Recent issues in brane theory and the gauge/gravity correspondence

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

2. p-Brane Actions

3. The parameter D and the harmonic function H(y)

4. Electric and magnetic p-branes “Elementary”

5. Conformal branes and AdS space

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

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

8. Randall Sundrum gravity trapping Kaluza Klein expansion in non compact space 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

9. Positivity of the Wall Tension

10. The “dual frame” of Boonstra, Skenderis and Townsend We learn that although the AdS x S8-p is not a solution of supergravity, we can notheless compactify on the sphere S8-p, or other compact manifold X8-p !!!

11. “Near brane” factorization in the dual frame

12. The transverse cone (D-p-1) - Cone An X8-p compact manifold is the base of the transverse cone C( X8-p ) p-brane In some sense In D=10 the p-brane splits the space into a d=p+1 world volume and a transverse cone C( X8-p ) that has the compact manifold X8-pas base.

13. Domain wall supergravity from “sphere reduction”

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

15. 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? • It is a “gauged supergravity” • But which “gauging” ? • Typically a non compact one. It is compact for AdS branes! • What are the possible gaugings? • These are classifiable and sometimes classified • How is the gauging determined and how does it reflect microscopic string dynamics? • ??? This is the research frontier!

16. Massless Multiplet Long massive multiplet The shadow of the graviton multiplet can be a gravitino multiplet, suggesting a superHiggs mechanism but…...

17. Lagrangian has N SUSY but not SU(N) symm. All vacua have N° < N SUSY Conjecture: Shadow SUGRAS It seems that there exist more general N-extended SUGRAS with SU(N0) symmetrywhere N0<N

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

21. Structure of M-theory AdS4 UIR relations

22. Differential Geometry of X7knows all the relations implied by SUSY in AdS4

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

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

27. The value E0=3 realized in the Kaluza Klein model is reached only at the boundary of moduli space in N=4 standard SUGRA Lines of constant E0 over the disk

28. Questions-Conclusions • Standard N=4 SUGRA realizes SuperHiggs with E0<3 • M-theory à la KK realizes SuperHiggs with E0=3 • This realization follows from a general shadowing mechanism • 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 Msh and an FDA describing superHiggs where E0=3 or bigger is allowed. • Shadow SUGRAS exist also for other N? • is shadowing true also in D=5? • We know the CFT interpretation of the universal multiplet but there is no time…….!

39. 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

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

48. Introduction • 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 • Transverse space is R2 x R4/G. • We have found a supersymmetric (BPS) D3-brane solution where transverse space is R2 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.

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

50. The solvable Lie algebra parametrization of the coset naturally introduces thedilatonand theRR 0-form The SL(2,R)~SU(1,1) structure