Holographic Models of Cosmological Singularities Ben Craps Vrije Universiteit Brussel & The International Solvay Institutes “New Perspectives in String Theory” Opening Conference GGI, Firenze, April 8, 2009
Plan • AdS cosmology: review of basic idea • ABJM theory and an unstable triple trace deformation • Beyond the singularity? Self-adjoint extensions • Summary and outlook 1
AdS cosmology: basic idea Starting point: supergravity solutions in which smooth, asymptotically AdS initial data evolve to a big crunch singularity in the future. Can a dual gauge theory be used to study this process in quantum gravity? Hertog, Horowitz 2
AdS cosmologies: basic idea • AdS: boundary conditions required • Usual supersymmetric boundary conditions: stable • Modified boundary conditions: smooth initial data that evolve into big crunch (which extends to the boundary of AdS in finite time) • AdS/CFT relates quantum gravity in AdS to field theory on its conformal boundary • Modified boundary conditions � potential unbounded below in boundary field theory; scalar field reaches infinity in finite time • Goal: learn something about cosmological singularities (in the bulk theory) by studying unbounded potentials (in the boundary theory) Hertog, Horowitz 3
AdS cosmology: the bulk theory Compactify 11d sugra on S 7 and truncate (consistently) to � � � d � x √− g 1 2 R − 1 1 de Wit, Nicolai 2( ∇ ϕ ) � + � � √ S = Duff, Liu R � 2 + cosh( 2 ϕ ) AdS This describes a scalar whose mass squared is negative but above the BF bound. In all solutions asymptotic to the AdS 4 metric � � dr � ds � = R � − (1 + r � ) dt � + 1 + r � + r � d � � AdS the scalar field decays at large radius as ϕ ( r ) ∼ α ( t, �) + β ( t, �) r � r Consider AdS invariant boundary conditions β = − hα � Hertog, Maeda 4
ϕ ( r ) ∼ α r + β AdS cosmology: bulk solution β = − hα � r � h = 0 Standard supersymmetric boundary conditions: h � = 0 For , there exist smooth asymptotically AdS initial data that evolve to a singularity that reaches the boundary of AdS in finite global time. Example: analytic continuation of Euclidean instanton ds � = dρ � ϕ ( ρ ) ∼ α ρ + β b � ( ρ ) + ρ � d � � with ρ � leads to Lorentzian cosmology: • inside the lightcone (corresponding to the origin of the Euclidean instanton): open FRW universe with scale factor that vanishes at t = π/ 2 some finite time . • outside the lightcone: asymptotic behavior − hα � ( t ) α ( t ) = α (0) ϕ ∼ α ( t ) with r � cos t r Hertog, Horowitz 5
ϕ ( r ) ∼ α r + β AdS cosmology: dual field theory β = − hα � r � R × S � dual M-theory on AdS 4 x S 7 M2-brane CFT on Maldacena β = 0 φ • With usual boundary conditions , the scalar field is dual to a dimension 1 operator O = 1 N Tr T ij φ i φ j Aharony, Oz, Yin α ↔ �O� O φ The expectation value of is determined by the asymptotic behavior of : h � = 0 • Boundary conditions with correspond to deforming the CFT by a triple trace operator: � S → S + h Witten; Berkooz, Sever, Shomer; O � Hertog, Maeda 3 �O� This corresponds to a potential that is unbounded from below, and becomes infinite in finite time: �O� = α ( t ) = α (0) cos t Hertog, Horowitz 6
AdS cosmology: toy model for the boundary theory O = 1 N Tr T ij φ i φ j O Ignore the non-abelian structure in and replace by the square of a single scalar field: O → φ � We find a scalar field theory with standard kinetic term and potential V = 1 8 φ � − h 3 φ � The quadratic term corresponds to the conformal coupling to the curvature of the S 2 . V φ Hertog, Horowitz 7
AdS cosmology: what happens when the field reaches infinity? φ = (3 / 8 h ) � / � V • Classical solution: cos � / � t t = π/ 2 Field reaches infinity at finite time φ • Semiclassically: field tunnels out of metastable minimum and reaches infinity at finite time. • Quantum mechanics of the homogeneous mode: theory of quantum mechanics with unbounded potentials. V = 1 8 φ � − h Self-adjoint extensions of Hamiltonian: field bounces 3 φ � back from infinity. • Quantum field theory with unbounded potentials: not much known. Particle creation may be important. φ � • Regularization by adding irrelevant operator to potential: big crunch replaced by M large black hole. Thermalization? Hertog, Horowitz; Elitzur, Giveon, Porrati, Rabinovici; Banks, Fischler 8
