Making Contact with Supersymmetric AdS 5 Solutions Jerome Gauntlett - PowerPoint PPT Presentation

Making Contact with Supersymmetric AdS 5 Solutions Jerome Gauntlett Maxime Gabella, Eran Palti, James Sparks, Dan Waldram

Supersymmetric AdS Solutions Consider most general AdS backgrounds of string/M-theory or equivalently the most general CFTs with gravity duals. Consider AdS × Y with warped product metric: ds 2 = e 2∆( y ) [ ds 2 ( AdS d +1 ) + ds 2 ( Y )( y )] with fluxes preserving SO ( d, 2) isometries of AdS . (Fermions=0). Try and solve supergravity equations IIB supergravity: g , φ , H 3 , F 1 , F 3 , F 5 = ∗ F 5 D = 11 supergravity: g , G 4

R µν − 1 2 Rg µν − T µν = 0 d ∗ ( fluxes ) = 0 d ( fluxes ) = 0 Hard! Also: if one finds solutions are they stable? Perturba- tive and non-perturbative instabilities. Interesting examples are known. Hence focus on supersymmetric solutions. δ ( bosons ) ≈ fermions = 0 so need to impose δ ( fermions ) = 0: ˆ ∇ µ ǫ ∼ [ ∇ µ + ( fluxes × Γ µ )] ǫ = 0 , M ( fluxes ) ǫ = 0 . i.e. solutions admitting a “Killing spinor” ǫ Typically Killing spinors ⇒ equations of motion

Goals: Precise characterisation of the geometry of Y and fluxes How does this geometry relate to the SCFT? Develop tools to calculate quantities of physical interest: eg central charge and spectrum of chiral primary operators. Explore landscape of examples.

Focus on N = 1 SCFTs in d = 4 that are dual to AdS 5 solutions of type IIB. All such SCFTs have an abelian R -symmetry that encodes im- portant properties of the SCFT: 1. Dimension of (chiral primary) operators ∆( O ) ≥ 3 2 | R ( O ) | 2. Central charge a = 3 32[3 TrR 3 − TrR ] 3. a-maximisation Intriligator, Wecht Should have analogues in the geometry of the dual AdS solutions

Plan: 1. AdS 5 solutions using Sasaki-Einstein metrics Much known. Focus on some formulae of Martelli, Sparks, Yau for central charge a and ∆( O ) for operators dual to supersymmetric wrapped D3-branes 2. General AdS 5 solutions of type IIB

Sasaki-Einstein solutions Calabi-Yau Spaces Complex I ij and I 2 = − 1 ahler: hermitian metric g with ω ij = g ik I kj = − ω ji and dω = 0 K¨ Ricci flat ⇔ ∇ µ ǫ = 0 ǫγ (2) ǫ and Ω = ǫ T γ (3) ǫ with dω = 0, d Ω = 0 ⇔ ω = i ¯

Calabi-Yau cone metric: CY = dr 2 + ds 2 ( SE 5 ) ds 2 where ds 2 ( SE 5 ) defines a five-dimensional Sasaki-Einstein met- ric. An intrinsic definition ∇ m ψ + i 2 γ m ψ = 0 Can then define bi-linears etc. Note: no known analogue of Calabi-Yau theorem.

Place D3-brane probes at the apex of the cone. This leads to a quantum field theory on the D3-brane that preserves N = 1 supersymmetry. For this case the back-reacted geometry can easily be con- structed: ds 2 = H − 1 / 2 [ ds 2 ( R 1 , 3 )] + H 1 / 2 [ dr 2 + r 2 ds 2 ( SE 5 )] with H = 1 + 1 /r 4 and in the “near horizon limit” ds 2 = r 2 [ ds 2 ( R 1 , 3 )] + 1 r 2 [ dr 2 + r 2 ds 2 ( SE 5 )] and so we get ds 2 = AdS 5 × SE 5 F 5 = V ol ( AdS 5 ) + V ol ( SE 5 )

