Survey

# Download p-brane

Document related concepts

Perturbation theory wikipedia , lookup

Equation of state wikipedia , lookup

History of quantum field theory wikipedia , lookup

Yang–Mills theory wikipedia , lookup

Time in physics wikipedia , lookup

Relativistic quantum mechanics wikipedia , lookup

Supersymmetry wikipedia , lookup

Nordström's theory of gravitation wikipedia , lookup

Mathematical formulation of the Standard Model wikipedia , lookup

Partial differential equation wikipedia , lookup

Transcript

Recent issues in brane theory and the gauge/gravity correspondence Four possible topics 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 P.Fré Bogolyubov Institute, Kiev Dec. 2001 p-Brane Actions The parameter D and the harmonic function H(y) Electric and magnetic p-branes “Elementary” Conformal branes and AdS space AdS is a special case of a Domain Wall ELECTRIC BRANE These two forms are related by a coordinate transformation Coordinate patches and the conformal gauge Conformal brane a=0 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 Positivity of the Wall Tension 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 !!! “Near brane” factorization in the dual frame The transverse cone (D-p-1) - Cone p-brane 2 dsbrane dx d In some sense An X8-p compact manifold is the base of the transverse cone C( X8-p ) 2 dsCone dr 2 r 2ds X2 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-p as base. Domain wall supergravity from “sphere reduction” The DW/QFT correspondence of Boonstra Skenderis & Townsend 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! The shadow of the graviton multiplet can be a gravitino multiplet, suggesting a superHiggs mechanism but…... Conjecture: Shadow SUGRAS It seems that there exist more general Nextended SUGRAS with SU(N0) symmetry where N0<N M 123 eigenvalue s of Laplacians on X 7 The spectrum is determined by eigenvalues of Laplace Beltrami operators Structure of M-theory AdS4 UIR relations Differential Geometry of X7 knows all the relations implied by SUSY in AdS4 There are 4 neutral scalars whose vev ‘s trigger superHiggs R symmetry * = modulus in N=4 N=3 This multiplet is Universal in N=3 KK This suggests consistent truncation! 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 Questions-Conclusions SU (1,1) USp(3,3) M sh W6 U (1) U (3) 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…….! 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 Type IIB Supergravity, D3 branes and ALE manifolds 1. 2. 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 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. 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 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 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) The rheonomic parametrizations~susy rules Fermionic Components are expressed in terms of space-time components Our task: the equations to be solved Note that these equations are written in flat indices and appear as constraints on the curvature components Fractional Branes= D3 branes on orbifolds C2/G In the dual gauge theory, x4+ix5 makes the complex scalar of the vector multiplet while Susy is halved in the bulk by restricted holonomy x6, x 7 , x8 , x9 constitute the scalar part of a hypermultiplet 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 ALE Manifolds: main relevant feature The ansatz: Note the essential use of the harmonic 2 forms dual to the homology cycles of ALE Holomorphic field 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 Killing spinors I: Killing spinors II: Killing spinors III: Two solutions because of SU(2) holonomy of ALE Harmonic equation in the Eguchi Hanson case Explicit form of the compact harmonic 2-form and “intersection function” Note factorization of near cycle metric The partial Fourier transform Makes Laplace Equation on ALE x R2 inhomogeneous Heun confluent equation D3 brane charge Fixed by boundary conditions 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 Power series and questions See picture 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 S2 The World Volume Action 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 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