Geometric flows and gravitational instantons Marios Petropoulos - PowerPoint PPT Presentation

Geometric flows and gravitational instantons Marios Petropoulos CPHT – Ecole Polytechnique – CNRS Galileo Galilei Institute for Theoretical Physics AdS 4 /CFT 3 and the Holographic States of Matter – October 2010

Highlights Motivations and summary Gravitational instantons: homogeneity and self-duality The view from the leaf: geometric flows Extensions Outlook

Framework The Ricci flow describes the parametric evolution of a geometry as ∂ g ij ∂ t = − R ij ◮ Introduced by R. Hamilton in 1982 as a tool for proving Poincaré’s (1904) and Thurston’s (late 70s) 3D conjectures ◮ In non-critical string theory Ricci flow is an RG flow [Friedan, 1985] – can mimic time evolution as UV → IR t = log 1 / µ

Basic features: a reminder ◮ Volume is not preserved along the flow d D x √ det gg ij ∂ g ij d D x √ det gR d V d t = 1 � ∂ t = − 1 � 2 2 Consequence: ◮ positive curvature → space contracts ◮ negative curvature → space expands ◮ Killing vectors are preserved in time: the isometry group remains unaltered – or grows in limiting situations

Example ◮ At initial time: R ( 0 ) = ag ( 0 ) with a constant ij ij ◮ Subsequent evolution: linear rescaling g ij ( t ) = ( 1 − at ) g ( 0 ) ij R ij ( t ) = R ( 0 ) ij ◮ Properties ◮ a > 0 ⇒ uniform contraction → singularity at t = 1 / a ◮ a < 0 ⇒ indefinite expansion

Gravitational instantons ◮ Useful for non-perturbative transitions in quantum gravity ◮ Appear in string compactifications e.g. in heterotic: C 2 / Γ → ALE spaces → Gibbons–Hawking multi-instantons as Eguchi–Hanson (blow-up of the C 2 / Z 2 A 1 singularity) ◮ Describe hyper moduli spaces e.g. in IIA: ◮ Taub–NUT ( SU ( 2 ) × U ( 1 ) , Λ = 0): tree-level ◮ Pedersen/Fubini–Study ( SU ( 2 ) × U ( 1 ) , Λ � = 0): supergravity ◮ Calderbank–Pedersen (Heisenberg × U ( 1 ) , Λ � = 0): string pert ◮ Calderbank–Pedersen ( U ( 1 ) × U ( 1 ) , Λ � = 0): string non-pert or in heterotic: Atiyah–Hitchin ( SU ( 2 ) , Λ = 0)

Geometric flows arise in gravitational instantons with time foliation ◮ In 4D self-dual gravitational instantons with homogeneous Bianchi spatial sections: time evolution is a Ricci flow of the 3D homogeneous space ◮ In non-relativistic gravity with invariance explicitly broken to foliation-preserving diffeomorphisms and with detailed-balance dynamics: time evolution is a geometric flow of the 3D space (valid actually in D + 1 → D ) Geometric flows might carry information on holographic evolution in some gravitational set ups – yet to be unravelled

Highlights Motivations and summary Gravitational instantons: homogeneity and self-duality The view from the leaf: geometric flows Extensions Outlook

Cartan’s formalism Metric and torsionless connection one-form ω a b and curvature two-form R a b in an orthonormal frame: d s 2 = δ ab θ a θ b bcd θ c ∧ θ d ◮ Riemann tensor: R a b = d ω a b + ω a c ∧ ω c b = 1 2 R a ◮ Torsion tensor: T a = d θ a + ω a b ∧ θ b = 1 bc θ b ∧ θ c 2 T a ◮ Cartan structure equations: ω ab = − ω ba , T a = 0 ◮ Bianchi identity: d R a b + ω a c ∧ R c b − R a c ∧ ω c b = 0 ◮ Cyclic identity: d T a + ω a b ∧ T b = R a b ∧ θ b = 0

Holonomy ◮ d s 2 = δ ab θ a θ b invariant under local SO ( D ) transformations θ a ′ = Λ − 1 a b θ b ◮ Connection and curvature transform ◮ ω a ′ b = Λ − 1 ac ω c d Λ d b + Λ − 1 ac d Λ c b ◮ R a ′ b = Λ − 1 ac R c d Λ d b Connection and curvature are both antisymmetric-matrix-valued two-forms ∈ D ( D − 1 ) / 2 representation of SO ( D )

