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Holographic duals for 4d N =4 SYM on space-times with boundaries

Holographic duals for 4d N =4 SYM on space-times with boundaries. Ofer Aharony Weizmann Institute of Science Sixth Crete Regional Meeting on String Theory , Milos, June 20, 2011 Mostly based on : OA, Marolf, Rangamani, arXiv:1011.6144;

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Holographic duals for 4d N =4 SYM on space-times with boundaries

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  1. Holographic duals for 4d N=4 SYM on space-times with boundaries Ofer Aharony Weizmann Institute of Science Sixth Crete Regional Meeting on String Theory,Milos, June 20, 2011 Mostly based on : OA, Marolf, Rangamani, arXiv:1011.6144; OA, Berdichevsky, Berkooz, Shamir, arXiv:1106.1870

  2. Outline • Motivations • Boundary conditions for 4dN=4 supersymmetric Yang-Mills (SYM) • Dual gravitational backgrounds : (a) Orientifolds and orbifolds; (b) 3-branes ending on 5-branes and related boundary conditions • Summary and future directions

  3. Motivations • Following the AdS/CFT correspondence, we have learned a lot about 4d N=4 SYM on R3,1 or on S3 x R. Can we learn more by putting the theory on other space-times ? Time-dependent space-times are obviously interesting but difficult. Much easier to study space-times that preserve a lot of supersymmetry. • A simple example of such a background is the half-line R2,1 x R+, or (t,x,y,z) with z > 0. Can put half-SUSY-preserving boundary conditions and try to understand the strongly coupled theory by S-duality and by the AdS/CFT correspondence.

  4. Another interesting example is to put the theory on anti-de Sitter spaceAdS4. In particular, in global coordinates AdS4behaves like a box in many ways; classical fields on this space all have energies #/LAdS with positive coefficients. So, anti-de Sitterspace provides an IR cutoff in a maximally symmetric fashion (SO(d-1,2) isometry), and is useful for studying strongly coupled theories without worrying about IR divergences. Can again preserve 16 supercharges (3d N=4 superconformal algebra) for appropriate boundary conditions. (Relevant for AdS/CFT; strong coupling in the bulk.)

  5. These two apparently different examples are actually equivalent, since the half-line is conformally equivalent to AdS4. This is evident in Poincare coordinates, • Thus, if we choose boundary conditions that preserve the conformal symmetry, the theories on these two spaces are equivalent, and will have the same S-duals and AdS/CFT-duals. Of course the interesting questions to ask about the two cases may be different, but have a simple translation. • I will describe the study of the 4dN=4 SYM theory on these two space-times, going freely between the two languages as convenient.

  6. Boundary conditions for N=4 SYM • The behavior of the theory on a space with a boundary, and in particular its string theory dual, depends on the choice of boundary conditions. • For free fields (e.g. U(1) N=4 SYM) there are two (linear) choices of boundary conditions preserving 3d conformal symmetry on the half-line : • Scalars : Dirichletf(z=0) = 0, Neumann dzf(z=0) = 0. • Gauge fields (in Az=0 gauge) : DirichletAi(z=0) = 0 (Fij(z=0) = 0), NeumanndzAi(z=0) = 0 (Fiz(z=0)=0). • Similarly for fermions. • The two boundary conditions for the gauge field are exchanged by electric-magnetic duality.

  7. Should map to AdS4 – what is interpretation there ? • For massless scalars naively on AdS4 only one possible boundary condition. But conformally coupled scalars have m2=-2/LAdS2 on AdS4, which is precisely in the range with two possible boundary conditions. After the conformal transformation to AdS4 get f(z)~az+bz2+…, and have Dirichlet a=0 (D=2 in CFT3) or Neumann b=0 (D=1). • Similarly for gauge fields in AdS4. Standard boundary condition is DirichletAi(z=0)=0, giving a global symmetry in AdS4/CFT3. But can also have Neumann, corresponding to gauging the global symmetry (integrating over Ai(z=0)). Still related by S-duality. New U(1) global symmetry from J=*F.

  8. Cannot preserve all SUSY, but can preserve half (3d N=4 superconformal algebra). Under this symmetry have vector multiplet (Ai, Yi, fermions) and hypermultiplet (Az, Xi, fermions) and must choose same boundary conditions in each multiplet. Have 2 choices : Dirichlet for vector, Neumann for hyper or Dirichlet for hyper, Neumann for vector. Exchanged by S-duality. First is realized by D3-brane ending on D5-brane, second by D3-brane ending on NS5-brane. • What changes in non-Abelian case (G=U(N)) ? One difference is that boundary conditions could break the gauge group. For instance, can choose Neumann for subgroup H in G (gauge part of global symmetry from AdS4 point of view).

  9. Naively expect that the two simplest choices are still S-dual, but this cannot be true : for instance, Dirichlet for vector has U(N) global symmetry, while Neumann has just U(1) global symmetry. • Have two extra possibilities not yet discussed (Gaiotto-Witten) : • Can couple to 3d N=4 SCFT on the boundary. • SUSY is preserved not just for (say) dzXi(z=0)=0, but whenever dzXi = ieijk[Xj,Xk]. So, can have generalized boundary conditions such that near the boundary Xi ~ ti/z, with ti an N-dimensional representation of SU(2). • Both are realized in brane constructions.