AdS cosmology: questions V • Can we perform computations in M2-brane theory? φ • How can we interpret the unstable potential? � Brane nucleation Bernamonti, BC • Do self-adjoint extensions make sense in field theory? • If so, how does a wavepacket evolve after it reaches infinity? • If so, what is the bulk interpretation? 9
Plan • AdS cosmology: review of basic idea • ABJM theory and an unstable triple trace deformation • Beyond the singularity? Self-adjoint extensions • Summary and outlook 10
ABJM theory: action N = 6 superconformal U(N) x U(N) Chern-Simons-matter theory with levels k and -k ˆ A � A � • Gauge fields and ( N, � Y A , A = 1 , . . . 4 N ) SU (4) R • Scalar fields in fundamental of and in of gauge group � k � 4 π ǫ �νλ Tr( A � ∂ ν A λ + 2 i A λ − 2 i 3 A � A ν A λ − ˆ A � ∂ ν ˆ A � ˆ ˆ A ν ˆ d � x S = A λ ) 3 � Tr( D � Y A ) † D � Y A + V ��� + terms with fermions � V ��� = − 4 π � Y A Y † A Y B Y † B Y C Y † C + Y † A Y A Y † B Y B Y † C Y C 3 k � Tr � +4 Y A Y † B Y C Y † A Y B Y † C − 6 Y A Y † B Y B Y † A Y C Y † C Aharony, Bergman, Jafferis, Maldacena 11
ABJM theory: brane interpretation and gravity dual � � � k ABJM theory is worldvolume action of N coincident M2-branes on orbifold of y A → exp(2 πi/k ) y A � k : Coupling constant of ABJM theory is 1/k � “’t Hooft” limit: large N with N/k fixed. ds � = R � AdS � × S � 4 ds � AdS � + R � ds � � k Gravity dual: orbifold of : S � F � ∼ N ′ ǫ � ( N ′ = kN ) R = (32 π � N ′ ) � / � l p S � = ( dχ + ω ) � + ds � ds � Can write � P � 2 π χ Orbifold identification makes periodicity . In ’t Hooft limit: weakly coupled IIA string theory. k Aharony, Bergman, Jafferis, Maldacena 12
A triple trace deformation of ABJM theory ϕ � k Scalar field of consistent truncation of sugra survives quotient � Bulk analysis extends to k>1. Will study ’t Hooft limit (large N with N/k fixed). ϕ SU (4) R Dimension 1 chiral primary operator with same symmetry properties as under : 1 N � Tr( Y � Y † � − Y � Y † O = � ) � � � V = − f Tr( Y � Y † � − Y � Y † � ) Triple trace deformation: N � Quantum corrections: is effective potential truly unbounded below? Elitzur, Giveon, Porrati, Rabinovici � Sensitive to UV behavior! (Does one need to turn on irrelevant operators?) Vertex in double line notation: Will find beta function at order 1/N 2 BC, Hertog, Turok 13
Warm-up: O(N) vector model � � � � � � φ − λ − ∂ � & φ � ∂ � & φ � & & d � x S = φ N � Perturbative beta function up to order 1/N: β ( λ ) = 9 λ � 9 λ � π � N − 32 π � N Stephen, McCauley; Stephen; Lewis, Adams; Pisarski ( λ > 0) Positive coupling : λ ∗ = 32 • Perturbative UV fixed point: λ c = 8 π � < λ ∗ N = ∞ • Non-perturbatively: UV fixed point at (for ) 3 Bardeen, Moshe, Bander λ > λ c “instability” for (masses of order the cutoff) ( λ < 0) Negative coupling : • UV fixed point at � asymptotic freedom, effective potential truly unbounded below λ = 0 Coleman, Gross 14
Renormalization of triple trace deformation of ABJM theory: simplified � � � V = − f Tr( Y Y † ) with f > 0 Consider simplified potential N � 9 f � Beta function β ( − f ) = 4 π � N � + . . . d* = − 9 f � * d f Callan-Symanzik: 4 π � N � 8 π � N � f � = Solution: 9 ln( * � /M � ) � � � 8 π � Tr( Y Y † ) V ( Y ) = − Coleman-Weinberg potential: 9 N � ln[Tr( Y Y † ) /M � ] � Reliable for large Tr( Y Y † ) � � � V = − f Tr( Y � Y † � − Y � Y † � ) Question: is this also true for ? N � 15
Warm-up: O(N) x O(N) vector model � � � � � � � � φ � − λ ��� − λ ��� − ∂ � & φ � � ∂ � & φ � − ∂ � & φ � � ∂ � & φ � � & & φ � � & & d � x S = φ � φ � N � N � � � � � � � � � � � � − λ ��� − λ ��� φ � � & & φ � � & & φ � � & & φ � � & & φ � φ � φ � φ � N � N � Rabinovici, Saering, Bardeen 16
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