2. Recent issues in brane theory and the gauge/gravity correspondence P.Fré Bogolyubov Institute, Kiev Dec. 2001 1.Domain Walls 2.Shadow Multiplets and the AdS/CFT in D=3 3.Algebraic Geometry, Cones and Conformal Field Theories in D=3 4.D3 branes and ALE manifolds Four possible topics

3. p-Brane Actions

4. The parameter  and the harmonic function H(y)

5. Electric and magnetic p-branes “Elementary”

6. Conformal branes and AdS space

7. AdS is a special case of a Domain Wall These two forms are related by a coordinate transformation ELECTRIC BRANE

8. Coordinate patches and the conformal gauge Conformal brane a=0

9. Randall Sundrum gravity trapping These potentials have a Volcano shape that allows the existence of normalizable zero mode describing the graviton in D-1 dimensions. The continuum Kaluza Klein spectrum contributes only a small correction to the D-1 dimensional Newton’s law Randall Sundrum Kaluza Klein expansion in non compact space

10. Positivity of the Wall Tension

11. The “dual frame” of Boonstra, Skenderis and Townsend We learn that although the AdS x S 8-p is not a solution of supergravity, we can notheless compactify on the sphere S 8-p, or other compact manifold X 8-p !!!

12. “Near brane” factorization in the dual frame

13. p-brane (D-p-1) - Cone An X 8-p compact manifold is the base of the transverse cone C( X 8-p ) In D=10 the p-brane splits the space into a d=p+1 world volume and a transverse cone C( X 8-p ) that has the compact manifold X 8-p as base. In some sense The transverse cone

14. Domain wall supergravity from “sphere reduction”

15. The DW/QFT correspondence of Boonstra Skenderis & Townsend

16. This raises some basic questions and we have some partial answers: Which supergravity is it that accommodates the Domain Wall solution after the “sphere” reduction? Which supergravity is it that accommodates the Domain Wall solution after the “sphere” reduction? It is a “gauged supergravity” It is a “gauged supergravity” But which “gauging” ? But which “gauging” ? Typically a non compact one. It is compact for AdS branes! Typically a non compact one. It is compact for AdS branes! What are the possible gaugings? What are the possible gaugings? These are classifiable and sometimes classified These are classifiable and sometimes classified How is the gauging determined and how does it reflect microscopic string dynamics? How is the gauging determined and how does it reflect microscopic string dynamics? ??? This is the research frontier! ??? This is the research frontier!

17. The shadow of the graviton multiplet can be a gravitino multiplet, suggesting a superHiggs mechanism but…...

18. Conjecture: Shadow SUGRAS It seems that there exist more general N- extended SUGRAS with SU(N 0 ) symmetry where N 0 <N

21. The spectrum is determined by eigenvalues of Laplace Beltrami operators

22. AdS 4 UIR relations Structure of M-theory

23. Differential Geometry of X7X7 knows all the relations implied by SUSY in AdS 4

24. * = modulus in N=4 N=3 There are 4 neutral scalars whose vev ‘s trigger superHiggs R symmetry

25. This suggests consistent truncation! This multiplet is Universal in N=3 KK

28. The value E 0 =3 realized in the Kaluza Klein model boundary of moduli is reached only at the boundary of moduli space in N=4 standard SUGRA Lines of constant E0E0 over the disk

29. Standard N=4 SUGRA realizes SuperHiggs with E 0 <3 Standard N=4 SUGRA realizes SuperHiggs with E 0 <3 M-theory à la KK realizes SuperHiggs with E 0 =3 M-theory à la KK realizes SuperHiggs with E 0 =3 This realization follows from a general shadowing mechanism This realization follows from a general shadowing mechanism Truncation to graviton + massive gravitino should be consistent since harmonics are constants Truncation to graviton + massive gravitino should be consistent since harmonics are constants This hints to the existence of a new N=4 shadow supergravity based on the scalar manifold M sh and an FDA describing superHiggs where E 0 =3 or bigger is allowed. This hints to the existence of a new N=4 shadow supergravity based on the scalar manifold M sh and an FDA describing superHiggs where E 0 =3 or bigger is allowed. Shadow SUGRAS exist also for other N? Shadow SUGRAS exist also for other N? is shadowing true also in D=5? is shadowing true also in D=5? We know the CFT interpretation of the universal multiplet but there is no time…….! We know the CFT interpretation of the universal multiplet but there is no time…….! Questions-Conclusions