  10. ND3-branes ending on a single D5-brane should have just a U(1) global symmetry : have Dirichlet for vector but Xigoes as N-dimensional representation of SU(2). (“Fuzzy funnel”.) S-dual to Neumann, realized by a single NS5-brane. • ND3-branes ending on ND5-branes have U(N) global symmetry : Xi(z=0)=0. S-dual to ND3-branes ending on NNS5-branes, but there have non-trivial 3d N=4SCFT at the boundary (with U(N) global symmetry). • Boundary conditions with Xi~ general matrix – can decompose into irreducible representations; each block of size L is a D5-brane with LD3-branes ending on it.

  11. Configurations of D3-branes ending on D5-branes characterized by how many D3-branes end on each D5-brane. Have mbD5-branes with LbD3-branes ending on each. When back-reaction taken into account, Lb determines bending of D5-branes. • These theories have a PU(mb) global symmetry. • D3-branes ending on more than one NS5-brane involve a coupling to a non-trivial 3d N=4 SCFT. This can be seen by slightly separating the NS5-branes, giving some 3d N=4 gauge theory on the D3-branes stretched between them, that flows to the SCFT in the IR.

  12. For D3-branes ending both on D5-branes and on NS5-branes, also have non-trivial 3d N=4 SCFT at the boundary, but the full story is more complicated, since D3-branes are created when shift branes. Configuration characterized by linking numbers – for each 5-brane, how many D3-branes end on it from one side, minus number of D3-branes ending from other side, plus the number of 5-branes of opposite type on the other side. This number is preserved by moving the 5-branes around; controls bending. NS5s NS5s D3s D5s D5s

  13. Strongly coupled behavior • How does the 4d N=4 SYM theory with the various boundary conditions we discussed behave like at strong coupling ? • At very strong coupling, gYM >> 1, can use S-duality in the bulk. The action of S-duality on the boundary conditions was understood by Gaiotto+Witten. In the brane construction one just exchanges D5-branes with NS5-branes. • Can we find a gravitational dual description that will be valid for large N and large ‘t Hooft coupling gYM2 N ? For which boundary conditions ?

  14. Two obvious problems with this : • The (conformal) boundary of the gravitational space-time should be the space that the field theory lives in, R2,1 times a half-line (or AdS4). How can a boundary have a boundary ? • For many of the boundary conditions we discussed have a large global symmetry in the field theory (coming e.g. from gauge transformations that do not vanish at the boundary). Should thus have a large gauge group in the bulk. How can this arise in gravity ? • Let’s look for appropriate solutions and see how these problems are resolved…

  15. Simple solutions • We expect the bulk to include an AdS5 x S5region that is dual to the field theory away from its boundary; but the boundary of the gravitational theory should be AdS4 instead of R3,1. • In fact, AdS5has an AdS4slicing (used e.g. for Randall-Sundrum). However, in this slicing the boundary of AdS5 is mapped to two copies of AdS4, connected at their boundary, so it is not quite what we want. • To get one copy of AdS4, we need to identify the two boundaries.

  16. The simplest way to get a holographic dual of the SYM theory on a single AdS4 is to perform an orbifold / orientifold that identifies the two copies of AdS4. This can preserve half of the SUSY for an orbifold / orientifold5-plane wrapping AdS4 x S2. • This gives simple solutions that are naturally identified with the field theory on D3-branes intersecting orbifold / orientifold 5-planes. The corresponding boundary conditions were analyzed by Gaiotto+Witten, and they depend on the precise choice of orbifold / orientifold. For O5- and O5+orientifold planes, one has Neumann boundary conditions for USp(N) and SO(N) subgroups of U(N), respectively (and Dirichlet for rest).

  17. Gaiotto+Witten also analyzed the action of S-duality on these boundary conditions, which is rather complicated. For example, for D3-branes ending on an O50orientifold, the boundary condition is again Neumann for a USp(N) subgroup of SU(N), and there is also a fundamental hypermultiplet living on the boundary. The S-dual of this is an orbifold plane involving Neumann boundary conditions for an SU(N/2) x SU(N/2) subgroup of SU(N). • This construction gives us gravitational (really stringy) duals for a large class of boundary conditions, as orbifolds / orientifolds of AdS5 x S5. “Boundary of boundary” is realized by identification; no large global symmetries in this class. What about 5-brane boundary conditions ?

  18. Gravitational solutions for D3-branes ending on 5-branes • To find the gravity solutions for D3-branes ending on 5-branes, recall that the most general solutions of type IIB supergravity with 3d N=4superconformal (OSp(4|4)) symmetry were already found in 2007 by D’Hoker, Estes and Gutperle. They were mainly interested in D3-branes intersecting 5-branes, or in “Janus solutions” in which the coupling constant of N=4 SYM changes as one crosses some 3d defect. However, precisely the same symmetries arise for D3-branes ending on 5-branes, so solutions (at least for some boundary conditions) should be included in their general classification.