Self-dual/anti-self-dual decomposition in 4D Duality supported by the fully antisymmetric symbol ǫ abcd ◮ Dual connection: b = 1 ω a 2 ǫ a bc ω c d ˜ d ◮ Dual curvature: b = 1 R a ˜ 2 ǫ a bc R c d d Curvature and connection ∈ 6 (antisymmetric) of SO ( 4 ) – reducible as ( 3 , 1 ) ⊕ ( 1 , 3 ) under SU ( 2 ) sd ⊗ SU ( 2 ) asd ∼ = SO ( 4 )

θ 0 , θ i � � to the action of SU ( 2 ) sd ⊗ SU ( 2 ) asd Adapting the frame ◮ Connection one-form � ω 0 i + 1 / 2 ǫ ijk ω jk � ( 3 , 1 ) Σ i = 1 / 2 � ω 0 i − 1 / 2 ǫ ijk ω jk � ( 1 , 3 ) A i = 1 / 2 ◮ Curvature two-form � R 0 i + 1 / 2 ǫ ijk R jk � ( 3 , 1 ) S i = 1 / 2 � R 0 i − 1 / 2 ǫ ijk R jk � ( 1 , 3 ) A i = 1 / 2 ◮ R a b = d ω a b + ω a c ∧ ω c b decomposes ◮ S i = d Σ i − ǫ ijk Σ j ∧ Σ k ◮ A i = d A i + ǫ ijk A j ∧ A k { Σ i , S i } vectors of SU ( 2 ) sd and singlets of SU ( 2 ) asd and vice-versa for { A i , A i }

Dynamics in 4D Einstein–Hilbert action in Palatini formalisms 1 � R cd ∧ θ c ∧ θ d ˜ S EH = 16 π G M 4 d ∧ θ d = 0 ˜ ◮ Vacuum equations: R c d ∧ θ d = 0 ◮ Cyclic identity for torsionless connection: R c Curvature (anti)self-duality guarantees vacuum solution ± ˜ R a R a = ⇒ Ricci flatness b b � A i = 0 S i = 0 or

The M 4 geometry Foliation and spatial homogeneity [textbook: Ryan and Shepley, 1975] ◮ Topologically M 4 = R × M 3 ◮ Bianchi 3D group G acts simply transitively on the leaves M 3 M 3 is locally G ◮ left-invariant Maurer–Cartan forms σ i : d σ i = 1 jk σ j ∧ σ k 2 c i ◮ 3 linearly independent Killing vectors tangent to M 3 : � = c i � ξ i , ξ j jk ξ k ◮ Classes A ( T 3 , Heisenberg, E 1 , 1 , E 2 , SL ( 2 , R ) , SU ( 2 ) ) & B

Self-dual vacuum solutions Geometry Foliation plus spatial homogeneity → ◮ Good ansatz for the metric ( g ij s functions of t ): d s 2 = d t 2 + g ij σ i σ j = δ ab θ a θ b ◮ Minimalistic (diagonal) ansatz: d s 2 = d t 2 + ∑ γ i σ i � 2 � i (the most general in most Bianchi classes)

Second-order equations: A i = d A i + ǫ ijk A j ∧ A k = 0 Solutions: anti-self-dual flat connections A i = λ ij 2 σ j λ i ℓ c ℓ jk + ǫ imn λ m [ j λ n k ] = 0 G → SU ( 2 ) homomorphisms [Bourliot, Estes, Petropoulos, Spindel, 2009] ◮ λ ij = 0 rank-0 (trivial) homomorphism: Class A, Class B ◮ λ ij � = 0 ◮ rank-1: I, II, VI h = − 1 , VII h = 0 & III, IV, V ,VI h � = − 1 , VII h � = 0 ◮ rank-3: VIII, IX