40. The list of supersymmetric homogeneous 7-dimensional cosets was determined in the eighties. We have a particular interest in this coset that yields N=3 supersymmetry and reveals the intriguing relation between shadowing and susy breaking

48. Type IIB Supergravity, D3 branes and ALE manifolds Based on recent work by: 1. M. Bertolini, G. Ferretti, (P. F.) M. Trigiante, L. Campos, P. Salomonson hep-th 0106186 2. M.Billo, (P.F.), C. Herman, L. Modesto, I. Pesando (to appear)

49. Introduction There are interesting solutions of type IIB theory, named fractional D3 branes. There are interesting solutions of type IIB theory, named fractional D3 branes.  The gauge duals are non-conformal N=2 gauge theories in d=4 Fractional branes are commonly viewed as 5-branes wrapped on a vanishing cycle of transverse space Fractional branes are commonly viewed as 5-branes wrapped on a vanishing cycle of transverse space  Transverse space is R 2 x R 4  We have found a supersymmetric (BPS) D3-brane solution where transverse space is R 2 x ALE We have found a supersymmetric (BPS) D3-brane solution where transverse space is R 2 x ALE  In the orbifold limit we recover fractional branes  The warp factor is determined by a harmonic equation on ALE  In Eguchi Hanson case the harmonic equation reduces to a confluent Heun equation  Open questions on the boundary action and the gauge dual.

50. Type IIB Sugra The D3 brane couples to C [4] whose field strength involves a Chern Simons of lower forms Castellani & Pesando (1991) established geometric formulation

51. The SL(2,R)  SU(1,1) structure The solvable Lie algebra parametrization of the coset naturally introduces the dilaton and the RR 0-form

52. The FDA defining the Theory As for all supergravities the algebraic structure is encoded in an FDA. (D’Auria, Fré, (1982), Castellani, P. Van Nieuwenhuizen, K. Pilch (1982) Note the Chern Simons

53. The rheonomic parametrizations~susy rules Fermionic Components are expressed in terms of space-time components

54. Our task: the equations to be solved Note that these equations are written in flat indices and appear as constraints on the curvature components

55. Fractional Branes= D3 branes on orbifolds C 2 /  In the dual gauge theory, x 4 +ix 5 makes the complex scalar of the vector multiplet while x 6, x 7, x 8, x 9 constitute the scalar part of a hypermultiplet Susy is halved in the bulk by restricted holonomy

56. ALE manifolds as orbifold resolutions ALE manifolds are related to the ADE classification of Lie algebras. They can be obtained as suitable Hyperkahler quotients of flat HyperKahler manifolds Kronheimer construction of ALE spaces is realized by String Theory

57. ALE Manifolds: main relevant feature

58. The ansatz: Note the essential use of the harmonic 2 forms dual to the homology cycles of ALE Holomorphic field

59. Pinpointing the sources: This is the real problem of interpretation: what is the source of B,C fields? Even without D3 brane charge there is effective source for 5-form

60. Killing spinors I:

61. Killing spinors II:

62. Killing spinors III: Two solutions because of SU(2) holonomy of ALE

63. Harmonic equation in the Eguchi Hanson case Note factorization of near cycle metric Explicit form of the compact harmonic 2-form and “intersection function”

64. The partial Fourier transform Makes Laplace Equation on ALE x R 2 inhomogeneous

65. Heun confluent equation Fixed by boundary conditions D3 brane charge

66. Asymptotics and the charges The D3 brane charge determines the coefficient of the irregular solution near the cycle Asymptotic flatness fixes the coefficient of the irregular solution at infinity

67. Power series and questions See picture

68. Interpolation and singularity Far from the cycle we just see the charge of a normal D3 in D=10 Near the cycle we see a D3 brane in D=8. It is as if we had compactified sugra on an S 2

69. The World Volume Action Red=Auxiliary fields Brown=Sugra bckg fields Blue=World volume fields The interactions of world—volume fields (fluctuations from vacuum values) feel all the bckg fields. Yet this action is source only for C 4.. Where is the source of A [2]  ? Kappa supersymmetric: the fermions are hidden in the p—forms that are superforms This is the open question answering the which will shed light on the real nature of fractional branes