A Supergravity Dual for 4d SCFT s Universal Sector

Transcription

1 SUPERFIELDS European Research Council Perugia June 25th, 2010 Adv. Grant no A Supergravity Dual for 4d SCFT s Universal Sector Gianguido Dall Agata D. Cassani, G.D., A. Faedo, arxiv: work in progress... related work: J. Liu, P. Szepietowski, Z. Zhao, arxiv: J. Gauntlett, O. Varela, arxiv: K. Skenderis, M. Taylor, D. Tsimpis, arxiv:

2 The Messages i) Any SCFT with at least N=1 has a finite Universal spectrum of pure gauge operators. ii) We constructed the dual 5d supergravity model that captures the full dynamics induced by and/or iii) This model is a consistent truncation of Type IIB supergravity compactified on a squashed Sasaki Einstein 5-manifold O Tr O Tr O

3 non-spherical horizons in AdS/CFT

4 non-spherical horizons in AdS/CFT M 4 CY

5 non-spherical horizons in AdS/CFT M 4 CY Spherical Horizon New Radial coordinate opening up (5th dimension) Near horizon geometry AdS 5 S 5

6 non-spherical horizons in AdS/CFT When the branes are at a singular point the resulting horizon is not spherical anymore Cone singularity Radial coordinate 6-dimensional Calabi-Yau manifold D3 5-dimensional base (Y 5 ) Near horizon geometry AdS 5 Y 5

7 non-spherical horizons in AdS/CFT D3-branes generate 5-form fluxes: F flux 5 =(1+ )2k vol(y 5 ) Conditions on the base manifold Supersymmetry SASAKI Equations of Motion EINSTEIN ( M + kγ M ) =0 R ab = 2k 2 g ab

8 non-spherical horizons in AdS/CFT Definitions of Sasaki Einstein manifold from: globally defined Killing spinor; existence of a constant-norm Killing vector (Reeb vector); 5-dimensional spaces whose cone is a (non-compact) Calabi Yau manifold; For our purposes, we use their contact structure: Globally defined 1-form and 2-forms, Reeb U(1) R-symmetry Locally: U(1) fibration over a Kæhler Einstein base. η J Ω

9 » non-spherical horizons in AdS/CFT Examples Same topology : S 2 S 3 Different metrics: SU(2) SU(2) U(1) T 1,1 Y p,q L p,q,r SU(2) SU(2) SU(2) U(1) U(1) U(1) Isometries = Flavour + U(1) R A i»» SU(N) > > >» B i» > Y 2,1... SU(N)

10 non-spherical horizons in AdS/CFT Which physics can we discuss? Predict anomalous dimensions of CFT spectrum Study deformations (RG flows and such...) Perturbative and non-perturbative effects on 4d effective theories Holographic superconductors / CM applications...

11 non-spherical horizons in AdS/CFT Problem: turning on operators and/or vevs W = W +tro forces additional operators and/or vevs. We need to solve 10d equations: 5d Kaluza Klein modes talk to each other If the OPEs close O i O j O k O i O j O k O io j O k we can keep a finite number of states O i Consistent Truncation

12 Consistent Truncations Consistent truncations are not always effective theories Examples: N =8 pure sugra from M-theory on massless modes T 7 S 7 N =8 SO(8) gauged sugra from M-theory on Coset manifold reductions keeping all left-invariant modes For a given G H one has a H-structure reduction massive truncations

13 N=1 Universal Sector

14 N=1 Universal Sector Chiral singletons 4-dimensional N=1 SCFTs Flavour indices W α Gauge superfields Universal S i Matter fields Y 5 -dependent Generic operators = words using letters above tr W tr (S i S j )+... tr (W α S i S j...s k )+...

15 N=1 Universal Sector Ceresole, GD, D AURIA, Ferrara The gauge superfield is fermionic. Hence there is only a finite number of primary states (also using superspace identities) W α singleton pure gauge tr W 2 chiral hypermultiplet tr (W α W α ) tr (W α W 2 ) tr (W 2 W 2 ) energy tensor semi-conserved long multiplet graviton Ist KK tower (gravitino) 2nd KK tower (breathing mode vector multiplet)

16 N=1 Universal Sector Gravity dual: use Sasakian contact structure ( η, J, Ω ) Algebraic conditions J Ω =0 Ω Ω =2J J η J = η Ω =0 Differential conditions dη =2J dω =3i η Ω

