Connection description of 3D Gravity joint work with Y. Herfray and C. Scarinci arXiv:1605.07510 Workshop on Teichmueller theory and geometric structures on 3-dimensional manifolds Kirill Krasnov (Nottingham)
We usually describe geometry using metrics Works in any dimension Natural constructions: Einstein metrics Ricci flow There are of course other types of geometric structures (reductions of the structure group of the frame bundle) and other types of geometry (e.g. complex, symplectic)
In specific dimensions other descriptions possible 2D - complex rather than conformal structure 3D - Cartan formalism (Chern-Simons) 4D - Penrose encodes metric into an almost complex structure on twistor space plus contact form Many have previously suggested that metric may not be the best “variable” to describe gravity
In the approach to be described 3D geometry is encoded by connection on space(time) rather than metric Discovered in: Connection formulation of (2+1)-dimensional Einstein gravity and topologically massive gravity Peter Peldan (Goteborg, ITP). Oct 1991. 33 pp. Published in Class.Quant.Grav. 9 (1992) 2079-2092 Cited by 9 records interpretation via a certain 3-form in the total space of the SU(2) bundle over space(time)
Plan Einstein-Cartan, Chern-Simons descriptions of 3D gravity 3D gravity in terms of connections Volume gradient flow on connections - torsion flow 6D interpretation
3D gravity in first order formalism Λ < 0 For concreteness - hyperbolic case Riemannian Let e be (co)frame field Metric ds 2 = − 2 Tr( e ⊗ e ) e ∈ Λ 1 ⊗ s u (2) The associated SU(2) spin connection w ∈ Λ 1 ⊗ s u (2) (locally) Compatibility equation (zero torsion) (1) r e = 0 r e ⌘ d e + w ^ e + e ^ w -unique solution of the algebraic equation for w w = w ( e )
3D Einstein equations Λ = − 1 Metric has constant curvature (2) f ≡ f ( w ) = d w + w ∧ w f = e ∧ e curvature 2-form f ∈ Λ 2 ⊗ s u (2) Both (1), (2) follow as EL equations from ✓ ◆ f ∧ e − 1 Z S [ e , w ] = Tr 3 e ∧ e ∧ e If substitute get EH Lagrangian for the metric w = w ( e )
Chern-Simons description (1),(2) arise as real, imaginary parts of f ( a ) = 0 a := w + i e connection SL(2 , C ) S [ e , w ] = Im( S CS [ a ]) where ✓ ◆ a ∧ d a + 2 Z S CS [ a ] = Tr 3 a ∧ a ∧ a Chern-Simons functional
“Pure connection” description only possible for non- zero scalar curvature! Instead of solving for the connection, can solve e = e ( w ) also possible algebraic equation for e f = e ∧ e in higher D must be special in order w there to be a solution To describe the solution, need some notions Given f ∈ Λ 2 ⊗ s u (2) and choosing a volume form v get a map φ f : Λ 1 → s u (2) φ f ( α ) := α ∧ f /v ∀ α ∈ Λ 1 Can apply this map to the curvature itself λ ( f ) := 4 Note that the sign here is 3Tr ( φ f ⊗ φ f ( f )) invariantly defined!
Definition: Connection w is called definite if map is an isomorphism φ f Connection w is called positive (negative) definite if λ ( f ) > 0 ( λ ( f ) < 0) Can be defined at a point, then everywhere For a connection that comes from a metric map is just the Ricci curvature φ f definiteness asks the Ricci to have no zero eigenvalues similarly λ ( f ) is a multiple of the determinant of Ricci note that our definiteness is a weak condition that does not require the eigenvalues to have the same sign
Proposition: Given a negative definite connection, can solve for f = e ∧ e e = e ( f ) For a positive definite connection, can similarly solve f = − e ∧ e In both cases, the associated metric is a Riemannian 3D metric Proof: p Define v f := − λ ( f ) v does not depend on v, only on its orientation The solution explicitly i ξ e f = ( f ∧ i ξ f − i ξ f ∧ f ) / 2 v f Can be checked to satisfy f = e f ∧ e f
For a connection that comes from frame a simple calculation shows that does not change under constant rescaling of the √ det R R − 1 e where R is Ricci e f = original frame gives a Riemannian metric, degenerate where R is degenerate forgets about the scale of the original metric! the scale is introduced f = e ∧ e when solving this equation Λ = − 1
Pure connection action Substituting e = e f into the first-order action gives Volume of the space Z computed using the metric S [ w ] = v f defined by the connection p v f := − λ ( f ) v λ ( f ) := 4 3Tr ( φ f ⊗ φ f ( f )) φ f ( α ) := α ∧ f /v Functional on the space of connections of definite sign Its critical points - “constant curvature” connections
