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

Making Contact with Supersymmetric AdS5 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: ds2 = e2∆(y)[ds2(AdSd+1) + ds2(Y )(y)] with fluxes preserving SO(d, 2) isometries of AdS. (Fermions=0). Try and solve supergravity equations IIB supergravity: g, φ, H3, F1, F3, F5 = ∗F5 D = 11 supergravity: g, G4

Rµν − 1 2Rgµν − Tµν = d ∗ (fluxes) = d(fluxes) = 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 ∇µǫ ∼ [∇µ + (fluxes × Γµ)]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 AdS5 solutions

• f 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[3TrR3 − TrR]

• 3. a-maximisation Intriligator, Wecht

Should have analogues in the geometry of the dual AdS solutions

Plan:

• 1. AdS5 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 AdS5 solutions of type IIB

Sasaki-Einstein solutions Calabi-Yau Spaces Complex Iij and I2 = −1 K¨ ahler: hermitian metric g with ωij = gikIkj = −ωji and dω = 0 Ricci flat ⇔ ∇µǫ = 0 ⇔ ω = i¯ ǫγ(2)ǫ and Ω = ǫTγ(3)ǫ with dω = 0, dΩ = 0

Calabi-Yau cone metric: ds2

CY = dr2 + ds2(SE5)

where ds2(SE5) defines a five-dimensional Sasaki-Einstein met- ric. An intrinsic definition ∇mψ + i 2γmψ = 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: ds2 = H−1/2[ds2(R1,3)] + H1/2[dr2 + r2ds2(SE5)] with H = 1 + 1/r4 and in the “near horizon limit” ds2 = r2[ds2(R1,3)] + 1 r2[dr2 + r2ds2(SE5)] and so we get ds2 = AdS5 × SE5 F5 = V ol(AdS5) + V ol(SE5)

R-symmetry ξi = Iij(r∂r)j It is tangent to SE5 and is a Killing vector. It is dual to the abelian R-symmetry of the dual SCFT. Can also directly define ξm = ¯ ψγmψ This R-symmetry can either be U(1) or R. Locally we can write ξ = ∂γ and ds2(SE5) = (dγ + a)2 + ds2(KE+

4 )

with da = 2ωKE Regular: U(1) R-symmetry, KE manifold Quasi Regular: U(1) R-symmetry, KE orbifold Irregular: R R-symmetry

Examples: S5, T 1,1 + 6 more regular, Y p,q, La,b,c Toric construction: Three Killing vectors vi that preserve ω: Lviω = 0 ⇒ iviω = dµi with moment maps µi : X → R3. Image of µi 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 SE5 space has a contact structure: σm = ¯ ψγmψ = gmnξn with σ ∧ dσ ∧ dσ = 8V ol(SE5) = 0 The contact structure gives a symplectic structure on the cone: ω = 1 2d(r2σ) = rdr ∧ σ + 1 2r2dσ and this is the K¨ ahler form of the CY3 Note that ξ is the “Reeb vector” for the contact structure: iξσ = 1, iξdσ = 0

Central Charge of SCFT Known that a ∝ 1/G5 ∝ V ol(SE5) Henningson, Skenderis Five-form flux quantisation N ∼

• SE5

F5 ∝ V ol(SE5) we find the simple formula aN =4 a = 1 (2π)3

• Y σ ∧ dσ ∧ dσ ,

where aN =4 = N2/4 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 aN =4 a = 1 (2π)3

• X e−r2/2 ω3

3! = 1 (2π)3

• X e−H eω

where H = r2/2 is the Hamiltonian for the Reeb vector field ξ: iξω = −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 − iξ)(e−Heω) = 0 to show that the integrand is exact

• utside F).

On CY3 cone, |ξ|2 = r2 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 SE5 aN =4 a = 1 (2π)3

• X e−H ω3

3! =

• vertices p∈P

3

• i=1

1 ξ, up

i ,

where up

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 XP of X.

Wrapped D3-branes Similar results can be obtained for the conformal dimensions of

• perators in the dual SCFT that are dual to D3-branes that

wrap Σ3 ⊂ SE5 and preserve supersymmetry. These can be characterised as saying that on the CY3 cone that they wrap holomorphic 4-cycles (divisors). ∆(OΣ3) = 2πN

• Σ3 σ ∧ dσ
• Y σ ∧ dσ ∧ dσ ,

We can write

• Σ3

σ ∧ dσ =

• Σ4

e−H ω2 2! , 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 Σ4.

