D-Branes and Giant Gravitons in AdS 4 xCP 3

1. 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 Andrea Prinsloo (UCT)

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

3. Introduction Andrea Prinsloo (UCT)

4. electrically magnetically 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. Andrea Prinsloo (UCT)

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. 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/0611108). Bagger & Lambert (hep-th/0711.0955); Gustavsson (hep-th/0709.1260). Andrea Prinsloo (UCT)

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 Andrea Prinsloo (UCT)

8. AdS4/CFT3 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 radius of circle small when k>>N1/5 Andrea Prinsloo (UCT)

10. 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 Andrea Prinsloo (UCT)

11. 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 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 Andrea Prinsloo (UCT)

13. Giant Gravitons Andrea Prinsloo (UCT)

14. 1st SU(N) 2nd SU(N) 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). Andrea Prinsloo (UCT)

15. totally symmetric and traceless totally anti-symmetric Balasubramanian et al (hep-th/0107119), Corley, Jevicki & Ramgoolam (hep-th/0111222), Berenstein (hep-th/0403110). • 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 Andrea Prinsloo (UCT)

16. p r x 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/0901.0009) Andrea Prinsloo (UCT)

17. 2-sphere with radius r angular direction of motion • More specifically, • Set Nishioka & Takayanagi (hep-th/0808.2691) Andrea Prinsloo (UCT)

18. We can integrate out the sphere degrees of freedom. supported by the 3-form • 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 Andrea Prinsloo (UCT)

19. point graviton giant graviton Andrea Prinsloo (UCT)

20. Describe the two 2-spheres embedded in CP3 in terms of cartesian coordinates Fluctuation Spectrum • Consider the spectrum of small fluctuations about the D2-brane solution. Das, Jevicki & Mathur (hep-th/0204013) Andrea Prinsloo (UCT)

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 Andrea Prinsloo (UCT)

22. 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. Andrea Prinsloo (UCT)

23. remove composite field and replace with word remove one scalar field Z or Z† and replace with word Attaching Words to Operators • We can attach words to the operators dual to the AdS giant graviton in various ways. Andrea Prinsloo (UCT)

24. 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. 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/0604123) Correa & Silva (hep-th/0608128) Andrea Prinsloo (UCT)

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 Andrea Prinsloo (UCT)

27. 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. Andrea Prinsloo (UCT)

28. describes trajectory on giant graviton with pj = L. 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. 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 Andrea Prinsloo (UCT)

30. Future Research Andrea Prinsloo (UCT)

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/0005172) Andrea Prinsloo (UCT)

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/0202150) Andrea Prinsloo (UCT)