Euler-Lagrange equations S [ w ] = − 2 Z Tr ( e f ∧ e f ∧ e f ) 3 Z δ S [ w ] = − Tr ( δ ( e f ∧ e f ) ∧ e f ) Z Z = � Tr ( δ ( f ) ^ e f ) = � Tr ( r δ w ^ e f ) second-order PDE on w r e f = 0 ⇒ says that w is the spin connection compatible with the frame defined by w once this equation is satisfied, the metric is automatically constant curvature since by construction f = e f ∧ e f
Associated gradient flow Recall the flow that plays role in Floer homology - the gradient flow of Chern-Simons ◆ ∗ ✓ δ S CS [ a ] da where need a metric to define the * dt = δ a dimensional reduction of the 4D self-duality equations plays role in Donaldson-Witten theory
The volume gradient flow similarly define the gradient flow for our connection functional ◆ ∗ where now the star is defined using the ✓ δ S [ w ] d w dt = − metric defined by the connection δ w Parabolic equation d w Alternatively dt = ( r e f ) ∗ For positive connection needs to change the sign If decompose w = ˜ w + t t-torsion with by definition ˜ r e f = 0 and ( r e f ) ∗ is basically torsion Then r e f = t ^ e f + e f ^ t Flow by torsion - Possibly useful as an alternative to Ricci flow
Homogeneous case 3 ds 2 = X If start with metric ( a i ) 2 ( e i ) 2 i =1 Get connection de 1 = 2 e 2 ∧ e 3 w 1 = g 1 e 1 g 1 = ( a 1 ) 2 − ( a 2 ) 2 − ( a 3 ) 2 with a 2 a 3 connection forgets about the scale! Note that in the round case a 1 = a 2 = a 3 g 1 = g 2 = g 3 = g = − 1 Now start with connection w 1 = g 1 e 1 f 1 = (2 g 1 + g 2 g 3 ) e 2 ∧ e 3 f 1 = − θ 2 ∧ θ 3 θ 1 = a 1 e 1 Get metric Λ = +1 s − (2 g 2 + g 3 g 1 )(2 g 3 + g 1 g 2 ) a 1 = 2 g 1 + g 2 g 3
The associated volume is vol 2 = − (2 g 1 + g 2 g 3 )(2 g 2 + g 3 g 1 )(2 g 3 + g 1 g 2 ) If parametrise g 1 = − 1 + x, g 2 = − 1 + y, g 3 = − 1 + z So that x = y = z = 0 is the round metric 3-sphere The contour plots of the volume function are vol 2 = 1 / 2 vol 2 = 0 concave function! so the flow will maximum at the origin return to the round metric 3-sphere
The gradient flow explicitly g 1 = − a 1 (2 a 1 + g 2 a 3 + g 3 a 2 ) ˙ 2 a 2 a 3 For the round sphere g = − 1 g 1 = g 2 = g 3 = g ≡ − 1 + x for connection that comes from the metric a 1 = a 2 = a 3 = p p − (2 g + g 2 ) ≡ 1 − x 2 vol( g ) = (1 − x 2 ) 3 / 2 metric sphere has the largest volume gradient flow x = − x ˙ returns to the metric sphere
Compare Ricci flow 3 ! ds 2 = a 2 X ( e i ) 2 i =1 dg Ricci flow gives Not a gradient flow! dt = Ricci g dt ( a 2 ) = − 2 d collapses in finite time a 2 Similarly, for negative curvature the Ricci flow will expand the manifold, while the volume flow just returns to the metric connection
Towards 6D interpretation: 3-forms in 6D Let P be a 6-manifold dim( Λ 3 P ) = 6 · 5 · 4 / 3! = 20 Consider a 3-form Ω ∈ Λ 3 P This form is called generic or stable of positive (negative) type if at each point it is in the orbit dim = 36 − 8 − 8 = 20 GL(6 , R ) / SL(3 , R ) × SL(3 , R ) Ω = α 1 ^ α 2 ^ α 3 + β 1 ^ β 2 ^ β 3 , α 1 ^ α 2 ^ α 3 ^ β 1 ^ β 2 ^ β 3 6 = 0 GL(6 , R ) / SL(3 , C ) Ω = 2 Re ( α 1 ∧ α 2 ∧ α 3 ) , α 1 , α 2 , α 3 ∈ T ∗ C P
Negative type case: almost complex structure Given a volume form v, can define an endomorphism K Ω : T ∗ P → T ∗ P i ξ ( K Ω ( α )) := α ∧ i ξ Ω ∧ Ω /v This endomorphism squares to a multiple of identity K Ω ( α ) 2 = λ ( Ω ) I For negative type (stable) 3-forms λ ( Ω ) < 0 1 Can define J Ω := J 2 K Ω Ω = − I p − λ ( Ω ) Almost-complex structure are (0,1) forms For in the canonical form Ω α 1 , 2 , 3
Hitchin functional invariantly defined form in p v Ω := − λ ( Ω ) v given orientation class Z S [ Ω ] = v Ω P S [ Ω ] under variations Theorem (Hitchin): Critical points of in a given cohomology class are integrable J Ω Proof: is the result of acting with v Ω = 1 where ˆ ˆ Ω Ω ∧ Ω in all 3 slots of Ω J Ω 2 Z Z ˆ ˆ d ˆ δ S [ Ω ] = Ω ∧ δ Ω = Ω ∧ dB Ω = 0 ⇒ Ω c = Ω + iˆ d Ω c = 0 closed ⇒ Ω Ω c is (0,3) form ⇒ integrable ACS
6D interpretation of the connection formulation of 3D gravity Let P be the principal SU(2) bundle over a 3-dimensional M P → M This is necessarily a trivial bundle P = SU(2) × M Let be an SU(2) invariant stable 3-form in P Ω Proposition: defines an SU(2) connection and metric on M Ω (with some genericity assumption) Proof: Take J Ω Define the image of vertical vector J Ω fields to be horizontal ⇒ connection in P Define the pairing of horizontal vector fields to be the Killing-Cartan pairing of their vertical images J Ω
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