R-symmetry ξ i = I ij ( r∂ r ) j It is tangent to SE 5 and is a Killing vector. It is dual to the abelian R -symmetry of the dual SCFT. Can also directly define ξ m = ¯ ψγ m ψ This R -symmetry can either be U (1) or R . Locally we can write ξ = ∂ γ and ds 2 ( SE 5 ) = ( dγ + a ) 2 + ds 2 ( KE + 4 ) with da = 2 ω KE Regular: U (1) R-symmetry, KE manifold Quasi Regular: U (1) R-symmetry, KE orbifold Irregular: R R-symmetry

Examples: S 5 , T 1 , 1 + 6 more regular, Y p,q , L a,b,c Toric construction: Three Killing vectors v i that preserve ω : L v i ω = 0 ⇒ i v i ω = dµ i with moment maps µ i : X → R 3 . Image of µ i is a convex poly- hedral cone. Toric data can be used to calculate eg central charge. Via work of Hanany et al dual SCFTs are quiver gauge theories.

Contact Structure The SE 5 space has a contact structure: ψγ m ψ = g mn ξ n σ m = ¯ with σ ∧ dσ ∧ dσ = 8 V ol ( SE 5 ) � = 0 The contact structure gives a symplectic structure on the cone: ω = 1 2 d ( r 2 σ ) = rdr ∧ σ + 1 2 r 2 dσ and this is the K¨ ahler form of the CY 3 Note that ξ is the “Reeb vector” for the contact structure: i ξ σ = 1 , i ξ dσ = 0

Central Charge of SCFT Known that a ∝ 1 /G 5 ∝ V ol ( SE 5 ) Henningson, Skenderis Five-form flux quantisation � N ∼ F 5 ∝ V ol ( SE 5 ) SE 5 we find the simple formula a N =4 1 � = Y σ ∧ dσ ∧ dσ , (2 π ) 3 a where a N =4 = N 2 / 4 If we know the contact structure (equivalently the symplectic structure on the cone) then we can calculate central charge.

This can be rewritten on the cone X X e − r 2 / 2 ω 3 1 1 a N =4 � � X e − H e ω = 3! = (2 π ) 3 (2 π ) 3 a where H = r 2 / 2 is the Hamiltonian for the Reeb vector field ξ : i ξ ω = − dH . This is a Duistermaat-Heckman integral. It localises on the fixed point set F of the flow generated by the Killing vector ξ (one uses ( d − i ξ )( e − H e ω ) = 0 to show that the integrand is exact outside F ). On CY 3 cone, | ξ | 2 = r 2 and hence only vanishes at the singular tip of the cone r = 0.

To proceed, one equivariantly resolves the conical singularity and then evaluates. For the special case of toric SE 5 3 X e − H ω 3 a N =4 1 1 � � � = 3! = i � , � ξ, u p (2 π ) 3 a vertices p ∈P i =1 where u p i , i = 1 , 2 , 3, are the three edge vectors of the moment polytope P at the vertex point p. The vertices of P correspond to the U (1) 3 fixed points of a symplectic toric resolution X P of X .

Wrapped D3-branes Similar results can be obtained for the conformal dimensions of operators in the dual SCFT that are dual to D3-branes that wrap Σ 3 ⊂ SE 5 and preserve supersymmetry. These can be characterised as saying that on the CY 3 cone that they wrap holomorphic 4-cycles (divisors). 2 πN � Σ 3 σ ∧ dσ ∆( O Σ 3 ) = Y σ ∧ dσ ∧ dσ , � We can write e − H ω 2 � � σ ∧ dσ = 2! , Σ 3 Σ 4 again depends only on the symplectic structure of ( X, ω ) and the Reeb vector field ξ . This again may be evaluated by localization, having appropriately resolved the tip of the cone Σ 4 .

