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

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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

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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.

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• Another interesting example is to put the theory on anti-de Sitter space AdS4. In particular, in global coordinates AdS4 behaves like a box in many ways; classical fields on this space all have energies #/LAdS 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.)

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• 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 4d N=4 SYM theory on these two space-times, going freely between the two languages as convenient.

./)( 2222222 zdzdydxdtLds AdS

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• 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 dz(z=0) = 0.• Gauge fields (in Az=0 gauge) : Dirichlet Ai(z=0) = 0 (Fij(z=0) = 0), Neumann dzAi(z=0) = 0 (Fiz(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

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• 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/LAdS

2 on AdS4, which is precisely in the range with two possible boundary conditions. After the conformal transformation to AdS4 get (z)~az+bz2+…, and have Dirichlet a=0 (=2 in CFT3) or Neumann b=0 (=1).

• Similarly for gauge fields in AdS4. Standard boundary condition is Dirichlet Ai(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.

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• 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).

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• 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) dzXi(z=0)=0, but whenever dzXi = iijk[Xj,Xk]. So, can have generalized boundary conditions such that near the boundary Xi ~i/z, with i an N-dimensional representation of SU(2).

• Both are realized in brane constructions.

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• N D3-branes ending on a single D5-brane should have just a U(1) global symmetry : have Dirichlet for vector but Xi 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 : Xi(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 Xi ~ general matrix – can decompose into irreducible representations; each block of size L is a D5-brane with L D3-branes ending on it.

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• Configurations of D3-branes ending on D5-branes characterized by how many D3-branes end on each D5-brane. Have mb D5-branes with Lb D3-branes ending on each. When

back-reaction taken into account, Lb determines bending of D5-branes.

• These theories have a U(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.

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• 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 D5s

NS5sNS5s

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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 gYM

2 N ? For which boundary conditions ?

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• Two obvious problems with this :

1) 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 ?

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…

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Simple solutions

• We expect the bulk to include an AdS5 x S5 region 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, AdS5 has an AdS4 slicing (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.

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• 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 / orientifold 5-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).

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• 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 O50 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 AdS5 x S5. “Boundary of boundary” is realized by identification; no large global symmetries in this class. What about 5-brane boundary conditions ?

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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.

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• Review of their solutions : symmetries imply that geometry should have an AdS4 factor and two S2 factors. So, geometry is a warped product of these with a Riemann surface ,

Also have some 3-forms (RR and NS-NS) on the S2’s (times a 1-form in ), a 5-form on both S2’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 AdS5 x S5 spikes.

.)()()( 2222

221

224

222

214 dsdswfdswfdswfds

SSAdS

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• 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 SYM SU(Ni) theories on a half-line, couplings gYM

2i ,

intersecting and interacting at a point.

AdS5xS5

AdS5xS5

AdS5xS5 AdS5xS5

SU(N1) SU(N2)

SU(N4) SU(N3)

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• 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 S3 instead of an S5), involving D5-branes or NS5-branes wrapping AdS4 x S2.

• In particular, can get solutions with two AdS5 x S5 spikes, and various 5-brane “throats”, interpreted as near-horizon of D3-branes intersecting 5-branes :

AdS5xS5 AdS5xS5

SU(N1) SU(N2)

D5sNS5s

NS5sNS5s

NS5s

D5s

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• 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.

AdS5xS5

D5sD5s

D5s

NS5s

NS5sNS5s

SU(N1)

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• 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.

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• 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).

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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, <O> = C/z. These can just be read off from the behavior of our solutions near boundary of AdS5xS5 region. For example, O2=tr(X1

2+X22+X3

2-Y12-Y2

2-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.

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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).

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• 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 / AdS4) : no supersymmetry or conformal symmetry… Study phase transitions for global AdS4.

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• 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 :

.||)(4)()()( 22222

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