to attach wilson lines to the dibaryon operators, and it proves possible to match the small fluctuations with the equivalent process on the gauge theory side, this is a non-trivial test, which depends on the CS nature of the ABJM theory.
19/02/09 (Imperial)
Andrea Prinsloo (UCT)