Bianchi IX: G ≡ SU ( 2 ) and M 3 ≡ S 3 Convenient parameterization: Ω i = γ j γ k σ 1 � 2 + Ω 3 Ω 1 σ 2 � 2 + Ω 1 Ω 2 d s 2 = Ω 1 Ω 2 Ω 3 d T 2 + Ω 2 Ω 3 σ 3 � 2 � � � Ω 1 Ω 2 Ω 3 General self-duality equations: A i = λ ij 2 σ j λ ij = 0 Lagrange system (Euler-top) [Jacobi] Ω 1 = − Ω 2 Ω 3 , Ω 2 = − Ω 3 Ω 1 , Ω 3 = − Ω 1 Ω 2 ˙ ˙ ˙ λ ij = δ ij Darboux–Halphen system [Darboux 1878; Halphen 1881] Ω 1 = Ω 2 Ω 3 − Ω 1 � Ω 2 + Ω 3 �  ˙   Ω 2 = Ω 3 Ω 1 − Ω 2 � Ω 3 + Ω 1 � ˙ Ω 3 = Ω 1 Ω 2 − Ω 3 � Ω 1 + Ω 2 �  ˙ 

Solutions with γ 1 = γ 2 → SU ( 2 ) × U ( 1 ) symmetry 1. Lagrange: Eguchi–Hanson [Eguchi, Hanson, April 1978] 2 + ( σ 2 ) 2 + 1 − a 4 2 � � ρ 4 + ρ 2 ( σ 1 ) ( σ 3 ) d ρ 2 d s 2 = ρ 4 1 − a 4 4 with a removable bolt at ρ = a 2. Darboux–Halphen: Taub–NUT [Newman, Tamburino, Unti, 1963] r 2 − m 2 � ( σ 1 ) 2 + ( σ 2 ) 2 d s 2 = r + m m σ 3 � 2 d r 2 + r − m � � 4 + r − m 4 r + m with a removable nut at r = m

Note: not the most general ◮ γ 1 = γ 2 = γ 3 → SU ( 2 ) × SU ( 2 ) : solution is flat space ◮ γ 1 � = γ 2 � = γ 3 → strict- SU ( 2 ) : solutions exist but have often naked singularities ◮ Lagrange system: ∃ naked singularities [Belisnky, Gibbons, Page, Pope, June 1978] ◮ Darboux–Halphen system: solvable in terms of quasi-modular forms [Halphen, 1881] , ∃ naked singularities except for one solution with a bolt [Atiyah, Hitchin, 1985] describing the configuration space of two slowly moving BPS SU ( 2 ) Yang–Mills–Higgs monopoles [Manton, 1981]

Reminder: bolts and nuts Fixed points of isometries generated by ξ - characterised by the rank of ∇ [ ν ξ µ ] - potential removable or non-removable singularities, depending on the precise behaviour of g µν - χ bolt = 2 , χ nut = 1 Around t = 0 ◮ rank 4: nut – removable if γ i ≃ t / 2 ∀ i ◮ rank 2: bolt – removable if γ 1 ≃ γ 2 ≃ finite and γ 3 ≃ nt / 2 Gravitational instantons of GR are classified according to bolts, nuts and asymptotic behaviours (Euclidean vs. Taubian) within the positive-action conjecture [Gibbons, Hawking, 1979]

Highlights Motivations and summary Gravitational instantons: homogeneity and self-duality The view from the leaf: geometric flows Extensions Outlook

Curvature for 3D homogeneous spaces ˜ M 3 : homogeneous 3D Bianchi IX space with metric s 2 = γ ij σ i σ j = δ ij ˜ θ i ˜ θ j d ˜ ( Γ ij inverse of γ ij ) ij = − ǫ ij ℓ n ℓ k Bianchi A classes: c k ◮ Cartan–Killing: C ij = − 1 2 ǫ ℓ im ǫ kjn n mk n n ℓ ◮ Ricci: tr ( n γ ) 2 Ric [ γ ] = C − 1 det γ γ + γ n γ n γ 2 det γ

Back to 4D: self-duality equations M 4 with d s 2 = d t 2 + g ij ( t ) σ i σ j Self-duality over M 4 with g ij = γ ik K k ℓ γ ℓ j A i ≡ 1 � ω 0 i − 1 � = λ ij 2 σ j ⇔ d γ ij d t = − R ij [ γ ] − 1 2 ǫ ijk ω jk 2tr ( α i α j ) 2 θ i SU ( 2 ) Yang–Mills connection over α = α i ˜ ˜ M 3 α i = ( C ij − λ ij ) t j with tr ( t i t j ) = − 2 δ ij ◮ t -independent: d α / d t = 0 ◮ flat: F ≡ d α + [ α , α ] = 0 ( ⇔ λ i ℓ c ℓ jk + ǫ imn λ m [ j λ n k ] = 0)