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 AdS5 solutions in type IIB ds2 = e2∆[ds2(AdS5) + ds2(Y )] F5 = f[V ol(AdS5) + V ol(Y )] and φ, H3 = dB2, F1 = da, F3 = dC2 − aH all non-zero on Y . ds2 = e2∆r2[ds2(R1,3)] + e2∆ r2 [dr2 + r2ds2(SE5)] Convenient to define P = 1 2dφ + i 2eφF1 G = −e−φ/2H3 − ieφ/2F3

Conditions for supersymmetry: two spinors ψ1, ψ2 = Dmψ1 + i

4

• fe−4∆ − 2
• γmψ1 + 1

8e−2∆Gmnpγnpψ2

= ¯ Dmψ2 − i

4

• fe−4∆ + 2
• γmψ2 + 1

8e−2∆G∗ mnpγnpψ1

and = (γm∂m∆ − i

4fe−4∆ + i)ψ1 − 1 48e−2∆γmnpGmnpψ2

= (γm∂m∆ + i

4fe−4∆ + i)ψ2 − 1 48e−2∆γmnpG∗ mnpψ1

and = γmPmψ2 + 1

24e−2∆γmnpGmnpψ1

= γmP ∗

mψ1 + 1 24e−2∆γmnpG∗ mnpψ2 .

where Dm = ∇m + i

4eφ(F1)m

These imply all type IIB equations of motion.

Unlike SE5 case we don’t have a picture of what configurations

• f geometry and probe branes give rise to these solutions.

Examples: Pilch-Warner solution; beta deformations of SE5 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 ξi. Locally we have an “identity structure”- a canonically defined orthonor- mal frame. New observation: Y has a contact structure when f = 0

R-symmetry ξm ≡ 1

2

• ¯

ξ1γmξ1 + ¯ ξ2γmξ2

• .

It is Killing and preserves all fluxes – it is dual to the R-symmetry. Contact structure f = 0 σm ≡

2e∆ f

• ¯

ξ1γmξ1 − ¯ ξ2γmξ2

• ,

and satisfies σ ∧ dσ ∧ dσ = 128e8∆ f2 V olY ξ is again the Reeb vector for the contact structure iξσ = 1, iξdσ = 0 But it is not, in general, compatible with the metric σm = gmnξn

The contact structure implies that the cone has a canonical symplectic structure ω = 1 2d(r2σ) = rdr ∧ σ + 1 2r2dσ and H = r2/2 is again Hamiltonian for ξ: iξω = −dH. Central charge: As before we need to calculate 1/G5 ∝

e8∆V olY .

We also need to take into account flux quantisation of five-form N ∼

• (F5 + H ∧ C2)

Interestingly the solutions don’t have any quantised three-form

• flux. We again find, as in SE5 case

aN =4 a = 1 (2π)3

• Y σ ∧ dσ ∧ dσ ,

We similarly find that for D3-branes wrapping supersymmetric cycles: ∆(OΣ3) = 2πN

• Σ3 σ ∧ dσ
• Y σ ∧ dσ ∧ dσ ,

Both of these can be expressed in terms of Duistermaat Heckman formulae and evaluated by localisation. For toric case we have previous formula. Applications:

• 1. If R symmetry is U(1) then a is a rational number
• 2. Pilch Warner solution: without knowing metric can calculate

a = 27/32. Interestingly, the metric is toric, but the fluxes are

• not. Nevertheless, can use toric formula to obtain a!

Generalized Geometry The cone X is symplectic. Recall that for SE5 case the CY3 cone is symplectic (Kahler) and complex. In fact X admits two “generalized complex structures” one if which is integrable. Generalized Geometry Hitchin and Gualtieri Focus on geometry of E = T(X) ⊕ T ∗(X). Allows one to in- corporate diffeos with gauge transformations of B-field via a Generalized Lie derivative. If V ∈ E with V = v + λ

LV g = Lvg, LV B = LvB − dλ

Note that E has a natural O(6, 6) metric: < V, V >= ivλ

Spinors Ω are polyforms: the Clifford action is simply V · Ω = ivΩ + iλΩ (V · W · +W · V ·)Ω = 2 < V, W > Ω Positive chirality - even forms; Negative chirality - odd forms Generalized almost complex structures: J 2 = −1 or equivalently pure spinors Ω. There is a notion of integrability. For general susy AdS5 solutions there are two complex structures Ω± and defining e2A = e2∆+φ/2r2 we have dΩ− = d(e−AReΩ+) = d(eAImΩ+) = 1 8e4Ae−B(∗F1 − ∗F3 + ∗F5)

For AdS5 × SE5 solutions: e2A = r2 and Ω− = Ω, Ω+ = ir3e

i r2ω

For general AdS5 solutions: Ω+: is not an integrable generalized complex structure. Contains symplectic structure ω ∼ eAΩ+|2-form Ω−: is an integrable generalized complex structure and expect that some properties (eg a, ∆(O) ) of the SCFT depend just on Ω− and the Reeb vector ξ. Another result: have elucidated a generalized analogue of Kahler- Einstein geometry

Conclusion 1. Most general supersymmetric AdS5 × Y5 solutions of type IIB SUGRA with F5 = 0 have a canonical contact structure or equivalently the cone has a canonical symplectic structure. When combined with the R-symmetry Killing Reeb vector, we obtain Duistermaat-Heckman integrals for central charge which can be evaluated by localisation.

• 2. Cone over Y5 is a special kind of generalized complex Calabi-

Yau geometry. Can we generalise other MSY results? 3. What are the analogues of all this for general AdS5 × Y6 solutions of D = 11 SUGRA?

• 4. MSY results also apply to AdS4 × SE7 solutions. How about

the most general AdS4 × Y7 solutions?