17 N=1 Universal Sector Gravity dual: use Sasakian contact structure ( η, J, Ω ) Metric Ansatz: ds 2 = e 2 3 (4U+V ) ds 2 (M) +e 2U ds 2 (B KE )+e 2V (η + A) (η + A) Forms reduction: B = b 2 + b 1 (η + A)+b J J +Re(b Ω Ω) 5d gauge symmetries inherited from form gauge and diffeos η η + dω B B + dλ δb 2 = dλ 1 + λ 0 da, δb Ω = 3iωb Ω, δb 1 = dλ 0, δb J = 2λ 0,

18 N=1 Universal Sector The resulting effective theory is a 5d N=4 gauged supergravity Content: 1 graviton + 2 vector multiplets GD, Herrmann, Zagermann (g µν, 4ψ µ, 6A µ, 4λ, Σ) Scalar manifold M scal =SO(1, 1) (A µ, 4λ, 5φ) SO(5, 2) SO(5) SO(2) Gauge group G = Heis 3 U(1) R

19 The Supergravity Model

20 The Supergravity Model: Content The N=4 structure follows from the existence of 2 globally defined spinors manifold on the SE internal Each IIB gravitino gives 2 5d pseudo-majorana gravitinos ζ 1,2 (y) Ψ α µ(x, y) =ψ α 1 µ (x) ζ 1 (y)+ψ α 2 µ (x) ζ 2 (y). where ψ µi (ψ i µ) γ 0 = Ω ij (ψ j µ) T C, Charged vectors become tensor fields (satisfying 1st order eoms) db M M M N B N

21 The Supergravity Model: Gauge Group The Gauge group is G = Heis 3 U(1) R Covariant curvatures include tensors F 5 =0 dη = 0 dω = 0 H M = da M X NP M A N A P + Z MN B N 4 vectors are in the adjoint of G, A Λ 4 vectors are in a non-trivial representation of G (dualized to tensors) Non-trivial mixing Γ fλσ (t Λ ) I Σ First stringy t Λ = realization A I 0 (t Λ ) J I R-charged

22 The Supergravity Model: The Potential The gauging induces a scalar potential V = 12 e 14 3 U 2 3 V +2e 20 3 U+ 4 3 V e 20 3 U 8 3 V φ b Ω e 20 3 U 8 3 V +φ c Ω C 0 b Ω 2 + e 32 3 U 8 3 V 3 Im b Ω c Ω + k 2 This potential has 2 vacua: Round SE metric N=2 G = U(1) N=0 G broken U = V = b Ω = c Ω =0 e 4U = e 4V = 2 3, bω = eiθ+φ/2 3 c Ω = b Ω τ, τ (C 0 + ie φ ) Squashed metric

23 The Supergravity Model: The Potential The masses at the susy vacuum match the CFT spectrum The N=0 vacuum scalars mix: Mass Eigenstate m 2 δu + δv Im δc Ω +Reδb Ω +8δU Im δc Ω Re δb Ω + δφ 27 2(1 + 7) 3 Im δb Ω +Reδc Ω δc (1 + 7) 3 Re δb Ω +Imδc Ω 4 δu 9 2(1 + 3) 3 Im δc Ω Re δb Ω 2 δφ Re δc Ω + δc Im δb Ω + δc 0 0 4

24 The Supergravity Model: Applications Supersymmetric deformation: This truncation includes as a subcase the Girardello Petrini Porrati Zaffaroni flow (turning on only the gaugino condensate operator tr W 2 ) Non-susy deformations including both vevs and the operator may reach the IR point Holographic superconductors Gubser-Pufu-Rocha Gubser-Herzog- Pufu-Tesileanu U(1) symmetry gets broken from UV to IR String theory backgrounds with non-relativistic conformal symmetry Maldacena- Martelli-Tachikawa Massive vectors can be used to obtain Schrödinger or Lifshitz symmetry

25 In Progress...

26 In Progress... Additional flavour singlets come from the cycles of the internal manifold: Betti Multiplets. U(1) Baryonic current J B Betti hyper Tr(W 2 1 W 2 2 ) This is Y-dependent. Hence we focused on the T 11 manifold. There is a new vacuum! New Einstein metric for the T11, not the standard one. (unfortunately, it s unstable) New applications: gaugings corresponding to fluxes on the cycles holographic RG-flows related to fractional branes...

27 Summary i) Any SCFT with at least N=1 has a finite Universal O spectrum of pure gauge operators. ii) We constructed the dual 5d supergravity model that captures the full dynamics induced by Tr O and/or Tr O iii) This model is a consistent truncation of Type IIB supergravity compactified on a squashed Sasaki Einstein 5-manifold iv) Countless applications... next time