  19. Review of their solutions : symmetries imply that geometry should have an AdS4factor and two S2 factors. So, geometry is a warped product of these with a Riemann surfaceS, Also have some 3-forms (RR and NS-NS) on the S2’s (times a 1-form in S), a 5-form on both S2’s (times a 1-form in S), and a dilaton F(w). • D’Hoker et al showed that the general SUSY solution of this type involves a genus g Riemann surface, and is determined by two harmonic functions on this Riemann surface. These functions can have singularities which give AdS5 x S5 spikes.

  20. AdS5xS5 SU(N1) SU(N2) AdS5xS5 • Boundary of these space-times includes (2g+2) half-lines intersecting at a point (or AdS4’s sharing a boundary). Solutions dual to (2g+2)N=4 SYMSU(Ni) theories on a half-line,couplings gYM2i , intersecting and interacting at a point. SU(N4) SU(N3) AdS5xS5 AdS5xS5

  21. D’Hoker et al analyzed a particular degeneration of these solutions that occurs when two AdS5 x S5 spikes come together. They showed that the spikes are then replaced by 5-brane “throats” (surrounded by an S3instead of an S5), involving D5-branes or NS5-branes wrapping AdS4 x S2. • In particular, can get solutions with two AdS5 x S5spikes, and various 5-brane “throats”, interpreted as near-horizon of D3-branes intersecting 5-branes : NS5s D5s NS5s SU(N2) SU(N1) AdS5xS5 AdS5xS5 NS5s D5s NS5s

  22. The parameters of these solutions include the two asymptotic 5-form fluxes = the ranks N1,N2. We showed that one can take the limit of N2 going to zero and obtain a smooth geometry with a single AdS5 x S5 region. This is dual to the SU(N1) theory on a half-line / AdS4, as we wanted ! • Boundary of boundary not an issue in 10d, global symmetries realized by gauge fields on 5-branes. NS5s D5s D5s SU(N1) AdS5xS5 D5s NS5s NS5s

  23. Discrete parameters – n = number of NS5-brane stacks, m = number of D5-brane stacks (n+m>0). • Up to SL(2,R) transformations of type IIB SUGRA, have 2(n+m) continuous parameters. Can think of them as the number of 5-branes in each stack, and the number of D3-branes that end on each of these 5-branes. More precisely, when have both D5s and NS5s the latter is replaced by the linking numbers. The 5-form flux in each throat is ambiguous (since dF5 = F3 ^ H3, no unique conserved 5-form), related to ambiguity in definition of linking numbers. • 5-branes separated geometrically according to linking numbers ~ bending, as expected. Exact match of SUGRA parameters to field theory boundary conditions.

  24. Solutions are weakly curved (except deep inside the 5-brane “throats”) as long as all the numbers of 5-branes and D3-branes are large. Natural to scale N5 ~ N1/2 for large N, but not necessary. • We can also start from solutions with more than 2 AdS5 x S5 regions + 5-branes, and take all asymptotic 5-form fluxes but one to zero. This gives similar non-singular solutions, but with additional 3-cycles not surrounding 5-branes (Riemann surface has higher genus). These solutions (like original ones) have no brane construction, and we do not yet know precisely which boundary conditions they correspond to – probably some complicated 3d N=4 SCFT at the boundary (always have both D5s and NS5s, no weak coupling limit).

  25. Computations • Solutions (for D3-branes ending on 5-branes) are explicit – can compute anything. • Would be interesting to compute spectrum, but this is quite hard (computation started for similar intersecting brane case by Bachas+Estes). • Simplest thing to compute is 1-point functions; generally for CFT with boundary, <OD> = CD/zD. These can just be read off from the behavior of our solutions near boundary of AdS5xS5 region. For example, O2=tr(X12+X22+X32-Y12-Y22-Y32) has an expectation value that we can easily compute. For D3s ending on D5-branes can compare to weak coupling – behavior as function of parameters is different.

  26. Summary and future directions • Found general SUGRA solutions for N=4 SYM on half-line or AdS4, with boundary conditions preserving maximal supersymmetry. Also have stringy solutions not in SUGRA, with orbifolds / orientifolds. Would be interesting to complete field theory understanding of boundary conditions which have SUGRA duals, and look for other possible string theory solutions. • Should do more computations, in particular compute full spectrum (from SUGRA, from 5-branes and from stretched strings/branes).

  27. Many possible generalizations : • M2-branes ending on M5-branes; should be limit of known solutions (D’Hoker, Estes, Gutperle and Krym) for M2-branes intersecting M5-branes. • Other Dp-branes ending on D(p+2)-branes and/or NS5-branes : more difficult since no conformal symmetry, need to solve equations in more variables and less BPS conditions. • D3-branes stretched between NS5-branes (or between NS and NS’ branes) : should locally look like our solutions near the edges of the interval, but more complicated elsewhere. • Finite temperature (for half-line / AdS4) : no supersymmetry or conformal symmetry… Study phase transitions for global AdS4.

  28. Explicit form of solution for D3-branes ending on 5-branes : Choose w to live on quarter of complex plane : Re(w) < 0, Im(w) > 0. Naively have boundary but will not be boundary of full 10d geometry. Then :

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