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Type IIB Supergravity, D3 branes and ALE manifolds

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 ). Introduction.

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Type IIB Supergravity, D3 branes and ALE manifolds

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  1. 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)

  2. 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.

  3. 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

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

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

  6. Supergravity People @ Stony Brook 1982

  7. Always running to get ahead ......... while Sergio relaxes in L.A! Ops!.. I was also there but do not tell anybody....

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

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

  10. In the dual gauge theory,x4+ix5makes the complex scalar of the vector multiplet while x6, x7 ,x8 , x9constitute the scalar part of a hypermultiplet Susy is halved in the bulk by restricted holonomy Fractional Branes= D3 branes on orbifolds C2/G

  11. 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

  12. ALE Manifolds: main relevant feature

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

  14. 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 Pinpointing the sources:

  15. Killing spinors I:

  16. Killing spinors II:

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

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

  19. Makes Laplace Equation on ALE x R2 inhomogeneous The partial Fourier transform

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

  21. 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 Asymptotics and the charges

  22. See picture Power series and questions

  23. Near the cycle we see a D3 brane in D=8. It is as if we had compactified sugra on an S2 Far from the cycle we just see the charge of a normal D3 in D=10 Interpolation and singularity

  24. Kappa supersymmetric: the fermions are hidden in the p—forms that are superforms Red=Auxiliary fields Brown=Sugra bckg fields Blue=World volume fields The World Volume Action The interactions of world—volume fields (fluctuations from vacuum values) feel all the bckg fields. Yet this action is source only for C4.. Where is the source of A[2]L ? This is the open question answering the which will shed light on the real nature of fractional branes

  25. The End Let us make an appointment for all supergravity people to be back here forSupergravity @ 50 Such people are always running when someone watches..........

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