Comment: MSY have shown that a is completely determined by complex structure I and ξ . Formulae in terms of counting holomorphic functions weighted by R symmetry charges. MSY have shown that if one goes “off-shell” by considering Sasaki metrics i.e. Kahler metrics on the cone that a ( ξ ) satis- fies a variational principle that extremises on the Sasaki-Einstein metrics.

General AdS 5 solutions in type IIB ds 2 e 2∆ [ ds 2 ( AdS 5 ) + ds 2 ( Y )] = F 5 = f [ V ol ( AdS 5 ) + V ol ( Y )] and φ, H 3 = dB 2 , F 1 = da , F 3 = dC 2 − aH all non-zero on Y . ds 2 = e 2∆ r 2 [ ds 2 ( R 1 , 3 )] + e 2∆ r 2 [ dr 2 + r 2 ds 2 ( SE 5 )] Convenient to define 1 2 dφ + i 2 e φ F 1 P = − e − φ/ 2 H 3 − ie φ/ 2 F 3 G =

Conditions for supersymmetry: two spinors ψ 1 , ψ 2 fe − 4∆ − 2 D m ψ 1 + i � � γ m ψ 1 + 1 8 e − 2∆ G mnp γ np ψ 2 0 = 4 fe − 4∆ + 2 D m ψ 2 − i � � γ m ψ 2 + 1 8 e − 2∆ G ∗ mnp γ np ψ 1 ¯ 0 = 4 and 4 fe − 4∆ + i) ψ 1 − 1 ( γ m ∂ m ∆ − i 48 e − 2∆ γ mnp G mnp ψ 2 0 = 4 fe − 4∆ + i) ψ 2 − 1 48 e − 2∆ γ mnp G ∗ ( γ m ∂ m ∆ + i 0 = mnp ψ 1 and γ m P m ψ 2 + 1 24 e − 2∆ γ mnp G mnp ψ 1 0 = γ m P ∗ 24 e − 2∆ γ mnp G ∗ m ψ 1 + 1 0 = mnp ψ 2 . where D m = ∇ m + i 4 e φ ( F 1 ) m These imply all type IIB equations of motion.

Unlike SE 5 case we don’t have a picture of what configurations of geometry and probe branes give rise to these solutions. Examples: Pilch-Warner solution; beta deformations of SE 5 SCFTs Are there more? Is there a toric construction? The supersymmetry conditions were examined directly by JPG, Martelli, Sparks, Waldram. They can be equivalently phrased in terms of a “ G -structure”: Can be constructed as algebraic and differential conditions on bi-linears constructed from ξ i . Locally we have an “identity structure”- a canonically defined orthonor- mal frame. New observation: Y has a contact structure when f � = 0

R-symmetry ξ m ≡ 1 � ξ 1 γ m ξ 1 + ¯ ξ 2 γ m ξ 2 � ¯ . 2 It is Killing and preserves all fluxes – it is dual to the R-symmetry. Contact structure f � = 0 2 e ∆ � � ξ 1 γ m ξ 1 − ¯ ¯ σ m ≡ ξ 2 γ m ξ 2 , f and satisfies σ ∧ dσ ∧ dσ = 128 e 8∆ V ol Y f 2 ξ is again the Reeb vector for the contact structure i ξ σ = 1 , i ξ dσ = 0 But it is not, in general, compatible with the metric σ m � = g mn ξ n

The contact structure implies that the cone has a canonical symplectic structure ω = 1 2 d ( r 2 σ ) = rdr ∧ σ + 1 2 r 2 dσ and H = r 2 / 2 is again Hamiltonian for ξ : i ξ ω = − dH . � e 8∆ V ol Y . Central charge: As before we need to calculate 1 /G 5 ∝ We also need to take into account flux quantisation of five-form � ( F 5 + H ∧ C 2 ) N ∼ Interestingly the solutions don’t have any quantised three-form flux. We again find, as in SE 5 case a N =4 1 � = Y σ ∧ dσ ∧ dσ , (2 π ) 3 a