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. 2 Outline 1)Motivations 2)Boundary conditions for 4d N =4 supersymmetric Yang-Mills (SYM) 3)Dual gravitational backgrounds : (a) Orientifolds and orbifolds; (b) 3-branes ending on 5-branes and related boundary conditions 4)Summary and future directions

3. 3 Motivations Following the AdS/CFT correspondence, we have learned a lot about 4d N =4 SYM on R 3,1 or on S 3 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 R 2,1 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. 4 Another interesting example is to put the theory on anti-de Sitter space AdS 4. In particular, in global coordinates AdS 4 behaves like a box in many ways; classical fields on this space all have energies #/L AdS with positive coefficients. So, anti-de Sitter space 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. 5 These two apparently different examples are actually equivalent, since the half-line is conformally equivalent to AdS 4. 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 4d N =4 SYM theory on these two space-times, going freely between the two languages as convenient.

6. 6 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 : Dirichlet  (z=0) = 0, Neumann d z  (z=0) = 0. Gauge fields (in A z =0 gauge) : Dirichlet A i (z=0) = 0 (F ij (z=0) = 0), Neumann d z A i (z=0) = 0 (F iz (z=0)=0). Similarly for fermions. The two boundary conditions for the gauge field are exchanged by electric-magnetic duality. Boundary conditions for N =4 SYM

7. 7 Should map to AdS 4 – what is interpretation there ? For massless scalars naively on AdS 4 only one possible boundary condition. But conformally coupled scalars have m 2 =-2/L AdS 2 on AdS 4, which is precisely in the range with two possible boundary conditions. After the conformal transformation to AdS 4 get  (z)~az+bz 2 +…, and have Dirichlet a=0 (  =2 in CFT 3 ) or Neumann b=0 (  =1). Similarly for gauge fields in AdS 4. Standard boundary condition is Dirichlet A i (z=0)=0, giving a global symmetry in AdS 4 /CFT 3. But can also have Neumann, corresponding to gauging the global symmetry (integrating over A i (z=0)). Still related by S-duality. New U(1) global symmetry from J=*F.

8. 8 Cannot preserve all SUSY, but can preserve half (3d N =4 superconformal algebra). Under this symmetry have vector multiplet (A i, Y i, fermions) and hypermultiplet (A z, X i, 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 AdS 4 point of view).

9. 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) : 1)Can couple to 3d N =4 SCFT on the boundary. 2)SUSY is preserved not just for (say) d z X i (z=0)=0, but whenever d z X i = i  ijk [X j,X k ]. So, can have generalized boundary conditions such that near the boundary X i ~  i /z, with  i an N-dimensional representation of SU(2). Both are realized in brane constructions.

10. 10 N D3-branes ending on a single D5-brane should have just a U(1) global symmetry : have Dirichlet for vector but X i goes as N-dimensional representation of SU(2). (“Fuzzy funnel”.) S-dual to Neumann, realized by a single NS5-brane. N D3-branes ending on N D5-branes have U(N) global symmetry : X i (z=0)=0. S-dual to N D3-branes ending on N NS5-branes, but there have non-trivial 3d N =4 SCFT at the boundary (with U(N) global symmetry). Boundary conditions with X i ~ general matrix – can decompose into irreducible representations; each block of size L is a D5-brane with L D3-branes ending on it.

11. 11 Configurations of D3-branes ending on D5-branes characterized by how many D3-branes end on each D5-brane. Have m b D5-branes with L b D3-branes ending on each. When back-reaction taken into account, L b determines bending of D5-branes. These theories have a  U(m b ) 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. 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. D3s D5s NS5s

13. 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, g YM >> 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 g YM 2 N ? For which boundary conditions ?

14. 14 Two obvious problems with this : 1)The (conformal) boundary of the gravitational space-time should be the space that the field theory lives in, R 2,1 times a half-line (or AdS 4 ). How can a boundary have a boundary ? 2)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. 15 Simple solutions We expect the bulk to include an AdS 5 x S 5 region that is dual to the field theory away from its boundary; but the boundary of the gravitational theory should be AdS 4 instead of R 3,1. In fact, AdS 5 has an AdS 4 slicing (used e.g. for Randall-Sundrum). However, in this slicing the boundary of AdS 5 is mapped to two copies of AdS 4, connected at their boundary, so it is not quite what we want. To get one copy of AdS 4, we need to identify the two boundaries.

16. 16 The simplest way to get a holographic dual of the SYM theory on a single AdS 4 is to perform an orbifold / orientifold that identifies the two copies of AdS 4. This can preserve half of the SUSY for an orbifold / orientifold 5-plane wrapping AdS 4 x S 2. 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. 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 O5 0 orientifold, 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 AdS 5 x S 5. “Boundary of boundary” is realized by identification; no large global symmetries in this class. What about 5-brane boundary conditions ?

18. 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 =4 superconformal (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. 19 Review of their solutions : symmetries imply that geometry should have an AdS 4 factor and two S 2 factors. So, geometry is a warped product of these with a Riemann surface , Also have some 3-forms (RR and NS-NS) on the S 2 ’s (times a 1-form in  ), a 5-form on both S 2 ’s (times a 1-form in  ), and a dilaton  (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 AdS 5 x S 5 spikes.

20. 20 Boundary of these space-times includes (2g+2) half-lines intersecting at a point (or AdS 4 ’s sharing a boundary). Solutions dual to (2g+2) N =4 SYM SU(N i ) theories on a half-line, couplings g YM 2 i, intersecting and interacting at a point. AdS 5 xS 5 SU(N 1 ) SU(N 2 ) SU(N 4 ) SU(N 3 )

21. 21 D’Hoker et al analyzed a particular degeneration of these solutions that occurs when two AdS 5 x S 5 spikes come together. They showed that the spikes are then replaced by 5-brane “throats” (surrounded by an S 3 instead of an S 5 ), involving D5-branes or NS5-branes wrapping AdS 4 x S 2. In particular, can get solutions with two AdS 5 x S 5 spikes, and various 5-brane “throats”, interpreted as near-horizon of D3-branes intersecting 5-branes : AdS 5 xS 5 SU(N 1 ) SU(N 2 ) D5s NS5s D5s

22. 22 The parameters of these solutions include the two asymptotic 5-form fluxes = the ranks N 1,N 2. We showed that one can take the limit of N 2 going to zero and obtain a smooth geometry with a single AdS 5 x S 5 region. This is dual to the SU(N 1 ) theory on a half-line / AdS 4, as we wanted ! Boundary of boundary not an issue in 10d, global symmetries realized by gauge fields on 5-branes. AdS 5 xS 5 D5s NS5s SU(N 1 )

23. 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 dF 5 = F 3 ^ H 3, 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. 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 N 5 ~ N 1/2 for large N, but not necessary. We can also start from solutions with more than 2 AdS 5 x S 5 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. 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, = C  /z . These can just be read off from the behavior of our solutions near boundary of AdS 5 xS 5 region. For example, O 2 =tr(X 1 2 +X 2 2 +X 3 2 -Y 1 2 -Y 2 2 -Y 3 2 ) 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. 26 Summary and future directions Found general SUGRA solutions for N =4 SYM on half-line or AdS 4, 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. 27 Many possible generalizations : 1)M2-branes ending on M5-branes; should be limit of known solutions (D’Hoker, Estes, Gutperle and Krym) for M2-branes intersecting M5-branes. 2)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. 3)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. 4)Finite temperature (for half-line / AdS 4 ) : no supersymmetry or conformal symmetry… Study phase transitions for global AdS 4.

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